Uniform excitation: applied field and optical generation. Non-uniform doping/excitation: diffusion, continuity

Transcription

1 Electronic Devices and Circuits Lecture 2 - Uniform Excitation; Non-uniform conditions Announcements Review Carrier concentrations in TE given the doping level What happens above and below room temperature? Drift and mobility - The full story. Uniform excitation: optical generation Generation/recombination in TE Uniform optical generation - external excitation Population excesses, p' and n', and their transients Low level injection; minority carrier lifetime Uniform excitation: applied field and optical generation Photoconductivity, photoconductors Non-uniform doping/excitation: diffusion, continuity Fick's 1st law; diffusion Diffusion current; total current (drift plus diffusion) Fick's 2nd law; carrier continuity Clif Fonstad, 9/15/09 Lecture 2 - Slide 1

2 Extrinsic Silicon, cont.: solutions in Cases I and II Case I - n-type: N d > N a :, (N d - N a ) >> n i "n-type Si" Define the net donor concentration, N D : N D " (N d # N a ) We find: " N D, p o = n i 2 (T) / " n i 2 (T) /N D >> n i >> p o In Case I the concentratiof electrons is much greater than that of holes. Silicon with net donors is called "n-type". Case II - p-type: N a > N d :, (N a - N d ) >> n i "p-type Si" Define the net acceptor concentration, N A : N A " (N a # N d ) We find: p o " N A, = n i 2 (T) / p o " n i 2 (T) /N A p o >> n i >> In Case II the concentratiof holes is much greater than that of electrons. Silicon with net acceptors is called "p-type". Clif Fonstad, 9/15/09 Lecture 2 - Slide 2

3 Variatiof carrier concentration with temperature (Note: for convenience we assume an n-type sample) Around R.T. Full ionization Extrinsic doping N + d " N d, N # a " N a N + # ( d # N a ) >> n i " ( N d # N a ), p o = n 2 i At very high T Full ionization Intrinsic behavior N d + " N d, N a # " N a n i >> N d + # N a # " p o " n i At very low T Incomplete ionization Extrinsic doping, but with carrier freeze-out N + d << N d N + d " N " a >> n i # N + " d " N a ( assuming n " type) ( ) << N d " N a ( ), p o = n i 2 Clif Fonstad, 9/15/09 Lecture 2 - Slide 3

4 Uniform material with uniform excitations (pushing semiconductors out of thermal equilibrium) A. Uniform Electric Field, E x, cont. Drift motion: Holes and electrons acquire a constant net velocity, s x, proportional to the electric field: s ex = " µ e E x, s hx = µ h E x At low and moderate E, the mobility, µ, is constant. At high E the velocity saturates and µ deceases. Drift currents: Net velocities imply net charge flows, which imply currents: J dr ex = "q s ex = qµ e E x J dr hx = q p o s hx = qµ h p o E x Note: Even though the semiconductor is no longer in thermal equilibrium the hole and electron populations still have their thermal equilibrium values. Clif Fonstad, 9/15/09 Lecture 2 - Slide 4

5 Conductivity, σ o : Ohm's law on a microscale states that the drift current density is linearly proportional to the electric field: J dr x = " o E x The total drift current is the sum of the hole and electron drift currents. Using our early expressions we find: J dr x = J dr ex + J dr hx = qµ e E x + qµ h p o E x = q ( µ e +µ h p o )E x From this we see obtaiur expression for the conductivity: ( ) [S/cm] " o = q µ e +µ h p o Majority vs. minority carriers: Drift and conductivity are dominated by the most numerous, or "majority," carriers: n-type >> p o " # o $ qµ e p-type p o >> " # o $ qµ h p o Clif Fonstad, 9/15/09 Lecture 2 - Slide 5

6 Resistance, R, and resistivity, ρ o : Ohm's law on a macroscopic scale says that the current and voltage are linearly related: v ab = R i D The question is, "What is R?" We have: with E x = v AB l Combining these we find: i D w " t = # o which yields: J x dr = " o E x and J x dr = i D w # t v AB l v AB = l w " t where Note: Resistivity, ρ o, is defined as the inverse of the conductivity: Clif Fonstad, 9/15/09 Lecture 2 - Slide 6 + v AB 1 # o i D = R i D R $ l w " t " o # 1 $ o [Ohm - cm] i D 1 # o = σ ο w l w " t % o = l A % o l t

