Inelastic electron tunneling through a vibrational modulated barrier in STM

Transcription

1 Romnn Reports n Physcs, olume 55, Numer 4, P , 3 Inelstc electron tunnelng through vrtonl modulted rrer n SM P. udu culty of Physcs, Unversty of uchrest, PO ox MG, Mgurele, Romn strct: Usng mny ody formlsm, we hve studed the effect of the modulton of the tunnelng rrer y the vrtng molecule, on the nelstc tunnelng current n SM. In our model, the domnnt nelstc term ppers n the frst order of the perturton theory. We hve consdered lso the second nd the thrd order elstc nd nelstc terms, whch cn contrute to the tunnelng current n the SM. Key words: Scnnng unnelng Mcroscopy, elstc nd nelstc tunnelng Recent experments proved the posslty to perform nelstc electron tunnelng wth SM, through vrtng molecules [,]. Prevous theores of nelstc electron tunnelng, model ths process y dpole couplng [3,4], or tunnelng through n dsorte resonnce [5-7]. Detled mcroscopc clcultons sed on Green uncton technque [8] or Densty unctonl heory [9], expln the selecton rules oserved n experments, y destructve nterferences of nelstc sctterng processes. In the present pper, we model the nelstc tunnelng of electrons usng smlr mltonn s used y Persson nd rtoff [5]. We consder tht the vrton of the molecule modultes the tunnelng rrer nd tht the couplng of the tunnelng electrons to ths vrton s the mn perturton. We model the followng physcl process: electrons tunnel from n electron resonnce loclzed on the SM tp to n electronc resonnce of vrtng molecule sored on metllc surfce (g.). he correspondng projected denstes of sttes on the sttes nd re denoted () nd ().

2 48 P.udu he process s modeled y the mltonn: k k k ( k k ) R R R R R kr kr ( k k ) hωυ υ ( Q)( ) L L kl kl kl kl () tp;, re the electronc levels close to the erm levels of the sustte nd the, - lel the energy levels of the one prtcle egensttes of the tp (L) nd k L k R the sustrte (R);, re the mtrx elements for the electron trnsfer etween the tp nd k L k L level, nd etween the sustrte nd level; h Ω - s the vrton energy of the dsorte;,,,,,,,,υ,υ re the nnhlton nd creton opertors for the,, hω sttes respectvely; of kl k L, kl k R k R Q s the dsplcement correspondng to the vrton mode of the dsorte. We neglect the dependence of nd on Q, ut we consder the dependence on Q ( ( Q) ) the vrton of the molecule.. s drectly relted to the SM rrer nd s modulted y y expndng to frst order n Q we hve: θ ( Q) ( O) Q ( υ υ ) () Q O he mltonn () cn e rewrtten usng (): ( ) ( )( υ ) υ (3)

3 Inelstc electron tunnelng through vrtonl modulted rrer n SM 49 s the qudrtc prt of the mltonn, nd cn e dgonlzed s n [5]. fter the dgonlzton we otn: Ω υ υ h ( ) ( ) ( ) υ υ (4) { }, { } denote the one prtcle electronc sttes of the dsortesustrte system, nd of the tp. he projected denstes of sttes on nd sttes re defned: ( ) ( ) ( ) ( ) (5) Let us consder the next expnson of the trnston opertor: I I I (6) We evlute the trnston mtrx elements n the frst, second nd thrd order etween the next ntl nd fnl sttes: fph ph n fn n n ; ; χ (7) denotes the occupton of the stte wth one electron; denotes the empty stte;

4 5 P.udu, re ntermedte sttes up (elow) the erm level; ndcte the numer of phonons n the ntl (fnl) stte. n, ph, n f, ph, ph f, ph or elstc trnston n n. or nelstc trnston wth emsson of, ph f, ph one phonon n, n, etc...he ove n nd fn sttes correspond to trnston (elstc or nelstc) for one stte electron. It s very esy to extend the notton, consderng two (or more) electrons processes. In the frst order of the development (6) we hve elstc processes nd nelstc processes wth one electron ndcted symolclly n g.. he mtrx elements correspondng to these processes re: ( ) elstc ( ) nelstc fn e fn n (8) ll the mtrx elements (elstc or nelstc) n the second order nvolvng oneelectron processes vnsh. In fct, ths sttement s vld for ll n th order mtrx elements wth even n (n ). hs s not vld n Persson nd rtoff model n whch the frst nonvnshng nelstc processes wth one tunnelng electron re n the second order. he explnton of ths fct s due to the presence of the couplng etween the resonnt level nd the vrton mode n Persson nd rtoff model. hs couplng term s not ncluded n our model. he mtrx elements n the second order re: j fn n j, wth, j, (9) he nonvnshng mtrx elements n the second order nvolve two electrons. he elstc nd nelstc processes n the second order re presented n g.3. he mtrx elements n the thrd order re:

