Lecture 3. Electron and Hole Transport in Semiconductors

Transcription

1 Lecture 3 lectro ad Hole Trasort i Semicoductors I this lecture you will lear: How electros ad holes move i semicoductors Thermal motio of electros ad holes lectric curret via lectric curret via usio Semicoductor resistors Review: lectros ad Holes i Semicoductors holes + electros A Silico crystal lattice + As + + As + Door imurity atoms + There are two tyes of mobile charges i semicoductors: electros ad holes I a itrisic (or udoed) semicoductor electro desity equals hole desity Semicoductors ca be doed i two ways: N-doig: to icrease the electro desity P-doig: to icrease the hole desity 1

2 Thermal Motio of lectros ad Holes I thermal equilibrium carriers (i.e. electros or holes) are ot stadig still but are movig aroud i the crystal lattice because of their thermal eergy The root-mea-square velocity of electros ca be foud by equatig their kietic eergy to the thermal eergy: 1 mvth v th 1 KT KT m I ure Silico at room temerature: 7 v th ~ 10 cm s Thermal Motio of lectros ad Holes I thermal equilibrium carriers (i.e. electros or holes) are ot stadig still but are movig aroud i the crystal lattice ad udergoig collisios with: vibratig Silico atoms with other electros ad holes with doat atoms (doors or accetors) ad other imurity atoms Mea time betwee collisios = c I betwee two successive collisios electros (or holes) move with a average velocity which is called the thermal velocity = v th I ure Silico, v c th cm s s 0. 1 s Browia Motio Mea distace traveled betwee collisios is called the mea free ath vth c I ure Silico, cm 0.01m

3 Drift: Motio of lectros Uder a Alied lectric Field Silico slab L + - V V L Force o a electro because of the electric field = F = -q The electro moves i the directio oosite to the alied field with a costat velocity equal to v d The electro velocity v d is roortioal to the electric field stregth vd vd The costat is called the electro mobility. It has uits: cm cm I ure Silico, 1500 Drift: Motio of Holes Uder a Alied lectric Field Silico slab L + - V V L Force o a hole because of the electric field = F = q The hole moves i the directio of the alied field with a costat velocity equal to v d The hole velocity v d is roortioal to the electric field stregth vd vd The costat is called the hole mobility. It has uits: cm I ure Silico, 500 cm 3

4 Derivatio of xressios for Mobility lectros: Force o a electro because of the electric field F q F Acceleratio of the electro q a m m Sice the mea time betwee collisios is c, the acceleratio lasts oly for a time eriod of c before a collisio comletely destroys electro s velocity q So i time c electro s velocity reaches a value a c c m This is the average velocity of the electro, i.e. Comarig with vd we get, Holes: Similarly for holes oe gets, q c m q c m v d q c m Secial ote: Masses of electros ad holes (m ad m ) i Solids are ot the 31 same as the mass of electros i free sace which equals kg Mobility Vs Doig More doig (-tye of -tye) meas more frequet collisios with charged door ad accetor imurity atoms ad this lowers the carrier mobility Mobility (cm /V-s) Doat cocetratio (1/cm 3 ) Note: Doig i the above figure ca either be -tye or -tye 4

5 Drift Curret Desity of lectros Cosider electros movig uder a alied electric field: v d Flux Desity: Flux desity is the umber of articles crossig a uit area surface er secod It has uits cm - -s -1 Desity: Velocity: v d Uit area surface Flux desity: v d Area Time Volume = 1 x (v d x 1) Drift Curret Desity of lectros lectros Drift Curret Desity: lectro flux desity from v d lectro curret desity J is, Check directios v d J J q electro flux desity q v d q vd J q J has uits: Coulombs Ams cm - s cm 5

6 Drift Curret Desity of Holes Holes Drift Curret Desity: The hole curret desity is J, J q hole flux q v d q Check directios vd v d J J q J has uits: Coulombs Ams cm - s cm Coductivity ad Resistivity Total Drift Curret Desity: The total curret desity J is the sum of J ad J J J q J The quatity is the coductivity of the semicoductor: q Coductivity describes how much curret flows whe a electric field is alied. Aother related quatity is the resistivity which is the iverse of the coductivity, 1 Uits of coductivity are: Ohm -1 -cm -1 or -1 -cm -1 or S-cm -1 Uits of resistivity are: Ohm-cm or -cm or S -1 -cm 6

