Electrons & Phonons. Thermal Resistance, Electrical Resistance P = I 2 R T = P R TH V = I R. R = f( T)

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1 lectrons & Phonons Ohm s & Fourier s Laws Mobility & Thermal Conductivity Heat Capacity Wiedemann-Franz Relationship Size ffects and Breadown of Classical Laws 1 Thermal Resistance, lectrical Resistance P = I R T = P R TH V = I R R = f( T) Fourier s Law (18) Ohm s Law (187)

2 Poisson and Fourier s quations ( V ) J V qdn drift diffusion ( T ) p JT T diffusion only Note: chec units! Some other differences: Charge can be fixed or mobile; Fermions vs. bosons 3 Mosquitoes on a Windy Day Some are slow Some are fast Some go against the wind n n(v )dv wind Also characterized by some spatial distribution and average n(x,y,z) dv Can be characterized by some velocity histogram (distribution and average) v 4

3 Calculating Mosquito Current Density Area A dr Current = # Mosquitoes through A per second N A = n(r,v)*vol = n(r,v)*a*dr Per area per second: J A = dn A /dt = n(dr/dt) = nv 5 Charge and nergy Current Density Area A dr What if mosquitoes carry charge (q) or energy () each? Charge current: J = qnv Units? nergy current: J = nv Units? 6

4 Particles or Waves? Recall, particles are also matter waves (de Broglie) Momentum can be written in either picture So can energy p m v * 1 * m v p v * * m m Acnowledging this, we usually write n() or n() 7 Charge and nergy Flux (Current) Total number of particles in the distribution: Charge & energy current density (flux): J q n( ) v( ) q g( ) f ( ) v( ) q Of course, these are usually integrals (Slide 1) J ( ) n( ) v( ) ( ) g( ) f ( ) v( ) N n( ) g( ) f ( ) 8

5 What is the Density of States g()? Number of paring spaces in a paring lot g() = number of quantum states in device per unit volume How big is one state and how many particles in it? L L n n L d vol, d 1 s L d d dimensions s spins or polarizations L d L L, A, or V 9 Constant nergy Surface in 1-, -, 3-D L y z x x y x For isotropic m *, the constant energy surface is a sphere in 3-D - space, circle in -D -space, etc. x Si z m l.91m m t.19m y llipsoids for Si conduction band Odd shapes for most metals 1

6 Counting States in 1-, -, or 3-D L y 3 L z x x y x System has N particles (n=n/v), total energy U (u=u/v) This is obtained by counting states, e.g. in 3-D: N g( ) f ( ) d U 1 u ( ) g( ) f ( ) d V V 11 What is the Probability Distribution? Probability ( f() 1) that a paring spot is occupied Probability must be properly normalized 1 f( ) Just lie the number of particles must add up N n( ) g( ) f ( ) But people generally prefer to wor with energy distributions, so we convert everything to instead of, and wor with integrals rather than sums 1

7 Fermi-Dirac vs. Bose-instein Statistics 1. f FD T= T=3 T=1 f B T= T=3 T=1 -. f FD ( ) - F (ev) 1 F exp 1 T B Fermions = half-integer spin (electrons, protons, neutrons) f..4.6 (ev) B ( ) 1 exp 1 T B Bosons = integer spin (phonons, photons, 1 C nuclei) In the limit of high energy, both reduce to the classical Boltzmann statistics: 13 Temperature and (Non-)quilibrium Statistical distributions (FD or B) establish a lin between temperature, quantum state, and energy level Particles in thermal equilibrium obey FD or B statistics Hence temperature is a measure of the internal energy of a system in thermal equilibrium What if (through an external process) I greatly boost occupation of electrons at =.1 ev? Result: hot electrons with effective temperature, e.g., at T=1 Non-equilibrium distribution f FD T=3 T= F (ev) 14

8 Counting Free lectrons in a 3-D Metal N d n g( ) f ( ) d f ( ) 3 V V L / L 3 T > spin # states in -space spherical shell F 3 z L d y 1/ g x Convert integral over to integral over since we now energy distribution (for electrons, Fermi-Dirac) 15 Counting Free lectrons Over At T =, all states up to F are filled, and above are empty This also serves as a definition of F f 1,, F F 3 1 g D 3/ 1/ m F 1 m n g f d g d F F F 3 3 n m /3 h 3 n 8m /3 /3 11 J 6 ev 891 T 4 F F / B few 1 K So F >> B T for most temperatures 3/ 3/ 1/ g F 16

