12 Solid-State Physics for Electronics 2)=n(2 from which it can be deduced that ka tan -ka aa an asymmetric solution in the form yu(x)=B sin kx. Just as before, the onditions of continuity make it possible to obtain the relationship written Equations [1.10] and [1. 11] can be combined in the form an ka= In addition, equations [1.10] and [lIl must be compatible with the equations that define a and k so that 2mn a Ya237 F
12 Solid-State Physics for Electronics 2 2 2 2 I II I II a a x x a a x x §· §· ¨¸ ¨¸ ©¹ ©¹ \ \ w\ w\ w w §· §· ½ ¨¸ ¨¸ ° ©¹ ©¹ ° ¾ ªº ª º ° «» « » ¬¼ ¬ ¼ ° ¿ [1.10] from which it can be deduced that 1 tan , 2 ka ka a D – an asymmetric solution in the form II \ ( ) sin . x B kx Just as before, the conditions of continuity make it possible to obtain the relationship written: 1 cotan . 2 ka ka a D [1.11] Equations [1.10] and [1.11] can be combined in the form: 2 tan . ² ² k ka k D D [1.12] In addition, equations [1.10] and [1.11] must be compatible with the equations that define D and k, so that: 2 ²²² . ² mW D J k = [1.13] ka Da 0 S Ja 2S 3S Figure 1.4. Solutions for narrow potential wells
The problem is normally resolved graphically. This involves noting the points here aa=f(ka)at the intersection of the curves described by equations [1.10] and [1. 11] with the curve given by equation [1.13](the quarter circle). The latter equation can be rewritten as ∞3+(ka0y=(m) [.14] The solutions for o and thus e(as E )correspond to the points where the circle of equation [1. 14] crosses with the deduced curves from equations [1.10 and [1.11 <π→ one symmetric solution→ one energy level 兀<ya<27→ two asymmetric solution two asymmetric solution → two energy levels 丌<ya<nπ→ n energy levels y(r) asymmetric level I asymmetric level Isymmetric level Figure 1.5. The first four energy levels along with the appearance of the corresponding wave functions in the narrow potential wells
Representations of Electron-Lattice Bonds 13 The problem is normally resolved graphically. This involves noting the points where Da = f (ka) at the intersection of the curves described by equations [1.10] and [1.11] with the curve given by equation [1.13] (the quarter circle). The latter equation can be rewritten as: 1/2 2 2 a ka a . ª º D J « » ¬ ¼ [1.14] The solutions for D and thus E (as ² ² 2 ) m E D = correspond to the points where the circle of equation [1.14] crosses with the deduced curves from equations [1.10] and [1.11]. If: Ja So one symmetric solution one energy level o 2 two asymmetric solution two energy levels two asymmetric solution S J So a ½ ¾ o ¿ S J So – 1 n an n energy levels Figure 1.5. The first four energy levels along with the appearance of the corresponding wave functions in the narrow potential wells Energy 1st symmetric level 2nd symmetric level 2nd asymmetric level 1st asymmetric level \( ) x
14 Solid-State Physics for Electronics (1)(2) (3)(4)(5) Figure 1.6. Scheme of the potential ene 4. 2. Solutions for two neighboring narrow potential wells Schrodingers equation, written for each of the regions denoted I to 5 igure 1.6 gives the following solutions(which can also be found in the problems later on in the book) V3 = A char Conditions of continuity for x=dn2 and for x=dn2 +an make it possible to state a tan ka 1+-tan ka tha Similarly, for the asymmetric solution we obtai
14 Solid-State Physics for Electronics Figure 1.6. Scheme of the potential energies of two narrow potential wells brought close to one another 1.4.2. Solutions for two neighboring narrow potential wells Schrödinger’s equation, written for each of the regions denoted 1 to 5 in Figure 1.6 gives the following solutions (which can also be found in the problems later on in the book): – Symmetric solution: 1 2 3 4 5 cos cos . x x Ce B kx A ch x B kx Ce D D \ \ M \ D \ M \ Conditions of continuity for x = d/2 and for x = d/2 + a/2 make it possible to state that: 2 2 tan ; 1 tan d k d k ka th k ka th D D D D D [1.15] – Similarly, for the asymmetric solution we obtain: 2 k 2 tan coth . 1 tan coth d k d ka k ka D D D D D [1.16] E x – W ad/2 d/2 d/2 a+d/2 (1) (2) (3) (4) (5)
