Sign conventions. (a) (b) E. H. Putley The Hall effect and related phenomena, Butterworths, London, 1960 Trans.Elec-83

Transcription

1 Sin convntions a b. H. Putly h Hall ct and rlatd phnomna, uttrworths, ondon, 96 rans.lc-8

2 h ONSAGR rlations ars Onsar, Phys. Rv and Phys. Rv Applid th nd principl o thrmodynamics to th irrvrsibl ntropy ddt> Dducd that th nralid conductivity tnsor is anti-symmtric q j y x y x y yy xy y yy xy x yx xx x yx xx y x y x y yy xy y yy xy x yx xx x yx xx y x y x y yy yx y yy yx x xy xx x xy xx y x y x y yy yx y yy yx x xy xx x xy xx rans.lc-84

3 h ONSAGR rlations ars Onsar, Phys. Rv and Phys. Rv h nralid conductivity tnsor is anti-symmtric Consquncs:. h conductivitis ar vn unctions o mantic ild. h Pltir and Sbc coicints ar rlatd Klvin rlation Π xx xx xx. h Nrnst and ttinshausn coicint ar rlatd ridman rlations xx xx xx xx N xx rans.lc-85

4 Solvin Onsar or alvanomantic and thrmomantic coicints j q y x y yy yx x xy xx α J J q,, α rans.lc-86

5 Common misconcptions: what Onsar dos NO imply. h Hall coicint is NO an odd unction o mantic ild xy ; xy xy. h Pltir and Sbc coicints ar NO vn unctions o xx xx yx Umhr ct Michnaud t al., Solid Stat Commun rans.lc-87

6 Crystal Symmtry => symmtry o transport proprtis Crystals ar anisotropic => physical proprtis ar too nsor notation alon orthonormal principal axs x,y,: Crystalloraphic axs ar not ncssarily orthonormal!! lctrical and thrmal conductivity : xx yx x Crystal class j xy yy y xx yy x y Cubic xx = yy = xy -=- xy, Rrnc: symmtris or all transport proprtis: Y. C. Ao & G. A. Saundrs: J. Phys. C: Solid Stat Phys., vol 8, pp ,975 and vol, 8, pp 96-97, 975. rional traonal Hxaonal riclinic Monoclinic Othrorhombic xx = yy xx yy rans.lc-88

7 xampls o symmtry in alvanomantic proprtis Aö and Saundrs, J. Phys. C. Solid Stat Phys rans.lc-89

8 xampls o symmtry in alvanomantic proprtis Aö and Saundrs, J. Phys. C. Solid Stat Phys rans.lc-9

9 xampls o symmtry in thrmomantic proprtis Aö and Saundrs, J. Phys. C. Solid Stat Phys rans.lc-9

10 xampls o symmtry in thrmomantic proprtis Aö and Saundrs, J. Phys. C. Solid Stat Phys rans.lc-9

11 Solvin oltmann q. in a mantic ild Ω.v v q Ω.v v j H v Ω H v A A v d d m m v v q v v j d d Rmmbr th quations in th absnc o mantic ild Not riorous at all rans.lc-9

12 Apply to transport: turn th cran d m m d m n n n n H K d m d K n n n vv Rmmbr in th absnc o mantic ild y x j j q q y x yy yx yy yx xy xx xy xx yy yx yy yx xy xx xy xx y x y x Assum only in-xy-plan transport in th prsnc o a ild alon rans.lc-94

13 A-coicints. H. Putly h Hall ct and rlatd phnomna, uttrworths, ondon, 96 rans.lc-95

14 Galvanomantic and hrmomantic cts rans.lc-96

15 Cas or most thrmolctric smiconductors t l t Many hav rmi suracs that ar llopsoids o rvolution Many ar cubic Anisotropy actor: A A K t t l l m m Non-parabolic disprsion rlations And i you hav a simpl band, just st = ' ' d d m m m D m t l DOS DOS Numbr o llispoids rans.lc-97

16 hrmomantic alvanomantic quations d d d K K K n R m n d Hall DOS m n m n Din th transport intrals =, =, = Carrir concntration Hall coicint rans.lc-98

17 hrmomantic alvanomantic quations DIRCION DOS d d d d m n m n Conductivity Mobility hrmopowr Carrir concntration rans.lc-99

