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lecture21 - 2.57 Nano-to-Macro Transport Processes Fall...

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2.57 Fall 2004 – Lecture 21 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 21 Last time we talked about the current density as 11 12 ' e d dT J L q L dx dx Φ = + , For electrons, q e = − and e e µ ϕ Φ = . Here e ϕ is electrostatic potential, which is related to the electrical field. Chemical potential µ is related to diffusion. Their combination Φ is electrochemical potential, indicating the total driving force of charges. The current density can be rewritten as 11 12 e d d J L L dx dx Φ Τ = + . Note: The second term 12 12 1 d d L L T dx T dx Τ Τ = is similar to dQ S T = and may be compared with entropy flux. The heat transferred is ( ) 21 22 2 x y z q x k k k d dT J v E f L L V dx dx µ Φ = = + ∑∑∑ For open circuits, Je=0. We obtain 12 11 / / h c L d dx V S dT dx T T L Φ = = = , where S is called the seebeck coefficient. Note: (1) S is dependent on the density of states. Therefore, it can be enhanced by using nanostructures such as thin films or nanowires. This effect is also the principle of thermal couples. (2) In the summation, we cannot use integral for quantized directions. x y Π 2 Π 1 1 2 q=J( Π 2 - Π 1 ) q=J( Π 1 - Π 2 ) J
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2.57 Fall 2004 – Lecture 21 2 (2) In / / d dx S dT dx Φ = − , T should be the electron temperature T e . In equilibrium cases, T e is close to phonon T p and we can use this effect to measure T p . However, for extreme cases such as laser ablation, the two temperatures are not in equilibrium. Cautions should be taken. (3) For an on-chip thermocouple, the measured temperature does not correspond to the junction point, but closer to the average temperature from 1 to 3. This is different from the normal thermocouples.
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