Introduction of Nano Science and Tech. Thermal and Electric Conduction in Nanostructures. Nick Fang

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1 Introduction of Nano Science and Tech Thermal and Electric Conduction in Nanostructures Nick Fang Course Website: nanohub.org Compass.illinois.edu ME Nick Fang, University of Illinois. All rights reserved. 1

2 First Midterm Friday, Sept PM Coverage: Scaling Quantum Effects Molecular Dynamics of Transport Nanoscale Solid Mechanics A Review Lecture on Monday, Sept 21 ME Nick Fang, University of Illinois. All rights reserved. 2

3 Quantum Fracture Mechanics V Applying Griffith s approach to atomic lattices, e.g. graphenes: U U a a a a U T a U E a crit From N. Pugno and R. S. Ruoff, Quantized fracture mechanics, Philosophical ME Nick Fang, University of Illinois. All rights reserved. 3 Magazine 84 (2004),

4 Quantized Critical Strength! From N. Pugno and R. S. Ruoff, Quantized fracture mechanics, Philosophical Magazine 84 (2004), ME Nick Fang, University of Illinois. All rights reserved. 4

5 Mechanical coupling at Nanoscale E.G. Piezoelectricity (i.e. electric potential in response to applied stress) S = strain (relative length change ΔL/L, dimensionless) L0= ceramic length [m] E = electric field strength [V/m] dij= piezoelectric coefficient of the material [m/v] See: ME Nick Fang, University of Illinois. All rights reserved. 5

6 Magneto-restrictive Effect Magnetostriction is the strain of a material in response to change of magnetization. giant magnetorestriction found in nanostructured materials Application: Flat panel speakers (e.g. sound bugs) ME Nick Fang, University of Illinois. All rights reserved. 6

7 Shape Memory Effect ME Nick Fang, University of Illinois. All rights reserved. 7

8 Mechanical Nanoresonators Putting Mechanics into Quantum Mechanics, Keith C. Schwab and Michael L. Roukes, Physics Today, 2005, 36-42) ME Nick Fang, University of Illinois. All rights reserved. 8

9 Thermal Noise in Resonators Cantilever total energy: Each are subject to thermal noise 1/2kT From We get ME Nick Fang, University of Illinois. All rights reserved. 9

10 Back to Constitutive Equations Hooke s Law Fourier s Law = q = -k Fick s Law of Diffusion J = -DC Newton s law on shear stress = -du/dy) Ohm s Law J= How are they correlated in the nanoscale? ME Nick Fang, University of Illinois. All rights reserved. 10

11 Look Into the Conducting Nanowires A Cloud of moving carriers - Some are slow - Some are fast - Some move against the mainstream Number of particles with momentum k V Momentum k ME Nick Fang, University of Illinois. All rights reserved. 11

12 Calculating Current Density Current = # electrons through A per second I e n( r, k) Adr / dt Average number of electrons Average velocity Current density: J e n( r, k) v( r, k) Power density: Q E( r, k) n( r, k) v( r, k) ME Nick Fang, University of Illinois. All rights reserved. 12

13 Net Charge/Energy Flux In order to find the net charge/energy flux (net current/power density), we need to consider all possible states at thermal equilibrium J e n( r, k) v( r, k) dk k Q E( r, k) n( r, k) v( r, k) dk k Note: n( r, k) dk DOS( r, k) p( r, k) dk Density of States Boltzmann Distributions ME Nick Fang, University of Illinois. All rights reserved. 13

14 Recall: For Quantum Particles Electrons: only two states possible (conduction, valence) p( E ) i exp(( EF Ei ) / kbt ) 1 exp(( E E ) / k T ) F i B Photons and Phonons: all possible states of energy nh p( ) 1 exp( h / k T ) 1 B ME Nick Fang, University of Illinois. All rights reserved. 14

15 Time Evolution in f(r, k) We Learned from fluidic mechanics: (r1, k1,t) (r2, k2,t) Since k is also changing over time, we add: r D F F F Dt t t r r D k p( r, k) p r p k p Dt t t t (Corrected for particle motion) (Since total number of states is unchanged over time) 0 Reaction Convection Acceleration ME Nick Fang, University of Illinois. All rights reserved. 15

16 Boltzmann Transport Equation t r r F p v r p k p h Convection Acceleration So how to estimate reaction? Locally, the system that is away from thermal equilibrium has a tendency to relax toward equilibrium state: t p p p 0 Equilibrium Distribution ME Nick Fang, University of Illinois. All rights reserved. 16

17 Steady State Boltzmann TE r p p0 r F v r p k p h In order to write p(r, k) explicitly, we further assume p p0 p( r, k) p( r, k) p 0 Further, we learned the trick from DOS: p E E k p k r r F p p p 0 0 v ( r p0 ) h E Velocity! ME Nick Fang, University of Illinois. All rights reserved. 17

18 Current Density and Mobility J e n( r, k) v( r, k) dk k r r F p0 J e D( r, E) v( r, E) ( v ( r p0 )) de h E E Assume F along z direction, we find: kbtne ene J ( e ) Fz z m m Electron diffusion Migration under external field ME Nick Fang, University of Illinois. All rights reserved. 18