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1、Chapter 1 Brief introduction to QuantumPhysics & Modern Theory of Solid1.1 Photons and electrons(光子和光子和电电子子)1.2 Schrdinger equation(薛定薛定谔谔方程方程)1.3 Application of Schrdinger equation(薛定薛定谔谔方程的运用方程的运用) FromPrinciplesofelectronicMaterialsDevices,SOKasap(McGraw-Hill,2005)1.1PhotonsandelectronsThe classi
2、cal view of light as an electromagnetic wave. An electromagnetic wave is a travelling wave which has time varying electric and magnetic fields which are perpendicular to each other and to the direction of propagation.Light as a waveThe electric field Ey at position x at time t may be described by:Wh
3、ere k is the wavenumber 波数波数 (k=2/, the wavelength), and the angular frequency 角角频频率率 (=2, the frequency). Particle-like properties of light are confirmed by many experiments:-Photoelectric effect(光光电电效效应应)-Compton scattering(康普康普顿顿散射散射)-Black body radiation(黑体黑体辐辐射射)Intuitive visualization of light
4、 consisting of a stream of photons From R. Serway etal, Modern Physics, Saunders College Publishing, 1989, p.56, Fig. 2.16(b)Scattering of an x-ray photon by a free electron in a conductor.Electron: particle! wave?Youngs double slit experiment with electrons (电电子的子的杨杨氏双氏双缝实验缝实验) involves an electron
5、 gun and two slits in a cathode ray tune (CRT) (hence in vacuum). Electrons from the filament are accelerated by a 50 kV anode voltage to produce a beam which is made to pass through the slits. The electrons then produce a visible pattern when they strike a fluoresecent screen (e.g. a TV screen) and
6、 the resulting visual pattern is photographed (pattern from C. Jnsson, etal, Am. J. Physics, 42, Fig. 8, p. 9, 1974.Electron diffraction fringes on the screenYes！De Broglie relationship(德布德布罗罗意关系意关系)(普朗克常量普朗克常量)Wave-particle duality(波粒二象性波粒二象性)(波矢波矢)(角角频频率率)Question (energy of blue photon): what is
7、the energy of a blue photon that has a wavelength of 450 nm?Question (X-ray energy and momentum): X-rays are photons with very short wavelengths that can penetrate or pass through objects, which is used in medical imaging, security scans at airport, x-ray diffraction studies of crystal structures. T
8、ypical X-rays have a wavelength of about 0.6 angstrom (1 = 10-10 m). Calculate the energy and momentum of an X-ray with this wavelength.Plane wave(平面波平面波)For the light wave, the electric field Ey at position x at time t is described by:A more generalized form is used to describe a plane wave in x di
9、rection.(振幅振幅)1.2 Schrdinger equation The wave equation of photonsPlane wave(平面波平面波):The wave equation of photonsPlane wave(平面波平面波):The wave equation of photonsThe wave equation of photonsThe wave equation of electronsThe wave equation of electrons与与时间有关的薛定有关的薛定谔方程方程哈密哈密顿函数函数The wavefunction is a so
10、lution of the time-dependent Schrodinger equation, which determines the wavefunction evolution in space and timeA particle (e.g. an electron) is described by a complex wavefunction(x,t)The wavefunction must be a continuous, single-valued function of position and time. 波函数波函数单值单值、延、延续续 Probability in
11、terpretation (几率解几率解释释 The probability of observing a particle within the interval x to x+dx at time between t and t+dt isProbability densityProbability interpretation (几率解几率解释释 At any time the particle must certainly be somewhere. The probability of finding the particle with x coordinate between mi
12、nus and plus infinity must be unity 1 . Hence the wavefunction must have its square modulus integrable and be normalized. 模的平方可模的平方可积积，归归一化一化 Probability interpretation (几率解几率解释释 The time-independent Schrodinger equationNormalization 归归一化一化 定定态scalar 标量量vector 矢量矢量Laplace operator 拉普拉斯算符拉普拉斯算符 Examp
13、le 1. Free electron 自在自在电电子子 : Solve the Schrdinger equation for a free electron whose energy is E.Since V = 0:Solving the differential equation:1.3 Application of Schrdinger equation Define k2=The probability distribution of the electron:Multiplying exp(-jEt/) and =E/:Example 2. Electron in a one-d
