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Handout 23

# Handout 23 - Handout 23 Electron Transport Equations and...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 23 Electron Transport Equations and the Thermoelectric Effect In this lecture you will learn: • Position dependent non-equilibrium distribution functions • The Liouville equation • The Boltzmann equation • Relaxation time approximation • Transport equations • Thermoelectric Seebeck effect and the Seebeck tensor William Schockley (1910-1989) ECE 407 – Spring 2009 – Farhan Rana – Cornell University Note on Notation In this handout, unless states otherwise, we will assume a conduction band with a dispersion given by: ( ) k M k E k E T c r r h r . . 2 1 2 + = ( ) k M k v r h r r . 1 =

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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Position Dependent Non-Equilibrium Distribution Function We generalize the concept of non-equilibrium distribution functions to situations where electron distributions could also be a function of position (as is the case in almost all electronic/optoelectronic devices): The local electron density is obtained upon integration over k-space: ( ) t r k f , , r r ( ) ( ) ( ) t r k f k d t r n d d , , 2 2 , FBZ r r r r × = π Local Equilibrium Distribution Function: Electrons at a given location are likely to reach thermal equilibrium among themselves much faster than with electrons in other locations. The local equilibrium distribution function is defined by a local Fermi-level in the following way: ( ) ( ) ( ) ( ) KT t r E k E o f e t r k f , 1 1 , , r r r r + = with the condition that the local Fermi level must be chosen such that: ( ) ( ) ( ) t r k f k d t r n o d d , , 2 2 , FBZ r r r r × = π ECE 407 – Spring 2009 – Farhan Rana – Cornell University Case of No Scattering: Liouville Equation Question: How does the non-equilibrium distribution function behave in time in the absence of scattering? ( ) t r k f , , r r In time interval “ t ” each electron would have moved in k-space according to the dynamical equation: E r ( ) E e dt t k d r r h = ( ) t t r k f + , , r r ( ) ( ) value momentum final value momentum initial = + = t t k t k r r Consider an initial non-equilibrium distribution 2 d dimensions at time “ t ”, as shown There is also an applied electric field, as shown k r r r But in the same time interval t ” each electron would have moved in real-space according to the equation: ( ) ( ) ( ) t k v dt t r d r r r = ( ) ( ) value position final value position initial = + = t t r t r r r
3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University The distribution at time “ t + t ” must obey the equation: ( ) ( ) ( ) ( ) ( ) ( ) t t r t k f t t t t r t t k f , , , , r r r r = + + + This is because in time “ t “ the electron with initial momentum and position would have gone over to the state with momentum and position ( ) t k r ( ) t t k + r ( ) t r k f , , r r E r ( ) t t r k f + , , r r k r r r Case of No Scattering: Liouville Equation ( ) E e dt t k d r r h = ( ) ( ) ( ) t k v dt t r d r r r = ( ) t r r ( ) t t r + r ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t r k f t t t r k f t dt t r d t r k f t dt t k d t r k f t r k f t t r k f t t t dt t r d r t dt t k d k f t t r t k f t t t t r t t k f r k , , , , . , , . , , , , , , , , , , , , r r r r r r r r r r r r r r r r r r r r r r r r =

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Handout 23 - Handout 23 Electron Transport Equations and...

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