1. Energy Band Theory with n=1, 2, 3, (1.1.9) The wave function is thus given by: yn(x)=An sin( (1.1.10) n22h2 and the energy of the electron is: En2mg2 (1.1.11) ls rthe e quite similar to that obtained for a free electron, in both cnergy is a function of the squared momentum. The difference and the energy E can take any value, while in the case of the particle-in- a-box problem, k and E can only take discrete values(replacing k by ni/a in Expression 1. 1.3 yields Equation 1. 1. 11). These values are fixed by the geometry of the potential well. Intuitively, it is interesting to note that if the width of the potential well becomes very large(a-)oo)the different values of k become very close to one another, such that they are no longer discrete values but rather form a continuum. as in the case for the free electron Which values can k take in a finite crystal of macroscopic dimensions? Let us consider the example of a one-dimensional linear crystal having a length L(Figure 1.3). If we impose Y(x=0)=0 and Y(x-L)=0 as in the case of the particle- in-the-box approach, Relationships 1.1.9 and 1.1.11 tell us that the permitted values for the momentum and for the energy of the electron will depend on the length of the crystal. This is clearly unacceptable for we know from experience that the electrical properties of a macroscopic sample do not depend on its dimensions Much better results are obtained using the Born-von Karman boundary conditions, referred to as cyclic boundary conditions. To obtain these conditions, let us bend the crystal such that x=0 and x=L become coincident. From the newly obtained geometry it becomes evident that for any value of x, we have the cyclical boundary conditions: P(x+L) Y(x). Using the free-electron wave function(Expression 1.1.2),and taking into account the periodic nature of the problem, we can write Y(x+L=A exp(k(x+LD)=A exp(ikr)expljkl)= A expljkx)=Y(x) which imposes ey0u=1→k=2x (1.1.12) where n is an integer number In the case of a three-dimensional crystal with dimensions (Lx, Ly, lz),the Born-von Karman boundary conditions can be written as follows
1. Energy Band Theory The wave function is thus given by: and the energy of the electron is: This result is quite similar to that obtained for a free electron, in both cases the energy is a function of the squared momentum. The difference resides in the fact that in the case of a free electron, the wave number k and the energy E can take any value, while in the case of the particle-ina-box problem, k and E can only take discrete values (replacing k by in Expression 1.1.3 yields Equation 1.1.11). These values are fixed by the geometry of the potential well. Intuitively, it is interesting to note that if the width of the potential well becomes very large the different values of k become very close to one another, such that they are no longer discrete values but rather form a continuum, as in the case for the free electron. Which values can k take in a finite crystal of macroscopic dimensions? Let us consider the example of a one-dimensional linear crystal having a length L (Figure 1.3). If we impose and as in the case of the particle-in-the-box approach, Relationships 1.1.9 and 1.1.11 tell us that the permitted values for the momentum and for the energy of the electron will depend on the length of the crystal. This is clearly unacceptable for we know from experience that the electrical properties of a macroscopic sample do not depend on its dimensions. Much better results are obtained using the Born-von Karman boundary conditions, referred to as cyclic boundary conditions. To obtain these conditions, let us bend the crystal such that x = 0 and x = L become coincident. From the newly obtained geometry it becomes evident that for any value of x, we have the cyclical boundary conditions: Using the free-electron wave function (Expression 1.1.2), and taking into account the periodic nature of the problem, we can write: which imposes: where n is an integer number. In the case of a three-dimensional crystal with dimensions the Born-von Karman boundary conditions can be written as follows: 5 with n = 1,2,3,... (1.1.9)
Che 2 2 2In Lx (1.1.13) where nx, ny, nz are integer numbers inear crystal bent crystal Figure 1.3: Bending of a crystal; Born-von Karman boundary conditions 1.1. 2. Energy bands of a crystal (intuitive approach) hopa n a single atom, electrons occupy discrete energy levels.What happens when a large number of atoms are brought together to form a crystal? Let us take the example of a relatively simple element with low atomic number, such as lithium(z=). In a lithium atom, two electrons of opposite spin occupy the lowest energy level(ls level), and the remaining third electron occupies the second energy level (2s level). The electronic configuration is thus Is2 2sI. All lithium atoms have exactly the same electronic configuration with identical energy levels. If an hypothetical molecule containing two lithium atoms is formed, we are now in the presence of a system in which four electrons wish"to have an energy equal to that of the ls level. But because of the Pauli exclusion principle, which states that only two electrons of opposite spins can occupy the same energy level, only two of the four Is electrons can occupy the ls level. This clearly poses a problem for the molecule. The problem is solved by splitting the ls level into two levels having very close, but nevertheless different energies(Figure 1. 4)
