physics of artificial nano-structures on surfaces

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Physics of artificial nano-structures on surfaces M. Tsukada a, *, N. Kobayashi b , M. Brandbyge c , S. Nakanishi a a Department of Physics, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan b Surface and Interface Laboratory, RIKEN (The Institute of Physical and Chemical Research), 2-1 Hirosawa, Wako-shi, Saitama, 351-0198, Japan c Mikroelektronik Centret (MIC), Technical University of Denmark (DTU), Building 345 east, DK2800, Lyngby, Denmark Abstract Quantum electron transport through nano-structures such as metal atomic wires or molecular bridges is investigated with various theoretical approaches. The dierence of the quantization feature between Na and Al atom wires is explained based on the eigenchannel analyses combined with the recursion-transfer matrix calculation. The eigenchannels are calculated self-consistently for Au atom wires at finite bias voltage and the nonlinear conductance is explored in relation to the oset energies of d band channels. As for molecular bridges, we find that a remarkable metalization is caused, if the coupling of the molecule with the metal electrode is enhanced. Internal current distribution within the molecular networks is discussed and exotic properties of the quantum transport is found. In particular, a strong induced loop current is revealed circulating the ring part of the molecule. The direction of the loop current is switched sharply when the electron incident energy sweeps a degenerate molecular level. 7 2000 Published by Elsevier Science Ltd. All rights reserved. 1. Introduction Recent development of scanning probe microscopy (SPM) opened a new research 0079-6816/00/$ - see front matter 7 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S0079-6816(00)00014-9 Progress in Surface Science 64 (2000) 139–155 www.elsevier.com/locate/progsurf * Corresponding author. Tel.: +81-3-3812-2111 extn 4223; fax: +81-3-3814-9717. E-mail address: [email protected] (M. Tsukada).

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Page 1: Physics of artificial nano-structures on surfaces

Physics of arti®cial nano-structures on surfaces

M. Tsukadaa,*, N. Kobayashib, M. Brandbygec, S. Nakanishia

aDepartment of Physics, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,

Tokyo, 113-0033, JapanbSurface and Interface Laboratory, RIKEN (The Institute of Physical and Chemical Research), 2-1

Hirosawa, Wako-shi, Saitama, 351-0198, JapancMikroelektronik Centret (MIC), Technical University of Denmark (DTU), Building 345 east, DK2800,

Lyngby, Denmark

Abstract

Quantum electron transport through nano-structures such as metal atomic wires ormolecular bridges is investigated with various theoretical approaches. The di�erence of thequantization feature between Na and Al atom wires is explained based on the eigenchannel

analyses combined with the recursion-transfer matrix calculation. The eigenchannels arecalculated self-consistently for Au atom wires at ®nite bias voltage and the nonlinearconductance is explored in relation to the o�set energies of d band channels. As for

molecular bridges, we ®nd that a remarkable metalization is caused, if the coupling of themolecule with the metal electrode is enhanced. Internal current distribution within themolecular networks is discussed and exotic properties of the quantum transport is found. In

particular, a strong induced loop current is revealed circulating the ring part of themolecule. The direction of the loop current is switched sharply when the electron incidentenergy sweeps a degenerate molecular level. 7 2000 Published by Elsevier Science Ltd. Allrights reserved.

1. Introduction

Recent development of scanning probe microscopy (SPM) opened a new research

0079-6816/00/$ - see front matter 7 2000 Published by Elsevier Science Ltd. All rights reserved.

PII: S0079 -6816 (00)00014 -9

Progress in Surface Science 64 (2000) 139±155

www.elsevier.com/locate/progsurf

* Corresponding author. Tel.: +81-3-3812-2111 extn 4223; fax: +81-3-3814-9717.

E-mail address: [email protected] (M. Tsukada).

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®eld of the science of arti®cial nano-structures, which will be an important basisof high technology in the 21st century. By means of SPM, one can observe theatomic details of the individual nano-structures, and fabricate them in a controlledway. Furthermore, various electronic, nano-mechanical, magnetic, and chemicalproperties of the nano-structures can be in principle investigated by the use ofSPM. For example the scanning tunneling microscopy (STM) is most advantageousto investigate the quantum transport through nano-structures.

