vs dagram p m m he parameter s called the crystal mometum ad s a parameter that results from applyg Schrödger wave equato to a sgle-crystal lattce. lectros travelg dfferet drectos ecouter dfferet potetal patters ad therefore dfferet -space boudares. he vs dagrams are geeral a fucto of the -space drecto a crystal. urso propedéutco de lectróca IAO 010 111
ffectve mass of carrers electros d d m m d / d he effectve mass of a electro a bad wth a gve (, ) relatoshp ( ) m e Whe the coducto bad edge occurs at = 0, we ca represet the bad-structure as a smple parabola, arrow parabola small effectve mass 1 m e d d 1 where: = mmum of the coducto bad eergy m e = effectve mass of a electro, GaAs m e = 0.063m 0 Ge m e = 0.55m 0 S m e = 1.09m 0 urso propedéutco de lectróca IAO 010 11
ffectve mass of carrers holes Smlarly, the eergy mometum relato for the valece bad ca be wrtte V m h where: V = maxmum of the valece bad eergy m h = effectve mass of a hole. here are actually two bads ear the top of the valece bad of dfferet wdths, leadg to heavy holes ad lght holes. GaAs m hh = 0.45m 0 m lh = 0.08m 0 S m hh = 0.49m 0 m lh = 0.16m 0 urso propedéutco de lectróca IAO 008 Dr. Joel Mola & Dr. Pedro Rosales 113
Itrsc materal A perfect semcoductor crystal wth o mpurtes or lattce defects s called a trsc semcoductor. I such materal there are o charge carres at = 0 K. At hgher temperatures HPs are geerated as valece bad electros are thermally excted across the badgap to the coducto bad. If a steady state carrer cocetrato s mataed, there must be recombato of HPs at the same rate at whch they are geerated r = g urso propedéutco de lectróca IAO 010 114
xtrsc materal I addto to the trsc carrers geerated thermally Whe a group V or III atom (As, ) s substtuted to the S lattce a electro s doated or accepted ad the semcoductor becomes -type or p-type respectvely. he crystal s extrsc whe the dopg s such that: ( 0, p 0 ) [ D ] I a extrsc semcoductor at ay temperature the carrers cocetrato have two cotrbutos: 1. hermal. Dopg [ D or A ] or a -type semcoductor: = D ad p = / D or a p-type semcoductor: p = A ad = / A [ A ] urso propedéutco de lectróca IAO 010 115
At 0K the extra electros assocated wth the door atoms are fxed to the door stes at a eergy level d. As the temperature creases there s eough thermal eergy to oze the door atoms.e. for a electro to mae the trasto to the coducto bad whch s oly a eergy jump of d where d << g. o create holes the valece bad a p-type semcoductor, electros eed oly a eergy of a to reach the acceptor level where a << g. urso propedéutco de lectróca IAO 010 116
Iozato level of dopats Approxmate eergy requred to excte the 5 th electro of a door atom to the coducto bad: D 0 S m m e 0 H hydroge eergy levels ohr Model 5 mev for S 7 mev for GaAs 50 mev for S, GaAs V t = /q at 300 K = 6 mev door levels acceptor levels I realty, dfferet dopats have dfferet ozato levels ad deep levels ( > 3 ), whch ca be mportat, but ths smple model gves the correct order of magtude. urso propedéutco de lectróca IAO 010 117
Iozato level of dopats D 0 S m m e 0 H urso propedéutco de lectróca IAO 010 118
lectros ad holes Quatum Wells A quatum well laser (QWL) s a laser dode whch the actve rego of the devce s so arrow that quatum cofemet occurs. he wavelegth of the lght emtted by a QWL s determed by the wdth of the actve rego rather tha just the badgap of the materal from whch t s costructed. hs meas that much shorter wavelegths ca be obtaed from QLW tha from covetoal laser dodes usg a partcular semcoductor materal. he effcecy of a QLW s also greater tha a covetoal laser dode due to the stepwse form of ts desty of states fucto. urso propedéutco de lectróca IAO 010 119
Desty of states fucto o obta the carrer desty per ut volume we must frst calculate the umber of allowed states (cludg sp) per eergy rage per ut volume. or electros the coducto bad where the - relato s of the form, m e Smlarly, for holes the valece bad where the - relato s of the form, V m h he desty of states s gve by: m h 3 1 e 4 he desty of states s gve by: m h 3 1 h 4 Accoutg for the cotrbuto from both lght ad heavy holes m 3 h 3 3 mlh mhh See appedx IV of Streetma urso propedéutco de lectróca IAO 010 10
