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Microscopic-macroscopic connection
Valérie Véniard
Laboratoire des Solides Irradiés
Ecole Polytechnique, Palaiseau - France
European Theoretical Spectroscopy Facility (ETSF)
Lyon,10 Dec 2007
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Outline
1
Introduction: Which quantities do we need?
2
Macroscopic average
Definition
Example
3
Dielectric tensor for cubic symmetries
4
Dielectric tensor for non-cubic symmetries
Properties
Principal axis
5
Summary
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Outline
1
Introduction: Which quantities do we need?
2
Macroscopic average
Definition
Example
3
Dielectric tensor for cubic symmetries
4
Dielectric tensor for non-cubic symmetries
Properties
Principal axis
5
Summary
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Linear response
Perturbation theory
For a sufficiently small perturbation, the response of the system can
be expanded 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.
6= strong field interaction (intense lasers for instance)
Example
R
R
Density response function: n1 (r, t) = dt 0 dr0 χ(r, t, r0 , t 0 )V (r0 , t 0 )
R 0R 0
Dielectric tensor D(r, t) = dt dr (r, t, r0 , t 0 )E(r0 , t 0 )
Microscopic-macroscopic connection
Valérie Véniard
Intro
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Introduction: Which quantities do we need?
Absorption coefficient
i(Kr−ωt)
The general solution of the Maxwell’s equations
.
√ is E(r, t) = E0 e
Defining the complex refractive index as n = = ν + iκ, the electric
field inside a medium is the damped wave:
E (x, t) = E0 e
iω
c xn
e −iωt = E0 e
iω
c νx
ω
e − c κx e −iωt
ν and κ are the refraction index and the extinction coefficient and they
are related to the dielectric constant as
1 = ν 2 − κ2
2 = 2νκ
The absorption coefficient α is the inverse distance where the intensity of
the field is reduced by 1/e
ω2
α=
νc
(related to the optical skin depth δ).
Microscopic-macroscopic connection
Valérie Véniard
Intro
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Introduction: Which quantities do we measure?
Reflectivity
Normal incidence reflectivity
z
Reflected beam
R=|
Transmitted beam
x
Incident beam
Microscopic-macroscopic connection
R=
ER 2
| <1
Ei
(1 − ν)2 + κ2
(1 + ν)2 + κ2
The knowledge of the optical
constant implies the knowledge of
the reflectivity, which can be
compared with the experiment.
Valérie Véniard
Intro
Macro
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Conclu
Introduction: Which quantities do we need?
Energy loss by a fast charged particle
Given an external charge density ρext , one can obtained the external
potential Vext
k 2 Vext (k, ω) = 4πρext (k, ω)
(Poisson equation)
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
Vtot (k, ω) = 1 + 2 χ(k, ω) Vext (k, ω) = −1 (k, ω)Vext (k, ω)
k
Microscopic-macroscopic connection
Valérie Véniard
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Introduction: Which quantities do we need?
Energy loss by a fast charged particle
Charge density of a particle (e − ) with velocity v: ρext (r, t) = −eδ(r − vt)
The total electric field is Etot (r, t) = −∇r Vtot (r, t) and the energy lost
by the electron in unit time is
Z
dW
= drj . Etot with the current density j = −evδ(r − vt)
dt
Z
dW
dk
e2
ω
We get
=− 2
Im
dt
π
k2
(k, ω)
Electron Energy Loss Spectroscopy
n
o
1
−Im (k,ω)
is called the loss function
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Outline
1
Introduction: Which quantities do we need?
2
Macroscopic average
Definition
Example
3
Dielectric tensor for cubic symmetries
4
Dielectric tensor for non-cubic symmetries
Properties
Principal axis
5
Summary
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Outline
1
Introduction: Which quantities do we need?
2
Macroscopic average
Definition
Example
3
Dielectric tensor for cubic symmetries
4
Dielectric tensor for non-cubic symmetries
Properties
Principal axis
5
Summary
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Maxwell’s equations
Maxwell’s equations in the vaccum
∇.Eext (r, t) = 4πρext (r, t)
∇.Bext (r, t) = 0
1 ∂Bext (r, t)
c
∂t
4π
1 ∂Eext (r, t)
jext (r, t) +
c
c
∂t
∇ × Eext (r, t) = −
∇ × Bext (r, t) =
ρext and jext are the external (free) charge and current density
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
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Maxwell’s equations
Maxwell’s equations in a medium
∇.Etot (r, t) = 4πρtot (r, t)
or
∇.D(r, t) = 4πρext (r, t)
∇.Btot (r, t) = 0
1 ∂Btot (r, t)
c
∂t
4π
1 ∂Etot (r, t)
jtot (r, t) +
c
c
∂t
∇ × Etot (r, t) = −
∇ × Btot (r, t) =
with ρtot = ρext + ρind , and jtot = jext + jind .
ρind and jind are the induced charge and current density.
It is often more convenient to use D = Etot + 4πP instead of Etot ,
as D is very close to the external field (∇.D = ∇.Eext )
Microscopic-macroscopic connection
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Macroscopic average
Macroscopic quantities
At long wavelength, external fields are slowly varying over the unit
cells.
2π
>> V 1/3
λ=
q
where V is the volume per unit cell of the cystal.
Example
Eext (r, t), Aext (r, t), Vext (r, t),...
Typical values:
dimension of the unit cell for silicon acell ' 0.5nm
Visible radiation 400nm ≤ λ ≤ 800nm
