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Transcript
December 22, 2010
13:5
World Scientific Book - 9.75in x 6.5in
Gavrilenko˙NanoOptics
Appendix A
Thomas–Fermi Approximation and Basics
of the Density Functional Theory
As stated at the beginning of section 2.7 the total energy is a key function describing the basic physical and chemical properties of materials: the ground state. It
consists of both kinetic (describing motion) and potential energy parts. To make
a theoretical model realistic it is very important to incorporate all important contributions to both the parts of the total energy. In view of the big number of the
particles involved into the model, this is very challenging for the first-principles
theory. Different approximations are applied in order to achieve a trade-off between
complexity and accuracy. Very successful in realistic modeling of the ground state
is the density functional theory, DFT. In this chapter we present the basic ideas
of the DFT and demonstrate both advantages and problems for optics within this
method.
Initially Thomas and Fermi (TF) in the 1920s suggested describing atoms as
uniformly distributed electrons (negatively charged clouds) around nuclei in a sixdimensional phase space (momentum and coordinates). This is enormous simplification of the actual many-body problem. It is instructive to consider the basic ideas
of the TF approximation before starting with a more accurate theory: the DFT.
The basic ideas and results of the TF model in application for atoms are provided
here.
Following the TF approach the total energy of the system could be presented
as a function (functional) of electron density [McQuarrie (1976); Parr and Yang
(1989)]. Each h3 of the momentum space volume (h is the Planck constant) is
occupied by two electrons and the electrons are moving in an effective potential
field that is determined by nuclear charge and by assumed uniform distribution of
electrons. The density of ΔN electrons in real space within a cube (nanoparticle)
with a side l is given by
ρ(r) =
ΔN
ΔN
= 3 .
v
l
(A.1)
The electron energy levels in this three-dimensional infinite well are given by:
E=
h2
h2 2
R̃ ,
(n2x + n2y + n2z ) =
2
8ml
8ml2
279
nx , ny , nz = 1, 2, 3, . . .
(A.2)
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Appendix A
280
Radius R = R̃max of the sphere in the space (nx , ny , nz ) covering all occupied
states determines the maximum energy of electrons: the Fermi energy F . The
number of energy levels within this maximum value at zero temperature is given by
3/2
π 8ml2 F
1 4πR3
=
.
(A.3)
NF = 3
2
3
6
h2
The density of states is defined as
g(E)dE = NF (E + dE) − NF (E) =
π
4
8ml2
h2
3/2
E 1/2 dE.
(A.4)
At zero temperature all energy levels below the Fermi energy are occupied:
1E≤F
f (E) =
.
(A.5)
0E>F
Consequently the total energy of the electrons in one cell will be given by
F
F
4π
3/2 3
Ef (E)g(E)dE = 3 (2m) l
E 3/2 dE
E=
h
0
0
3
π 2l
(2m)3/2 F 5/2 .
=
5 h
(A.6)
The Fermi energy F can be obtained from the total number of electrons ΔN in a
cell:
3
F
π 2l
ΔN = 2
f (E)g(E)dE = 3
(2m)3/2 F 3/2 .
(A.7)
h
h
0
Combining Eqs. (A.6) and (A.7) the energy of the electrons in one cell is given by
l3 5/3
l3
3
(3π 2 )2/3
ρ
= CF
,
2
10
(2π)
(2π)5/3
3 2 2/3
3π
CF =
= 2.871.
10
E=
(A.8)
In Eq. (A.8) we reverted to atomic units e = h = m0 = 1. The electron density is
a smooth function in a real space. For systems without translational symmetry it
is different for different cells. However, for spatially periodic systems only consideration within the unit cell is required since all unit cells are equivalent. Now adding
the contributions from all cells with energies within F , we obtain
(A.9)
TTF [ρ] = CF ρ5/3 (r)d3 r.
