Nonequilibrium Excess Carriers in Semiconductors

Transcription

1 Lecture 8 Semicoductor Physics VI Noequilibrium Excess Carriers i Semicoductors

2 Noequilibrium coditios. Excess electros i the coductio bad ad excess holes i the valece bad Ambiolar trasort : Excess electros ad excess holes diffuse, drift, ad recombie with the same effective diffusio coefficiet, drift mobility, ad lifetime. Ambiolar trasort equatio Quasi-Fermi eergy for electros ad quasi- Fermi eergy for holes 2

3 3 CARRIER GENERATION AND RECOMBINATION The Semicoductor i Equilibrium The electros ad holes are created i airs, so the thermal-geeratio rates of electros ad holes are equal. G = G 0 0 The electros ad holes recombie i airs, so the recombiatio rates of electros ad holes are equal. R = R 0 0 The cocetratios of electros ad holes are ideedet of time; therefore, the geeratio ad recombiatio rates are equal: G = G = R = R

4 Uder oequilibrium coditios Whe high-eergy hotos are icidet o a semicoductor, electros i the valece had may be excited ito the coductio bad. => Electro-hole airs are geerated. => The additioal electros ad holes created are called excess electros ad excess holes. The geeratio rate of excess electros ad holes are equal ' ' g = g 4

5 Fig Creatio of excess electro ad hole desities by hotos. = +δ 0 = + δ 0 Excess electro ad hole cocetratios 5 = i

6 As i the case of thermal equilibrium, a electro i the coductio bad may "fall dow" ito the valece bad, leadig to the rocess of excess electro-hole recombiatio. The recombiatio rate of excess electros ad holes ' ' are equal R = R 6 Fig Recombiatio of excess carriers reestablishig thermal equilibrium

7 The carrier cocetratio is deedet o time. The et rate of chage i the electro cocetratio d( t) dt = α 2 r[ i ( t) ( t)] Sice 0 ad 0 are ideedet of time ad δ( t) = δ ( t) d ( δ ( t )) dt = α + δ + δ 2 r[ i ( 0 ( t))( 0 ( t))] = α δ ( )[( + ) + δ ( ))] r t 0 0 t 7 If we cosider a -tye material ( 0 >> 0 ) uder low-level ijectio ( δ ( ) << ) d t ( δ ( t) ) dt 0 = α δ ( t) r 0

8 8 αr t δ( t) = δ(0) e = δ(0) e t/ τ 0 0 The excess carrier cocetratio is a exoetial decay from the iitial excess cocetratio 1 0 = ( r 0) a costat for the low-level ijectio τ α called the excess miority carrier lifetime The recombiatio rate R ' d( δ( t)) δ( t) = = + αr0δ( t) = dt τ For the direct bad-to-bad recombiatio, the excess majority carrier holes recombie at the same rate, so that for the -tye material ' ' δ( t) R = R = τ 0 0

9 I the case of a -tye material ( 0 >> 0 ) uder lowlevel ijectio δ ( t ) << 0 R ' = R = ' δ ( t) τ 0 The excess miority carrier lifetime τ ( α = ) 0 r 0 1 ' ' I summary, R = R = f ( ( t) ) ' ' ' ( ( )) G = G f t G = G = f ' ( time, sace) 9

10 CHARACTERISTICS OF EXCESS CARRIERS The excess carriers behave with time ad i sace i the resece of electric fields ad desity gradiets is of equal imortace. The excess electros ad holes diffuse ad drift with the same effective diffusio coefficiet ad with the same effective mobility. This heomeo is called ambiolar trasort. 10

11 Cotiuity Equatios A oe-dimesioal hole article flux is eterig the differetial elemet at xad is leavig the elemet at x + dx. The et icrease i the umber of holes er uit time i the differetial volume elemet is: + FP dxdydz = dxdydz + gdxdydz dxdydz t x τ Fig Differetial volume showig x comoet of the hole-article flux t 11 The et icrease i the umber of holes er uit time due to x-comoet hole flux τ t the icrease i the umber of holes er uit time due to the geeratio of holes the decrease i the umber of holes er uit time due to the recombiatio of holes. icludes the thermal equilibrium carrier lifetime ad the excess carrier lifetime.

