Transport theory

8/14/2019 Transport theory

1/75

SERC School on Condensed Matter Physics 06

VBS Transport Theory 0

Transport Theory

Vijay B. Shenoy([email protected])

Centre for Condensed Matter Theory

Indian Institute of Science


2/75



Overview

Motivation Why do this?

Mathematical and Physical Preliminaries

Linear Response Theory

Boltzmann Transport Theory

Quantum Theory of Transport


3/75



What is Transport Theory ?

Are we thinking of this?


4/75



What is Transport Theory (in Materials)?

Material Atoms arranged in a particular way

Stimulus takes material awayfrom thermal

equilibriumMaterial responds possibly by transferring energy,charge, spin, momentum etc from one spatial part to

anotherTransport theory: Attempt to construct a theory thatrelates material response to the stimulus

...Ok..., so why bother?


5/75



Why Bother (Taxpayer Viewpoint)?

ALL materials are used for their response tostimulus

Eg. Wool (sweater), Silicon (computer chip), Copper(wire), Carbon (writing) etc...

Key materials question: What atoms and how should I

arrange them to get a desired response to a particulartype of stimulus

...

Transport theory lays key foundation of theoreticalmaterials design

...

Blah, blah, blah...


6/75



Why Bother (Physicists Viewpoint)?

The way a material responds to stimulus is acaricature of its state

Transport measurements probe excitations above aground state

Characteristic signatures for transport are

universal can can be used to classify materials(metals, insulators etc.)

...

Ok, convinced? So what do we need to studytransport theory?


7/75



Prerequisites

A working knowledge of Fourier transforms

Basic quantum mechanics

Equilibrium (quantum) statistical mechanics

Band theory of solids

Some material phenomenology transport

phenomenology in metals, mainly

...

Our focus: Electronic transport


8/75



Fourier Transforms

Function f(r, t) is a function of space and time

Its Fourier transform f(k, ) is defined as

f(k, ) =

d3r

dt f(r, t) ei(krt)

We will write f(k, ) as f(k, ) (without the hat!)

Inverse Fourier transform

f(r, t) = 1(2)3

d3k 12 d f(k, ) ei(krt)


9/75



Some Useful Results

FT of delta function is 1

Step function (t) = 1 t 00 t


10/75



Transport Theory: Introduction

Example: A capacitor with a dielectric layer

Stimulus: Voltage applied V

Response: Charge stored Q

In general, we expect the response to be acomplicated function of the stimulus

Make life simple (although unreal in many systems),consider only cases where response is linearfunction ofthe stimulus

Focus on Lineartransport theory part of the generalLinear Response Theory

What is the most general form of linear response?


11/75


12/75



Linear Response

Keep aside spatial dependence: =(t t), responseto spatially homogeneous, time varying stimulus

In Fourier language Q() =()V() another way tosee it independent linear response for differentfrequencies of stimulus!

What can we say about () (or (t t

)) on generalgrounds?

Clearly phase of the response may differ from that ofstimulus consequence: response function is complexin general () =() +i()

Looks like linear response is characterised by two realvalued functions () and ()

SERC S h l C d d M Ph i 06


13/75



Causal Response

We know that the future does not affect the present(usually) response must be causal

Another way to say this (t t

) = 0 ift t


14/75



Kramers-Kronig Relations

Real and imaginary parts of response function are notindependent of each other, in fact one of the completelydetermines the other:

() = 1

Pd

()

, () =

1

Pd

()

Important experimental consequences: example, onecan obtain conductivity information from reflectancemeasurements!

SERC S h l C d d M tt Ph i 06


15/75



Its nice when response is linear...



16/75



But nature has many nonlinear responses...

(Slap!)



17/75



Nonlinear Response

(Jain, Raychaudhuri (2003))



18/75



What now?

We posited response to be linear

Reduced the problem to obtaining (say) the real part

of the response based on very general causalityarguments!

