Download Link. Of course, the entire text was critically reviewed, updated, and improved. In other words, it is important that students become comfortable with SI units in Download free solution manual of Electronic Properties of Materials 4th edition written by Hummel eBook in pdf format.
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In addition, pirated copies of the first and second editions have surfaced in Asian countries. Further, a translation into Korean appeared of course, I feel that one should respect the rights of the owner of intellectual property. It is quite satisfying for an author t6 learn that his brainchild has been favorably accepted by students as well as by professors and thus seems to serve some useful purpose. Authors: Rolf E. The book has now gone through several reprinting cycles among them a few pirate prints in Asian countries.
The calculation of the remaining branches bands is left to the reader, see Problem 5. One important question has remained essentially unanswered: What do these E versus k curves really mean? Simply stated, they relate the energy of an electron to its k-vector, i. All these diagrams relate in graphic form one parameter with another variable in order to provide an easier interpretation of data. We shall eventually learn to appreciate complete band diagrams in later chapters, from which we will draw important conclusions about the electronic properties of materials.
In Figs. We call the allowed bands, for the time being, the n-band, or the m-band, and so forth. In later sections and particularly in semiconductor physics see Chapter 8 we will call one of these bands the valence band because it contains the valence electrons and the next higher one the conduction band. As a rule this does not cause any confusion.
Finally, we need to stress one more point: The wave vector k is inversely proportional to the wavelength of the electrons see equation 4. We will show in Section 5. Fundamentals of Electron Theory reciprocal of the interplanar distance. The tips of all such vectors from sets of parallel lattice planes form the points in a reciprocal lattice. An X-ray diffraction pattern is a map of such a reciprocal lattice.
In other words, the lowest band shown in Fig. Now, we learned above that the individual branches in an extended zone scheme Fig. A reduced zone scheme as shown in Fig. Actually, we projected the second Brillouin zone into the first Brillouin zone. The same can be done with the third Brillouin zone, etc. We now consider the behavior of an electron in the potential of a twodimensional lattice. The electron movement in two dimensions can be described as before by the wave vector k that has the components kx and ky, which are parallel to the x- and y-axes in reciprocal space.
One obtains, in the two-dimensional case, a two-dimensional field of allowed energy regions which corresponds to the allowed energy bands, i. We shall illustrate the construction of the Brillouin zones for a twodimensional reciprocal lattice Fig.
For the first zone one constructs the perpendicular bisectors on the shortest lattice vectors, G1. For the following zones the bisectors of the next shortest lattice vectors are constructed. It is essential that for the zones of higher order the extended limiting lines of the zones of lower order are used as additional limiting lines.
The first four Brillouin zones are shown in Fig. Note that all the zones have the same area. The first four shortest lattice vectors G1 through G4 are drawn in Fig. Cubic primitive crystal structure. The first four Brillouin zones of a two-dimensional, cubic primitive reciprocal lattice. The significance of the Brillouin zones will become evident in later sections, when the energy bands of solids are discussed. A few words of explanation will be given here, nevertheless.
The Brillouin zones are useful if one wants to calculate the behavior of an electron which may travel in a specific direction in reciprocal space. The consequence of 5. We will learn later that these considerations can be utilized to determine the difference between metals, semiconductors, and insulators.
This will be done briefly here because of its immediate intuitive power. We consider an electron wave that propagates in a lattice at an angle y to a set of parallel lattice planes Fig. The corresponding rays are diffracted on the lattice atoms. At a certain angle of incidence, constructive interference between rays 10 and 20 occurs. It has been shown by Bragg that each ray Figure 5.
Overlapping of allowed energy bands. Bragg reflection of an electron wave in a lattice. The angle of incidence is y. This is always the case when the path difference 2a sin y is an integer multiple of the electron wavelength l, i. With 4. Equation 5. At this critical k-value the transmission of an electron beam through the lattice is prevented.
Then, the incident and the Bragg-reflected electron wave form a standing wave. Three-Dimensional Brillouin Zones In the previous section, the physical significance of the Brillouin zones was discussed. It was shown that at the boundaries of these zones the electron waves are Bragg-reflected by the crystal. Fundamentals of Electron Theory 46 seen to have the unit of a reciprocal length and is therefore defined in the reciprocal lattice.
