Lecture 18 - Semiconductors - continued

Transcription

1 Lctur 18 - Smiconductors - continud Lctur 18: Smiconductors - continud (Kittl C. 8) + a - Donors and accptors Outlin Mor on concntrations of lctrons and ols in Smiconductors Control of conductivity by doping (impuritis) Mobility and conductivity Trmolctric ffcts Transport of carg and nrgy U, U, Carrirs in a magntic fild Cyclotron rsonanc Hall ffct (Rad Kittl C 8) Pysics 460 F 2006 Lct 18 1 Pysics 460 F 2006 Lct 18 2 Law of Mass Action (from last tim) Product n p = 4 (k T/ 2 π 2 ) 3 (m c m v ) 3/2 xp( -( c - v )/k T) is indpndnt of t Frmi nrgy vn toug n and p vary by ug amounts, t product np is constant! Wy? Tr is an quilibrium btwn lctrons and ols! Lik a cmical raction, t raction rat for an lctron to fill a ol is proportional to t product of tir dnsitis. If on crats mor lctrons by som procss, ty will tnd to fill mor of t ols laving fwr ols, tc. Control of carrirs by doping Impur crystals may av addd lctrons or ols tat cang t balanc from an intrinsic idal crystal. If an impurity atom adds an lctron, it is calld a donor If an impurity atom subtracts an lctron, it is calld a accptor (it adds a ol) T Frmi nrgy cangs (n and p cang) ut (Law of mass action ) t product n p = 4 (k T/ 2 π 2 ) 3 (m c m v ) 3/2 xp( -( c - v )/k T) dos not cang! vn toug n and p vary by ug amounts, t product np is constant! Pysics 460 F 2006 Lct 18 3 Pysics 460 F 2006 Lct 18 4 Wat dos it man to say an impurity atom adds or subtracts an lctron? Considr rplacing an atom wit on t tat as on mor lctron (and on mor proton),.g., P in Si, As in G, Zn rplacing As in GaAs,. Qustion: Is tat lctron bound to t impurity sit? Or is it fr to mov and count as an lctron carg carrir? T probability tat it scaps dpnds on t crystal and t impurity --- ut if it scaps from t impurity, tn it acts as an addd lctron indpndnt of t natur of t impurity Similar argumnt for ols Pysics 460 F 2006 Lct 18 5 Substitution Impuritis in Diamond or Zinc-blnd crystals Zinc-blnd structur crystal (.g., GaAs) Diamond (.g., Si) if pink and gry atoms ar t sam Impurity substituting for ost atom,.g., Donors: P in Si S on As sit in GaAs Accptors: in Si Zn on Ga sit in GaAs Pysics 460 F 2006 Lct

2 Lctur 18 - Smiconductors - continud inding of lctron to impurity Simplst approximation accurat in many cass - qualitativly corrct in otrs (Kittl p 210) lctron around impurity is xactly lik a ydrogn atom -- xcpt tat t lctron as ffctiv mass m* and t Coulomb intraction is rducd by t dilctric constant ε m m*; 2 2 /ε T binding is (s back insid covr of Kittl) binding = ( 4 m*/ 2 ε 2 2 ) = (1/ ε 2 )(m*/m) 13.6 V T radius is: a binding = (ε 2 / m* 2 ) = ε (m/m*) a or = ε (m/m*).053 nm and binding Pysics 460 F 2006 Lct 18 7 inding of lctron to impurity Typical valus in smiconductors m* ~ m; ε ~ 5-20 Tus binding nrgis ar binding ~ V ~ 5 K - 5,000 K Sizs a ~ nm + a - In many cass t binding can b vry wak and t siz muc gratr tan atomic sizs Hols ar similar (but oftn m* is largr) Pysics 460 F 2006 Lct 18 8 Trmal ionization of donors and accptors Suppos w av donors wit binding nrgy muc lss tan t band gap (t usual cas). T fraction of ionizd donors can b workd out simply if t dnsity of donor atoms N d is muc gratr tan t dnsity of accptors and intrinsic dnsity of ols and lctrons (otrwis it is mssy) Tn t dnsity of ionizd donors N d+ quals t dnsity n of lctrons tat scap, wic can b found by t sam approac as t dnsity of lctrons and ols for an intrinsic crystal. Trmal ionization of donors and accptors Assuming k T << binding t rsult is (Kittl p 213) n = 2(m c k T/ 2 π 2 ) 3/2 N d 1/2 xp( - binding /k T) Frmi nrgy and n N d + binding Pysics 460 F 2006 Lct 18 9 Pysics 460 F 2006 Lct Wn is a dopd smiconductor a mtal? If t dnsity of donors (or accptors) is larg tn ac impurity is not isolatd T pictur of an isolatd ydrogn-lik bound statdos not apply Wat appns if t stats ovrlap? a - T systm bcoms mtallic! Similar to Na mtal in t sns tat t lctrons ar dlocalizd and conduct lctricity vn at T=0 Tis is a mtal if t distanc btwn t impurity atoms is comparabl to or lss tan t radius a Tr ar also spcial cass s latr Pysics 460 F 2006 Lct Conductivity wit lctrons and ols ot lctrons and ols contribut to conductivity Currnt dnsity j = dnsity x carg x vlocity = n q v + p q v = - n v + p v Not: = carg of lctron >0 Pysics 460 F 2006 Lct

