Chapter 33、1If were to increase, the bandgap energywould decrease and the material would begin to behave less like a semiconductor and more like a metal、 If were to decrease, thebandgap energy would increase and thematerial would begin to behave more like an insulator、_______________________________________ 3、2Schrodinger's wave equation is:Assume the solution is of the form:Region I: 、 Substituting theassumed solution into the wave equation, we obtain:which beesThis equation may be written asSetting for region I, the equationbees:whereQ、E、D、In Region II, 、 Assume the sameform of the solution:Substituting into Schrodinger's waveequation, we find:This equation can be written as:Setting for region II, thisequation beeswhere againQ、E、D、_______________________________________ 3、3We haveAssume the solution is of the form:The first derivative isand the second derivative beesSubstituting these equations into thedifferential equation, we findbining terms, we obtainWe find thatQ、E、D、 For the differential equation in and theproposed solution, the procedure is exactly the same as above、_______________________________________ 3、4We have the solutionsfor andfor 、The first boundary condition iswhich yieldsThe second boundary condition iswhich yieldsThe third boundary condition iswhich yieldsand can be written asThe fourth boundary condition iswhich yieldsand can be written as_______________________________________3、5(b) (i) First point:Second point: By trial and error,(ii) First point:Second point: By trial and error,_______________________________________3、6(b) (i) First point:Second point: By trial and error,(ii) First point:Second point: By trial and error,_______________________________________ 3、7Let ,ThenConsider of this function、We findThenFor ,So that, in general,AndSoThis implies thatfor_______________________________________ 3、8(a)JFrom Problem 3、5JJor eV(b)JFrom Problem 3、5,JJor eV_______________________________________3、9(a)At ,JAt , By trial and error,JJor eV(b)At ,JAt 、 From Problem 3、5,JJor eV_______________________________________3、10(a)JFrom Problem 3、6,JJor eV(b)JFrom Problem 3、6,JJor eV_____________________________________3、11(a)At ,JAt , By trial and error,JJor eV(b)At ,JAt , From Problem 3、6,JJor eV_______________________________________ 3、12 For K,eVK, eVK, eVK, eVK, eVK, eV_______________________________________ 3、13The effective mass is given byWe haveso that_______________________________________ 3、14The effective mass for a hole is given byWe have thatso that_______________________________________ 3、15Points A,B: velocity in -x directionPoints C,D: velocity in +x directionPoints A,D:negative effective mass Points B,C:positive effective mass _______________________________________ 3、16For A:At m, eVOr JSoNowkgorFor B:At m, eVOr JSoNowkgor_______________________________________ 3、17For A:kgorFor B:kgor_______________________________________3、18(a)(i)orHz(ii)cmnm(b)(i)Hz(ii)cmnm_______________________________________3、19(c)Curve A: Effective mass is a constantCurve B: Effective mass is positivearound , and is negativearound 、_______________________________________ 3、20ThenandThenor_______________________________________3、21(a)(b)_______________________________________3、22(a)(b)_______________________________________ 3、23For the 3-dimensional infinite potential well, when , , and、 In this region, the wave equationis:Use separation of variables technique, so let Substituting into the wave equation, we have Dividing by , we obtainLetThe solution is of the form:Since at , thenso that 、Also, at , so that、 Then whereSimilarly, we haveandFrom the boundary conditions, we findandwhereandFrom the wave equation, we can writeThe energy can be written as_______________________________________ 3、24The total number of quantum states in the3-dimensional potential well is given(in k-space) bywhereWe can then writeTaking the differential, we obtainSubstituting these expressions into the density of states function, we haveNoting thatthis density of states function can besimplified and written asDividing by will yield the density ofstates so that_______________________________________ 3、25For a one-dimensional infinite potential well, Distance between quantum statesNowNowThenDivide by the "volume" a, soSomJ_______________________________________3、26(a) Silicon,(i) At K, eVJThenmor cm(ii) At K,eVJThenmor cm(b) GaAs,(i) At K, Jmor cm(ii) At K, Jmcm_______________________________________ 3、27(a)Silicon,(i)At K, Jmor cm(ii)At K, Jmor cm(b)GaAs,(i)At K, Jmor cm(ii)At K, Jmor cm_______________________________________3、28(a)For ;eV; mJeV; mJeV; mJeV; mJ(b)For ;eV; mJeV; mJeV; mJeV; mJ_______________________________________3、29(a)(b)_______________________________________ 3、30Plot_______________________________________3、31(a)(b)(i)(ii)_______________________________________3、32(a),(b),(c),_______________________________________ 3、33or(a),(b),(c),_______________________________________3、34(a);;;;;(b);;;;;_______________________________________ 3、35andSo ThenOr_______________________________________ 3、36For , Filled stateJor eVFor , Empty stateJor eVTherefore eV_______________________________________3、37(a)For a 3-D infinite potential wellFor 5 electrons, the 5th electron occupiesthe quantum state ; soJor eVFor the next quantum state, which is empty, the quantum state is 、 This quantum state is at the same energy, soeV(b)For 13 electrons, the 13th electronoccupies the quantum state; soJor eVThe 14th electron would occupy the quantum state 、 This state is at the same energy, soeV_______________________________________ 3、38The probability of a state atbeing occupied isThe probability of a state atbeing empty isorso Q、E、D、_______________________________________3、39(a)At energy , we wantThis expression can be written asorThenor(b)At ,which yields_______________________________________3、40(a)(b) eV(c)ororwhich yields K_______________________________________3、41(a)or 0、304%(b)At K, eVThenor 14、96%(c)or 99、7%(d)At , for all temperatures_______________________________________3、42(a)ForThenFor , eVThenor(b)For eV,eVAt ,orAt ,or_______________________________________3、43(a)AtorAt , eVSoor(b)For ,eVAt ,orAt ,or_______________________________________ 3、44soor(a)At K, ForAt(b)At K, eVFor ,For ,At ,(eV)(c)At K, eVFor ,For ,At ,(eV)_______________________________________3、45(a)At ,Si: eV,orGe: eVorGaAs: eVor(b) Using the results of Problem 3、38, theanswers to part (b) are exactly the same asthose given in part (a)、_______________________________________3、46(a)oreVso K(b)or K_______________________________________3、47(a)At K,eVeVBy symmetry, for ,eVThen eV(b)K, eVFor , from part (a),eVThen eV_______________________________________。