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Some pictures are taken from the UvA-VU Master Course: Advanced Solid State Physics by Anne de Visser (University of Amsterdam), Solid State Course by Mark Jarrel (Cincinnati University), from Ibach and Lth, from Ashcroft and Mermin and from several sources on the web.

the study of transport phenomena in

physics is related with the exchange of

mass, energy, and momentum studied

systems.

fluid mechanics, heat transfer, and mass transfer

in solid state physics, the motion and interaction of electrons, holes and

phonons are studied under "transport phenomena".

http://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Momentumhttp://en.wikipedia.org/wiki/Physical_systemhttp://en.wikipedia.org/wiki/Fluid_mechanicshttp://en.wikipedia.org/wiki/Heat_transferhttp://en.wikipedia.org/wiki/Mass_transferhttp://en.wikipedia.org/wiki/Solid_state_physicshttp://en.wikipedia.org/wiki/Phonons

Content:

1. Introduction. General. Fick Law. Boltzmann Eq. Relaxation time.

2. Electronic transport in conductors. Electron-phonon scattering.

3. Electron-imperfection scattering

4. Electrical conductivity. Bloch-Gruneisen

5. Magnetic scattering

6. Thermal conductivity

7. Thermoelectric phenomena

8. Electrical conductivity in magnetic fields.

9. Anomalous Hall effect

10. Magnetoresistance : AMR, CMR

11. Magnetoresistance : GMR TMR.

12. Strongly correlated electron systems

1. C. Kittel, H. Kroemer, Thermal Physics, W.H. Freeman Co. New York

2. C. Kittel, Introduction to Solid State Physics (7-8 ed., Wiley, 1996)

3. N. W. Ashcroft, N. D. Mermin, Solid State Physics, 1976. 4. U. Mizutani, Introduction to the Electron Theory of Metals, Cambridge University

Press 2001.

5. Ch. Enss, S. Hunklinger, Low-Temperature Physics, Springer-Verlag Berlin

Heidelberg 2005.

6. M. Coldea, Magnetorezistenta. Efecte si Aplicatii, Presa Universitara Clujeana, 2009.

7. E. Dagotto, Nanoscale Phase Separation and Colossal Magnetoresistance, Springer-

Verlag Berlin Heidelberg 2002.

8. J.M deTeresa, New magnetic materials and their functions 2007, Cluj-Napoca, Romania.

Summer School (http://esm.neel.cnrs.fr/2007-cluj)

9. UvA-VU Master Course: Advanced Solid State Physics

10. H. Ibach and H. Lth: Solid State Physics 3rd edition (Springer-Verlag, Berlin, 2003) ISBN

3-540-43870-X 11. J. M. ZIMAN, ELECTRONS AND PHONONS ,The Theory of Transport Phenomena in Solids,

UNIVERSITY OF CAMBRIDGE , OXFORD AT THE CLARENDON PRESS 1960

Some pictures are taken from the UvA-VU Master Course: Advanced Solid State Physics by Anne de Visser (University of Amsterdam), Solid State Course by Mark Jarrel (Cincinnati University), from Ibach and Lth, from Ashcroft and Mermin and from several sources on the web.

http://esm.neel.cnrs.fr/2007-cluj/http://esm.neel.cnrs.fr/2007-cluj/http://esm.neel.cnrs.fr/

Introduction. Transport Processes

T1 T2

Entropy (reservoir 1+reservoir 2+ System) INCREASES

Nonequilibrium steady state

Driving force=Temperature gradient Transport of Energy

Flux = (coeficient) x (driving force)

Linear phenomenological law (if the force is not to large)

e.g. Ohms law for the conduction of electricity

System

JA= flux density of A = net quantity of A transported

across area in unit time

Net transport = the transport in one direction the transport in oposite direction

Summary of phenomenological transport laws A grad = DJ A

from Enss

Particle diffusion

1 2

particle flow

Entropy (reservoir 1+reservoir 2+ System) INCREASES

T= constant

Presume: the difference of chemical potential is caused by a difference in particle concentration

Jn- the number of particels passing through a unit area in unit time

nDJn grad=Ficks law:

D particle diffusion constant - diffusivity

from Kittel, Thermal Physics

http://upload.wikimedia.org/wikipedia/commons/1/12/Diffusion.svg

l the mean free path

At position z the particles come into a local equilibrium condition (z) and n(z)

z z+l

z-l

[ ] zzzzzzn lcdzdnc)lz(n)lz(nJ =+=

21

We express czlz in terms of lc

cosllz = cosccz =

The average is taken over the surface of a hemisphere

The element of surface area is dsin 2

lcdsincos

lclc zz 31

2

22

0

2

=

=

dzdnlcJ zn 3

1= lcD

31

=

Particle diffusion is the model for other transport probelms

Particle diffusion transport of particles Thermal conductivity transport of energy by particles Viscosity transport of momentum by particles Electrical conductivity transport of charge by particles

The linear transport coefficients that describes the

processes are proportional to the particles diffusivity D

So that

A the concentration of the physical quantity A.

-the flux density of A in the z direction is:

zAzA vJ =

zv is the mean drift velocity of the particles in the z direction (drift velocity is zero in thermal equilibrium)

If A (e.g. energy, momentum) depends on the velocity of a molecule:

zAAzA vfJ =

fA is a factor with magnitude of the order of unity it depends on the velocity dependence of A and may be calculated (e.g. by using the Boltzmann transport equation)

A grad = DJ A

let

By analogy with Fick law

Thermal conductivity

Fourier laws TKJ A grad=

Describes the energy flux density Jn in terms of the thermal conductivity K and the temperature gradient

1 2

System

This form assumes that there is a net transport of energy, but not particles

Another term must be added if additional energy is transported by means of particle flow (as when electrons flow under the influence of an electric field.)

zuzu vJ

The energy flux density in the z direction is:

u is the energy density

By analogy with the diffusion equation:

)dxdT)(T(DdxdD uu = Diffusion of energy

Tu Is the heat capacity per unit volume, CV.

TCDJ Vu grad=

lcCDCK VV == 31

The thermal conductivity of a gas is independent of

pressure until very low pressure when the mean free

path becomes limited by the dimensions of the apparatus

rather then by intermolecular collision.

Hot Th

Cold Tc

L

Q (heat flow)

dxdTkA

LTTkAQ ch ==

Thermal conductivity

Viscosity

Viscozity is a measure of the diffusion of momentum parallel to the flow

velocity and transverse to the gradient of the flow velocity

x

vx

Viscosity coefficient:

)p(JdzdvX xzxz == z

from Enss

The particle flux density in the z direction: dzDdnvnJ zzn ==

The transverse momentum density: xnMv

Its flux density in the z direction: zx v)nMv(

This flux density ( ) dznMvDd x( ) A grad = DJ A

nM=

( ) dzdvdzdvDvvpJ xxxxxz ===

Mass density:

lcD 31

==CGS- poise

SI unit: Pa.s

Kittel, Thermal physics

http://en.wikipedia.org/wiki/SI_unit

The mean free path: nd/l 21 =

Viscozity: 23 dcM = independent of gas pressure

The independence fails at very high pressures when the molecules are always is contact, or at very low pressures when the mean free path is longer than the dimension of the apparatus.

Robert Boyle 1660

air

=D /CK v=Kinematic viscosity

Generalized Forces

The transfer of entropy from one part of a system to another is a consequence of any transport process.

