1、半导体材料与技术全册配套半导体材料与技术全册配套 完整精品课件完整精品课件2 Chapter 1 Brief introduction to Quantum Physics Thus, c). When P has a finite value, The energy of electrons be characterized by a series of allowed and forbidden regions. A typical energy versus k plot The energy as a function of k. The discontinuities occur a
2、t k = n/a, n = 1, 2, 3, . One-dimensional energy band diagram in a reduced zone scheme. The KronigPenney model is a simple, analytically solvable 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 i
3、n 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 comparison. Occupied states and band structures giving (a) an insulator, (b) a metal Band structure of (a) Si and (b) Ge Band structure of GaN and CuAlO2 En
4、ergy Band Theory The energy band structure of a solid can be constructed by solving the Schrdinger equation for electrons in a crystalline solid which contains a large number of interacting electrons and atoms. Complexity: many-body problem - motion of atomic nuclei - many electrons Many method is u
5、sed to solve many-body problem: Tight-binding approximation, The cellular (Wigner-Seitz) method, The augmented-plane wave (APW) method, fi rst- principles method, the density-functional theory (DFT), etc. Chapter 1 Brief introduction to Quantum Physics Thus, c). When P has a finite value, The energy
6、 of electrons be characterized by a series of allowed and forbidden regions. A typical energy versus k plot The 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, analytical
7、ly solvable 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
8、drawn for comparison. Occupied states and band structures giving (a) an insulator, (b) a metal Energy Band Theory The energy band structure of a solid can be constructed by solving the Schrdinger equation for electrons in a crystalline solid which contains a large number of interacting electrons and
9、 atoms. Complexity: many-body problem - motion of atomic nuclei - many electrons Many method is used to solve many-body problem: Tight-binding approximation, The cellular (Wigner-Seitz) method, The augmented-plane wave (APW) method, fi rst-principles method, the density-functional theory (DFT), etc.
10、 Band structure of (a) Si and (b) Ge Band structure of GaN and CuAlO2 Go over Chapter 1 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) (振幅振幅) A more generalized form is used to describe a plane wave in x direction. （与时间有关的薛定谔方程）（与时间有关的薛定谔方程） The minimum energy corresponds to n=1. This is called
11、the ground state. Electron in a one-dimensional infinite PE well. The energy of the electron is quantized. Possible wavefunctions and the probability distributions for the electron are shown. Example 2. Electron in a one-dimensional infinite PE well（一维无限深势阱）（一维无限深势阱） Example 3. Electron confined in
12、three dimensions by a three dimensional infinite “PE box“(三维无限深势阱三维无限深势阱) V=0 in 0 x a, 0 y b and 0 z c V = , outside For a square box for which a=b=c, the energy is Where N2 =n12 +n22 +n32 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
13、 the electron from the metal at the Fermi level. 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 Typical electron energy band diagram for a metal. Example 4. Kronig-Pen
14、ny 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 = V0. The lattice becomes a square well array. A typical energy versus k plot The energy as a function of k. The discontinuities occur
15、at k = n/a, n = 1, 2, 3, . 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 comparison. Occupied states and band structures giving (a) an insulator, (b) a metal Band s
16、tructure of (a) Si and (b) Ge Exercise 1 Exercise 2 质子 Chapter 2 Electrical and Thermal Conduction in Solid 2.1 Classical theory: The Drude model(德鲁特模型德鲁特模型) 2.2 Temperature dependence of resistivity: ideal pure metals (电阻对时间的依赖性：理想纯金属电阻对时间的依赖性：理想纯金属) 2.3 Matthiessens and Nordheims rules(马西森和诺德海姆定则马
17、西森和诺德海姆定则) 2.4 Resistivity of mixtures and porous materials (混合物和孔洞材料的电阻率混合物和孔洞材料的电阻率) 2.5 The Hall effect and Hall devices(霍尔效应和霍尔器件霍尔效应和霍尔器件) 2.6 Thin metal films(金属薄膜金属薄膜) 2.7 Thermal conduction(热传导热传导) 2.8 Electrical conductivity of nonmetals(非金属的电导非金属的电导) From Principles of electronic Materials
18、 Devices, SO Kasap (McGraw-Hill, 2005) Content Electrical conduction involves the motion of charges in a material under the influence of an applied field. A material can generally be classified as a conductor if it contains a large number of free or mobile charge carriers. In metals, the valence ele
19、ctrons that are free to move within the metal are called as conduction electrons. Objectives of electrical conduction: conduction electrons; acceleration of free charge carriers; drift velocity(漂移速漂移速 度度); electron collisions(碰撞碰撞) with lattice vibrations(晶格振晶格振 动动), crystal defects, impurities(杂质)