18 hrmomantic alvanomantic quations d d d d d d d d N Isothrmal Nrnst Coicint ornt ratio rans.lc-

19 MHOD o th OUR COICINS Unnown paramtrs. Dnsity o carrirs n. Mobility o carrirs. ctiv mass m * DOS n + m * DOS => 4. nry dpndnc o scattrin, q = scattrin paramtr V=>S V =>N Sbc Nrnst ach. Rsistivity. Hall coicint R H => n + R H =>. Sbc coicint S =>, 4. ransvrs Nrnst- ttinshausn N N => n + => m * DOS V => V =>R H Rsistivity Hall I, j rans.lc-

20 Whn you hav mor than poct to th rmi surac j q POCKS y x Cas or isotropic pocts, and : Conductivity Hall Sbc Nrnst hrm cond rans.lc-

21 7. Gnraliation: No band structur, only a dnsity o stats vs nry SiO Cristobalit Amorphous quart N.. Mott and. A. Davis, lctronic procsss in noncrystallin matrials, nd d., Clarndon, Oxord 979 rans.lc-

22 h Io-Rl rul Dirnc btwn insulator and mtal? Wav oriin o th lctrons can b propaatin wav or standin wav Dlocalid stats ocalid stats l 6 a - cm a = intratomic distanc Å Io-Rl Critrion A.. Io and A.R. Rl, Pro. Smicond. 4, rans.lc-4

23 yps o Dos s in disordrd smiconductors and tails Stats in nry ap Ascrib to an nry lvl a dnsity o stats It has an lctrical conductivity h total lctrical conductivity is dtrmind by thos lvls that ar within o th rmi nry. h total conductivity is thn: d h amount o hat carrid by an lctrical currnt, i.. th Pltir hat lux, is q i Apply Onsar s rlation S= h MO RAION x ID S d d rans.lc-5

24 Variations to th Mott Rlation. or mtals and havily-dopd smiconductors: d. or th vry dilut lctron as, r-lctron li disprsion d S S ln 5 S Const S. ln. or smiconductors with band conduction, vry dilut lctron concntrations rans.lc-6

25 Variations on th Mott ormula: th His ormula 4. or char transport than involvs th motion o chars usually localid on spciic atoms: Din x as th ratio o th numbr o lctrons that can mov to th numbr o sits S x ln x h His ormula xampl in a V-lass, thr ar V 4+ and V 5+ atoms. Conduction mchanism: th motion o char btwn V-atoms,. h thrmopowr is thn S 4 V V ln 5 rans.lc-7

26 Polarons: movmnt o char ollowd by lattic distorsion h concntration o lctrons is activatd, with activation nry a ut transport o char can involv a local dpnin o th wlls, by an additional nry W S xp a a W const. rans.lc-8

27 h xprimntal tst S xp a a W const. Plot both ON H SAM SCA rans.lc-9

28 8. Gnraliation: ransport in mantic matrials = = rans.lc-

29 Gnralid Onsar or spin dpndnt conductanc = N c char chmical potntial s spin accumulation Gc G G char conductanc G G conductanc P G G spin polariation Jc J J char currnt n, m nm Js J J spin currnt rans.lc- J G J G Jc G P c J s P s t nm

30 Add tmpratur radints to spin dpndnt and spin conductanc == N,, J c P S c Js G P PS s J Q S PS PG P G rans.lc-

31 Gnralid luxs Gnralid Onsar j q ju j n m j S s J is antisymmtric J Q IRRVRSI m s m m mm sm j, q s s ms ss V n Gnralid orcs xampls: Diaonal trms: = lctrical conductivity, = thrmal conductivity, mm = diusion O-diaonal trms:, rlatd to Pltir and Sbc coicints m, m = lctrophorsis m, m = thrmal diusion S, S = spin-sbc ct C. M. Jaworsi, J. Yan, S. Mac, D. D. Awschalom, J. P. Hrmans, and R. C. Myrs, Obsrvation o th Spin-Sbc ct in a rromantic Smiconductor, Natur Matrials, ; C. M. Jaworsi, t al., Giant Spin Sbc in non-mantic smiconductors, Natur 487, - rans.lc-