Herre van der Zant. Molecular Electronics and Devices group

Transcription

1 transport through (magntic) nanoscal objcts Hrr van dr Zant Molcular Elctronics and Dvics group

2 transport mchanisms: Ohms law lctrons ar viwd as particls in a pinball machin, bouncing around rsistanc rsults from (back) scattring and can b charactrizd by th mobility (a matrial dpndnt proprty). How fast dos an lctron mov in an lctric fild? mobility µ τ * m Drud conductivity

3 ballistic transport: 1D channl or a singl atom lctrons ar viwd as wavs that ar transmittd in channls; th transmission dtrmins th conductanc which is quantizd with a valu indpndnt on th matrial G = T h T i i : transmission of i channl i For onchannl and prfct transmission G = h = (1.9 kω) Which physical quantity dtrmins th numbr of channls? How fast do th lctrons go? 1

4 tunnling in on dimnsion: singl-barrir cas B * 1) : strong barrir ( ) ( sinh 1 1 probability : tunnl Φ = + >> + + = = m a a G G k k t a a k k TT t γ γ γ γ γ γ γ Φ Β = work function = 5.3 V (= Frmi-nrgy gold) d = a 0 0 ) ( :, ) ( ) ( : ) ( : me k T x u a x V E E m V B A x u a x a me k R x u a x ikx x x ikx ikx = = > < = + = < < = + = < γ γ γ gold-vacuum-gold junction: dirct, through spac tunnling Φ Β -a a

5 voltag dpndnc: Simmons formula for tunnling d = barrir width Φ B = barrir hight J. Simmons, J. Appl. Phys. 34 (1963) Φ Φ = + Φ Φ B B B B 4 V m d V m d V V d A I π xpanding th squar root (Φ B >> V): ) sinh( V B 0 Vd d A I d β π β Φ B 0 B V Φ = Φ = m m β β Φ B =4.8 V; d=0.5 nm

6 linar rspons (zro-bias conductanc) 0 sinh( βvvd ) βvvd G = G0 β d gold-vacuum-gold junction Φ Β = 5.3 V m = m 0 = kg β V 0 = β = m Φ B mφ B = 1.1 =.34 (nm) o -1 (A) ths ar th maximum valus whn working with gold: in practic lowr du to imag chargs tc. -1 β 0 : dcay lngth

7 an xampl: braking of a gold contact PhD. Thsis Roman Hubr (1,,3): quantizd conductanc (G 0 ) in mtallic rgim Landaur formalism: displacmnt of th pizo G=G 0 ΣT i G 0 = /h (1.9 KΩ or 77mS) (4): rlaxation of th contact (5): tunnling Why is th scond stp not xactly on G 0?

8 What is th rsistanc of a singl-molcul?

9 transmolcular transport: doubl-barrir problm Th rsrvoirs (sourc and drain) ar bulk-lik rgions whr th lctrons ar in quilibrium. Ths rgions ar kpt at som tmpratur, and th numbr of lctrons is variabl as thy ar connctd to th voltag sourc. Th lctrons in ths rsrvoirs ar distributd according to Frmi functions. Th molcular island is th part of th systm, which is small in all dirctions hnc, this is th part whr th Coulomb intraction (E C ) plays an important rol. It is usful to tak as a rfrnc th isolatd island. In that cas w hav a st of quantum stats with discrt nrgis (lvls, nrgy ). Th molcul-lad coupling is charactrizd in trms of th rat, Γ, at which lctrons cross th tunnl barrirs sparating th island from th lads. Th transport through th barrirs is a tunnling procss. Γ also sts th lif-tim of a charg on a molcular stat and hnc dtrmins th broadning of th lvl; at prsnt th molcul-lad coupling can still not b controlld and vn prdictd. molcular island ssntial for undrstanding transport: location of th lvls with rspct to th Frmi nrgy of th lctrods th molcul-lad coupling (Γ) sourc gat V drain V G

