TRANSIENT EFFECTS IN ELECTRON TRANSPORT ACROSS A QUANTUM WIRE INTERACTING WITH A SUBSTRATE

Transcription

1 Aca Physicae Superficierum Vol. XII 2012 TRANSIENT EFFECTS IN ELECTRON TRANSPORT ACROSS A QUANTUM WIRE INTERACTING WITH A SUBSTRATE R. TARANKO Insiue of Physics, Marie Curie-Sk lodowska Universiy, Lublin, Poland address: ryszard.aranko@umcs.pl T. KWAPIŃSKI Insiue of Physics, Marie Curie-Sk lodowska Universiy, Lublin, Poland address: omasz.kwapinski@umcs.pl Absrac. Time-dependen elecron ranspor hrough a linear chain of quanum dos is sudied heoreically using he model igh-binding Hamilonian and he equaion of moion for he appropriae correlaion funcions. The wire is addiionally coupled wih he insulaor, semiconducor or meallic subsrae. The ransien effecs, which are induced by a sudden chemical poenial drop, are analyzed. We have found ha he ransien curren oscillaions occur also in he case of a wire coupled wih he surface bu he curren values decrease especially for he subsrae wih srongly localized elecrons. Moreover, depending on he subsrae and he wire-subsrae couplings he elecron occupaions of he wire sies oscillae afer he poenial drop and hey form a kind of charge wave along he wire. 1. Inroducion Recenly ime-dependen elecron ranspor in he low-dimensional quanum sysems has been he subjec of boh heoreical and experimenal invesigaions. Such sysems reveal ineresing physical possibiliies for elecron ranspor which are ofen quie differen in comparison wih he saionary case. For a single quanum do (QD) coupled wih wo elecron reservoirs and driven by exernal ac signal many new effecs have been found like he phoon assised unnelling, urnsile effecs or elecron pumping [1, 2, 3]. The pumping curren flowing be-

2 2 Aca Physicae Superficierum ween unbiased leads appears for a periodic change of one or more device-conrol parameers [4, 5]. In comparison wih a single QD more ineresing are complex sysems consising of a larger number of QDs, e.g. T-shape geomery of QDs, Aharonov-Bohm ring sysems or one-dimensional linear chain of coupled QDs (quanum wire, QW). In he saionary case he conducance of hese sysems is quanized and depends on he QD geomery as well as on he oal number of QD sies. For linear aomic wires i urns ou ha he conducance depends on wheher he number of sies is even or odd (even-odd conducance oscillaions) [6] bu also he oscillaions wih larger periods may occur [7]. One-dimensional aomic wires can be fabricaed on vicinal surfaces and invesigaed by means of scanning unnelling microscopes [8]. These srucures are very sable and he measuremen can be repeaed many imes, however, i is difficul o conrol/change all wire parameers and he wire-subsrae couplings. The bes mehod for invesigaions of linear chains seems o be a row of coupled QDs (series of QDs) where all QD parameers are conrolled by he addiional exernal elecrodes. For such real wires elecrons flow no only beween QD sies bu also hey can inerac wih he subsrae. The wire-subsrae couplings are especially imporan for conducing (meallic or semiconducor) surfaces because he conducance depends on wheher or no he surface elecrons are localized or delocalized [9]. To adop such nanosysems in pracical devices, i is essenial o undersand he behaviour of ransien currens appearing, for example, in response o an abrup drop of he source-drain volage applied across he sysem. Especially, bearing in mind he emerging field of he quanum compuing, i would be desirable o perform he simulaions of ranspor properies also in a ransien imescale when some parameers of he QD sysem are suddenly changed. In his paper we sudy heoreically he ranspor properies of a linear chain of quanum dos beween he lef and righ elecron reservoirs and addiionally coupled wih he subsrae focusing on he ransien effecs. The spaial separaion of he QDs allows us o consider, beyond an insulaor surface, wo model subsraes: (i) each QD is coupled individually o a subsrae wih localized elecrons and (ii) all QDs are coupled wih a single surface elecrode wih delocalized elecrons. Our goal is o invesigae he role of he elecron localizaion in he subsrae on he he ransien curren and he elecron occupaion probabiliies of he wire sies. The paper can be reaed as generalizaion of our earlier works concerning he saionary ranspor hrough a wire-subsrae sysem and he ransien effecs in he QD sysems [9, 10]. I is known ha he abrup change of he sysem parameer generaes he ransien curren wih dumped oscillaions [11]. These oscillaions of he curren or he elecron occupaion provide much useful informaion abou he sysem under consideraion. For example, a sudden change of he chemical poenials leads o he curren oscillaions he period of hese

