aper Figure 1.7, and the following notations will be used: the inter-atomic distance is a+b, the potential energy near an atom is VI, and the potential energy between atoms is Vo. Both VI and Vo are negative with respect to an arbitrary reference energy, V=0, taken outside the crystal. We will study the behavior of an electron with an energy e lying between v/ and Vo(Vo>E>Vi). This case is similar to a ls electron previously shown for lithium V(x) position of an atom region II region I Figure 1.7: Periodic potential of the Kronig and Penney model In region I (O<x<a), the potential energy is v(x)=vI, and the time- independent Schrodinger equation can be written as x)+E·v(x)=0 In region Il (-b<x<0), the potential energy is v(x)=vo, and independent Schrodinger equation becomes ds (x) m d (11.18) The solution to these homogenous second-order differential equations are yI(x)=A expli Bx)+b exp(-jBx) with B= 2m(E-vu (1.1.19) yn(x)=C exp(ax)+ Dexp(-aox)with a= m(0-E Note that a and B are real numbers. The periodic nature of the crystal lattice suggests that the wave function satisfies the Bloch theorem (1. 1. 16) and can be written in the following form: Y(x)=uk(x)expljkx)
10 Chapter 1 Figure 1.7, and the following notations will be used: the inter-atomic distance is a+b, the potential energy near an atom is and the potential energy between atoms is Both and are negative with respect to an arbitrary reference energy, V=0, taken outside the crystal. We will study the behavior of an electron with an energy E lying between and . This case is similar to a 1s electron previously shown for lithium. In region I (0<x<a) , the potential energy is and the timeindependent Schrödinger equation can be written as: In region II (-b<x<0), the potential energy is and the timeindependent Schrödinger equation becomes: The solution to these homogenous second-order differential equations are: and Note that and are real numbers. The periodic nature of the crystal lattice suggests that the wave function satisfies the Bloch theorem (1.1.16) and can be written in the following form:
1. Energy Band Theory here uk(x) is a periodic function with period a+b, which imposes uk(xtn(a+b))=uk(x). One can thus write yI(x+n((a+b))= yn(x) explink(a+b)) (1.1.21) and Yu(x+nla+b)= yn(x) explink(a+b) (11.22) Boundary conditions must be used to calculate the four integr constants A, B, C and D of Equations 1.1.19 and 1. 20. This can be by imposing the condition that the wave function, y(x), and its first derivative, dy(x)/dx, are continuous at x=0 and x=a. By doing so one obtains the following equations: o Y(x)is continuous at x=. Thus 1(0)= yuo), which yields A+B=C+D (11.23) o dY(x)/dx is continuous at x=0. Therefore, dy,(0)dx= dyu()ld. jB(A-B)=a(C-D) (1.1.24) o y(x) is continuous at x=a giving Yr(a)=Yu(a). Using the Bloch theorem( Equation 1. 1. 16)at x=a we have yr(a)=yi-b) explk(a+b)), exp(ik(a+b))/A exp(-jBb)+B exp(pb))=C exp(aa)+D exp(-ata) (1.1.25) o dy(x)/dx is continuous at x=a giving d y (a)/dx= dyn(a)/dx. Usin Bloch's theorem: Yr(a)=yr-b) exp(jk(a+b)we obtain explik(a+b))jB/lA exp(-jBb)-B exp(iBb) =a/C exp(aa)-D exp(-aa)(1.1.26) Equations (1. 1. 23)to(1. 1. 26)form a system of four equations with four unknowns: A, B, C and D. This system can be written in a matrix form: exp(k(a+b))exp(-jBb)exP(ik(a+b))exp(Bb) -exp(aa)-exp(-aa)c=o exp(k(a+b)JjBexp(-jBb)-explik(a+b) Bexp(Bb)-aexp(aa)aexp(-aa)LD Lo (1.1.27) In order to obtain a non-trivial solution for A. B. C and d. i.e. a solution different from A=B=C=D=0. the determinant of the 4x4 matrix must be equal to zero, which is equivalent to writing(see Problem 1.5) 2aB sinh(aa) sin(Bb)+ cosh(aa) cos(Bb)=cos(k(atb))(1.1.28)
1. Energy Band Theory 11 where is a periodic function with period a + b, which imposes One can thus write: and Boundary conditions must be used to calculate the four integration constants A, B, C and D of Equations 1.1.19 and 1.1.20. This can be done by imposing the condition that the wave function, and its first derivative, are continuous at x=0 and x=a. By doing so one obtains the following equations: is continuous at x=0. Thus which yields: is continuous at x=0. Therefore, is continuous at x=a giving Using the Bloch theorem (Equation 1.1.16) at x=a we have which yields: is continuous at x=a giving Using Bloch's theorem: exp(jk(a+b)) we obtain: Equations (1.1.23) to (1.1.26) form a system of four equations with four unknowns: A, B, C and D. This system can be written in a matrix form: In order to obtain a non-trivial solution for A, B, C and D, i.e. a solution different from A=B=C=D=0, the determinant of the 4×4 matrix must be equal to zero, which is equivalent to writing (see Problem 1.5):
12 Chapter I The right-hand term of this equation depends only on E, through a and (Expressions 1. 1. 19 and 1. 1. 20). Let us call this term P(E) and rewrite Expression 1. 28 in the following form: P(E)= cos(k(a+b)) (1.1.29) The right-hand side of Equation 1. 1. 29 is sketched as a function of energy in Figure 1. 8. Because the argument in the exponential term of (1.1.16) must be imaginary, k must be real. Therefore, simultaneous solution of both left-and right-hand side of Equation 1.1.29 imposes that-IsP(E)S 1. This defines permitted values of energy forming the energy bands, and forbidden values of energy constituting forbidden energy bands. This important result is the same to that intuitively unveiled in Section 1.1.2: in a crystal there are bands of permitted energy values separated by bands of forbidden energy values Note: In the case when the electron energy is greater than Vo, E-vo has a positive value and Equation 1. 1. 20 becomes Yin()=Cexpiax)+D exp(iac)with a=\/2mvo-E) In that case the Kronig- Penney model yields an equation different from relationship 1. 1.28; however, the same general conclusion can be drawn, i.e., the existence of permitted and for bidden energy bands Permitted Permitted Permitted 1.5 1 10121416 Energy(ev) Figure 1.8: P(E)as a function of the electron energy, E, for silicon The shaded areas correspond to the permitted energy bands, where here is a solution to Equation 1. 1. 29