20、etc. Thermal conduction in solid 2.1 Classical theory: the Drude model The electric current density J is defined as: Drift velocity in the x direction (average over N electrons): 漂移速度 Drift of electrons in a conductor in the presence of an applied electric field. 2.1 Classical theory: the Drude mode
21、l The number of electrons per unit volume n: Electrons drift with an average velocity vdx in the x direction.(Ex is the electric field.) (a) A conduction electron in the electron gas moves about randomly in a metal (with a mean speed u) being frequently and randomly scattered by thermal vibrations o
22、f the atoms. In the absence of an applied field there is no net drift in any direction. (b) In the presence of an applied field, Ex, there is a net drift along the x-direction. This net drift along the force of the field is superimposed(叠加) on the random motion of the electron. After many scattering
23、 events the electron has been displaced by a net distance, x, from its initial position toward the positive terminal vxi: the velocity in the x direction of the electron i uxi: the velocity after collision (initial velocity) Ex; applied field in the x direction me: the mass of an electron ti: the la
24、st collision time (relaxation time(弛豫时 间) Velocity gained in the x-direction at time t from the electric field (Ex) for three electrons. There will be N electrons to consider in the metal. Drift velocity vdx (average velocity for all such electrons along x): Suppose that is the mean free time (or me
25、an time between collisions): Drift mobility(漂移迁移率) d: where Ohms law: I =V / R where is conductivity Summation operator (求和符号) Example (Suppose each Cu atom donates one electron.) Example (Suppose each Cu atom donates one electron.) Example (drift velocity and mean speed): What is the applied electr
26、ic field that will impose a drift velocity equal to 0.1 percent of the mean speed u (106 m/s) of conduction electrons in copper? What is the corresponding current density through a Cu wire of a diameter of 1 mm? Electric field: Current density: A current through a 1mm-diameter copper wire: When an e
27、lectric field is applied to a conductor, for all practical purposes, the mean speed is unaffected. 2.2 Temperature dependence of resistivity: ideal pure metals - Since the scattering cross sectional area is S, in the volume Sl there must be at least one scatterer, Ns(Su)=1. NS: the number of scatter
28、ing centers per unit volume. mean free path Where u is the mean speed - Scattering of an electron from the thermal vibrations of the atoms. - The electron travels a mean distance l = u between collisions. The mean free time is given as: An atom covers a cross-sectional area a2 with the vibration amp
29、litude a. The average kinetic energy of the oscillations is given as: Where is the oscillation frequency. C: constant A: temperature independent constant Example (temperature dependence of resistivitiy): what is the percentage change in the resistance of a pure metal wire from Saskatchewans summer (
30、20C) to winter (-30C), neglecting the changes in the dimensions of the wire? Example (drift mobility and resistivity due to lattice vibrations): Given that the mean speed of conduction electrons in copper is 1.5x106 m/s and the frequency of vibration of the copper atoms at room temperature is about
31、4x1012 S-1, estimate the drift mobility of electrons and the conductivity of copper. The density of copper is 8.96 g/cm3 and the atomic mass Mat is 62.56 g/mol. 2.3 Matthiessens and Nordheims rules 2.3.1 Matthiessens rule and the temperature coefficient of resistivity () If we assume the two scatter
32、ing mechanisms are independent. We now effectively have two types of mean free times: T from thermal vibration only and I from collisions with impurities. The net probability of scattering 1/ is given as: The theory of conduction that considers scattering from lattice vibrations only works well with
33、 pure metals. In a metal alloy, an electron can be scattered by the impurity atoms due to unexpected change in the potential energy PE because of a local distortion. Strained region by impurity exerts a scattering force F = - d(PE) /dx Two different types of scattering processes involving scattering
34、 from impurities alone and thermal vibrations alone. The drift mobility: The effective (or overall) resistivity (Matthiessens rule): Considering other scattering effects (dislocations, grain boundaries and other crystal defects), the effective resistivity of a metal may be written as: Where R is the
