Semicoductor evices Prof. Rb Robert tat A. Taylor The aim of the course is to give a itroductio to semicoductor device physics. The syllabus for the course is: Simple treatmet of p- juctio, p- ad p-i- structures as photodetectors, light-emittig diodes ad lasers (excludig optical gai ad cavity properties). Semicoductor heterojuctios, quatum wells ad aostructures. Low dimesioal semicoductor devices, e.g. quatum well laser. (No-examiable) trasistors ad their uses.
Semicoductor devices * p juctio: (i) at equilibrium, (ii) uder bias * photodiodes * solar cells (photovoltaics) * light emittig diodes * semicoductor lasers * bipolar juctio trasistors * field effect trasistors * heterojuctios * quatum wells ad dots, quatum well lasers * sigle electro trasistors
Carrier drift i applied field Electro drift velocity: qτ v = E = μe m * where τ is the mea time betwee scatterig evets, ad μ is the electro mobility. There is a similar relatio for holes. The mobility depeds o field at high field due to ielastic scatterig processes, ad the drift velocity saturates; the maximum value of v is ~10 5 ms -1 i Si. The et curret desity is: ( μ p μ ) J = J + J = q + E p 0 0 p
Carrier diffusio Whe there is a spatial variatio i the electro desity, there will be a diffusio curret. I oe dimesio: J = d q dx where is the electro diffusivity. ad μ are related via the Eistei relatio: kt B ktτ B = μ = * q m The hole mobility ad diffusivity are usually much smaller tha the electro values because of the hole s heavier mass.
esity depedece of,μ 1E14 1E16 1E18 1E0 ENSITY (cm -3 )
Carrier recombiatio The recombiatio rate R is proportioal to the umber of electros ad the umber of holes : R = β p I thermal equilibrium the recombiatio rate is equal to the geeratio rate: Gth = Rth = β0p0 If there is a excess of carriers of a particular type, e.g. caused by illumiatio of a doped semicoductor, the excess carriers will recombie. If Δp holes are ijected ito -type material: dp β t 0 = βp 0 pt ( ) = p0 +Δpe dt (β 0 ) -1 is the miority carrier lifetime τ p.
Cotiuity The cotiuity equatios for electros ad holes are: 1 J p 1 J p = + G R = + G R t q x t q x p p Uder steady state coditios: p p p p0 = 0 = p t x τ p Usig the boudary coditio p(x ) = p 0 yields: x L p p x p p(0) p e ( ) = + [ ] 0 0 where L p = ( p τ p ) is the diffusio legth, ad (p(0)-p 0 ) is the excess hole populatio at x = 0.
epletio Regio or Space Charge Regio p- juctio at equilibrium p p p - E
p- juctio at equilibrium The electrostatics of the juctio at equilibrium are described by the Poisso equatio: dv de ρ q dx dx εε 0 r εε 0 r ( N N p ) = = = + where we have assumed that all doors ad acceptors are ioized. Far away from the juctio we have: dv Assume the juctio is abrupt. I the depletio regio the free carriers are totally depleted, so that : ( N N p ) 0 0 A dx = + = dv dx dv dx qn A = da x < 0 εε 0 0 r qn = 0 < x d εε r A
The electric field is the: The potetial differece across the juctio is: d 0 qn A( x + da) qn( x d) VC = E( x)dx = dx dx εε εε d A 0 epletio layer qn A( x + da) Ex ( ) = da x< 0 εε 0 r qnx Ex ( ) = -EM 0 < x d εε 0 r qnd qn AdA where EM = = εε εε 0 r 0 0 q 1 = N d + N d = E W E εε ad W is the width of the juctio: d A ( A A ) M (i.e. the area uder ) r W = d + d = A 0 r 0 0 r d εε ( N + N ) V r A C qn N A r
cotact potetial V C ca be obtaied from the coditio that the drift curret balaces the diffusio curret: dv d ( x) xq ( ) μ + q = 0 dx dx Itegrate this equatio from -d A to d to obtai: V = V V = C p kt B q l( ) p where is the electro desity i the -regio, ad p is the electro desity i the p-regio. N.B. = / N, where i is the itrisic carrier desity. p i A
p- juctio uder bias
voltage-curret relatio for a p- diode The ideal diode is assumed to operate uder the followig coditios: (i) the depletio layer is abrupt, ad there is charge eutrality outside of the layer, (ii) the charge desities at the boudaries are give by the electrostatic potetial, (iii) the miority carrier ijectio is weak, much less tha the majority desities, (iv) there are o geeratio or recombiatio currets i the depletio layer.
p- juctio uder bias The relatio for V C ca be re-writte: = e ad p = p e qvc / kbt qvc / kbt p p We expect these relatios to hold for o-equilibrium carrier desities whe a bias V is applied: ˆ = ˆ e ad pˆ = pˆ e qv ( C V)/ kt B qv ( C V)/ kt B p p Sice ˆ we the have : qv / kbt ˆ = ( e 1) at x = d p p p A qv / kbt pˆ p = p( e 1) at x = d I the eutral - ad p-regios there is o electric field: d pˆ ˆ p p = pˆ ( p = p e dx pτ p 1) e dˆ ˆ = ˆ ( p p = p e dx τ 1) e qv / k T ( x d )/ L B p p qv / k T ( x+ d )/ L B A A
curret-voltage characteristics The total curret flowig is: J = J ( d ) + J ( d ) p A dpˆ ˆ dp = qp q dx + d dx qv / kbt = JS( e 1), q p q where JS = + L L p p p The curret icreases expoetially uder forward bias, but saturates at J S for egative bias. e d p = qi + LpN LeNA A Juctio breakdow Forward Si Reverse Si Theory forward (Ge) Theory reverse (Ge)
curret-voltage characteristics J/J s 6 ( / e qv k 1) B T J = J s 4 0-5 -4-3 - -1 0 1 3 4 5 - qv/kt
Web resources http://jas.eg.buffalo.edu/ http://ece-www.colorado.edu/~bart/book/book/title.htm http://cx.org/cotet/m1004/latest/ http://people.deas.harvard.edu/~joes/ap16/images/p_juctio/p_juctio.html http://www.aohub.org/simulatio_tools/pjuctio_tool_iformatio http://www.mtmi.vu.lt/pfk/fukc_dariiai/diod/idex.html