10 rsonant and off-rsonant transport Schmatics of th lctrochmical potntials of an island connctd to two rsrvoirs, across which a small (ngativ) bias voltag V is applid. A gat voltag is usd to shift th lctrostatic potntial of th nrgy lvl. first ordr procsss blockad rsonanc: currnt flows µ µ(n +1) (N µ ) ( N) µ (N ) µ S S µ(n-1) +/C V Γ L µ Γ R S µ µ (N) µ ( N) µ S D µ D Rsonant transport (first-ordr transport): rsonant transport bcoms possibl whn th gat voltag pushs on of th lvls within th bias window V. Th µ(n) lvl is alignd with µ S and th numbr of lctrons on th dot altrnats btwn N and N-1 (squntial tunnling). V G V G scond ordr procsss µ(n +1) Ε m S DE S m(n) m D V µ µ µ S D µ (Ν) µ(n +1) m S DE m(n) m D µ (Ν) µ D Off-rsonant transport: th lvls ar not alignd. Coulomb blockad fixs th numbr of lctrons on th dot to N. Transport, howvr, is possibl through a virtual co-tunnl procss in which a unoccupid lvl is brifly occupid. A similar procss xists for th occupid lvl, µ(n), which can b brifly unoccupid. In contrast to rsonant transport th lvl is mpty (full) most of th tim. V G V G

11 off-rsonant tunnling through molcular junctions th furthr th molcular lvl (ithr th HOMO or th LUMO) is away from Frmi nrgy of th lctrods ( E), th largr th zro-bias rsistanc sam xponntial distanc dcay as th singl-barrir cas but S S Φ B m m* E F - E LUMO or E F E HOMO β 0 αβ 0 (α taks into account nonrctangular barrirs) β 0 charactrizs lctron transfr molculs Φ B co-tunnling LUMO HOMO alkans: β Å -1 polyns: β Å -1 oligophnyls: β Å -1 conjugatd molculs: β Å -1 0 G = G 0 β d

12 off-rsonant tunnling through dithiol-alkans conducting AFM xponntial lngth dpndnc zro-bias rsistanc IV charactristics 0.88 Å -1 Englks, Bb, Frisbi, JACS 16 (004) 1487

13 rsonant transport: thr-trminal dvics first ordr procsss: squntial tunnling (incohrnt) blockad µ µ(n +1) (N µ ) ( N) µ (N ) µ S S µ(n-1) V G +/C V rsonanc: currnt flows Γ L µ Γ R S µ µ (N) µ ( N) µ S D µ D V G island thr-trminal dvics: lvls can b brought in rsonanc with th lads (if th gat coupling is larg nough)! molcul can b oxidizd or rducd (chang its charg stat) and th proprtis of th chargd molculs can b studid in dtail: this maks it a compltly diffrnt story!! sourc gat drain V V G

14 chmical potntial and addition nrgy From non-quilibrium thrmodynamics, w know that a currnt is drivn by a chmical potntial diffrnc hnc w should compar th chmical potntial on th dvic, µ( N ) = U ( N) U ( N 1) = ( N 1/ ) V xt + E N C with that of th sourc and drain in ordr to s whthr a currnt is flowing through th dvic. In xprimnts, w ground th drain lctrod(µ D = 0). V S R S,C S R D,C D island V I C G V G V D Th distanc btwn th two subsqunt chmical potntial lvls, th addition nrgy (= th nrgy it costs to put an additional charg on th island) is givn by µ( N + 1) µ ( N) = + E N +1 E N C Not that th diffrnc in nrgy lvls occurring in this xprssion (E N+1 E N ) is nothing but th lvl splitting.

15 lctronic lvl structur isolatd C 60 optical transitions Enrgy lvls calculatd by th dnsity functional mthod by Grn t al., J of Phys. Chm. 100, (1996) 1489.

