Where V(x) is the periodic potentia

Where V(x) is the periodic potential with period of a. According to the Bloch- Floquet theorem discussed above, the general solution of Eq. (4.28) is given by
()()ikxkkxuxe⋅=rrφ (4.29)
Note that between the potential barriers (i.e., 0 < x < a), V(x) = 0, and Eq. (4.28) becomes
2220kokkxφφ∂+=∂ (4.30)
Where
222omEk=h (4.31)
is the wave vector of free electrons. Since the solution of electron wave functions given by Eq. (4.29) is valid everywhere in the periodic lattice, one can substitute Eq. (4.29) into Eq. (4.30) to obtain an equation that contains only the Bloch function uk(x), namely,
()222220kkokduduikkkudxdx++−= (4.32)
This is a second order differential equation with constant coefficients, and the roots of its characteristic equation are equal to –i(k ± ko). Thus, the general solution of Eq. (4.32) for uk(x) can be expressed as
(4.33) ()(cossinkoikxuxeAkxBkx−⋅=+
Where A and B are constants, which can be determined from the periodic boundary conditions. The first boundary condition can be obtained by noting the fact that both uk(x) and φk(x) are invariant under translational operation. Thus, one can write
(4.34) 0()()kkuu=
Where a is the period of the crystal potential (i.e.,V(x)=V(x+a)). To calculate the change in the slope of electron wave functions across the infinitely thin potential barrier at the atomic site, one can integrate Eq. (4.28) from x = 0–
on the left-hand side of the potential barrier to x = 0+ on the right-hand side of the potential barrier at x = 0. This yields
()2220020kkmEVxdxxφφ+−⎧⎫∂⎪⎛⎞+−⎡⎤⎨⎜⎟⎣⎦∂⎝⎠⎪⎪⎩⎭ =∫h (4.35)
Or
()()()2200kkkmC+−−=⎛⎞′′⎜⎟⎝⎠φφφh (4.36)
Equation (4.36) is obtained using the fact that as x→ 0 inside the potential barrier, integration of 0Edx=∫over the barrier width is equal to zero, and the change in the slope of electron wave functions (φk’=dφk/dx) across the potential barrier is given by Eq. (4.36). From Eq. (4.29) and Eq. (4.36), one obtains the derivative of uk as
()()()2200kkkmCuuu+−⎛⎞′′−=⎜⎟⎝⎠h (4.37)
Now, replacing 0+ = 0 and 0– = a in Eq. (4.37), the second boundary condition for uk(x) is given by
()()()220kkkmCuuau⎛⎞′′=+⎜⎟⎝⎠h (4.38)
Note that the first derivative of uk(x) is identical on the left-hand side of each potential barrier shown in figure 4.1b. Next, substituting the two boundary conditions given by Eqs. (4.34) and (4.38) into Eq. (4.33), one obtains two simultaneous equations for A and B:
()()cos1sin0ikaikaoAekaBeka−− −+ (4.39)
()221cossinikaikaooomCAikekaekka−−⎡⎤⎛⎞−−+−⎜⎟⎢⎥(sincos0ikaooooBkeikkakka−⎡ ++−⎣ (4.40)
In order to have a nontrivial solution for Eqs. (4.39) and (4.40) the determinant of the coefficients of A and B in both equations must be set equal to zero, which yields
()()21cos1sin02cossinsincosikaikaooikaikaikaoooooooekaekamCikekaekkakeikkakka−−−−−⎡⎤−⎣⎦=⎛⎞⎡⎤−−+−+−⎜⎟⎣⎦⎝⎠h (4.41)
Solving Eq.(4.41) one obtains
cossincosooPkakakaka⎛⎞=+⎜⎟⎝⎠ (4.42)
Where P = mCa/ħ2, and C is defined by
(4.43) 0()00()limVxdxCVxdx+−∞→→=∫
Equation (4.42) has a real solution for the electron wave vector krif the value of the right-hand side of Eq. (4.42) lies between –1 and +1. Figure 4.2 shows a plot of the right-hand side term of Eq. (4.42) versus koa for a fixed value of P. It is noted that the solution of Eq. (4.42) consists of a series of alternate allowed and forbidden regions with the forbidden regions becoming narrower as the value of koa becomes larger. The physical meanings of Figure 4.2 are discussed as follows.
