Elements of the conductivity matrix

Transcription

1 Eleents of the conductivity atrix e v ( ) D( ) f ( ) d e f e E 4 T ( ) v ( ) ( ) v ( ) B E E( ) v ( ) T d B, T, e ( ) ( ) f ( ) 4 v E E d j is not necessarily parallel to E set E xˆ j E elec n n to calculate xx, xy, and xz

2 Free electrons e v ( ) D( ) f ( ) d v ( ) E( ) D( ) spin E ( ) j elec e ( ) 4 f d probability that the states are occupied

3 Oh's law (free electrons) f 4 T e e / ( ) B E T d B, T, e f ( ) 4 E d Choose E to be in the z-direction cos d * 4 e E f

4 A useful integral d f 4 cos F f cos ˆ ˆ x x y y z zˆ d The x and y coponents are zero: odd functions over even intervals. d sin dj d d cos z f cos sin djd d The j integration contributes a factor of. f d d 4 cos sin

5 4 cos sin f d d A useful integral () The J integration contributes a factor of /. 4 4 f d * F F f 4 cos f F d

6 Oh's law (free electrons) e E f e E jz d * F cos * 4 F For free electrons: n F Drude result: ne E j ne E z z * ob z Project: Write a progra that will calculate the integral over.

7 Boltzann equation The general for of the electrical current density is: e v ( ) D( ) f ( ) d e f e E 4 T ( ) v ( ) ( ) v ( ) B E E( ) v ( ) T d

8 Theroelectric current (free electrons) e f e E( ) 4 T ( ) B E ( ) E T d,,, B E E e f E( ) 4 T ( ) Td 4 e T f e T f cos d cos d * * 4 T 4 T Siilar to the useful integral Useful integral

9 Theroelectric current e T 4 e T 4 4 T 4 T 4 5 F F * * j elec F e F T * * T F e E * FT T E F E F For free electrons: n F ne T E T F E * FT n ob T

10 Diffusion current (free electrons) e f e E( ) 4 T ( ) B E ( ) E T d B, T, E e f ( ) 4 d Choose in the z direction e f d cos 4 j elec e n n ob Useful integral

11 Seebec effect e f e E 4 T ( ) v ( ) v ( ) B E E( ) v ( ) T d B, j, E elec The electric current due to an electrootive force is cancelled by the electric current due to a theral gradient f e E( ) ( ) ( ) ( ) T v E E v T d Theropower (Seebec effect): Q n E T / e dx n hot + + E - - cold

12 Theroelectric effects Seebec effect: A theral gradient causes a theral current to flow. This results in a voltage which sends the low entropy charge carriers bac to the hot end. V QT electrocheical Q is the absolute theral power. The sign of the voltage (electrocheical potential, electrootive force) is the sae as the sign of the charge carriers. The Seebec effect can be used to ae a theroeter. The gradient of the teperature is the sae along both wires but the gradient in electrocheical potential differs. V Therocouple T ref T sense

13 Theroelectric effects Intrinsic Q is negative because electrons have a higher obility.

14 Theroelectric effects I Peltier effect: driving a through a bietallic junction causes heating or cooling. Cooling taes place when the electrons ae a transition fro low entropy to high entropy at the junction.

15 Hall effect e f e E( ) 4 T ( ) B E ( ) E T d I x B Bz, jx, j y, jz, T A e f e ( ) ( ) 4 B E E d Hall coefficient: diffusive etals: R l n R H El j B e 1 ne n

16 Nernst effect e f e E( ) 4 T ( ) B E ( ) E T d B B, T z e f e E( ) ( ) ( ) 4 T B E E T d Nernst coefficient: N ln B El T n Open circuit and easure voltage

17 Ettingshausen effect e f e E( ) 4 T ( ) B E ( ) E T d B B, E, T z e f e E( ) 4 T ( ) B ( ) E T d Ettingshausen coefficient: P ln 1 j B T el n short circuit and easure current

18 Boltzann Group Albert von Ettingshausen, Prof. at TU Graz. Nernst was a student of Boltzann and von Ettingshausen. He won the 19 Nobel prize in Cheistry. (Standing, fro the left) Walther Nernst, Heinrich Streintz, Svante Arrhenius, Hiece, (sitting, fro the left) Aulinger, Albert von Ettingshausen, Ludwig Boltzann, Ignacij Kleencic, Hausanninger (1887).

19 Annalen der Physi, vol. 65, pp. 4 47,

20 Theroelectric effects ( ) e v B E f f f f ( ) f ( ) ( ) v T T Electrical current: Particle current: Energy current: Heat current: e v( ) f ( ) d 4 1 jn v( ) f ( ) d 4 1 je v( ) E( ) f ( ) d 4 1 jq v( ) E( ) f ( ) d 4

21 Theral conductivity f e E( ) jq E( ) ( ) B E E( ) T d 4 T B, j elec f E( ) jq E( ) ( ) Td 4 T n j T Q n