We will now attempt to construct three-dimensional Brillouin zones for two important crystal structures, namely, the facecentered cubic fcc and the body-centered cubic bcc crystals. Since the Brillouin zones for these structures have some important features in common with the so-called Wigner—Seitz cells, it is appropriate to discuss, at first, the Wigner—Seitz cells and also certain features of the reciprocal lattice before we return to the Brillouin zones at the end of Section 5.
Wigner—Seitz Cells Crystals have symmetrical properties. For its construction, one bisects the vectors from a given atom to its nearest neighbors and places a plane perpendicular to these vectors at the bisecting points. Wigner—Seitz cell for the body-centered cubic bcc structure.
All crystal structures can be traced to one of the 14 types of Bravais lattices see textbooks on crystallography. Energy Bands in Crystals 47 In the fcc lattice, the atoms are arranged on the corners and faces of a cube, which is equivalent to the center points of the edges and the center of the cell Fig.
The Wigner—Seitz cell for this structure is shown in Fig. Conventional unit cell of the fcc structure. In the cell which is marked black, the atoms are situated on the corners and faces of the cubes. In the white cell, the atoms are at the centers of the edges and the center of the cell. Wigner—Seitz cell for the fcc structure. It is constructed from the white cell which is marked in Fig.
Energy Bands in Crystals can be defined. Using this translation vector it is possible to reach, from a given lattice point, any other equivalent lattice point. For this, the factors n1, n2, n3 have to be integers. In Fig. The factor 2p is introduced for convenience. In X-ray crystallography, this factor is omitted. We therefore write11 Figure 5. Plane formed by t2 and t3 with perpendicular vector b1. As an example of how these transformations are performed, we calculate now the reciprocal lattice of a bcc crystal.
We note immediately an important result. The end points of the reciprocal lattice vectors of a bcc crystal are at the center of the edges of a cube. This means that points of the reciprocal lattice of the bcc structure are identical to the lattice points in a real lattice of the fcc structure, see Fig.
Conversely, the reciprocal lattice points of the fcc structure and the real lattice points of the bcc structure are identical. In Section 5. Similarly, a threedimensional Brillouin zone can be obtained by bisecting all lattice vectors b and placing planes perpendicular on these points.
As has been shown in Section 5. A comparison of the fundamental lattice vectors b and t gives the striking result that the Wigner—Seitz cell for an fcc crystal Fig. Fundamentals of Electron Theory 52 Figure 5. Lattice vectors in reciprocal space of a bcc crystal. The primitive vectors in the reciprocal lattice are because of 5. First Brillouin zone of the bcc crystal structure. Thus, a Brillouin zone can be defined as a Wigner—Seitz cell in the reciprocal lattice. Free Electron Bands We mentioned in Section 5.
In other words, the energy Ek0 for k0 outside the 53 5. We proceed now to three-dimensional zone pictures. We might correctly expect that the energy bands are not alike in different directions in k-space. We explain the details using the bcc crystal structure as an example.
In three dimensions the equation analogous to 5. They are the [] direction from the origin G to point H, the [] direction from G to N, and the [] direction from G to P. The sequence of the individual subgraphs is established by convention and can be followed using Fig.
We now show how some of these bands are calculated for a simple case. To start with, let G be 0. Then 5. The set of numbers thus gained is inserted into square brackets; see textbooks on materials science. This can be convincingly seen by comparing Figs. Fundamentals of Electron Theory 54 Figure 5. Energy bands of the free electrons for the bcc structure.
The numbers given on the branches are the respective hi values see the calculation in the text. The curve which represents 5. Then we obtain, by using 5.
Similarly, all bands in Fig. The free electron bands are very useful for the following reason: by comparing them with the band structures of actual materials, an assessment is possible if and to what degree the electrons in that material can be considered to be free. First Brillouin zone of the fcc structure.