Among a variety of arti®cial nano-structures, the bridge systems sandwichedbetween the tip of STM and sample surface are most attractive since the contactof the nano-structures with the electrode pad is established in a natural way. Infact, there have been many of the experimental studies for the metallic wire bridgesystems (atom bridge hereafter) and remarkable properties, such as the quantumstaircase of the conductance have been reported [1,33]. On the other hand, themolecular systems are promising to a�ord ideal nano-scale electronic circuitsystems, which might realize various new functions of the quantum devices.Though at the moment various technical problems such as the way of socketingthe molecule to conductive electrodes are to be developed, theoretical explorationof the novel properties as quantum devices would be important.

Therefore, in the present article, we study the quantum transport features ofmetallic atom bridges and molecular bridges sandwiched between the metallicelectrodes. As for the atom bridges, we clarify the details of eigenchannels [2,29]and their relation with the atomic/molecular orbitals of the bridge part. From thisview point, how the quantum transport properties depend on the atom kind andthe microscopic shape of the atom wires might be clari®ed. Moreover, the natureof the quantization of the conductance and detailed transmission spectra re¯ectingresonant scattering inside the bridge part will be discussed.

As for the molecular bridge, the important issue is to achieve the metallicconduction through the bridge part. This might be realized by the resonanttunneling, i.e., tuning the level of HOMO, or LUMO close to the Fermi level. Butthis may be possible only for a rather special case. The other method to achievethe high transmission is to use the metallization of the molecule due to the closecoupling to metal electrodes. We will discuss the possibility of this in the presentarticle. One of the remarkable ®ndings we obtained in this article is a strong loopcurrent inside the molecular part induced by the source-drain current. Such a loopcurrent emerges when the electron energy approaches close to the degeneratelevels of the molecules and changes its direction and distribution by the relativelocation of the level and the electron energy. By the use of this remarkablefeature, one might control the phase of the induced quantum state of themolecular part, which might be promising for realizing various quantum functionsof the molecular bridges.

2. Al and Na atomic wires

First we present ®rst-principles calculations of the transmission channels of

M. Tsukada et al. / Progress in Surface Science 64 (2000) 139±155140

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atom wires for two materials; one is an s valence metal, Na, and the other is an s-p valence metal, Al [3]. Detailed features of the transmission channels areelucidated for several atomic con®gurations, and a remarkable di�erence of thebehaviors is found between Na and Al bridges. We will discuss how the behaviorsof the channels are in¯uenced by geometrical changes for the atom wires, and whyquantized conductance is observed more clearly for a Na atom wire.

The channels are investigated by the ®rst-principles calculations combined witheigenchannel decomposition [4]. The electronic states are calculated using the ®rst-principles recursion-transfer matrix method [5,32,31], and the conductance [6] andthe local density of states (LDOS) [7] are decomposed into the eigenchannels[2,29]. The model consists of three Na or Al atoms between two semi-in®nitejellium electrodes as shown in Fig. 1 (upper inset). The atom±atom and the atom±jellium bond lengths are taken to be 7.0 (5.4) and 3.0 (2.6) bohrs, and the middleatom is displaced in the x direction by d, keeping the same bond length.

Fig. 1 shows the conductance and the channel transmissions of the wires as afunction of the displacement d. The conductance, the sum of individual channeltransmissions is expressed in the quantized unit, 2e 2/h. For the Na wire, theconduction is contributed by a single channel which is almost fully open. Thevalue of the conductance is close to 2e 2/h, and does not change with thedisplacement, d. On the other hand, for the Al wire, three channels contribute tothe conduction; one almost fully open and the other two half open. With increaseof d, each channel transmission decreases, and thus the conductance alsodecreases. The detailed mechanism of causing such behaviors is described in Ref.[3].

