介观物理与纳米电子学导论第五章量子输运的DFT模拟

㄀ϔゴDŽᓩ䆎:෎ᴀὖᗉ,⡍ᕕ䭓ᑺ,ҟ㾖⠽⧚,㒇㉇ᡔᴃ,㒇㉇ሎᑺ݇䬂ᅲ偠ᡔᴃ,㒇㉇ㅵ--⬉ᄤ఼ӊⱘᮄ⾔,ফ䰤䞣ᄤԧ㋏㄀ѠゴDŽ䞣䞣ᄤ⬉ᇐ:䞣ᄤ⚍᥹㾺ᅲ偠㒧ᵰ,ϔ㓈ᬷᇘ䯂乬,Landauer-Buttiker݀ᓣˈϔ㓈䕀⿏ⶽ䰉ˈ⃻ྚᅮᕟ䯂乬ˈѠ㓈䕀⿏ⶽ䰉ᮍ⊩㄀ϝゴDŽᬷᬷᇘⶽ䰉⧚䆎˖Buttiker໮ッ݀ᓣˈ⡍ᕕ࢓ˈᔅ䴲㒓ᗻ⬉ᇐˈԢ乥⬉ᇐˈ䞣ᄤ⬉ᆍˈ䞣ᄤ⛁Ӵᇐˈ㞾ᮟ䱻こ㄀ಯゴDŽḐḐᵫߑ᭄Ϣᬷᇘⶽ䰉˖䞣ᄤ࡯ᄺḐᵫߑ᭄ˈFISHER-LEE݇㋏ˈッষ㞾㛑ˈሔඳ䚼ߚᗕᆚᑺ㄀ѨゴDŽ㒇㒇㉇ሎᑺ䕧䖤ⱘ㄀ϔॳ⧚῵ᢳ˖෎ᴀḚᶊ,䴲ᑇ㸵GREENߑ᭄ˈᆚᑺ⊯ߑˈᅲぎ䯈ᮍ⊩ˈᓔᬒԧ㋏ⱘ㞾⌑ˈ⬉⌕⬉य़᳆㒓ҟ㾖⠽⧚Ϣ㒇㉇⬉ᄤᄺᇐ䆎:Ⳃᔩ㄀Ѩゴ:㒇㉇㒧ᵘЁⱘ䞣ᄤ䕧䖤੠Ң༈䅵ㅫ1.῵ൟϢϔ㠀⧚䆎ᶊᵘ2.䞣ᄤ࡯ᄺЁⱘḐᵫߑ᭄3.䳊⏽Ḑᵫߑ᭄4.䴲ᑇ㸵Ḑᵫߑ᭄˄NEGF˅ঞᑨ⫼5.ᑨ⫼NEGFѢ䞣ᄤ䱻こ䯂乬6.ᑧӺቯ䯂乬7.ᑨ⫼NEGFࠄᯊ䯈Ⳍ݇ⱘ䞣ᄤ䱻こ8.ᅲぎ䯈TBᮍ⊩9.NEGF+DFTⱘᑨ⫼ҟ㾖⠽⧚Ϣ㒇㉇⬉ᄤᄺᇐ䆎1.೉ᅛናህֱ஛౅਒ࣕҟ㾖䕧䖤䯂乬ⱘ῵ൟLRC᳝䰤㣗ೈⱘッষϢ఼ӊऎⱘ㗺ড়ッষ໘Ѣᑇ㸵⢊ᗕˈッষᰃञ᮴䰤ⱘഛࣔԧ㋏Ёᖗ఼ӊऎৃҹ᳝৘⾡໡ᴖⳌѦ԰⫼ᑇ㸵ᯊⱘ఼ӊ῵ൟLeftReservoirRightReservoirDeviceLCRRCRCLCLHHHHHH䴲ᑇ㸵ᯊⱘ఼ӊ῵ൟLeftReservoirRightReservoirDeviceRLLCRRCRCLCLHHHHHHbiasVgateV䕧䖤䯂乬•䕧䖤䯂乬ᰃᡒࠄ఼ӊⱘ⬉⌕⬉य़݇㋏ेI(V)=?᳈ϔ㠀ഄ,I(V,t)=?�߽⫼䴲ᑇ㸵Ḑᵫߑ᭄(NEGF)੠䕀⿏જᆚ乓䞣,I(V)ⱘ݀ᓣৃҹ᥼ߎᴹ,Ԛ䳔㽕⊼ᛣ䆹݀ᓣⱘ䗖⫼ᴵӊঞ䰤ࠊ.�߽⫼ᆚᑺⶽ䰉,⬉⌕⬉य़݀ᓣгৃ᥼ߎ.�䳔㽕䅸䆚ࠄ㒇㉇ԧ㋏ⱘ᳝䰤ሎᑺ,NEGF੠ᆚᑺⶽ䰉䛑䳔㽕೼ᅲぎ䯈䅵ㅫ.ेՓ⫼NEGFᮍ⊩㞾⌑гᰃᖙ㽕ⱘI-V᳆㒓˖I(V)=?dEGGffTrheIar)])([( 䴲ᑇ㸵Ḑᵫߑ᭄ᑊϡ㞾ࡼֱ䆕⬉⌕ᅜᘦ੠㾘㣗ϡবᗻʽҡ✊䳔㽕㞾⌑∖㾷ʽᑇ㸵ᯊⱘ㞾⌑}{}{effCUTHSolveKohn-ShamEq.Getelectrondensity䴲ᑇ㸵ᯊⱘ㞾⌑{}{}{,,}CeffLRHTUESolveKohn-ShamEq.()(,,)2irGrrEdEᗢḋ㦋ᕫᆚᑺ()r䳔㽕ᇍ⊶ⶶ∖੠ˈ៪ᇍ㛑䞣⿃ߚˈ䅵ㅫ䞣䕗໻2.ோᏳலᆓ᎒ݨࢯ௢ऩབྷGreen'sFunctionsinQuantumPhysics,E.N.Economou3.௯႔ࢯ௢ऩབྷᇍऩ㉦ᄤᚙᔶ,Ϟ䗄ᅮНⱘḐᵫߑ᭄੠䞣ᄤ࡯ᄺЁᅮНⱘ᭄ᄺ⠽⧚ᛣНϞⱘḐᵫߑ᭄Ⳍৠ.