The materials science of semiconductors bonding and antibonding combinations and finally bands of states, which we leave for Chapter 5. To see why it should be possible to mix electronic states in linear combinations. it is useful to consider some direct solutions equation, which governs the motion of electrons in an arbitrary potential. As we will see,this is a second-order linear differential equation. It therefore has particular solutions that can be constructed from linear combinations of any set of solutions of the general equation. Consequently, we may suspect that, at least in some cases, a linear combination of atomic orbitals should describe the general problem of a solid In the following discussion we will assume that the material is a periodic crystal Some of the results turn out to be applicable in most respects to aperiodic (amorphous)structures as well Electrons behave as both waves and particles. The consequences of their wave and particle nature are derived through the formalism of quantum mechanics. The requirement for conservation of energy and momentum forces the electrons to select specific states described by"quantum numbers, analogous to resonant vibrations of a string on a musical instrument. The "resonant"states associated with each set of quantum numbers results in a set of wave functions"which describe the probability of finding an electron around a given location at a given time. The wave functions the resonant states are found as follows The total energy, Etot, of an electron is the sum of its potential and kinetic energies In classical terms one could express this relationship as where U(r) describes the local potential energy of the particle at position r and the kinetic energy is given by the classical expression Ekin="/2m in which m is the particle mass and p is the momentum. This equation applies to any classical bod When the object in question is of a scale small enough that quantum mechanical behaviors become dominant, we need to rewrite Equation 2. 1 in quantum mechanical terms. Under such conditions, the exact energy(E)at any specific time(t)can not be described to within an accuracy better than AEAt=h where h is Plank's constant divided by 2T. To account for this uncertainty the particle must be described by a probability distribution(its wave function) rather than by indicating a specific position. Its total energy is given by the change in the wave function, f(r, t), per unit time multipled by ih, where i'=-1. The momentum of a quantum particle is likewise, the spatial derivative of the wave function multiplied by ih. Based on the classical behavior and using the mathematics of operators, the kinetic energy, p/2m becomes the second derivative of the wave function times -/2m. Similarly, the potential energy represents a weighted average potential using y as the weighting function. Substituting these expressions in Equation 2. 1 yields d n ⅴ+U(ryH dt 2m
22 bonding and antibonding combinations and finally bands of states, which we leave for Chapter 5. To see why it should be possible to mix electronic states in linear combinations, it is useful to consider some direct solutions of the Schrödinger equation, which governs the motion of electrons in an arbitrary potential. As we will see, this is a second-order linear differential equation. It therefore has particular solutions that can be constructed from linear combinations of any set of solutions of the general equation. Consequently, we may suspect that, at least in some cases, a linear combination of atomic orbitals should describe the general problem of a solid. In the following discussion we will assume that the material is a periodic crystal. Some of the results turn out to be applicable in most respects to aperiodic (amorphous) structures as well. Electrons behave as both waves and particles. The consequences of their wave and particle nature are derived through the formalism of quantum mechanics. The requirement for conservation of energy and momentum forces the electrons to select specific states described by “quantum numbers,” analogous to resonant vibrations of a string on a musical instrument. The “resonant” states associated with each set of quantum numbers results in a set of “wave functions” which describe the probability of finding an electron around a given location at a given time. The wave functions of the resonant states are found as follows. The total energy, Etot, of an electron is the sum of its potential and kinetic energies. In classical terms one could express this relationship as Etot = p2 /2m + U(r), 2.1 where U(r) describes the local potential energy of the particle at position r and the kinetic energy is given by the classical expression Ekin= p2 /2m in which m is the particle mass and p is the momentum. This equation applies to any classical body. When the object in question is of a scale small enough that quantum mechanical behaviors become dominant, we need to rewrite Equation 2.1 in quantum mechanical terms. Under such conditions, the exact energy (E) at any specific time (t) can not be described to within an accuracy better than ∆E∆t = = where = is Plank’s constant divided by 2π. To account for this uncertainty the particle must be described by a probability distribution (its wave function) rather than by indicating a specific position. Its total energy is given by the change in the wave function, Ψ(r,t), per unit time multipled by i =, where i2=-1. The momentum of a quantum particle is, likewise, the spatial derivative of the wave function multiplied by i =. Based on the classical behavior and using the mathematics of operators, the kinetic energy, p2 /2m becomes the second derivative of the wave function times - =2 /2m. Similarly, the potential energy represents a weighted average potential using Ψ as the weighting function. Substituting these expressions in Equation 2.1 yields: i= dΨ dt = − =2 2m ∇2 Ψ + U(r → )Ψ, 2.2 The Materials Science of Semiconductors
