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Comlemeti di Fisica Lectures 25-26 Livio Laceri Uiversità di Trieste Trieste, 14/15-12-2015

i these lectures Itroductio No or quasi-equilibrium: excess carriers ijectio Processes for geeratio ad recombiatio of carriers Cotiuity equatios for carriers Cotiuity equatios: three imortat secial cases Steady-state ijectio from oe side diffusio legth L Miority carriers recombiatio at the surface diffusio legth ad surface recombiatio velocity S lr The Hayes-Shockley exerimet Evidece for simultaeous diffusio, drift ad recombiatio Are we describig the behaviour of miority carriers aloe? What about majority carriers? Why are miorities imortat? Some examles Built-i electric field (Gauss!) ad ambiolar trasort equatios 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 2

I these lectures Referece textbooks D.A. Neame, Semicoductor Physics ad Devices, McGraw- Hill, 3 rd ed., 2003,.189-230 ( 6 Noequilibrium excess carriers i semicoductors ) R.Pierret, Advaced Semicoductor Fudametal, Pretice Hall, 2d ed.,.134-174 ( 5 Recombiatio-Geeratio Processes ) J.Nelso, The Physics of Solar Cells, Imerial College Press,. 79-117 (Ch.4, Geeratio ad Recombiatio ) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 3

Ijectio of excess carriers No-equilibrium! (i some cases: quasi-equilibrium)

Carrier ijectio - itroductio carrier ijectio = rocess of itroducig excess carriers i a semicoductor, so that: > i 2 Otical excitatio: shie a light o a semicoductor crystal; if the eergy of the hotos is hν > E g, the Photos absorbed excess electro-hole airs are created: Δ = Δ Other methods: Forward-bias a juctio I a extrisic semicoductor, the relative effect of Δ = Δ is very differet for majority ad miority carriers, sice Let us work out a examle (-tye Si, 0 > 0 at equilibrium) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 5

Carrier ijectio thermal equilibrium low ijectio high ijectio Examle: -tye Si at 300K thermal equilibrium 0 i 0 0 0 = 2 i 2 i = 1.45 10 N D = 10 N D 10 15 cm cm 3 3 = 2.1 10 5 cm 3 majority carriers itrisic cocetratio miority carriers 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 6

Carrier ijectio thermal equilibrium low ijectio high ijectio Examle: -tye Si at 300K excess carriers low-ijectio Δ = 10 = = = 10 = Δ 15 5 0 0 > 2 i + 10 + 10 = 10 + Δ 12 + Δ 12 12 0 10 10 cm Δ 15 12 3 cm cm << 3 3 N D majority carriers itrisic cocetratio miority carriers Large icrease i miority carriers 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 7

Carrier ijectio thermal equilibrium low ijectio high ijectio Examle: -tye Si at 300K excess carriers high-ijectio Δ = Δ > = = 0 0 > 2 i N D + Δ Δ + Δ Δ majority carriers itrisic cocetratio miority carriers Large icrease for both carriers Similar cocetratios 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 8

Carrier ijectio - summary carrier ijectio = rocess of itroducig excess carriers i a semicoductor Several methods (otical, etc.) Low-level ijectio: relative effect o cocetratio Negligible o majority carriers Imortat for miority carriers (also called miority carriers ijectio ) High-level ijectio If very high, both cocetratios become comarable Sometimes ecoutered i device oeratio 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 9

Quasi-equilibrium Examle: Ijectio or Geeratio of a excess of electros ad holes by absortio of hotos i a very short time, about 10-14 s Excess electros ad holes relax searately to thermal equilibrium i about 10-12 s ad remai for a much loger time i this quasi-equilibrium state Recombiatio of electros ad holes via several ossible mechaisms takes tyically about 10-6 s; lety of time to do somethig useful with stable electros ad holes! 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 10