35、 residual resistivity. The residual resistivity shows very little temperature dependence. Where A and B are temperature independent constants. The temperature coefficient 0 is defined as: Where 0 is the resistivity at the reference temperature T0, usually 273K (or 293K), and =-0, is the change in th
36、e resistivity due to a small increase in temperature T=T-T0. When 0 is constant over a temperature range T0 to T: Frequently, the resistivity versus temperature behavior of pure metals can be empirically represented by a power law: n: the characteristic index =AT+B is oversimplified. As the temperat
37、ure decreases, typically below 100K for many metals, the resistivity becomes =DT5+R, where D is a constant. -Tin melts at 505 K whereas nickel and iron go through a magnetic to non-magnetic (Curie) transformations at about 627 K and 1043 K respectively. -The theoretical behavior ( T) is shown for re
38、ference. From Metals Handbook The resistivity of various metals as a function of temperature above 0 C. - Above about 100 K, T - At low temperatures, T 5 - At the lowest temperatures approaches the residual resistivity R . - The inset shows the vs. T behaviour below 100 K on a linear plot ( R is too
39、 small on this scale). The resistivity of Cu from lowest to highest temperatures (near melting temperature, 1358 K) on a log-log plot. Typical temperature dependence of the resistivity of annealed and cold worked (deformed) copper containing various amount of Ni in atomic percentage (data adapted fr
40、om J.O. Linde, Ann. Pkysik, 5, 219 (1932). Example (Matthiessens rule Cu alloys) 2.3.2 Solid solutions and Nordheims rule The temperature-independent impurity contribution I increases with the concentration of solute atoms. This means that as the alloy concentration increases, the resistivity increa
41、ses and becomes less temperature dependent as I overwhelms T, leading to 1/273. For example: Nichrome (80% of Ni and 20% of Cr) has a resistivity, that increases almost 16 times compared to that of pure Ni. The alloy (Nichrome) has a very low value of . Example (Cu-Ni system) (a) Phase diagram of th
42、e Cu-Ni alloy system. Above the liquidus line only the liquid phase exists. In the L + S region, the liquid (L) and solid (S) phases coexist whereas below the solidus line, only the solid phase (a solid solution) exists. (b) The resistivity of the Cu-Ni alloy as a function of Ni content (at.%) at ro
43、om temperature. from Metals Handbook-10th Edition and Constitution of Binary Alloys -An isomorphous binary alloy system (one phasefcc). -Solid solution phase exists in the whole composition range. -The maximum of is at around 50% of Ni. An important semiempirical equation that can be used to predict
44、 the resistivity of an alloy is Nordheims rule which relates the impurity resistivity pI to the atomic fraction X of solute atoms in a solid solution, as follows: Where C is the constant termed the Nordheim coefficient. For dilute solutions, Nordheims rule predicts the linear behavior, that is, I =
45、CX for X 10c: Where d is the volume fraction of the dispersed phase d. Case 2: if d 10c: Example (combined Nordheim and mixture rules): Brass is an alloy composed of Cu and Zn. Consider a brass component made from sintering 90at% Cu and 10at% Zn brass powder. The component contains dispersed air por
46、es at 15vol%. The Nordheim coefficient C of Zn in Cu is 300 nm. Predict the effective resistivity of this brass component, if the resistivity of pure Cu is 16nm at room temperature. The resistivity of the brass alloy: The effective resistivity of the component: 2.4.2 Two-phase alloy (Ag-Ni) resistiv
47、ity and electrical contacts -Nordheims rule can be used in the composition ranges 0-X1 and X2-100%B. -Mixture rule between X1 and X2. (a) The phase diagram for a binary, eutectic forming alloy. (b) The resistivity vs composition for the binary alloy. When we apply a magnetic field in a perpendicular
48、 direction to an applied electric field (which is driving the electric current), we find there is a transverse electric field in the sample that is perpendicular to the direction of both the applied electric field Ex and the magnetic field Bz because of Lorentz force (F = qvxB). 2.5 The Hall effect
49、and Hall devices Illustration of the Hall effect. The z-direction is out from the plane of paper. The externally applied magnetic field is along the z-direction. A moving charge experiences a Lorentz force in a magnetic field. (a) A positive charge moving in the x direction experiences a force downw
50、ards. (b) A negative charge moving in the -x direction also experiences a force downwards. Lorentz force: Where q is the charge The accumulation of electrons near the bottom results in an internal electric field EH (Hall field). When this happened, the magnetic-field force evdBz that pushes the elec