We can relate the rate of change of entropy to flux density of particles and of energy

For V= const. dNT

dUT

dS = 1

The entropy current density : nus JT

JT

J

=1

Entropy density S

The net change of entropy density at a fixed position sS JgtS

div=tS

Eq. of continuity Rate of production of entropy

(*)

C. Kittel, Thermal Physics

In a transfer process U and N are conserved

The equations of continuity:

uJdtu div=

nJdtn div=

Divergence of SJ

( ) ( ) ( )TJJTTJJT

J nnuuS graddivgraddivdiv +=

11

tn

Ttu

TtS

= 1

(*)

( ) ( )TJTJg nuS += gradgrad

1

nnuuS FJFJg

+=

Generalized forces

( )TFu 1grad

( )TFn grad

Thermodynamics of irreversible processes:

nuu FLFLJ

1211 +=

nun FLFLJ

2221 +=

Onsager relation:

=

BLBL jiij

In magnetic field

Coupled effects

Avanced Treatment: Boltzmann Transport Equation

We work in the 6 D (six-dimensional space of Cartesian coordinates r and v).

The classical distribution function:

( ) vdrdvdrdv,rf in particles of number =

The effect of time displacement dt on the distribution function:

( ) )v,r,t(fvdv,rdr,dttf =+++

In the absence of collisions

With collisions:

( ) collisions)t/f(dt)v,r,t(fvdv,rdr,dttf =+++

With a series development:

collisionsvr )t/f(dtfgradvdfgradrd)tf(dt =++

dtvda

collisionsvr )t/f(fgradafgradvtf =++

Boltzmann transport equation

Relaxation time aproximation:

This is based on the assumption that a nonequilibrium distribution gradually returns to its equilibriun value within a characteristic time, the relaxation time , by scattering of particles with the velocity into states , and vice versa.

)v,r,t(f

)v,r(c

v

'v

ccollisions /)ff()t/f( 0=

Suppose that a nonequilibrium distribution of velocities is set up by external forces which are suddenly removed.

The decay of the distribution towards equilibrium is then obtained:

c/)ff(t)ff( 00 =

from Kittel, Thermal physics

)/t()ff()ff( ctt = = exp000

00 = tf

( )v,rcc

=Generally

cvr

fffgradafgradvtf

0=++

In the steady state: 0= t/fby definition

Particle Diffusion

Consider an isothermal system with a gradient of particle concentration

The steady-state Boltzmann transport equation in the relaxation time approximation:

cx /)ff(dxdfv 0=

dxdfvff cx 001 First order approximation

dxdf

dxdf 0

Second order approximation

20

22200102 dxfdvdxdfvfdxdfvff cxcxcx +=

The iteration is necessary for the treatment of nonlinear effects

Classical Distribution

]Tk/)exp[(f B =0

)dx/d)(Tk/f()dx/d)(d/df(dx/df B 000 ==

)dx/d)(Tk/fv(ff Bcx 00 =

The first order solution for the nonequilibrium distribution becomes:

The particle flux density in the x direction:

d)(fDvJ xxn =

The density of orbitals per unit volume per unit energy range:

2123

222

41

=

M)(D

Presume c constant, independent of velocity nd)(Df = 0

)dx/dn)(M/Tk()dx/d)(M/n(J cBcxn ==

const.log += nTkB

cBc vM/TkD 2

31

==

Diffusivity:

If we presume

vl

c = clD31

=

finally,

because

Fermi-Dirac distribution 1

10 +

=Tk/)exp[(

fB

)(ddf 0 )(Fd)()(F +

=

dx/d)(dxdf =0

The particle flux density

d)(D)(v)dx/d(d)(fDvJ xcxxn == 2

F/n)(D 23=3222 32 )n)(m/( =

dx/dnvdx/dn)m/(J cFFcxn

2

3132 ==

We know

cFvD 2

31

=

Electrical conductivity

We multiply the particle flux density by the particle charge q

dx/d xqEdx/qd =

EE)m/qn()dx/d)(m/qn(J ccxq ===

2

m/nq c2=

Electrical conductivity

For a classical gas (Drude)

For the Fermi-Dirac distribution (Sommerfeld)

We will discuss this in more detail, later.

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Some pictures are taken from the UvA-VU Master Course: Advanced Solid State Physics by Anne de Visser (University of Amsterdam), Solid State Course by Mark Jarrel (Cincinnati University), from Ibach and Lüth, from Ashcroft and Mermin and from several sources on the web.

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