Quasi-equilibrium ad quasi-fermi levels F P F N I quasi-equilibrium coditios: Two differet Quasi-Fermi Levels F N ad F P, describe the searate quasi-equilibrium cocetratios of electros ad holes, each oulatio corresodig to a searate Fermi-Dirac df (oe for electros, aother for holes) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 11

Quasi-Fermi levels: defiitio I quasi-equilibrium coditios: Two differet Quasi-Fermi Levels F N ad F P, describe the searate quasi-equilibrium cocetratios of electros ad holes: Electros: ( i e F N E i ) kt = N C e ( E C F N ) kt F N E i + kt l( ) i = E C kt l( N C ) Holes: ( i e E i F P ) kt = N V e ( F N E V ) kt F P E i kt l( ) i = E V + kt l( N V ) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 12

Modified mass actio law Fermi level E F at equilibrium: ( 0 = i e E F E i ) kt ( 0 = i e E i E F ) kt 0 0 = i 2 Quasi-Fermi levels at quasi-equilibrium: ( i e F N E i ) kt ( i e E i F P ) kt = 2 i e ( F N F ) kt 2 > i Differece i total chemical otetials or quasi-fermi levels 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 13

Quasi-Fermi levels: a examle Thermal equilibrium Fermi level E F -tye -tye equilibrium No-equilibrium No-equilibrium Quasi-Fermi levels F N, F P 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 14

Quasi-Fermi levels ad currets At (quasi-)equilibrium, (quasi-)fermi levels are costat Why? Because from the thermodyamic oit of view the Fermi level is the total chemical otetial, icludig the iteral chemical otetial ad exteral otetial eergy cotributios, like for istace the electrostatic otetial eergy. Off-equilibrium, the et movemet of carriers is related to the chagig total chemical otetial or (quasi-)fermi level by: J,x µ F N x J,x µ F P x From the defiitios of F N, F P by substitutio oe obtais: k J,x = qµ E x + qµ B T q x J,x = qµ E x qµ k B T q x drift diffusio D drift diffusio D NB: A quasi-fermi level that varies with ositio i a bad diagram immediately idicates that curret is flowig i the semicoductor! (see exercise 1 for a alicatio) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 15

Geeratio ad Recombiatio Charge carriers: electros ad holes

Electros ad holes Geeratio rate G : G = umber of free carriers geerated (searatig electros from holes) er secod ad er uit volume G is usually a fuctio of the available eergy (temerature, etc.) Recombiatio rate R: R = umber of free carriers disaearig due to recombiatio er secod ad er uit volume R is usually roortioal to the roduct of cocetratios of carriers ad recombiatio ceters ad to a cature coefficiet defied as c = v th σ, where v th is the thermal velocity ad σ is the recombiatio rocess cross-sectio Net recombiatio rate: U = R - G 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 17

Geeratio rocesses

Geeratio rocesses (for quatitative details: see bibliograhy) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 19

Radiative (light) geeratio Bad structure for direct ad idirect semicoductors Ge: direct (+ idirect) Si: idirect (+ direct) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 20

Photo absortio: igrediets direct trasitios idirect trasitios 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 21

Photo absortio coefficiet α idirect direct idirect direct I ( 0) x x + dx di dx = αi ( x) I ( x) = I ( 0)e α x x 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 22

Recombiatio rocesses

Recombiatio rocesses Recombiatio via tras Auger recombiatio ofte domiat i Silico devices 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 24

Recombiatio via tras Recombiatio: ofte domiated by idirect rocesses through recombiatio ceters or tras (direct recombiatio is egligible for Si) Examle: i a -tye semicod., uder low-ijectio coditios: for the miority-carriers (holes!) excess-recombiatio, the bottleeck is hole cature, that determies the hole lifetime τ Oce catured, the hole recombies quickly, sice there are may electros available U v th σ N t ( 0 ) 1 τ v th σ N t 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 25