Equation Eq. (A.9) represents the well-known Thomas–Fermi kinetic energy functional, which is a function of the local electron density. The functional Eq. (A.9)
could be applied to electrons in atoms encountering the most important idea of the
modern DFT, the local density approximation (LDA) [Martin (2004)]. Adding to
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Appendix A
Gavrilenko˙NanoOptics
281
Eq. (A.9) classical electrostatic energies of electron–nucleus attraction and electron–
electron repulsion we arrive at the energy functional of the Thomas–Fermi theory
of atoms:
ρ(r) 3
ETF [ρ(r)] = CF ρ5/3 (r)d3 r − Z
d r
r
1
ρ(r 1 )ρ(r 2 ) 3
+
d r 1 d3 r 2 .
(A.10)
2
|r 1 − r 2 |
Note that the nucleus charge Z is measured in atomic units. The energy of the
ground state and electron density can be found by minimizing the functional (A.10)
with the constrain condition
ρ(r)d3 r.
N=
(A.11)
The electron density in Eq. (A.10) has to be calculated in conjunction with
Eq. (A.11) from the following equation for chemical potential, defined as the variational derivative according to
5
ρ(r 2 ) 3
δETF [ρ]
Z
5/3
= CTF ρ (r) − +
d r2 .
(A.12)
μTF =
δρ(r)
3
r
|r 1 − r 2 |
The Thomas–Fermi model provides reasonably good predictions for atoms. It has
been used before to study potential fields and charge density in metals and the
equation of states of elements [Feynman et al. (1949)]. However, this method is
considered rather crude for more complex systems because it does not incorporate the actual orbital structure of electrons. In view of the modern DFT theory
the Thomas–Fermi method could be considered as an approximation to the more
accurate theory.
For systems like molecules and solids, much better predictions are provided by
the DFT. Search for the ground state within the DFT follows the rule that the
electron density is a basic variable in the electronic problem (the first theorem of
Hohenberg and Kohn [Hohenberg and Kohn (1964)]) and another rule that the
ground state can be found from the energy variational principle for the density (the
second theorem of Hohenberg and Kohn [Kohn (1999)]). According to the DFT the
total energy could be written as
E[ρ] = T [ρ] + U [ρ] + EXC [ρ],
(A.13)
where T is the kinetic energy of the system of noninteracting particles and U is
the electrostatic energy due to Coulomb interactions. The most important part
in the DFT is EXC , the exchange and correlation (XC) energy that includes all
many-body contributions to the total energy. The charge density is determined by
the wave functions, which for practical computations could be constructed from
single orbitals, φj (e.g. antisymmetrized product—the Slater determinant, atomic
or Gaussian orbitals, linear combinations of plane waves). The charge density is
given by
2
|φj (r)| ,
(A.14)
ρ(r) =
j
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Appendix A
282
where the sum is taken over all occupied j orbitals. In the spin-resolved case there
will be orbitals occupied with spin-up and spin-down electrons. Their sum gives
the total charge density and their difference gives the spin density. In terms of the
electron orbitals the energy components are given in atomic units as
1 ∗
T =−
φj (r)|∇2 |φj (r)d3 r,
(A.15)
2
j
U =−
N n α
j
+
+
1
2
φ∗j (r) Zα
φj (r)d3 r
(Rα − r) φ∗i (r 1 )φ∗j (r 2 )
i,j
N
N α β<α
1
φi (r 1 )φj (r 2 )d3 r 1 d3 r 2
(r 1 − r 2 )
Zα − Zβ
.
|Rα − Rβ |
(A.16)
The first term in potential energy (A.16) stands for the electron–nucleus attraction,
the second term describes for electron–electron repulsion, and the third term represents nucleus–nucleus repulsion. In Eq. (A.16) Zα refers to the charge on nucleus
α of N −atom system.
The third term in Eq. (A.13) describes the exchange and correlation energy.