12 The et icrease i the hole cocetratio er uit time is + F = + g t x τ t the cotiuity equatio for holes Similarly, the oe-dimesioal cotiuity equatio for electros is + F = + g t x τ t 12

13 13 Time-Deedet Diffusio Equatios The hole ad electro curret desities are J = eµ E ed J x = eµ E + ed x The hole ad electro flux are J + J = F = µ E D = F = µ E D + e x e Substitute them ito the cotiuity equatio where 2 ( E) = µ E + D + g 2 t x x τ t 2 ( E) = µ E + D + g 2 t x x τt ( ) E E = E + x x x x

14 The time-deedet diffusio equatios for holes ad electros are E D E g x x x t 2 µ ( ) = τ t E D E g x x x t 2 µ ( ) = τt The thermal equilibrium cocetratios, o ad oare ot fuctios of time. For the secial case of a homogeeous semicoductor, o ad o are also ideedet of the sace coordiates. Therefore, the time-deedet diffusio equatios ca be ( δ ) ( δ ) E ( δ ) D E g x x x t 2 µ ( ) = τ t 14 ( δ) ( ( δ) E ) ( δ) D E g 2 µ + + = 2 x x x τt t Which describe the sace ad time behavior of the excess carriers

15 AMBIPOLAR TRANSPORT Fig The creatio of a iteral electric field as excess electros ad holes ted to searate With a alied electric field, the excess holes ad electros are created. This searatio will iduce a iteral electric field betwee the two sets of articles. E = E + E The egatively charged electros ad ositively charged holes the will drift or diffuse together with a sigle effective mobility or diffusio coefficiet. - called ambiolar trasort. a it 15

16 To relate the excess electro ad hole cocetratios to the iteral electric field, Poisso's equatio is itroduced e( δ δ) Eit Eit = = If ( δ = δ), Eit = 0 εs x where is the ermittivity of the semicoductor material Sice ε s g = g = g R = = R = = R τ τ the charge eutrality coditio δ = δ t t 16 The diffusio equatios ca be writte as ( δ ) ( ( δ ) E ) ( δ ) D E g R 2 µ + + = 2 x x x t ( δ) ( ( δ) E ) ( δ) D E g R 2 µ + + = 2 x x x t

17 17 Combiig the two diffusio equatios yields 2 ( δ) ( δ) ( µ D + µ D) + ( µ )( ) 2 µ E x x ( δ) + ( µ + µ )( g R) = ( µ + µ ) t The ambiolar trasort equatio is derived as 2 ( δ ) ( δ ) ( δ ) D' + µ ' E + g R = 2 x x t where the ambiolar diffusio coefficiet µ D + µ D D' = µ + µ ad the ambiolar mobility is µ µ ( ) µ ' = µ + µ

18 The Eistei relatio relates the mobility ad diffusio coefficiet µ µ = = e D D kt The ambiolar diffusio coefficiet may be writte i the form D' = D D ( + ) D + D Sice both ad cotai the excess-carrier cocetratio, which are deedet o time ad sace, the coefficiet i the amhiolar trasort equatio are ot costats. ( δ) ( δ) ( δ) D' E g R 2 + µ ' + = 2 x x t The ambiolar trasort equatio is a oliear differetial equatio. 18

19 Limits of Extrisic Doig ad Low Ijectio D' = D D [( + δ) + ( + δ)] 0 0 D ( + δ) + D ( + δ) 0 0 µ ' = µ µ ( ) µ + µ I a -tye semicoductor with low-level ijectio 0 >> 0 δ << 0 D' = D µ ' = µ I a -tye semicoductor with low-level ijectio << δ << D' = D µ ' = µ ad The ambiloar arameters reduce to a miority carrier value, which are costats. 19 The equivalet ambiolar article is egatively charged.