Recombiatio via tras: termiology Cature, emissio : From the oit of view of the tra! I articular (figure, ext slide): (a) electro cature = (3) (b) electro emissio = (2) (c) hole cature = (4) (d) hole emissio = (1) Detailed treatmet: beyod our scoe! Shockley-Read-Hall model See back-u slides ad referece texts 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 26

Recombiatio via tras: lifetime aroximatio -tye mostly emty -tye semicoductor: electro lifetime domiated by electro cature (3) i emty RG ceters U v th σ N t 0 τ 1 v th σ N t ( ) 1.0µs (Si) -tye mostly full -tye semicoductor: hole lifetime domiated by hole cature (4) i full RG ceters U v th σ N t ( 0 ) τ 1 v th σ N t 0.3µs (Si) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 27

Cotiuity equatios Overall coservatio of charge! Detailed accoutig of local carrier desity as a fuctio of time: Geeratio, recombiatio, drift, diffusio

Summary of (B) (A) From: The Feyma Lectures o Physics, vol.ii 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 29

(A) Coservatio of charge: cotiuity equatios Ay et flow of charge must come from some suly!! J ˆ ds = V! J! dv = d S dt! J! = J x x + J y y + J z z = ρ t ρ dv The flux of a curret from a closed surface is equal to the decrease of the charge iside the surface ρ is the et charge desity (egative ad ositive, algebraic sum) Let us cosider electros ad holes, searately, i a semicoductor, i a simle oe-dimesioal case V = dq dt 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 30

Cotiuity for electros Exteral voltage V Volume elemet Adx How fast does the umber of electros chage i A dx? 1 q ρ t Substitutig: t Adx = $ J x ( )A J x + dx ( )A' & ) + ( G R )Adx % q q ( J ( x + dx) = J ( x) + J x dx +... Net carriers er secod through the walls + geeratio - recombiatio ad dividig by A dx see ext age 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 31

Cotiuity for electros ad holes t = 1 q Oe-dimesioal t = 1 q J x + ( G R ) J x + G R ( ) Three-dimesioal t = 1 q t = 1 q! J! + ( G R )! J! + ( G R ) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 32

Cotiuity for electros ad holes Miority carriers: Electros i -tye J,x = q µ E x + D x holes i -tye J,x = q µ E x D x Oe-dimesioal, uder low-ijectio coditios, for miority carriers: Electros: i -tye t = µ E x x + µ E x (electric field) x + D 2 x 2 + G 0 τ miority carrier excess holes: i -tye t = µ E x x µ E x x + D 2 x 2 Simly substitute J=J(drift)+J(diffusio) + G 0 τ Recombiatio rate R miority carrier lifetime 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 33

Cotiuity: geeratio, recombiatio, drift, diffusio Summary ad real-life alicatios

Geeratio, Recombiatio, Cotiuity - 1 Net recombiatio rate (2) U U = = R R G G For istace Photogeer. G L miority excess Δ (Δ ); aroximatios: 1-dimesioal, Electric field~0, Uiform doig 0 0 (x), 0 0 (x), Low-ijectio, hotogeeratio GL oly Geeral miority diffusio arox. lifetime arox. (1) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 35

Geeratio, Recombiatio, Cotiuity - 2 Eistei relatioshis miority carriers diffusio legths (el.) (h.) miority carrier lifetimes bad bedig Quasi-Fermi 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 36

Device simulatios I real life, device desigers use rograms erformig umerical itegratios i discrete sace ad time stes, to obtai (*): Process simulator doig N, N D A rofiles (*) carrier cocetratios, fields, currets i: (1) equilibrium, (2) steady state, (3) trasiets Device simulator, el. field!! J, J + boudary coditios, exteral fields ad excitatios!! 2 ρ E = V = ε ñ = + N A + N D!!! J = qµ V + qd!!! J = qµ V qd 1!! J = ( R G) + q t 1!! J = ( R G) + q t 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 37

Equilibrium vs steady state Equilibrium : detailed balace, for each rocess steady state : overall balace 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 38