12 Chapter 1 The right-hand term of this equation depends only on E, through and (Expressions 1.1.19 and 1.1.20). Let us call this term P(E) and rewrite Expression 1.1.28 in the following form: The right-hand side of Equation 1.1.29 is sketched as a function of energy in Figure 1.8. Because the argument in the exponential term of (1.1.16) must be imaginary, k must be real. Therefore, simultaneous solution of both left- and right-hand side of Equation 1.1.29 imposes that 1. This defines permitted values of energy forming the energy bands, and forbidden values of energy constituting forbidden energy bands. This important result is the same to that intuitively unveiled in Section 1.1.2: in a crystal there are bands of permitted energy values separated by bands of forbidden energy values. Note: In the case when the electron energy is greater than has a positive value and Equation 1.1.20 becomes: In that case the Krönig-Penney model yields an equation different from Relationship 1.1.28; however, the same general conclusion can be drawn, i.e., the existence of permitted and for bidden energy bands
1. Energy Band Theory Using Expression 1. 1. 28 the e(k) diagram can be plotted as well. Figure 1.9 presents the energy of the electron as a function of the wave number k. The e(k) diagram for a free electron is also shown. It can be observed that the energy of the electron in a crystal coarsely represents the same dependence on k as that of a free electron. The main differences reside in the existence of forbidden energy values and curvatures of each segment of the e(k) curves. Permitted band Forbiden band Figure 1.9: Energy versus k in a one-dimension crystal. The dotted line parabola represents the e(k)relationship for a free electron(from are 1.1) AAA period. The shaded area highlights the first Brillouin zone(Bz).6 igure 1. 10: E(k) diagram of Figure 1.9, repeated with a 2r/( Fi
1. Energy Band Theory 13 Using Expression 1.1.28 the E(k) diagram can be plotted as well. Figure 1.9 presents the energy of the electron as a function of the wave number k. The E(k) diagram for a free electron is also shown. It can be observed that the energy of the electron in a crystal coarsely represents the same dependence on k as that of a free electron. The main differences reside in the existence of forbidden energy values and curvatures of each segment of the E(k) curves
Because of the periodicity of the crystal lattice (period a+b), the periodicity of the reciprocal lattice(k-space)is atb. The E(k) curve can be extended from k with a periodicity of a+b yields the permitted energy values for the entire one-dimensional crystal The e(k) curves shown in Figure 1.10 can be limited to k-values ranging fror 1 to a+b without any loss of information. This particular region of the k-space is called the first Brillouin zone. The second a+b a+b and from atb a+b eg* the Brillouin zone extends from atb o a+b and from、x third zone extends from 23 Applying the Born-von Karman boundary conditions(Expression 1.1.12) the tensional crystal yields the values for k exp0Na+b)=1→k=-2nm=D+1,+2,+3…)(11.30) where N is the number of lattice cells in the crystal (or the number of atoms in the case of a one-dimension crystal). The length of the crystal is equal to N(a+b). Since we limit our study to the first Brillouin zone, the k- values which have to be considered are given by the following relationshipatb sk<atb e value k=_I k= a+b is excluded because it duplicate of the k=lk wave number). The corresponding values for n range from -N/2 to(N/2-1) Therefore, the values of k to consider are: k n=0,土l 1),-N2) (1.1.31) There are thus wave numbers in the first brillouin zone. which corresponds to the number of elementary lattice cells. For every wave number there is a permitted energy value in each energy band. By virtue of the Pauli exclusion principle, each energy band can thus contain a maximum of 2N electrons The one-dimensional volume of the first Brillouin zone is equal 27/(a+b). Since it contains N k-values, the density of k-values in the first Brillouin zone is given by (k)=density ofk =number of k-v
14 Chapter 1 Because of the periodicity of the crystal lattice (period = a + b), the periodicity of the reciprocal lattice (k-space) is The E(k) curve can be extended from with a periodicity of which yields the permitted energy values for the entire one-dimensional crystal (Figure 1.10). The E(k) curves shown in Figure 1.10 can be limited to k-values ranging from to without any loss of information. This particular region of the k-space is called the first Brillouin zone. The second Brillouin zone extends from to and from to the third zone extends from to and from to etc. Applying the Born-von Karman boundary conditions (Expression 1.1.12) to the one-dimensional crystal yields the values for k: where N is the number of lattice cells in the crystal (or the number of atoms in the case of a one-dimension crystal). The length of the crystal is equal to N(a+b). Since we limit our study to the first Brillouin zone, the kvalues which have to be considered are given by the following relationship: (the value is excluded because it is a duplicate of the wave number). The corresponding values for n range from -N/2 to (N/2-1). Therefore, the values of k to consider are: There are thus N wave numbers in the first Brillouin zone, which corresponds to the number of elementary lattice cells. For every wave number there is a permitted energy value in each energy band. By virtue of the Pauli exclusion principle, each energy band can thus contain a maximum of 2N electrons. The one-dimensional volume of the first Brillouin zone is equal to Since it contains N k-values, the density of k-values in the first Brillouin zone is given by: