Simulating atomic structures of metallic nanowires in the grand canonical ensemble
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Transcript of Simulating atomic structures of metallic nanowires in the grand canonical ensemble
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.