erm-drac dstrbuto fucto he probablty that a electro occupes a electroc state wth eergy s gve by the erm-drac dstrbuto fucto: he dstrbuto of electros over a rage of allowed eergy levels at thermal equlbrum. 1 ( ) / 1 or = the ()=0.5 he erm eergy s the eergy for whch the probablty of occupato by a electro s exactly ½ urso propedéutco de lectróca IAO 010 11
he erm dstrbuto fucto s smplfed for a electro the coducto bad sce, 3 e ad for a hole the valece bad sce, 3 1 e or S at 300 K: =p 10 10 cm -3 Desty of avalable states at v ad c: 10 19 cm -3 ecause of the relatvely large desty of states each bad, small chages f() ca result sgfcat chages carrer cocetrato. urso propedéutco de lectróca IAO 008 Dr. Joel Mola & Dr. Pedro Rosales 1
he dstrbuto of electros the coducto bad s gve by the desty of allowed quatum states tmes the probablty that a state s occuped by a electro. he cocetrato of the electros the coducto bad s: f ( ) ( ) d c he cocetrato of the holes the valece bad s: c p [1 f ( )] ( ) d urso propedéutco de lectróca IAO 010 13
lectro cocetrato he electro desty the coducto bad s gve by, top top 0 d 0 d tag the bottom of the coducto bad to be =0 ag the prevous smplfed resso for (), 4 m h e 3 0 1 d urso propedéutco de lectróca IAO 010 14
dx e x h m x e 0 1 3 3 4 Let x = / h m e 3 he effectve desty of states the coducto bad, 15 urso propedéutco de lectróca IAO 010 lectro cocetrato
lectro cocetrato ag the bottom of the coducto bad as rather tha =0, Where: c 3/ m e h or S (300 K) =.8 x 10 19 cm -3 or GaAs (300 K) = 4.7 x 10 17 cm -3 urso propedéutco de lectróca IAO 010 16
Hole cocetrato Smlarly for holes the valece bad, p V V Where: v 3/ m h h or S (300 K) V = 1.04 x 10 19 cm -3 or GaAs (300 K) V = 7 x 10 18 cm -3 urso propedéutco de lectróca IAO 010 17
Law of Mass Acto p V g hs resso s depedet of ad s vald for extrsc (doped) semcoductors too. V g or S (300 K) = 9.65 x 10 9 cm -3 or GaAs (300 K) =.5 x 10 6 cm -3 urso propedéutco de lectróca IAO 010 18
p he product p s the, p p So the law of mass acto holds also for extrsc semcoductors. 19 urso propedéutco de lectróca IAO 010 Law of Mass Acto. Itrsc Semcoductors
alculato of extrsc erm level D D l At 300 K there s usually eough thermal eergy to completely oze the dopat atoms, so for -type semcoductor = D (door cocetrato) So as the cocetrato of door atoms creases the erm level moves closer to the bottom of the coducto bad 130 urso propedéutco de lectróca IAO 010
So as the cocetrato of acceptor atoms creases the erm level moves closer to the top of the valece bad A V V l V V A p V V Smlarly for p-type semcoductors, p = A (acceptor cocetrato) 131 urso propedéutco de lectróca IAO 010 alculato of extrsc erm level
It s ofte useful to ress the carrer desty terms of the trsc carrer cocetrato ad the trsc erm level Smlarly for holes, p or electros, 13 urso propedéutco de lectróca IAO 010 alculato of extrsc erm level
Itrsc carrer destes as a fucto of temperature Itrsc carrer destes S, Ge ad GaAs as a fucto of the recprocal of temperature. V g g (Ge) = 0.66 ev g (S) = 1.1 ev g (GaAs) = 1.4 ev or S (300 K) = 9.65 x 10 9 cm -3 or GaAs (300 K) =.5 x 10 6 cm -3 urso propedéutco de lectróca IAO 010 133
ergy adgap (ev) ergy gap g as a fucto of temperature he temperature depedece of the eergy badgap, g, has bee ermetally determed yeldg the followg resso for g as a fucto of the temperature, : 1.6 1.4 g (GaAs) = 1.4 ev where g (0), a ad b are the fttg parameters. hese fttg parameters are lsted for germaum, slco ad gallum arsede able. 1. 1 0.8 0.6 g (S) = 1.1 ev g (Ge) = 0.66 ev 0.4 0. 0 emperature (K) 0 00 400 600 800 1000 100 urso propedéutco de lectróca IAO 010 134
lectro desty 0 as a fucto of temperature At low temperatures the thermal eergy s suffcet to oze all door atoms so < D At hgher temperatures the thermal eergy s suffcet to oze all door atoms so = D At some temperature the trsc carrer desty becomes comparable to the door cocetrato ad beyod ths pot the semcoductor becomes trsc. urso propedéutco de lectróca IAO 010 135