6256
J. Phys. Chem. C 2009, 113, 6256–6260
First-Principles Study on Even-Odd Conductance Oscillation of Pt Atomic Nanowires Tomoya Ono† Department of Precision Science and Technology, Osaka UniVersity, Suita, Osaka 565-0871, Japan ReceiVed: December 19, 2008; ReVised Manuscript ReceiVed: February 17, 2009
The electron-transport property of single-row Pt atomic nanowires is examined by first-principles calculations based on the density functional theory. I found that oscillation patterns with a period longer than two-atom length are dominant in the conductance of Pt nanowire, although the transmission of the s-dz2 channel still exhibits even-odd oscillatory behavior. When the nanowires are deformed into zigzag configurations from the straight configuration, the oscillation behavior of the patterns with long periods changes and the oscillations eventually disappear. On the other hand, the even-odd oscillatory behavior of the s-dz2 channel still survives even in the deformed nanowire. The even-odd oscillation in the conductance of Pt nanowire is interpreted to be due to the low sensitivity of the oscillation of the s-dz2 channel to the spatial deformation of the nanowires. I. Introduction Metallic nanowires consisting of several atoms have been generated by using scanning tunneling microscopes or mechanically controllable break junctions.1 Since they have the simplest possible structure and represent the ultimate limit of the miniaturization of electronic devices, many theoretical and experimental studies on their electron-transport properties have been carried out. In such minute systems, the electron transport becomes ballistic, in contrast to the diffusive transport in macroscopic systems. In the case of monovalent atoms, such as Na and Au, the conductance is quantized in the unit of G0 ()2e2/h),2-4 where e is the electron charge and h is Planck’s constant. Moreover, several theoretical studies report oscillatory behavior with a period of two-atom length in the conductance of single-row nanowires made of monovalent atoms,5-14 which is known as even-odd oscillation. In 2003, Smit et al.15 experimentally found oscillatory behavior with a period of two-atom length in the conductance of Ir and Pt atomic nanowires in addition to that of the Au nanowire. The following year, de la Vega et al.16 examined the electron-transport properties of Ir, Pt, and Au nanowires by a tight-binding calculation and claimed that additional oscillation patterns with a longer period and larger amplitude than those obtained in ref 15 can be found in the conductance of Ir and Pt nanowires. Although it is intuitively expected that patterns with a large amplitude are dominant in the conductance trace obtained by experiments, there have been no experimental studies reporting the existence of such oscillation patterns. From the theoretical point of view, oscillations with a period longer than two-atom length can be observed because the oscillatory behavior of the conductance is caused by the quantummechanical wave character of the electrons. Thygesen and Jacobsen17 in 2003 found using first-principles calculations that the conductance trace of Al nanowires exhibits the oscillation with the period of four-atom length although it has not been observed by experiments up to now. Garcı´a-Sua´rez et al.18 computed the even-odd oscillation in the conductance trace of Pt nanowires with three-, four-, and five-atom length by firstprinciples simulations. However, the reason for the even-odd conductance oscillation of the Pt nanowire and the absence of †
E-mail address:
[email protected].
