Exploring Boltzmann Equations: Derivations, Fermi Golden Rule, Electron-Phonon Interactions
Delve into the realm of Boltzmann equations with a focus on Fermi-Golden Rule, electron-phonon interactions, conservation laws, and the H-theorem. Uncover the derivation process, linearized Boltzmann equations, and Kubo linear response, along with significant insights into the electron-phonon system. Explore the intricate dynamics of electron-phonon coupling and the Fermi Golden Rule application. Understand the single-mode relaxation time approximation and solve Boltzmann equations in a uniform system to elucidate the electric current density.
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Week 9, Boltzmann equation Fermi-golden rule, derivation of the Boltzmann equations for electron-phonon system, conservation laws, H-theorem, linearized Boltzmann equation, Kubo linear response
Fermi golden rule P 2 ' H ( ) 2 + H f T i = = | '| W f H i E E 0 H fi f i H 0 0 = ( ) t T c n = ( 0) t i n here H n ' H = + H 0 Tt 0 = = , H i E i H f E f 0 0 i f = = ( ) t ( ) ( 0) solve for ( ) c t n c t c t T n n ni n n 2 = ( 0 i H t P c f i f = ) ' H + ' ' n H m H nm
Derive the Fermi-golden rule iE t dc dt = + = n ' , ( ) ( ) , then n i E c H nm m c c t A t e n n n n m = (0) A n ni ( ) ) iE E T f i T ( ) i i 1 i 1 i 1 dA dt e i ( ) ) ) E E t E E t m n f i = = ' ' ' , n i H e A A H e dt H f i nm m f fi fi ( ) E E m 0 f i ( ) E E T f i 2 4sin 2 2 2 = = ' P A H f i f fi 2 ( ) E E f i 2 sin ( ) xT = use for , ( ) x T 2 x T ?? ??
Problem setup : electron- phonon interaction ,q n,k 1 2 ( ) = + + + T j 2 H c Hc p u Ku c M c u j j g 1 2 k = + + n c c k a a q k k q q n n q n 1 N ( ) + + q m,k+q m ( ) k g c c a a + k q k q q mn n , , , , m n k q c ' H = + + 0 e 0 p H H k q , : electron band indices, : electron wavevector, m n : phonon branch index, : phonon wavevector, : electron band struct : phonon dispersion q ure k n + q k k k q ( ): electron-phonon coupling matrices, is associated with , : number of repeating unit cells for g N n m mn c
Graphene band structure ??? and phonon dispersion ???
Boltzmann equations Df Dt f t f r f f t + + = = v F v , nk k e e k ( ) colli DN Dt N t N r N q N t q q + r + q = = r , colli = = = = n r r ( , ) t , ( , ) t f f c c k N N a a q k k q q n n = = k q short-hand notation: ( ), ( ) k n q
Single-mode relaxation time approximation 0 Df Dt f t f r f f t f f = + v + = = F v k , n k k ( ) colli 1 1 = = = 0 F E , ( ) f e + ( ) 1 e k T k n B `solve' the Boltzmann equation in uniform sys tem and steady state: 0 0 f f f = + = + 0 0 0 F E E v f f f e f e k k ( ) ( ) k n vv electric current density: : 3 by 3 tensor or dyadic 3 3 k k f d d = = e = E 2 j v vv E ( ) 0 e f ( ) ( ) 3 3 2 2 n k n 1BZ 1 BZ
?? ???: electron Collision rate scattering out, absorbing a phonon ( for electron, for phonon) k k q n n n k = i q ' k = whatever the scattering leads to (and will summed over) f q 1 N + + is Kroneker delta = 2 q k k k k q ( ' ) g c c a i ' ' k k q c 1 N = + + q k k k k q ( ' ) 1 1 1 1 g n n n n n n ' ' ' k k q k k q c k ( ) 2 = Apply Fermi golden rule ' Tr f H i 1 ( ) 2 = q k k (1 ) (1 ) 1 (1 ') g n n n n n n n n n f f N ' ' ' ' k k q k k q k k q N c , ' , f n f n N n ' k k q 2 1 2 + q k k k k q ( ' ) (1 f ') ( ) g f N ' ' k k q N ' k q c
Recall the properties of creation/annihilation operators boson: = = + + = 1 , 1 1 , 0,1,2, a n n n a n n n n fermion: = = + = 1 , 1 1 , 0,1 c n n n c n n n n
