Corey FlackDepartment of Physics, University of Arizona

Thesis Advisors:Dr. Jérôme Bürki

Dr. Charles Stafford

OverviewMotivationModeling the nanowireMonte-Carlo simulated annealingSimulated annealing in the grand canonical ensembleResults: Equilibrium structuresConclusions

Nanowires are of principal interest for applications in nanotechnology

What is their atomic structure?Early simulations predicted non-crystalline structures of

either icosahedral packing or a helical multishell TEM video by Takayanagi ‘sgroup suggests helical structuresClassical structural models lead to

Rayleigh instabilityNeed quantum mechanics!

Atomic scale TEM image of a gold nanowire. Diagram courtesy of Ref. [2]

Cylindrical nanowires are found to be stable with a number of conductance channels equal to magic conductance values

Predicted by nanoscale free-electron model (NFEM)Confinement potential generated by the electron

gas

Conductance quanta:

24

1RkG

GN Fo

C

h

eGo

22

Stability diagram for cylindrical nanowires. Diagram courtesy of Ref. [4]

The total energy of the ions

j ji

iji

iii REzUrE int2

1,

Confinement Interaction Energy zUR

RAzU end

22

),(

otherwise

RLz

Rz

RLz

Rz

R

AzU S

S

s

S

send 4

4

0

2

2

o

LRm

ijij

sije

R

aRE

4)(int

•Solution to Poisson’s equation using NFEM electronic density

•Phenomonological, hard-core repulsion

•Screened Coulomb force

•Kinetic energy is neglected for an equilibrium state

Monte-Carlo simulated annealing methods use random displacements with a slow cooling method

Attempt to reach a minimum energy configurationBeginning at high temperatures – high thermal

mobilityAs T is lowered, atoms are frozen into a minimum

energy configurationMetropolis algorithm: new configurations are

generated from random displacements of the ionsAccepted with a probability of:

kTEp /exp

Simulated annealing in the canonical ensemble

V, N, and T are externally controlled parametersInitial random configuration of uniform densityRandom, isotropic displacement of one atom

imitating Maxwellian velocity distribution

Acceptance of moves according to Boltzman factor:Decreases in energy automatically acceptedIncreases accepted with finite probability

rnormkT

kTdr

oR

Conductance G=1Go – zigzag structure

Arbitrary orientation

Torsional stiffness

Conductance G=3Go – helical hollow shell with four atomic strands

Equilibrium structures in the canonical ensemble

Canonical ensemble does not represent the physical reality of a wire suspended between two contacts

Canonical ensemble: difficult to anneal out defects at the wire ends

Grand canonical ensemble allows for atom interchange with the contacts

V, T, and μ are externally controlled parameters

Conductance G=3Go – trapped defect at wire end

Implementation of the grand canonical ensemble allows for two new Monte-Carlo moves: addition and removal of atoms

Probability of move acceptance is given by the Gibbs factor:

Probability of trying removal is dependent on position

Placement of additional atoms determined by:

kTNEp /exp

otherwise

RLz

Rz

RLzC

RzC

zp S

S

S

S

4

4

0

exp

exp

)(

oS

oS

ZRL

ZRz

1ln

1ln

0

0.5

1

1.5

2

2.5

30 40 50 60 70 80 90 100

Final number of atoms(b)

Ch

emic

al p

ote

nti

al

0

0.5

1

1.5

2

2.5

0 20 40 60 80

Final number of atoms(a)

Ch

emic

al p

ote

nti

al

Simulations were run for various constant chemical potentials

Rise at N=60: region of canonical ensembleDisposition to atom removal:

Kinetic effects Atom addition method Non self-consistent confining potential

Final number of atoms in 3Go wire with constant chemical potential. A) Initial number of atoms: 60. b) Ion addition radial range changed to [0,R+2RS]. Initial number of atoms: 60.

Simulations with constant chemical potential also generated underfilled and overfilled structures for

wires of conductance G = 3Go

Underfilled wire: 4 atomic strands No = 60; Nfinal = 48

Overfilled wire: helical structure of 5 atomic strands, line of atoms through axis No = 60; Nfinal = 93

ConclusionsEquilibrium wire structures with G = 1 and 3Go were

obtained within canonical and grand canonical simulationsAdvantages of grand canonical simulation:

Correct physical ensemble Defects not trapped at wire ends

Difficulties of grand canonical ensemble: μ not known a priori Achieving detailed balance between atom addition and removal

Further directions: Further experimentation within the grand canonical ensemble Exploring equilibrium structures for higher conductance wires

AcknowledgementsDr. Bürki Dr. Stafford

All atomic structure images generated by Jmol: an open-source Java viewer for chemical structures in 3D. http://www.jmol.org/

References:[1] O. Gülseren, F. Ercolessi, E. Tosatti, Phys. Rev. Lett. 80, 3775 (1998)[2] Y. Kondo, K. Takayanagi, Sci. 289, 606 (2000)[3] C.A. Stafford, D. Baeriswyl, J. Bürki, Phys. Rev. Lett. 79, 2863 (1997)[4] J. Bürki, C.A. Stafford, Appl. Phys. A 81, 1519 (2005)[5] N.W. Ashcroft, N.D. Mermin. Solid State Physics. (1976)[6] Numerical Recipes in C, 10.9, pp.444-455[7] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, J. Chem. Phys. 21, 1087

(1053)[8] Numerical Recipes in C, 7.2, pp. 289-290[9] D. Conner, Master Thesis (2006), N. Rioradan, Independent studies with C.A. Stafford

(2007)[10]D. Frenkel. Introduction to Monte Carlo Methods.