The Physics of solids where v is referred to as the laplacian and is the second spatial derivative of the function it operates on(in this case the wave function). Equation 2.2 is the full time dependent Schrodinger Equation and describes not only the steady-state behavior an electron but also the way in which the electron changes energy as a function of time. Whenever the potential that the electron experiences does not change with time, the time variable can be separated from the space variable. In this case, the energy of the particle cannot change with time, and the spatial-portion of the Schrodinger Equation becomes Ey(r)=(-h/2m)V-y(r)+U(r)y(r) 2.3 where v(r) is the time-independent wave function. The electronic structure of solids is derived by the solution of this equation under the boundary conditions appropriate to the solid being modeled. We will now consider some solutions to Equation 2.3 2.1.1 Free electrons in solids The simplest form of Equation 2.3 is the special case of U(r=0, where there is no ential affecting the motion of electrons. For simplicity we will make the further restriction of considering only a one-dimensional problem. In this case h2 d2 v(x) 2m dx The general solution to this equation, obtained by Fourier transform methods, is a linear combination of two waves moving in the positive and negative x directions v(x)=Ae*±Ae-k where A+ and A. are the amplitudes of the two waves, and wavenumber of the waves(electrons) with wavelength A. The energies of these waves are determined by substituting Equation 2.5 into 2. 4. The second derivative of v(x)from Equation 2.5 is dw(x)dx=kv(). Thus v(x)+Ey(x)=0 from which E 方2k2 his holds for any A, and a but requires A++A-=l for a wave with unit amplitude. The y(x)are known as eigenvectors of Equation 2. 4 and the energies are the eigenvalues. The momentum of this wave is p=hk, thus k represents the electron momentum to within a factor of h. This energy vs wavenumber ["E(k)] relationship is illustrated in Figure 2.1
The Physics of Solids 23 where ∇2 is referred to as the Laplacian and is the second spatial derivative of the function it operates on (in this case the wave function). Equation 2.2 is the full timedependent Schrödinger Equation and describes not only the steady-state behavior of an electron but also the way in which the electron changes energy as a function of time. Whenever the potential that the electron experiences does not change with time, the time variable can be separated from the space variable. In this case, the energy of the particle cannot change with time, and the spatial-portion of the Schrödinger Equation becomes: E ψ(r) = (- =2 /2m) ∇2 ψ(r) + U(r)ψ(r), 2.3 where ψ(r) is the time-independent wave function. The electronic structure of solids is derived by the solution of this equation under the boundary conditions appropriate to the solid being modeled. We will now consider some solutions to Equation 2.3. 2.1.1 Free electrons in solids The simplest form of Equation 2.3 is the special case of U(r)=0, where there is no potential affecting the motion of electrons. For simplicity we will make the further restriction of considering only a one-dimensional problem. In this case, Eψ(x) + =2 2m d2 ψ dx2 = 0. 2.4 The general solution to this equation, obtained by Fourier transform methods, is a linear combination of two waves moving in the positive and negative x directions: ψ(x) = A+e ikx ± A−e−ikx , 2.5 where A+ and A- are the amplitudes of the two waves, and k = 2π λ is the wavenumber of the waves (electrons) with wavelength λ. The energies of these waves are determined by substituting Equation 2.5 into 2.4. The second derivative of ψ(x) from Equation 2.5 is d2 ψ(x)/dx2 = k2 ψ(x). Thus, − =2 k2 2m ψ(x) + Eψ(x) = 0 2.6 from which E = =2 k2 2m + - + 2 +A– 2 =1 for a wave with unit amplitude. The ψ(x) are known as eigenvectors of Equation 2.4 and the energies are the eigenvalues. The momentum of this wave is p = =k, thus k represents the electron momentum to within a factor of = . This energy vs. wavenumber [“E(k)”] relationship is illustrated in Figure 2.1. . This holds for any A and A but requires A
