Institute of Solid State PhysicsTechnische Universität Graz

24. Transport

Jan. 16, 2020

Thermoelectric effects

Peltier effect: driving a through a bimetallic junction causes heating or cooling.

Cooling takes place when the electrons make a transition from low entropy to high entropy at the junction.

Bismuth chalcogenides Bi2Te3 and Bi2Se3

I

http

://en

.wik

iped

ia.o

rg/w

iki/P

eltie

r_ef

fect

#Pel

tier_

effe

ct

Hall effect

0

r llmn

m n

Rej B

0rT

Nerst effect

r llmn

m n

Ne T B

0, , elecj T B

0, , r T B

Ettingshausen effect

http

://dx

.doi

.org

/10.

1002

/and

p.18

8626

5101

0

Annalen der Physik, vol. 265, pp. 343–347, 1886

Albert von Ettingshausen, Prof. at TU Graz.

(Standing, from the left) Walther Nernst, Heinrich Streintz, Svante Arrhenius, Hiecke, (sitting, from the left) Aulinger, Albert von Ettingshausen, Ludwig Boltzmann, Ignacij Klemencic, Hausmanninger (1887).

Boltzmann Group

Nernst was a student of Boltzmann and von Ettingshausen. He won the 1920 Nobel prize in Chemistry.

Thermoelectric effects

33 ( ) ( )

4elecej v k f k d k

Electrical current:

33

1 ( ) ( )4nj v k f k d k

Particle current:

33

1 ( ) ( ) ( )4Ej v k E k f k d k

Energy current:

33

1 ( ) ( ) ( )4Qj v k E k f k d k

Heat current:

Thermoelectric effects

emmn

n

jE

Electrical conductivity: 0, 0T B

Qmmn

n

jT

Thermal conductivity: 0B

Qmmn

en

jj

Peltier coefficient: 0, 0T B

mmn

n

ST

Thermopower (Seebeck

effect):0, 0ej B

ll mn

em n

ERj B

Hall effect: 0, 0elT j

Nerst effect: llmn

m n

ENB T

0elecj

Student Projects

Calculate some transport property for a free electron gas or for a semiconductor.

Numerically calculate a transport property for a one dimensional material.

Hall effect

Velocity of k-states

1 dEvdk

Institute of Solid State PhysicsTechnische Universität Graz

Crystal Physics

Institute of Solid State PhysicsTechnische Universität Graz

Crystal Physics

Crystal physics explains what effects the symmetries of the crystal have on observable quantities.

An Introduction to Crystal Physics Ervin Hartmannhttp://ww1.iucr.org/comm/cteach/pamphlets/18/index.html

International Tables for Crystallographyhttp://it.iucr.org/

Kittel chapter 3: elastic strain

Strain

A distortion of a material is described by the strain matrix

ˆ ˆ ˆ(1 )ˆ ˆ ˆ(1 )ˆ ˆ ˆ(1 )

xx xy xz

yx yy yz

zx zy zz

x x y z

y x y z

z x y z

x

y

z

x

y

z

Stress

9 forces describe the stress

Xx, Xy, Xz, Yx, Yy, Yz, Zx, Zy, Zz

Xx is a force applied in the x-direction to the plane normal to x

Xy is a sheer force applied in the x-direction to the plane normal to y

yx z

x y z

yx z

x y z

yx z

x y z

XX XA A A

YY YA A A

ZZ ZA A A

stress tensor:

Stress is force/m2

Xx Xy

Xz

x

y

z

Stress and Strain

The stress - strain relationship is described by a rank 4 stiffness tensor. The inverse of the stiffness tensor is the compliance tensor.

ij ijkl kls

ij ijkl klc

Einstein convention: sum over repeated indices.

xx xxxx xx xxxy xy xxxz xz xxyx yx xxyy yy

xxyz yz xxzx zx xxzy zy xxzz zz

s s s s s

s s s s

Statistical Physics

Microcannonical Ensemble: Internal energy is expressed in terms of extrinsic quantities U(S, M, P, , N).

ij ij k K l ldU TdS d E dP H dM

ij K lij k l

U U U UdU dS d dP dMS P M

The normal modes must be solved for in the presence of electric and magnetic fields.