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Microscopic-macroscopic
connection
Valérie Véniard
ETSF France
Laboratoire des Solides Irradiés,
Ecole Polytechnique, CEA-DSM, CNRS,
91128 Palaiseau, France
Outline
1. Introduction: which quantities do we need
2. Macroscopic average
- Definition
- Examples
•  Dielectric tensor for cubic symmetries
•  Dielectric tensor for non-cubic symmetries
- Properties
- Principal axis
•  Summary
Linear response
Perturbation theory
 For a sufficiently small perturbation, the response of the system can be extended
into a taylor series, with respect to the perturbation.
  We will consider only the first order (linear) response, proportional to the
perturbation.
 The linear coefficient linking the response to the perturbation is called a response
function. It is independent of the perturbation and depends only on the system.
 ≠ Strong field interaction(laser field for instance)
Examples
Density response function
Dielectric tensor

  

n1 (r , t ) = ∫ dt ' ∫ dr ' χ (r , t , r ' , t ' )V (r ' , t ' )
 
 
   
D(r , t ) = ∫ dt ' ∫ dr ' ε (r , t , r ' , t ' ) E(r ' , t ' )
Which quantities do we need?
Absorption coefficient
The general solution of the Maxwell's
equations for the electric field is
! !
! i(kx!!t )
E(r, t) = E0 e
Defining the complex refractive index as n =
field inside a medium is the damped ωwave:
k=
!
"
c
ε = ν + iκ , the electric
 
 i xn −iωt  i ωνx −ω κx −iωt
E(r , t ) = E0e c e = E0e c e c e
ν  and κ are the refraction index and the extinction coefficient and
they are related to the dielectric constant (ε=ε1+iε2) as
2
ε1 = ν − κ
2
ε 2 = 2νκ
The absorption coefficient α is the inverse distance where the intensity
of the field is reduced by 1/e
(related to the optical skin depth δ).
c
!=
"#
2"# "$ 2
!=
=
c
%c
Which quantities do we need?
!
E
!
k
Reflected beam
Reflectivity
Normal incidence reflectivity
z
R=
Transmitted beam
x
Incident beam
ET
Ei
2
<1
Using the continuity of the tangential
component of the electric field at the surface
(1 −ν ) 2 + κ 2
R=
(1 +ν ) 2 + κ 2
The knowledge of the optical
constant implies the knowledge
of the absorption and of the
reflectivity, which can be
compared with the experiment.
Which quantities do we need?
Energy loss by a fast particle
Given an external charge density ρext, one can obtained the external
potential Vext


2
(Poisson equation)
k Vext (k , ω) = 4πρext (k , ω)
The response of the system is an induced density, defined by the
response function χ



ρind (k , ω ) = χ (k , ω )Vext (k , ω )
and the total (induced + external) potential acting on the system is

 



 4π
−1
Vtot (k , ω ) = 1 + 2 χ (k , ω Vext (k , ω ) = ε (k , ω )Vext (k , ω )
 k




Vext (k , ω) = ε (k , ω)Vtot (k , ω)
Dielectric function:
Which quantities do we need?
Energy loss by a fast particle
 
Charge density of a particle (e-) with velocity v : ρext = eδ (r − v t )



Etot = −∇rVtot (r , t )
The total electric field is
and the energy lost by the electron in unit time is
 
dW
= ∫ dr j .Etot
with the current density
dt
   
j = ev δ (r − v t )

 ω 
dW
e dr
= − 2 ∫ 2 Im  
dt
π k
 ε (k , ω ) 
2
We get
 1 
− Im 

Electron Energy Loss Spectroscopy:
 ε (k , ω ) 
is called the loss function
Outline
1. Introduction: which quantities do we need
2. Macroscopic average
- Definition
- Examples
•  Dielectric tensor for cubic symmetries
•  Dielectric tensor for non-cubic symmetries
- Properties
- Principal axis
•  Summary
Maxwell’s equations in vacuum
! ! !
!
!. Eext (r, t) = 4!"ext (r, t)
! ! !
!. Bext (r, t) = 0
!
! ! !
1 $Bext
! " Eext (r, t) = #
c $t
!
! ! !
4! ! !
1 $Eext
! " Bext (r, t) =
jext (r, t) +
c
c $t
ρext and jext are the (external) free charge and current density
They can be arbitrary, but they have
 
to fulfill the continuity equation
E: electric field
B: magnetic field
(induction)

∂ρ ext
∇. jext (r , t ) +
=0
∂t
Maxwell’s equations in a medium
! ! !
! ! !
!
!
It is often more
!. Etot (r, t) = 4!"tot (r, t) or !. D(r, t) = 4!"ext (r, t)
convenient to use
! ! !
 

