1、Chapter 1 Brief introduction to QuantumPhysics & Modern Theory of Solid1.1 Photons and electrons(光子和电子光子和电子)1.2 Schrdinger equation(薛定谔方程薛定谔方程)1.3 Application of Schrdinger equation(薛定谔方程的应用薛定谔方程的应用) From Principles of electronic Materials Devices, SO Kasap (McGraw-Hill, 2005)1.1 Photons and electro
2、nsThe classical 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 des
3、cribed by:Where k is the wavenumber（波数）（波数）(k=2/, the wavelength), and the angular frequency （角频率）（角频率）(=2, the frequency). Particle-like properties of light are confirmed by many experiments:-Photoelectric effect(光电效应光电效应)-Compton scattering(康普顿散射康普顿散射)-Black body radiation(黑体辐射黑体辐射)Intuitive visua
4、lization of light 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 (电子的杨氏双缝实验电子的杨氏双缝实验) inv
5、olves an electron 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.
6、 a TV screen) and 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(德布罗意关系德布罗意关系)(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)(角频率角频率)Question (energy of blue
7、photon): what is 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 crys
8、tal structures. Typical 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 p
9、lane wave in x direction.(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（与时间有关的薛定谔方程）（与时间有关的薛定谔方程）哈密顿函数哈密顿函
10、数The wavefunction is a solution 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.（波函数单值、
11、连续）（波函数单值、连续）Probability interpretation (几率解释）几率解释）The probability of observing a particle within the interval x to x+dx at time between t and t+dt isProbability densityProbability interpretation (几率解释）几率解释）At any time the particle must certainly be somewhere. The probability of finding the particle
12、 with x coordinate between minus and plus infinity must be unity （1）. Hence the wavefunction must have its square modulus integrable and be normalized.（模的平方可积，归一化）（模的平方可积，归一化）Probability interpretation (几率解释）几率解释）The time-independent Schrodinger equationNormalization （归一化）（归一化）（定态）（定态）（scalar 标量）标量）
13、（vector 矢量）矢量）Laplace operator（拉普拉斯算符）（拉普拉斯算符）Example 1. Free electron（自由电子）（自由电子）: 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:Multiply
14、ing exp(-jEt/) and =E/:Example 2. Electron in a one-dimensional infinite PE well（一维无限深势阱）（一维无限深势阱）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
15、with j2 = -1The Schrdinger equation in the region 0 xa:The general solution is:Eulers formula (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 a
16、re quantized. n is called a quantum number. For each n, there is a special wavefunction (called eigenfunction(本证函数本证函数)n=1,2,3.From kn = n/a, eigenenergies(能量本征值能量本征值)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
17、 condition: The total probability of finding the electron 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
18、 energy of the electron is quantized. Possible wavefunctions and the probability distributions for the electron are shown.Example 3. Electron confined in three dimensions by a three dimensional infinite “PE box“(三维无线深势阱三维无线深势阱)V=0 in0 x a,0 y b and 0 z cV = , outsideEverywhere inside the box, V = 0,
19、 but outside, V = . The electron cannot escape from the box. 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 =
20、c:andandWhere n1, n2 and n3 are quantum numbers.The eigenfunctions 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 = (
21、2/a)3/2 for a square box (a=b=c).The energy as a function 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
22、the same as E121 and E112, so there are three states (i.e., 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 low
23、est energy for the electron is equal to E111, not zero. Question (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 thi
24、rd energy level?How can this energy be provided?Take N Li (lithium) atoms from infinity(无限远处无限远处) 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 s
25、eparated energy levels, forming an energy band.Band structure of metals（金属的能带结构）（金属的能带结构）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 (betw
26、een the top and bottom of the band) The 2p energy level, as 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.The various bands overlap to produce a single band in which the energy is nearly continuous.As solid
27、atoms are brought together from infinity, the atomic orbitals overlap and give rise to bands. Outer orbitals overlap first. The 3s orbitals give rise to the 3s band, 2p orbitals to the 2p band and so on. 无限远处无限远处Electron band structure of metalsThere are states with energies up to the vacuum level w