Rather simple for computations but surprisingly good approximation is the local
density approximation, LDA, which assumes that the charge density varies slowly
on the atomic scale, i.e., the effect of other electrons on the given (local) electron
density is described as a uniform electron gas. The XC energy can be obtained by
integrating with the uniform gas model (see e.g. [Ceperley and Adler (1980)]):
∼
(A.17)
EXC = ρ(r)ẼXC [ρ(r)]d3 (r),
where ẼXC [ρ(r)] is XC energy per particle in a uniform electron gas. For many
systems a good approximation provides analytic expression for ẼXC [ρ(r)] suggested
by Perdew and Wang (1992). In practical calculations through minimization of the
total energy Eq. (A.13) one determines self-consistently the electron density and
the actual XC part. A variational minimization procedure leads to a set of coupled
equations proposed by Kohn and Sham [Kohn and Sham (1965)]:
1 2
(A.18)
− ∇ − VN + Ve + μXC (ρ) φj = Ej φj ,
2
with
μXC =
∂
(ρEXC ) ,
∂ρ
(A.19)
The solution of the Kohn–Sham equation provides the equilibrium geometry
and the ground-state energy of the system. However, eigen functions and eigen
energies of the Kohn–Sham equation cannot be interpreted as the quasiparticle
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Appendix A
Gavrilenko˙NanoOptics
283
quantities needed for optics. The term quasiparticle refers to a particle-like entity
arising in certain systems of interacting particles. If a single particle moves through
the system, surrounded by a cloud of other interactiong particles, the entire entity
moves along somewhat like a free particle (but slightly different). The quasiparticle
concept is one of the most important in materials science, because it is one of the few
known ways of simplifying the quantum mechanical many-body problem describing
excitation state and is applicable to an extremely wide range of many-body systems.
Calculation of the ground state from the Kohn–Sham equation does not result
automatically in correct prediction of excitation energies required for optics. For
example, in nonmetallic systems the predicted value of the energy difference (energy
gap) between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) in most cases is underestimated (gap problem).
Special corrections (quasiparticle, QP corrections) are required to get more accurate excitation energies [Onida et al. (2002)]. Without corrections in semiconductors
and insulators the local density approximation, LDA, substantially underestimates
forbidden gap values. In this chapter we present LDA results for optics with different QP corrections avoiding, however, detailed analysis of theoretical methods.
For advanced reading of the DFT one can recommend original papers [Hohenberg
and Kohn (1964); Kohn (1999); Ceperley and Adler (1980)] and the monographs
[McQuarrie (1976); Parr and Yang (1989); Martin (2004); Michaelides and Scheffler
(2008)].
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Appendix B
Evaluation of Optical Functions within
the Perturbation Theory
In this section we describe evaluation of the light-filled induced charge within perturbation theory using plane wave representation, which is used in this chapter for
calculations of optical functions (see section 2.7). Equilibrium electron charge density is defined through the density operator (using definition of Trace, T r, as a sum
of the diagonal elements):
neq (r) = eT r[ρ0 , δ(r − r 0 )].
(B.1)
Without illumination if the system is periodic (at least in one dimension) the density operator could be defined in energy representation on a set of Bloch functions
according to [Davydov (1980)]
ρ|s = ρ0 |k, l = f (Ek,l )|k, l,
where the equilibrium Fermi distribution function is given by
F −Es
−1
f (Es ) = e kT − 1
,
(B.2)
(B.3)
The Bloch functions
|s = |k, l = uk,l (k)eikr ,
(B.4)
are solutions of an undisturbed Schrödinger equation with periodic potential:
1 2
(B.5)
H0 |k, l = − ∇ + V0 (r) |k, l = Ek,l |k, l.
2
In an external optical field when light quanta strike electrically neutral atoms the
equilibrium is broken through the deformation of electron clouds. Time-dependent
changes of the electron charge density can be represented as Taylor expansion. The
number of the terms to be included into the Taylor sum for the induced part of the
charge depends on the excitation intensity:
n(r, t) = neq (r) + nind (r, t) = eT r[ρ, δ]
= eT r[ρ0 , δ] + eT r[ρ(1) , δ] + eT r[ρ(2) , δ] + . . .