Free electron bands of the fcc structure. The letters on the bottom of the graphs correspond to letters in Fig. Fundamentals of Electron Theory 5. Other directions in k-space are likewise seen. These specific symmetry points and directions are selected by convention from a much larger number of possible directions.
They sufficiently characterize the properties of materials, as we will see below. We inspect now some calculated energy-band structures. They should resemble the one shown in Fig. In the present case, however, they are depicted for more than one direction in k-space.
Additionally, they are displayed in the positive k-direction only, similarly as in Fig. We start with the band diagram for aluminum, Fig. The band diagram for aluminum looks quite similar to the free electron bands shown in Fig.
This suggests that the electrons in aluminum behave essentially free-electronlike which is indeed the case. We also detect in Fig. Note, however, that the individual energy bands overlap in different directions in k-space, so that as a whole no band gap exists.
This is in marked difference to the band diagram of a semiconductor, as we shall see in a moment. The lower, Figure 5. Energy bands for aluminum. Adapted from B. Segal, Phys. The meaning of the Fermi energy will be explained in Section 6. Energy Bands in Crystals 57 Figure 5. Band structure of copper fcc. The calculation was made using the l-dependent potential. For the definition of the Fermi energy, see Section 6.
The origin of the energy scale is positioned for convenience in the lower end of this s-band. Next, we discuss the band structure for copper, Fig. We notice in the lower half of this diagram closely spaced and flat running bands.
Calculations show that these can be attributed to the 3d-bands of copper see Appendix 3. They superimpose the 4s-bands which are heavily marked in Fig. The band which starts at G is, at first, s-electronlike, and becomes d-electronlike while approaching point X. The first half of this band is continued at higher energies. It is likewise heavily marked.
It can be seen, therefore, that the d-bands overlap the s-bands. Again, as for aluminum, no band gap exists if one takes all directions in k-space into consideration.
As a third example, the band structure of silicon is shown Fig. Of particular interest is the area between 0 and approximately 1 eV in which no energy bands are shown. For semiconductors, the zero point of the energy scale is placed at the bottom of this energy gap, even though other conventions are possible and in use. Finally, the band structure of gallium arsenide is shown in Fig. The so-called III—V semiconductor compounds, such as GaAs, are of great technical importance for optoelectronic devices, as we will discuss in later sections.
They have essentially the same crystal structure and the same total 58 I. Calculated energy band structure of silicon diamond-cubic crystal structure.
Adapted from M. Cohen and T. Bergstresser, Phys. See also J. Chelikowsky and M. Cohen, Phys. B14, Calculated energy band structure of GaAs. Adapted from F. Herman and W. Spicer, Phys. Energy Bands in Crystals 59 number of valence electrons as the element silicon. Again, a band gap is clearly seen. It should be mentioned, in closing, that the band structures of actual solids, as shown in Figs. Experimental investigations, such as measurements of the frequency dependence of the optical properties, can help determine which of the various calculated band structures are closest to reality.
Curves and Planes of Equal Energy We conclude this chapter by discussing another interesting aspect of the energy versus wave vector relationship. In the two-dimensional case, i. This leads to curves of equal energy, as shown in Fig. For a two-dimensional square lattice and for small electron energies, the curves of equal energy are circles. However, if the energy of the electrons is approaching the energy of the boundary of a Brillouin zone, then a deviation from the circular form is known to occur.
It is of particular interest that the energy which belongs to point K in Fig. Electron energy E versus wave vector k two-dimensional. This figure demonstrates various curves of equal energy for free electrons. Curves of equal energy inserted into the first Brillouin zone for a twodimensional square lattice.
Consequently, the curves of equal energy for the first Brillouin zone may extend into the second zone. This leads to an overlapping of energy bands as schematically shown in Fig. For copper and aluminum the band overlapping leads to quasi-continuous allowed energies in different directions of k-space.
For semiconductors the band overlapping is not complete, which results in the already-mentioned energy gap Figs. In three-dimensional k-space one obtains surfaces of equal energy.