Fig. 2 shows the channel resolved LDOS distributions on the plane normal to

Fig. 1. Conductance and channel transmission of NA (upper) and Al (lower) atom wires as a function

of displacement of d. Upper inset: schematic representation of the model used. Three Na or Al atoms

are sandwiched between jellium electrodes.

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the wire axis for the straight Na and Al wires. A single channel of the Na wirehas an s orbital character. On the other hand, the ®rst channel of the Al wire hasa character of s and pz orbitals, while the second and the third channel, which aredegenerate, have a character of px or py orbitals. For the bent Al wire, threechannels contribute to the conduction as well as the straight case, by the sÿ pzcharacter and the px character are mixed, and only the py character is similar tothe straight case.

Fig. 3 shows the channel resolved DOS of the center atom and the channeltransmissions. The position of the onset energy of each channel relative to theFermi energy is important to understand the behavior of the channel. For the Nawire, the Fermi energy cuts just after the onset of the ®rst channel. Therefore, theconductance is close to the quantized value, and does not change even for thebent wire. On the other hand, for the Al wire, the Fermi energy cuts the onset ofthe second and the third channels. This is the reason why the channeltransmission does not take the quantized value. We found that the conductance ofthe Al wire is sensitive to the geometrical change, and is reduced by the bending.

Experimentally, quantization of the conductance of Na is clearly observed [8].In actual observation of the conductance during the elongation process, theatomical con®guration is changing. For the Na atom wire, the conductance isinsensitive to the geometrical changes, as has been seen above. This fact impliesthat the observation of the conductance quantization is easier for the Na wire.

The present results indicate the decrease of the bending angle causes asigni®cant increase of the conductance for the Al wire. This is consistent with thepositive slope of the conductance experimentally observed [9].

Fig. 2. Channel resolved LDOS of Na (upper) and Al (lower) atom wire on the plane including the

center atom and normal to the wire-axis. The contour plots are in units of 2 � 10ÿ2 hartreeÿ1 bohrÿ3.The broken lines correspond to 1 � 10ÿ3 hartreeÿ1 bohrÿ3. The solid circles indicate the atomic

positions. An area with a side of 20 bohrs is displayed.

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3. Au atomic wires

Fig. 4 shows the energy bands for Pb, Al, Nb and Au monoatomic wiresobtained using the tight binding model parametrized by ®ts to ab initio bandstructures [12,32]. In experiments using superconducting electrodes transmissionsof individual conductance channels have been extracted for contacts down to thesingle atom [10]. An agreement of the number of the bands with the number ofchannels at the Fermi energy is found. For Au, the top of the d band touches theFermi energy from the lower energy side. The transport through the d orbitals istherefore expected for a ®nite bias. In this section, we focus on the Au atombridges and discuss the details of the transmission properties.

Eigenchannel properties of the Au atom wires are calculated at ®nite biasvoltage by a non-orthogonal tight-binding model with the non-equilibrium Greensfunction approach [13]. We assume local charge neutrality and include an on-sitepotential at each atom which is adjusted self-consistently in order to achieve theneutrality [11]. We apply the local neutrality also in the ®nite bias which enablesus to calculate the voltage drop through the contact.

In Fig. 5, we consider a 3-atom long Au wire attached to layers of 4 and 9atoms in both ends which again are connected to the (100) faces of two semi-in®nite gold electrodes. All interatomic distances are assumed as the same as inthe bulk. We present the calculated self-consistent on-site potential (layeraveraged) and the change in this potential with applied voltage (voltage drop).

In Fig. 6, we present the eigenchannel transmissions for the self-consistentcalculations of the 3-atom Au chain. For a ®nite bias an energy window (vertical

Fig. 3. Channel resolved DOS of the center atom and channel transmissions for the Na (upper) and Al

(lower) wires as a function of energy measured from the Fermi level. The left panels correspond to the

straight wires, and the right to the displacement of d=2.

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Fig. 4. Band structures of in®nite Pb, Al, Nb and Au monatomic wires.

Fig. 5. The layer averaged atomic potential shifts (hfiiLayer) for di�erent bias voltages. Layer ÿ1correspond to the ®rst (100) surface layer of the semi-in®nite electrode. The voltage drop

(hfiiLayer(V)ÿhfiiLayer(0)) is shown to the right. The largest voltage drop is seen in the entrance of the

contact between layer 2 and 3.