Cotiuity equatios: alicatios Three examles (1) Steady state ijectio from oe side (2) Recombiatio at the surface (3) Hayes-Shockley exerimet

System of differetial equatios (A) Cotiuity (trasort) equatios for miority carriers, 1-d case (Sze otatios): t = µ E x x + µ E x x + D 2 x 2 + G 0 τ t = µ E x x µ E x (B) Gauss law, relatig the divergece of the electric field with the local charge desity, 1-d case: E x x = ρ ε x + D 2 x 2 ( ) q + N D N A ρ = q + N D + N A N = N D N A ( ) + G 0 τ Globally eutral, locally ca be ubalaced! To be solved with give boudary coditios! 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 40

(ex.1) Steady-state ijectio from oe side -tye semicoductor miority carriers: holes x = cocetratio ( )? Cotiuity equatio i this case: 2 = 0 t E = 0 G x 0 = 0 ( 0) 0 ( ) x 2 steady state o alied field o geeratio i the bulk at thermal equilibrium 0 excess ijected at x = 0 (boudary coditio) 1 = ( 0 ) Dτ Diffusio legth x L Solutio: ( x) = 0 + ( ( 0) 0) e L = Dτ 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 41

(ex.1) Diffusio legth - tyical values Diffusio legth L = D τ L = D τ µ [cm 2 /Vs] D [cm 2 /s] µ [cm 2 /Vs] D [cm 2 /s] Si 1350 35 480 12.4 GaAs 8500 220 400 10.4 Ge 3900 101 1900 49.2 examle L = D τ = ( 12.4) ( 5 10 7 ) = 25 µm 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 42

(ex.2) Recombiatio at the surface = 0 t E = 0 ( 0) 0 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 43 G x L 0 0 steady state o alied field Uiform geeratio i the bulk!!! at thermal equilibrium boudary coditio Here the boudary coditio is fixed by the rate at which carriers disaear with surface recombiatio velocity S lr = v th σ N st deedig o the surface tra desity N st cm s -1 cm s -1 cm 2 cm -2 Equatio to be solved (x > 0, bulk): 2 Δ 2 x Δ D τ + G D L = ( x) 0 0 Δ =

(ex.2) Solutio with boudary coditios Geeral solutio: Δ Boudary coditios: Δ ( x) %% G x + L τ A = 0 Δ ( x) % % Δ x 0 0 x ( ) = Ae x L + Be x L + G L τ comlemetary (homegeeous) L = τ D ( ) Δ ( 0) = B + G L τ B = Δ ( 0) G L τ after some algebra, substitutig A ad B, our solutio: % ( x) = 0 + G L τ 1+ Δ (0) G Lτ ' & G L τ e x L articular S lr ( * Δ (0) =??? ) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 44

(ex.2) Surface boudary coditio surface bulk A l + l Cosider a thi volume ( A 2l) x eclosig the surface: [ ( )] Al v th σ N st J x ( x = l) A = G L ( v th σ N t )Δ 0 diffusio curret Geer. recomb. (bulk) l 0 ( )Δ 0 Recombiatio (surface) ( ) A I the limit l : J ( 0) = ( v σ N ) Δ ( 0) 0 x th st 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 45

(ex.2) Solutio with surface recomb. velocity J x ( 0) = ( v th σ N ) st Δ 0 # % $ dδ dx Δ 0 & ( ' x= 0 = B L ( ) = G L τ + B dδ ( ) D % # $ dx & ( ' x= 0 = S lr Δ ( 0) cm 2 s -1 cm -4 cm s -1 cm -3 from the geeral solutio ad boudary coditios: $ D B ' & % L ) = S lr G L τ + B ( B = S G lr Lτ D L + S lr ( ) Solutio exressed i terms of the surface recombiatio velocity: $ ( x) = 0 + G L τ & 1 % S lr τ L + S lr τ e x L ' ) ( 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 46

