Chapter 2 Motion and Recombination of Electrons and Holes

Transcription

1 Chapter 2 Motio ad Recombiatio of Electros ad Holes 2.1 Thermal Motio Average electro or hole kietic eergy kt mv th 2 2 v th 3kT m eff JK kg K m/s cm/s Slide Thermal Motio Zig-zag motio is due to collisios or scatterig with imperfectios i the crystal. Net thermal velocity is zero. Mea time betwee collisios is m ~ 0.1ps Slide

2 Hot-poit Probe ca determie sample doig type Hot-poit Probe distiguishes N ad P type semicoductors. Thermoelectric Geerator (from heat to electricity ) ad Cooler (from electricity to refrigeratio) Slide Drift Electro ad Hole Mobilities Drift is the motio caused by a electric field. Slide

3 2.2.1 Electro ad Hole Mobilities m v qe p mp v q E m p mp v p E q p m mp p v E q m m p is the hole mobility ad is the electro mobility Slide Electro ad Hole Mobilities v = E ; has the dimesios of v/e cm/s V/cm 2 cm. V s Electro ad hole mobilities of selected semicoductors Si Ge GaAs IAs (cm 2 /V s) p (cm 2 /V s) Based o the above table aloe, which semicoductor ad which carriers (electros or holes) are attractive for applicatios i high-speed devices? Slide

4 Drift Velocity, Mea Free Time, Mea Free Path EXAMPLE: Give p = 470 cm 2 /V s, what is the hole drift velocity at E = 10 3 V/cm? What is mp ad what is the distace traveled betwee collisios (called the mea free path)? Hit: Whe i doubt, use the MKS system of uits. Solutio: = p E = 470 cm 2 /V s 10 3 V/cm = cm/s mp = p m p /q =470 cm 2 /V s kg/ C = m 2 /V s kg/c = s = 0.1 ps mea free path = mh th ~ s cm/s = cm = 220 Å = 22 m This is smaller tha the typical dimesios of devices, but gettig close. Slide Mechaisms of Carrier Scatterig There are two mai causes of carrier scatterig: 1. Phoo Scatterig 2. Ioized-Impurity (Coulombic) Scatterig Phoo scatterig mobility decreases whe temperature rises: phoo phoo phoo desity carrier 1 1 3/ 2 T 1/ 2 thermal velocity T T = q/m T v th T 1/2 Slide

5 Impurity (Dopat)-Io Scatterig or Coulombic Scatterig - Boro _ Electro - Electro + Arseic Io There is less chage i the directio of travel if the electro zips by the io at a higher speed. impurity N a v 3 th N d T N a 3 / 2 N d Slide Total Mobility Mobility (cm 2 V -1 s -1 ) Electros Holes 1 1 phoo 1 1 phoo 1 impurity 1 impurity 0 1E14 1E15 1E16 1E17 1E18 1E19 1E20 Total Impurity N Coceratio (atoms cm -3 a + N d (cm -3 ) ) Slide

6 Temperature Effect o Mobility Questio: What N d will make d /dt = 0 at room temperature? Slide Velocity Saturatio Whe the kietic eergy of a carrier exceeds a critical value, it geerates a optical phoo ad loses the kietic eergy. Therefore, the kietic eergy is capped at large E, ad the velocity does ot rise above a saturatio velocity, v sat. Velocity saturatio has a deleterious effect o device speed as show i Ch. 6. Slide

7 Hall Effect Slide Slide

8 Slide Slide

9 2.2.3 Drift Curret ad Coductivity E J p + uit area + Hole curret desity J p = qpv A/cm 2 or C/cm 2 sec EXAMPLE: If p = cm -3 ad v = 10 4 cm/s, the J p = C cm cm/s 2 2 = 1.6 C/s cm 1.6 A/cm Slide Drift Curret ad Coductivity J p,drift = qpv = qp p E J,drift = qv = q E J drift = J,drift + J p,drift = E =(q +qp p )E coductivity (1/ohm-cm) of a semicoductor is = q + qp p 1/ = is resistivity (ohm-cm) Slide