...

How do we calculate ()?This is a major chunk of what we will do obtainresponse functions



19/75



What we plan to do...

Goal: Study transport in metals

Focus on zero frequency electrical response (DC

response)Review: Drude theory

Review: Bloch theory and semiclassical approximation

Boltzmann transport theory

...

But before all this, lets see what we need to explain



20/75



Resistivity in Metals

(Ibach and Luth)

Almost constant at low temperatures...all way tolinear at high temperatures



21/75



Resistivity in Metals...Theres More!

(Ibach and Luth)

Increases with impurity contentHas some universal features



22/75

y


Transport in Metals

Wiedemann-Franz Law: Ratio of thermal() toelectrical conductivities () depends linearly on T

/= (Const)T, (Const) 2.3 108

watt-ohm/K

2

(Ashcroft-Mermin)



23/75

y


Magneto-transport! Levitating!

Hall effect

Nernst effect

Righi-Leduc effectEttingshausen effect

...

Things are getting to be quite effective

Goal: Build a reasonable theoretical framework toexplain/calculate all this



24/75


Drude Theory Review

Electrons: a classical gas

Collision time , gives the equation of motion

dp

dt =

p

+ F

p momentum, F external forceGives the standard result for conductivity

=

ne2

m

(all symbols have usual meanings)

All is, however, notwell with Drude theory!



25/75


Bloch Theory

We do need quantum mechanics to understand metals(all materials, in fact)

In the periodic potential of the ions, wave functionsarek(r) =eikruk(r) (uk is a lattice periodic

function), k is a vector in the 1st Brillouin zone

The Hamiltonian expressed in Bloch languageH=k(k)|kk| (one band), (k) is the banddispersion (set aside spin throughout these lectures!)

Average velocity in a Bloch state v(k) = 1

k

Occupancy of a Bloch state f0(k) = 1

e((k)) + 1,

= 1/(kBT), chemical potential



26/75


So, what is a metal?

Chemical potential determined from electronconcentration

Try to construct a surface in the reciprocal space suchthat (k) =

If such a surface exists (at T = 0) we say that the

material is ametal

A metal has a Fermi surface

Ok, so how do we calculate conductivity?

Need to understand how electron moves under theaction of external forces



27/75


Semi-classical Electron Dynamics

Key idea: External forces (F; electric/magnetic fields)cause transition of electronic states

Rate of transitions dk

dt =F Quantum version ofNewtons law

By simple algebra, we see the acceleration

dv

dt =M1F, M1 =

12

2kk

Electron becomes a new particle in a periodic

potential! Properties determined by value ofM at thechemical potential

But, what about conductivity? If you think about this,

you will find a very surprising result! (Essentiallyinfinite!)



28/75


Conductivity in Metals

What makes for finiteconductivity in metals?

Answer: Collisions

Electrons may scatter from impurities/defects,electron-electron interactions, electron-phononinteraction etc...

How do we model this? Brute force approach ofsolving the full Schrodinger equation is highlyimpractical!

Key idea: The electron gets a life-time i.e., anelectron placed in a Bloch state k evolves according to

(t) kei(k)t t

k ; lifetime is k!

Conductivity could plausibly be related to k; how?



29/75


Boltzmann Theory

Nonequilibrium distribution function f(r,k, t):

Occupancy of state k at position r and time t

r in f(r,k, t) represents a suitable coarse grainedlength scale (much greater than the atomic scale)such that each r represents a thermodynamicsystem

Idea 1: The (possibly nonequilibrium) state of asystem is described by a distribution functionf(r,k, t)

Idea 2: In equilibrium,f(r,k, t) =f0(k)! Externalforces act to drive the distribution function away fromequilibrium!

Idea 3: Collisions act to restore equilibrium try to

bring f back to f0



30/75


Time Evolution off(r,k, t)

Suppose we know f at time t= 0, what will it be at alater time t if we know all the forces acting on thesystem?

