# Presentation: 523054

## Ballistic conductance calculation of

- atomic-scale nanowires of Au and Co
**Peter Bennett, Arizona State University, ****ECS-0304682**

- State-of-the-art electron transport calculations have been made for metallic nanowires consisting of few-atom metal chains, in a zig-zag configuration that allows a range of inter-atom spacing while preserving bond-lengths. Cobalt and gold were chosen to investigate the effect of localized states (*d* and *f* levels) and to investigate the role of spin, both of which are outstanding issues in the topic of quantum point contacts. It was found that Co conductance is quantized, but only in the majority channel. In contrast, Au is not spin polarized, and shows well-behaved quantization, but only for chains less than 3 atoms wide. In addition, the zigzag Au chain can show a Peierls instability, making it insulating. These results help explain recent measurements for quantum point contacts.

*Ke , van Schilfgaarde , Kotani & Bennett*, Nanotechnology 18 (2007) 095709

- Top: Conductance quantization as a function of wire cross section. Bottom: Band structure and ball-and-stick model of the Au chains with different surface roughness: (a) planar geometry; (b) rotates every Au pair by 90◦ around the *z* axis; (c) rotates every Au pair by 180◦.

**Notes:**
Aim of the project:
     In the past couple of decades a great deal of attention has been   devoted to one-dimensional nanowire structures because of their technological potential for creating nanoscale electronic devices. Experimentally, it is possible to synthesize single chains of gold atoms and observe the effects of the quantization of conductance for different applied strains. Experimental results of measured conductance in noble metals Au are more consistent than in transition metals.  The objective of our project is to study the ballistic conductance of the infinite zigzag Au and Co chain by using the Landauer-Büttiker formalism within the local-density  approximation (LDA).

Research results:
      Calculations were performed for infinite zigzag chains as well as  monatomic chains when the wire was highly stretched. By using a generalization of the standard method of linear muffin-tin orbitals (LMTO), calculations were carried out within the LDA .  we use the Landauer-Büttiker formula to study the transport conductance  during the process of stretching the zigzag chain into monatomic chain in the Au and Co cases.  The latter has a large density of d states at the Fermi level, and thus its behaviour is rather different from Au.  
       We also investigate cross section of Au chain and transport quantization. It is well known that for bulk materials, the conductivity is well predicted by ohm's law, with quantum effects important when the dimensions becomes small.  A key question is then, ``how small must the wire be to exhibit quantized conductance?''  In the case of Co, we have seen that it should never do so.  It turns out that for Au nanowires, which have a well-behaved quantized conductance observed experimentally and in our infinite chain model, the quantization is lost rather quickly as the wire width increases. 
        A further complication is the fact that the wire itself may not be ideal, but can flex and bend. Here we consider one such example: what happens if we introduce surface roughness to these wires?  As a possible variation in wire geometry, we consider a rotational deformation mode of the zigzag chain. Finally, we show that the zigzag chain of Au can have a Peierls instability, making the chain insulating.

Significance of the results:
      (1) Au chains (which are found to be not spin polarized) show well-behaved transport conductivity quantization in the case of infinite zigzag chain, consistent with experiments.  The conductance is quantized as 2G0 or 4G0, and changes abruptly with stretching, induced by the change in the number of bands crossing EF, rather than from discontinuous change in the atomic structure. The energy splitting between two subbands near Fermi level increases as we stretch the zigzag wire, as explain in terms of the infinite quantum well model. 
      (2) By increasing the dimension of the cross section of the Au chain, the quantization is rather rapidly lost.  The calculation predicts that it is possible to realize the lowest four conductance quanta  only for wires of order two atoms thick.
      (3) In the case of infinite zigzag Cobalt chains, the chain was found  to be spin polarized, and the (s-like) majority channel exhibits quantized similar to that of Au.  Because of the close spacing of the $d$ levels at EF of the minority channel, that channel does not show quantized conductance.  As a consequence the only way to observe a single quantum of conductance is to spin-resolve the conductance measurement.
      (4) We have examined a rotational deformation mode for the zigzag Au chain.  The mode cause the Peierls instability, which make the wire insulating. This is competitive with the level inversion effect expected for zigzag chain.

Future plans:
       As shown in next slides, we did some further studies.
      (1) The soft phonon of the monatomic chained between two electrodes. By solving the eigensystem of the dynamical matrix,  we can obtain the eigenvector of the lowest phonon modes.
       (2) Calculate the energy barrier of the deformation pathway suggested by these obtained eigenvector of the lowest soft modes.  These energy barrier play an important part to determine the stability of the wire.

