Muffin Tins, Green s Functions, and Nanoscale Transport [ ] Derek Stewart CNF Fall Workshop Cooking Lesson #1

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1 Muffn Tns, Geen s Funcons, and Nanoscale Tanspo G [ ] E H Σ 1 Deek Sewa CNF Fall Wokshop Cookng Lesson #1

2 Talk Ovevew A moe localzed appoach Ogns: Mulple Scaeng Theoy & KK Lnea Muffn Tn Obals Geen s funcons and Tanspo Geen s Funcon bascs G [ ] E H Σ 1 Tgh bndng models Tanspo

3 Bass expanson aound aoms In he pevous alk hs monng, you leaned abou a delocalzed bass se (plane waves) Muffn n appoaches (KK & LMTO) Sphecal poenals aound each aom k e Wavefuncon expanded n sphecal waves (s, p, d, f chaace) Poenal s zeo n space beween aoms Soluon of dffeen ses conneced ogehe (mulple scaeng, cancellaon of obal als) bondng aom elecon V() V() V()? V V V() V() V()

4 Mulple Scaeng Theoy (MST) Fom Pon Scaees o Solds Mulple scaeng echnques deemne eleconc sucue by accounng fo he scaeng evens an elecon wavefuncon expeences whn a sold. Ths s oughe han looks sngle scaee, sngle scaeng even analyc soluon wo o moe scaees, nfne numbe of possble scaeng evens, ecusve soluons equed fo wavefuncons e k e kz! Sngle se Mulple ses

Sho Hsoy of MST Lod aylegh (1892) On he Influence of Obsacles n ecangula Ode upon he Popees

Kasen (1897) exends MST o Helmholz equaon (scaeng of sound waves by collecon of sphees)

ask) Kohn and osoke edscove n 195 s (Phys. ev.

5 Sho Hsoy of MST Lod aylegh (1892) On he Influence of Obsacles n ecangula Ode upon he Popees of a Medum Phl Mag. Laplace Equaon N. Kasen (1897) exends MST o Helmholz equaon (scaeng of sound waves by collecon of sphees) Konga (Physca, 1947) fs use o fnd eleconc saes n solds (compuaonal facles howeve no up o he ask) Kohn and osoke edscove n 195 s (Phys. ev.) Ths leads o he Konga Kohn osoke appoach - aka -KK 196 s fs seous calculaons usng he appoach compues begn o cach up wh he heoy! *Images couesy of Emlo Segé Vsual Achves (hp://

6 So you sysem has poenal. ψ [ + V ] ψ ( ) Eψ ( ) H o -H o s he fee space Hamlonan -V s he peubng poenal Ψ s he elecon wavefuncon o 3 ( ) χ ( ) + G (, ) V ( ) ψ ( ) d We can expess he wavefuncon a some poson as a sum of he fee space wavefuncon, χ, wh no peubng poenal, and conbuons fom he peubng poenal, V, a dffeen ses. In hs case, G o s he fee elecon popagao and descbes moon n egons whee no scaeng fom he poenal occus.

7 Leng Geen do he expanson In analogy o he pevous wave funcon equaon, we can do a smla expanson fo he sysem Geen funcon. G G + o GoVG Geoge Geen s Mll We can expand hs equaon ou o nfny Nongham, England G Go + GoVGo + GoVGoVGo + GoVGoVGoVGo +... The oal Geen funcon acs as he sysem popagao. Ths expanson shows he nfne numbe of scaeng evens ha can occu hough poenal neacons. Elecon popagaon n fee space s descbed by G o.

8 Inoducng he T max We can eaange he las equaon o solae he effecs of he poenal. G G o o o + G TG o ( V + VG V + VG VG V + ) G + G... o o o o G o whee T ( V ) V + VG V + VG VG V +... o o The scaeng max, T, compleely descbes scaeng whn he poenal assembly. I conans all possble scaeng pahs. o

9 Mulple Scaeng Ses Assume he poenal s made up of a sum of ems due o dffeen cells o aoms. V V The T max n hs case becomes: T + T V V V G o V j j +... We can sepaae ou he sequences whee he scaeng always nvolves he same cell o aom no he cell max. V + V G V o + V G V o G V o +...