9 Density of States in 1-, -, 3-D (with the nearly-free electron model and quadratic energy bands!) * 3/ 3D 1 m 1/ g g D m * m g *3/ 1/ 1D 1/ van Hove singularities 17 lectronic Properties of Real Metals Metal n (1 8 m -3 ) m * /m v F (1 6 m/s) Cu Ag Au Al Pb source: C. Kittel (1996) Fermi equi-energy surface usually not a sphere In practice, F and m * must be determined experimentally m* m due to electron-electron and electron-ion interactions (electrons are not entirely free ) Fermi energy: F = m * v F / 18

10 nergy Density and Heat Capacity U d u ( ) g( ) f ( ) d ( ) f ( ) 3 V V L / L 3 similar to before, but don t forget () u f ( ) CV g( ) d T T C V n B T B F For nearly-free electron gas, heat capacity is linear in T and << 3/n B we d get from equipartition Why so small? And what are the units for C V? 19 Heat Capacity of Non-Interacting Gas Simplest example, e.g. low-pressure monatomic gas Bac to the mosquito cloud From classical equipartition each molecule has energy 1/ B T per degree of motion (here, translational) 3 Hence the heat capacity is simply CV nb Why is C V of electrons in metal so much lower if they are nearly free? F B T

11 Current Density and nergy Flux Current density: J q n( ) v( ) q g( ) f ( ) v( ) d q v 1 What if f() is a symmetric distribution? nergy flux: J n( ) v( ) ( ) g( ) f ( ) v( ) ( ) d Chec units? 1 Conduction in Metals Balance equation for forces on electrons (q < ) dv v m m q( ε vb) dt Balance equation for energy of electrons d IV dt Current (only electrons near F contribute!) J q( n) v F

12 Boltzmann Transport quation (BT) The particle distribution (mosquitoes or electrons) evolves in a seven-dimensional phase space f(x,y,z, x, y, z,t) The BT is just a way of booeeping particles which: move in geometric space (dx = vdt) accelerate in momentum space (dv = adt) scatter Consider a small control volume drd Rate of change of particles both in dr and d = Net spatial inflow due to velocity v() + Net inflow due to acceleration by external forces + Net inflow due to scattering f t F vr f f f t scat 3 Relaxation Time Approximation (RTA) f t F vr f f f t scat Where recall v = and F = In the relaxation time approximation (RTA) the system is not driven too far from equilibrium f f f f t ( ) ( ) '( ) scat scat scat Where f () is the equilibrium distribution (e.g. Fermi-Dirac) And f () is the distribution departure from equilibrium 4

13 BT in Steady-State ( f/ t = ) F f ' vr f f Still in RTA, approximate f on left-side by f Furthermore by chain rule: F F f f So we can obtain the non-equilibrium distribution f ' v r f f F v And now we can directly calculate the current & flux 5 Current Density and Mobility Jq qg( ) f '( ) v( ) d qg( ) vr f F v vd f This may be converted to an integral over energy Assume F is along z: v v / d (dimension d=1,,3) z q q Jq, z n F * z BT n * m z m (Lundstrom, ) (Ferry, 1997) Assumption: position-independent relaxation time (τ) 6

14 nergy Bands & Fermi Surface of Cu Fermi surface of Cu is just a slightly distorted sphere nearly-free electron model is a good approximation 7 Why the nergy Banding in Solids? ( x) u ( x) e ix u ( x) u ( x a) Key result of wave mechanics (Felix Bloch, 198) Plane wave in a periodic potential Wave momentum only unique up to π/a lectron waves with allowed (,) can propagate (theoretically) unimpeded in perfectly periodic lattice 8

15 What Do Bloch Waves Loo Lie? u ( x) u ( x a) Periodic potential has a very small effect on the plane-wave character of a free electron wavefunction xplains why the free electron model wors well in most simple metals 9 Semiconductors Are Not Metals empty states F empty conduction band filled valence band electron states in isolated atom electron states in metal electron states in semiconductor Filled bands cannot conduct current What about at T > K? Where is the F of a semiconductor? ne e pe h m m e h 3

16 Semiconductor nergy Bands Si GaAs qui-energy surfaces at bottom of conduction band Si GaAs Silicon: six equivalent ellipsoidal pocets (m l*, m t* ) GaAs: spherical conduction band minimum (m*) 31

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