3 Lctur 18 - Smiconductors - continud Trmopowr and Pltir ffct ot lctrons and ols contribut to conductivity and conduct at T Pltir ffct is t gnration of a at currnt u du to an lctric currnt q in t absnc of a trmal gradint lctrons and ols tnd to cancl - can giv itr sign - on way to dtrmin wtr lctrons or ols dominat t transport! U, Trmopowr and Pltir ffct Quantitativ dfinition: Pltir cofficint is t ratio of nrgy to carg transportd for ac carrir Surprising? T nrgy for an lctron is c - µ + (3/2) K T; and for a ol is µ - v + (3/2) K T Π = ( c - µ + (3/2) K T) / q = - ( c - µ + (3/2) K T) / Π = (µ - v + (3/2) K T) / q = + (µ - v + (3/2) K T) / U, U, U, Pysics 460 F 2006 Lct Pysics 460 F 2006 Lct How a solid stat rfrigrator works T Pltir ffct is t gnration of a at currnt u du to an lctric currnt q in t absnc of a trmal gradint Wy smiconductors? caus Π is so larg du to t larg valu of t nrgy pr carrir ( c - µ + (3/2) K T) or (µ - v + (3/2) K T) Dmonstration Wat dtrmins U,total? dirction? U, U, Pysics 460 F 2006 Lct Trmopowr and Pltir ffct Rcall: ot lctrons and ols contribut to conductivity and conduct at T trmolctric ffct is t gnration of an lctrical voltag by a at currnt u in t absnc of an lctric currnt. ust as in Pltir ffct, lctrons and ols tnd to cancl - can giv itr sign U, -dt/dx U, Pysics 460 F 2006 Lct Trmopowr If tr is a trmal gradint but no lctrical currnt, tr must b an lctric fild to prvnt t currnt T logic is vry similar to t Hall ffct and lads to t xprssion for t lctric fild ndd to prvnt lctrical currnt Wat dtrmins? dirction? U, -dt/dx U, Pysics 460 F 2006 Lct Trmopowr Tis lads to trmopowr: gnration of powr from at flow (by allowing t currnt to flow troug a curcuit) Wir U, Motor, tc. -dt/dx q,total U, Pysics 460 F 2006 Lct

4 Lctur 18 - Smiconductors - continud Mobility Caractrizs t quality of a smiconductor for lctron and ol conduction sparatly Rcall: Currnt dnsity j = dnsity x carg x vlocity = n q v + p q v = - n v + p v Dfin mobility µ = spd pr unit fild = v/ = = (n µ + p µ ) Not: t symbols µ and µ dnot mobility (Do not confus wit t cmical potntial µ ) Pysics 460 F 2006 Lct xprimnts: How do w know ols ar positiv? How do w know tat lctrons act lik ty av ffctiv masss? xprimnts in magntic filds Hall ffct Cyclotron rsonanc Pysics 460 F 2006 Lct Hall ffct I From our analysis bfor Adding a prpndicular magntic fild causs t lctrons and ols to b pusd t sam dirction wit forc -- but sinc tir cargs ar opposit, t currnt in t y dirction tnds to cancl Hall ffct II In ordr to av no currnt in t y dirction, w must av lctric fild in t y dirction, i.., j y = (n µ + p µ ) y + (- n µ v + p µ v ) z = 0 Tus y = z (- n µ v + p µ v ) / (n µ + p µ ) x Pysics 460 F 2006 Lct z y R Hall = Hall / j = (1/) (- n µ 2 + p µ 2 ) / (n µ + p µ ) 2 (Kittl problm 3) Hall z y x Pysics 460 F 2006 Lct Cyclotron rsonanc Masurs ffctiv mass dirctly Subtl points THIS IS XTRA MATRIAL NOT RQUIRD FOR HOMWORK OR TH XAM Motion of carrir in Magntic fild Forc: q ( v x ) = dk /dt lctron movs on constant nrgy surfac, wit only cang in dirction of k Tus dk /dt = - v x / = - (/m*) k Isotropic bands (sam in all dirctions lik for fr lctrons): priod of rvolution in k spac is 2πk/(dk/dt) = 2π/ω c and ω c = q/m* dk/dt k Cyclotron Rsonanc Pysics 460 F 2006 Lct Pysics 460 F 2006 Lct