(B.6)
The first- and higher-order corrections to the density operator are determined from
the standard perturbation theory:
dρ
= [H, ρ] = (Hρ − ρH),
(B.7)
i
dt
285
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Appendix B
286
with
H = H0 + V (1) + . . . ,
ρ = ρ0 + ρ(1) + ρ(2) + . . .
(B.8)
In Eq. (B.7) for simplicity we neglected the effect of energy dissipation, which could
be included through the relaxation time. Plugging (B.8) in (B.7) and choosing terms
of the same order on both the left and right side of the equation of motion for the
density operator, Eq. (B.7) splits into a series of equations for zero, first, second,
etc., orders of perturbations, respectively:
iρ̇0 = [H0 , ρ0 ],
iρ̇(1) = H0 , ρ(1) + V (1) , ρ0
iρ̇(2) = H0 , ρ(2) + V (1) , ρ(1)
(B.9)
Dynamic optical response is described through the time-dependent density operator.
In an external electromagnetic field the perturbation is harmonic, i.e.,
ρ(t) = ρ(0)eiωt ,
iρ̇(ω) = −ωρ(ω).
(B.10)
It is convenient now to switch to the matrix representation in Eq. (B.10) by projecting the relevant quantities on a set of Bloch functions in Eq. (2.68). To this
end one should multiply every term in Eq. (B.10) with the function Eq. (2.68);
the complex conjugate of Eq. (2.68) is then multiplied on the left and right side of
the relevant equation. Through integration over the entire space and by taking into
account orthonormality conditions for Bloch functions, this leads to the following
expression for the first-order terms:
(1)
(1)
(1)
(1)
Vst ρ0ts −
ρ0sp Vps ,
(B.11)
−ωρss = (Es − Es )ρss +
t
p
The density operator defined as Eq. (B.2) in matrix presentation has the form
(0)
ρss = f (Es )δss .
(B.12)
Equation (B.11) is now transformed into
(1)
(1)
−ωρss = (Es − Es )ρss + [f (Es ) − f (Es )] Vss .
(B.13)
At zero temperature, optical excitations occur between completely filled and empty
states with Fermi functions equal to either 1 or zero, respectively. Consequently,
f (Es ) − f (Es )
(1)
Vss = (Es − Es − ω)−1 Vss |T =0 .
(B.14)
ρss (ω) =
Es − Es − ω
For the second-order perturbation one needs to use the first-order solution
Eq. (B.14). Plugging it into Eq. (B.10) after some algebra leads to the following expressions at T = 0:
(1) (1)
(1) (1)
(2)
(2)
Vss ρs s −
ρss Vs s ,
(B.15)
−ωρss = (Es − Es )ρss +
s
s
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Appendix B
or
(2)
ρss (ω) =
Gavrilenko˙NanoOptics
287
(1) (1)
1
1
[Vss Vs s (
Es − Es − ω Es − Es − ω
s
1
)].
−
Es − Es − ω
(B.16)
Equations (B.14) and (B.16) can be used now to obtain induced charge density from
Eq. (B.6) within the first and second order of external perturbation, respectively.
The first- and second-order contributions to the induced charge density in Eq. (B.6)
follow from (B.14) and (B.16), respectively. In a system with a periodicity the
perturbation potential is given by
∞
V (q + G, ω)ei(q+G)r dω,
(B.17)
V (r, t) =
qG
−∞
where G is the reciprocal lattice vector. For Fourier transform of the potential
∞
V (r, ω)eiωt ,
(B.18)
V (r, t) =
−∞
the expansion of the potential is given by
V (r, ω) =
V (q + G, ω)ei(q+G)r .
(B.19)
qG
In a periodic system with all equivalent atoms separated by Ri one has for the
charge:
neq (r 0 + Ri ) = neq (r 0 ),
with
n(r) =
eiqr n(q) =
q
eiqr
q
(B.20)
δqG n(G).
(B.21)
G
For induced charge density in (B.6) we have
nind (r, ω) =
nind (q + G, ω)ei(q+G)r .