the additional oscillation patterns in the experimental observation has not been uncovered yet. In this paper, I implemented a first-principles electrontransport calculation for single-row Pt atomic nanowire within the framework of the density functional theory.19 The aims of this study are to explore whether the oscillation patterns with a long period and large amplitude are observed in the conductance trace of the Pt nanowire when they are computed by firstprinciples calculation and to reveal why the even-odd oscillation is dominant in the experimentally obtained conductance traces. My results indicate that the even-odd conductance oscillation is insensitive to the structural deformation of the nanowire, while oscillation patterns with a longer period than two-atom length and with a large amplitude are easily affected by structural deformation. This implies that only the even-odd oscillation can survive and the other oscillation patterns are canceled out by being averaged over the thousands of scans in the experiment.15 The rest of this paper is organized as follows. In Section II, I briefly describe the method used in this study. My results are presented and discussed in Section III, and I summarize my findings in Section IV. II. Computational Methods Our first-principles calculation method for obtaining the electron-transport properties is based on the real-space finitedifference approach,20-22 which enables us to determine the selfconsistent electronic ground state with a high degree of accuracy by a time-saving double-grid technique21,22 and the direct minimization of the energy functional.23 Moreover, the realspace calculations eliminate the serious drawbacks of the conventional plane-wave approach, such as its inability to describe nonperiodic systems accurately. I examined the electrontransport properties of the nanowires suspended between semiinfinite electrodes by computing the scattering wave functions continuing from one electrode to the other using the overbridging boundary-matching formula21,24 under the semi-infinite boundary condition in the z direction. The conductance of the nanowire system at the limits of zero temperature and zero bias was described by the Landauer-Bu¨ttiker formula, G ) tr(T†T)G0,25 where T is the transmission matrix. To investigate the states
10.1021/jp811214k CCC: $40.75 2009 American Chemical Society Published on Web 03/24/2009
Even-Odd Conductance Oscillation of Pt Atomic Nanowires
J. Phys. Chem. C, Vol. 113, No. 15, 2009 6257
Figure 1. Schematic description of the scattering region of atomic Pt nanowire with pyramidal bases suspended between semi-infinite jellium electrodes. The size of the sphere is varied according to the distance from the viewpoint.
actually contributing to electron transport, the eigenchannels were computed by diagonalizing the Hermitian matrix T†T.8 Figure 1 shows the computational model, in which the nanowire is connected at both ends to the pyramidal bases made of atoms, modeled after the [111] strands, and all of these components intervene between the electrodes. The distance between the atomic plane of the bases and the surface of the jellium electrode is 2a0/6, where a0 is the lattice constant of Pt fcc bulk and is chosen to be 7.43 au The interatomic distance of the Pt nanowire was chosen to be equal and d ()a0/2). The grid spacing h was set to 3a0/34 and a denser grid spacing of h/3 is employed in the vicinity of nuclei by the augmentation of the double-grid technique.21,22 The Wigner-Seitz radii of the jellium electrodes were taken to be rs ) 1.35 au so that the elements of the electrodes correspond to Pt bulk. The jellium approximation gives conduction properties corresponding to those with crystalline bulk as electrodes when the materials of the nanowire and electrodes are the same.8,9,26 To determine the Kohn-Sham effective potential, a conventional supercell was employed under the periodic boundary condition in all directions; the length of the supercell in the z direction is Lz ) nd + 122h.27 The norm-conserving pseudopotentials28 of Troullier and Martins29 were employed to describe the electron-ion interaction, and exchange correlation effects were treated by a local density approximation.30 III. Results and Discussion Figure 2 shows the conductance and the transmissions of the eigenchannels of the Pt nanowire. The quantum numbers of the eigenchannels are assigned according to their charge density distributions. The transmissions of the channels not shown in Figure 2 are smaller than 10-3. Although oscillatory behavior is not observed in the conductance, even-odd oscillation emerges in the transmission of the upper s-dz2 channel. In our previous study,31 we found that the period of oscillation coincides with the least common multiple of π/kz and the geometric period of the nanowire, i.e., the interatomic distance. Figure 3a shows the energy band structures of the infinite straight Pt wire with equal interatomic distance of d. Since the upper s-dz2 band of the infinite wire gets across the Fermi level at kz ≈ π/2d, the period of the oscillation becomes π divided by π/kz (≈2d).14 The two-atom period of the oscillation of the upper s-dz2 channel and the long period of the oscillation of the other channels are consistent with the conclusion in the previous study. The oscillation of the lower s-dz2 channel cannot be observed in the experiments because of its long oscillation period. However, the dxz and dyz channels are expected to contribute significantly to the experimentally obtained conductance trace, since these channels exhibit a rather short period ( V1 is
[
T) 1+
|2E1 - k2 | 2sin2 kL 8E1k2
]
-1
(1)
where k ) [2(E1 - V1)]1/2. It is obvious that the amplitude of the oscillation becomes larger with the decrease of the kinetic energy in the potential barrier k. In the case of the twodimensional nanowire connected to electrodes, the transmission probability of the lth channel is Figure 4. Charge density distributions of (a) upper s-dz2, (b) lower s-dz2, (c) dxz,(d)dyz, (e) dx2 - y2, and (f) dxy bands of infinite straight Pt wire with the interatomic distance of d. Each contour represents twice or half the density of the adjacent contour lines. The lowest contour line corresponds to 3.13 × 10-5 electrons/bohr3. (g) Schematic description of the plane where the distributions are shown.
