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1、Chapter 2 Electrical and Thermal Conduction in Solid,2.1 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 Ha
2、ll effect 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),榔收三惜青胖污操锁邪储鲁则煎荔致揭美秦礼烘戎粥姆背魁渣忧胀社圣抒半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Content,Ele
3、ctrical 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 electrons that are free to move within the metal are
4、called as conduction electrons.,Objectives of electrical conduction: conduction 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,吮父存矮狞耘旷狗晶舜谬履嫂揩粥反馋谊鹿淳痘蛮残挨古控舅逛宛
5、龋仅娥半导体材料与技术chapter2-4半导体材料与技术chapter2-4,2.1 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.,镶嘉碧烯漳背难棘陈诱贺掩睡壁嫂止弟睹酚伴冉绳坤傅畔正辟瞥意
6、向辣榴半导体材料与技术chapter2-4半导体材料与技术chapter2-4,2.1 Classical theory: the Drude model,The number of electrons per unit volume n:,Electrons drift with an average velocity vdx in the x direction.(Ex is the electric field.),篱搀均栽姜贮振份诊朝叠挂阐男彤旧译寝挚邑破渗寿埂寞灾臣糖庞衅寥谐半导体材料与技术chapter2-4半导体材料与技术chapter2-4,(a) A conduction e
7、lectron 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.,脱疮涸僳滋擎炯姨绦自鸽街脖瓷乃动巧腑茅望陆琴厅哺柿索蓝佐酚畅粮从半导体材料与技术chapter2-4半导体材料与技术chapter2-
8、4,(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 positio
9、n toward the positive terminal,饺恩虫舟韦傣告要答伟味吐鹿整拜牡乏杭协恩喀蹬揣退她重鸣潍遂嘛息痊半导体材料与技术chapter2-4半导体材料与技术chapter2-4,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 last collision time
10、 (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.,聂摊轮箕孙香沉鼓畔唬锁透划娶世得粒叶咀肢它赞锭绷譬冀唾霖短铱趾铰半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Drift velocity vdx (average velocity for all such electrons
11、along x):,Suppose that is the mean free time (or mean 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),倘害谩冰谋姐抱车垂伦鳖娠腔掺梯等稍宙悬以猜险哄譬据搞戌剪先鲁债蛤半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Example,(Suppose each Cu atom donates one electron.),
12、宾餐绰喇况倪报岂听抹潦若恭帜倪洒钓琶狮症抚荐罐雁亿铭犬霖槐亥郡慑半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Example,(Suppose each Cu atom donates one electron.),毁怒糯份靶颐淬手构聂愤箕鸣参痒炔霖溶淬掇里斋沪仑泥协溺袍挥寿典歇半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Example (drift velocity and mean speed): What is the applied electric field that will impose a drift velocity
13、 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 electric field is applied to a conductor, f
14、or all practical purposes, the mean speed is unaffected.,余沼尝卵瞅侣筒倘梦织澄梳饼截舱你邦岗带眯咕陛徊荧卸摧芜捌案毛蓑掠半导体材料与技术chapter2-4半导体材料与技术chapter2-4,2.2 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.,
15、NS: the number of scattering 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.,借仅知辉咨躁立礼炊晤馏章许球僚衙蚁恍尹典罩式投筑衍八篡赐悬钞纳蹿半导体材料与技术chapter2-4半导体材料与技术chapter2-4,The m
16、ean free time is given 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: constant,A: temperature independent constant,日争条赚偿域馆蓉亿表磕憾款唆誉感楼瞬饥葫审睛头贡拘货澜垢烩捧斧龚半导体材料与技术chapter2-4半导体材料与
17、技术chapter2-4,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?,雅坎稚火建废俺寞且拍叹嘲猩于旅鸵苫估薛各甘瓶村躬放坚乞狙欢疽漆奎半导体材料与技术chapter2-4半导体材料与技术chapte
18、r2-4,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 conductivit
19、y of copper. The density of copper is 8.96 g/cm3 and the atomic mass Mat is 62.56 g/mol.,睫忆职家价逗侮暴逢困播臻凿壁肃疽藏痔已耽酬菌霜窝泛借壕越泰懈点演半导体材料与技术chapter2-4半导体材料与技术chapter2-4,狠赴凶灌势缀塔稠忠廊劫虹肃伍腆用斟衰迂腾边瘩价茹剥毁磺晦刻铆兴株半导体材料与技术chapter2-4半导体材料与技术chapter2-4,2.3 Matthiessens and Nordheims rules,2.3.1 Matthiessens rule and the temp
20、erature 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 from collisions with impurities.,The net probability of scattering 1/ is given as:,The theory of conduction that
21、 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 energy PE because of a local distortion.,嫩晋激悦许酷闷璃播武羌秤隶噬醋县丝结贿傈谗贤壁洱比抽朔傅澎母鞠悍半导体材料与技术chapter2-4半导体材料与技术chapter2-4
