SUMMARY
Chapt 3 What a scnducts s usful? Macscpc ppts - Hw d scnducts hav such a wd ang f cnductvty? vn f th sa scnduct cpund,. g. S? - Th natu f - Hw d s hav pstv Hall ffcts (p-typ), and s hav ngatv Hall ffcts (n-typ)? - Hw ds th cnductvty vay s uch vs. tpatu, th typ and cncntatn lvl f puts? - Why ds th ca cncntatn xhbt xp(-/t) bhav? - Hw d chag cas hav such a wd ang f blty? Whch vas vs. tpatu, ca cncntatn, ca typs, and scnduct cpstn? Mcscpc thy lctnc bndng and ts ngy stuctu n SC cystal - Th ffcts f dpant - Th quantu chancal bhav f lctns - Th statstcal chancs bhav f th cas
asc lctcal tanspt cncpts S: Cnductvty: s; sstvty: 1/s Macscpcally: th sstanc f a pc f atal s lnaly pptnal t ts lngth and nvsly pptnal t ts css sctn: s a spcfc sstanc, calld sstvty, can b dfnd: l R A RA l c unt : c Mcscpcally: Cunt dnsty J ca/unt vlu x chag x avag ca vlcty: J qnv R c Stady stat avag ca vlcty s call dft vlcty. v Dft vlcty wh s dfnd as ca blty, ts unt s : (c/s)/(v/c) c /V sc V V l RA 1 I AJ Aqn Aqn ; R ; s qn l I Aqn l qn Why ds cnductvty vay s uch f scnduct? Ths s hw (#1) l A
Classcally, a fly vng chag has a vlcty: Fc v t ass q whch ncass wth t ndfntly. ut t can't, bcaus f scattng as shwn. Th s a tnal "tavagd" vlcty. If yu asu th nt tavld dstanc f a lng t, th patcl wll hav an avag vlcty calld dft vlcty. W can dfn a laxatn t t as th t f whch: v d q t S, blty can b xpssd as: q q v d t t What s "dft vlcty"? Hw ds blty vay vs. tpatu, cystal qualty (puts & dfcts), ca typs, and scnduct cpstns? t
Hall ffcts lcttv (Lntz) fc: F qv : Fy qvx. z (In atal H) At qulbu, th ust b a fc that cunts ths Lntz fc: an lctc fld fd by th gadnt f th ca dstbutn: q qv v F lctds wdth w, th Hall vltag s: V H w y. Dfn Hall cffcnt R H : y RH JxHz RH y / JxH Masud Hall vltag Sgn ndcats ca typs (n p) Masud Hall vltag Hall fld(hall) dft vlcty: J s s x V v H x x vx RHs Dft vlctyca cncntatn: s Jx wh zjx J xwh J z x qnvx n J x / qvx Dft vlctyhall blty: qv VH w y v x H z v x y V H wh z x z y H x z z
Ky cncpts
Hw t slv th scnduct puzzl
and fatn Wav functns A ~ ~ A ~ A C ~ a A ac A b b b; ub; ~ A b: band (cnd., val., ) C : cdnats x, y, z u b; : lch wavfunctn : lattc ntu
and fatn (cnt.) ach stat n an at ylds an nfnt nub f stats n an nfnt cystal cndnsd att A dsct lvl f ngy usually (but nt ncssaly) bcs an nfntly dns ngy band. (f th ats a nfntly apat, all th stats a stll dgnat nt n ngy lvl) A y fsh: ach quantu stat has a dfnt ngy lvl. ut any stats can hav th sa ngy lvl, a stuatn calld dgnacy. Th stats a sad t b dgnat-usually a cnsqunc f s gty. xapl: atc angula ntu l has l+1 dgnat stats. S, th nub f ngy lvls s always sall qual t th nub f stats. A pdc cystal has th tanslatnal syty: t ans that yu can shft th cystal by an ntgal nub f ts lattc dnsn and yu can nt tll th dffnc. Th lattc ntu s th quantu nub f ths syty. All valus f f a 3-D zn calld llun zn.
Cpasn f quantu dscptn lctn n an at lctn n a cystal lag cndnsd att (aclcul) Radal quantu nub (QN) and (cnductn, valnc, spn-bt, ) adal QN Angula ntu QN and angula ntu QN (s-l, p-l, d-l, ) Magntc QN Magntc QN Lattc ntu QN
ttc ntu & lctn tn L at th lch wavfunctn: Th pat b; ub; s a plan wav. What ds t d t th pdc lch functn? 0.75 1 0.5 0.5-10 -5-0.5 5 10-0.5 1 0.75 0.5 0.5-10 -5-0.5 5 10-0.5 0.5-10 -5 5 10-0.5-1 1 0.5-10 -5-0.5 5 10-1 Tp: 0; nd : 0.; 3 d : 0.5; btt: 1 (ab. unt) It dulats th "lcal" lch functn. Analgy: Chstas lght stng: ach lght bulb s a lch functn, dulatn s t tun bulbs n-ff n squnc. Th lght appas t v: ths s hw lctn vs n a cystal lattc.