!. Btot (r, t) = 0
D = Etot + 4πP
!
! ! !
1 $Btot
! " Etot (r, t) = #
instead of Etot , as D can
c $t
be very close to
!
! ! !
4! ! !
1 $Etot
the external field
! " Btot (r, t) =
jtot (r, t) +
c
c $t
   
∇.D = ∇.Eext
(see the previous slide)
With ρtot =ρext+ρind and jtot =jext+jind ,
ρind and jind are the induced charge and current density:
they are not arbitrary, but reflect the spatial structure of the medium on a
microscopic level and the motion of the particle in it, including also the response
to an external field.
Microscopic spatial fluctuations
•  Infinite crystals → microscopic inhomogeneities (atomic scale)
•  Semi-infinite crystals → presence of the surface
•  Desorded medium → liquid
•  Rough surfaces
Macroscopic average
We consider spatial fluctuations whose characteristic length scale
is much smaller than the wavelength of light
Macroscopic quantities
Quantities that are slowly varying over the unit cells.
where V is the volume per unit cell of the crystal.
Examples
Eext(r, t), Aext(r, t), Vext(r, t),…
Typical values:
• dimension of the unit cell for silicon acell ≈0.5nm
• Visible radiation 400 nm < λ < 800 nm
! >> V 1/3
Macroscopic average
Microscopic quantities
Total and induced fields are rapidly varying. They include the contribution
from electrons in all regions of the cell.
The contribution of electrons close to or far from the nuclei will be very different.
⇒ Large and irregular fluctuations over the atomic scale.
Examples
Etot(r, t), jind(r, t), ρind(r, t),...
Macroscopic average
Measurable quantities
One measures quantities that vary on a macroscopic scale.
In the long wavelength limit,
the macroscopic neighbourhood contains many particles
We have to average over distances :
  large compared to the cell diameter
  small compared to the wavelength of the external perturbation
Macroscopic average
General definition:
We have to define two operators P̂a and P̂f which extract the average component
and the fluctuation component of any function F
P̂f = 1̂! P̂a
Fa = P̂a F
Ff = P̂f F
P̂a and P̂f have the following properties:
1)
2)
P̂a2 = P̂a
P̂f2 = P̂f and P̂a P̂f = P̂f P̂a = 0
P̂f
⇒ Projectors
P̂a commutes with the time and space differential operators
The average part of the field must obey
the macroscopic Maxwell’s equations
Macroscopic average
The differences between the microscopic fields and the averaged (macroscopic)
fields are called the local fields
Complexity of the problem:
• Macroscopic external field ⇒ induced fields
• The macroscopic procedure must take into account the fact that all the
components of the induced fields will create the response.
Procedure:
•  model for the system expressed in terms of an hamiltonian
•  microscopic response of the system (linear-response theory for instance)
! !
! !!
!
⇒ Definition of the microscopic dielectric tensor D(r, ! ) = ! dr '! (r, r ', ! )E(r ', ! )
• Averaging: definition of εM which relates the average parts of D and E
Macroscopic average
Infinite crystals
Functions having the crystal symmetries V(r+ R)=V(r), where R is any vector
of the Bravais lattice, can be represented by the Fourier series
  

 
i ( q +G ) r
V (r , ω ) = ∑ V (q + G, ω )e

qG
It can be also written as



iqr
V ( r , ω ) = ∑ V ( r ; q , ω )e

q
where

 
 
iGr
V (r ; q , ω ) = ∑
V ( q + G , ω )e

G
is a periodic function, with respect to the Bravais lattice.
Varies strongly even if the original wave is a long wave and nearly
constant within each cell (contains all the G-harmonics of the field).
Macroscopic average
Infinite crystals
Spatial average over a cell of the periodic part