20 Cosider the geeratio ad recombiatio terms i the ambiolar trasort equatio. For electros G For holes, = R 0 0 g R = g R = ( G + g ) ( R + R ') ' 0 0 δ g R = g' R' = g' τ g R = g R = G + g R + R ' ( 0 ) ( 0 ') 20 G = R 0 0 δ g R = g' R' = g' τ The geeratio rate for excess electros must equal the geeratio rate for excess holes. g ' = g ' = g' The miority carrier lifetime is essetially a costat for low ijectio.

21 The ambiolar trasort equatio ca be writte i terms of miority carrier arameters. For a -tye semicoductor uder low ijectio, ( δ) ( δ) δ ( δ) D E g x x t 2 + µ ' 2 + = τ 0 0 For a -tye semicoductor uder low ijectio, ( δ ) ( δ ) δ ( δ ) D E g x x t 2 + µ ' 2 + = τ 0 The coditio of charge eutrality: δ = δ The behavior of excess majority carriers is determied by the miority carrier arameters. 21

22 22 Table Commo ambiolar trasort equatio simlificatio

23 Examle1 Cosider a ifiitely large, homogeeous -tye semicoductor with zero alied electric field. Assume that at time t= 0, a uiform cocetratio of excess carriers exists i the crystal, but assume that g' = 0 for t> 0. If we assume that the cocetratio of excess carriers is much smaller tha the thermalequilibrium electro cocetratio, the the low ijectio coditio alies. Calculate the excess carrier cocetratio as a fuctio of time for t> 0. Solutio 23 For the -tye semicoductor, we eed to cosider the ambiolar trasort equatio for the miority carrier holes ( δ ) ( δ ) δ ( δ ) D E g x x t 2 + µ ' 2 + = τ 0 We are assumig uiform cocetratio of excess holes so that ( δ )/ ( δ )/ x = x =

24 For t>0, we are also assumig that g =0 So the biolar trasort equatio reduces to The solutio is d( δ ) dt δ = τ 0 t/ e τ δ ( t) = δ (0) where δ (0) is the uiform cocetratio of excess carriers that exists at time t = 0. From the charge-eutrality coditio, we have that So the excess electro cocetratio is give by t/ e τ δ ( t) = δ (0) 0 0 δ = δ 24

25 25 Examle2 Cosider a ifiitely large, homogeeous -tye semicoductor with a zero alied electric field. Assume that, for t < 0, the semicoductor is i thermal equilibrium ad that, for t>0, a uiform geeratio rate exists i the crystal. Calculate the excess carrier cocetratio as a fuctio of time assumig the coditio of low ijectio. Solutio The coditio of a uiform geeratio rate ad a homogeeous semicoductor agai imlies that The equatio, for this case, reduces to 2 2 ( δ )/ x = ( δ )/ x = 0 δ d( δ ) g' = τ dt The solutio to this differetial equatio is t/ δ t g τ e τ 0 ( ) = ' (1 ) 0 0

26 26 Examle3 Cosider a -tye semicoductor that is homogeeous ad ifiite i extet. Assume a zero alied electric field. For a oedimesioal crystal, assume that excess carriers are beig geerated at x=0 oly. The excess carriers beig geerated at x = 0will begi diffusig i both the +xad -xdirectios. Calculate the steady-state excess carrier cocetratio as a fuctio of x. Solutio Sice E=0, g =0 for x 0, ad for steady state ( δ) t = 0 2 d ( δ) δ D = 0 dx 2 τ0 Dividig by the diffusio coefficiet δ δ δ δ = = 0 dx D τ dx L 2 2 d ( ) d ( ) The arameter L, has the uit of legth ad is called the miority carrier electro diffusio legth.

27 The geeral solutio to the equatio is δ x Ae Be t/ L x/ L ( ) = + The miority carrier electro cocetratio will the decay toward zero at both x = + ad x = These boudary coditios mea that B = 0 for x > 0 ad A = 0 for x < 0. The solutio is δ x δ x = δ x 0 x/ L ( ) (0) e = δ x 0 + x/ L ( ) (0) e The steady-state excess electro cocetratio decays exoetially with distace away from the source at x = 0. 27

28 28 Fig Steady-state electro ad hole cocetratios for the case whe excess electros ad holes are geerated at x= 0.

