1、Chapter 2 Electrical and Thermal Conductionin Solid2.1 Classical theory: The Drude model(德鲁德鲁特模型特模型)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(马西马西森和诺德海姆森和诺德海姆定则定则)2.4 Resistivity of mixtures and porous materials
2、(混合物和混合物和孔孔洞材料的电阻率洞材料的电阻率)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 Devices, SO Kasap (McGraw-Hill, 2005)ContentElectrical conducti
3、on 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 electrons that are free to move within the metal are called as conducti
4、on 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) etc.Thermal conduction in solid2.1 Classical theory: the Drude modelThe
5、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 modelThe number of electrons per unit volume n:Electrons drift with an average vel
6、ocity 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 of the atoms. In the absence of an applied field there is no net drift in any dire
7、ction.(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 events the electron has been displaced by a net distance, x, from its initial pos
8、ition toward the positive terminalvxi: the velocity in the x direction of the electron i uxi: the velocity after collision (initial velocity)Ex; applied field in the x directionme: the mass of an electronti: the last collision time (relaxation time(弛豫时间)Velocity gained in the x-direction at time t f
9、rom 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 mean time between collisions):Drift mobility(漂移迁移率) d:whereOhms law:I =V / Rwhere is conduct
10、ivitySummation 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 electric field that will impose a drift velocity equal to 0.1 percent of the mean speed u (106 m/s) of cond
11、uction 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 electric field is applied to a conductor, for all practical purposes, the mean speed is unaffected.2.2 Tem
12、perature 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 scattering centers per unit volume.mean free pathWhere u is the mean speed- Scattering of an electron from the ther
13、mal vibrations of the atoms. - The electron travels a mean distance l = u between collisions. The mean free time isgiven as:An atom covers a cross-sectional area a2 with the vibration amplitude a. The average kinetic energy of the oscillations is given as:Where is the oscillation frequency.C: consta
14、ntA: temperature independentconstantExample (temperature dependence of resistivitiy): what is the percentage change in the resistance of a pure metal wire from Saskatchewans summer (20C) to winter (-30C),neglecting the changes in the dimensions of the wire?Example (drift mobility and resistivity due
15、 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 4x1012 S-1, estimate the drift mobility of electrons and the conductivity of copper. The density of copper is 8.96 g/cm3
16、 and the atomic mass Mat is 62.56 g/mol.2.3 Matthiessens and Nordheims rules2.3.1 Matthiessens rule and the temperature coefficient of resistivity ()If we assume the two scattering mechanisms are independent.We now effectively have two types of mean free times: T from thermal vibration only and I fr
17、om 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 pure metals.In a metal alloy, an electron can be scattered by the impurity atoms due to unexpected change in the potential en
18、ergy PE because of a local distortion.Strained region by impurity exerts a scattering force F = - d(PE) /dxTwo different types of scattering processes involving scattering from impurities alone and thermal vibrations alone.The drift mobility:The effective (or overall) resistivity (Matthiessens rule)
19、: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 residual resistivity.The residual resistivity shows very little temperature dependence.Where A and B are temperature independent con
20、stants.The temperature coefficient0 is defined as:Where 0 is the resistivity at the reference temperature T0, usually 273K (or 293K), and =-0, is the change in the resistivity due to a small increase in temperature T=T-T0.When 0 is constant over atemperature range T0 to T:Frequently, the resistivity
21、 versus temperature behavior of pure metals can be empirically represented by a power law:n: the characteristicindex=AT+B is oversimplified. As the temperature decreases, typically below 100K for many metals, the resistivity becomes =DT5+R, where D is a constant.-Tin melts at 505 K whereas nickel an
22、d 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 reference. From Metals HandbookThe resistivity of various metals as a function of temperature above 0 C. - Above about 100 K, T- At low temperatures
23、, 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 small on this scale).The resistivity of Cu from lowest to highest temperatures (near melting temperature, 1358 K) on a log-log plot. Typical temper
24、ature dependence of the resistivity of annealed and cold worked (deformed) copper containing various amount of Ni in atomic percentage (data adapted from J.O. Linde, Ann. Pkysik, 5, 219 (1932).Example (Matthiessens rule Cu alloys)2.3.2 Solid solutions and Nordheims ruleThe temperature-independent im
25、purity contribution I increases with the concentration of solute atoms. This means that as the alloy concentration increases, the resistivity increases 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 i
26、ncreases 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 the 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
27、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 room temperature. from Metals Handbook-10th Edition and Constitution of Binary Alloys-An isomorphous binaryalloy system (one phasefcc).-Solid solution phase
28、existsin the whole compositionrange.-The maximum of is ataround 50% of Ni.An important semiempirical equation that can be used to predict 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:W
29、here C is the constant termed the Nordheim coefficient.For dilute solutions, Nordheims rule predicts the linear behavior, that is, I = 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
