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
Introduction of Nano Science and Tech. Basics of Solid Mechanics in Nanostructures. Nick Fang
Introduction of Nano Science and Tech Basics of Solid Mechanics in Nanostructures Nick Fang Course Website: nanohub.org Compass.illinois.edu ME 498 2006-09 Nick Fang, University of Illinois. All rights
More informationIntroduction of Nano Science and Tech. Thermal and Electric Conduction in Nanostructures. Nick Fang
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