16 lctronic nrgis and lctrochmical potntials µ N U(N) U(N-1) U = total nrgy µ 1 µ 0 = E +E c E = 1.65 V E c : charging nrgy ( /C) stimat E c mtal sphr: C = 4πε 0 R R = 0.4 nm E c = 3.6 V

17 analytical solution IV charactristic singl-lvl modl γ 1 γ currnt shows stps as rsonanc crosss bias window T = 0: f 1 -f = 1 for 0< E <V and zro othrwis including spin and for symmtric barrirs: V / ε γ 1γ I = de h ( E ε ) + ( γ / ) V / V / ε γ 1γ 1 I = d( E ε ) h ( E ε ) + ( γ / ) V / ε 4 γ γ V ε V ε 1 I = arctan arctan h γ 1 + γ γ γ currnt proportional to th broadning γ currnt in units Γ 1 /(R Q ) di/dv in units of /h Γ 1 =Γ Γ=0.01 Γ=0.05 Γ=0.1 ε = 0.5 V Why dos th stp occur at ε? Voltag (V) di/dv shows Lorntzian shap Γ 1 =Γ Γ=0.01 Γ=0.05 Γ= Voltag (V)

18 complication: at low bias th modl prdicts th sam dpndnc as singl-barrir tunnling currnt in units Γ 1 /(R Q ) tunnling I = 0.01*sinh(3.3V) Voltag (V) currnt in units Γ 1 /(R Q ) Γ 1 =Γ Γ=0.01 Γ=0.05 Γ=0.1 ε = 0.5 V rsonant pictur Voltag (V) This can thrfor not b usd as a proof for molcular transport:

19 an xampl: a spin-crossovr molcul T = 6 K mchanically controlld brak junction tchniqu Why is th zro-bias conductanc not zro? Why do th stps not flattn out at high bias voltags?

20 stability diagram: vary bias and gat voltags V bias 0 currnt N N+1 E add / αe add / Th currnt is supprssd whn all chmical potntial lvls li outsid of th bias window. As w can tun th location of ths lvls using th gat voltag. W can calculat th lin in th plan which sparats a rgion of supprssd currnt from a rgion with finit currnt. This lin is dtrmind by th condition that th chmical potntial of th sourc (or drain) is alignd with that of a lvl on th island. V ( N 1/ ) β = C /( C + C ) D G G C C = + γ = C G / C S E N V G For th chmical potntial to b alignd to th sourc (µ(n) = µ S = -V) and with kping th dot s charg constant): ( V CV C ) V = β / G C G If th chmical potntial is alignd with th drain (µ(n) = µ D = 0), w hav th linar rlation: V ( CV C V ) = γ / C G G S also: J.M. Thijssn and H.S.J. van dr Zant, Phys. stat. sol. (b) 45 (008)

21 masurd stability diagram carbon nanotubs lvl splitting and charging nrgis can b dtrmind V = µ( N) µ ( N + 1) = E add Draw th corrsponding chmical potntial of th lvls and of th nrgis of th orbitals 1: E C 4 mv : E C mv 3: E C 4 mv 4: E C + 13 mv

22 most xprimnts thus far: off-rsonant transport molcular wirs bhav as smiconductors Only if clar stps in th currnt-voltag curvs ar obsrvd, can w xclud that dirct tunnling dtrmins transport. What about molcular signaturs? (addition nrgis will not work bcaus..) Whr to look for molcular signaturs: - Choos for molculs whos (intrnal) orbital structur dos not allow a dscription in trms of a singl tunnl barrir (.g. quantum intrfrnc molculs) - Prform dtaild spctroscopy on th molculs looking for molculspcific proprtis (vibrations, spin-rlatd xcitations, ) Why do you want to prform th spctroscopy at low tmpraturs?