distribution behaves like two thin zigzag wires, the transmissions of the dxz channel are easily affected by the structural deformation of the nanowire in the x direction. The oscillations of the channels whose charge density distributions are divided by the plane perpendicular to the nanowire axis are removed because the experimentally obtained conductance trace is averaged over thousands of scans. In addition, since the deviation from the straight nanowire can arise in various directions and at various positions of the nanowire, the oscillations of the dyz and dxz channels disappear and only the oscillation from the s-dz2 channel can survive. Moreover, the reduction of the transmission
[
|(β¯ l)2 - (klz)2 | 2 sin2 klzLeff Tl ) 1 + (β¯ + β¯ *)2(kl )2 l
l
z
]
-1
(2)
according to eq A11 of ref 14. The definitions of a wave vector jl in the electrode klz, a modified wave vector in the nanowire β and the effective wire length Leff are the same with those in ref 14. When the width of the wire W is infinity, the oscillation j l)2. Figure 5 shows the coefficients disappears because (klz)2 ) ( β of the square of the sine as a function of the energy of the incident electrons. The z [x] component of the kinetic energy, (klz)2 [(klx)2], decreases [increases] with increasing the number of the transverse nodes since the number of transverse nodes in the charge density distribution corresponds to l - 1. Therefore, the coefficient becomes larger as l increases. The x component of the kinetic energy in the nanowire region plays the role of the height of the potential barrier of the one-dimensional case,
Even-Odd Conductance Oscillation of Pt Atomic Nanowires
Figure 5. Coefficients of the square of the sine as a function of energy of incident electrons, |( βj l)2 - (klz)2|2/( βj l + βj l*)2(klz)2. Solid, broken, and dotted curves are the coefficients for the first, second, and third channels, respectively. There are l - 1 transverse nodes in the lth channel. Here, W is the width of the nanowire.
and the amplitude of the oscillation in the conductance of the channels with the transverse nodes is larger than that of the channel without the transverse nodes. There also remains the concern that the arbitrary chosen structural parameters might affect my conclusion. Panels b-e in Figure 3 plot the energy band structure of the infinite Pt wire by changing the interatomic distance and bond angle. The model in which two Pt atoms are contained in a supercell was employed in the case of zigzag wires, while that in which one atom exists was adopted in the case of the straight one. The energy band structure drastically changes depending on the bond angle while it is insensitive to the interatomic distance. In addition, the d bands of the zigzag wire split and the crossing point between the d bands and the Fermi level significantly changes at ∆x ) 0.5 au, which causes the small phase shift of the transmission oscillation of the dyz channel shown in Figure 2d. Note that the s-dz2 band still crosses the Fermi level at ∼π/2d even when these structural parameters were changed. Since the relation between the conductance oscillation and the crossing point of the Fermi level has already been established in our previous study,14 the s-dz2 band, whose characteristics are insensitive to the structural deformation, causes the even-odd oscillation in the conductance trace of the Pt nanowires. IV. Conclusion I have implemented a first-principles investigation of the