Carriers in a semiconductor diffuse in a carrier gradient by random thermal motion and scattering from the lattice and impurities.

Transcription

1 Diffusio of Carriers Wheever there is a cocetratio gradiet of mobile articles, they will diffuse from the regios of high cocetratio to the regios of low cocetratio, due to the radom motio. The diffusio reresets a imortat charge trasort rocess i semicoductors. Carriers i a semicoductor diffuse i a carrier gradiet by radom thermal motio ad scatterig from the lattice ad imurities. As the electros (or holes) move with a thermal velocity v th they udergo radom collisios. I the absece of a electric field they have a eual robability of movig i ay directio i betwee collisios. Average distace travelled betwee collisio is the mea free ath, l Average time betwee collisios is the mea free time, c l th c

2 Let s cosider a -tye semicoductor with a carrier cocetratio which varies i the x directio A arbitrary cocetratio gradiet i 1D. To calculate the diffusio curret, oe must determie the et flow of electros er uit time er uit area crossig the lae at x = 0. Due to their radom motio, electros at (1) have a eual robability of movig i either directio so withi the mea free time oe half of them will cross the lae at x = 0. The et umber of electros assig x 0 from left to right i oe mea free time: 1 1 ( la) ( 1 la ) divisio of (x) ito segmets of legth eual to a mea free ath for the electros exaded view of two of the segmets cetered at x 0. The rate of electro flow i the +x directio er uit area is give by: l ( x0) ( 1 ) t the former e. ca be writte i terms of the carrier gradiet d(x)/: ( x) l t d( x) The uatity (l /t) is called the electro diffusio coefficiet D [cm /s]. Curso roedéutico de Electróica INAOE 010 Dr. oel Molia & Dra. Claudia Reyes 160

3 Diffusio Curret Desity Rate of electro flow i the +x directio Rate of hole flow i the +x directio ( x) D d( x) ( x) D d( x) A curret desity ca flow i the absece of a electric field due to the diffusio of holes ad electros, therefore: diff diff diff Curret desity is simly the roduct of the charge ad article flux, therefore: diff D d D d Electros ad holes move together i a carrier gradiet but the resultig currets are i oosite directios because of the oosite charge of electros ad holes. Curso roedéutico de Electróica INAOE 010 Dr. oel Molia & Dra. Claudia Reyes 161

4 Diffusio ad Drift of Carriers If a electric field is reset i additio to the carrier gradiet, the curret desity will each have a drift comoet ad a diffusio comoet: drift diffusio x xfx D x xfx D The total curret desity is the sum of the cotributios due to electros ad holes: x x x Curso roedéutico de Electróica INAOE 010 Dr. oel Molia & Dra. Claudia Reyes 16

5 This relatio betwee articles flow ad their currets ca be better visualized: The total curret may be due rimarily to the flow of electros or holes, deedig o the relative cocetratios ad the relative magitudes ad directios of electric field ad carrier gradiets. Curso roedéutico de Electróica INAOE 008 Curso roedéutico de Electróica INAOE 010 A imortat result is that the miority carries ca cotribute sigificatly to the curret desity through diffusio. Sice the drift terms are roortioal to the carrier cocetratio, miority carrier seldom rovide much drift curret desity. I the other had, the diffusio curret desity is roortioal to the gradiet cocetratio. Dr. oel Molia & Dr. Pedro Rosales Dr. oel Molia & Dra. Claudia Reyes 163

6 I discussig the motio of carriers i a electric field, the ifluece of the field o the eergies of electros i the bad diagrams take great imortace. Sice electros drift i a directio oosite to the field, the otetial eergy for electros icrease i the directio of the field. From the defiitio of electric field: F x dv x We ca relate F(x) to the electro otetial eergy i the bad diagram by choosig some referece i the bad for electrostatic otetial. Choosig E i as a coveiet referece, we ca relate the electric field to this referece by: F x dv x d E ( ) i 1 de i Curso roedéutico de Electróica INAOE 008 Dr. oel Molia & Dr. Pedro Rosales 14

7 Eistei s Relatioshi At euilibrium, o et curret desity flows i a semicoductor. Thus ay fluctuatio which would begi a diffusio curret desity also sets u a electric field which redistributes carriers by drift. A examiatio of the reuiremets for euilibrium idicates that the diffusio coefficiet ad mobility must be related. F x x x d xf x D 0 D by usig: ( x ) E ex i i E kt F F( x) D 1 kt dei de F The euilibrium Fermi level does ot vary with x, ad the derivative of Ei is give by: F( x) 1 dei Therefore: D kt Eistei s relatio Curso roedéutico de Electróica INAOE 010 Dr. oel Molia & Dra. Claudia Reyes 165

8 Mobilities ad diffusivities i Si ad GaAs at 300 K as a fuctio of imurity cocetratio. ( D / K ). Curso roedéutico de Electróica INAOE 010 Dr. oel Molia & Dra. Claudia Reyes 166