29 29 Examle4 Assume that a fiite umber of electro-hole airs is geerated istataeously at time t = 0 adat x = 0. But assume g' = 0 for t > 0. Assume we have a -tye semicoductor with a costat alied electric field equal to E 0. which is alied i the +xdirectio. Calculate the excess carrier cocetratio as a fuctio of x ad t. Solutio The oe-dimesioal ambiolar trasort equatio is D ( ) ( ) δ ( ) x x t 2 + µ 2 E = τ 0 The solutio to this artial differetial equatio is of the form t/ x t e τ δ ( x, t) = '(, ) usig Lalace trasform techiques 0 ( x µ E t) '(, ) = ex[ ] 1/2 (4 πdt) 4Dt x t

30 The total solutio is ( x E t) t/ τ 0 2 e µ 0 (, ) = ex[ ] 1/2 (4 πdt) 4Dt δ x t 30 Excess-hole cocetratio versus distace at various times for zero alied electric field Excess-hole cocetratio versus distace at various times for a costat alied electric field.

31 Dielectric Relaxatio Time Costat Fig The ijectio of a cocetratio of holes ito a small regio at the surface of a -tye semicoductor. Iitially,a cocetratio of excess holes is ot balaced by a cocetratio of excess electros. How is charge eutrality achieved ad how fast? 31 Poisso's equatio is E = ρ ε The curret equatio, Ohm's law, is The cotiuity equatio, eglectig the effects of geeratio ad recombiatio, is J = ρ t ρ is the et charge desity ad the iitial value is e ( δ ) J = σe

32 Takig the divergece of Ohm's law ad usig Poisso's equatio, J = σ E = σρ ε Substitutig it ito the cotiuity equatio σρ ρ d ρ = = ε t dt d ρ dt σ + ρ = ε 0 Its solutio is where ε τd = σ ( t/ τ ) ρ( t) = ρ(0) e called the dielectric relaxatio time costat. It meas the time to charge eutrality. 0 32

33 Examle Assume a -tye semicoductor with a door imurity cocetratio of N d = cm -3. Calculate the dialectic relaxatio time costat. Solutio The coductivity is foud as σ e µ Nd = ( )(1200)(10 ) = 1.92( Ω cm) The ermittivity of silico is ε ε ε = r 0 = (11.7)( ) F/ cm The dielectric relaxatio time costat is the τ d 14 ε (11.7)( ) = = = σ s 33

34 QUASI-FERMI ENERGY LEVELS The thermal-equilibrium electro ad hole cocetratios are fuctios of the Femi eergy level. EF EFi EFi EF 0 = iex( ) 0 = iex( ) kt kt If excess carriers are created i a semicoductor, we are o loger i thermal equilibrium ad the Fermi eergy is strictly o loger defied. However, a quasi-fermi level ca be defied for oequilibrium. EF EFi EFi EF 0 + δ = iex( ) 0 + δ = iex( ) kt kt 34

35 Examle Cosider a -tye semicoductor at T= 300 K with carrier cocetratios of = 10 cm, = 10 cm, ad = 10 cm 0 i 0 I oequilibrium, assume that the excess carrier cocetratios 13 3 are δ = δ = 10 cm Calculate the quasi-fermi eergy levels. Solutio The Fermi level for thermal equilibrium is The quasi-fermi level for electros i oequilibrium is The quasi-fermi level for holes i oequilibrium is 0 EF EFi = kti( ) = eV 1 0 EF EFi = kti( ) = eV 1 +δ +δ 0 EFi EF = kti( ) = 0.179eV 1 35

36 Thermal-equilibrium eergy-bad diagram Quasi-Fermi levels for electros ad holes Note: EF > E F 36

37 Summary The excess carrier geeratio rate ad recombiatio rate. Ambiolar trasort: Excess electros ad holes do ot move ideedetly of each other, but move together. The ambiolar trasort equatio: the excess electros ad holes diffuse ad drift together with the characteristics of the miority carrier uder low ijectio ad extrisic doig. Excess carrier behavior is a fuctio of time ad a fuctio of sace. The quasi-fermi level for electros ad the quasi-fermi level for holes were defied to characterize the total electro ad hole cocetratios i a semicoductor i oequilibrium. 37