Cpasn f ntu f a f patcl wth that n a cystal ; ; b b u p ˆ p ; ; ; ; ; ˆ b b b b b u u u u u p ; ; ˆ b u b u Wavfunctn: Mntu: (th nd t usually vanshs bcaus f paty ag syty) F patcl Patcl n cystal Cnclusn: th xpctd valu f ntu f a patcl wth lattc ntu vct s, whch s th sa as that f a f patcl wth th sa ntu vct. ut: t s nt a quantu stat f th ntu pat.
ngy and Lattc Mntu ach quantu stat n a cystal wth a lattc ntu has an ngy Ths ngy s a functn f, als calld as ngy - dspsn latn xapl f a f patcl: ˆ p z y x z y x z y x ˆ / In a al cystal, th ngy functn f has a cplcatd shap: - Th ngy functn shap ( stuctu) s unqu f ach band (cnductn, valnc, splt-ff, hgh bands, ) - Th nly way t nw f su s by asunts - Lag cputatn pgas hav bn usd t btan th ngy band stuctu
Ppts na and xta lctns absnc f lctns a usually cncntatd at ngy band xta ngy at band xta can b xtaplatd 1 b b j! If stpc: b 1! b j b 1 j j b / d / d s dfnd as th ffctv ass ffctv ass (M) appxatn s th st ptant and usful appxatn f all th sults f th band thy. It allws us t tat cas as f patcls. Many classcal tanspt cncpts a vald as lng as th ass s M. F th st f ths cus, vythng wll b dn wth ffctv ass appxatn. ffctv asss n vaus scnducts a asud. A hug aunt f w and publshd ltatu s cncnd wth asung ths valus.
An ptant M sults: hls pty p v J J nt all ( ) v; v; pty v lct v hl p hl v Th absnc f an lctn n th valnc band xtu (wth ngatv ffctv ass) bhavs l a pstv chag patcl wth valnc ffctv ass, dfnd as hl.
Hw t slv th scnduct puzzl
Hw d scnducts hav chag cas? Intnsc: thal xctatn f th valnc band t th cnductn band xtnsc: cas f dpants, puts, dfcts Dn (D): dnats xta lctns t th cnductn band; sngl ultpl; D + : nzd dn. Accpt (A): accpts valnc lctns, lav bhnd hls; sngl ultpl. A - : nzd accpt. Aphtc dpant: th D A, dpndng n st. Shallw dn/accpt: lctn/hl ngy lvl cls t cnductn band/valnc band dg. Dp lvls, dp taps: D A h ngy lvl na d gap. Lghtly/havly dpd: sult n nndgnat/dgnat ca ppulatn Majty/nty ca, abpla ppulatn
Hw t slv th scnduct puzzl
Statstcal chancs cncpts Dstbutn functns xapl 1: : Factn f ats n and +D, wth vlcty btwn v and v+dv f, v,t DD at t t: glbal dstbutn. v xapl : f s th pbablty f havng patcls ccupyng stats wth ngy : als calld ccupatn nub: latv dstbutn.
Hw t us ccupatn nub Su f Occ. Pb. x Nub f stats Ttal nub f patcls
Dstbutn f dns cntnuus ngy n D f d
F-Dac dstbutn functn f ; f / T 1 wh: s th ltzann s cnstant0.08617 V/Klvn, T s th abslut tpatu, and f s th ngy that spaats th flld and unflld stats at z T, calld F lvl 1 f In a gvn syst, f s dtnd by th quatn: n f ; D d f
Tnlgy and unts 1. lctn/hl dnsty: # f lctn/hl p unt vlu,. g. c 3. (Inzd) Dn/Accpt dnsty: # f D/A / unt vl. (nzd nt) 3. Scnduct unts f lngth and ngy Lngth Dn h adus a a ngy Dn Rydbg Ry Ry ngy s th lctn ffctv ass, s th dlctc cnstant Rlatn f ths n ts f atc unts a a 1 a Ry 1 R R a a a
Chag nutalty: n, p a lctn/hl dnsts at thal qulbu, D + /A - a nzd dn/accpt dnsts A p D n At thal qulbu: T v T c v f f c N p N n / / ; wh 3/ 3 4 Ry T a g N c C C h c v h v V N g g Ry T a g N 3/ 3/ 3 3/ 4 N c, N v fd t as ffctv dnsty f stats a NOT ally DOS, but nly quantts f appxatn cnvnnc; g c, g v a dngacy facts f cnductn and valnc band. T v c g N N p n / n p n p n Appxatn: ntnsc n, p at lw ntnsc dnsts Appxatn at lw dpng dnsts:
F lvl acss dssla atals Rat f lctn gng f 1 (lft) t (ght): Rat f lctn gng f (ght) t 1 (lft): f N f N1 1 1 f N f N 1 1 1 At qulbu, th tw ats ust b qual; canclng all f f1 ts nvlvng N: T 1 / T 1 1 / 1 f 1 f f 1 f