10 Aomc max uncoveed Solve he adal Schodnge s equaon fo an solaed muffn n poenal and deemne he egula and egula soluons, Z and S. The aomc max s dagonal n he angula momenum epesenaon. α l snδ e l δ l The phase shf, δ, can be found fom he aomc wavefuncon.

11 All he possble pahs We can now we he T max n ems of he sngle se scaeng max,. T V + j G o j +... Ths equaon shows ha he scaeng max of an scaeng assembly s made up of all possble scaeng sequences. Each scaeng sequence nvolves scaeng a ndvdual cells wh fee elecon popagaon beween. T j T j whee T j δ j + G o k T kj

12 Geng he Band Togehe In he MT fomalsm, he T max becomes: T j δ j + k Thee exss a max M such ha T j ae he elemens of s nvese. The max m s jus he nvese of he cell max. M j ~ ~ G j m δj G 1 k T kj ( δ ) The nvese of he T max s cleanly sepaaed no poenal scaeng componens, m, and sucual componens, G j. The poles of M deemne he egeneneges fo he sysem fo a gven k hough he followng equaon: ~ de [ m G( k) ] Ths allows us o calculae he sysem band sucue. j

13 Coheen Poenal Appoxmaon (CPA) Bes sngle-se soluon fo descbng scaeng n subsuonal alloys Scaeng popees of alloy can be epesened by an effecve medum Tea scaeng by aom as an mpuy n he effecve medum. Inoducon of aom should gve no scaeng n he coec effecve medum (eave soluon). Aom n bnay alloy Aom n equvalen effecve medum

14 FeC Alloys C magnec momen LKK-CPA (D. Sewa, unpublshed) KK-CPA (Kulkov e al., 1997) Expemenal (Alded e al., 1976) Fe magnec momen FeC Alloy Magnec Momen

15 Poblems wh he KK appoach Lnkng nesal egon (V) wh sphecal egons wh muffn n poenals can be dffcul Deemnan used o fnd band sucue s a nonlnea funcon of enegy (enegy dependence caed n he se maces) hs can no be educed o a sandad max egenvalue poblem The Soluon Lneaze he equaon LMTO appoach (Andesen, PB, caons)

16 Lnea Muffn Tn Obals Muffn Tn Sphee Two ses of soluons (1) Solves S. Eq. n sphee (2) Solves Laplace Eq. n nesal S Obals based on angula momenum chaace s, p, d, f Small bass se! Need obals and 1 s devaves o mach a sphee bounday E Poenal Man challenges (1) Machng condons a sphee bounday (2) Need an equaon ha s lnea n enegy

17 Makng Lfe Ease wh ASA Aomc Sphee Appoxmaon Many cysals ae close-packed sysems (fcc, bcc, and hcp) Mos of he space s flled by aomc sphees Wha f we chea a lle and have he sphees ovelap. Dong hs, we emove he nesal egon and ou negaon ove space becomes an negaon of aomc sphees. Ths appoach woks bes when he sysem s close packed, Ohewse we have o pack he sysem wh empy sphees o fll space

18 Solvng fo he Inesal egon ( ) ( ) [ ] ( ),, 2 + E E V E ϕ ϕ Poenal n nesal egon s zeo Inesal egon has no space, elecon knec enegy n egon zeo as well ( ) ) ( ˆ ) ( Y L ϕ ( ), 2 E ϕ L(l,m) l,1,2, m <l () () () () ( ) l l l l + w J Y J J L L , ˆ Take advanage of sphecal symmey expess wavefuncon n ems of sphecal hamoncs and adal poon () () () () 1, ˆ + l l l w K Y K K L L We ge wo soluons fo Laplace s equaon an egula one, J L () (goes o zeo a ) and egula one, K L () (blows up a )