 
V ( R, ω ) = V ( r ; q , ω )
R


 
1
iGr
= ∫ dr ∑
V ( q + G , ω )e

Ω
G


= V (q + 0, ω )
The macroscopic average corresponds to the G=0 component.
⇔  Τruncation that eliminates all wave vectors outside the first Brillouin zone
(wave-vector truncation)
Macroscopic quantities have all their G components equal to 0,
except the G=0 component.
→ Satisfies the two criteria previously defined
Macroscopic average
•  If the external applied field is not macroscopic, this averaging procedure
for the response function of the material has no meaning.
• One has to consider an average procedure based on the statistical
and quantum mechanical sense (beyond the scope of this lecture)
Exemples:
X-ray spectroscopy (very short wavelength)
EEL Spectroscopy with atomic resolution
Outline
1. Introduction: which quantities do we need
2. Macroscopic average
- Definition
- Examples
•  Dielectric tensor for cubic symmetries
•  Dielectric tensor for non-cubic symmetries
- Properties
- Principal axis
•  Summary
Macroscopic average
A simple example: the longitudinal case
All the fields can be expressed in terms of potentials (E=-∇V)
!
! !!
!
(Real space)
V
(
r,
!
)
=
d
r
'
!
(
r,
r
',
!
)V
(
r
',
!
)
!
ext
tot
The longitudinal dielectric
function is defined as
! !
! ! ! !
! !
Vext (q + G, ! ) = !" (q + G, q + G ', ! )Vtot (q + G ', ! )
G'
 
Vext is a macroscopic quantity : Vext (q + G, ω ) = Vext (q, ω )δ G 0
 
This is not the case for Vtot (q + G, ω )
(Reciprocal space)
Macroscopic average of Vext :



Vext (q, ω) = ∑ ε 0G ' (q, ω)Vtot (q + G' , ω) ≠ ε 00 (q, ω)Vtot (q, ω)
G'
The average of the product is not the product of the averages
Macroscopic average
A simple example: the longitudinal case
We have also
 
−1 
Vtot (q + G, ω ) = ∑ ε GG
' (q , ω )Vext (q + G ' , ω )
G'
where is ε-1GG the inverse dielectric function :


∑ε GG'' (q, ω)ε G−1''G' (q, ω) = δGG'
G ''
Macroscopic average of Vtot :
 

Vext is macroscopic ⇒ Vtot (q + G, ω ) = ε G−10 (q, ω )Vext (q, ω )

−1 
Vtot (q , ω ) = ε 00 (q , ω )Vext (q, ω )
Macroscopic average
A simple example: the longitudinal case
Macroscopic dielectric constant:




Vext (q , ω ) = ε M (q , ω )Vtot (q , ω ) ⇒ ε M (q , ω ) =
1
−1 
ε 00 (q , ω )

−1 
• Inversion of the full dielectric matrix ε GG ' (q , ω ) → ε GG ' (q , ω )
−1 
• We take the G=G =0 component of ε GG ' (q , ω )
Interpretation
All the microscopic components of the induced field will couple together to
produce the macroscopic response
Macroscopic average
Another example: the Lorentz model
For non interacting dipoles (dilute media)
α: polarisability of the dipole (atoms or molecules)


N: number of dipoles per unit volume (density)
P = Nα Eloc
Eloc: local field acting on a given dipole
If Eloc=Etot, we have
! !
! !
!
D = Etot + 4! P = Etot + 4! N" Etot


D = ε M Etot ⇒ ε M = 1+ 4π Nα
Mosotti (1850)
Claisius (1879)
Lorentz (1909)
Macroscopic average
The Lorentz model
In fact, Eloc≠Etot

P=


4 
Eloc = Etot + π P
3

Nα
Etot
4π
1−
Nα
3
εM
in the Lorentz model
(for isotropic or cubic media)
4πNα
= 1+
4π
1−
Nα
3
 Based on classical physics, does not include microscopic
(quantum-mechanical) effects
 Increase of the static dielectric function
 Red shift of the transition frequency
 Not the case, when including microscopic effects
f 012
α (ω ) = 2
⇒
ω01 − ω 2
4πNf 012
ε M = 1+
4π
2
2
ω01 − ω −
Nf 012
3
Macroscopic average
Summary
•  We have defined microscopic and macroscopic fields
•  Microscopic quantities have to be averaged to be compared to experiments
• The dielectric function has
- a microscopic expression (related to quantum mechanics)
- a macroscopic expression (classical scheme - Maxwell's equations)
• Absorption ↔ Im {εM} and EELS ↔ -Im {1/εM}
Outline
1. Introduction: which quantities do we need
2. Macroscopic average
- Definition
- Examples
•  Dielectric tensor for cubic symmetries
•  Dielectric tensor for non-cubic symmetries
- Properties
- Principal axis
•  Summary
Dielectric tensor for cubic symmetries
Useful definitions
Longitudinal fields
! ! !
! " E(r ) = 0
or
  
k × E (k ) = 0
E(k) propagates along k
Examples:
•  plasmon oscillations
•  sceening
•  electron energy loss
Some definitions:
with
Transverse fields
  
