Conference-ISSS-4-First-principles study of conduction through a sodium atomic sheet

Electronic conduction in an atomic sheet is examined theoretically from first principles as an example of a typical two-dimensional system. The conductance per atomic chain, calculated assuming a bias voltage of 0.1 V, is 0.87 G0 for the atomic sheet, which is lower than for the corresponding single chain (1.0 G0; G0 = 2e /h). The difference is attributed to a reduction of the local density of states around the Fermi level, particularly in the region where the bridge comes into contact with the electrode. [DOI: 10.1380/ejssnt.2006.507]


I. INTRODUCTION
The rapid progress in microelectronics engineering in recent years has spurred interest in the characteristics of conduction through nanometer-size bridge structures.Quantum effects are expected in such bridge structures, and the nature of electronic transport through these nanostructures has been studied actively from both experimental and theoretical approaches [1].Most experimental research on nanoscale conduction has been conducted for quantum point contacts made using a scanning tunneling microscope (STM) and/or the break junction technique [1].This kind of contact is point-like (zero dimensional) or linear (one dimensional) in structure, involving either a single molecule or a single atomic chain between interelectrodes.Most theoretical studies on nanoscale conduction have thus considered low-dimensional, singlebridge structures of this type.
Recently, however, multi-bridge structures have also been investigated [2][3][4].In a pioneering experiment on Au atomic contacts, Ohnishi et al. revealed that the number of conduction channels is proportional to the number of bridges [2].This result suggests that inter-bridge interaction has little effect on conduction.Theoretical studies, on the other hand, have predicted a different response.In one study on alkanes chains [3], the conductance was found to change linearly with the number of chains, and the slope of the change was suggested to be related to the degree of inter-bridge interaction.A first-principles study of parallel carbon-atom wires also concluded that the conductance is dependent on the distance between the wires [4].These theoretical studies thus suggest that inter-bridge interactions may have a considerable effect on conduction.The origin of the discrepancy between the experiment and theory, and in the various influences of inter-bridge interaction on conduction seen in the theoretical studies, has yet to be clarified.
In the present study, conduction in an atomic sheet is examined theoretically from first principles as a typical two-dimensional system, and the results compared with those reported for atomic chains.Atomic sheets represent an extreme case of two-dimensional system in which interbridge interactions are most remarkable, and are thus appropriate for first-principles analysis.

II. CALCULATIONS
Calculations were performed using a boundarymatching scattering-state density functional (BSDF) method developed by our group [5][6][7].The BSDF method takes explicit account of semi-infinite electrodes by dividing the system into three regions; the deep regions inside each of the two electrodes, and the scattering region in between the electrodes.Wave functions in the scattering region are determined so as to be connected properly to wave functions in the electrode regions.The effective potential and electron density are determined selfconsistently so as to maintain the difference in Fermi level between the two electrodes according to an applied bias voltage.Local density approximations [8,9] are employed for the exchange-correlation potential, and the local pseudopotential is adopted for the ionic potential [10].
The model considered in this analysis (Fig. 1) consists of an atomic sheet (4 Na atoms) sandwiched between two jellium electrodes.Periodic boundary conditions are defined in the x and y directions.The Wigner-Seitz radius of jellium, 4.0 a.u., is equal to the average valence-electron density of the Na crystal [11,12].The distance between the jellium edges and the neighboring atoms is fixed at 3.0 a.u., and the inter-atomic distances in the bridge are fixed at 7.0 a.u., corresponding to the length between the adjacent atoms in the Na bulk.
The thickness of the left and right jellium electrodes in the scattering region is set at 20 a.u.The cross-section of the supercell is rectangular, with dimensions of 7 a.u. and 35 a.u. in the x and y directions, respectively.The wave functions are expanded as plane waves in the x and y directions, and the cutoff energy of the expansion is set at 5.5 Ry.The number of k points in the first Brillouin Zone is 10 × 2 along both the k x and k y directions.The sampling k -point spacing is 0.898 a.u.−1 in both the k x and k y directions.
For comparison, calculations for a sodium atomic chain were also performed (Fig. 1(c)), using a 21 × 21 (a.u.) 2 supercell with the same boundary conditions.