The materials science of semiconductors igure 2. 1: The energy vs. momentum diagram for a free electron in the absence of a periodic 2.1.2 Free electrons in a periodic potential In a solid there is a regular spacing of atoms. In a crystal, this spacing is defined by he translation vectors of the Bravais lattice. (See Chapter 4 for a description of semiconductor crystal lattices. )In an amorphous material the spacing is the average distance from one atom to its nearest neighbors. The result is the same- there is an imposed periodicity on the wave functions. The wave function must have the same value at equivalent positions in the solid. These positions are separated by lattice translation vectors R, thus w(r)=v(r+R)in a crystal. For a wave of the form given in Equation 2.5, this imposes an additional condition that v(x)=v(x+Lx) where Lx is the lattice spacing along the x direction. Likewise, the potential energy of a particle will be periodic(at least locally) such that U(r)=U(r+R). The periodicity of y requires that ere cnk(r)cnk(r+R)are the Fourier components for wave vector k of wave on w(r)[the proof is called"Bloch's Theorem ]. It can further be shown that the electron wave vector is given by kb/n where b is a reciprocal lattice vector of the crystal lattice(see Chapter 4 )and N is the number of unit cells in the [real-space] lattice. Therefore, any change in k must be by a unit vector of the reciprocal lattice as in any diffraction problem. Electron waves in a solid are susceptible to scattering as one would have for x-rays and the problem can be represented with, for example, an Ewald sphere construction as for normal diffraction of x-rays in a periodic crystal The primary consequence of Equation 2.7 is that electron wave behaviors are reproduced whenever the wave vector k is changed by a translation vector of the reciprocal lattice, 2T/a, where a is the one dimensional lattice constant along a given direction. This means we can replicate Figure 2. 1 every 2/a units along the wave
24 Figure 2.1: The energy vs. momentum diagram for a free electron in the absence of a periodic potential. 2.1.2 Free electrons in a periodic potential In a solid there is a regular spacing of atoms. In a crystal, this spacing is defined by the translation vectors of the Bravais lattice. (See Chapter 4 for a description of semiconductor crystal lattices.) In an amorphous material the spacing is the average distance from one atom to its nearest neighbors. The result is the same – there is an imposed periodicity on the wave functions. The wave function must have the same value at equivalent positions in the solid. These positions are separated by lattice translation vectors R, thus ψ(r) = ψ(r+R) in a crystal. For a wave of the form given in Equation 2.5, this imposes an additional condition that ψ(x)=ψ(x+Lx) where Lx is the lattice spacing along the x direction. Likewise, the potential energy of a particle will be periodic (at least locally) such that U(r) = U(r+R). The periodicity of ψ requires that: ψ( G r ) = ei G k • G r ck ( G r ), 2.7 where cnk(r)=cnk(r+R) are the Fourier components for wave vector k of the wave function ψ(r) [the proof is called “Bloch’s Theorem”]. It can further be shown that the electron wave vector is given by k=b/N where b is a reciprocal lattice vector of the crystal lattice (see Chapter 4) and N is the number of unit cells in the [real-space] lattice. Therefore, any change in k must be by a unit vector of the reciprocal lattice, as in any diffraction problem. Electron waves in a solid are susceptible to scattering as one would have for x-rays and the problem can be represented with, for example, an Ewald sphere construction as for normal diffraction of x-rays in a periodic crystal. The primary consequence of Equation 2.7 is that electron wave behaviors are reproduced whenever the wave vector k is changed by a translation vector of the reciprocal lattice, 2π/a, where a is the one dimensional lattice constant along a given direction. This means we can replicate Figure 2.1 every 2π/a units along the wave The Materials Science of Semiconductors
The Physics of solids E(k 4π2π 0 2兀4兀 Figure 2. 2: The periodic structure of the free electron energy vs. wave number in a periodic solid. The minimum section of the plot needed to provide a complete description of the relationship of E to k is shaded gray vector axis as shown in Figure 2.2. Note that if you consider the symmetry of this plot you will find that all of the necessary information is contained within the space between 0<k<T/a. Thus, we will represent electron energy vs wave vector plots in this reduced zone of k values hereafter 2.1. 3 Nearly free electrons A somewhat more realistic picture is the case where U(r) is not zero or constant but varies weakly with position. This is the case referred to as nearly free"electron behavior. The solutions to the Schodinger equation can be constructed from the same set of plane waves we had in Equation 2. 7 with the proviso that U(r) is not too large In this case, a general solution to Equation 2.3 is still given, at least approximatel by Equation 2.7 but now the cnk must account for the effect of the periodic potential This imposes the following constraint on the coefficient k2-EF4+∑Uc=0 2
The Physics of Solids 25 vector axis as shown in Figure 2.2. Note that if you consider the symmetry of this plot you will find that all of the necessary information is contained within the space between 0 < k < π/a. Thus, we will represent electron energy vs. wave vector plots in this reduced zone of k values hereafter. 2.1.3 Nearly free electrons A somewhat more realistic picture is the case where U(r) is not zero or constant but varies weakly with position. This is the case referred to as “nearly free” electron behavior. The solutions to the Schödinger equation can be constructed from the same set of plane waves we had in Equation 2.7 with the proviso that U(r) is not too large. In this case, a general solution to Equation 2.3 is still given, at least approximately, by Equation 2.7 but now the cnk must account for the effect of the periodic potential. This imposes the following constraint on the coefficients: =2 2m k 2 − E ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ck + Uk' ck' k'≠k ∑ = 0, 2.8 Figure 2.2: The periodic structure of the free electron energy vs. wave number in a periodic solid. The minimum section of the plot needed to provide a complete description of the relationship of E to k is shaded gray