∇.E (r ) = 0
or
  
k .E (k ) = 0
E(k) propagates perpendicular to k
Examples:
•  photons
•  optical properties of solids
 
 
E (r ) ↔ E (k ) (Fourier transform)
  
k = q + G for crystals
(q is in the first Brillouin zone and G is
a reciprocical lattice vector)
Dielectric tensor for cubic symmetries
Transverse-longitudinal decomposition:
Any vector field can be split into longitudinal and transverse components
! ! L !T
E =E +E
with
!
!L !
! !
k
E (k ) = k̂ !"k̂. E(k )#$ and k̂ =
k
! !L
!"E = 0
and
! !T
!. E = 0
(In real space, the relations are nonlocal)
Macroscopic dielectric tensor
The relation D(q,ω) = εM(q,ω) Etot(q,ω) can be written in terms
of the longitudinal and transverse components
 D L   ε MLL
 T  =  TL
 D  ε
   M
ε MLT  EtotL 


TT  T 
ε M  Etot 
Dielectric tensor for cubic symmetries
Question:
How can we make the link between
•  the microscopic dielectric tensor
  
    
 
D(q + G, ω ) = ∑ ε (q + G, q + G' , ω ) Etot (q + G' , ω )
Microscopic
components
of D and Etot
G'
•  the macroscopic dielectric tensor
 
 

D(q, ω) = ε M (q, ω) Etot (q, ω)
Macroscopic
components
of D and Etot
Dielectric tensor for cubic symmetries
No symmetry
! LL
LT
!
!
! !
! M (q, " ) = # MTL MTT
# !
" M !M
Cubic symmetry with q→0
$
&
&
%
A longitudinal (transverse)
perturbation induces longitudinal
and transverse responses.
 ε MLL

ε M (q, ω ) = 
 0

0 

TT 
εM 
•  A longitudinal perturbation induces
a longitudinal response only.
•  A transverse perturbation induces
a transverse response only.
• Independent of the direction of q
This holds only for macroscopic quantities
The microscopic dielectric tensor has off-diagonal elements
! LT and ! TL
Cubic symmetries with q→0
Longitudinal dielectric function
ε MLL (ω ) = lim
q →0
1
1+

4π
χ
(
q
,ω)
ρρ
2
q
where χρρ(q,ω) is the density-density response function (TDDFT),
defined as



ρ ind (q , ω ) = χ ρρ (q , ω )Vext (q , ω )
Transverse dielectric function
!
lim ! MTT (q, ! ) = " MLL (! )
q!0
Dielectric tensor
The tensor is diagonal and contains only one quantity
ε MLL (ω )
Cubic symmetries with q≠0
Longitudinal dielectric function
One can show that the relation
!
! (q, " ) =
1
LL
M
1+
!
q
!0
holds also when
!
depends on q

Transverse dielectric function
TL

!
4#
$
(
q,
")
%%
2
q
LT

ε MTT (q, ω ) ≠ ε MLL (q, ω )

We have also ε M (q, ω ) ≠ 0 ε M (q, ω ) ≠ 0
These quantities are much more complicated
and need further approximation to be computed
(cannot be expressed in terms of TDDFT).
Cubic symmetries
Summary
•  We have defined the longitudinal and transverse components
of the dielectric tensor.
•  In the long wavelength limit q→0 ,only one quantity is needed (optical isotropy)
ε MLL (ω ) = ε MTT (ω ) = lim
1

4π
χ
(
q
,ω)
ρρ
2
q
• For q≠0, only εMLL has a simple expression in terms of the response function.
q →0
1+
Cubic symmetries
Some references
  H. Ehrenreich, in the Optical Properties of Solids, Varenna Course XXXIV,
edited by J. Tauc (Academic Press, New York, 1966), p. 106
  R. M. Pick, in Advances in Physics, Vol 19, p. 269.
  D. L. Johnson, Physical Review B, 12 3428 (1975).
  S. L. Adler, Physical Review, 126 413 (1962).
  N. Wiser, Physical Review, 129 62 (1963).
  R. Del Sole and E. Fiorino, Phys. Rev. B 29, 4631 (1984).
 W. L. Mochan and R. Barrera, Phys. Rev. B 32 4984 (1985); ibid 4984 (1985).
Outline
1. Introduction: which quantities do we need
2. Macroscopic average
- Definition
- Examples
•  Dielectric tensor for cubic symmetries
•  Dielectric tensor for non-cubic symmetries
- Properties
- Principal axis
•  Summary
Non-cubic symmetries
Properties of the macroscopic quantities
LL 
LT 