Use semi-classical dynamics: An electron at r in state

k at time t was at r vt in the state k F

t attime t t

Thus, we get the Boltzmann transport equation

f(r,k, t) =f(r vt,k F

t, t t) +

f

t coll.t

=f

t + v

f

r+F

f

k =

f

t

coll.

If we specify the forces and the collision term, we havean initial value problem to determine f(r,k, t)



31/75


Boltzmann Theory

So what if we know f(r,k, t)?

f(r,k, t) is determined by the external forces F

the stimulus (and, of course, the collisions which wetreat as part of our system)

If we know f(r,k, t) we can calculate the responses,

eg.,

j(r, t) = 1

(2)3

d3k (ev) (f(r,k, t) f0(k))

Intuitively we know that f(r,k, t) f0(r,k, t) F, sowe see that we can calculate linear response functions!



32/75


Approximations etc.

We know the forces F, eg., F = e(E+ v B)

What do we do about f

t coll.?Some very smart folks have suggested that we can set

ft coll. = f f0

k

the famous relaxation time appoximation...

In general, k is notsame as the electron lifetime(more later)...this is really a phenomenologicalapproach it embodies experience gained by looking

at experiments



33/75


Electrical Conductivity

BTE becomes

f

t +v

f

r +

F

f

k =

f f0

k

Homogeneous DC electric field F = eE

We look for the steady homogeneousresponseF

f

k =

f f0

k= f=f0

kF

f

k

Approximate solution (Exercise: Work this out)

f(k) f0 +ekE

f0

k

f0k+ekE



34/75


Solution of BTE

f0(k)

ky

kx

eE

f(k)

Fermi surface shifts (Exercise: estimateorder of magnitude of shift)



35/75


Conductivity from BTE

Current

j = 1

(2)3 d

3k (ev)ekE

f0

kConductivity tensor

=

1

(2)3e2

d3k k v f0kFurther, with spherical Fermi-surface (free electron

like), k roughly independent ofk (Exercise: Show this)

=ne2

m 1

This looks strikingly close to the Drude result, but thephysics could not be more different!



36/75


What about experiments?

Well, we now have an expression for conductivity; weshould compare with experiments?

What determines the Tdependence of conductivity?Yes, it is essentiallythe T dependence of (only inmetals)

But we do not yet have !!

Need a way to calculate ...

...

Revisit the idea of electron-lifetime...how do wecalculate life time of an electron?



37/75


Lifetime due to Impurity Scattering

Impurity potential VI, causes transitions from oneBloch state to another

Rate of transition from k k

Wkk =2

|k|VI|k|

2((k) (k))

Total rate of transition, or inverse lifetime

1

I

k

= 1

(2)3 d3k Wkk

Can we use Ik as the in the Boltzmann equation?

Ok in order of magnitude, but not alright! Why?



38/75


How to calculate ?

Look back at the collision term, can write it moreelaborately as

ftcoll.

= 1(2)3

d3k Wkk f(k)(1 f(k)) f(k)(1 f(k=

1

(2)3 d3k Wkk f(k) f(k)Note that k and k are of the same energy

Take k to depend only on (k)Now, (f(k) f(k)) e

f0

v(k) v(k)

E



39/75


Calculation of contd

Putting it all together

e

f0

v(k) E =

1

(2)3

e

f0

d3k Wkk v(k) v(k) =

1

=

1

(2)3 d3k Wkk 1

v(k) E

v(k

) E

=1

=

1

(2)3

d3k Wkk

1 cos(

k,k)

Note is differentfrom the quasiparticle life time!

Key physical idea: Forward scattering does notaffectelectrical conductivity!