28、here the electron is free.In a metal the various energy bands overlap to give a single band of energies that is only partially full of electrons. 真空能级真空能级部分满的电子部分满的电子At 0K, all energy levels up to the Fermi level EF0 is full.The work function (功函数功函数) is required to liberate the electron from the me
29、tal at the Fermi level.Typical electron energy band diagram for a metal. All the valence electrons are in an energy band which they only partially fill. The top of the band is the vacuum level where the electron is free from the solid (PE = 0).费米能级费米能级The electrons in the energy band of a metal are
30、loosely bound valence electrons which becomes free in the crystal and therefore form a kind of electron gas.The electrons within a band do not belong to any specific atom, but to the whole solid.These electrons are constantly moving around in the metal. Their wavefunctions must be of the traveling w
31、ave type.We can represent each electron with a wavevector k so that its momentum p is k.束缚很弱的价电子束缚很弱的价电子电子气电子气波矢波矢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）The square wells have a width of a with V = 0, and the square barriers have a width of b with V =
32、 V0. The lattice becomes a square well array.Bloch theorem:The solutions of the Schrdinger equation for a periodic potential V(r) = V(r + R) must be of a special form:where uk(r) has the periodicity of the crystal lattice withThe eigenfunctions of the wave equation for a periodic potential are the p
33、roduct of a plane wave exp(ik r) times a function uk(r) with the periodicity of the crystal lattice.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 fun
34、ctions mustalso be period and be of Bloch form (the Bloch waves).Hence has the following formSubstituting it into Schrdinger equations in region I and II, therefore u(x) will follows (0 x a)(-b x 0)u1(x) - the value of u(x) in the interval (0 x a)u2(x )- the value of u(x) in (-b x 0).Assume E V0, an
35、d are defined asThe solutionsBoundary conditions:(The requirement of continuity for the wave function andits derivative demands that the functions u(x) satisfy thesame continuity conditions.)At x = 0:At x = a:The non-trivial solution (a solution other thanA=B=C=D=0): Further assumption to simplify t
36、he result The potential is represented by the periodic function b0 and V0 But the product V0b remains constantThe function potential.The non-trivial solution reduces towhere P = 2ba/2 maV0b/2, and V0b is a constant remains finite a = (2mE/2)1/2a is in 1, +1, an electron that moves in a periodically
37、varying potential can only occupy certain allowed energy regions. Out of 1, +1, there is no valid and k. This means there are disallowed regions of energy, i.e., energy gaps. The solution consists of series of alternate allowed and forbidden regions. The forbidden regions become smaller as the value
38、 of a becomes larger.cos(ka) is only defined between +1 and 1, : 1, +1The magnitude of P is closely related to the binding energy of electrons in the crystal. free-electron case.In this case any energy is allowed, i.e., the energy of electrons is continuous in k space.b). If P, then the energy of el
39、ectrons becomes independent of k. Isolated atom.as P approaches infinity, sin(a) must be 0, which implies a = n. For Electrons are completely bound to the atom and their energy levels become discrete. At the boundary of an allowed band cos (ka) = 1; Thus,c). When P has a finite value, The energy of
40、electrons be characterized by a series ofallowed and forbidden regions.A typical energy versus k plotThe energy as a function of k. The discontinuities occur at k = n/a, n = 1, 2, 3, .One-dimensional energy band diagram in a reduced zone scheme.The KronigPenney model is a simple, analytically solvab
41、le model that visualizes the effect of the periodic potential on the electrons, and the formation of a band structure.Example 5. Wave equation of electron in a general periodic potential(电子在周期势场中的波动方程电子在周期势场中的波动方程)The solution in a general periodic zone scheme. The free electron curve is drawn for c
42、omparison.Occupied states and band structures giving (a) an insulator, (b) a metalBand structure of (a) Si and (b) GeBand structure of GaN and CuAlO2Energy Band TheoryThe energy band structure of a solid can be constructed by solving the Schrdinger equation for electrons in a crystalline solid which
43、 contains a large number of interacting electrons and atoms.Complexity: many-body problem- motion of atomic nuclei- many electronsMany method is used to solve many-body problem: Tight-binding approximation, The cellular (Wigner-Seitz) method, The augmented-plane wave (APW) method, first-principles method, the density-functional theory (DFT), etc.