ε M (q , ω ) ε M (q , ω ) 
 

ε M (q, ω ) =  TL 
TT 

ε
(
q
,
ω
)
ε
(
q
,
ω
)
M
 M

Non-cubic symmetries
Dielectric tensor - General case
  
   

qˆα (q , q , ω ) 
ε M (q , ω ) = 1 + 4πα (q , q , ω ) 1 + 4π qˆ

LL  
1
−
4
πα
(
q
,
q
,
ω
)




  
    
 
Quasipolarisability α : jind (q + G, ω ) = ∑ α (q + G, q + G' , ω ) E pert (q + G' , ω )
G'
But one can show that the relation :
holds also for the non-cubic symmetries.
 
 
1
and we have α LL (q, q, ω ) = − 2 χ ρρ (q, q, ω )
q

ε MLL (q, ω ) =
1
 
1 − 4πα (q, qω )
LL
Longitudinal-longitudinal
dielectric function
Outline
1. Introduction: which quantities do we need
2. Macroscopic average
- Definition
- Examples
•  Dielectric tensor for cubic symmetries
•  Dielectric tensor for non-cubic symmetries
- Properties
- Principal axis
•  Summary
Non-cubic symmetries q→0
!
q
Main general result concerning εM : εM is an analytic function of
The limit q→0 does not depend on the direction of q.
!
!
(
"
)
=
!
(
q,
lim M " )
⇒ We can define εM(ω) as
M
!
q!0
! MLL is not analytic in the general case
Depending on the symmetry of the system, one can define the
3 principal axis, if they exist, (n1, n2,n3) defining a frame
in which εM(ω) is diagonal.
!
!
!
!
If Etot is parallel to one of these axis ni ! M (" )Etot (" ) = !i (" )Etot (" )
εi (ω) can be calculated as a longitudinal dielectric function

ε i (ω ) = ε MLL (ni , ω )
Non-cubic symmetries q→0
The distinction between longitudinal and transverse is not meaningful
The only important direction is the direction of the electric field
If q→ 0, the fields do not propagate
Existence of the principal frame ? εM is symmetric but complex …
No general answer
Use of geometrical arguments
Symmetries
•  Cubic
•  Hexagonal
•  Orthorombic
•  Monoclinic
•  Triclinic
Shorter wavelength q≠0
Alternative:
  
ˆ
 
   
qα (q , q , ω ) 
ε M (q , ω ) = 1 + 4πα (q , q , ω ) 1 + 4π qˆ
We can use

LL  
1
−
4
πα
(
q
,
q
,
ω
)






  
 


j
(
q
+
G
,
ω
)
=
α
(
q
+
G
,
q
+
G
'
,
ω
)
E
(
q
+ G' , ω )
where ind
∑
pert
G'
The induced current can be evaluated through the
(Time-Dependent)-Density-Current Functional Theory (TD-DCFT)
Which quantities do we need?
" 1 %
Electron Energy Loss Spectroscopy: ! Im # ! &
$ ! (q, " ) '
!
LL !
In that case, ! (q, ! ) = " M
!
!
!
(q, ! )
with Vext (q, ! ) = " (q, ! )Vtot (q, ! )
Is this correct? The perturbation is longitudinal.
What about the transverse response?
One can show that
!T !
!2 !T !
E (q, ! ) = 2 2 D (q, ! ) and
cq
!2
v2
! 2
2 2
cq
c
In the nonrelativistic approximation, the transverse fields are negligible
and the LL component of the dielectric tensor describes the energy
loss of charged particles
Outline
1. Introduction: which quantities do we need
2. Macroscopic average
- Definition
- Examples
•  Dielectric tensor for cubic symmetries
•  Dielectric tensor for non-cubic symmetries
- Properties
- Principal axis
•  Summary
Summary
 The key quantity is the dielectric tensor.
 Relation between microscopic and macroscopic fields.
 For cubic crystals, the longitudinal dielectric function εM(ω) defines entirely
the optical response in the long wavelength limit (q→0).
 For non-cubic crystals, the dielectric functions calculated along the principal
axis can be used to define entirely the optical response in the long
wavelength limit.
 For non-vanishing momentum, the situation is not so simple: εMLL (q,ω) only
can be defined in a simple way.