7 Velocity saturation The breakdowf Ohm's law at large electric fields. 108 Silicon GaAs (electrons) Carrier drift velocity (cm/s) Ge Si T = 300K Electrons Holes Above: Velocity vs. field plot at R.T. for holes and electrons in Si (loglog plot). (Fonstad, Fig. 3.2) Left: Velocity-field curves for Si, Ge, and GaAs at R.T. (log-log plot). (Neaman, Fig. 5.7) Electric field (V/cm) Clif Fonstad, 9/15/09 Figure by MIT OpenCourseWare. Lecture 2 - Slide 7

8 Variatiof mobility with temperature and doping 10 4 N D = cm µ n T = 300 K Log µ n (T) 3/2 (T) -3/ µ p Ge µn (cm 3 /V - S) 10 3 Si Impurity scattering Log T Lattice scattering Mobility (cm2/v-s) Si µ n µ p µ n GaAs T(K) Impurity concentration (cm -3 ) µ p Figure by MIT OpenCourseWare. Figure by MIT OpenCourseWare. µ e vs T in Si at several doping levels µ vs doping for Si, Ge, and GaAs at R.T. (Neaman, Fig. 5.3) Clif Fonstad, 9/15/09 Lecture 2 - Slide 8

9 Having said all of this, it is good to be aware that the mobilities vary with doping and temperature, but in we will 1. use only one value for the hole mobility in Si, and one for the electron mobility in Si, and will not consider the variation with doping. Typically for bulk silicon we use µ e = 1600 cm 2 /V-s and µ h = 600 cm 2 /V-s 2. assume uniform temperature (isothermal) conditions and room temperature operation, and 3. only consider velocity saturation when we talk about MOSFET scaling near the end of the term. Clif Fonstad, 9/15/09 Lecture 2 - Slide 9

10 Uniform material with uniform excitations (pushing semiconductors out of thermal equilibrium) B. Uniform Optical Generation, g L (t) The carrier populations, n and p: The light supplies energy to "break" bonds creating excess holes, p', and electrons, n'. These excess carriers are generated in pairs. Thus: Electron concentration : " + n'(t) # $ with n'(t) = p'(t) Hole concentration : p o " p o + p'(t) % Generation, G, and recombination, R: In general: $ dn = dp & G > R # dn = G " R % dt dt dt & G < R # dn ' dt In thermal equilibrium: G = R G = g o R = p o r " # $ G = R % g o = p o r = n i 2 r = dp dt = dp dt > 0 < 0 Clif Fonstad, 9/15/09 Lecture 2 - Slide 10

11 B. Uniform Optical Generation, g L (t), cont. With uniform optical generation, g L (t): G = g o + g L (t) R = n p r = ( + n' )( p o + p' ) r thus = G " R = g o +g L (t) " ( + n')(p o + p')r dn dt = dp dt The question: Given N d, N a, and g L (t), what are n(t) and p(t)? To answer: Using dn (1) = dp = dn' dt dt dt (2) g o = p o r (3) n' = p' = dp' dt gives one equation ine unknown*: dn' = g L (t) " (p o + + n')n'r dt * Remember: and p o are known given N d, N a Clif Fonstad, 9/15/09 Lecture 2 - Slide 11

12 B. Uniform Optical Generation, g L (t), cont. This equation is non-linear: It is in general hard to solve dn' dt = g L (t) " (p o + + n')n'r Special Case - Low Level Injection: LLI: n' << p o assume p-type, p o >> When LLI holds our equation becomes linear, and solvable: dn' " g L (t) # p o n'r dt = g L (t) # n' with $ min %1 p o r $ min This first order differential equation is very familiar to us. The homogeneous solution is: n'(t) = Ae "t # min Clif Fonstad, 9/15/09 Lecture 2 - Slide 12