1.4.2. 1. Neighboring potential wells that are well separated If d is very large, equations [1. 15] and [1. 16become k 1+-tan ka and tend to give the same solutions as those obtained above for narrow wells. In effect, by making -=tan 0, equation [1. 17] is then written as tan 0= tan(ka-0) for which the solution is 0=-(ka+ nT). This in turn gives if n is odd then In effect, we again find the solutions of equations [1.10] and [1. 11 for isolated wells, which is quite normal because when d is large the wells are isolated. Here though with a high value of d, the solution is degenerate as there are in effect two identical solutions. ie those of the isolated wells 1.4.2.2. Closely placed neighboring wells fd is small. we have e<< I and anh a equations [1. 15] and [1.16 give cotham=1+2e-ad ka-2[1-2e k 1+-(1-2e ad)tanBa 1+-(1+2e
Representations of Electron-Lattice Bonds 15 1.4.2.1. Neighboring potential wells that are well separated If d is very large, equations [1.15] and [1.16] become: k tan 1+ tan k ka k ka D D D [1.17] and tend to give the same solutions as those obtained above for narrow wells. In effect, by making tan k D T , equation [1.17] is then written as tan tan T T ka for which the solution is 1 2 T S ka n . This in turn gives: – if n is even then 2 tan ; ka k D – if n is odd then 2 cotan . ka k D In effect, we again find the solutions of equations [1.10] and [1.11] for isolated wells, which is quite normal because when d is large the wells are isolated. Here though with a high value of d, the solution is degenerate as there are in effect two identical solutions, i.e. those of the isolated wells. 1.4.2.2. Closely placed neighboring wells If d is small, we have 1 d e D and: 2 2 tanh 1 2 equations [1.15] and [1.16] give: coth 1 2 d d d d e e D D D ½ ° ¾ D ° ¿ - d 1 - 2 - d 1 - 2e k tan e 1+ tan a k ka k D D D D ª º D ¬ ¼ E [1.18] and - d 1 + 2 - d 1 + 2e k e 1+ tan a k tg a k D D D D ª º D E ¬ ¼ E [1.19]
16 Solid-State Physics for Electronics For the single solution (oo) in equation [1.17(if the wells are infinitely separated) there are now two solutions: one is as from equation [1. 18] and the other is aa from equation [1. 19]. For isolated or well separated potential wells, all states ( symmetric or asymmetric)are duplicated with two neighboring energy states(as as and aa are in fact slightly different from ao). The difference in energy between the symmetric and asymmetric states tends towards zero as the two wells are separated (d -oo). In addition, we can show quite clearly that the symmetric state is lower han the asymmetric state as in Figure 1.7 asvmm svmm 2 wells in proximity olated wells Figure 1.7. Evolution ofenergy levels and electronic states on going from one isolated well to two close wells The example given shows how bringing together the discrete levels of the isolated atoms results in the creation of energy bands. The levels permitted in these bands are such that two wells induce the formation of a"band of two levels n wells induce the formation of a"band"of n levels
16 Solid-State Physics for Electronics For the single solution (D0) in equation [1.17] (if the wells are infinitely separated) there are now two solutions: one is Ds from equation [1.18] and the other is Da from equation [1.19]. For isolated or well separated potential wells, all states (symmetric or asymmetric) are duplicated with two neighboring energy states (as Ds and Da are in fact slightly different from D0). The difference in energy between the symmetric and asymmetric states tends towards zero as the two wells are separated ( ). d o f In addition, we can show quite clearly that the symmetric state is lower than the asymmetric state as in Figure 1.7. Figure 1.7. Evolution of energy levels and electronic states on going from one isolated well to two close wells The example given shows how bringing together the discrete levels of the isolated atoms results in the creation of energy bands. The levels permitted in these bands are such that: – two wells induce the formation of a “band” of two levels; – n wells induce the formation of a “band” of n levels. \( ) x 1st symm. 1st symm. 1st asymm. 1st asymm. 2nd asymm. 2nd symm. isolated wells 2 wells in proximity