(ex.2) Miority carriers at the surface 2 Δ x 2 Δ D τ + G L D = 0 cm 2 s -1 s cm -3 s -1 cm 2 s -1 S lr = v th σ N st cm s -1 cm s -1 cm 2 cm -2 cm L = D τ cm G L τ S lr τ L + S lr τ cm cm -3 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 47

(ex.2) Limitig cases Neglectig surface recombiatio: S lr τ << L x ( ) = 0 + G L τ ( ) = 0 + G L τ 0 as exected! Large ( immediate ) surface recombiatio: S lr τ >> L x ( ) ( ) = 0 + G L τ 1 e x L ( 0) = 0 as exected! 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 48

(ex.3) The Hayes-Shockley exerimet alied el. field localized light ulse Exerimetal set-u diffusio, recombiatio excess carrier distributios at successive times t 1 ad t 2, o alied field drift, diffusio, recombiatio excess carrier distributios at successive times t 1 ad t 2, with a costat alied field 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 49

(ex.3) The Hayes-Shockley exerimet After a light ulse: G L = 0 E x x = 0 o bulk geeratio costat alied field Trasort equatio for excess miority carriers (-tye semicoductor): Δ t = µ E x Δ x + D Solutio, o alied field: Δ ( x,t) = 2 Δ x 2 N 4πD t ex & x 2 4D t t ) ( ' τ + * Δ τ Δ = 0 diffusio, recombiatio Solutio, with alied field: Δ ( x,t) = & N 4πD t ex ( x µ E t x ( 4D t ' ( ) 2 t τ ) + + * drift, diffusio, recombiatio 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 50

The role of Gauss law Dielectric relaxatio Ambiolar trasort

The role of Gauss law Divergece of the electric field, 1-d case: E x x = ρ ε ( ) q + N D N A ρ = q + N D + N A ~ comlete ioizatio ( ) divergece-less fields ca be o-zero! (due to exteral charges) E x x = 0 E = 0 x N = N D N A Examle: curret i a semicoductor E x = cost. Uiform resistivity: uiform el.field, o local charge No-uiform resistivity: o-uiform field, local charge 0! E x1 ρ 1 = J x = E x2 ρ 2 E x1 > E x2 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 52

Dielectric relaxatio time costat -tye! Poisso (Gauss) E! = ρ ε! Ohm J = σ E!! Cotiuity J! = ρ t! J! = σ! E! = σρ ε = ρ t Examle: i a short time iject a excess of holes Δ i a small regio of a semicoductor crystal, that will exeriece a local ubalace of electric charge. How fast will be electrical eutrality restored? dρ dt + ( ) * σ + ε, - ρ = 0 ρ( t) = ρ( 0) e t τ d τ d = ε σ Dielectric relaxatio time costat 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 53

Dielectric relaxatio: Debye legth Debye legth L D (~ 10-5 cm): % ε ( L D D ' * = D & σ ) τ d τ d ε σ dielectric relaxatio time τ d (~ 10-12 s) Exect o sigificat deartures from electrical eutrality, over distaces greater tha about 4 L D to 5 L D i uiformly doed extrisic material, at thermal equilibrium (also true off-equilibrium!); this rocess is much faster tha the tyical excess carrier lifetime (10-7 s) Numerical examle for -tye Si, doed with door cocetratio N D = 10 16 cm -3 τ d = ε σ ε rε 0 q e µ N D = = 5.4 10 13 s = 0.54 s ( 11.7) 8.85 10 14 ( ) ( 1.6 10 19 )( 1200) ( 10 16 ) F cm ( Ω cm) 1 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 54

ambiolar trasort Two examles: (1) Biolar diffusio (2) Shockley exerimet

Ambiolar trasort - equatios Combiig the trasort equatios for electros ad holes with Gauss law, uder some simlifyig assumtios, (see back-u slides ad referece texts):! 0! 0!! E << it E a equatios of couled cotiuity for excess cocetratios 2 " D" + µ " E 2 x x " x + g R = " t ambiolar trasort equatio No-liear! With ambiolar diffusio coefficiet ad ambiolar mobility : D" = µ D µ + µ + µ D µ " = µ µ µ + µ ( ) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 56