23 lvl spctroscopy: xcitd stats (Γ<<U) incohrnt, rsonant transport: non-linar rgim µ(n +1) µ (N ) E mµs S µ(n) DE m(n) mµ D µs m (N) mµ D D two knobs : V SD and V gat stat contributs to transport if μ s > μ(n) > μ d V sd 0 E N N+1 xcitd stats rsult in xtra lins in stability diagram (rd) xcitations can prob lctronic spctrum, spin or vibrational stats V G

24 xcitation lins: lctronic fingrprints of molcular vibrations (Γ<<U) N-1 N sourc drain gat di/dv E symmtric with rspct to th bias voltag xcitations th sam for diffrnt charg stats mor than 15 lins sparatd by a fw mv: unlikly lctronic xcitations in such a small molcul E.A. Osorio t al., Adv. Mat. 19 (007) 81

25 molcular spin transistor: addrssing magntic proprtis on a singl molcul lvl using lctrons contacts ar mtals (Au, Pt) V b sourc gat V g drain molcul with dsird magntic proprtis can b dsignd by chmists but dos it rtain its proprtis whn contactd? spacr plus anchoring: tunnl coupling spacr alligator clip mchanical anchoring magntic cor of th molcul alligator clip

26 th molcul: singl-molcul magnt F 4 H = DS gµ SB z B Iron(III)) Oxygn Carbon nrgy barrir U = S D ZFS = (S-1)D synthsis: Andra Cornia high stability of F 4 ; rtains its magntic proprtis whn disprsd on a mtallic surfac (Nat. Mat. 8 (009) 194; Natur 468 (010) 417) bulk zro-fild splitting: ZFS = (S-1)D = 0.6 mv; nrgy barrir: U = 1.4 mv modrat quantum tunnling (H = DS z +E(S + + S - )/); bulk: E= mv How to obsrv anisotropy in an xprimnt (as Γ in gnral largr than ZFS)? What ar th magntic paramtrs for an individual molcul in a solid-stat dvic? asy axis hard axis

27 rol of th molcul-lad coupling spacr plus anchoring: tunnl coupling xprimntally, th coupling (Γ) can b controlld to som xtnt; full control rquirs fixing all atoms positions in th junction long spacr: wak coupling Dlta function DOS currnt is blockd to first ordr involvs two (singl-particl) charg stats Γ L Γ R short spacr: intrmdiat to strong coupling Lorntzian DOS highr-ordr tunnling procsss (cohrnt) inlastic spin-flip, Kondo involvs only on charg stat Γ L Γ R V V

28 highr-ordr procsss in Coulomb blockad inlastic tunnling spctroscopy (IETS) Kondo-ffct (lastic): spin filling µ(n +1) µ(n +1) µ(n +1) µ(n +1) mµ S µ (N) DE m Ε m(n) mµ µ DE D S Ε m(n) mµ D µ (Ν) µ (Ν) m S DE m(n) mµ D µ S µ (Ν) m S DE m(n) m md µ D µ S (IETS) V sd 0 E N N+1 V sd 0 vn odd V G off-rsonant transport V G

29 Kondo: signatur of a larg quantum spin conductanc vs Fild [ns] sampl 1 inlastic co-tunnling Conductanc N N+1 Kondo rsonanc at V=0 in two adjacnt charg stats Bias voltag V conductanc vs Fild sampl [ns] unpaird high quantum spin at last on stat has spin > 1/! N N+1 T K ~ 10's of Klvin A. Zyazin t al., Synth. Mtals 161 (011) 591 zro-fild splitting maskd by strong Kondo (Γ, T k > ZFS) wakr coupling

30 inlastic cotunnling: signaturs of zro-fild splitting magntic transitions (i.., from S=5 to S=4) rd: co-tunnl rgim ZFS ZFS ZFS ZFS = (S-1)D xpctd valu nutral molcul: mv (bulk masurmnts) obsrvd in adjacnt charg stats T =1.5 K N

31 magntic fild dpndnc: nrgy splitting incrass with B and is symmtric around B=0 N N+1 S = 5 D = mv thta ~ 0 g =.1 S B consistnt with xpctations for ZFS!! masurmnts indicat that th angl is btwn 0 and 30 dgrs D-paramtr in th rducd stat largr than in th nutral stat (50%)