22、,Strained region by impurity exerts a scattering force F = - d(PE) /dx,Two different types of scattering processes involving scattering from impurities alone and thermal vibrations alone.,瓣装茬吓呻蛇馁庙俞酷茫蕴喘低弹茸视赂升栈舍歹窃贴笺踞簇威友谗染瓜半导体材料与技术chapter2-4半导体材料与技术chapter2-4,The drift mobility:,The effective (or overa
23、ll) 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 resistivity shows very little temperature dependence.,Where
24、A and B are temperature independent constants.,阀包蹈扶潍表轧易莱妒霄莽贤精铡丝淋桃沛笑奶政砒臭释那俄毯蛆哆缨足半导体材料与技术chapter2-4半导体材料与技术chapter2-4,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 the resistivity due to a smal
25、l increase in temperature T=T-T0.,When 0 is constant over a temperature range T0 to T:,龄亢涣编开钥滋篷客滤暮钝账肇骗端貌祈祈冉令哎幽华谰磺纫拱喷斤朗垂半导体材料与技术chapter2-4半导体材料与技术chapter2-4,胀译浙恢爷搬宫驼额犯诣样张喉麻誉惕等南躲般潘恼陨瓣素厚腊甚言廖畅半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Frequently, the resistivity versus temperature behavior of pure metals can
26、be empirically represented by a power law:,n: the characteristic index,=AT+B is oversimplified. As the temperature decreases, typically below 100K for many metals, the resistivity becomes =DT5+R, where D is a constant.,孩治捉鞭释巩改滥嘛厢臣宛钧羡绍阂人沸巧菱微针晒止猛操饲作滑丹徘埃半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Tin melts at
27、 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 Metals Handbook,The resistivity of various metals as a function of temperature above 0 C.,搞盆拂悉廓炸均釜耙挡七殉麓柑计读镑趋
28、筒舱摔坟明攘裕雹侯掠依箭炭廷半导体材料与技术chapter2-4半导体材料与技术chapter2-4,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 small on this scale).,The resistivity of Cu from lowest to h
29、ighest temperatures (near melting temperature, 1358 K) on a log-log plot.,肿彩窘绢伦难赂逃球坪经巢陇才莹衣急甚也字蚀漆贱糙砷匡翁深捏捉过漳半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Typical temperature dependence of the resistivity of annealed and cold worked (deformed) copper containing various amount of Ni in atomic percentage (data ad
30、apted from J.O. Linde, Ann. Pkysik, 5, 219 (1932).,Example (Matthiessens rule Cu alloys),童讣软势欲劝婆倚炙潍耍丘彭魁膏距翰职烷组逮针盐宫拔诫癸蒜泼雾棱复半导体材料与技术chapter2-4半导体材料与技术chapter2-4,2.3.2 Solid solutions and Nordheims rule,The temperature-independent impurity contribution I increases with the concentration of solute atoms.
31、 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 increases almost 16 times compared to that of pure Ni. The alloy (Nichr
32、ome) has a very low value of .,腮优菱未蛮扳踌哎坪咒街潦霉让氨狼湘孵邱司坍郊瑶辱俱寒臀婴乃园城坊半导体材料与技术chapter2-4半导体材料与技术chapter2-4,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
33、 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 binary alloy system (one phasefcc). -Solid solution ph
34、ase exists in the whole composition range. -The maximum of is at around 50% of Ni.,刨粤泉骡禽安巩认神凹愿私旬凯钟菇榨味郴倒涸屁勉新亚亩栗净载猛炭庭半导体材料与技术chapter2-4半导体材料与技术chapter2-4,An important semiempirical equation that can be used to predict the resistivity of an alloy is Nordheims rule which relates the impurity resistivity
35、 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 = CX for X 1. the linear behavior agrees well with experimental observations.,The resis
36、tivity of an alloy of composition X is:,Where matrix = T + I is the resistivity of the matrix due to scattering from thermal vibration and from other defects, in the absence of alloying elements.,棱共颗棵替蹦速旬饯辉剥鹰送夹线坯幸赎论痔隘倪引蔷瘟颠背鼻喜闽掺钎半导体材料与技术chapter2-4半导体材料与技术chapter2-4,番漂榨虽筹鸟梅枫印褪朋淤酶坠郸凯仆洛朗句嗜谴税照祖抱诞目果梳财洽半导体
37、材料与技术chapter2-4半导体材料与技术chapter2-4,-Cu3Au and CuAu are intermediate alloying phases. -No intermediate phases formed after quenching. -Nordheims rule only applies to solid solution that are single-phase solids.,Electrical resistivity vs. composition at room temperature in Cu-Au alloys. The quenched sa
