Plan of the lectures - master-mcn.u-strasbg.fr

Master Lecture 5 G. Manfredi, IPCMS, Strasbourg1

Plan of the lectures

1. Introductory remarks on metallic nanostructures

• Relevant quantities and typical physical parameters

• Applications

2. Linear electron response: Mie theory and generalizations

3. Nonlinear response

• Survey of various models from N-body to macroscopic

• Mean-field approximation (Hartree and Vlasov equations)

4. Beyond the mean-field approximation

• Hartree-Fock equations

• Time-dependent density functional theory (DFT) and local-density approximation (LDA)

5. Macroscopic models: quantum hydrodynamics

6. Linear theory and comparison of various models

7. Illustration: the nonlinear electron dynamics in thin metal films


Synopsis of classical and quantum models

N-body Schrödinger N-body Liouville

TD HartreeWigner Vlasov

Quantum hydrodynamics

Classicalhydrodynamics

BBGKY (gC<<1)

ħ → 0

ħ→ 0

kλD << 1

N-body

Mean

field

Macro-scopic

DFTxc

Hartree-Fock

x

kλTF << 1

gQ << 1


Why do we need macroscopic models?

• Even mean-field model are sometimes too complex

– Classical : 6D Vlasov equation in phase space

– Quantum : Hartree or DFT equations: N three-dimensional

Schrödinger-like equations (N >> 1)


Classical fluid (hydrodynamic) models – I.

Vlasoveq.


Classical fluid models II.

♦

♦

♦


Classical fluid models III.


Classical fluid models IV.


Classical fluid models V.

• We have obtained a set of just 2 evolution equations in real space (3D)

– Electric field comes from Poisson’s equation + external fields

• They replace the more complex Vlasov equation in phase space (6D)

nTkP

qN

N

nnP

B : closure isothermal : 1

)0( closure adiabatic : 2

const.)(

==γ•

=+=γ•

×= γ


Quantum hydrodynamics I.

• Hydrodynamic (“fluid”) equations are obtained by taking moments of the relevant

kinetic equation (Vlasov or Wigner):

• Starting with either the Vlasov or Wigner equations, we obtain the same fluid equations:

• What happened to quantum effects? They’re hidden in the pressure term…

density (0) average velocity (1) pressure (2)

Continuity equation

Euler equation


Quantum hydrodynamics II.

• We rewrite the pressure using the expression of the Wigner function in terms of N wave functions ψ

α:

• We obtain:

• We express the wave functions in terms of their amplitude A and phase S

• We can split the pressure into a “classical” and a “quantum” part:

P = PC + PQ

Mean velocity for each ψα


Quantum hydrodynamics III.

• In order to close the system, we need a relation between the pressure and the density:

– PC = PC(n) : classical equation of state

– valid for λ >> λTF (long wavelengths)

• We finally obtain the conservation equation for u

coupled to the continuity and Poisson’s equations

Uncertainty due to standard velocity dispersion

Uncertainty due to quantum effects

⇒→ nAα


Stationary quantum hydrodynamics

• ∂ / ∂t = 0 : stationary

• u = 0 : no center-of-mass motion

• PQ = 0 : no explicit quantum effects

• Then we obtain

• For PC we take the pressure of a uniform fermion gas at zero temperature

• It is easy to show that:

• Thus, we obtain:

( ) 01 =

∂∂−

∂∂

x

P

ne

x

C

φ

( ) 3/53/222

325

2

5

2n

mnEPP FF

C πh===

x

E

x

P

n

FF

∂∂=

∂∂1

( ) 0=∂

∂−∂∂

x

Ee

x

Fφ µφ ==− const.)(nEe F Thomas-Fermi model!

Quantum hydrodynamic models are sometimes called time-dependent Thomas-Fermi models

equation sPoisson' obeys φ

Including PQ(n) would improve the kinetic energy functional (beyond EF)

Hypotheses


Schrödinger equation in hydrodynamic form − I.

• What is the origin of PQ ?

• Schrödinger equation:

• Separate amplitude and phase of the wave function (“Madelung transformation”)

• By using:

• The equations for the imaginary part yields the continuity equation:

ψψψV

xmti +

∂∂−=

∂∂

2

2

2

hh

h/),(),(),( txiSetxAtx =ψ

{ } h

h

h

h

/22

/

)2(/)(

)/(

iSxxxxxxxxxx

iSttt

eSAASiASA

eAiSA

++−=

+=

ψψ

0)( =∂∂+

∂∂

nuxt

n

m

Su

An

x=

= 2


Schrödinger equation in hydrodynamic form − II.

• The real part of the Schrödinger equation yields:

• It can be written as an Euler hydrodynamic equation

• The Bohm potential is related to the “quantum pressure”

• Remember also:

+−

∂∂−=

∂∂

A

A

mtxV

x

S

mt

S xx

2),(

2

1 22h

Bohm potential

Hamilton-Jacobi equation

∂∂∂+

∂∂−=

∂∂+

∂∂

n

n

xmx

V

mx

uu

t

u xx2

2

2

2

1 h

x

V

x

P

n

BohmQ

∂∂=

∂∂1

x

E

x

P

n

FC

∂∂=

∂∂1

m

Su

An

x=

= 2


Equations of state

• Zero-temperature 3D electron gas. “Classical” pressure at equilibrium ,

computed from the Fermi-Dirac distribution

• What happens in a dynamical situation?

The density n(x,t) will differ from the equilibrium value n0(x).

We write for the pressure

• Adiabatic closure:

• Not appropriate for wave propagation, which is essentially a 1D phenomenon

• In that case, one has to take the 1D value (γ = 3) even for a 3D situation.

3/5 /)2( =+= NNγ

3/50~ n


Synopsis of classical and quantum models

N-body Schrödinger N-body Liouville

TD HartreeWigner Vlasov

Quantum hydrodynamics

Classicalhydrodynamics

BBGKY (gC<<1)

ħ → 0

ħ→ 0

kλD << 1

N-body

Mean

field

Macro-scopic

DFTxc

Hartree-Fock

x

kλTF << 1

gQ << 1

Plan of the lectures - master-mcn.u-strasbg.fr

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