Formation and Self-Assembly of Quantum Dots N anostructure - - PowerPoint PPT Presentation

Formation and Self-Assembly of Quantum Dots Nanostructure Interdisciplinary Research Team

• M. Asta, P.W. Voorhees

Materials Science and Engineering, Northwestern S.H. Davis, M.J. Miksis and A. Golovin Applied Mathematics, Northwestern B.J. Spencer Applied Mathematics, SUNY Buffalo G.B. McFadden NIST

Interdisciplinary Research

• Collaboration between materials scientists and

applied mathematicians

• Jointly supervised students and postdocs
• Students will take courses in both departments
• Postdocs will spend time at a NIST
• Funded by both Div. Materials Research and Div.

Mathematical Sciences at NSF

• Research was initiated 2001

Projects Underway

• First-principles calculation of surface stress and

the stress-dependence of surface diffusion

• Equilibrium shape of quantum dots on surfaces
• Morphological stability of nanowires
• Dynamics of surface evolution during deposition,

both numerical and analytical

• Stability of interfaces under stress with highly

anisotropic surface energy

Thin Film Deposition

• Deposit one material on another from the vapor

How do dots form?

• Misfitting film is always unstable to the formation of

ripples

• Spencer and Meiron, Chiu

and Gao, Yang and Srolovitz – Nonlinear evolution – Isotropic surface energy – Finite-time singularity (Spencer and Meiron (1994))

Quantum Dots in Ge-Si

• Medeiros et al (1998)
• Ge islands on Si substrate
• Islands are faceted: pyramids and domes

Quantum Wires/Nanowires

From Chen et al. STM topographs showing ErSi From Chen et al. STM topographs showing ErSi2

2

(011) nanowires grown on a flat Si(001) substrate. The (011) nanowires grown on a flat Si(001) substrate. The Si terraces increase in height from deep blue to green. Si terraces increase in height from deep blue to green.

Stability of Nanowires

• Quantum wires with

isotropic surface energy would tend to bead up rather than persist as wires.

• Isotropic Case: Perturbation:

H(φ) sin (kz) Stable for λ < 2π R

• Anisotropic surface energy, but no missing orientations
• No elastic stress (at the moment)

Rod Stability for Cubic Materials

• n=0 cases are shown, n>1 are all stable
• [001] Orientation: four-fold symmetry axis

−1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5 x y

Cross Cross-

• sections of the

sections of the unperturbed rod for the [001] unperturbed rod for the [001] case: from the inner to the outer case: from the inner to the outer curve, curve, ε ε4

4 =

= -

• 0.0556,

0.0556, -

• 0.0278, 0,

0.0278, 0, 0.0278, 0.0556, and 0.0833. 0.0278, 0.0556, and 0.0833.

0.5 1 1.5 2 −1 −0.5 0.5 1 1.5 2 κ σ −0.06 −0.04 −0.02 0.02 0.04 0.06 0.08 0.1 0.5 1 1.5 2 2.5 3 3.5 ε4 κc 2

• Eigenvalues

Eigenvalues for n =0 for n =0

• Isotropic:

Isotropic: k kc

c= 1

= 1

• Anisotropy can stabilize or

Anisotropy can stabilize or destabilize the system destabilize the system

• Rod always unstable

Rod always unstable The dotted curve The dotted curve represents an asymptotic represents an asymptotic solution. solution.

Rod Stability for Cubic Materials: [001]

ε ε4

4 =

= -

• 0.0556

0.0556

• 0.0278

0.0278 0.0000 0.0000 0.0278 0.0278 0.0556 0.0556 0.0833 0.0833

−0.1 −0.05 0.05 0.1 1 2 3 4 5 ε4 kc 2

• [111] Orientation: three

[111] Orientation: three-

• fold symmetry axis

fold symmetry axis

The [111] orientation The [111] orientation has no plane of has no plane of symmetry normal to symmetry normal to cylinder axis cylinder axis

Rod Stability for Cubic Materials: [111]

Si(001) c2x4 Dimer Reconstruction Top view:

Bulk Atoms Surface Dimers

] 1 1 [ ] 110 [

First Principles Calculations: Surface Stress and Surface Diffusion

Ge Adatom Diffusion Calculations

Diffusion Path First-Principles Calculations

• Local Density Functional Theory

– Plane-Wave Pseudopotential Calculations (VASP code)

• NPACI Resources

Adatom

Dynamics: Coarsening of Growing Anisotropic Surfaces

Deposition flux Surface Diffusion Growth

• Same regularization as used in the equilibrium shape work
• No elastic stress

Dynamics: Convective Cahn-Hilliard

• Long wave equation in one dimension:
• D deposition rate, and u slope of the surface
• D < Dc
• D > Dc

Dynamics: Convective Cahn-Hilliard

• Integrated long wave equation in two dimensions
• D < D1
• D > D1

Dynamics: Asymptotics

• Convective Cahn-Hilliard equation, same anisotropic

surface energy

• Asymptotic solutions in the limit of small corner

curvature

• Evolution of surface given in terms of the motion of

kinks and antikinks

V

Projects Underway

• First-principles calculation of surface stress and

the stress-dependence of surface diffusion

• Equilibrium shape of quantum dots on surfaces
• Morphological stability of nanowires
• Dynamics of surface evolution during deposition,

both numerical and analytical

• Stability of interfaces under stress with highly

anisotropic surface energy