10 Relatioship betwee Resistivity ad Dopat Desity DOPANT DENSITY cm -3 N-type P-type RESISTIVITY (cm) = 1/ Slide EXAMPLE: Temperature Depedece of Resistace (a) What is the resistivity () of silico doped with cm -3 of arseic? (b) What is the resistace (R) of a piece of this silico material 1m log ad 0.1 m 2 i crosssectioal area? Solutio: (a) Usig the N-type curve i the previous figure, we fid that = cm. (b) R = L/A = cm 1 m / 0.1 m 2 = cm 10-4 cm/ cm 2 = Slide

11 EXAMPLE: Temperature Depedece of Resistace By what factor will R icrease or decrease from T=300 K to T=400 K? Solutio: The temperature depedet factor i (ad therefore ) is. From the mobility vs. temperature curve for cm -3, we fid that decreases from 770 at 300K to 400 at 400K. As a result, R icreases by Slide Diffusio Curret Particles diffuse from a higher-cocetratio locatio to a lower-cocetratio locatio. Slide

12 J, diffusio 2.3 Diffusio Curret qd d dx J p, diffusio qd p dp dx D is called the diffusio costat. Sigs explaied: p x x Slide Total Curret Review of Four Curret Compoets J TOTAL =J +J p J = J,drift + J,diffusio = q E + J p = J p,drift + J p,diffusio = qp p E qd qd p d dx dp dx Slide

13 2.4 Relatio Betwee the Eergy Diagram ad V, E V(x) 0.7eV 0.7V + N- N type Si 0 x E c ad E v vary i the opposite directio from the voltage. That is, E c ad E v are higher where the voltage is lower. E(x)= dv dx 1 de q dx c 1 dev q dx x E + - E c (x) E f (x) E v (x) 0.7V Slide Eistei Relatioship betwee D ad Cosider a piece of o-uiformly doped semicoductor. N-type semicoductor -type semicoductor Decreasig door cocetratio E c (x) E f E v (x) ( E Nce d Nc ( E e dx kt dec kt dx kt qe c E c E f f ) / kt ) / kt de dx c Moder Semicoductor Devices for Itegrated Circuits (C. Hu) Slide

14 2.5 Eistei Relatioship betwee D ad d dx kt qe D J q E qd d dx qd 0 q E q kt kt q 0 E Similarly, at equilibrium. D p kt p q These are kow as the Eistei relatioship. Slide EXAMPLE: Diffusio Costat What is the hole diffusio costat i a piece of silico with p = 410 cm 2 V -1 s -1? Solutio: D p kt p q (26 mv) 410 cm 2 V s cm 2 /s Remember: kt/q = 26 mv at room temperature. Slide

15 2.6 Electro-Hole Recombiatio The equilibrium carrier cocetratios are deoted with 0 ad p 0. The total electro ad hole cocetratios ca be differet from 0 ad p 0. The differeces are called the excess carrier cocetratios ad p. p 0 ' p 0 p' Slide Charge Neutrality Charge eutrality is satisfied at equilibrium ( = p = 0). Whe a o-zero is preset, a equal p may be assumed to be preset to maitai charge equality ad vice-versa. If charge eutrality is ot satisfied, the et charge will attract or repel the (majority) carriers through the drift curret util eutrality is restored. ' p' Slide

16 Recombiatio Lifetime Assume light geerates ad p. If the light is suddely tured off, ad p decay with time util they become zero. The process of decay is called recombiatio. The time costat of decay is the recombiatio time or carrier lifetime,. Recombiatio is ature s way of restorig equilibrium ( = p = 0). Slide Recombiatio Lifetime rages from 1s to 1ms i Si ad depeds o the desity of metal impurities (cotamiats) such as Au ad Pt. These deep traps capture electros ad holes to facilitate recombiatio ad are called recombiatio ceters. E c Direct Recombiatio is ufavorable i silico E v Recombiatio ceters Slide

