ECE606: Solid State Devices Lecture 12 (from17) High Field, Mobility Hall Effect, Diffusion

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1 ECE66: Solid State Devices Lecture 1 (from17) High Field, Mobility Hall Effect, Diffusio Gerhard Klimeck gekco@purdue.edu Outlie 1) High Field Mobility effects ) Measuremet of mobility 3) Hall Effect for determiig carrier cocetratio 4) Physics of diffusio 5) Coclusios REF: ADF, Chapter 5, pp. 19-

2 Mobility at High Fields? υe ( ) x x x x τ υ = µ µ E 1 + EC N * N 1 m E E E N What causes velocity saturatio at high fields? E c E Where does all the mobility formula i device simulator come from? 3 Velocity Saturatio i Si/Ge E = = + = 1 E Ec = > + 1 E E = > c 3 + E Ec =

3 Velocity Overshoot & Iter-valley Trasfer Larger m* Smaller m* What type of scatterig would you eed for iter-valley trasfer? 5 Outlie 1) High Field Mobility effects ) Measuremet of mobility 3) Hall Effect for determiig carrier cocetratio 4) Physics of diffusio 5) Coclusios REF: ADF, Chapter 5, pp. 19-6

4 Problem of mobility measuremet ukow dopig cocetratio Four-Probe Measuremet 1 measures voltage of device without measurig drop i curret carryig wires V Ca we fid out the dopig cocetratio ad type by a electrical measuremet without ay kowledge of how the sample was prepared? 1 - Problem of mobility measuremet ukow dopig cocetratio Four-Probe Measuremet 1 measures voltage of device without measurig drop i curret carryig wires V E = ρ = ( µ + µ p) E 1 ρ = ( µ + µ p) (i the low voltage limit where fieldvoltage curve is liear) p p = 1 µ N D for -type 1 = for p-type µ p N A 1 -

5 Outlie 1) High Field Mobility effects ) Measuremet of mobility 3) Hall Effect for determiig carrier cocetratio 4) Physics of diffusio 5) Coclusios 9 Trivia - FYI Discovered i 1879 by Edwi Herbert Hall while he was workig o his doctoral degree at ohs Hopkis Uiversity i Baltimore, Marylad. Doe 18 years before the electro was discovered. 4 Nobel Prizes associated directly with it. Read the origial article (O a New Actio of the Maget o Electric Currets) at de9.html for a fasciatig accout of the discovery of the effect. ad 1

6 y z x - VH + B d w Set up for Hall Measuremet Force actig o a charged particle i a magetic field z I Applyig a magetic field oe pushes carriers towards oe face This iduces a voltage across the two faces perpedicular to curret flow. We will relate this voltage to the curret ad magetic field ad deduce the desity of carriers. L UIC system: 4-3K, -1.5 T 11 Simple classical Newto s law expressio m effective mass v drift velocity w/ scatterig * m υ E υ B = τ * m υ = τe τυ B τ τ τ E E B m* τ = τe + E B * m τe τ υ = + E B * * m m Drude Model for electros * Weak B field * m υ E τ τe υ = * m Perturbatio works iff. τ B τ B τω 1 * * c m τ = m << * Same model works for holes, but with + istead of - 1

7 Drude Model & Hall Effect x y z = υ τ τ τ = E E B * * * m m m = σe σ µ E B y V H E x E y E z B x B y B z z B Field I x σ σ µ B E z x = σ µ σ y Bz Ey 13 ~, sice voltmeters have a very large iteral resistace so very little curret flows through x σ σ µ B = σ µ Bz σ y x σ σ µ B y z z Ex σ E y Ex E y y V H Hall Resistace I the limit of small B z B Field I x σ Ex = σ E µ Bz σ y R H = Ey Bz 1 = x x is measured, B z is kow, V h is measured so E y is kow 14