19 Solvng nsde he Aomc Sphee E ϕ (, E) ϕ(, E) YL ( ˆ ) S 2 ϕ (, E) + [ V ( ) E] ϕ(, E) We need o mach adal amplude up wh nesal soluons, J and K, a S ϕ (, E) N 1 l [ Kl () P l ( E) J l() ] ( E) Nomalzaon funcon Poenal funcon

20 Muffn Tn Obals We can defne he oal wavefuncon as a supeposon of muffn n obals as ψ () a Ψ (, E) L L L Whee he muffn n obals ae gven by: Ψ (, E) N ( E) ϕ (, E) + P ( E) J ( ) L L l L fo < S Muffn-n Head K L ( ) fo > S Muffn-n Tal We also need o make sue soluons wok n ohe aomc sphees Expanson heoem used o lnk soluons ceneed a dffeen sphees K L ( ) S J ( ) L, L L L Sucue consans lace nfo

21 Cancelng Muffn Tn Tals Ψ (, E) N ( E) ϕ (, E) + P ( E) J ( ) L L l L < S MT head K L ( ) > S MT al n nesal L S L ( ), L J L < S ( ) MT al a ohe sphees The fom of he muffn-n obals does no guaanee ha solves he Schodnge equaon. We mus nsue ha does ψ () a Ψ (, E) L L L Tal Cancellaon needed [ P ( E) δ S ] P ( E) de[ δ S ] al L L, L L, L L L, L L, L L Fo peodc sysems, we can we hs n k-space and ge he band sucue!

22 The Lnea Appoxmaon Taylo expanson of he obal ϕ l (, E) ϕ (, E ) + ( E E ) & ϕ (, E ) l v v L v Ths allows us o expess he sysem n ems of lnea muffn n obals ha depend on ϕ and ϕ& n a gh bndng fom (TB-LMTO) Ψ whee + ( ) ϕ L( ) & ϕ L ( ) L L L h, L H L, L Ev, Lδ L, L + h L, L Speed Impovemen: emoval of non-lneay n deemnan equaon, acceleaes calculaons. Accuacy: Egenvalues coec up o hd ode n (E-E v ) Lmaons: Can un no poblems wh sem-coe d-saes ousde of he effecve enegy wndow.

23 Makng eveyhng self-conssen Inal guess n n () Calculae V eff [n] n Solve Schodnge Equaon ecalculae n ou () Mx n & ou n n () n ou ()? No Yes Calculae Toal Enegy *Dagam couesy Xaoguang Zhang (ONL)

24 Comng up hs afenoon LMTO commands unnng LMTO calculaons Slcon ole of empy sphees Magnec popees Nckel Densy of saes, band sucue, ec

25 An Inoducon o Geen s Funcons

26 Move ove Wavefuncons G [ E ] 1 H Off dagonal ems gve you elecon popagaon, [ ] A Γ G Γ G T T L N N1 1 ρ n 1 π 1 π Dagonal elemens gve () f ( E ε ) ImG(,, E) ( E) ImG(,, E) F d 3 de Chage densy Densy of saes Fom he chage, we can calculae he poenal and pefom self-conssen calculaons

27 Inegaon n he Complex Plane Chage densy s deemned by negang he Geen s funcon ove enegy. Howeve, on he eal axs, Geen s funcon s a vey shap funcon. If we move off he eal axs, he Geen s funcon becomes much smoohe. 3 daa pons able o do he wok of 1 s! ImE Coe saes Valence Bands E F ee Zelle e al., Sold Sae Comm., 44, 993 (1982)

28 Tgh Bndng Models Tgh-bndng model (esuls genealze o any fs pncples appoach wh sceenng o sho ange neacons) Take an nfne chan ε -2 ε -1 ε ε 1 ε 2 ε ,-1-1,,1 1,2 2,3 H... 2, 3 3, 2 2 1, 2 2, 1 1, 1 ε ε 1, ε 1,,1 ε 1 2,1 1,2 ε 2 3,2 2,3... dagonal max (vey nce fo nveson)