30、 and Zn. Consider a brass component made from sintering 90at% Cu and 10at% Zn brass powder. The component contains dispersed air pores 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
31、 temperature.The resistivity of the brass alloy:The effective resistivity of the component:2.4.2 Two-phase alloy (Ag-Ni) resistivity and electrical contacts-Nordheims rule canbe used in thecomposition ranges 0-X1 and X2-100%B.-Mixture rulebetween X1 and X2.(a) The phase diagram for a binary, eutecti
32、c forming alloy. (b) The resistivity vs composition for the binary alloy.When we apply a magnetic field in a perpendicular 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
33、 of both the applied electric field Ex and the magnetic field Bz because of Lorentz force (F = qvxB).2.5 The Hall effect and Hall devicesIllustration 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 exp
34、eriences a Lorentz force in a magnetic field. (a) A positive charge moving in the x direction experiences a force downwards. (b) A negative charge moving in the -x direction also experiences a force downwards.Lorentz force:Where q is the chargeThe accumulation of electrons near the bottom results in
35、 an internal electric field EH (Hall field). When this happened, the magnetic-field force evdBz that pushes the electrons down just balance the force eEH that prevents further accumulation.In the steady state:From Jx = envdx:Hall coefficient RH:For metals:Note: From =end d = /(en) Hall mobility H =
36、| RH |Example (Hall-effect Wattmeter)Wattmeter based on the Hall effect. Load voltage and load currenthave L as subscript. C denotes the current coils. for setting up amagnetic field through the Hall effect sample (semiconductor)VH=wEH=wRHJxBzIxBzVLILW is the thickness.Example (Hall mobility): The H
37、all coefficient and conductivity of copper at 300K have been measured to be -0.55x10-10 m3A-1s-1 and 5.9x107 -1m-1, respectively. Calculate the drift mobility of electrons in copper.From H = | RH |Example (conduction electron concentration in copper)Since the concentration of copper atoms is 8.5x102
38、8 m-3, the average number of electrons contributed per atom is (1.15x1029)/(8.5x1028) = 1.36.2.6 Thin metal films(a ) Grain boundaries cause scatte ring of the electron and there fore add to the re sistivity by Matthiessens rule.(b) For a very grainy solid, the electron is scattered from grain bound
39、ary to grain boundary and the mean free path isapproximately equal to the mean grain diameter.Polycrystalline films and grain boundary scatteringThe mean free path l: mean free path in the single crystald: grain size.From crystal 1/ and 1/l:Mayadas-Shatkez formula:Where R is a parameter, which is be
40、tween 0.24 to 0.40 for copperFor example: the predicted /crystal 1.20 for a Cu film, if R = 0.3 and d 3 = 120 nm (since the bulk crystal 40 nm).Surface scatteringConduction in thin films may be controlled by scattering from the surfaces.D is the filmthicknessFrom a more rigorous calculation (Fuchs-S
41、ondheimer equation):The value of p is dependent on the preparation conduction and microstructure. p = 0.9-1 for most epitaxial thin films, unless very thin (D).a) film of the Cu polycrystalline films vs. reciprocal mean grain size (diameter), 1/d. Film thickness D=250nm- 900nm does not affect the re
42、sistivity. The straight line is film=17.8 nm+(595nmnm)(1/d), (b) film of the Cu thin polycrystalline films vs. filmthickness D. In this case, annealing (heat treating) the films to reduce the polycrystallinity does not significantly affect the resistivity because film is controlled mainly by surface
43、 scattering.From (a) Microelec. Engin. and (b) Appl. Surf. Sci.2.7 Thermal conductionMetals are both good electrical and good thermal conductors. Free conduction electrons in a metal play an important role in heat conduction. When a metal piece is heated at one end, the amplitude of the atomic vibra
44、tion and thus the average kinetic energy of the electrons in the region increases. Electrons gain energy from energetic atomic vibrations when the two collide. By virtue of their increased random motion, these energetic electrons then transfer the extra energy to the colder regions by colliding with
45、 the atomic vibrations there. Thus, electrons act as “energy carriers”Note: In nonmetals, the thermal conduction is due to lattice vibrations.Thermal conduction in a metal involves transferring energy from the hot region to the cold region by conduction electrons. More energetic electrons (shown wit
46、h longer velocity vectors) from the hotter regions arrive at cooler regions and collide there with lattice vibrations and transfer their energy. Lengths of arrowed lines on atoms represent the magnitudes of atomic vibrations.The thermal conductivity measures the ability of heat transportation throug
47、h the medium.T/x: the temperature gradientA: the cross-sectional area The sign “-”: indicates the heat form hot end to cold end. (Fouriers law) : thermal conductivity(Fouriers law)Heat flow in a metal rod heated at one end. Consider the rate of heat flow, dQ/dt, across a thin section x of the rod. T
48、he rate of heat flow is proportional to the temperature gradient T/ x and the cross sectional area A.In metals, electrons participate in the process of charge and heat transport, which are characterized by (electrical conductivity) and k, respectively.Therefore, it is no surprising to find that the
49、two coefficients are related by the Wiedemann-Franz-Lorenz law.Wiedemann-Franz-Lorenz law:Where CWFL = 2k2/2e2= 2.44x10-8 WK-2 is the Lorenz number (or the Wiedemann-Franz-Lorenz coefficent).Experiments show that the Wiedemann-Franz-Lorenz law is reasonably obeyed at close to room temperature and ab
50、ove.Thermal conductivity, vs. electrical conductivity for various metals (elements and alloys) at 20 C. The solid line represents the WFL law with CWFL 2.44108 W K-2.Thermal conductivity vs. temperature for two pure metals (Cu and Al) and two alloys (brass and Al-14%Mg). Data extracted from Thermoph