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lines) of width eV contributes to the conductance. In the lower panels of Fig. 6,we show the projected density of states (PDOS) onto the atomic orbitals of themiddle atom. The PDOS is resolved into its eigenchannel components [13,14].

The transmissions presented in Fig. 6 change with increased bias: the localcharge neutrality forces the (almost degenerate) dzx=dyz states down in energy anddelays the onset of the derived eigenchannels somewhat. We observe that thedzx=dyz-derived channels ®rst begin to contribute inside the voltage window from avoltage bias of about 1.5 V.

The transmission of the s=pz=dz 2 �lz � 0� channel within the voltage window inlowered with increasing bias. The partially open dzx=dyz transmission channels aswell as the lowering of the s=pz=dz 2 transmission in the high bias regime lead to anincreased electron-ion momentum transfer (electron wind) where the currentcarrying electrons are backscattered. The backscattering takes place mainlybetween layers 2 and 3 according to the voltage drop in Fig. 5. An increasedelectron wind could be part of the reason for the apparent decrease in stability ofthe one-atom gold contacts found in Ref. [15].

We have also considered shorter and longer chains [13]. In contrast to the resultfor the 3-atom long chain, we ®nd for a 2-atom long chain that the px=py orbitalsplay a more pronounced role in the 2nd and 3rd eigenchannels and the modelpredict a noticeable contribution from the 2nd and 3rd eigenchannels of about0.05 around EF: For a 6-atom long chain we ®nd that the dzx=dyz contributions lie

Fig. 6. Upper panel: the eigenchannel transmissions of the 3-atom Au chain for 0 (left) and 2.0 (right)

volts bias. An almost fully transmitting single channel (solid line) is seen inside the voltage window and

two near degenerate channels are seen just below (dotted lines). For the higher bias voltages these

degenerate channels begin to enter the voltage window. Lower panel: the corresponding eigenchannel

density of states projected onto the atomic orbitals of the middle atom (denoted by 3 in Fig. 5).

M. Tsukada et al. / Progress in Surface Science 64 (2000) 139±155 145

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closer to EF: For long chains the dzx=dyz derived electronic states inside the chainwill approach the 1D band states (Fig. 4) with the sharp onset in thecorresponding transmission channels close to Fermi energy.

The large non-linear conductance found in Ref. [16] is not seen in the results ofthe present calculations. We ®nd an almost linear behavior up to 2 Vcorresponding to a conductance close to G0 � 2e2=h: A less e�cient screening(assumed to be complete in the present calculations of Au wires) and in the limitof long chains one could imagine a signi®cant non-linearity due to the onset of dchannels. However, for the shorter chains other explanations of the experimentshave to be found.

4. Molecular bridges

Quantum transport of electrons through molecules sandwiched between metallicelectrodes is one of very attractive problems, which would be the bases ofdesigning arti®cial nano-structures. In this section we will investigate how thequantum transport is realized and what novel properties can be expected. Inparticular we will focus on the topics such as metalization of the molecules,internal current distribution, induced large loop current, and magnetic control ofthe transmission. Recently, there have been a number of reports [17±19], in whichelectronic transport is measured through molecular systems as SAM membrane,though most of the works concentrate on the total conductance [20±25]. On theother hand, the current distribution inside the molecule has not been investigatedso far, because it was considered to be premature to discuss the quantum devicescomposed by individual molecules. However, as we will show below so manyremarkable features can be expected for the quantum phenomena related with theinternal current distributions.

In the present article we introduce a simple molecular bridge system sandwichedbetween metallic electrodes as shown in Fig. 7 and investigate electrontransmission and eigenchannel features by the quantum scattering theory based ona tight binding model [26,27]. As a case study the molecular bridge system shown

Fig. 7. The model of molecular bridge. The hopping integral indicated in solid line is set to be t=ÿ1and that indicated in dotted line is set to be t '.