14、imensional infinite PE well 一一维维无限深无限深势势阱阱 Consider the behavior of the electron when it is confined to a certain region, 0 x a. Its PE is zero inside that region and infinite outside. The electron cannot escape.From (0)=0Note: ej = cos + j sin with j2 = -1The Schrdinger equation in the region 0xa:T
15、he general solution is:Eulersformula(欧拉公式)SubstituteinSince no PE(potential energy = 0) The momentum px may be in the +x direction or the x direction, so that the average momentum is actually zero, pav = 0.The solution is sin(ka) = 0K and E are quantized. n is called a quantum number. For each n, th
16、ere is a special wavefunction (called eigenfunction(本本证证函数函数)n=1,2,3.From kn = n/a, eigenenergies(能量本征能量本征值值)are:n=1,2,3.ka=n, where n=0,1,2,3,. an Integer (but n=0 is excluded)Boundary condition: =0 at x=a: (a) = 2 Aj sin ka = 0Normalization condition: The total probability of finding the electron
17、in the whole region 0 x a is unity (1).Carrying out the integration:The resulting wavefunction for the electron is thusThe minimum energy corresponds to n=1. This is called the ground state. Electron in a one-dimensional infinite PE well. The energy of the electron is quantized. Possible wavefunctio
18、ns and the probability distributions for the electron are shown.Example 3. Electron confined in three dimensions by a three dimensional infinite “PE box“(三三维维无无线线深深势势阱阱)V=0 in0 x a,0 y b and 0 z cV = , outsideEverywhere inside the box, V = 0, but outside, V = . The electron cannot escape from the bo
19、x. What is the energy and wavefunction of the electron?The three-dimensional version of Schrdinger equation:The total wavefunction is a simple product:If (x,y,z) = 0 at x=a, kxa = n1, with n = 1,2,3.Similarly, if (x,y,z) = 0 at y = b and z = c:andandWhere n1, n2 and n3 are quantum numbers.The eigenf
20、unctions of electron, denoted by the quantum numbers n1, n2 and n3, are given by:Each possible eigenfunction can be labeled a state for the electron. Thus, 111 and 121 are two possible states.Normalization of |n1n2n3(x,y,z)|2 results in A = (2/a)3/2 for a square box (a=b=c).The energy as a function
21、of kx, ky and kz:For a square box for which a=b=c, the energy isWhere N2 =n12 +n22 +n32There are three quantum numbers, each one arising from boundary condition along one of the coordinates.The next energy level corresponds to E211, which is the same as E121 and E112, so there are three states (i.e.
22、, 211, 121, 112) for the energy. The number of states that have the same energy is termed the degeneracy of the energy level. The second energy level E211 is thus three-fold degenerate.The energy is dependent on three quantum numbers. The lowest energy for the electron is equal to E111, not zero. Qu
23、estion (electron confined within atomic dimensions): Consider an electron in an infinite potential well of 0.1 nm (typical size of an atom). What is the ground energy of the electron? What is the energy required to put the electron at the third energy level?How can this energy be provided?Take N Li
24、(lithium) atoms from infinity(无限无限远处远处) and bring them together to form the Li metal. N (1023) The atomic 1s orbital is close to the Li nucleus and remains undisturbed in the solid. The single 2s energy level E2s splits into N (1023) finely separated energy levels, forming an energy band.Band struct
25、ure of metals 金属的能金属的能带带构造构造 There are N 2s-electrons but 2N states in the band. The 2s-band therefore is only half full. As ET - EB is on the order of 10 eV, but there are 1023 atoms, the energy band is practically continuous.ET - EB (between the top and bottom of the band) The 2p energy level, as
26、well as the higher levels at 3s and so on, also split into finely separated energy levels. The energy band of 2s overlaps with the 2p band.Thevariousbandsoverlaptoproduceasinglebandinwhichtheenergyisnearlycontinuous.Assolidatomsarebroughttogetherfrominfinity,theatomicorbitalsoverlapandgiverisetoband
27、s.Outer orbitals overlap first. The3sorbitalsgiverisetothe3sband,2porbitalstothe2pbandandsoon.无限无限远处Electron band structure of metalsTherearestateswithenergiesuptothevacuumlevelwheretheelectronisfree.Inametalthevariousenergybandsoverlaptogiveasinglebandofenergiesthatisonlypartiallyfullofelectrons.真空