3 Insrucions for Auhors 3 Fig. 1: The skech of he considered sysems of N sie quanum wire (series of QDs) coupled wih he lef and righ leads. In he upper panel he QD wire is coupled wih a common subsrae elecrode S and he boom panel depics he case of each wire QD coupled wih he individual subsrae elecrode, S i. oscillaions is deermined by relaive disances beween he appropriae elecron energy levels in he QD sysem and boh chemical poenials. Calculaions of he ime-dependen ranspor properies of nanosysems are usually performed using he Keldysh formalism of he non-equilibrium Green s funcions or wihin he mehods based on he Liouville equaion. In our sudies we apply he equaion of moion mehod for he appropriae correlaion funcions and obain he currens flowing from he lef, righ and surface elecrodes as well as beween he QD sies. Addiionally, he elecron occupaions a all QDs are also calculaed and discussed in he paper. The paper is organized as follows. In Sec. 2 he model Hamilonian and he heoreical descripion of a quanum wire on he subsrae are presened. In Sec. 3 he numerical resuls are presened and discussed. Sec. 4 is devoed o conclusions. 2. Hamilonian and formalism We consider wo QDs sysems skeched in Fig. 1 which consis of a linear quanum wire coupled wih wo leads and also ineracing wih he subsrae surface. The firs case, A, corresponds o he siuaion when he subsrae is meallic and he QDs elecrons unnel o a delocalized orbial or in oher words, he subsrae is a common elecrode for all QDs. The second case, B, describes he subsrae elecrode wih raher a shor mean free pah (like in a semi-conducor or in an insulaor surface) and corresponds o a model of individual N elecrodes coupled wih a corresponding QD. The oal Hamilonian wrien in he sandard second-quanized noaion akes he following form: H = H 0 + H QDs + H in, where H 0 = ε kα n kα + ε kα n kα (1) k α=l,r k α=s 1,...,S N

4 4 Aca Physicae Superficierum describes he elecrons in all elecrodes (L, R and S - lef, righ and subsrae, respecively), H QDs = N i=1 N 1 ε i n i + V i,i+1 c + i c i+1 + h.c. (2) i=1 and H in = Hin L + HR in + HS in where HL in = k V kl,1c + kl c 1 + h.c., H in R = k V kr,n c + kr c N +h.c. and Hin S = N k j=1 V ks j,jc + ks j c j +h.c.. Here he operaors c kα (c + kα ) and c i(c + i ) denoe he annihilaion (creaion) ones for he elecrons in he α-elecrode wih he wave vecor k and for he QD elecrons, respecively. The elemens V kα,j and V i,j are responsible for he elecron ransfer beween he α-elecrode and j h QD and beween he i-h and j-h QDs, respecively. The energy levels of he QDs and elecrodes are denoed by ε i and ε kα, respecively. To calculae he ime dependence of he QDs occupaion probabiliies, n j (), and he curren flowing in he sysem in response o he abrup change of he sysem parameers we use he equaion of moion for he appropriae correlaion funcions. As he models considered here do no include many-body erms hen he required quaniies will be accuraely calculaed. Inroducing he funcions: f n,m () = c + n ()c m (), g j,kα () = c + j ()c kα(0) and G n,m α () = ( k V kα,n exp i ) 0 d ε kα ( ) g m,kα () he elecron curren flowing from he α-h elecrode can be obained from he ime evoluion of he occupaion number operaor of his elecrode (e.g. [2, 10]) and wrien as follows: ( J α () = 2Im G j,j α () i Γ ) α 2 n j(), (3) where Γ α = 2π k V kα,j 2 δ(ε ε kα ) and we assume ha all pars of he sysem were swiched on a = 0. Here... sands for he quanum-saisical average, n j () denoes he j-h QD occupaion probabiliy (he index j idenifies he QD coupled wih he α-h lead) and e = = 1 unis were used. To calculae he curren we should know he correlaion funcions g j,kα and he occupancies of all QDs, n j (). We find hese funcions solving heir equaions of moion. For he wire of N QDs we have o solve he sysem of N(N+1)/2+3N nk coupled differenial equaions for n j (), g j,kα and f n,m (). Here nk denoes he number of k vecors aken in he calculaions of he corresponding summaion over hese vecors (see he definiion of G n,m α ()) and usually i exends from 200 for small source-drain volages o over 2000 for a larger bias volage window. For example, for he case of a QW on he meallic subsrae we have (for 1 < j < N): { } n j() = 2Im G j,j S () + i(v j,j+1f j,j+1 () V j 1,j f j 1,j ()) i 2 N n=1 Γ j,n S f j,n() (4)