(1)(2)H=U()(,'),rUrr݊Ёߚ߿Ўऩԧㅫヺ੠ϸԧㅫヺ䆕ᯢऩԧ䚼ߚϸԧ䚼ߚᇍऩ㉦ᄤᚙᔶ,ϸ⾡ᅮНⱘḐᵫߑ᭄ㄝӋHartree-Fock䖥Ԑ᳝䰤⏽ᑺḐᵫߑ᭄4.ߢ࿻ქဧݨࢯ௢ऩབྷೈ䘧ϞᅮНⱘNEGFtt2t1tt1t2t4t3t1t2t1t2t3t4OrdinaryTimeOrderingClosedPathTimeOrderingNEGFᅮНGreen’sFunctions:t2t1t2t1t1t1t2t2),(21ttGt),(21ttGt),(21ttG),(21ttGDysonᮍ⿟䯁䏃ϞⱘḐᵫߑ᭄Ϣ䗮ᐌⱘḐᵫߑ᭄⒵䎇ৠḋⱘDysonᮍ⿟DŽ಴ℸˈϸ⾡Ḑᵫߑ᭄ⱘᖂᡄ䅵ㅫᮍ⊩෎ᴀⳌৠDŽLangrethᅮ⧚(㾷ᵤᓊᢧ)12ACBCCCt'tCC'tt'ttt't't)',(ttC঺ϔ㉏Langrethᅮ⧚Langrethᅮ⧚ⱘϔѯ䖤⫼Kadanoff-Baym݀ᓣᅲ䰙䅵ㅫᡔᎻ䴲ᑇ㸵Ḑᵫߑ᭄ⱘᅲ䰙䅵ㅫᡔᎻ˖1.߽⫼ᑇ㸵ᗕḐᵫߑ᭄ⱘᖂᡄ䆎2.߽⫼㾷ᵤᓊᢧⶽ䰉ᔶᓣⱘNEGF�ݡϔ⃵ⳟࠄˈⶽ䰉ᔶᓣⱘ䴲ᑇ㸵Ḑᵫߑ᭄Ϣೈ䘧Ϟⱘ䯁䏃Ḑᵫߑ᭄ᰃᅠܼϔḋⱘDŽ�Ԛᰃᇍ⬉ᄤໄᄤⳌѦ԰⫼ᯊⱘ䴲ᑇ㸵Ḑᵫߑ᭄⫼ೈ䘧ϞⱘḐᵫߑ᭄໘⧚ᆍᯧѯDŽᆚᑺ੠ᗕᆚᑺ48NotesonNEGFᑇ㸵Ϣ䴲ᑇ㸵Ḑᵫߑ᭄೼㒧ᵘϞᰃㄝӋⱘˈଃϔⱘᏂ߿ᰃᅲ䕈⿃ߚ⫼䯁䏃⿃ߚ᳓ᤶDŽ↨བˈDysonᮍ⿟ᔶᓣᰃⳌৠⱘㄝDŽ,,,raG݋ԧᑨ⫼ᯊˈা䳔㽕ᅲ䕈ϞᅮНⱘ䖭ৃ⬅ܜ೼䯁䏃䖯㸠䅵ㅫˈ✊ৢ⬅㾷ᵤᓊᢧ(Langrethᅮ⧚)ᕫࠄᅲ䕈Ϟⱘ䞣DŽ,,rrrraCABCABCABAB㢹߭,,rrraararrDABCABCDABCABCABCU㢹߭'߽⫼Ḑᵫߑ᭄ⱘ䖤ࡼᮍ⿟⊩ˈৃҹ᳓ҷ೒ᔶᖂᡄᮍ⊩5.቎ቫ1(*)ኂோᏳྥۯႜ࿸LRCLLHRRHCCHCRHLCHipi01234aaa⬉⌕I䖤ࡼᮍ⿟⊩ッষḐᵫߑ᭄LeftSemi-InfiniteProbe00H11H33H44H55H22H10H21H32H43H54H65H102132435465001122334455,HHHHHHHHHHHH⬉⌕IIㅔ࣪⬉⌕ᕫࠄDattaкЁⱘ݀ᓣ6.ଲుݞႜ࿸ҟ㾖䕧䖤䯂乬ⱘ῵ൟLRC᳝䰤㣗ೈⱘッষϢ఼ӊऎⱘ㗺ড়ッষ໘Ѣᑇ㸵⢊ᗕˈッষᰃञ᮴䰤ⱘഛࣔԧ㋏Ёᖗ఼ӊऎৃҹ᳝৘⾡໡ᴖⳌѦ԰⫼䖤ࡼᮍ⿟⊩Hartree-Fock䖥Ԑ៾ᮁ໘⧚ᑇഛഎⱘ঺ϔϾᮍ⊩㞾⌑䯂乬17ContourdZZrrGirn),,(2)(computetoHowEnergyContourTemperature=0EnergyatbottomofthebandFermiEnergyᆚᑺⱘ䅵ㅫ⍝ঞᇍḐᵫߑ᭄ⱘ㛑䞣⿃ߚˈ㗠Ḑᵫߑ᭄೼ᅲ䕈Ϟᰃ᳝㛑䞣༛⚍ⱘˈ಴ℸ⿃ߚ䳔㽕Փ⫼໡বߑ᭄Ꮉ݋DŽϞ䗄೒⼎㒭ߎњ䳊⏽ⱘ䅵ㅫ῵ᓣˈᇍ䴲䳊⏽䖬䳔㽕㗗㰥೼䌍㉇䴶໘⊓㰮䕈ⱘ༛⚍DŽⳌѦ԰⫼੠㞾㛑䯂乬7.