It is noted that the magnitude of P is closely related to the binding energy of electrons in the crystal. For example, if P is zero, then one has the free-electron case, and the energy of electrons is a continuous function of wave vector k, as is given by Eq. (4.31). On the other hand, if P approaches infinity, then the energy of electrons becomes independent of k. This corresponds to the case of an isolated atom. In this case, the values of electron energy are determined by the condition that sin koa in Eq. (4.42) must set equal to zero as P approaches infinity, which implies koa = nπ. Thus, the electron energy levels are quantized for this case, and is given by
22222222onoknEmmaπ==hh (4.44)
Where n=1,2,3…. In this case, the electrons are completely bound to the atom and their energy levels become discrete. If P has a finite value, then the energy band scheme of electrons is characterized by the alternate allowed- and forbidden energy regions, as is shown in Figure 4.2. The allowed regions are the regions in which the magnitude of the right-hand side in Eq. (4.42) lies between –1 and +1, while the forbidden regions are the regions in which the absolute magnitude of the right-hand side is greater than one. It is further notice from this figure that the forbidden region should become smaller and the allowed region becomes larger as the value of koa increases.
Figure 4.3 shows the plot of electron energy as a function of P. As shown in this figure, at the origin where P = 0 corresponds to the free-electron case, and the energy of electrons is continuous in k-space. In the region where P has a finite value, the energy states of electrons are characterized by a series of allowed (shaded area) and
forbidden regions. As P approaches infinity, the energy of electrons becomes discrete (or quantized), which corresponds to the case of an isolated atom with atomic spacing a → ∞.
Based on the Kronig-Penney model discussed above, a schematic energy band diagram for the 1-D periodic lattice is illustrated in Figure 4.4, which is plotted in the extended zone scheme. The values of wave vector, k, are given by –nπ/a,..., – π/a, 0, + π/a,..., nπ/a. The first Brillouin zone, known as the unit cell of the reciprocal lattice, is defined by the wave vectors with values varying between –π/a and +π/a. Figure 4.4 illustrates two important physical aspects of the energy band diagram: (i) at the zone boundaries where k = ±nπ/a and n = 1, 2, 3,..., there exists an energy discontinuity, and (ii) the width of allowed energy bands increases with increasing electron energy, and the width of forbidden gaps decreases with increasing electron energy.
If the energy band diagram (i.e., E vs. k) is plotted within the first Brillouin zone, then it is called the reduced zone scheme. The reduced zone scheme (i.e., –π/a ≤ k ≤ π/a) is more often used than the extended zone scheme because for any values of wave vector 'kurin the higher zones there is a corresponding wave vector kr in the first Brillouin zone, and hence it is easier to describe the electronic states and the related physical properties using the reduced zone scheme. The relation between 'kurand krcan be obtained via the translational symmetry operation, which is given by
2'/kknaπ=±rr (4.45)
Where represents the wave vector in the higher zones, 'kurkris the corresponding wave vector in the first Brillouin zone, n = 1, 2, 3...., and a denotes the lattice constant of the crystal. Thus, the reduced zone scheme contains all the information relating to the electronic states in the crystalline solids.
The Kronig-Penney model described above can be employed to construct the energy band diagrams of an isolated silicon atom and an artificial 1-D periodic silicon lattice. Figure 4.5a and b show the discrete energy level schemes for such an isolated silicon atom and the energy band diagram for a 1-D silicon lattice, respectively. As shown in Figure 4.5a, electrons in the 3s and 3p shells are known as the valence electrons while electrons in the 1s, 2s, and 2p orbits are called the core electrons. When the valence electrons are excited into the conduction band, the conductivity of a semiconductor increases. It is noted that as the spacing of silicon atoms reduces to a few angstroms, the discrete energy levels shown in figure 4.5a broaden into energy bands, and each allowed energy band is separated by a forbidden band gap. In this energy band scheme the highest filled band (i.e., 3s and 3p states for
silicon) is called the valence band, while the lowest empty band is called the conduction band. In a semiconductor, a forbidden band gap always exists between the conduction and the valence bands while in metals the energy bands are usually continuous. For most semiconductors, the band gap energies may vary between 0.1eV and 6.2 eV.
The main difference in the energy band scheme between the 1-D and 2- or 3-D crystal lattice is that, in the 1-D case, an energy discontinuity always exists at the zone boundary, and hence the energy band is characterized by a series of alternate allowed and forbidden bands. However, in the 3-D case, the energy band discontinuity may or may not exist since the values of kmax at the zone boundaries along different crystal orientations may be different, as is clearly illustrated in figure 4.6. This will lead to an overlap of energy states at the zone boundaries and hence the possible disappearance of the bandgap in the 3-D energy band diagram. It should be mentioned that the electron wave functions in a 3-D periodic crystal lattice are of the Bloch type and can be described by Eq. (4.17). In the next section we shall describe the nearly- free electron (NFE) approximation for constructing the energy band scheme of valence electrons in a semiconductor. It is noted that the NFE approximation can only provide a qualitative description of the energy band schemes for the valence electrons in a 3-D crystal lattice. To obtain true energy band structures for semiconductors and metals, more rigorous and sophisticated methods such as the pseudopotential and orthogonalized plane wave techniques must be used in calculations of the energy band structures for these materials.