8 ~, sice voltmeters have a very large iteral resistace so very little curret flows through x σ σ µ B = σ µ Bz σ y z Ex E y y V H Hall Resistace I the limit of small B z B Field I x σ Ex y σ µ Bz σ E y By a simple electrical measuremet, we ow kow the cocetratio of electros i the sample. From the x σ Ex x is measured, B z is itrisic = cocetratio of carriers i the semicoductor σ E µ Bz σ kow, V h is measured so ad temperature we y ca deduce the dopig E y is kow cocetratio i the sample R H = Ey Bz 1 = x 15 Hall Resistace ~, sice voltmeters have a very large iteral resistace so very little curret flows through x σ σ µ B = σ µ Bz σ y x σ σ µ B y z z Ex σ E y Ex E y y V H I the limit of small B z B Field I x Why Bz σ set to zero Ei x oe e x ad is measured, ot the other? B z is = => σ E µ Bz σ aalytical simplicity. kow, V h is measured so The system is also explicitly y solvable E y is i kow terms of Bz^ ad Ey.. Oce Bz^z/Ey is small compared to 1/ the oe ca eglect Ey Bz 1 R = higher H = orders ad come to the same result x 16

9 Temperature-depedet Cocetratio N D Freeze out 1 Extrisic Itrisic i N D Temperature 17 Outlie 1) High Field Mobility effects ) Measuremet of mobility 3) Hall Effect for determiig carrier cocetratio 4) Physics of diffusio 5) Coclusios 18

10 Now the Diffusio term 1 = r + g N N N = µ E + D N N N I p 1 = r + g P P P = pµ E D p P P P + ( D A ) D = p + N N V 19 Diffusio Flux Assumig idepedet radom motio - o average, half of the electros (at each x) will move to the left, half to the right NET movemet of electros to left i.e. agaist the gradiet. Opposite sig for holes. x dp dp p() + p() l p() + p( ) + l = + l l l υth lυ dp dp = D

11 Diffusio Flux This looks like a completely classical derivatio. Assumig idepedet x Where is the Quatum Mechaics? radom motio - o average, half of the electros (at each x) Quatum will move Mechaics to the left, is half i to the Diffusio coefficiet ad the Drift Velocity. the Determies right NET the movemet available states of electros to left i.e. agaist Determies the gradiet. the capability to carry curret Opposite sig for holes. Scatterig is built i here! Iteractios of may, may, may electros ad he surroudig. dp dp Without scatterig p() + () p() + p( ) + l oe p could ot l explai the eual partitioig ito = x + l directios! d l l υth lυ dp dp = D 1 Eistei Relatioship D µ = lυ τ m * ( υ τ ) υ 1 * mυ = = τ m * = kbt because scatterig domiates both pheomea

12 Eistei Relatioship The derivatio is based o the basic otio that carriers at o-zero temperature (Kelvi) have a additioal thermal eergy, which euals kt/ per degree of freedom. It is the thermal eergy, which drives the diffusio process. At T = K there is o diffusio. The reader should recogize that the radom ature of the thermal eergy would ormally reuire a statistical treatmet of the carriers. Istead we will use average values lυ to describe ( υ τ ) the υ process. 1 * Such approach is justified o the basis that D µ = a more elaborate statistical approach mυ yields = = kbt the same results. To further simplify the derivatio, we will derive = the diffusio curret for a oe-dimesioal τ semicoductor τ i which carriers ca oly move alog oe directio. * * m m We ow itroduce the average values of the variables of iterest, amely the thermal velocity, vth, the collisio time, tc, ad the mea free path, l. The thermal velocity is the average velocity of the carriers goig i the positive because or egative scatterig directio. The domiates collisio time both is the pheomea time durig which carriers will move with the same velocity before a collisio occurs with a atom or with aother carrier. The mea free path is the average legth a carrier will travel betwee collisios. Klimeck ECE66 Fall 1 otes adopted from Alam 3 A alterate derivatio ν th = l τ RMS velocity of carriers is decided by the average mea free path ad average scatterig time Carriers that arrive at x= do so by travellig oe mea free path from right to left or vice versa. Net curret from left to right (at x=) = charge o carrier*et flux of carriers from left to right Figures ad derivatio from 4

13 Derivatio for electro diffusio curret Electro flux at x= from left to right φ,left right = 1 ν th(x = l) The factor half appears because the other half at x=-l travels towards the left Electro flux at x= from right to left φ,right left = 1 ν th(x = l) Net Flux φ = φ,left right φ,right left = 1 υ th [(x = l) (x = l)] 5 Derivatio cotiued φ = lυ th [(x = l) (x = l)] l If the mea free path is small eough, the we ca write this as d φ = lυ th The egative sig arises because we take the gradiet is usually measured for icreasig values of x. The curret desity is the give by. d = φ = lυ th 6

