1. Mechanics of Nanowires
Davide Cadeddu
Mechanics of Nanowires
Introduction to Nanomechanics - Fall 2018

2. Fluctuation-Dissipation Theorem
When limited by thermal fluctuations:
Gives a minimum detectable force:
mechanics of nanowires

3. Optimal Cantilever Geometry
For a given temperature we need to minimize mechanical dissipation:
mechanics of nanowires

4. Bottom-up NWs are ideal force sensors:
Bottom-up Nanowires
Bottom-up NWs are ideal force sensors:
nearly perfect crystalline structure
singly-clamped cantilever geometry
high aspect ratio and nanometer-scale dimensions
high resonance frequencies
mechanics of nanowires

5. MBE-grown GaAs/AlGaAs nanowires
mechanics of nanowires

6. Equation of motion 𝑥 (𝑡)+𝛾 𝑥 (𝑡)+ 𝜔 0 𝑥(𝑡)=𝐹(𝑡)
Linear Response
Equation of motion 𝑥 (𝑡)+𝛾 𝑥 (𝑡)+ 𝜔 0 𝑥(𝑡)=𝐹(𝑡)
mechanics of nanowires

7. 𝑥 𝑡 +𝛾 𝑥 𝑡 + 𝑥 𝑡 (𝜔 0 +𝛼 𝑥 2 (𝑡))=𝐹(𝑡)
Duffing Oscillator
Duffing Equation 𝑥 𝑡 +𝛾 𝑥 𝑡 + 𝜔 0 𝑥 𝑡 +𝛼 𝑥 3 (𝑡)=𝐹(𝑡) positive (negative) α could be seen as a hardening (softening) of the spring constant
𝑥 𝑡 +𝛾 𝑥 𝑡 + 𝑥 𝑡 (𝜔 0 +𝛼 𝑥 2 (𝑡))=𝐹(𝑡)
𝑥 𝑡 =𝑍 cos⁡(𝜔𝑡−𝜓)
mechanics of nanowires

8. Duffing Oscillator
Duffing Equation 𝑍 2 𝜔 2 − 𝜔 0 2 − 3 4 𝛼 𝑍 𝛾𝑍 𝜔 2 = 𝐹
Bistable solution!
The jumping point depends on the history of the resonator
mechanics of nanowires

9. Nanowire mode doublet
MODE I
MODE II
Slightly asymmetric nanowire gives two non-degenerate flexural modes
ω 0 = 𝛽 𝑛 5𝐸𝐴 24𝑚 d L 2
mechanics of nanowires

10. External Force
mechanics of nanowires

11. External Force
mechanics of nanowires

12. External Force
mechanics of nanowires

13. External Force
mechanics of nanowires

14. External Force
mechanics of nanowires