13 Important facts about τ min and recombination: The minority carrier lifetime is a gauge of how quickly excess carriers recombine in the bulk of a semiconductor sample. Recombination also occurs at surfaces and contacts. (Problem x in P.S. #2 deals with estimating the relative importance of recombination in the bulk relative to that at surfaces and contacts.) In silicon: the minority carrier lifetime is relatively very long, the surface recombination can be made negligible, and the only significant recombinatioccurs at ohmic contacts (Furthermore, the lifetime is zero at a well built ohmic contact, and any excess carrier reaching a contact immediately recombines, so the excess population at ahmic contact is identically zero.) In most other semiconductors: both bulk and surface recombination are likely to be important, but it is hard to make any further generalizations Clif Fonstad, 9/15/09 Lecture 2 - Slide 13

14 Uniform material with uniform excitations (pushing semiconductors out of thermal equilibrium) C. Photoconductivity - drift and optical generation When the carrier populations change because of optical generation g L (t) " n(t) = + n'(t), p(t) = p o + n'(t)...the conductivity changes: i D (t) = with [" o + "'(t)] w # d V AB l i d (t) = "'(t) w # d l "(t) = q[µ e n(t) + µ h p(t)] = I D + i d (t) V AB = q[µ e + µ h p o ] + q[µ e + µ h ]n'(t) = " o +"'(t) This change is used in photoconductive detectors to sense light: The current varies in response to the light g L (t) " i d (t) Clif Fonstad, 9/15/09 Lecture 2 - Slide 14 + V AB g L (t) σ(t) w i D = I D + i d (t) Used : p'(t) = n'(t) l d

15 An antique photoconductor at MIT: A Stanley Magic Door with a lensed photoconductorbased sensor unit. Do you know where it is on campus? Clif Fonstad, 9/15/09 Lecture 2 - Slide 15

16 Modern photoconductors - mid-infrared sensors, imagers* Electron energy mid-infrared: λ = 5 to 12 µm, hν = 0.1 to 0.25 ev Electron energy Conducting states Conducting states E d E d 25 mev E d E d ev E g ev µm Bonding states Density of states E g 1.1 ev µm Bonding states Intrinsic, band to band option: small energy gap; n i very large n' (= p') small relative to signal very weak Density of states Extrinsic, donor to band option: large energy gap; n i very small n' (= N D+ ) large relative to signal much stronger Clif Fonstad, 9/15/09 Lecture 2 - Slide 16 * Night vision, deep space imaging, thermal analysis

17 Photoconductors - quantum well infrared photodetectors QWIPs Photoexcited electron Photocurrent Emitter Energy Distance Donor atoms Collector Lockheed Martin and BAE Systems. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Figure by MIT OpenCourseWare. Above: Schematic illustratiof QWIP structure and function. Upper right: A QWIP imager photograph Lower right: Spectral response of a typical IR QWIPs Clif Fonstad, 9/15/09 Ref: Lockheed-Martin (now BAE Systems), Nashua, N.H. Responsivity (ma/w) Blue QWIP -3V -1V Wavelength (microns) Note: 5 µm 0.25 ev Red QWIP T = 77K, T BB = 800K, f/4-1v -3V Figure by MIT OpenCourseWare. Lecture 2 - Slide 17 8 µm 0.15 ev

18 Non-uniform doping/excitation: Diffusion When the hole and electron populations are not uniform we have to add diffusion currents to the drift currents we discussed before. Diffusion flux (Fick's First Law): Consider particles m with a concentration distribution, C m (x). Their random thermal motion leads to a diffusion flux density: #C F m = " D m m #x where D m is the diffusion constant of the particles. Note that the diffusion flux is down the gradient. Diffusion current: Diffusion depends only on the random thermal motiof the particles and has nothing to do with fact that they may be charged. However, if the particles carry a charge q m, the particle flux is also an electric current density: J m = " q m D m #C m #x [ particles/cm 2 - s] [ A/cm 2 ] Clif Fonstad, 9/15/09 Lecture 2 - Slide 19