9 Diffusio ad Recombiatio: The Cotiuity Euatio A simle statemet of coservatio of articles emerges Rate of article flow = Particle flow due to curret + Particle gai due to geeratio - Particle loss due to recombiatio Curso roedéutico de Electróica INAOE 010 Dr. oel Molia & Dra. Claudia Reyes 167

10 Diffusio ad Recombiatio: The Cotiuity Euatio For the discussio of excess carriers, we have thus far eglected the imortat effects of recombiatio. These effects must be icluded i a descritio of coductio rocesses sice recombiatio ca cause a variatio i the carrier distributio. The hole curret desity leavig the volume, (x+x), ca be larger or smaller tha the curret desity eterig, (x), deedig o the geeratio ad recombiatio of carriers takig lace withi the volume. Geeratio: the rocess whereby electros ad holes (carriers) are created. Recombiatio: the rocess whereby electros ad holes (carriers) are aihilated. Curso roedéutico de Electróica INAOE 010 Dr. oel Molia & Dra. Claudia Reyes 168

11 Diffusio ad Recombiatio: The Cotiuity Euatio The et icrease i hole cocetratio er uit time, /t, is the differece betwee the hole flux er uit volume eterig ad leavig, mius the recombiatio rate. Hole flow rate ito the slice at x is simly the curret at x divided by the charge of a hole: x A Hole flow rate out of the slice at x+ is simly the curret at x+ divided by the charge of a hole: x A The overall rate of chage i the umber of holes i the slice is: t A xa x A G R A G= hole geeratio rate. R= hole recombiatio rate. The secod term i the euatio ca be exaded ito a Taylor series to: x x x cotiuity euatio for holes cotiuity euatio for electros 1 t x G R 1 t x G R 147

12 Rate of Recombiatio Electros i Ec ca recombie with holes i Ev ad geerate a hoto. For a -tye semicoductor (>>), excess electros ijected by some meas (e.g. the absortio of light) will recombie with the majority carriers (holes) with a recombiatio rate give by: R Excess electro desity Recombiatio lifetime o electro desity euilibrium electro desity mea time the electro is free before recombiig with a hole. Back to the cotiuity euatio ad itroducig the recombiatio rate factor: 1 t x G R 1 t x G R 1 t x G o 1 t x G o Curso roedéutico de Electróica INAOE 010 Dr. oel Molia & Dra. Claudia Reyes 170

13 Where we have defied a imortat uatity called the diffusio legth, Thigs could start to get really comlicated whe we substitute the drift ad diffusio currets i our earlier exressios istead we will look at the secial case where the curret is carried oly by the diffusio rocess ad there is o geeratio. This is ofte the case whe cosiderig trasort i - juctio diodes ad biolar trasistors whe there are o otical excitatios. x D diff Diffusio ad Recombiatio: The Cotiuity Euatio o G x t 1 o x D t I the steady state the time derivative is zero so: o x D o o L D x D L D L L/L is the average distace a electro/hole diffuses before recombiig. 171 Curso roedéutico de Electróica INAOE 010 Dr. oel Molia & Dra. Claudia Reyes

14 Diffusio Legth Cosider a -tye semicoductor with steady state ijectio o oe side W >> L This is the case for examle i a log - diode where the carriers are ijected at the origi ad the excess desity decays exoetially to zero dee withi the bulk of the semicoductor. x 0 o x L e o Miority carrier desity decays e with a characteristic legth give by L W << L This is the case for examle i a biolar trasistor with a arrow base regio. I this case the carrier desity varies essetially liearly from oe boudary value to the other. x o 0 o sih W x L sih W L For a small W (x) decays liearly. Semicoductor Devices, /E by S. M. Sze 150 Coyright 00 oh Wiley & Sos. Ic. All rights reserved.

15 Diffusio Legth Cosider a -tye semicoductor with steady state ijectio o oe side t 0 D x o Boudary coditios are, x 0 0 x o Solutio of (x) is, x 0 o x L e o Miority carrier desity decays with a characteristic legth give by L Semicoductor Devices, /E by S. M. Sze Coyright 00 oh Wiley & Sos. Ic. All rights reserved. 173

16 Diffusio legth If all excess carriers are extracted at W (the thickess of the samle), Boudary coditios are, x 0 0 W o Solutio of (x) is, x o 0 o sih W x L sih W L For a small W (x) decays liearly Semicoductor Devices, /E by S. M. Sze Coyright 00 oh Wiley & Sos. Ic. All rights reserved. 174

17 Homework Read sectio The Hayes-Shockley Exerimet Read sectio Gradiets i the Quasi-Fermi Levels Curso roedéutico de Electróica INAOE 010 Dr. oel Molia & Dra. Claudia Reyes 175