M. Tsukada et al. / Progress in Surface Science 64 (2000) 139±155146

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in Fig. 7 is studied below. The electrode is a cubic lattice with (001) surface and10 � 10 supercell structures are assumed for the calculation. Inside electrodes anda molecule, the hopping integral is set to be a common value t=ÿ1, the hoppingintegral connecting electrodes and a molecule is set to be t '. The site energy is alsotaken commonly over the whole system. The eigenchannel analysis is applied toobtain the total transmission and the internal current distribution [2,4,6,29].

The total conductances for t 0 � 0:5t and t 0 � t are shown in Fig. 8. There areup to two open eigenchannels out of 100 channels in the electrode. The criterionfor the open channel is that the transmission probability for this channel exceedsthe threshold 10ÿ12.

In Fig. 8 we ®nd that the positions of the peaks of the conductance correspondto the eigen-energies of the isolated molecule. The width of the peak is determinedby the value of the parameter t '; if the absolute value of t ' is increased, the peaksget wider and some merge each other. For the larger values of vt 'v, signi®cantvalues of the conductance are realized for general values of the electron incidentenergy E. To see this, the total conductance G is represented in gray scale in Fig. 9in the 2D space of the electron energy E and electrode±molecule coupling t '. It isclearly shown that the conductance peak appears at the molecular level and hencethe resonant transmission takes place at the molecular level, when vt 'v is small. Onthe other hand, for the increased values of vt 'v, the values of the transmissionspread out even in the regions between molecular levels, so the conductance haslarge values over the wide ration of the electron energy. This feature has somerelevance with the appearance of the metal induced gap states in the gap of the

Fig. 8. The conductance of the molecular bridge shown in Fig. 7. The conductance G is shown in the

unit of G0 � 2e 2=h: The solid line shows the case t 0 � t and the dotted line shows the case t 0 � 0:5trespectively. The arrows indicate the positions of molecular energy levels and the numbers above them

denote the degeneracy of the levels.

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semiconductors attached to metals. Namely this is a manifestation of themetalization phenomena of the molecule.

The current distribution inside the molecule at an energy near the conductancepeak, and at an energy far from the conductance peak are shown in Figs. 10 and11, respectively. In the case of these ®gures, the electrode±molecule coupling t ' isset of 0.5t. The most noteworthy feature in Fig. 10 is that a loop current isinduced in the ring part of the molecule, for each eigenchannel as well as the totalcurrent. The magnitude of the loop current is several times larger than thatentering the molecule and leaving the molecule. The loop current circulates at theboundary of the whole molecule.

Such a signi®cant induced loop current is found at the energies near thedegenerate molecular levels. Fig. 12 illustrates the variation of the inducedmagnetic moment M� �

Cj�x� � x dl (C: molecular bonds) due to the loop current

Fig. 9. The conductance G of the molecular bridge shown in Fig. 7. G is illustrated in the 2D space of

the electron energy E and the molecule±electrode coupling t '. The arrows indicate the positions of the

molecular levels and the numbers beside them describe the degeneracy.

Fig. 10. The internal current distribution at the energy E=ÿ1.73 near which the degenerate molecular

levels exist. The current decomposed into eigenchannels and the total current is shown.

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in the 2D map of the electron energy E and the coupling strength t '. The largeloop current appears at the region close to the degenerate levels of the molecule.Another netable feature is that the direction of the loop current changes its signsteeply when energy sweeps from below to above the degenerate level.

The origin of the induced loop current is discussed below. The wavefunction ofthe channel can be expanded inside the molecular region by the molecular orbitals(MO) ffmg: cn � Smamfmn: Here, the MO is taken to be real and n indicates thesite. The current from the site n to m is given by

jmn �X

Im�c�mHmncn ��Xvi, vi

tmnfvimfvjnj avikavj j sin�yvj ÿ yvi �: �1�

The phase ym of the coe�cient am of the MO fm is de®ned by am �j am j eiym :The product of the absolute values of the coe�cient controls the upper bound

of the current, but the phase di�erence determines the magnitude and the

Fig. 11. The internal current distribution at the energy E=ÿ2.00 which is located far from the

molecular levels. The current decomposed into eigenchannels and the total current is shown.