28、能真空能级部分部分满的的电子子At0K,allenergylevelsuptotheFermilevelEF0isfull.Theworkfunction(功函数)isrequiredtoliberatetheelectronfromthemetalattheFermilevel.Typicalelectronenergybanddiagramforametal.Allthevalenceelectronsareinanenergybandwhichtheyonlypartiallyfill.Thetopofthebandisthevacuumlevelwheretheelectronisfr
29、eefromthesolid(PE=0).费米能米能级Theelectronsintheenergybandofametalarelooselyboundvalenceelectronswhichbecomesfreeinthecrystalandthereforeformakindofelectrongas.Theelectronswithinabanddonotbelongtoanyspecificatom,buttothewholesolid.Theseelectronsareconstantlymovingaroundinthemetal.Theirwavefunctionsmustb
30、eofthetravelingwavetype.Wecanrepresenteachelectronwithawavevectorksothatitsmomentumpisk.束束缚很弱的价很弱的价电子子电子气子气波矢波矢Example 4. Kronig-Penny model of the square well periodic potential 克勒尼希克勒尼希-彭宁模型彭宁模型 ThesquarewellshaveawidthofawithV=0,andthesquarebarriershaveawidthofbwithV=V0.Thelatticebecomesasquarewe
31、llarray.Bloch theorem:ThesolutionsoftheSchrdingerequationforaperiodicpotentialV(r)=V(r+R)mustbeofaspecialform:whereuk(r)hastheperiodicityofthecrystallatticewithTheeigenfunctionsofthewaveequationforaperiodicpotentialaretheproductofaplanewaveexp(ikr)timesafunctionuk(r)withtheperiodicityofthecrystallat
32、tice.In region I (0 x a):In region II (-b x 0):In region I (0 x a):In region II (-b x 0):The potential V(x) is periodic with period of (a + b)V(x) = Vx + (a + b)According to the Bloch theorem, the wave functions mustalso be period and be of Bloch form (the Bloch waves).Hence has the following formSu
33、bstituting it into Schrdinger equations in region I and II, therefore u(x) will follows (0xa)(-bx0)u1(x)-thevalueofu(x)intheinterval(0xa)u2(x)-thevalueofu(x)in(-bx0).AssumeEV0,andaredefinedasThesolutionsBoundaryconditions:(Therequirementofcontinuityforthewavefunctionanditsderivativedemandsthatthefun
34、ctionsu(x)satisfythesamecontinuityconditions.)Atx=0:Atx=a:Thenon-trivialsolution(asolutionotherthanA=B=C=D=0):FurtherassumptiontosimplifytheresultThepotentialisrepresentedbytheperiodicfunctionb0andV0ButtheproductV0bremainsconstantThefunctionpotential.Thenon-trivialsolutionreducestowhereP=2ba/2maV0b/
35、2,andV0bisaconstantremainsfinitea=(2mE/2)1/2aisin1,+1,anelectronthatmovesinaperiodicallyvaryingpotentialcanonlyoccupycertainallowedenergyregions.Outof1,+1,thereisnovalidandk.Thismeanstherearedisallowedregionsofenergy,i.e.,energygaps.Thesolutionconsistsofseriesofalternateallowedandforbiddenregions.Th
36、eforbiddenregionsbecomesmallerasthevalueofabecomeslarger.cos(ka)isonlydefinedbetween+1and1,:1,+1ThemagnitudeofPiscloselyrelatedtothebindingenergyofelectronsinthecrystal.free-electroncase.Inthiscaseanyenergyisallowed,i.e.,theenergyofelectronsiscontinuousinkspace.b).IfP,thentheenergyofelectronsbecomes
37、independentofk.Isolatedatom.asPapproachesinfinity,sin(a)mustbe0,whichimpliesa=n.ForElectronsarecompletelyboundtotheatomandtheirenergylevelsbecomediscrete.Attheboundaryofanallowedbandcos(ka)=1;Thus,c).WhenPhasafinitevalue,Theenergyofelectronsbecharacterizedbyaseriesofallowedandforbiddenregions.Atypic
38、alenergyversuskplotTheenergyasafunctionofk.Thediscontinuitiesoccuratk=n/a,n=1,2,3,.One-dimensionalenergybanddiagraminareducedzonescheme.TheKronigPenneymodelisasimple,analyticallysolvablemodelthatvisualizestheeffectoftheperiodicpotentialontheelectrons,andtheformationofabandstructure.Example 5. Wave e
39、quation of electron in a general periodic potential(电电子在周期子在周期势场势场中的中的动摇动摇方程方程)The solution in a general periodic zone scheme. The free electron curve is drawn for comparison.Occupiedstatesandbandstructuresgiving(a)aninsulator,(b)ametalBandstructureof(a)Siand(b)GeBandstructureofGaNandCuAlO2Energy Ba
40、nd TheoryTheenergybandstructureofasolidcanbeconstructedbysolvingtheSchrdingerequationforelectronsinacrystallinesolidwhichcontainsalargenumberofinteractingelectronsandatoms.Complexity:many-bodyproblem-motionofatomicnuclei-manyelectronsManymethodisusedtosolvemany-bodyproblem:Tight-bindingapproximation,Thecellular(Wigner-Seitz)method,Theaugmented-planewave(APW)method,first-principlesmethod,thedensity-functionaltheory(DFT),etc.