## Atomic Calculation of Au Nanowire Break-Junction

**Peter Bennett, Arizona State University, ****ECS-0304682**

- Nanoscale break-junctions are widely used to study the electrical properties of single molecules by briefly capturing them between nanoscale electrodes during rupture of the junction. In this project, state-of-the-art calculations have been used to explore the mechanical behavior of a gold nanowire break-junction. Unlike most calculations, lattice motions are included. It is found that atom motions perpendicular to the wire axis are significant at room temperature. The lattice motion causes wire breakage via a thermal-assisted process, which occurs at a lower energy than for the normally assumed linear chain. The magnitude of this “dynamic bond-breaking” process has been calculated and agrees favorably with experimental measurements from the same group (see next Highlight).

*Ke , van Schilfgaarde , Kotani & Bennett*, Nanotechnology (accepted).

**Figure**. (a) Ball-and-stick model of the Au chain between two electrodes for an electrode spacing Z = 20.77 Å , following the constrained energy minimization and deformation path described in the text. The position profile of the central chain atom (labeled *B*), is plotted. Other atoms plotted are in their initial relaxed condition before the pulling down process. (b) Eigenvector of the lowest phonon mode. (c) Total energy as a function of *z* coordinate displacement of
- atom *B* from its initial position *z*0.

**Notes:**
Aim of the project:
     The stability and lifetime of one-dimensional nanowire structures are of great interest in any potential application they may have.  A better understanding of them could be helpful to fabricate longer chains or even more complicated structures. Usually, barrier heights of breaking a wire are calculated by assuming an essentially linear deformation of atoms along wire axis.  This is reasonable when the system is far from equilibrium.  However, such a deformation is unlikely when thermal activation is important.  In the equilibrium regime, the most likely motion will be along the softest normal modes, at least near the (metastable) equilibrium. Our objective is to study the soft phonon modes to obtain the collective motions of the chain atoms and then calculate the energy barrier of the deformation pathway suggested by this obtained eigenvector of the lowest soft modes.
Research results:
     Unlike the conductance, which is independent of the loading rate, the breakdown force is very sensitive to it until a threshold is reached, at which point the force becomes independent of the loading rate. This behaviour can be understood based on the following considerations.  At finite temperature, thermal fluctuations can drive a nanowire to escape its (metastable) configuration.  Thus, if the wire is static or when the loading rate is sufficiently low (equilibrium regime), the breakdown is controlled by a thermally activated mechanism. 
     The dissociation path and the corresponding activation barrier plays an important role in determining stability of the Au wire.  However, it is very difficult to find the optimal pathway which nature presumably would follow.  Usually, barrier heights are calculated by assuming an essentially linear deformation of atoms along wire axis.  This is reasonable when the system is far from equilibrium.  However, such a deformation is unlikely when thermal activation is important.  In the equilibrium regime, the most likely motion will be along the softest normal modes, at least near the (metastable) equilibrium.
      Our calculations were performed to simulate finite monatomic chains between two electrodes.  They were carried out within the LDA using a generalization of the standard method of linear muffin-tin orbitals (LMTO).  We use the frozen phonon method to calculate the phonon normal modes of the wire. We restricted the dynamical matrix to atomic displacements of 9 atoms: 3 in the wire chain and the 6 atoms contacted with them on each end.  Thus, this restricted dynamical matrix contains 27 normal modes.  By solving the phonon eigensystem, 27 phonon eigenstates were obtained.  The lowest phonon modes involve collective atom motion in the xy plane, which means atoms initially move most easily in the plane perpendicular to the wire axis.  Actually, the lowest two modes are nearly degenerate and the collective motion associated with them are dominated by the motion of the central chain atom (labelled as atom $B$ in Figure) along the xy plane. Figure (right) shows the eigenvector of the lowest phonon mode.  The eigenvector of the second phonon is the same as that depicted in Figure, rotated by 90 degrees around z.  This shows that we should consider a deformation pathway which is not the wire-elongated pathway as generally assumed.
      We calculate the energy barrier of breaking the wire along the fracture path, in which the central atom of the chain shifts laterally as eigenvector suggested and then move to the top electrode.

Significance of the results:
     (1) The relaxed geometries of the Au nanostructure with one straight chain between two electrodes were studied.
     (2) By using the dynamical matrix, soft modes of the Au chain were analyzed, and it was found that the lowest energy modes correspond to lateral motion in the  xy plane. This suggests that we should consider a deformation pathway which is not the wire-elongated pathway as generally assumed.
     (3) Fracture involving initial deformations in the lateral dimension are somewhat lower in energy than fracture where the deformation of the Au chain takes place essentially along the wire axis.  The associated energy barrier has an upper limit of  ≈1.1eV.
Future plans:
     (1) Since the collective motions of the soft phonon modes are dominated by the motions of the central atom of on  the chain. We can move the atoms along the eigenvector of the lowest phonon mode to obtain the potential profile. By solving the wavefunction, we can compare the quantum effect and classical effect.
     (2) We can study the ballistic transport conductance of this nano-contact which is more like to happen in the real life than the infinite zigzag chain.