Fig. 12. The magnitude of magnetic moment shown in 2D space of E and t '.

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direction of the current. From Eq. (1), it is clear that if there is only one MOwhich mainly contributes to the resonant transmission, the magnitude of thecurrent cannot be large, since the large current ¯ows always with the help of theadmixture of the other (nearly) resonant MO states. When the resonant currentdescribed as above takes a large value, and exists only inside the molecule, it mustcirculate around the ring part of the molecule due to the continuity of the current.And this is what we con®rmed by the numerical calculation.

The direction of the resonantly induced loop current changes in a very sharpway when the electron energy is increased across the degenerate level. This isbecause the phase di�erence between the avs changes its sign. The mechanism ofthe behavior is explained below.

For the sake of simplicity, we assume that the molecule is connected to the one-dimensional source (drain) electrode at the site a�b�: Let us partition the systemand the Hamiltonian as shown in Fig. 13. The coe�cients av for the resonantMOs v=1,2 are

a1 � �V�LC ÿ l2�f1a

Eÿ E0 ÿ U11, a2 � �V

�LC ÿ l2�f2a

Eÿ E0 ÿ U22, �2�

U11 � lj VLC j2

VL��f1a�2 � �f1b�2�, U22 � l

j VLC j2VL

��f2a�2 � �f2b�2�: �3�

Here, we de®ned the phase factor l=exp(ik ) (k is a wave number). U11 and U22

are the modi®ed self-energy which is generated by the connection to the electrode.VLC is the element of the Hamiltonian which connects the electrode and themolecule �VLC � tL�: VL is the hopping integral inside the electrodes. fva(b)

indicates the value of the vth MO at the site a�b�:The phase di�erence of the coe�cients dy 0 arg(a2)ÿarg(a1) comes from the

denominator of the av in Eq. (2). The schematic ®gure of the denominators of avsare shown in Fig. 14. The phase di�erence dy increased when the electron energy

Fig. 13. The way of partitioning the system. Each symbol shows the matrix element of the Hamiltonian

(VR=VL, VRC=VLC, HRk=VLk).

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E approaches to the degenerate energy level E0, reaches its maximum (less than p/2), rapidly decreases its value toward 0 (at E � E0�, changes its sign, rapidlydecreases its value, achieves its minimum (more than ÿp/2), and graduallyincreases towards 0 with the E departs from E0.

5. Internal current of benzene ring array

In the next model we adopt, the source and drain part is connected by only a

Fig. 14. The origin of the phase di�erence between the coe�cients a1 and a2.

Fig. 15. The model of a simple molecular bridge. The ladder-type molecule is attached to the metallic

electrode via long one-dimensional wire.

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single atom, a and b, respectively. In this case we treat the electrodes as a simpleheat bath and the electron reservoir. The molecular part is a column of thebenzene rings and the site a and b are located on the same bottom ring as shownin Fig. 15.

The behavior of transmission probability T�E � and the internal current for thetypical system is shown in Fig. 16. At some energies, the transmission probabilityT�E � and all the internal current drop to zero. Some of them correspond to theenergy levels of the molecule but the others do not correspond to the energylevels. In the following the condition for T�E � � 0 is clari®ed.

The general condition for T�E � � 0 is that one of the following condition issatis®ed [28]. (A) The interference condition is satis®ed. The electron injected tothe site a cannot exit the molecule from the site b due to the electron interference.(B) The electron energy E corresponds to the degenerate energy levels of themolecule. In this case, the additional condition is required: (B '). The ratio betweenthe values of a component eigenfunction at the source (a ) and the drain (b ) site isdi�erent from that for the other component eigenfunction.

Let us consider the case that the electron energy E is set to the degenerateenergy level of the molecule but the condition (B ') is not satis®ed. When magnetic®eld is applied to the system, condition (B ') becomes satis®ed. Therefore a sharpdependence of the applied magnetic ®eld on the transmission is expected [28].

