《半导体材料与技术》chapter24课件

1、Chapter 2 Electrical and Thermal Conductionin Solid2.1 Classical theory: The Drude model(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)2.3 Matthiessens and Nordheims rules(马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)2.5 The Hall eff

2、ect and Hall devices(霍尔效应和霍尔器件)2.6 Thin metal films(金属薄膜)2.7 Thermal conduction(热传导)2.8 Electrical conductivity of nonmetals(非金属的电导) From Principles of electronic Materials Devices, SO Kasap (McGraw-Hill, 2005)第1页，共69页。ContentElectrical conduction involves the motion of charges in a material under t

3、he 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 conduction electrons.Objectives of electrical conduction: condu

4、ction electrons;acceleration of free charge carriers; drift velocity(漂移速度); electron collisions(碰撞) with lattice vibrations(晶格振动), crystal defects, impurities(杂质) etc.Thermal conduction in solid第2页，共69页。2.1 Classical theory: the Drude modelThe electric current density J is defined as:Drift velocity

5、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. 第3页，共69页。2.1 Classical theory: the Drude modelThe number of electrons per unit volume n:Electrons drift with an average velocity vdx in the x direction.(Ex is the electri

6、c field.)第4页，共69页。(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 direction.第5页，共69页。(b) In the presence of

7、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 position toward the positive ter

8、minal第6页，共69页。vxi: 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 from the electric fie

9、ld (Ex) for three electrons. There will be N electrons to consider in the metal.第7页，共69页。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 conductivitySummat

10、ion operator (求和符号)第8页，共69页。Example(Suppose each Cu atom donates one electron.)第9页，共69页。Example(Suppose each Cu atom donates one electron.)第10页，共69页。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

11、(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 electric field is applied to a conductor, for all practical purposes, the mean speed is u

12、naffected.第11页，共69页。2.2 Temperature dependence of resistivity: ideal pure metalsSince 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 speedScattering of an

13、 electron from the thermal vibrations of the atoms. The electron travels a mean distance l = u between collisions. 第12页，共69页。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

14、oscillation frequency.C: constantA: temperature independentconstant第13页，共69页。Example (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?第1

15、4页，共69页。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 4x1012 S-1, estimate the drift mobility of electrons and the conducti

16、vity of copper. The density of copper is 8.96 g/cm3 and the atomic mass Mat is 62.56 g/mol.第15页，共69页。第16页，共69页。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 ha

17、ve 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 pure metals.In a metal alloy, an electron can be scatt

18、ered by the impurity atoms due to unexpected change in the potential energy PE because of a local distortion.第17页，共69页。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 al

19、one.第18页，共69页。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 residual resistivity.The residual resisti

20、vity shows very little temperature dependence.Where A and B are temperature independent constants.第19页，共69页。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

21、 in temperature T=T-T0.When 0 is constant over atemperature range T0 to T:第20页，共69页。第21页，共69页。Frequently, the resistivity 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, typicall

22、y below 100K for many metals, the resistivity becomes =DT5+R, where D is a constant.第22页，共69页。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 reference. From Met

23、als HandbookThe resistivity of various metals as a function of temperature above 0 C. 第23页，共69页。Above about 100 K, TAt 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 small on this sc

24、ale).The resistivity of Cu from lowest to highest temperatures (near melting temperature, 1358 K) on a log-log plot. 第24页，共69页。Typical temperature dependence of the resistivity of annealed and cold worked (deformed) copper containing various amount of Ni in atomic percentage (data adapted from J.O.

25、Linde, Ann. Pkysik, 5, 219 (1932).Example (Matthiessens rule Cu alloys)第25页，共69页。2.3.2 Solid solutions and Nordheims ruleThe temperature-independent impurity contribution I increases with the concentration of solute atoms. This means that as the alloy concentration increases, the resistivity increas

26、es 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 .第26页，共69页。Example (Cu-Ni system)(a) Phase diagram

27、 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 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.%)

28、 at room temperature. from Metals Handbook-10th Edition and Constitution of Binary Alloys-An isomorphous binaryalloy system (one phasefcc).-Solid solution phase existsin the whole compositionrange.-The maximum of is ataround 50% of Ni.第27页，共69页。An important semiempirical equation that can be used to

29、 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:Where C is the constant termed the Nordheim coefficient.For dilute solutions, Nordheims rule predicts the linear behavior, that is