Microscopic-macroscopic connection
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Intro
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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.
Example
Etot (r, t), jind (r, t), ρind (r, t),...
Microscopic-macroscopic connection
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Intro
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Macroscopic average
Measurable quantities
One measures quantities that vary on a macroscopic scale.
We have to average over distances
large compared to the cell diameter
small compared to the wavelength of the external perturbation
Procedure
Average over a unit cell whose origin is at point R
Regard R as the continuous coordinate appearing in the
Maxwell’s equations
The differences between the microscopic fields and the averaged
(macroscopic) fields are called the local fields
Microscopic-macroscopic connection
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Macroscopic average
Procedure
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
V (r, ω) =
X
V (q + G, ω)e i(q+G)r
qG
It can be also written as
V (r, ω) =
X
V (r; q, ω)e iqr
q
P
where V (r; q, ω) = G V (q + G, ω)e iGr is a periodic function,
with respect to the Bravais lattice.
Microscopic-macroscopic connection
Valérie Véniard
Intro
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Macroscopic average
Procedure
Spatial average over a cell of the periodic part
V (R, ω) = < V (r; q, ω) >R
Z
X
1
=
dr
V (q + G, ω)e iGr
Ω
G
= V (q + 0, ω)
The macroscopic average corresponds to the G = 0 component.
⇐⇒ Truncation that eliminates all wave vectors outside the first
Brillouin zone.
Macroscopic quantities have all their G components equal to 0,
except the G = 0 component.
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Outline
1
Introduction: Which quantities do we need?
2
Macroscopic average
Definition
Example
3
Dielectric tensor for cubic symmetries
4
Dielectric tensor for non-cubic symmetries
Properties
Principal axis
5
Summary
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Macroscopic average
A simple example: the longitudinal case
All the fields can be expressed in terms of potentials (E = ∇V )
and the longitudinal dielectric function is defined as
Vext (q + G, ω) =
X
GG0 (q, ω)Vtot (q + G0 , ω)
G0
Vext is a macroscopic quantity : Vext (q + G, ω) = Vext (q, ω) δG 0
This not the case for Vtot (q + G, ω).
Macroscopic average of Vext :
Vext (q, ω) =
X
0G0 (q, ω)Vtot (q + G0 , ω) 6= 00 (q, ω)Vtot (q, ω)
G0
The average of the product is not the product of the averages
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Macroscopic average
The longitudinal case
We have also
Vtot (q + G, ω) =
X
0
−1
GG0 (q, ω)Vext (q + G , ω)
G0
where −1
GG0 (q, ω) is the inverse dielectric function:
P
−1
G00 GG00 (q, ω)G00 G0 (q, ω) = δGG0 ,ω
Macroscopic average of Vtot :
Vext is macroscopic ⇒ Vtot (q + G, ω) = −1
G0 (q, ω)Vext (q, ω)
Vtot (q, ω) = −1
0O (q, ω)Vext (q, ω)
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
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Conclu
Macroscopic average
Macroscopic dielectric constant
Vext (q, ω) = M (q, ω)Vtot (q, ω) ⇒ M (q, ω) =
1
−1
00 (q, ω)
Inversion of the full dielectric matrix GG0 (q, ω) → −1
GG0 (q, ω)
We take the G = G0 = 0 component of −1
GG0 (q, ω)
Interpretation
All the microscopic components of the induced field will couple
together to produce the macroscopic response.
Microscopic-macroscopic connection
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Is it always meaningful?
If the external applied field is not macroscopic (very short
wavelength), the averaging procedure for the response
function of the material has no meaning.
When dealing with surfaces, the definition is unclear due to
the lack of periodicity in the direction perpendicular to the
surface.
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
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)
macroscopic expression (classical scheme - Maxwell’s
equations)
Absorption ↔ Im {M } and EELS ↔ −Im
Microscopic-macroscopic connection
n
1
M
o
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Outline
1
Introduction: Which quantities do we need?
2
Macroscopic average
Definition
Example
3
Dielectric tensor for cubic symmetries
4
Dielectric tensor for non-cubic symmetries
Properties
Principal axis
5
Summary
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
General properties
Longitudinal fields
∇ × E(r) = 0 or (k) × E(k) = 0
E(k) propagates along k.
Transverse fields
∇.E(r) = 0 or (k).E(k) = 0
E(k) propagates perpendicular to k.
Longitudinal field : plasmon oscillations, screening, electron
energy loss
Transverse field : photons, optical properties of solids
Microscopic-macroscopic connection
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Intro
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General properties
Transverse - longitudinal decomposition
Any vector field can be split into longitudinal and transverse
components
V(k) = VL (k) + VT (k)
with k × VL (k) = 0 and k.VT (k) = 0
Macroscopic dielectric tensor
↔
The relation D(q, ω) = M (q, ω)Etot (q, ω) can written in terms of
the transverse and longitudinal components
 L   LL
 L 
M
LT
D
Etot
M