40/75


T dependence of

We now need to obtain T dependence of

Tdependence strongly depends on the mechanism of

scatteringCommon scattering mechanisms

Impurity scattering

ee scatteringephonon scatting

More than one scattering mechanism may be

operative; one has an effective (given by theMatthiesens rule)

1

= i

1

i



41/75


from Impurity Scattering

Essentially independent of temperature

Completely determines the residual resistivity

(resistivity at T = 0)1directly proportional to concentration of impurities

(Matthiesens rule!)

Well in agreement with experiment!



42/75


from ee Scattering

One might suspect that the effects ofee interactionsare quite strong...this is not actually so, thank to Pauli

ee scattering requires conservation ofbothenergy andmomentum

Phase space restrictions severely limit ee scattering

Simple arguments can show 1

kBT2

Also called as Fermi liquid effects

Can be seen in experiments on very pure samples atlow temperatures

At higher temperatures other mechanisms dominate


S


43/75


from ePhonon Scattering

There is a characteristic energy scale for phonons TD, the Debye temperature

Below the Debye temperature, the quantum nature ofphonons become important

Natural to expect different Tdependence above andbelow TD

ephonon scattering is, in fact, notelastic in general

Study two regimes separately : T TD and T TD


f Ph S i (T T )


44/75


from ePhonon Scattering (T TD)

Scattering processes are definitely inelastic

Electron can change state k to k by absorption or

emission of phononThe matrix element of transition rate in a phononemission with momentum q

Wkkq |Mqk q, nq+ 1|aq|k, nq|2

|nq+ 1|aq|nq|

2 nq kBT

1varies linearly with temperature!


f Ph S i (T T )


45/75


from ePhonon Scattering (T TD)

Scattering process is approximately elastic since onlyvery long wavelength phonons (acoustic) are present

Using expression for

1

|q|< kBTcd3qWkkq

1 cos( k,k q)

|q|2

T

TD5

Bloch-Gruneisen Law!

Phonons give a resistivity ofT at T TD and T5

forT TD

The key energy scale in the system is TD universal

features are not surprising


E i t Fi ll !


46/75


Experiments, Finally!

Our arguments show

Impurity resistivity does not depend ontemperature and is approximately linear withconcentration of impurities

At very low temperatures an in pure enoughsamples, we will see a T2 behaviour in resistivity

This is followed by a T5 at low T (T TD) goingover to T (T TD), and this behaviour withappropriate rescale should be universal

All of these are verified experimentally in nice metals!


Hi h T S i


47/75


High Tc Surprise

Resistivity in high Tc normal state

Looking for a research problem?


48/75


Thermogal anic Transport


49/75


Thermogalvanic Transport

Stimuli: Both E and T, Response : j and jQ

Cannot ignore spatial dependence off!

Steady state satisfies

v f

r

eE

f

k=

f f0

Approximate solution (Exercise: Work this out)

f f0 = f0

( )

T T+eE

v

Heat current jQ is given by (Question: Why( )?)

jQ=

1

(2)3 d3k ( )v (f(k) f0(k))




50/75



Transport relations can be expressed in compact from

j = e2A0E+ e

T

A1(T)

jQ = eA1E+ 1

TA2(T)

where matrices A= 1(2)3

d3k( ) vvf0

For nearly free electrons j

jQ

=

n

m

e2 12ekB

kBT

12ekBT

kBT

13k

2BT

E

T




51/75



Experimentally more useful resultE = j+QT

jQ = j T

Thermoelectric properties

= m

ne

2

Resistivity 108 ohm m

Q=1

2

kBe

kBT

Thermoelectric power

108TV/K (check factors!)

=QT Peltier coefficient

=2

3

n k2BT

m Electronic thermal conductivity

100 watt/(m2 K)


Widemann Franz!


52/75


Widemann-Franz!

We see the Lorenz number

T =

2

3

k2B

e2

amazingly close to experiments (makes you wonderif something is wrong!)

Actually, Widemann-Franz law is valid strictly onlywhen collisions are elastic...