቎ቫ1(*)ݡ་ਚᄐࣴݨோᏳྥۯᑇഛ⬉⌕ᆑᏺᵕ䰤ঠᯊFOURIERবᤶԡ⿏⬉⌕8.༏ଥਚ7%ࠝࠈ䴲ℷѸ෎ᑩ䯂乬䴲ℷѸ෎ᑩⱘᅲぎ䯈Ḑᵫߑ᭄ᆚᑺⶽ䰉བԩ䅵ㅫGབԩ䅵ㅫ⬉⌕བԩ䅵ㅫ(r)g ञ᮴䰤ッষḐᵫߑ᭄ⶽ䰉ܗⱘ䗁ҷ䅵ㅫ15,biasgateVVFEn,initialgeometry,device(,',)raGrrZGGKSHH()()nZContourdZZrrGirn),,(2)(econvergencCheckdataCollectrishpublish/penostartFlowchart),(gbVVIKSH(),FnZE()ZyesTBᚙᔶⱘ⺕എ9.1(*)')7ݨ቎ቫMcDcalTranSiestaCatomistix㔥キ˖+NEGFSmeagol㔥キ˖˖+ABINIT+NEGFWanTϝϾ՟ᄤAlCAl60)100(@)100(803AlCNScAlAlchainCAl)(7DeviceModelEquilibriumChargeDensityContourPlot,3extraelectronFlowsontoC60Conductancev.s.GateVoltage.Inset,transferredchargesv.s.GateVoltageI-VCurvefortwogatevoltageShowstheswitchpropertiesChargeonprobetoC80=4.7eChargesonScloses1.7eperatomChargesonNgets0.9eperatomC80gets8.9etotally1(*)ݨࠂጝ�ࠄ⦄೼ЎℶˈᏆ᳝໮Ͼⷨおᇣ㒘ᅲ⦄њᆚᑺ⊯ߑϢ䴲ᑇ㸵Ḑᵫߑ᭄ⱘ㞾⌑∖㾷DŽ�ሑㅵᅲ偠ᰃᬃᣕ䖭ḋⱘⷨお㒧ᵰⱘˈৃᰃᆚᑺ⊯ߑ↩コᰃϔϾᑇ㸵ᗕⱘ⧚䆎ˈᇍ䴲ᑇ㸵ᚙᔶᆚᑺ⊯ߑˈ䖬≵᳝ϔϾϹḐⱘ䆕ᯢDŽ�Ԛᰃ䆹ᮍ⊩㦋ᕫњᑓ⊯ⱘᑨ⫼੠ϡᮁⱘᬍ䖯DŽϟ䴶՟Вϔѯ䕗䖥ⱘ᭛⤂ˈկ໻ᆊ䖯ϔℹᄺдDŽ䴲ᑇ㸵Ḑᵫߑ᭄ᬭ⿟෎ᑩ໻ᇣᇍ䕧䖤ᕅડ໻⾡᳝ᬜⱘ䴲㞾⌑ⱘᮍ⊩೼޴Ͼ՟ᄤЁˈ㞾⌑Ϣ䴲㞾⌑㦋ᕫњᅮ䞣ਏড়Ѹᤶ݇㘨ᇍ䕧䖤ⱘᕅડ,205413(2007)ッষሎᇌᇍ䕧䖤ⱘᕅડJ.Chem.Phys.123,114701(2005)ッষ໻ᇣⱘᕅડ㒇㉇㒓䭓ⷁⱘᕅડO(N)ᮍ⊩䅵ㅫ䕧䖤ᔎ݇㘨ᇍ䕧䖤ⱘᕅડ0710.0482v1㞾ᮟⳌ݇ⱘ㄀ϔॳ⧚䅵ㅫׂℷᇍ䕧䖤ⱘᕅડ˄2008˅NEGF+TDDFTᮍ⊩J.Phys.:Condens.Matter20,083203(2008)ᔎ݇㘨䕧䖤˖DMFTᅲぎ䯈ܼ⬉ᄤ+TDDFT+NEGFarXiv:0902