5 Insrucions for Auhors 5 where Γ j,n S = 2π k V ks,jv n,ks δ(ε ε ks ) and he funcions g j,ks () saisfy he following equaion: g j,ks() = iε j g j,ks () + i(v j 1,j g j 1,kS () + V j,j+1 g j+1,ks ()) (5) ( ) + iv ks,j exp i d 1 ε ks ( 1 ) n ks (0) 1 N Γ j,n S 2 g n,ks() 0 Equaions 4 and 5 ogeher wih hose for f n,m () (no shown here) form he exac se of coupled equaions which should be solved for given realizaion of he iniial condiions. 3. Numerical resuls and discussion In he calculaions we se e = = k B = 1 and Γ L = Γ R = Γ = 1 is assumed as he energy uni. The curren and ime are expressed in he unis of 2eΓ/ and /Γ, respecively. The chemical poenials of he lef, righ and subsrae elecrodes aken a vanishing bias volage serve as he reference energy poin, µ α = 0. We prepare he sysem in he equilibrium sae swiching on all ineracions beween each par of he sysem a = 0. Nex, waiing unil he sysem achieves is equilibrium sae, say a = 0 = 30, he source-drain bias volage is abruply applied across he QD wire. This leads o a sudden change of he QDs energy levels and he chemical poenial of he righ elecrode, µ R = V SD, ε i = V SD /2. As a resul, he coheren oscillaions of he ransien curren are observed. In our sudies we concenrae on he role of he wire-subsrae ineracions and show he numerical resuls only for a wire consising of N = 5 QD sies. The generalizaion on he oher wire lenghs is obvious (he main conclusions of he paper are valid also for oher N). The resuling ransien curren J L () is shown in Fig. 2 for differen couplings beween he wire and he subsrae. The lef panel corresponds o he case of a common subsrae elecrode (delocalized elecrons in he subsrae) and he righ one, panel B, for he case of individual elecrodes for all QD sies elecrons are localized (cf. Fig. 1). Before he volage drop all currens, J L, J R and J S, do no flow hrough he sysem as in his case µ L = µ R = µ S = 0. The ransien curren appears for > 30 and srongly depends on he subsrae parameers. For an insulaing subsrae, Γ S = 0 he ransien curren is characerized by wo kinds of oscillaions which reflec he molecular srucure of he QD wire (high-frequency oscillaions) and he source-drain volage drop (low-frequency oscillaions) [10, 11]. Ineresingly, in he presence of he subsrae hese oscillaions sill remain (wih he same periods). Noe, however, ha he oscillaion ampliudes decrease wih he wire-surface coupling which is well visible in he righ panel for Γ S = 1. Moreover, as one can see, he larges curren flows from he lef elecrode in he case of an insulaor subsrae (Γ S = 0). For nonzero Γ S he value of he ransien n=1

6 6 Aca Physicae Superficierum J L () A B Γ S =0 Γ Si = Fig. 2: The ransien curren flowing from he lef elecrode, J L (), hrough he QD wire consising of N = 5 sies as a funcion of ime for differen wire-subsrae couplings Γ S = 0, 0.2, 0.5 and 1.0, respecively. The lef A (righ, B) panel corresponds o he case of he common subsrae elecrode (separae elecrodes, Γ Si = Γ S, i = 1 5), see Fig. 1 upper panel (boom panel). The oher parameers are: V = 4, 0 = 30, Γ L = Γ R = 1. For < 0 : ε i = 0, µ L = µ R = µ S = 0 and for 0 : ε i = 40, µ L = µ S = 0, V SD = µ R = 80 curren is lower because in his case he subsrae curren appears, J S. Thus elecrons flow hrough he wire from he lef elecrode and from he subsrae simulaneously (he wire capaciy for elecrons is spli ono charges from he L and S elecrodes). In consequence, he lef curren decreases wih Γ S > 0. I is also ineresing o analyze he role of he elecron localizaion in he subsrae. For srongly localized elecrons, panel B, he ransien curren decreases wih he wire-subsrae coupling much faser han in he case of delocalized elecrons, panel A. In he presence of separae subsrae elecrodes here are wo pahs for elecrons flowing o he righ elecrode: form he lef one (hrough he wire) and from he subsrae (also hrough he wire). The laer way depends on he wire-subsrae coupling and for he same Γ parameers (Γ L = Γ Si = 1) he currens flowing from he subsrae dominae. In his case he lef curren rapidly decreases wih Γ Si. For he case A (common subsrae elecrode) an elecron can unnel e.g. from he lef elecrode o he firs QD sie, nex o he subsrae and afer some ime i can appear wih he same probabiliy a every oher wire sie. During his process elecrons can sill flow from he lef elecrode and he lef curren does no decrease very fas wih he wire-subsrae coupling. In order o sudy furher he role of he wire-subsrae coupling in Fig. 3 we show he occupaions of all QDs, n i (), for a wire consising on N = 5 sies. The upper (boom) panels correspond o he case of he common subsrae elecrode (individual elecrodes) and he lef (righ) panels represen srong (weak) wiresubsrae coupling. Before he source-drain volage drop he occupaions of all QD sies are consan and equal o 0.5 (symmerical model). For > 30 hese