Reason: Roughly, inelastic forward scattering cannot

degrade an electrical current, but it doesdegrade thethermal current (due to transfer of energy to phonons)

Not expected to hold at T TD


Amazing Cobaltate Na CoO2


53/75


Amazing Cobaltate NaxCoO2

High thermoelectric power!!Another research problem!


Magneto-Transport


54/75


Magneto-Transport

Transport maxim: When you think you understandeverything, apply magnetic field!

Think of the Hall effect; the Hall coefficient is strictlynot a linear response fucntion... We will not worryabout such technicalities; take that the magnetic fieldB is applied and the response functions depend

parametrically on B in our original notation=(,B).

Let us start with an isothermal system and understand

how electrical transport is affected by a magnetic field Hall effect

But before that we will investigate semi-classical

dynamics in presence of a magnetic field


55/75


BTE with Magnetic Field


56/75


BTE with Magnetic Field

We will work with closed Fermi surfaces in the weakmagnetic fieldregime c 1...an electron undergoesmany collisions before it can complete one orbit

Boltzmann equation becomes

e(E+ v B) f

k=

f f0

With a bit of (not-so-interesting) algebra (bB E) = 0

f f0 = e1 + (c)2

E+ (c)B E vf0


And we attain the Hall of fame!


57/75


And we attain the Hall of fame !

Setting B=Bez, we get in plane response

jxjy = 0(1 + (c)2 1 cc 1 ExEy 0 =

ne2m

In the Hall experiment, jy = 0, thus

jx = 0Ex

Ey = c Ex= RH= EyjxB

= 1ne

Our model predicts a vanishing magnetoresistance!


Magnetoresistance


58/75


Magnetoresistance

There is weak magnetoresistance present even in nicemetals (0)B2 (this form arises from timereversal symmetry)

For nice metals there is something called the Koehlersrule

(B, T) (0, T)(0, T) = FrefB(0, T)The key idea is that magnetoresistance is determined

by the ratio of two length scales the mean free pathand the Larmour radius

For metals with open orbits etc. magnetoresponse can

be quite complicated!Research problem: Magnetoresponse of highTc normal


Manganites: Colossal Responses


59/75


Manganites: Colossal Responses

Colossal magnetoresistance in LCMO!


60/75


61/75


Righi-Leduc Effect


62/75


g

A temperature gradient is applied Tx along thex-direction

jx=jy = 0 and (jQ)y = 0A temperature gradient Ty develops

Response determined by

Ty

B Tx=

L


Ettingshausen Effect


63/75


g

Current jx flows, Tx = 0 along the x-direction

jy = 0 and (jQ)y = 0

A temperature gradient Ty develops

Response determined by Ettingshausen coefficient

Ty

Bjx= K

K is related to the Nernst coefficient K=N T


Thank You, Boltzmann!


64/75


,

This is how far we will go with Boltzmann theory...

Of course, one can do many more things...its left toyou to discover

...

Key ideas I : Distribution function, semiclassicalequation of motion, collision term,...

Key ideas II : Relaxation time, quasi-Bloch-electronslife-time, transportation life-time

Boltzmann theory deals with expectation value ofoperators, and does not worry about quantumfluctuations it of course takes into account thermalfluctuations, but cold shoulders quantum

fluctuationsOur next task is to develop a fully quantum theory of


Quantum Transport Theory


65/75


p y

There are many approaches...

Our focus: Green-Kubo theory

What we will seeTheory of the response function (Green-Kuborelations)

Fluctuation-dissipation theorem

Onsagers principle

Our development will be formal and realcalculations in this framework require (possibly)advanced techniques such as Feynman diagrams


The System


66/75


Our system: A (possibly many-particle) system withHamiltonian H0

Eigenstates H0|n =En|n

Time evolution: Schrodinger i|

t =H0| (set to

1)

Also write as: |(t) =eiH0t|(0)

In presence of a perturbation (stimulus), Hamiltonianbecomes H=H

0+V

One can study the time evolution in differentpictures : Schrodinger picture, Heisenberg picture,Dirac (interaction) picture


Dirac (interaction) picture


67/75


State evolve according to |I(t) =eiH0teiHt|(0)

Operators evolve according to AI(t) =eiH0tAeOH0t

Time evolution: i|It

=VI|I

Expectation value of operator A:

A(t) = I(t)|AI(t)|I(t)Interaction picture reducesto the Heisenberg picturewhen there is no stimulus!