For the free electron case and for a cubic lattice they are spheres. For a nonparabolic E- k behavior these surfaces become more involved. This is demonstrated in Fig. A particular surface of equal energy Fermi surface, see Section 6. Adapted from A. Pippard, Phil. London, A , Energy Bands in Crystals 61 Problems 1. What is the energy difference between the points L20 and L1 upper in the band diagram for copper? Hint: Consult the band diagram for silicon. Calculate how much the kinetic energy of a free electron at the corner of the first Brillouin zone of a simple cubic lattice three dimensions!
Construct the first four Brillouin zones for a simple cubic lattice in two dimensions. Plot the bands in k-space. Compare with Fig. Calculate the main lattice vectors in reciprocal space of an fcc crystal. This electron was in most cases an outer, i. However, in a solid of one cubic centimeter at least valence electrons can be found. In this section we shall describe how these electrons are distributed among the available energy levels. It is impossible to calculate the exact place and the kinetic energy of each individual electron.
We will see, however, that probability statements nevertheless give meaningful results. Many of the electronic properties of materials, such as optical, electrical, or magnetic properties, are related to the location of EF within a band.
This can be compared to a vessel, like a cup, the electron band into which a certain amount of water electrons is poured. The top surface of the water contained in this vessel can be compared to the Fermi energy.
The Fermi energies for aluminum and copper are shown in Figs. Numerical values for the Fermi energies for some materials are given in Appendix 4. They range typically from 2 to 12 eV.
The above-stated definition, even though convenient, can occasionally be misleading, particularly when dealing with semiconductors. Therefore, a more accurate definition of the Fermi energy will be given in Section 6. An equation for the Fermi energy is given in 6.
In three-dimensional k-space the one-dimensional Fermi energy is replaced by a Fermi surface. The energy surface shown in Fig. Fermi Distribution Function The distribution of the energies of a large number of particles and its change with temperature can be calculated by means of statistical considerations. EF is the Fermi energy which we introduced in Section 6.
This Figure 6. Electrons in a Crystal 65 Figure 6. This is best seen from 6. As can be seen from 6. This serves as a definition for the Fermi energy, as outlined in Section 6. Density of States We are now interested in the question of how energy levels are distributed over a band. We restrict our discussion for the moment to the lower part of the valence band the 3s-band in aluminum, for example because there the electrons can be considered to be essentially free due to their weak binding force to the nucleus.
The dimensions of this potential well are thought to be identical to the dimensions of the crystal under consideration. Then our problem is similar to the case of one electron in a potential well of size a, which we treated in Section 4. By using the appropriate boundary conditions, the 66 I. Now we pick an arbitrary set of quantum numbers nx, ny, nz. An energy state can therefore be represented by a point in quantum number space Fig. All points within the sphere therefore represent quantum states with energies smaller than En.
Since the quantum numbers are positive integers, the n-values can only be defined in the positive octant of the n-space. Thus, with 6. Representation of an energy state in quantum number space. Density of states Z E within a band. The electrons in this band are considered to be free. Note, that the density of states, as shown in this figure, is only parabolic for the threedimensional case solids. Z E looks different for the two-dimensional case quantum well , one-dimensional case quantum wire , or zero-dimensional case quantum dot.
See for example Fig. However, since we are discussing here only solids, the representation as shown above is correct and sufficient. The density of states plotted versus the energy gives, according to 6.
Figure 6. One can compare the density of states concept with a high-rise apartment building in which the number of apartments per unit height e. To stay within this analogy, only a very few apartments are thought to be available on the ground level. However, with increasing height of the building, the number of apartments per unit height becomes larger.
The area within the curve in Fig. In actual crystals, however, I. Schematic representation of the complete density of states function within a band. Let us consider, for example, the curves of equal energy depicted in Fig. For low energies, the equal energy curves are circles. Thus, the electrons behave free-electron like for these low energies.
The density of states curve is then, as before, a parabola. For larger energies, however, fewer energy states are available, as is seen in Fig.
Thus, Z E decreases with increasing E, until eventually the corners of the Brillouin zones are filled. At this point Z E has dropped to zero. The largest number of energy states is thus found near the center of a band, as shown schematically in Fig. Population Density The number of electrons per unit energy, N E , within an energy interval dE can be calculated by multiplying the number of possible energy levels, Z E , by the probability for the occupation of these energy levels.