Examle 1: Ambiolar diffusio Excess electros ad holes roduced by light close to the surface, i large cocetratios comared to the equilibrium (dark) oes. Electros have larger mobility ad move faster: electros ad holes artly searate (et charge ositive close to the surface, egative iside) The resultig electric field is directed so as to comesate for the differet mobilities (electros slowed dow, holes accelerated) + - The couled motio is called ambiolar diffusio. Sice electros ad holes move with the same velocity i the same directio, there is o et charge curret associated with this motio! 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 57

Examle 2: Heyes-Shockley exerimet V A + V B < V A - Delayed curret ulse see by robes at differet distaces: What is really movig??? 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 58

Examle 2 - qualitative iterretatio majority-carriers, locally slowed dow the miority-carrier cocetratio bum drags alog a majority-carrier cocetratio bum 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 59

Lecture 32 - exercises Exercise 1: A itrisic Si samle is doed with doors from oe side such that N D =N 0 ex(-ax). (a) Fid a exressio for the built-i electric field E(x) at equilibrium over the rage for which N D >> i. (b) Evaluate E(x) whe a = 1µm -1. Exercise 2: A -tye Si slice of thickess L is ihomogeeusly doed with hoshorous door whose cocetratio rofile is give by N D (x) = N 0 +(N L N 0 )(x/l). What is the formula for the electric otetial differece betwee the frot ad the back surfaces whe the samle is at thermal ad electric equilibria regardless of how the mobility ad diffusivity vary with ositio? What is the formula for the equilibrium electric field at a lae x from the frot surface for a costat diffusivity ad mobility? 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 60

Lecture 33 - exercises Exercise 1: Calculate the electro ad hole cocetratio uder steadystate illumiatio i a -tye silico with G L =10 16 cm -3 s -1, N D =10 15 cm -3, ad τ =τ =10 µs. Exercise 2: A -tye silico samle has 2x10 16 arseic atoms/cm 3, 2x10 15 bulk recombiatio ceters/cm 3, ad 10 10 surface recombiatio ceters/ cm 2. (a) Fid the bulk miority carrier lifetime, the diffusio legth, ad the surface recombiatio velocity uder low-ijectio coditios. The values of σ ad σ s are 5x10-15 ad 2x10-16 cm 2, resectively. (b) If the samle is illumiated with uiformly absorbed light that creates 10 17 electro-hole airs/(cm 2 s), what is the hole cocetratio at the surface? Exercise 3: The total curret i a semicoductor is costat ad is comosed of electro drift curret ad hole diffusio curret. The electro cocetratio is costat ad equal to 10 16 cm -3. The hole cocetratio is give by (x)=10 15 ex(-x/l) cm -3 (x>0), where L = 12µm. The hole diffusio coefficiet is D =12cm2/s ad the electro mobility is µ =1000cm 2 /(Vs). The total curret desity is J = 4.8 A/cm 2. Calculate (a) the hole diffusio curret desity as a fuctio of x, (b) the electro curret desity versus x, ad (c) the electric field versus x. 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 61

Lecture 34 - exercises Exercise 1: Excess electros have bee geerated i a semicoductor so that at t = 0 the excess cocetratio is Δ(0) = 10 15 cm -3. Assumig a excess-carrier lifetime τ = 10-6 s, calculate the excess electro cocetratio ad the recombiatio rate for t = 4µs. Exercise 2: Excess electros ad holes are geerated at the ed of a silico bar (at x = 0); the silico bar is doed with hoshorus atoms to a cocetratio N D = 10 17 cm -3. The miority lifetime is 10-6 s, the electro diffusio coefficiet is D = 25 cm 2 /s, ad the hole diffusio curret is D = 10 cm 2 /s. Determie the steady-state electro ad hole cocetratios as a fuctio of x (for x >0) ad their diffusio currets at x = 10µm. 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 62

Back-u slides (toics ot icluded i the stadard rogram!)

