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Transport Theory
Vijay B. Shenoy([email protected])
Centre for Condensed Matter Theory
Indian Institute of Science
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Overview
Motivation Why do this?
Mathematical and Physical Preliminaries
Linear Response Theory
Boltzmann Transport Theory
Quantum Theory of Transport
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What is Transport Theory ?
Are we thinking of this?
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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?
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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...
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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?
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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
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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)
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Some Useful Results
FT of delta function is 1
Step function (t) = 1 t 00 t
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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?
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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 ()
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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
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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!
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Its nice when response is linear...
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But nature has many nonlinear responses...
(Slap!)
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Nonlinear Response
(Jain, Raychaudhuri (2003))
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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
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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
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Resistivity in Metals
(Ibach and Luth)
Almost constant at low temperatures...all way tolinear at high temperatures
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Resistivity in Metals...Theres More!
(Ibach and Luth)
Increases with impurity contentHas some universal features
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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)
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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
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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!
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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
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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
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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!)
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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?
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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
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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)
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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!
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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
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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
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Solution of BTE
f0(k)
ky
kx
eE
f(k)
Fermi surface shifts (Exercise: estimateorder of magnitude of shift)
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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!
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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?
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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?
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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
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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!
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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
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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!
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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
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S
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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
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f Ph S i (T T )
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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!
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f Ph S i (T T )
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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
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E i t Fi ll !
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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!
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Hi h T S i
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High Tc Surprise
Resistivity in high Tc normal state
Looking for a research problem?
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Thermogal anic Transport
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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))
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Thermogalvanic Transport
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Thermogalvanic Transport
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
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Thermogalvanic Transport
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Thermogalvanic Transport
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)
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Widemann Franz!
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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
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Amazing Cobaltate Na CoO2
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Amazing Cobaltate NaxCoO2
High thermoelectric power!!Another research problem!
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Magneto-Transport
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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
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BTE with Magnetic Field
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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
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And we attain the Hall of fame!
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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!
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Magnetoresistance
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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
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Manganites: Colossal Responses
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Manganites: Colossal Responses
Colossal magnetoresistance in LCMO!
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Righi-Leduc Effect
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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
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Ettingshausen Effect
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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
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Thank You, Boltzmann!
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,
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
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Quantum Transport Theory
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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
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The System
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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
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Dirac (interaction) picture
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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?
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What about Equilibrium?
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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?
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Evolution of the Density Matrix
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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
SERC School on Condensed Matter Physics 06
Evolution of the Density Matrix
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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
SERC School on Condensed Matter Physics 06
Linear Response
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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.
SERC School on Condensed Matter Physics 06
Linear Response
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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
SERC School on Condensed Matter Physics 06
Fluctuation Dissipation Theorem
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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
SERC School on Condensed Matter Physics 06
Whats more?
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Lots!
Semiconductors/Ionic solids
Phonon TransportDisordered systems
Correlated systems
Nanosystems Landauer ideas...