6 Chapter 1 where are integer numbers. In a single atom, electrons occupy discrete energy levels. What happens when a large number of atoms are brought together to form a crystal? Let us take the example of a relatively simple element with low atomic number, such as lithium (Z=3). In a lithium atom, two electrons of opposite spin occupy the lowest energy level (1s level), and the remaining third electron occupies the second energy level (2s level). The electronic configuration is thus All lithium atoms have exactly the same electronic configuration with identical energy levels. If an hypothetical molecule containing two lithium atoms is formed, we are now in the presence of a system in which four electrons "wish" to have an energy equal to that of the 1s level. But because of the Pauli exclusion principle, which states that only two electrons of opposite spins can occupy the same energy level, only two of the four 1s electrons can occupy the 1s level. This clearly poses a problem for the molecule. The problem is solved by splitting the 1s level into two levels having very close, but nevertheless different energies (Figure 1.4). 1.1.2. Energy bands of a crystal (intuitive approach)
1. Energy Band Theor If a crystal of lithium containing N number of atoms is now formed, the system will contain N number of ls energy levels. The same consideration is valid for the 2s level. the number of atoms in a cubic centimeter of crystal is on the order of 5x1022. As a result, each energy level is split into 5x1022 distinct energy levels which extend throughout the crystal Each of these levels can be occupied by two electrons by virtue of the Pauli exclusion principle. In practice, the energy difference between the highest and the lowest energy value resulting from this process of split an energy level is on the order of a few electron-volts; therefore, energy difference between two neighboring energy levels is on the order of 10-22 eV. This value is so small that one can consider that the energy levels are no longer discrete, but form a continuum of permitted energy values for the electron. This introduces the concept of energy bands in a crystal. Between the energy bands(between the ls and the 2s energy bands in Figure 1. 4) there may be a range of energy values which are not permitted. In that case, a forbidden energy gap is produced between permitted energy bands. The energy levels and the energy bands extend throughout the entire crystal. Because of the potential wells generated by the atom nuclei, however, some electrons(those occupying the ls levels) are confined to the immediate neighborhood of the nucleus they are bound to. The electrons of the 2s band on the other hand can overcome nucleus attraction and move throughout the crystal Molecule ystal Figure 1. 4: Permitted energy levels an atom, an hypothetical molecule, and a crystal of lithium 1.1.3. Kronig-Penney model Semiconductors, like metals and some insulators, are crystalline materials. This implies that atoms are placed in an orderly and periodic manner in the material(see Annex A4). While most usual crystalline materials are polycrystalline, semiconductor materials used in the
1. Energy Band Theory 7 If a crystal of lithium containing N number of atoms is now formed, the system will contain N number of 1s energy levels. The same consideration is valid for the 2s level. The number of atoms in a cubic centimeter of a crystal is on the order of As a result, each energy level is split into distinct energy levels which extend throughout the crystal. Each of these levels can be occupied by two electrons by virtue of the Pauli exclusion principle. In practice, the energy difference between the highest and the lowest energy value resulting from this process of splitting an energy level is on the order of a few electron-volts; therefore, the energy difference between two neighboring energy levels is on the order of eV. This value is so small that one can consider that the energy levels are no longer discrete, but form a continuum of permitted energy values for the electron. This introduces the concept of energy bands in a crystal. Between the energy bands (between the 1s and the 2s energy bands in Figure 1.4) there may be a range of energy values which are not permitted. In that case, a forbidden energy gap is produced between permitted energy bands. The energy levels and the energy bands extend throughout the entire crystal. Because of the potential wells generated by the atom nuclei, however, some electrons (those occupying the 1s levels) are confined to the immediate neighborhood of the nucleus they are bound to. The electrons of the 2s band, on the other hand, can overcome nucleus attraction and move throughout the crystal. 1.1.3. Krönig-Penney model Semiconductors, like metals and some insulators, are crystalline materials. This implies that atoms are placed in an orderly and periodic manner in the material (see Annex A4). While most usual crystalline materials are polycrystalline, semiconductor materials used in the