30、, I = CX for X 10c:Where d is the volume fraction of the dispersed phase d.Case 2: if d 10c:第40页，共69页。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 disp

31、ersed 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 temperature.The resistivity of the brass alloy:The effective resistivity of the component:第41页，共69页。2.4.2 Two-phase al

32、loy (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, eutectic forming alloy. (b) The resistivity vs composition for the binary alloy.第42页，共69页。When we apply a magnetic f

33、ield 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 of both the applied electric field Ex and the magnetic field Bz because of Lorentz force (F = qvxB

34、).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.第43页，共69页。A moving charge experiences a Lorentz force in a magnetic field. (a) A positive charge moving in the x direc

35、tion experiences a force downwards. (b) A negative charge moving in the -x direction also experiences a force downwards.Lorentz force:Where q is the charge第44页，共69页。The accumulation of electrons near the bottom results in an internal electric field EH (Hall field). When this happened, the magnetic-f

36、ield 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:第45页，共69页。Note: From =end d = /(en) Hall mobility H = | RH |第46页，共69页。Example (Hall-effect Wattmeter)Wattmeter based on the

37、 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.第47页，共69页。Example (Hall mobility): The Hall coefficient and conductivity of copper at 300

38、K 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 8.5x1028 m-3, the average number of electrons contribute

39、d per atom is (1.15x1029)/(8.5x1028) = 1.36.第48页，共69页。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 boundary to grain boundary and the mean free

40、 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.第49页，共69页。From crystal 1/ and 1/l:Mayadas-Shatkez formula:Where R is a parameter, which is between 0.24 to 0.40 for copper

41、For 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).第50页，共69页。Surface scatteringConduction in thin films may be controlled by scattering from the surfaces.D is the filmthicknessFrom a more rigorous calculation (Fuchs-Sondheimer equation)

42、: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).第51页，共69页。第52页，共69页。a) film of the Cu polycrystalline films vs. reciprocal mean grain size (diameter), 1/d. Film thickness D=250nm- 900nm does not affect the r

43、esistivity. 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 surfac

44、e scattering.From (a) Microelec. Engin. and (b) Appl. Surf. Sci.第53页，共69页。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 a

45、tomic vibration 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 col

46、liding with the atomic vibrations there. Thus, electrons act as “energy carriers”Note: In nonmetals, the thermal conduction is due to lattice vibrations.第54页，共69页。Thermal conduction in a metal involves transferring energy from the hot region to the cold region by conduction electrons. More energetic

47、 electrons (shown with 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.第55页，共69页。The thermal conductivity measures the abilit

48、y of heat transportation through 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

49、 a thin section x of the rod. The rate of heat flow is proportional to the temperature gradient T/ x and the cross sectional area A.第56页，共69页。In metals, electrons participate in the process of charge and heat transport, which are characterized by (electrical conductivity) and k, respectively.Therefo

50、re, it is no surprising to find that the 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 o

51、beyed at close to room temperature and above.第57页，共69页。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.第58页，共69页。Thermal conductivity vs. temperature for two pure metals (Cu and Al) and

52、two alloys (brass and Al-14%Mg). Data extracted from Thermophysical Properties of Matter第59页，共69页。Nonmetals do not have free electrons. The energy transfer involves lattice vibration. The springs couple the vibrations to neighboring atoms and thus allow the large amplitude vibrations to propagate, a

53、s a vibrational wave, to the cooler regions of the crystal.The coefficient of heat transfer depends not only on the efficiency of coupling, and hence on the nature of interatomic bonding, but also on the propagation in the crystal. The stronger the coupling, the greater will be the thermal conductiv

54、ity, for example 1000 W/mK in diamond.Conduction of heat in insulators involves the generation and propogation of atomic vibrations through the bonds that couple the atoms. (An intuitive figure.)第60页，共69页。第61页，共69页。Example (thermal conductivity): A 95%Cu-5%Sn bronze bearing made of powdered metal co

55、ntains 15vol% porosity. Calculate its thermal conductivity at 300K, give that the electrical conductivity of the 95/5 bronze is 107 -1m-1.From /T = CWFL:The effective conductivity is:Note: 1/ could be considered as the thermal resistivity.第62页，共69页。2.8 Electrical conductivity of nonmetalsAll metals

56、are good conductors because of they have a very large number of conduction electrons.Based on typical values of conductivity, it is possible to empirically classify various materials into conductors, semiconductors and insulators.There is no well-defined sharp boundary between insulators andsemiconductors.