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Computational Solid State Physics 計算物性学特論 第9回 9. Transport properties I: Diffusive transport Electron transport properties le: mean free path of electrons lφ: phase coherence length λF: Fermi wavelength Examples of quantum transport key quantities le : mean free path of electrons lφ: phase coherence length λF: Fermi wavelength single electron charging Point contact: ballistic quantum conductance 2e 2 Go h Aharanov Bohm effect: phase coherent quantum magnetic flux 0 h / e Quantum dot: single electron charging Shubnikov-de Haas oscillations and quantum Hall effect Diffusive transport Equation of motion for electrons dk F k: wave vecot of Bloch electron dt d 1 r Vg (k ) dt k Scattering Rate 2 2 Pkk k (r ) Vscat (r ) k (r ) (k ) (k ) 1 Pkk' k k' How to solve equations of motion for electrons with scattering? Relaxation time approximation for scattering Direct numerical solution: Monte Carlo simulation Boltzmann equation for distribution function of electrons Relaxation time approximation 2k 2 (k ) c 2m m: effective mass 1 dk d 2 (k ) 1 F 2 dt dt dk m dvg equation of motion m dv g F m vg dt F e( E v g B ) j env g E: electric field B: magnetic field current density n: electron concentration Drude model: B=0 m dvg dt eE m vg E E0 exp( it ) e 1 : drift velocity vg E m 1 i ne 2 1 j envg E j E m 1 i ne 2 1 conductivity ( ) m 1 i Drude model: steady state solution in magnetic field m dv g F m vg dt F e( E v g B ) eB c m : cyclotron frequency : B is assumed parallel to z. F e( E x v y B , E y v x B , E z ) drift velocity v gx 1 e 1 v gy 2 2 c m 1 c v 0 gz c 1 0 E x E y 1 c2 2 E z 0 0 Conductivity tensor in magnetic field j env g E 1 ne 2 1 2 2 c m 1 c 0 eB c m c 0 1 0 2 2 0 1 c Hall effect B // z I // x 1 ne 1 j 2 2 c m 1 c 0 2 eB c m c 0 Ex 1 0 E y 2 0 1 c 2 Ez jx B ne jy 0 E y cE x ne 2 jx Ex m no transverse magneto-resistance Hall effect Monte Carlo simulation for electron motion dk F dt d 1 r Vg (k ) dt k 2 Pkk k (r ) Vscat (r ) k (r ) (k ) (k ) 2 dk F dt Scattering d 1 r Vg (k ) dt k Drift 2 Pkk k (r ) Vscat (r ) k (r ) (k ) (k ) Drift Scattering 2 Drift velocity as a function of time Vx 1 t Vx (t )dt t 0 jx en vx : current Boltzmann equation Motion of electrons in r-k space during infinitesimal time Interval Δt k r v g t r F k t Equation of motion for distribution function equation of motion for electron distribution function fk(r,t). f k (r , t dt ) f k (r , t ) f k (r , t dt ) f k (r , t ) f k | force f k |diff dt dt f Fdt (r , t ) f k (r , t ) f k (r v g dt , t ) f k (r , t ) k dt dt f F f k v g k r k Boltzmann equation f k f k |diff f k | force f k |scatt Steady state f k |diff f k | field f k |scatt F k f k v g r f k f k |scatt Boltzmann equation Electron scattering f k |scatt 1 (2 ) 3 1 [ f (1 f k ) Pkk ' f k (1 f k ' ) Pk 'k ]dk ' (2 ) 3 [ f f k ]Pkk ' dk ' k' k' detailed balance condition for transition probability Pkk ' Pk 'k Scattering term assume: elastic scattering, spherical symmetry 1 [ f f k ' ]Pkk ' dk ' (2 ) 1 0 0 [( f f ) ( f f k k k' k ' )] Pkk ' dk ' 3 (2 ) 1 0 ( fk fk ) 3 k k 1 k 1 (2 ) 3 P kk ' (1 cos kk ' )dk ' Pkk ' P(k , kk ' ) Transport scattering time 1 k 1 (2 ) 3 P kk ' Pkk ' P(k , kk ' ) (1 cos kk ' )dk ' k’ Θkk’ k Contribution of forward scattering is not efficient. Contribution of backward scattering is efficient. Linearized Boltzmann equation f k0 f k0 1 0 vk T vk F ( fk fk ) T k 0 0 f f f k f k0 vk ( k T k F ) k T 1 1 P (1 cos kk ' )dk ' 3 kk ' k (2 ) Pkk ' P(k , kk ' ) Fermi sphere is shifted by electric field. Current density and conductivity j 2 (2 )3 3 evf d k k f k0 3 j ev ( v eE ) d k k k k 3 (2 ) 2 j x xx E x xx 0 f e2 2 3 vx k k d 3k 4 Electron mobility in GaAs Energy flux and thermal conductivity f k0 3 U ( k ) v ( v T ) d k k k k 3 (2 ) T 2 f k0 3 ( k ) v v d k T k k k 3 (2 ) T 2 thermal conductivity U KT f k0 3 K ( k ) v v d k k k k 3 (2 ) T 2 Problems 9 Calculate both the conductivity and the resistivity tensors in the static magnetic fields, by solving the equation of motion in the relaxation time approximation. Study the temperature dependence of electron mobility in n-type Si. Calculate the electron mobility in n-type silicon for both impurity scattering and acoustic phonon scattering mechanisms, by using the linearized Boltzmann equation.