Collision rate electron scattering out, emitting a phonon = ( for electron, for phonon) k i n n n q ' k k q = k whatever the scattering leads to (and will summed over) f q 1 N + q k k k k k q ( ' ) g c c a i ' ' k q c 1 N = + + + + q k k k k q ( ' ) 1 1 1 1 1 g n n n n n n ' ' ' k k q k k q c k ( ) 2 = Apply F ermi golden rule ' Tr f H i 1 ( ) 2 + + + = + q k k (1 )( 1) (1 )( 1) 1 1 (1 ')( 1) g n n n n n n n n n f f N ' ' ' ' k k q k k q k k q q N c , ' , f n f n N n ' k k q q 2 1 2 + + 1) ( + q k k k k q ( ' ) (1 f ')( ) g f N ' ' q k k q N ' k q c
electron scattering in, absorbing a phonon = ( for electron, for phonon) k i n n n q ' k k q = whatever the scattering leads to (and will summed over) f k q 1 N + + = 2 q k k ( ' k k q ) g c c a i ' ' k k q c 1 N = + + q k k ( ' k k q ) 1 1 1 1 g n n n n n n ' ' ' k k q k k q c 2 Apply Fe rmi golden rule ' f H i k 1 2 q kk (1 ) (1 ) (1 ) ' f f N g n n n n n n ' ' ' k k q k k q N c , ' , f n f n N n ' k k q 2 1 2 + + q kk ( ' k k q )(1 ) ' f f N ( ) g ' ' k k q N ' k q c
electron scattering in, emitting a phonon = ( for electron, for phonon) k i n n n q ' k k q = whatever the scattering leads to (and will summed over) f k q 1 N + + q k k ( ' k k q ) g c c a i ' ' k k q c 1 N = + + + + q k k ( ' k k q ) 1 1 1 1 1 g n n n n n n ' ' ' k k q k k q c 2 Apply F ermi golden rule ' f H i k 1 2 + + + q kk (1 ) ( 1) (1 ) ( 1) (1 ) '( f f N 1) g n n n n n n ' ' ' k k q k k q q N c , ' , f n f n N n ' k k q q 2 1 2 + + + 1) ( + q kk ( ' k k q )(1 ) '( f f N ) g ' ' q k k q N ' k q c
Electron collision rate, add four terms 1 f t ( ) = (1 ) (1 ) S f f S f f ' ' ' ' kk k k k k k k N ' k colli c 2 ( ) 2 = ) ( + + 1) ( + q kk k k q ( ' ) ( ) S g N N ' ' ' ' kk q k k q q k k q q = = ( , ), = ( , k q q short-hand notation: ( , ), n ) k q q
Collision rate for phonon k, f k q q k , 1 f k N 2 1 2 ) (1 q = + ) ( + 1) (1 q kk ( ' k k q ) ( ) g f f N f f N ' ' ' ' k k q k k q k k q t N ' kk c colli Show that both collision rates are 0 at thermal equilibrium when 1 1 = = = = 0 0 q , f f N N k k q + ( ) 1 e 1 e q k
Conservation laws Electron numbers are conserved, ??= 1 ? ??3? ??? Total momentum is not conserved, as the wave vector is conserved only modulo a reciprocal lattice vector ? Energy is conserved not counting the EP interacting H term, ? = 1 ? ??3? ?????+ ? ???? H-theorem holds, entropy must increase
Prove the H-theorem 1 V ( ) ( ) = + + + + 3 r ln (1 )ln(1 ) ln ( 1)ln( 1) H d f f f f N N N N k q V show that dH dt 0 when Boltzmann equation is satisfied
Linearized Boltzmann equation 0 0 f N = = , 0 0 , , ' , f f N N ' k k q ( ) 0 0 f N = = + 0 0 0 0 (1 ), (1 ) , f f N N k q ( ) f t f dT dz f t ( ) + + = = ( ) + z q ' v eE P ' kk T N = ' , 1 k q colli c dT dz N t N N t ( ) p + = = ( 1) + + z q q ' v P ' kk T N ' kk colli c 2 2 ( 1) + = ) ( 0 0 0 q q kk k k q ( ' ' ) (1 ) ' f P g N f ' ' kk 2 2 ( 1) = + ) ( + + 0 0 0 q q kk k k q ( ' ' )( 1)(1 ) ' f P g N f ' ' kk
Relaxation time approximation (ignore off-diagonal terms in the linearized collision rates) 1 2 N 2 = + ') ( ) ( + + ') ( + q kk k k q ( ' ) (1 ' ' ) g N f N f ' ' k q c 1 = ( work out the expression for phonon ) p k : number of unit cells = number of points in 1st BZ N c
Linearized Boltzmann equation, linear nonequilibrium thermodynamics + + = = , , are known vectors F X F X F X H X 1 1 2 2 3 3 i : electric current : electron heat current : phonon heat current 1 T i X V J J J 1 dT dz 1 T dz 1 dT 2 p = = = , , F E F F 1 2 3 T 3 p = J i 3 = = , Onsager reciprocal relation: J L F L L i ik k ik ki = 1 k 1 V = = 1 1 T i T ( , is pseudo-inverse) L X H X H H H ik k 1 T 1 3 dS dt = = T entropy production: 0 F J H i i VT = 1 i
Kubo linear response theory H+F H + + t , 0 0 H H Fe F t t = ( ) t H tot , t 0 + 1 i = = t (0) (0); ( ) , j t ; [ , ] a b j dte F a b 0 1 b d H H Kubo correlation: ; Tr a b e ae eq 0 H 1 e = , = eq H k T Tr e B