5 Lctur 18 - Smiconductors - continud Cyclotron Rsonanc xprimntal way to masur ffctiv masss Magntic fild dfins particular dirct in spac lctron rotat in plan prpndicular to wit a priod of rvolution ω c = q/m* Obsrvd xprimntally by t absorption of lctromagntic wavs at frquncy ω c Intrprtation: wav causs lctron buncs to mov in circl - rsonanc occurs wn lctrons ar wav ar in pas at frquncy ω c Pysics 460 F 2006 Lct k &M Wav dk/dt Ral ands in a Smiconductor - Si Frmi nrgy Minimum 3,4 2 1 X = (2,0,0) π/a ,4 L = (1,1,1) π/a Filld lowr bands if tr ar 8 lctrons pr cll k Pysics 460 F 2006 Lct Wat if minimum is not at k = 0? Multipl quivalnt minima Anisotropic mass Conduction and Multipl minima Conduction band of Si - 6 minima along (,0,0), (0,,0), (0, 0, ) dirctions Conduction and minima gap All bands k 2 Havy ol Valnc ands Ligt ol 0 k In G, 8 minima along dirctions wit = = Pysics 460 F 2006 Lct Pysics 460 F 2006 Lct Anisotropic Mass Considr only on minimum at k = (k min,0,0) Anisotropic mass: d 2 d 2 d 2 d 2 = d 2 d 2 Constant nrgy surfacs Around ac of t t minima, t surfacs of constant nrgy in k spac ar circls or llipss xampl of Si ((k min,,0)) [or ((k min, 0, ))] ((,0,0)) (k min,0,0) ky [or ] Pysics 460 F 2006 Lct Pysics 460 F 2006 Lct

6 Lctur 18 - Smiconductors - continud Cyclotron Rsonanc Dpndnc upon dirction of magntic fild sows t anisotropy of t mass xampl: In Si all along any cubic axis is t sam. In ac dirction tr ar two rsonanc frquncis ω c = q/m* corrsponding two diffrnt masss for motion prpndicular to Larg orbit, larg mass Small orbit, small mass Pysics 460 F 2006 Lct Summary for Today Control of conductivity by doping (impuritis) Donors and accptors Hydrognic quations for binding Important tat binding b wak for carrir to scap and b abl to mov Conductivity and Mobility Trmolctric ffcts Pltir ffct Trmopowr Sign of carrir important Carrirs in a magntic fild Hall ffct Cyclotron rsonanc (xtra not rquird) (Rad Kittl C 8) Pysics 460 F 2006 Lct Summary of Smiconductors I Typical bands - undrstanding from narly fr lctron pictur Optical proprtis - (dirct vs indirct gap) Motion of wav packts F = dk /dt Group vlocity 1 1 d 2 ffctiv mass m*: m* = 2 d 2 k m* tnds to b small if t gap is small Ngativ lctrons; positiv ols Law of mass action: np = constant Doping and concntrations of lctrons, ols Donors, accptors inding of carrir to impurity sit Pysics 460 F 2006 Lct Summary of Smiconductors II Trmolctric ffcts: Pltir; Trmopowr Sign of carrir important Carrirs in a magntic fild Hall ffct Cyclotron rsonanc (xtra not rquird) Sign of carrir important (Rad Kittl C 8) LATR: Inomgnous Smiconductors -.g., variations in dopin in spac, p-n junctions,. Pysics 460 F 2006 Lct Nxt tim Smiconductor dvics Cratd by inomognous matrial or doping Variation in concntrations of lctrons and ols by controlld doping profils p-n junctions - rctification- forward - rvrs bias Mtal-smiconductor junctions Scottky barrirs - rctification Solar Clls Ligt mitting diods ipolar transistor n-p-n p- n-p (Kittl C. 17, p xtra class nots) Pysics 460 F 2006 Lct