=


T
T
TL
TT
Etot
D
M
M
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
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General properties
Question
How can we make the link between
the microscopic dielectric tensor
X↔
(q + G, q + G0 , ω)Etot (q + G0 , ω)
D(q + G, ω) =
G0
the macroscopic dielectric tensor
↔
D(q, ω) = M (q, ω)Etot (q, ω)
Microscopic-macroscopic connection
Valérie Véniard
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Cubic symmetries
Properties of Macroscopic tensor for cubic symmetries
→
D(q, ω) = ←
M (q, ω)Etot (q, ω)
Cubic symmetry with q → 0
No symmetry

↔
M
(q, ω) = 
LL
M
TL
M
LT
M
TT
M



↔
M
LL
M
0
0
TT
M
(ω) = 


Macroscopic quantities only:
A longitudinal pertubation induces a longitudinal response only
A transverse pertubation induces a transverse response only
Microscopic-macroscopic connection
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Cubic symmetries with q → 0
Longitudinal dielectric function
LL
M (ω) = lim
q→0
1
1+
4π
χ (q, ω)
q 2 ρρ
where χρρ (q, ω) is the density-density response function (TDDFT),
relating the induced density to the external potential
ρind (q, ω) = χρρ (q, q, ω)Vext (q, ω)
Transverse dielectric function
LL
lim TT
M (q, ω) = M (ω)
q→0
Dielectric tensor
The dielectric tensor is diagonal and contains only LL
M (ω)
Microscopic-macroscopic connection
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Cubic symmetries with q 6= 0
Longitudinal dielectric function
One can show that the relation
LL
M (q, ω) =
1
1+
4π
χ (q, ω)
q 2 ρρ
holds also when q 6= 0.
Transverse dielectric functions
LL
TT
M (q, ω) 6= M (q, ω)
TL
We have also LT
M (q, ω) 6= 0 and M (q, ω) 6= 0
These quantities are much more complicated and need further
approximations to be computed.
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
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)
TT
LL
M (ω) = M (ω) = lim
q→0
1
1+
4π
χ (q, ω)
q 2 ρρ
For q 6= 0, only LL
M (q, ω) has a simple expression in terms of
the response functions
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
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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).
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Outline
1
Introduction: Which quantities do we need?
2
Macroscopic average
Definition
Example
3
Dielectric tensor for cubic symmetries
4
Dielectric tensor for non-cubic symmetries
Properties
Principal axis
5
Summary
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Outline
1
Introduction: Which quantities do we need?
2
Macroscopic average
Definition
Example
3
Dielectric tensor for cubic symmetries
4
Dielectric tensor for non-cubic symmetries
Properties
Principal axis
5
Summary
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Non-Cubic symmetries
Properties - Macroscopic quantities
 LL
M