7 For a resistor we kow that, V I R We also kow that, I J A A V A A V L L xamle: A Semicoductor Resistor 1 Area A L L 1 R where 1 q A A L I Silico slab - V + Lessos: Kowig electro ad hole desities ad mobilities, oe ca calculate the electrical resistace of semicoductors -doig or -doig ca be used to chage the coductivity of semicoductors by orders of magitudes Diffusio Diffusio of ik i a glass beaker Why does usio hae? 7

8 Diffusio ad Diffusivity There is aother mechaism by which curret flows i semicoductors. Suose the electro desity iside a semicoductor is ot uiform i sace, as show below (x) electro flux i +x directio electro flux i -x directio d x sloe x Sice the electros move about radomly i all directios (Browia motio), as time goes o more electros will move from regios of higher electro desity to regios of lower electro desity tha the electros that move from regios of lower electro desity to regios of higher electro desity d x Net electro flux desity i +x directio d x D The costat D is called the usivity of electros (uits: cm -s -1 ) Diffusio Curret Desity lectros Diffusio Curret Desity: lectro flux desity from usio lectro usio curret desity J J q electro flux desity d x q D d D is, Holes Diffusio Curret Desity: d x Hole flux desity from usio D Hole usio curret desity J q hole flux d x q D J is, x Check directios x lec. flux Check directios x Hole flux J x J x J ad J Coulombs Ams has uits cm - s cm 8

9 istei Relatios istei worked o other thigs besides the theory of relativity.. We itroduced two material costats related to carrier trasort: 1) Mobility ) Diffusivity Both are coected with the trasort of carriers (electros or holes) It turs out that their values are related by the istei relatioshis istei Relatio for lectros: D K T q istei Relatio for Holes: D K T q xamle: I ure Silico, 1500 cm 500 cm This imlies, D 37.5 cm s D 1.5 cm s K is the Boltzma costat ad its value is: 1.38x10-3 Joules K KT has a value equal to Volts at room temerature (at 300 o K) q Total lectro ad Hole Curret Desities Total electro ad hole curret desities is the sum of ad usive comoets J lectros: x J x J x q x x q D d x Holes: J x J x J x q x x q D d x lectric currets are drive by electric fields ad also by carrier desity gradiets 9

10 Thermal quilibrium - I There caot be ay et electro curret or et hole curret i thermal equilibrium what does this imly?? Cosider electros first: x J x J x q x x q D J 0 d o x o d log ca also be writte as: o x q x KT Sice the electric field is mius the gradiet of the otetial: x We have: d logo x q KT qx The solutio of the above eretial equatio is: o x costat e KT But what is that costat i the above equatio??? We have: o x Thermal quilibrium - II qx costat e KT Note: oe ca oly measure otetial ereces ad ot the absolute values of otetials Covetio: The otetial of ure itrisic Silico is used as the referece value ad assumed to be equal to zero. q x So for itrisic Silico, costat KT o x e costat x But we already kow that i itrisic Silico, o i So it must be that, costat i Ad we get the fial aswer, o x qx KT i e Cosider Holes Now: Oe ca reeat the above aalysis for holes ad obtai: o x qx KT i e Check: o x o x i 10

11 Potetial of Doed Semicoductors What are the values of otetials i N-doed ad P-doed semicoductors?? N-doed Semicoductors (doig desity is N d ): The otetial i -doed semicoductors is deoted by: x o Nd q x Nd i e KT KT N d log q i P-doed Semicoductors (doig desity is N a ): KT N a log q i xamle: Suose, 17-3 N d 10 cm ad 10 i 10 cm KT N log d 0.4 Volts q i The otetial i -doed semicoductors is deoted by: o x N xamle: a q x Suose, N KT 17-3 a i e N a 10 cm ad i cm KT N log a 0.4 Volts q i