The materials science of semiconductors where the Uk terms are the Fourier components of the potential U for wavevector k. he free electron behavior is what is given inside the square brackets of Equation 2.8. Because the periodic potential is assumed to be small, the individual Ux terms are modest and the second term in Equation 2. 8 represents only a minor perturbation result. Furthermore because the c terms are Fourier coefficients, the ck onents can be obtained from the Fourier expansion of the Ux in the free electron waves. In other words, because both the wave function and the periodic potential can be expanded in the same Fourier terms, the ck terms are related to the UK terms via Equation 2.8. To construct a nearly free electron wave function, as modified by the Uk, from purely free electron waves we multiply each possible free electron wave by Cx and add the results to produce the new wave function solution in he presence of the periodic potential, as in Equation 2.7 The Uk terms serve to mix the free electron plane waves producing interference effects. The closer the plane wave is to the periodicity of the lattice the more strongly it will interact with the crystal and, likewise, the stronger the component of Fourier transform of Uk. Consider the interaction of two waves with the same or nearly the same wave vectors and energies. Graphically, the interactions occur near the points of intersection of curves in Figure 2.2. When the energy difference between different branches of the e(k) diagram(different curves in Figure 2.2)is large on a scale of the potential energy, then the behavior is essentially free electron like. However, near the intersection of two curves the energies are modified Approximating the periodic potential with only its first Fourier component, then Equation 2.8 yields two equations for the two curves, which can be represented matrix form as E-E -U E-E 0, where E,=hki d e 2m are the free-electron-like behaviors for the two (subscripts 1 and 2)near the meeting point, E is the energy at the meeting and U is the first Fourier component of the periodic potential. Note that at the intersection, k=k, so that e=E. This condition has the solution hk 2m See Figure 2.3 for an illustration of this situation. The result has two implications that are important Waves interact with each other to raise or lower their combined energies. Graphically, when curves on the e(k) diagram intersect they may interfere resulting in local changes in their energies. Note that
26 where the Uk terms are the Fourier components of the potential U for wavevector k. The free electron behavior is what is given inside the square brackets of Equation 2.8. Because the periodic potential is assumed to be small, the individual Uk terms are modest and the second term in Equation 2.8 represents only a minor perturbation on the result. Furthermore, because the ck terms are Fourier coefficients, the ck components can be obtained from the Fourier expansion of the Uk in the free electron plane waves. In other words, because both the wave function and the periodic potential can be expanded in the same Fourier terms, the ck terms are related to the Uk terms via Equation 2.8. To construct a nearly free electron wave function, as modified by the Uk, from purely free electron waves we multiply each possible free electron wave by ck and add the results to produce the new wave function solution in the presence of the periodic potential, as in Equation 2.7. The Uk terms serve to mix the free electron plane waves producing interference effects. The closer the plane wave is to the periodicity of the lattice the more strongly it will interact with the crystal and, likewise, the stronger the component of the Fourier transform of Uk. Consider the interaction of two waves with the same or nearly the same wave vectors and energies. Graphically, the interactions occur near the points of intersection of curves in Figure 2.2. When the energy difference between different branches of the E(k) diagram (different curves in Figure 2.2) is large on a scale of the potential energy, then the behavior is essentially free electron like. However, near the intersection of two curves the energies are modified. Approximating the periodic potential with only its first Fourier component, then Equation 2.8 yields two equations for the two curves, which can be represented in matrix form as: E − E1 −U −U* E − E2 = 0 , 2.9 where E1 = =2 k1 2 2m and E2 = =2 k2 2 2m curves (subscripts 1 and 2) near the meeting point, E is the energy at the meeting point, and U is the first Fourier component of the periodic potential. Note that at the intersection, k1=k2 so that E1=E2. This condition has the solution E = =2 k2 2m ± U . 2.10 See Figure 2.3 for an illustration of this situation. The result has two implications that are important: • Waves interact with each other to raise or lower their combined energies. Graphically, when curves on the E(k) diagram intersect they may interfere resulting in local changes in their energies. Note that are the free-electron-like behaviors for the two The Materials Science of Semiconductors