1. Stochastic Field Theory of Metal Nanostructures Seth Merickel Mentors: Dr. Charles Stafford and Dr. Jérôme Bürki May 3, 2007 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A AA A A A AA A

2. Stability of Metal Nanowires  Y is the maximum stress that a cylindrical wire can sustain before it undergoes plastic flow. Stress due to surface tension  s / R implies R min =  s /  Y where R min is the minimum radius. Experiments have shown that there are stable wires with much smaller radii.

3. The Goal To understand nanowire stability: How stable are atomically-thin wires? To develop a multi-component stochastic field theory describing the dynamics of nanowires with quadrupolar cross sections. Use the equations of motion to look for stationary energy configurations, in particular saddle points.

4. Kramer’s Formula Kramer’s formula describes the rate at which a nanowire will transition between metastable energy configurations with an energy barrier  E.

5. Electronic Energy Functional The Electronic energy acts like a potential energy for the metal ions in the Born-Oppenheimer approximation. The Energy  e is given by the Weyl expansion with the addition of a shell correction. The curvature term will be neglected in the following.

6. Quadrupolar Nanowires The radial function depends on z and. Where  ( z, t ), ( z, t ), and  ( z, t ) are dynamical fields. The volume of the wire does not depend on the parameters ( z, t ) or  ( z, t ).

7. Volume Integrating over r and yields:

8. Surface Area Where g is the Riemannian metric:

9. Shell Potential s 0 L dzV shell The quantum energy correction has been calculated by solving the Schrödinger equation numerically. The parameters  s, and  are material dependent; V shell is universal when expressed in natural units for the Fermi gas.

10. Ionic Energy Atoms are allowed to enter and leave the wire through the metal contacts. The grand canonical ensemble must be used for both the ionic and electronic energies.  a is the chemical potential of a surface atom.

11. Structural Dynamics The parameters ,, and  are perturbed from equilibrium when the wire fluctuates:

12. Chemical Potential  a is calculated for an equilibrium configuration and is defined by: The factor arises from the requirement that the change of radius adds one atom to the system.

13. Energy Expansion The energy is now Taylor-expanded in We keep terms up to second order  z  i. Where:

14. Equations of Motion The equations of motion are calculated by taking the functional derivative of. is white noise modeling thermal fluctuations. Where :

15. Results I have derived the stochastic Ginzburg- Landau equations describing the structural dynamics of metastable nanowires. These equations can be solved to obtain stationary solutions; i.e., saddle configurations. With the stationary solutions known, we can begin answering the question: How stable are atomically-thin wires?

16. References D.F. Urban, J. Bürki, C.A. Stafford, Hermann Grabert, Phys. Rev. B 74, 245414 (2006) J. Bürki, C.A. Stafford, D.L. Stein, Phys. Rev. Lett. 95, 090601 (2005) J. Bürki, C.A. Stafford, Appl. Phys. A 81, 1519-1525 (2005) Acknowledgments Special thanks to Dr. Stafford and Dr. Bürki for all of their help and guidance this semester.