We have to note, however, that because of the Pauli principle, each energy state can be occupied by one electron of positive spin and one of negative spin,16 i. Electrons in a Crystal or, with 6. We see immediately that for T! We consider the simple case T! Population density N E within a band for free electrons.
Consequences of the Band Model We mentioned in Section 6. If the highest filled s-band of a crystal is occupied by two electrons per atom, i. An electron has to absorb energy in order to move.
Keep in mind that for a completely occupied band higher energy states are not allowed. We exclude the possibility of electron jumps into higher bands. Solids in which the highest filled band is completely occupied by electrons are, therefore, insulators Fig. In solids with one valence electron per atom e.
An electron drift upon application of an external field is possible; the crystal shows metallic behavior Fig. Bivalent metals should be insulators according to this consideration, which is not the case. The reason for this lies in the fact that the upper Figure 6.
Simplified representation for energy bands for a insulators, b alkali metals, c bivalent metals, and d intrinsic semiconductors. Electrons in a Crystal 71 bands partially overlap, which occurs due to the weak binding forces of the valence electrons on their atomic nuclei see Fig.
If such an overlapping of bands occurs, the valence electrons flow in the lower portion of the next higher band, because the electrons tend to assume the lowest potential energy Fig.
As a result, bivalent solids may also possess partially filled bands. Thus, they are also conductors. We shall see in Chapter 8 that the valence as well as the conduction bands of semiconductors can accommodate 4N electrons. Because germanium and silicon possess four valence electrons, the valence band is completely filled with electrons.
Intrinsic semiconductors have a relatively narrow forbidden energy zone Fig. A sufficiently large energy can, therefore, excite electrons from the completely filled valence band into the empty conduction band and thus provide some electron conduction. This preliminary and very qualitative discussion on electronic conduction will be expanded substantially and the understanding will be deepened in Part II of this book. Effective Mass We implied in the previous sections that the mass of an electron in a solid is the same as the mass of a free electron.
Experimentally determined physical properties of solids, such as optical, thermal, or electrical properties, indicate, however, that for some solids the mass is larger while for others it is slightly smaller than the free electron mass. The cause for the deviation of the effective mass from the free electron mass is usually attributed to interactions between the drifting electrons and the atoms in a crystal.
On the other hand, the electron wave in another crystal might have just the right phase in order that the response to an external electric field is enhanced. We shall now attempt to find an expression for the effective mass. For this, we shall compare the acceleration of an electron in an electric field calculated by classical as well as by quantum mechanical means.
At first, we write an expression for the velocity of an electron in an energy band. Fundamentals of Electron Theory 72 We introduced in Chapter 2 the group velocity, i. Then, the group velocity is, according to 2. Forming the first derivative of 4. When inspecting band structures Fig. These regions might be found, particularly, near the center or near the boundary of a Brillouin zone. At points in k-space for which more than one electron band is found G-point in Fig.
We shall demonstrate the k-dependence of the effective mass for a simple case and defer discussions about actual cases to Section 8. Electrons in a Crystal 73 Figure 6. From this curve, both the first derivative and the reciprocal function of the second derivative, i. These functions are shown in Fig. We notice in Fig. We likewise observe in Fig.
It is, however, common to ascribe to the hole a positive effective mass and a positive charge instead of a negative mass and a negative charge. Electron holes play an important role in crystals whose valence bands are almost filled, e. The effective mass is a tensor in this case.
Fundamentals of Electron Theory It should be noted here for clarification that an electron hole is not identical with a positron. The latter is a subatomic particle like the electron, however with a positive charge. Positrons are emitted in the b-decay or are found in cosmic radiation.
When positrons and electrons react with each other they are both annihilated under emission of energy. Conclusion The first part of this book is intended to provide the reader with the necessary tools for a better understanding of the electronic properties of materials. We learned that the distinct energy levels which are characteristic for isolated atoms widen into energy bands when the atoms are moved closer together and eventually form a solid.