19 Non-uniform doping/excitation, cont.: Diffusion Hole Diffusion Fluxes and Currents: The hole concentration is p, each hole carries a charge +q, and the hole diffusion constant is D h. The hole diffusion flux and current densities are : #p #p F h = " D h J h = " qd h #x Electron Diffusion Fluxes and Currents: Similarly for electrons using n, -q, and D e : Total Current Fluxes: Adding the diffusion and drift current densities yield the total currents: Holes: Electrons: F e = " D e #n #x J h = qµ h pe " qd h #p #x Clif Fonstad, 9/15/09 Lecture 2 - Slide 20 #x J e = qd e #n #x J e = qµ e ne + qd e "n

20 Non-uniform doping/excitation, cont.: Diffusion... Continuity Total Current Fluxes, cont.: An important different between the drift and diffusion currents: Holes: Electrons: #p J h = qµ h pe " qd h #x "n J e = qµ e ne + qd e Drift depends on total carrier concentration Diffusion depends on the concentration gradient Continuity Relationships (Fick's Second Law): Another consequence of non-uniform doping/excitations is that fluxes can vary in space, leading to concentration increases or decreases with time: "F m = # "C m "t This effect must be added to generation and recombination when counting carriers. Clif Fonstad, 9/15/09 Lecture 2 - Slide 21

21 Non-uniform doping/excitation, cont.: Continuitity Continuity, cont.: For holes and electrons, Fick's Second Law translates to: Holes: Electrons: "F h "F e = 1 q "J h = 1 #q "J e = # "p "t = # "n "t With these factors the total expressions for the dp/dt and dn/dt are: Holes: Electrons: "p "t "n "t These can also be written as: "p "t + 1 q "J h = G # R # 1 q = G # R + 1 q = "n "t "J h "J e Clif Fonstad, 9/15/09 Lecture 2 - Slide 22 # 1 q "J e We'll come back to this in Lectures 3 and 6. = G # R

22 Non-uniform doping/excitation, cont.: Summary What we have so far: Five things we care about (i.e. want to know): Hole and electron concentrations: Hole and electron currents: Electric field: p and n J hx and J ex E x And, amazingly, we already have five equations relating them: Hole continuity: Electron continuity: Hole current density: Electron current density: Charge conservation: "p "t "n "t + 1 q # 1 q "J h "J e J h = qµ h pe # qd h "p J e = qµ e ne + qd e "n % = " [ &(x)e x ] So...we're all set, right? No, and yes... 2 = G # R $ G ext # [ n p # n i ]r(t) 2 = G # R $ G ext # [ n p # n i ]r(t) $ q[ p # n + N d (x) # N a (x)] Clif Fonstad, 9/15/09 Lecture 2 - Slide 23 We'll see next time that it isn't easy to get a general solution, but we can prevail.

23 Electronic Devices and Circuits Lect 2 - Excitation; Non-Uniform Profiles - Summary Uniform excitation: optical generation In TE, g o (T) = p o r(t) Uniform illumination adds uniform generation term, g L (t) Populations increase: + n', p o p o + p', and n' = p' dn'/dt = dp'/dt = g o (T) + g L (t) np r(t) = g L (t) [np p o ]r(t) focus is on minority g L (t) n'/τ min with τ min [p o r(t)] -1 if LLI holds Uniform excitation: both optical and electrical bb Photoconductivity: σ o σ o + σ' = q [µ e ( + n') + µ h (p o + p')] = σ o + q (µ e + µ h ) p' Photoconductors: an important class of light detectors Non-uniform doping/excitation: diffusion added bb Fick's first law: F mx = - D m dc m /dx [J mx = -q m D m dc m /dx] Diffusion currents: J ex,df = qd e dn/dx, J hx,df = -qd h dp/dx, Total currents: J ex = J ex,dr + J ex,df = qnµ e E x + qd e dn/dx J hx = J hx,dr + J hx,df = qpµ h E x qd h dp/dx Fick's second law: dc m /dt = -df mx /dx [dc m /dt = -(1/q m )dj mx /dx] Continuity: dn/dt - (1/q)dJ ex /dx = dp/dt + (1/q)dJ hx /dx = G - R Clif Fonstad, 9/15/09 Lecture 2 - Slide 24

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