Recombiatio via tras Shockley-Read-Hall model

Low-ijectio miority lifetime aroximatio -tye mostly emty -tye semicoductor: electro lifetime domiated by electro cature (3) i emty RG ceters U v th σ N t 0 τ 1 v th σ N t ( ) 1.0µs (Si) -tye mostly full -tye semicoductor: hole lifetime domiated by hole cature (4) i full RG ceters U v th σ N t ( 0 ) τ 1 v th σ N t 0.3µs (Si) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 65

Shockley-Read-Hall lifetimes - 1 What haes if these aroximatios are ot valid?, may be comarable (o loger true that >> or >> ) carrier lifetime o loger domiated by availability of: -tye: emty tras for electro cature (N t 0 = N t (1-F) N t ) -tye: full or ioized tras for hole cature (N t - = N t F N t ) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 66

Shockley-Read-Hall lifetimes - 2 All four idirect rocesses must be take ito accout (see also SZE 2.4.2, idirect recombiatio, or Neame 6.5.1) 1=d hole emissio (from a tra) 2=b electro emissio (from a tra) 3=a electro cature (i a tra) 4=c hole cature (i a tra) Net recombiatio rates for electros ad holes searately: R d = e N t ( 1 F) R b = e N t F R a = c N t 1 F R c = c N t F ( ) U = R a R b U = R c R d 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 67

Shockley-Read-Hall lifetimes - 3 From equilibrium coditios G L =0; detailed balace: R a - R b = R c - R d = 0) emissio coefficiets (e, e ) i terms of: cature coeff. (c = v th σ, c = v th σ ) e e = c = c 1 1 1 = 1 = e i e i ( E E ) i kt ( E E ) kt t t i 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 68

Shockley-Read-Hall lifetimes - 4 o-equilibrium steady-state (G L = costat 0, ad: U = R a - R b = U = R c - R d 0 ): see SZE eq. (63) U = U = U = i 2 1 ( + 1 ) + 1 ( + 1 ) c N t c N t = i 2 τ ( + 1 ) + τ ( + 1 ) This is a geeral result, usually imlemeted i device simulatios A secial case: the revious Low-ijectio miority lifetime result For very high doig cocetratios, direct trasitios become likely: this ca be modeled by makig τ ad τ cocetratio-deedet 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 69

ambiolar trasort Two examles: (1) Biolar diffusio (2) Shockley exerimet

Ambiolar trasort - equatios Secial case: homogeeous semicoductor Thermal equilibrium cocetratios 0, 0 costat (time ad sace) 2 " D x µ $ " 2 E % & x 2 " D x + E x x x + µ $ " 2 E x x + E x % & x!! E = E x ' ( ) + g ' ( ) + g = " τ x = " τ x x = q ( " " ) " 0 " 0 ε Assume: - Small iteral electric field, with resect to the alied field - Almost comlete balace of electro ad hole cocetratios - Geeratio, recombiatio g = E it << E a!! g g τ t = τ t R 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 71

Ambiolar trasort - equatios We get the : 2 " D x µ $ " 2 E x x + E x ' % & x ( ) + g R = " t 2 " D x + µ $ " 2 E x x + E x ' % & x ( ) + g R = " t µ µ Multily (see above), add ad divide by µ + µ 2 " D" + µ " E 2 x x " x + g R = " t ambiolar trasort equatio No-liear! With ambiolar diffusio coefficiet ad ambiolar mobility : D" = µ D µ + µ + µ D µ " = µ µ µ + µ ( ) 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 72

14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 73 Ambiolar trasort I a extrisic semicoductor uder low ijectio, the ambiolar mobility coefficiets reduce to the miority-carrier arameter values, that are costat -tye miority: electros -tye miority: holes t g x E x D t g x E x D x x " = " " + " " " = " " + " + " 0 2 2 0 2 2 τ µ τ µ The behaviour of excess majority carriers follows that of miority!!!