III. RESULTS AND DISCUSSION
The conductance per atomic chain, calculated for a bias voltage of 0.1 V, is 0.87 G 0 for the sheet and 1.0 G 0 (G 0 = 2e 2 /h) for the chain.This result suggests that inter-bridge interaction reduces conductance.
The origin of the difference in conductance can be inferred from a plot of the calculated local density of states (LDOS) in the region between the two edges of the jellium electrodes (Fig. 2).The LDOS around the Fermi level, where electrons contribute most to conduction, is lower in the sheet than in the chain.The origin of the difference in conductance can therefore be understood as being due to the LDOS at the Fermi level, as found previously for carbon chains [4].Furthermore, changing the integration region for calculation of the LDOS reveals that the LDOS at the Fermi level differs in the region around the edge of the left electrode (Fig. 2(b)), but not around the center of the bridge (Fig. 2(c)).Therefore, the difference in the LDOS at the Fermi level arises from a difference in electronic states in the region where the edge of the electrode contacts the bridge.
Although the present results are consistent with those of Lang and Avouris [4] in terms of relating the difference in conductance to the difference in LDOS near the Fermi level, Lang and Avouris examined parallel C 6 wires with a separation of 3.5 a.u. and found that its conductance is increased by inter-bridge interaction, which in the present case appears to reduce the conductance.Lang and Avouris pointed out that charge accumulation due to contact between wires and electrodes occurs near the contact region and that this charge accumulation has a substantial effect on the conductance of the wires.This is consistent with the present results showing that the difference in electronic states near the edge of the electrode affects the conductance.The discrepancy between the present results of Lang and Avouris can therefore be attributed to the difference in charge accumulation.Clarification of the details of accumulation, particularly with regard to the difference between sheets and chains, remains a topic for future research.The peaks in the LDOS profiles were investigated by calculating the density of states (DOS) for isolated bridges using a simple tight-binding approximation.In the calculation, one orbital was assumed per atom, the overlap in-tegrals between different orbitals were neglected, and only hopping of electrons between nearest-neighbor atoms was considered.The calculated energy band structures and DOS for the Na 4 atomic chain and sheet are shown in Fig. 3.In the case of the chain, there are four states of electrons, corresponding to the number of atoms.When the bridge is connected to the electrodes, these states are broadened by interaction between the bridge and electrodes.Considering these features, in conjunction with the feature that the DOS of an infinite one-dimensional chain diverges at the edges of the band (dashed line in Fig. 2(a)), the peak marked by an arrow can be tentatively assigned to the lower edge of the Na 3s band.In the case of the sheet, interactions in the x direction should be considered in addition to those in the z direction.The onedimensional energy band of the infinite chain is formed in the x direction, while there are a finite number of electronic states in the z direction due to the finite number of atoms in the latter direction.Therefore, the DOS of the sheet becomes the sum of the DOS for the finite onedimensional bands (Fig. 3(b)).It should be noted that the energy range of the DOS is wider for the sheet than for the chain, which causes the shift of the lower edge of the LDOS to lower energy (Fig. 2(a)).

IV. CONCLUSION
Electronic conduction through a sodium atomic sheet and chain was investigated theoretically by first-principles calculations.The conductance of a bridge in the sheet was found to be 0.87 G 0 , which is lower than that in the chain (= 1.0 G 0 ).The difference in conductance was found to be related to a difference in the LDOS at the Fermi level, mainly in the region where the jellium edge contacts the bridge.This is consistent with previous observations for parallel C chains [4].

FrontFIG. 1 :FIG. 2 :
FIG.1: Schematic models of (a) an atomic sheet, (b) the atomic sheet in front and lateral cross-section, and (c) an atomic chain in front and lateral cross-section.All values are in atomic units (a.u.).

FIG. 3 :
FIG.3: Energy band structure and DOS for (a) an isolated chain and (b) a sheet calculated using a simple tight-binding approximation.α and β denote on-site and hopping parameters, respectively.