## Current-induced Local Heating in Single Molecule Junctions 

**Peter Bennett, Arizona State University, ****ECS-0304682**

*Huang , Chen, D’ Agosta, Bennett, Di Ventra & Tao*.

**Figure**. Effective temperature (Teff) in single alkanedithiol junctions (HS-(CH2)n-SH, or Cn, n=6, 8, 10) vs. bias voltage. The experiment was performed in toluene at the room temperature. The stretching rate was fixed at 20 nm/s. The Inset is a schematic illustration of the formation of single alkanedithiol junction (Au/S-(CH2)n-S/Au) by STM break junction approach.

- Current-induced local-heating in a molecular device is a fundamentally and technologically important problem. We have measured local-heating in single molecules (n-alkanedithiol, n=6, 8 and 10) covalently attached to two gold electrodes as a function of applied bias voltage and molecular length. The effective temperature of the molecular junctions, due to local heating, increases with the applied bias and then decreases after reaching a maximum. At a given bias, the effective temperature decreases with the molecular length. These experimental findings are in agreement with the theoretical prediction based on a hydrodynamic approach including both electron-phonon and electron-electron interactions.

**Notes:**
Aim of the project:
     The stability and lifetime of one-dimensional nanowire structures are of great interest in any potential application they may have.  A better understanding of them could be helpful to fabricate longer chains or even more complicated structures. Usually, barrier heights of breaking a wire are calculated by assuming an essentially linear deformation of atoms along wire axis.  This is reasonable when the system is far from equilibrium.  However, such a deformation is unlikely when thermal activation is important.  In the equilibrium regime, the most likely motion will be along the softest normal modes, at least near the (metastable) equilibrium. Our objective is to study the soft phonon modes to obtain the collective motions of the chain atoms and then calculate the energy barrier of the deformation pathway suggested by this obtained eigenvector of the lowest soft modes.
Research results:
     Unlike the conductance, which is independent of the loading rate, the breakdown force is very sensitive to it until a threshold is reached, at which point the force becomes independent of the loading rate. This behaviour can be understood based on the following considerations.  At finite temperature, thermal fluctuations can drive a nanowire to escape its (metastable) configuration.  Thus, if the wire is static or when the loading rate is sufficiently low (equilibrium regime), the breakdown is controlled by a thermally activated mechanism. 
     The dissociation path and the corresponding activation barrier plays an important role in determining stability of the Au wire.  However, it is very difficult to find the optimal pathway which nature presumably would follow.  Usually, barrier heights are calculated by assuming an essentially linear deformation of atoms along wire axis.  This is reasonable when the system is far from equilibrium.  However, such a deformation is unlikely when thermal activation is important.  In the equilibrium regime, the most likely motion will be along the softest normal modes, at least near the (metastable) equilibrium.
      Our calculations were performed to simulate finite monatomic chains between two electrodes.  They were carried out within the LDA using a generalization of the standard method of linear muffin-tin orbitals (LMTO).  We use the frozen phonon method to calculate the phonon normal modes of the wire. We restricted the dynamical matrix to atomic displacements of 9 atoms: 3 in the wire chain and the 6 atoms contacted with them on each end.  Thus, this restricted dynamical matrix contains 27 normal modes.  By solving the phonon eigensystem, 27 phonon eigenstates were obtained.  The lowest phonon modes involve collective atom motion in the xy plane, which means atoms initially move most easily in the plane perpendicular to the wire axis.  Actually, the lowest two modes are nearly degenerate and the collective motion associated with them are dominated by the motion of the central chain atom (labelled as atom $B$ in Figure) along the xy plane. Figure (right) shows the eigenvector of the lowest phonon mode.  The eigenvector of the second phonon is the same as that depicted in Figure, rotated by 90 degrees around z.  This shows that we should consider a deformation pathway which is not the wire-elongated pathway as generally assumed.
      We calculate the energy barrier of breaking the wire along the fracture path, in which the central atom of the chain shifts laterally as eigenvector suggested and then move to the top electrode.

Significance of the results:
     (1) The relaxed geometries of the Au nanostructure with one straight chain between two electrodes were studied.
     (2) By using the dynamical matrix, soft modes of the Au chain were analyzed, and it was found that the lowest energy modes correspond to lateral motion in the  xy plane. This suggests that we should consider a deformation pathway which is not the wire-elongated pathway as generally assumed.
     (3) Fracture involving initial deformations in the lateral dimension are somewhat lower in energy than fracture where the deformation of the Au chain takes place essentially along the wire axis.  The associated energy barrier has an upper limit of  ≈1.1eV.
Future plans:
     (1) Since the collective motions of the soft phonon modes are dominated by the motions of the central atom of on  the chain. We can move the atoms along the eigenvector of the lowest phonon mode to obtain the potential profile. By solving the wavefunction, we can compare the quantum effect and classical effect.
     (2) We can study the ballistic transport conductance of this nano-contact which is more like to happen in the real life than the infinite zigzag chain.