Examle 1: Ambiolar diffusio Charge currets for electros ad holes: j x, = q D x + q µ E x j x, = q D x + q µ E x The et curret desity vaishes! Associated electric field: j x = j x, + j x, = 0 E x = D x D x µ + µ The article currets are therefore equal for electros ad holes: j e = j h = D µ + D µ µ + µ x = D amb D amb = D µ + D µ µ + µ is the ambiolar diffusio coefficiet 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 75 x

A steady-state examle: locally illumiated semicoductor bar

Igrediets ad qualitative exectatios -tye; o-equilibrium; oe-circuit; Local steady illumiatio Diffusio of excess carriers (, ) $ $ 0 = Δ 0 Diffusio currets, but also drift currets due to the electric field E x d! d! J h = qµ h Ex qdh J e = qµ eex + qde dx dx J = J + J = 0 Electric field E x (charge ubalace!) de x q = ( " " ) 0 dx ε The local charge ubalace is small!!!!! <<1!! 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 77 e h = Δ

Igrediets ad qualitative exectatios holes (h): miority electros (e): majority drift (e,h): >> q µ E >> qµ J J J h e e = qµ J h = = h J x qµ e h qµ E E e + x qd E J h << h x x h qd + qd = 0 qd d! dx E x e h d! dx d! dx -tye; o-equilibrium; oe-circuit; Local steady illumiatio Diffusio of excess carriers (, ) $ $ 0 = Δ 0 Diffusio currets, but also drift currets due to the electric field E x Diffusio (e,h): oosite currets, comarable sizes Electric field E x (charge ubalace!) d! de x q = ( " " ) 0 dx ε dx The local charge ubalace is small! i this case!!!! miority carriers flow <<1!! maily by diffusio h 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 78 = Δ

Qualitative results 2 d $ 2 dx $ D τ h h = = 0 ; g D L h ; 0 < x < δ x > δ 2 2 The cotiuity equatios above ca be solved aalytically to obtai (x) ad J h (x) J e (x) = - J h (x) J e + J = 0 h J e = J h If If D D e e = D E = 0, > D h h x " = " Majority diffusio larger Majority drift curret: same directio as J h (x) Electric field E x ( 0 + " ) Ex q e Ex J e(drift) = qµ e µ 0 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 80

Uder coditios of: - comarable mobilities - small ijectio i uiform extrisic material the miority-carrier curret will be comarable to the majority-carrier curret oly if miority carriers flow maily by diffusio The large suly of majority carriers effectively shields the miority oes from roducig ay sigificat sace charge. The small fields that are geerated by slight deartures from eutrality serve to adjust the majority-carrier curret to the geeral coditios of the roblem, without roducig sigificat effects o miority carriers. A aroximate calculatio of J h, J e, E x,, i the quasi-eutral aroximatio (without eforcig Gauss law with - = 0) will be quite satisfactory; of course, the small - will ot be very accurately determied from E x foud i this way 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 81

Aroximate quatitative solutio J J J E Assumig quasi-eutral behaviour: d! d!!! dx dx (well justified i most cases) e = J e(drift) + J e(diffusio) e(diffusio) e(drift) x = J = J d" d" De = qde qde = J dx dx Dh & D # e = J $ 1! e(diffusio) J h J h % Dh " J 0 h ( D D ) e(drift) e(drift) J h e h 1 qµ e qµ e qµ Aroximate charge ubalace (de x /dx) e 0 h 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 82

Nearly exact solutio A more accurate solutio, ot usig the arox. to evaluate J e (diffusio) I this examle: δ L h >> L D light beam width δ hole diffusio legth L h Debye legth L D 14/15-12-2015 L.Laceri - Comlemeti di Fisica - Lectures 25-26 83

Complementi di Fisica Lectures 25-26