7 Insrucions for Auhors A1 Γ S =1 A2 Γ S =0.1 n 1-5 () n 1-5 () B1 Γ S =1 n 1 n 2 n 3 n 4 n 5 B2 Γ S = Fig. 3: The probabiliy occupaions a each QD sie for N = 5 and for differen wiresubsrae couplings Γ S = 1 (lef panels) and 0.1 (righ panels). The upper (boom) panels correspond o he case of he common subsrae elecrode (separae elecrodes, Γ Si = Γ S). The oher parameers are he same as in Fig. 2. occupaions increase because he QW energy levels are shifed below he surface chemical poenial (µ S = 0, ε i = 40). As one can see for very weak wire-subsrae coupling, Γ S = 0.1, he asympoic values of n i () do no differ from each oher, panels B2 and A2. A more ineresing case is observed for srong Γ S here are no common feaures a boh panels A1 and B1. If elecrons in he subsrae are delocalized (one surface elecrode) han he middle wire sie is he mos occupied one (in comparison wih n 1 or n 5 occupaions, panel A1). On he oher hand, for srongly localized elecrons, panel B1, he occupaion of he las wire sie is maximal and he values of n i slighly differ from each oher (he wire is almos fully occupied). Moreover, he sysem achieves is equilibrium sae very fas which is refleced on he occupaion curves which almos do no oscillae in ime, panel B1. In his case elecrons which leave he wire and unnel o he subsrae elecrode come back exacly o he same sie and hey do no disurb oher QD occupaions. The differences in he QD charges are relaed o he charge densiy waves which appear e.g. in aomic chains for a specific relaion beween he on-sie elecron energies and he leads chemical poenials [7, 12]. Noe ha he sequence of he QD occupaions for he cases shown in Fig. 3 changes wih he wire-surface coupling.

8 8 Aca Physicae Superficierum 4. Conclusions Using he model igh-binding Hamilonian and he equaion of moion for he appropriae correlaion funcions we have obained he ransien currens flowing hrough a linear series of quanum dos (quanum wire) and he elecron occupaion probabiliies a all QW sies. We have sudied he role of he subsrae elecrode underneah he wire which has been considered as an insulaor, a semiconducor or a meallic one. A sudden change of he chemical poenials (volage drop) has induced he ransien curren oscillaions in he sysem for an insulaor subsrae bu hese oscillaions have survived also for he wire coupled wih he semiconducor or meallic surface. However, he ransien curren values have decreased very fas wih he wire-subsrae coupling especially for he subsrae wih srongly localized elecrons. We have also found ha depending on he subsrae and he wire-subsrae ineracion he elecron occupaions of he wire sies oscillae in ime and hey form a kind of charge wave along he wire. Acknowledgemens The work was suppored by he Minisry of Science and Higher Educaion Gran No. N N References [1] L. P. Kouwenhowen, P. L. McEuen, Single elecron ranspor hrough a quanum do, Nanoechnology, eds. G. Timp, (Springer, New York, 1998). [2] R. Taranko, T. Kwapiński, and E. Taranko, Phys. Rev. B 69 (2004) [3] S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406 (2005) 379 [4] T. Kwapiński, R. Taranko, J. Phys.:Condens.Ma. 23 (2011) [5] L. Foa Torres, Phys. Rev. B 72 (2005) [6] R. H. M. Smi, C. Unied, G. Rubio-Bollinger, R. C. Segers, and J. M. van Ruienbeek, Phys. Rev. Le. 91 (2003) [7] T. Kwapiński, J. Phys.:Condens.Ma. 17 (2005) 5849 [8] M. Krawiec, T. Kwapiński and M. Ja lochowski, Phys. Rev. B 73 (2006) [9] T. Kwapiński, S. Kohler and P. Hänggi, Eur. Phys. J. B 78 (2010) 75 [10] E. Taranko, M. Wierel, R. Taranko, J.App.Phys. 111 (2012) [11] F.M. Souza, Phys. Rev. B 76 (2007) [12] T. Kwapiński, J. Phys.:Condens.Ma. 18 (2006) 7313