cha electronics industry are single-crystal. These single crystals are almost perfect and defect-free, and their size is much greater than any of the microscopic physical dimensions which we are going to deal with in this chapter In a crystal each atom of the crystal creates a local potential well which attracts electrons, just like in the lithium crystal described in Figure 1.4 The potential energy of the electron depends on its distance from the atom nucleus. Electrostatics provides us with a relationship establishing the potential energy resulting from the interaction between an electron carrying a charge -q and a nucleus bearing a charge +qZ, where Z is the atomic number of the atom and is equal to the number of protons in the nucleus v() (1.1.14) 4丌6x In this relationship x is the distance between the electron and the nucleus V(x) is the potential energy and E is the permittivity of the material under consideration. Equation 1. 1. 14 ignores the presence of other electrons, such as core electrons "orbiting"around the nucleus. These electrons actually induce a screening effect between the nucleus and outer shell electrons, which reduces the attraction between the nucleus and higher energy electrons. The energy of the electron as a function of its distance from the nucleus is sketched in Figure 1.5 distance. x Figure 1.5: Energy of an electron as a function of its distance from the atom nucleus(V=0 when x=oo).[2] How will an electron behave in a crystal? In order to simplify the problem, we will suppose that the crystal is merely an infinite, one-
8 Chapter 1 electronics industry are single-crystal. These single crystals are almost perfect and defect-free, and their size is much greater than any of the microscopic physical dimensions which we are going to deal with in this chapter. In a crystal each atom of the crystal creates a local potential well which attracts electrons, just like in the lithium crystal described in Figure 1.4. The potential energy of the electron depends on its distance from the atom nucleus. Electrostatics provides us with a relationship establishing the potential energy resulting from the interaction between an electron carrying a charge -q and a nucleus bearing a charge +qZ, where Z is the atomic number of the atom and is equal to the number of protons in the nucleus: In this relationship x is the distance between the electron and the nucleus, V(x) is the potential energy and is the permittivity of the material under consideration. Equation 1.1.14 ignores the presence of other electrons, such as core electrons "orbiting" around the nucleus. These electrons actually induce a screening effect between the nucleus and outer shell electrons, which reduces the attraction between the nucleus and higherenergy electrons. The energy of the electron as a function of its distance from the nucleus is sketched in Figure 1.5. How will an electron behave in a crystal? In order to simplify the problem, we will suppose that the crystal is merely an infinite, one-
1. Energy Band Theory 9 dimensional chain of atoms. This assumption may seem rather coarse, but it preserves a key feature of the crystal: the periodic nature of the position of the atoms in the crystal. In mathematical terms, the expression of the periodic nature of the atom-generated potential wells can be written as V(xta+b)=v(x) (1.1.15) where a+b is the distance between two atoms in the x-direction(Figure AV(x) Aanr Figure 1.6: Periodic potential in a one-dimensional crystal The periodic nature of the potential has a profound influence on the wave function of the electron. In particular, the electron wave function must satisfy the time-independent Schrodinger equation whenever x+a+b is substituted for x in the operators that act on Y(x).[] This condition is obtained if the wave function satisfies the bloch theorem. which can be formulated as follows If V(x) is periodic such that V(x+a+b)=v(x), then Y(xta+b)=y(x)elk(a+ (1.1.16) a second formulation of the theorem is If V(x) is periodic such that V(x+a+b)=v(x), then Y()=u(x)elkx with u(x+a+b)= u(x) These two formulations are equivalent since Hax+a+b)=u(x+a+b)ekx+a+b)=u(x)hxeka+b)=y(x)eka+b Since the potential in the crystal, V(x), is a rather complicated function of r, we will use the approximation made by Kronig and Penney in 1931, in which V(x) is replaced by a periodic sequence of rectangular potential wells.[] This approximation may appear rather crude, but it preserves the periodic nature of the potential variation in the crystal while allowing a closed-form solution for Y(x). The resulting potential is depicted in
1. Energy Band Theory 9 dimensional chain of atoms. This assumption may seem rather coarse, but it preserves a key feature of the crystal: the periodic nature of the position of the atoms in the crystal. In mathematical terms, the expression of the periodic nature of the atom-generated potential wells can be written as: where a+b is the distance between two atoms in the x-direction (Figure 1.6). The periodic nature of the potential has a profound influence on the wave function of the electron. In particular, the electron wave function must satisfy the time-independent Schrödinger equation whenever x+a+b is substituted for x in the operators that act on This condition is obtained if the wave function satisfies the Bloch theorem, which can be formulated as follows: Since the potential in the crystal, V(x), is a rather complicated function of x, we will use the approximation made by Krönig and Penney in 1931, in which V(x) is replaced by a periodic sequence of rectangular potential the periodic nature of the potential variation in the crystal while allowing a closed-form solution for The resulting potential is depicted in If V(x) is periodic such that V(x+a+b) =V(x), then (1.1.16) A second formulation of the theorem is: If V(x) is periodic such that V(x+a+b) =V(x), then with u(x+a+b) = u(x). These two formulations are equivalent since wells.[4 ] This approximation may appear rather crude, but it preserves