5 Inelstc electron tunnelng through vrtonl modulted rrer n SM 5 n fn k j jk, wth,,, k j () he mtrx elements n the thrd order nvolve one or three electron processes. hese processes cn e elstc processes or nelstc processes wth one, two or three phonons (g.4). ll the two electron processes n the thrd order vnsh. Other nonvnshng mtrx elements n the thrd order re nelstc processes wth emsson of one, two or three phonons, ut the electronc events re the sme s n g.4, d. Now we evlute the tunnelng currents correspondng to the frst order terms ndcted n g.. or the elstc tunnelng current we hve: ( ) ( ) ( ) ( ) [ ] ( ) Θ Θ e I elstc elstc () where ( ) Θ s the erm Drc dstruton t zero temperture. he correspondng elstc conductnce s: ( ) ( ) ( ) ( ) ( ) elstc h e h e I π π 4 / / 4 () Where e he nelstc conductnce n the frst order s: ( ) ( ) ( ) ( ) ( ) ( ) ( ). / / Ω Ω Θ Θ Ω e e I nelstc (3)

6 5 P.udu he normlzed totl conductnce n the frst order s: I η ( ) ( ) ( ) elstc I nelstc I / ( Ω) ( ) elstc η Θ( e Ω). (4) We estmte tht ths formul gves the domnnt term n the tunnelng current nd predct pek n the second dervtve of the tunnelng current. Our result s dfferent from the result otned usng the Persson rtoff formlsm, whch predcts dp n the second dervtve of the current. We hve evluted the tunnelng currents nd conductnce [], correspondng to ll ndvdul processes ndcted n g. 4.he relevnt terms n the totl conductnce nclude the nterference terms etween the frst nd second order nelstc terms, ut lso nterference etween the frst nd thrd order elstc terms. In the Persson rtoff formlsm only the nterference terms etween the frst nd thrd order elstc terms pper. In our model, the nterference terms gve complcted formul, ut we pprecte tht for usul expermentl condtons (5 - o seprtons etween the tp nd the dsorte), (4) gves the domnnt term. or low seprton the nterctons cn modfy the reltve strength n the correcton terms n perturton theory nd the correcton terms cn chnge the sgn of η. n expermentl result n whch we oserve dp n the nelstc tunnelng spectr s otned y the rtz er Insttut group []. lso n the prevous estmtons, the contrutons to the tunnelng current of the modulton of the rrer y the vrtng molecules re neglected, we consder tht n the cse of nonresont electronc tunnelng, ths effect cn e domnnt nd cn expln the expermentl results.

7 Inelstc electron tunnelng through vrtonl modulted rrer n SM 53 cknowledgements: hs work enefted from the fnncl support of rtz er Insttute der Mx Plnck Gesellschft, erln, Germny. he uthor would lke to thnk Dr.. onrd, Prof. K. orn nd Dr.. Sptru for mny useful dscussons. References:. Stpe, Reze M nd o W 998 Scence Stpe, Reze M nd o W 999 Phys. Rev. Lett nng G, Grc N nd Rohrer 985 Phys. Rev Persson N J nd Demuth J E 986 Sold Stte omm Persson N J nd rtoff 987 Phys. Rev. Lett Gt M nd ntonewcz P R 993 Phys, Rev Sptru nd udu P 997 J. of Phys.: ond. Mtter Mngo N nd Mkosh K Phys. Rev. Lett Lorente N nd Persson M Phys. Rev. Lett udu P (unpulshed results). Psqul J I, Jckw J J, Song Z, Wess P S, onrd nd Rust P Phys. Rev. Lett. 86 5

8 54 P.udu gure cptons: g. Electrons tunnel from the left (L) electrode (tp) to the rght (R) electrode. he correspondng projected denstes of sttes re denoted () nd (). g. Dgrmmtc representtons of the mtrx elements contrutng to the resonnt tunnelng current n the frst order. he crcles denotes (s n [5 ]) the erm se of the tp ( sttes) nd the erm se of the dsorte sustrtes system ( sttes). g.3 Dgrmmtc representtons of the mtrx elements contrutng to the resonnt tunnelng current n the second order. g.4 Elstc thrd order processes wth one or three tunnelng electrons. rcled numers ndcte the tme orderng of the events.

9 Inelstc electron tunnelng through vrtonl modulted rrer n SM 55 () () -e g.

10 56 P.udu Elstc processe of order Inelstc processe of order g.

11 Inelstc electron tunnelng through vrtonl modulted rrer n SM 57 Elstc processes Inelstc processes g. 3

12 58 P.udu Elstc processes wthout phonons Elstc processes wth emsson nd resorpton of one phonon () (e) 3 () (f) (c) (g) (d) (h) g. 4