Fig. 16 shows the internal current distribution inside the molecule. Here we also®nd a large loop current under certain conditions. To investigate the nature of thelarge induced current, the analytical expression for the internal current is obtainedfor a model system of the network as shown in Fig. 16. The current distributionj�E, n, d� is given by the product of the ``envelope'' term jenv�E, n� and the spatialdistribution term jsp�E, d �: j�E, n, d � � jenv�E, n�jsp�E, d �: Here E is the electronenergy, n, the total number of the rings, and d the distance from the end of themolecule. The expressions of these quantities are given in Ref. [28].

Fig. 16. The transmission probability T�E � and the current j�E,d � at the position d of a simple

molecular bridge shown in Fig. 15. The bold solid line indicated above the graph shows the position of

the energy level of the molecule.

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The relative spatial dependence of the current to the terminal current isdetermined by jsp�E, d�: The result is shown in Fig. 17(a). The absolute value ofthis term is less than 1.5, thus a large induced current emerges from the``envelope'' term (Fig. 17(b)). The energy where the envelope term is stronglyenhanced is related with the edge of the overlapping region of the two conductionchannels [28].

6. Summary

In the present article, quantum transport properties of atom bridges andmolecular bridges are clari®ed based on various theoretical approaches. As for themetallic atom bridges, we investigated the simple metal system of Na and Al bythe ®rst-principles recursion transfer matrix method, and the noble metal systemof AU by a self-consistent tight binding method. Eigenchannel analyses turned outto be e�ective to see the mechanism of the conductance and its relation with themicroscopic electronic state of the system. For the case of Na, the value of thetotal conductance is unity in the quantized unit irrespective of bending. This isexplained by the fact that the conductance is always contributed by a singlecomplete open s channel. On the other hand, for the case of Al atom bridge theconductance is contributed by three channels; one is the open s and pz channeland the others are the half open px=py channels. Since the Fermi level is located atthe onset of the latter channels, the conductance changes sensitively with thebending.

Compared to the simple metal atom bridge, the atom bridge of Au is muchmore complicated, since the upper edge of the d orbital channels is close to theFermi level. The relative position of the o�set energy of the d channels has beeninvestigated with the use of the self-consistent tight-binding calculation. Sensitivityof this energy to the bias voltage and wire length is observed. The remarkable

Fig. 17. (a) The decomposed current jsp�E,d � and (b) jenv�E � for the simple molecular bridge shown in

Fig. 15. The current can be reorganized by j�E,d � � jsp�E,d �jenv�E �:

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non-linear feature seen in experiments [16] cannot be reproduced within thepresent model.

For the case of the molecular bridges, we adopted two molecular systems forthe case studies. With the increase of the coupling of the molecule to the metallicelectrodes, signi®cant conductance is expected even for the general electron energybetween the molecular levels. As a matter of course, the favorable condition of theenhanced conductance is the narrower HOMO±LUMO gap and/or tunability ofthe molecular levels by the gate potential.

The most remarkable ®nding is the induced large loop current circulating ringpart of the molecules. This feature seems rather general and appears near thedegenerated molecular levels. The direction of the internal loop current would besharply switched when the level crosses the Fermi level. The coupling of the loopcurrent with the embedded magnetic moment is expected to cause some exoticfunctions of the molecular bridge.

Various features of quantum electron transport discussed in this article might bedrastically changed, when bottleneck parts are introduced in between the electronchannel of atomic/molecular bridges. In such cases, electron±electron interactionsincluding Coulomb blockade and e�ects of the coupling with the vibration modestend to prevail. In this regime of the transport, where the phase of electron wavesis completely destroyed within the bridge region, aspects of electron transmissionwould be quite di�erent from those revealed in this article. The studies of theseproblems are interesting, but beyond the reach of the present theoretical methods,and other appropriate methods should be developed.

Acknowledgements

This work is supported in part by a Grant-in-Aid from the Ministry ofEducation, Science and Culture of Japan, and by the Core Research forEvolutional Science and Technology (CREST) of the Japan Science andTechnology Corporation (JST).

References

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