electron-transport properties of nanowires consisting of Pt atoms. Oscillation patterns with a period longer than two-atom length and a larger amplitude are found in the conductance of the straight Pt nanowire. These oscillations are caused by the channels whose charge density distributions are divided by the node of the wave functions. However, the transmission properties of these channels are easily affected by the structural deformation of the nanowires in the directions perpendicular to the nanowire axis. Therefore, only the even-odd oscillation of the s-dz2 channel remains in the experimentally obtained conductance traces while the oscillations of the other channels are washed out by being averaged over many scans. At this moment, the first-principles electron-transport calculation is not a suitable tool for averaging results over numerous implementations for the variations of the atomic configuration of the interface as well as the nanowires. It is expected that these averaging effects can be confirmed by simpler tools such as tight-binding approximations. Acknowledgment. The author would like to thank Professor Kikuji Hirose of Osaka University for reading the entire text in
J. Phys. Chem. C, Vol. 113, No. 15, 2009 6259 its original form. This research was partially supported by a Grant-in-Aid for the 21st Century COE “Center for Atomistic Fabrication Technology”, by a Grant-in-Aid for Scientific Research in Priority Areas “Development of New Quantum Simulators and Quantum Design” (Grant No. 17064012), and also by a Grant-in-Aid for Young Scientists (B) (Grant No. 20710078) from the Ministry of Education, Culture, Sports, Science and Technology. The numerical calculation was carried out with the computer facilities at the Institute for Solid State Physics at the University of Tokyo, the Research Center for Computational Science at the National Institute of Natural Science, and the Information Synergy Center at Tohoku University. References and Notes (1) (a) Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press: New York, 1995. (b) van Ruitenbeek, J. M. In Metal Clusters at Surfaces Structure, Quantum Properties, Physical Chemistry; Meiwes-Broer, K. H., Ed.; Springer: Berlin, Germany, 2000. (c) Agrat, N.; Yeyati, A. L.; van Ruitenbeek, J. M. Phys. Rep 2003, 377, 81, and references cited therein. (2) Krans, J. M.; van Ruitenbeek, J. M.; Flsun, V. V.; Yanson, I. K.; de Jongh, L. J. Nature (London) 1995, 375, 767. (3) Yanson, A. I.; Rubio Bollinger, G.; van den Brom, H. E.; Agrait, N.; van Ruitenbeek, J. M. Nature (London) 1998, 395, 783. (4) Ohnishi, H.; Kondo, Y.; Takayanagi, K. Nature (London) 1998, 395, 780. (5) (a) Lang, N. D. Phys. ReV. Lett. 1997, 79, 1357. (b) Lang, N. D. Phys. ReV. B 1997, 55, 4113. (c) Lang, N. D.; Avouris, Ph. Phys. ReV. Lett. 1998, 81, 3515. (d) Lang, N. D.; Avouris, Ph. Phys. ReV. Lett. 2000, 84, 358. (6) Emberly, E. G.; Kirczenow, G. Phys. ReV. B 1999, 60, 6028. (7) Kobayashi, N.; Brandbyge, M.; Tsukada, M. Surf. Sci. 1999, 433435, 854. (8) Kobayashi, N.; Brandbyge, M.; Tsukada, M. Phys. ReV. B 2000, 62, 8430. (9) Tsukamoto, S.; Hirose, K. Phys. ReV. B 2002, 66, 161402(R). (10) (a) Sim, H. S.; Lee, H. W.; Chang, K. J. Phys. E 2002, 14, 347. (b) Sim, H. S.; Lee, H. W.; Chang, K. J. Phys. ReV. Lett. 2001, 87, 096803. (11) Lee, Y. J.; Brandbyge, M.; Puska, M. J.; Taylor, J.; Stokbro, K.; Nieminen, R. M. Phys. ReV. B 2004, 69, 125409. (12) Egami, Y.; Sasaki, T.; Tsukamoto, S.; Ono, T.; Inagaki, K.; Hirose, K. Mater. Trans., JIM 