17 Direct ad Idirect Bad Gap Trap Direct bad gap Example: GaAs Direct recombiatio is efficiet as k coservatio is satisfied. Idirect bad gap Example: Si Direct recombiatio is rare as k coservatio is ot satisfied Slide Rate of recombiatio (s -1 cm -3 ) d dt p d dt p dp dt Slide

18 EXAMPLE: Photocoductors A bar of Si is doped with boro at cm -3. It is exposed to light such that electro-hole pairs are geerated throughout the volume of the bar at the rate of /s cm 3. The recombiatio lifetime is 10s. What are (a) p 0, (b) 0, (c) p, (d),(e) p, (f), ad (g) the p product? Slide Solutio: EXAMPLE: Photocoductors (a) What is p 0? p 0 = N a = cm -3 (b) What is 0? 0 = i2 /p 0 = 10 5 cm -3 (c) What is p? I steady-state, the rate of geeratio is equal to the rate of recombiatio /s-cm 3 = p / p = /s-cm s = cm -3 Slide

19 (d) What is? = p = cm -3 EXAMPLE: Photocoductors (e) What is p? p= p 0 + p = cm cm -3 = cm -3 (f) What is? = 0 + = 10 5 cm cm -3 ~ cm -3 sice 0 << (g) What is p? p ~ cm cm -3 = cm -6 >> i 2 = cm -6. The p product ca be very differet from i2. Slide Thermal Geeratio If is egative, there are fewer electros tha the equilibrium value. As a result, there is a et rate of thermal geeratio at the rate of /. Slide

20 2.8 Quasi-equilibrium ad Quasi-Fermi Levels Wheever = p 0, p i2. We would like to preserve ad use the simple relatios: ( Ec E f )/ kt N e p N But these equatios lead to p = i2. The solutio is to itroduce two quasi-fermi levels E f ad E fp such that N c p N ( E f Ev )/ kt ve ( Ec E f )/ kt ce ( E fp Ev )/ kt ve Eve whe electros ad holes are ot at equilibrium, withi each group the carriers ca be at equilibrium. Electros are closely liked to other electros but oly loosely to holes. Slide EXAMPLE: Quasi-Fermi Levels ad Low-Level Ijectio Cosider a Si sample with N d =10 17 cm -3 ad =p =10 15 cm -3. (a) Fid E f. = N d = cm -3 = N c exp[ (E c E f )/kt] E c E f = 0.15 ev. (E f is below E c by 0.15 ev.) Note: ad p are much less tha the majority carrier cocetratio. This coditio is called low-level ijectio. Slide

21 EXAMPLE: Quasi-Fermi Levels ad Low-Level Ijectio Now assume = p = cm -3. (b) Fid E f ad E fp. = cm -3 = N c e ( E E c f ) / kt E c E f = kt l(n c / cm -3 ) = 26 mev l( cm -3 / cm -3 ) = 0.15 ev E f is early idetical to E f because 0. Moder Semicoductor Devices for Itegrated Circuits (C. Hu) Slide EXAMPLE: Quasi-Fermi Levels p = cm -3 = N e v ( E E ) / kt fp E fp E v = kt l(n v /10 15 cm -3 ) = 26 mev l( cm -3 /10 15 cm -3 ) = 0.24 ev v E c E f E f E fp E v Slide

22 2.9 Chapter Summary J J v p v pe -E qp pe p, drift, drift qe J J, diffusio p, diffusio D D p qd qd kt q kt p q d dx p dp dx Slide Chapter Summary is the recombiatio lifetime. ad p are the excess carrier cocetratios. = 0 + p = p 0 + p Charge eutrality requires = p. rate of recombiatio = / = p / E f ad E fp are the quasi-fermi levels of electros ad holes. ( Ec E f )/ kt N e p N c ( E fp Ev ) / kt ve Slide