14 For holes, we repeat the derivatio ad fid p = φ p = l holes υ th,holes dp Lump together the secod ad third terms to form a diffusio costat p = D p dp = D d 7 For thermally distributed carriers 1 m* υ = k T B th Euipartitio theorem states that i euilibrium each carrier has thermal eergy of kt/ per degree of freedom. Our derivatio is i oe dimesio oe degree of freedom Use this to get a ew isight ito the relatio betwee drift ad diffusio 8

15 Alterate way to obtai the diffusio costats D = lυ th = τυ = τ m * υ th m * D µ = k T B D µ = D p µ p = k BT th = µ k BT Eistei relatios valid for electros ad holes AT EQULIBRIUM The euatio shows how thermal uatities goverig diffusio ca be related to those goverig drift (mobility depeds o the effective mass which describes motio of carriers i the presece of a field ad scatterers ) 9 A alterate derivatio of Eistei s relatioship No curret i euilibrium d = = µ E + D 1 d µ E = D E C E C1 1 E C E F L E D = 1e = 1 µ µ V D e E V L 1 ( EC EF ) / kt NCe = = e = e ( EC1 EF ) / kt N e C ( EC EC 1) / kt V / kt V µ V µ kt D kt D = = Similar to relatioship betwee c ad e discussed i Chapter 5 3

16 Coclusio 1) Measuremet of mobility ad carrier cocetratio is particularly importat for aalysis of semicoductor devices. ) Drift, diffusio, ad recombiatio-geeratio costitute the elemetal processes i semicoductor device physics. 3) We will put the pieces together i the ext class. 31 ECE66: Solid State Devices Lecture (from18) Cotiuity Euatios Gerhard Klimeck gekco@purdue.edu

17 Outlie 1) Cotiuity Euatio ) Example problems 3) Coclusio Ref. Advaced Semicoductor Fudametals, pp Now the Cotiuity Euatios These euatios have bee state of the art i device modelig util recetly (1-15 years ago ) 1 = r + g N N N = µ E + D N N N p 1 = r + g P P P = pµ E D p P P P + ( D A ) D = p + N N Cotiuity e. for electros Cotiuity e. for holes Poisso euatio 34

18 Some critical remarks about these euatios I Cotiuity euatios are always valid (regardless of the detailed physical model describig the device) because they describe a coservatio law. Poisso euatio as give does ot accout for explicit electro-electro repulsio. Might eed to be modified for strogly correlated systems. V Drift ad Diffusio euatios get modified whe devices are so small that essetially o scatterig takes place withi the device. 35 Cotiuity Euatio preuel: A Good Aalogy Wabash River Rate of icrease of water level i lake = (i flow - outflow) + rai - evaporatio 1 = + gn r N 36

19 Now the Cotiuity Euatios for electros Cosider a arbitrarily shaped semicoductor with cotacts that pump i curret. Divide this arbitrary semicoductor ito small cubic boxes of cross sectioal area A ad legth x. Boxes are large eough that cocepts like effective mass, mobility etc. are valid iside these boxes. ( x) Electros comig i are give by (x), goig out (x+ x) Total umber of electros per cm 3 g N r N x A=1 ( x + ) Now the Cotiuity Euatios g N ( x) x A=1 ( x+ ) g N -> geeratio rate i (per cm 3 per sec) from exteral processes such as light. r N -> recombiatio rate i (per cm 3 per sec) i the cetral box. r N We wish to relate all of these factors that affect the cocetratio of carriers i the cetral box. Our strategy (remember the aalogy) The rate of chage of umber of electros INSIDE the cetral box should be eual to No. of electros comig i MINUS No. of electros goig out per sec (govered by curret desity (x) ad (x+ x)) PLUS No. of electros gettig geerated from exteral processes per sec (govered by geeratio rate g N ) MINUS No. of electros lost by recombiatio per sec (govered by recombiatio rate r N ) 38

20 Let s just write the euatios dow g N ( x ) x A=1 ( x + d x ) r N ( A x ) A ( x) A ( x + ) = ( x) ( x + ) = + gn r x N + A g x A r x 1 = + g r N N N N 39 Cotiuity Euatios for Electro/Holes ( x) p ( x) p g N r N rp gp ( x + ) 1 = + g r N N Usually geeratio ad recombiatio rates for electros ad holes are the same sice the same processes create/destroy a electro/hole ( ) p x + p 1 = + g r P P P 4