29 Isolaed and Peodc Sysems Isolaed sysem (fne Hamlonan shap enegy levels) [ ] ε ε ε E E E H E G Applcaons: molecules, quanum wells, fne nanowes/ubes Infne peodc sysem sll fne max! (Peod N4 hee, couplng beween laye 1 and laye N) [ ] ε ε ε ε E E E E H E G Applcaons: Mullayes, Bulk Sysems

30 Devce Geomey Lef Lead Devce egon gh Lead Sem-nfne leads couplng beween layes n leads mus be dencal o peodc Devce egon no consans on couplng beween layes Couplng beween devce and leads hs couplng deemnes how easy s fo elecons o ene and leave he devce egon. Ths s ccal fo devce pefomance.

31 Geen s Funcon fo Open Sysem How do we ake an nfne sysem and educe o a manageable sze? H H L DL H LD D D D H No neacon beween leads 1 N,N+1 Lef Lead Devce gh Lead -3,-2,-1, 1,2,3.N N+1,N+2, We can fold he nfomaon abou he leads no self eneges n he devce egon. G ( E) [ E H Σ Σ ] 1 L Σ B B 1,1 1,g,,1 Σ N,N N, N + 1g N + 1, N + 1 N + 1, N Leads povde a souce and snk fo elecons. Suface Geen s funcon The self eneges gve he elecons n he devce egon a fne lfeme and boaden he enegy levels (no longe an solaed quanum box).

32 Solvng fo Suface Geen s Funcons Sem-nfne chan of aoms g 1 We need he Geen s funcon a he end of chan [ ] 1 ( ) E E ε g ( E) 1 Seveal appoaches fo deemnng he suface Geen s funcon have been devsed (dec eave, eave wh mxng, ec) 1 Mos obus echnque uses enomalzaon appoach known as laye doublng. Wh each eaon, he algohm doubles he sze of he laye. Afe n eaons, he effecve laye hckness s 2 n lage han he ognal hckness! apd convegence a he pce of moe max opeaons: M.P. Lopez-Sancho, J.M. Lopez Sancho, and J. ubo, J. Phys. F: Meal Phys. 15, 851 (1985). 1 1

33 Ballsc Tanspo Tanspo on lengh scales less han he scaeng lengh fo elecons, no dffusve anspo, concep of poenal a posons n devce s dffcul T [ Γ ] A G G T Tansmsson L L Γ L I de ( f f ) [ G G ] [ G ( E) ] A < T Γ Γ L L e{ T L,L 1 L 1, L }de + + Equlbum Non-equlbum G G <

34 Spn Polazed Tunnelng fom Co suface (LM Sue) Tunnelng fom oxdzed and unoxdzed Co sufaces o Al pobe. Oxygen monolaye on Co flps he spn polazaon of unnelng fom negave (mnoy caes) o posve (majoy caes). ox ox Fgue: k esolved ansmsson fom clean Co fo (a) majoy and (b) mnoy caes and fom oxdzed Co fo (c) majoy and (d) mnoy caes. Uns 1-11 fo (a,b) and 1-14 fo (c,d). Belashchenko e al., PB, 69, (24)

35 Tanspo n Molecula Juncons (all-elecon ab-no calculaon) Au S H C 195kΩ T(E,) T(E,V) Faleev e al., PB, 71, (25)

36 Benefs of Geen s Funcon Appoach Capable of Handlng Open Sysems (somehng peodc DFT codes have ouble wh) Sysem Popees (eleconc chage, densy of saes, ec) whou usng wavefuncons Ably o Handle Dffeen Scaeng Mechansms hough Self Enegy Tems (no dscussed hee) Naual Fomalsm fo Tanspo Calculaons

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