...Ok, how does one describe the thermodynamic(possibly nonequilibrium) state of a quantum system?


68/75


What about Equilibrium?


69/75


Well, clearly the equilibrium density matrix

0=

neEn

Z |nn|, partition function Z=

neEn

Exercise: Work out expressions for internal energy, entropy, etc

So far fixed particle number(canonical ensemble)

Treat |n to count states with different particlenumber state |n has Nn particles, and move over tothe grand canonical ensemble by introducing achemical potential

0=n

e(EnNn)

Z |nn|, Z=

n e

(EnNn)

Question: How does one get Fermi distribution out of this?


Evolution of the Density Matrix


70/75


Suppose I know the density matrix at some instant oftime... what will it be at a later instance?

Now (t0) =p||...if there system where in thestate |, it will evolve to |(t) =eiH(tt0)|...Thismeans (t) =

p|(t)(t)|, or

(t) =eiH(tt0)(t0)eiH(tt0) = it

+ [, H] = 0 !!!!

This is the quantum Louisville equation!

In thermal equilibrium (no perturbations), 0 isstationary! Question: Why? all this fits very well with ourearlier understanding


Evolution of the Density Matrix


71/75


Time evolution in the interaction representation

iI

t

+ [I, VI] = 0

Perturbation was slowly switched on in the distantpast t0

I=0+ I, the piece of interest is I

Clearly, I() = 0, and we have

I(t) =i t

dt[0, VI(t)]

We know the evolution of the density matrix to linear

order in the perturbation...we can therefore calculatethe linear response


Linear Response


72/75


The stimulus V(t) =f(t)B where B is someoperator (e.g. for an AC electric potentialV(t) = e(t)N where N is the number density

operator, (t) is a time dependent potentialAny response (observable) A of interest can now becalculated

A(t) = tr(I(t)AI(t))

= i t

dttr([0, BI(t)]AI(t))f(t

)

=

dt i(t t)[A(t), B(t)]0

AB(tt)f(t)!

Note that we have dropped all the Is in the last eqn.


Linear Response


73/75


Completely solved any linear response problem inprinciple!

AB(t t) = i(t t)[A(t), B(t)]0 is called

Green-Kubo relation

Key physical idea: Linear response to stimulus isdetermined by an equilibrium correlation function

(indicated by subscript 0)Causality is automatic!

In systems with strong interaction/correlations,

response calculation using Green-Kubo relation is adifficult task


Fluctuation Dissipation Theorem


74/75


The imaginary part of is related to the dissipation

Going back to the motivating capacitor example, the dielectric

response function will (t t) i(t t)[N(t),N(t)]0

The leakage current loss will be determined by the imaginary

part of ()

One can then go on to show that the imaginary part of() is

directly proportional to the autocorrelator of the density operator

(i.e., FT of{N(t),N(t)}0) Exercise: Do this, not really difficult

The autocorrelator is a measure of the fluctuations in equilibrium

The key physical idea embodied in the Fluctuation Dissipation

Theorem: Fluctuations in equilibrium (how they correlated in time)

completely govern the dissipation when the system is slightly

disturbed


Whats more?


75/75


Lots!

Semiconductors/Ionic solids

Phonon TransportDisordered systems

Correlated systems

Nanosystems Landauer ideas...

Date post:	04-Jun-2018
Category:	Documents
Upload:	faqrul-alam-chowdhury
View:	218 times
Download:	0 times

Transport theory

Documents