21 Euatios i place, learig to solve them 1 = r + g p 1 = r + g N N N = µ E + D N N N P P P = pµ E D p P P P + ( D A ) E = p + N N I V Two methods of solutio: Numerical ad Aalytical 41 Aalytical Solutios + ( D A ) D = p + N N 1 = r + g p 1 = r + g N N N = µ E + D N N N P P P = pµ E D p P P P Bad-diagram Diffusio approximatio, Miority carrier trasport, Ambipolar trasport 4

22 Outlie 1) Cotiuity Euatio ) Example problems 3) Coclusio 43 x Cosider a complicated real device example Upassivated surface Metal cotact 1 3 Acceptor doped Light tured o i the middle sectio. The right regio is full of mid-gap traps because of daglig bods due to u-passivated surface. Iterface traps at the ed of the right regio (That s where the daglig bods are ) The left regio is trap free. The left/right regios cotacted by metal electrode. 44

23 Recall: Aalytical Solutio of Schrodiger Euatio 1) d ψ + k ψ = N ukows for N regios ) ψ ( x = ) = ψ ( x = + ) = Reduces ukows 3) ψ dψ = ψ + x= xb x= xb dψ = + x= xb x= xb Set N- euatios for N- ukows (for cotiuous U) 4) Det(coefficiet matix)= 5) ψ ( x, E) = 1 Ad fid E by graphical or umerical solutio for wave fuctio 45 Recall: Boud-levels i Fiite well E U(x) 1) ψ = Asi kx + B cos kx ψ = Me + Ce α x + α x ψ = De + Ne α x + α x ) Boudary Coditios ψ ( x = ) = ψ ( x = + ) = a 46

24 Aalogously, we solve for our device Solve the euatios i differet regios idepedetly. Brig them together by applyig boudary coditios. 47 Regio : Trasiet, Uiform Illumiatio, Uiform dopig (uiform) 1 = N r + N gn ( + ) = + G τ (uiform) p 1 = r + g p P p = µ E + D N N N Recall Shockley- Read-Hall = pµ E D p p p P Acceptor doped ( p + p) p = + G Majority carrier τ p Electric field still zero becaus ew carriers balace + + D = p + N N = p + p + N N = ( D A ) ( D A ) 48

25 Example: Trasiet, Uiform Illumiatio, Uiform dopig, No applied electric field ( ) = + G τ ( x, t) = A + Be t τ Acceptor doped t =, ( x,) = A = B t, ( x, ) = Gτ = A t ( ) ( x, t) = Gτ 1 e τ No carriers yet geerated Steady state, o chage i carriers with time time 49 Regio 1: Oe sided Miority Diffusio at steady state = (steady-state) r N = (trap free) = (ogeeratio) g N E = Steady state Acceptor doped 1 Trap-free d = D N d D N (due to isertio of electros from cetral regio) 5

26 Example: Oe sided Miority Diffusio Metal cotact = DN d υ = m m ( x, t) = C + Dx' x a x = a, (x ' = a ) = C = Da (Metal has high electro desity as compared to semicoductor) x = ', (x ' = ') =C ust substitute x= i above e. ' (, ) ( ') x x t = x = 1 a x 51 = (steady-state) r N (ot trap free) = (ogeeratio) g N Regio 3: Steady state Miority Diffusio with recombiatio E = d D N (due to isertio of electros from cetral regio) = D N = D d ( + ) τ N d τ Flux Steady state Acceptor doped 3 Trap-filled 5

27 Diffusio with Recombiatio D N d = τ Fuctioally similar to Schrodiger e. b Metal cotact x L x L ( x, t) = Ee + Fe 3 x = b, ( x = b) = F = Ee b L x =, ( x = ) = E + F = ( x = ) X () x t = e e e (1 e ) x L b L x L (, ) ( ) b L x 53 ( x) = Gτ = ( ) = (') 1 ( x ') = ( x = ) 1 x a ' (') x L b L x L ( ) ( ) b L x = e e e (1 e ) x Combiig them all. 1 3 Match boudary coditio x' = Gτ 1 a d Calculatig curret N = µ N E + DN b Gτ e e e = (1 e ) x L b L x L ( ) b L 54

28 Coclusio 1) Cotiuity Euatios form the basis of aalysis of all the devices we will study i this course. ) Full umerical solutio of the euatios are possible ad may commercial software are available to do so. 3) Aalytical solutios however provide a great deal of isight ito the key physical mechaism ivolved i the operatio of a device. 55