We further learned that some of these energy bands are filled by electrons, and that the degree of this filling depends upon whether we consider a metal, a semiconductor, or an insulator. Finally, the degree to which electron energy levels are available within a band was found to be nonuniform. We discovered that the density of states is largest near the center of an electron band. All these relatively unfamiliar concepts will become more transparent to the reader when we apply them in the chapters to come.
What velocity has an electron near the Fermi surface of silver? Are there more electrons on the bottom or in the middle of the valence band of a metal? Calculate the Fermi energy for silver assuming 6. Assume the effective mass equals the free electron mass. Note: Take 1 eV as energy interval. The density of states at the Fermi level 7 eV was calculated for 1 cm3 of a certain metal to be about energy states per electron volt.
Someone is asked to calculate the number of electrons for this metal using the Fermi energy as the maximum kinetic energy which the electrons have. He argues that because of the Pauli principle, each 6.
Electrons in a Crystal 75 energy state is occupied by two electrons. Hint: Consider equation 7. What fraction of the 3s-electrons of sodium is found within an energy kBT below the Fermi level? Take room temperature, i.
Explain why, in a simple model, a bivalent material could be considered to be an insulator. Also explain why this simple argument is not true.
We stated in the text that the Fermi distribution function can be approximated by classical Boltzmann statistics if the exponential factor in the Fermi distribution function is significantly larger than one. Ashcroft and N. Askeland and P. Blakemore, Solid State Physics, W. Saunders, Philadelphia, PA Bube, Electrons in Solids, 3rd edn.
Datta, Quantum Phenomena, Vol. Fundamentals of Electron Theory D. Kasap, Principles of Electronic Materials and Devices, 2nd edn. Kittel, Solid State Physics, 8th edn. Mott and H. Hummel will consistently provide you good worth if you do it well. Hummel to read will not come to be the only goal.
The goal is by obtaining the positive value from guide until completion of guide. Hummel This is not only exactly how fast you review a book as well as not just has the amount of you finished the books; it has to do with just what you have obtained from the books.
Hummel to review is also required. You can pick guide based upon the favourite motifs that you such as. Hummel It can be also about the requirement that binds you to read the book. Hummel, you could discover it as your reading book, even your favourite reading publication. So, find your preferred publication here as well as obtain the connect to download and install the book soft file. Written for engineers, this fourth, updated edition of the popular text differs from other books on solid-state physics in stressing concepts, not mathematical formalism.
The new edition emphasizes applications and covers next-generation electronic materials. This book is divided into five distinct and self-contained parts, which makes it easier for the reader to find information on a particular area of interest.
It would also be useful for someone looking to gain an overall concept of device physics. From the Back Cover This book on electrical, optical, magnetic, and thermal properties of materials differs from other introductory texts in solid-state physics. First, it is written for engineers, particularly materials and electrical engineers, who what to gain a fundamental understanding of semiconductor devices, magnetic materials, lasers, alloys, and so forth.
Second, it stresses concepts rather than mathematical formalism, which should make the presentation relatively easy to read. Third, it is not an encyclopedia: The topics are restricted to material considered to be essential and that can be covered in one week semester.
The book is divided into five parts. The first part, "Fundamentals of Electron Theory," introduces the essential quantum mechanical concepts needed for understanding materials science; the other parts may be read independently of each other.
Many practical applications are discussed to provide students with an understanding of electronic devices currently in use. The solutions to the numerical problems are given in the appendix. Previous editions have been well received by students and teachers alike.
This Fourth Edition has again been thoroughly revised and brought up to date to take into account the explosive developments in electrical, optical, and magnetic materials and devices. Specifically, new topics have been added in the "applied sections," such as energy saving light sources, particularly compact fluorescence light fixtures, organic light-emitting diodes OLEDs , organic photovoltaics OPV cells , optical fibers, pyroelectricity, phase-change memories, blue ray disks, holographic versatile disks, galvanoelectric phenomena emphasizing the entire spectrum of primary and rechargeable batteries , graphene, quantum Hall effect, iron-based semiconductors pnictides , etc.
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