↔
M (q, ω) = 
 TL
M
LT
M

TT
M



Microscopic and macroscopic quantities
Even for q → 0:
A longitudinal pertubation induces a longitudinal and a transverse
response
A transverse pertubation induces a longitudinal and a transverse
response
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
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Non-Cubic symmetries
Dielectric tensor - General case
←
→
M (q, ω) = 1 + 4π α̃(q, q, ω) 1 + 4πq̂
q̂α̃(q, q, ω)
1 − 4π α̃LL (q, q, ω)
But one can show that the relation holds also for the non-cubic
symmetries.
1
LL
M (q, ω) =
1 − 4π α̃LL (q, q, ω)
R. Del Sole and E. Fiorino, Physical Review B, 29 4631 (1985).
Quasipolarisability α̃:
jind (q + G) =
X
α̃(q + G, q + G0 , ω)Epert (q + G0 , ω)
G0
with α̃LL (q, q, ω) = − q12 χρρ (q, ω)
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Outline
1
Introduction: Which quantities do we need?
2
Macroscopic average
Definition
Example
3
Dielectric tensor for cubic symmetries
4
Dielectric tensor for non-cubic symmetries
Properties
Principal axis
5
Summary
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Non-cubic symmetries with q → 0
Principal axis
One of the main general result concerning M is:
M (q) is an analytic function of q
The limit q → 0 does not depend on the q ⇒ M (ω)
Depending on the symmetry of the system, one can use the 3
principal axis n1 ,n2 ,n3 , defining a frame in which M is diagonal.
If Etot (q, ω) is parallel parallel to one of these axis ni
←
→
M (ω) : Etot (q, ω) = i (ω)Etot (q, ω)
whatever the direction of q. i (ω) is an eigenvalue:
⇒ i can be seen as a longitudinal dielectric function
i = LL
M (ni , ω)
but can also be used as a transverse dielectric function.
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
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Noncubic
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Non-cubic symmetries with q → 0
Longitudinal and transverse dielectric functions
Along these directions ni , a longitudinal perturbation induces a
longitudinal response through the usual relation
lim LL
M (q, ω) = lim
q→0
q→0
1
1+
4π
χ (q, ω)
q 2 ρρ
and if a transverse field is along ni , it will induce only a transverse
response.
Keep in mind that the principal frame is not always orthogonal and
so q could be different from ni !
Consequences
If the principal frame is known, on can deduce the optical
properties from a longitudinal calculation performed in this frame.
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Outline
1
Introduction: Which quantities do we need?
2
Macroscopic average
Definition
Example
3
Dielectric tensor for cubic symmetries
4
Dielectric tensor for non-cubic symmetries
Properties
Principal axis
5
Summary
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Summary
The key quantity is the dielectric tensor.
Relation between microscopic and macroscopic fields.
For cubic crystals, the longitudinal dielectric function LL
M (ω)
defines entirely the optical response in the long wavelength
limit.
For non-cubic crystals, the longitudinal 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:
LL
M (q, ω) defines only the longitudinal response
Microscopic-macroscopic connection
Valérie Véniard
Intro
Macro
Cubic
Noncubic
Conclu
Thank you for your attention
Microscopic-macroscopic connection
Valérie Véniard