.2306v1Wepresentforstaticdensityfunctionaltheoryandtime-dependentdensityfunctionaltheorycalculationsanall-electronmethodwhichemployshigh-orderhierarchicalfiniteelementbases.Ourmeshgenerationscheme,inwhichstructuredatomicmeshesaremergedtoanunstructuredmolecularmesh,allowsahighlynonuniformdiscretizationofthespace.Thusitispossibletorepresentthecoreandvalencestatesusingthesamediscretizationscheme,i.e.,nopseudopotentialsorsimilartreatmentsarerequired.Thenonuniformdiscretizationalsoallowstheuseoflargesimulationcells,andthereforeavoidsanyboundaryeffects.ࡼ䞣ぎ䯈+TDDFT+NEGFarXiv:0902.1489v1Weproposeafirst-principlestime-dependentdensityfunctionaltheoretical(TDDFT)approachinmomentum(P)spaceforquantitativestudyofelectrontransportinmoleculardevicesunderarbitrarybiases.Inthisapproach,thebasicequationofmotionisatime-dependentintegrodifferentialequationobtainedbyFouriertransformofthetime-dependentKohn-Sham(TDKS)equationinspatialcoordinate(R)space.Itisformallyexactandincludesalltheeffectsandinformationoftheelectrontransportinmoleculardevices.TheelectronwavefunctioniscalculatedbysolvingthisequationinaclosedfiniteP-spacevolume.Thisapproachisfreeofself-ener