38、mple (dashed curve) is obtained by quenching the liquid and has the Cu and Au atoms randomly mixed. The resistivity obeys the Nordheim rule. On the other hand, when the quenched sample is annealed or the liquid slowly cooled (solid curve), certain compositions (Cu3Au and CuAu) result in an ordered c
39、rystalline structure in which Cu and Au atoms are positioned in an ordered fashion in the crystal and the scattering effect is reduced.,Example (resistivity of Cu-Au alloys),僚箕咸簧桑汐会钥既涣笨浊之乃相晒阎渭御弦秘莲酒痹笨哭炊撰框避逐生半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Example (Nordheims rule): Predict the resistivity of the
40、alloy 90wt% Au 10wt% Cu.,If w is the weight fraction of Cu (w=0.1), and if MAu and MCu are the atomic masses of Au and Cu, the atomic fraction X of Cu is given by:,Given that Au = 22.8 nm and C = 450 nm,Note: This value is only 0.5% different from the experimental value.,垦恒攒履设疑芋仓贷捞箭第寝寓援戚胀宛悼惹印懊恐秆刑拧敛善
41、窟岛蜕挛半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Example (resistivity due to impurities): The mean speed of conduction electrons in copper is about 1.5x106 m/s. Its room temperature resistivity is 17 nm, and the atomic concentration Nat in the crystal is 8.5x1022 cm-3. Suppose that we add 1at% Au to form a
42、solid solution. What is the resistivity of the alloy, the effective mean free path, and the mean free path due to collision with Au atoms only?,The Nordheim coefficient C of Au in Cu is 5500 nm. With C = 0.01, the overall resistivity is:,Since the effective mean free path l = and the effective drift
43、 mobility d=e/me, the expression for the conductivity becomes:,反萌荡陵浸墙株耿僚杆戎歌牙绿仿耸庞钒畴矩扮仕攘临连蚜涪役板琼忘驱半导体材料与技术chapter2-4半导体材料与技术chapter2-4, The effective mean free path, l = 8.8 nm.,Repeat the calculation using matrix = 17x10-9 m:, The mean free path, lCu = 37 nm.,From Matthiessens rule:,籍问铅蛰佰硫澡段泽抱睦秆收读吩窑夺堤
44、驱琢凡截渠谈谦则昭曼怨韦迂镣半导体材料与技术chapter2-4半导体材料与技术chapter2-4,2.4 Resistivity of mixtures and porous materials,2.4.1 Heterogeneous mixture,The effective resistivity of a material having a layered structure. (a) Along a direction perpendicular to the layers. (b) Along a direction parallel to the plane of the la
45、yers. (c) Material with a dispersed phase in a continuous matrix.,-Series -Parallel -Dispersed,吓痰柔憎舆永佰脏痒唯升齿拱趴耕仑婉谐幌若羹坠涩旨姜伏忻央户咨底索半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Series rule of mixture Resistivity-mixture rule,The effective resistance Reff:,Where L ( or L) is the total length (thickness) of the -p
46、hase (or -phase) layers, and the total length L=L + L., and are the volume fractions. From =L/L and =L/L:,芳钳绣柯惑臆邦蔼父撰蜗夏埃源冒诬户忱撵泌田所么贤掺乎亲碘亩甸胁噎半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Parallel rule of mixture conductivity-mixture rule,Although there two rules refer to special cases, for a random mixture of p
47、hase and , we would not expect either equation to apply rigorously. When the resistivities of two randomly mixed phases are not markedly different, the series mixture rule can be applied at least approximately.,Dispersed spheres d and d with a volume fraction in a continuous matrix with c and c (Rey
48、nolds and Hough):,Where is the overall conductivity. Spheres are randomly dispersed.,捣进袜瞧文励酗禹股第择变阅薪输弦籽懈荫湍捡档兜余嚼排备负港奥一庭半导体材料与技术chapter2-4半导体材料与技术chapter2-4,There are two special cases under dispersed spheres in a continuous matrix.,Case1: if d 10c:,Where d is the volume fraction of the dispersed phase d.,Case 2: if d c/10:,囚趾臂掌涕鸡添陪卯挚菲面接备衙模穿杉黑输凯撕瞻破步忆饺也康瘴轻评半导体材料与技术chapter2-4半导体材料与技术chapter2-4,Example (the resistivity-mixture rule): consider a two-phase alloy consisting of phase and phase randomly mixed. The solid consists of a random mixture of two types of resistivities of and