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1、Allowed and Forbidden Energy Bands Formation of Energy Bands 02Energy level and band03Pauli exclusion principle01The KronigPenney Model 04The k-Space Diagram 05Pauli exclusion principleThe Pauli（泡利）exclusion principle was proposed by Pauli on January 16,1925.It is impossible for two or more electron
2、s in an atom to be in the same state.The state of an electron can be represented by four quantum numbers(n,m,l,s),then it is impossible for two or more electrons in an atom to have exactly the same four quantum numbers.Pauli（泡利）Spin upSpin downBand formationProbability density function of an isolate
3、d hydrogen atom.（d）Overlapping probability density functions of two adjacent hydrogen atoms.The wave functions of two electrons overlapThe splitting of the n=1 state.Split into energy bandsEnergy level and bandEnergy level will split into a band of energy levelsForbidden band Conduction bandSchemati
4、c of an isolated silicon atomValence band The KronigPenney Model Potential function of a single atomic Overlapped potential functionsPotential function of unidimensional atomic Bloch theoremjkxexux)()(The KronigPenney Model 0)()()(21221212xukdxxdujkdxxudIn region I,0 xa,V(x)=0 In region II，-bxV 0,th
5、e parameter is real,whereas if E V 0,then is imaginary.The KronigPenney Model x=0,The periodicity and the continuity condition mean that the function u 1,as x a,is equal to the function u 2,as x -b.u1(0)=u2(0)A+B-C-D=00201xxdxdudxdu(-k)A-(+k)B-(-k)C+(+k)D=0u1(a)=u2(-b)0)()()()(bkjbkjakjakjDeCeBeAebx
6、axdxdudxdu210)()()()()()()()(bkjbkjakjakjDekCekBekAekThe KronigPenney Model According to the boundary conditions,solve the system of homogeneous equations)(cos)(cos(cos)(sin(sin2)(22bakbabaFocusing on EV0,the parameter is then an imaginary quantity,define=j)(cos)(cos(cos)(sin(sin2)(22bakbabaLet the potential barrier width b0 and the barrier height V0 ,but such that the product bV0 remains finite.kaaaaPcoscossinThe KronigPenney Model The k-Space DiagramThe k-Space DiagramThe E versus k diagram showing 2?displacements of several sections of allowed energy bands.The k-Space Diagram