2004, 45, 1433. (13) Khomyakov, P. A.; Brocks, G. Phys. ReV. B 2004, 70, 195402. (14) Egami, Y.; Ono, T.; Hirose, K. Phys. ReV. B 2005, 72, 125318. (15) Smit, R. H. M.; Untiedt, C.; Rubio-Bollinger, G.; Segers, R. C.; van Ruitenbeek, J. M. Phys. ReV. Lett. 2003, 91, 076805. (16) de la Vega, L.; Martı´ n Rodero, A.; Yeyati, A. L.; Sau´l, A. Phys. ReV. B 2004, 70, 113107. (17) Thygesen, K. S.; Jacobsen, K. W. Phys. ReV. Lett. 2003, 91, 146801. (18) Garci a Sua´rez, V. M.; Rocha, A. R.; Bailey, S. W.; Lambert, C. J.; Sanvito, S.; Ferrer, J. Phys. ReV. Lett. 2005, 95, 256804. (19) Hohenberg, P.; Kohn, W. Phys. ReV. 1964, 136, B864. (20) (a) Chelikowsky, J. R.; Troullier, N.; Saad, Y. Phys. ReV. Lett. 1994, 72, 1240. (b) Hoshi, T.; Arai, M.; Fujiwara, T. Phys. ReV. B 1995, 52, R5459. (c) Seitsonen, A. P.; Puska, M. J.; Nieminen, R. M. Phys. ReV. B 1995, 51, 14057. (d) Fattebert, J.-L.; Bernholc, J. Phys. ReV. B 2000, 62, 1713. (e) Bertsch, G. F.; Iwata, J.-I.; Rubio, A.; Yabana, K. Phys. ReV. B 2000, 62, 7998. (f) Beck, T. L. ReV. Mod. Phys. 2000, 72, 1041. (g) Khomyakov, P. A.; Brocks, G. Phys. ReV. B 2004, 70, 195402. (h) Mortensen, J. J.; Hansen, L. B.; Jacobsen, K. W. Phys. ReV. B 2005, 71, 035109. (21) Hirose, K.; Ono, T.; Fujimoto, Y.; Tsukamoto, S. First-Principles Calculations in Real-Space Formalism; Imperial College Press: London, UK, 2005. (22) (a) Ono, T.; Hirose, K. Phys. ReV. Lett. 1999, 82, 5016. (b) Ono, T.; Hirose, K. Phys. ReV. B 2005, 72, 085105. (c) Ono, T.; Hirose, K. Phys. ReV. B 2005, 72, 085115. (23) Hirose, K.; Ono, T. Phys. ReV. B 2001, 64, 085115. (24) Fujimoto, Y.; Hirose, K. Phys. ReV. B 2003, 67, 195315. (25) Bu¨ttiker, M.; Imry, Y.; Landauer, R.; Pinhas, S. Phys. ReV. B 1985, 31, 6207. (26) Lang, N.; Di Ventra, M. Phys. ReV. B 2003, 68, 157301. (27) The Fermi level shifts when the Kohn-Sham effective potential is computed by the conventional supercell. The difference in the Fermi level from the case of Lz ) nd + 112h is ∼0.04 eV in the case of the 3-atom Pt nanowire. According to Figure 3a, the crossing point between the s-dz2 band and the Fermi level does not change significantly when the Fermi
6260 J. Phys. Chem. C, Vol. 113, No. 15, 2009 level shifts by 0.04 eV. Therefore, the electrode region is chosen to be sufficiently large to lead the conclusion of this study. (28) We used the norm-conserving pseudopotentials NCPS97 constructed by Kobayashi: Kobayashi, K. Comput. Mater. Sci. 1999, 14, 72. (29) Troullier, N.; Martins, J. L. Phys. ReV. B 1991, 43, 1993. (30) Perdew, J. P.; Zunger, A. Phys. ReV. B 1981, 23, 5048. (31) Egami, Y.; Aiba, S.; Hirose, K.; Ono, T. J. Phys.: Condens. Matter 2007, 19, 365201. (32) (a) Sa´nchez-Portal, D.; Artacho, E.; Junquera, J.; Ordejo´n, P.; Garcı´a, A.; Soler, J. M. Phys. ReV. Lett. 1999, 83, 3884. (b) Brandbyge, M.; Mozos, J.-L.; Ordejo´n, P.; Taylor, J.; Stokbro, K. Phys. ReV. B 2002,
Ono 65, 165401. (c) Ribeiro, F. J.; Cohen, M. L. Phys. ReV. B 2003, 68, 035423. (d) Ferna´ndez-Seivane, L.; Garcı´a-Sua´rez, V. M.; Ferrer, J. Phys. ReV. B 2007, 75, 075415. (33) Since the latest theoretical study using the empirical parameters of Zoubkoff et al (Zoubkoff, R.; de la Vega, L.; Martı´n-Rodero, A.; Yeyati, A. L.; Sau´l, A. Phys. B 2007, 398, 309. ) reported that the finite nanowire between electrodes does not form the zigzag structure, the existence of the zigzag structure of the nanowire between electrodes might be an unsettled question.
JP811214K