(B.22)
qG
Where Fourier transform of the induced charge is given by Fourier integral by
(B.23)
nind (q + G, ω) = nind (r, ω)e−i(q+G)r d3 r,
Linear part of the induced charge in (B.23) follows from (B.6)
nind (q + G, ω) = e T r ρ(1) (ω), δ(r − r) e−i(q+G)r d3 r = eT r ρ(1) (ω), e−i(q+G)r ,
(B.24)
The trace of the operator product is calculated according to
T r ÂB̂ =
m|ÂB̂|m =
Amn Bnm .
(B.25)
m
m
n
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Appendix B
288
Eq. (B.24) projected on the plane wave basis 2.68 can be represented as
nind = nind (q + G, ω)
−i(q+G)r =e
l , k + q|ρ(1)
ω |k, ll, k|δ(r − r)|k + q, l e
k+q k,l
=e
l , k + q|ρ(1)
ω |k, ll, k|
=e
ei(r−r )G e−i(q+G)r |k + q, l G
k+q k,l
−i(q+G)r
l , k + q|ρ(1)
|k + q, l .
ω |k, ll, k|e
(B.26)
k+q k,l
In (B.26) the bracket notation means space integration. We also used the definition
of the δ function:
ei(r−r )G .
(B.27)
δ (r − r ) =
G
Now Eq. (B.14) can be written as
f [El (k + q)] − f [El (k)] l , k + q|V (r, ω)|k, l
El (k + q) − El (k) − ω − iη
f [El (k + q)] − f [El (k)]
=
El (k + q) − El (k) − ω − iη
×
V (q + G, ω)l , k + q|ei(q+G)r |k, l.
(B.28)
l , k + q|ρ(1)
ω |k, l =
q,G
The complex part of the energy in the denominator of Eq. (B.28) is introduced
to prevent unphysical divergences at resonance frequencies. Plugging (B.28) into
(B.26) we arrive at the following expression for the induced charge:
nind (q + G, ω) = e
PG,G V (q + G, ω).
(B.29)
G
Using the notation (k = k + q) the polarization function is defined as
PG,G (ω) =
l , k |ei(q+G)r |k, ll, k|ei(q+G )r |k , l k ,k l ,l
×
f [El (k + q)] − f [El (k)]
.
El (k + q) − El (k) − (ω + iη)
(B.30)
Evaluation of the full polarization function is given in Appendix C. Full potential
in materials can be separated into two parts, the external and induced potential:
V (q + G, ω) = Vext (q + G, ω) + Vind (q + G, ω).
(B.31)
The Eq. (B.31) can be understood as a reduction (screening) of external potential
through the induced charge in materials. This can be presented in terms of dielectric
function:
ε−1
(B.32)
V (q + G, ω) =
G,G Vext (q + G , ω),
G
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Appendix B
or
Vext (q + G , ω) =
εG,G V (q + G, ω),
Gavrilenko˙NanoOptics
289
(B.33)
G
Equations (B.32) and (B.33) can be considered as the microscopic definition of the
dielectric function. The described computation of ε presents a transformation from
the microscopic (atom-related) quantities to the macroscopic values used in classical
electrodynamics theory. For advanced reading related to the definition of the optical
function within the first-principles theory, one can recommend [Martin (2004); Yu
and Cardona (2010)] or original papers (see e.g. [Onida et al. (2002); Gavrilenko
and Bechstedt (1997)] and references therein).
The induced potential satisfies the Poisson equation:
Vind (q + G, ω) =
4π
nind (q + G, ω).
|q + G|2
From Eqs. (B.29), (B.31), (B.33), and (B.34) we obtain
4π
PG,G V (q + G, ω)
V (q + G, ω) =
εG,G (ω) +
|q + G|2
G
δG,G V (q + G, ω).
=
(B.34)
(B.35)
G
The dielectric function can now be expressed in terms of the polarization function:
εG,G (ω) = δG,G −
4π
PG,G .
|q + G|2
(B.36)
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Appendix C
Local Field Effect in Optics of Solids from
the First Principles
Local field (LF) effect plays a key role in the optics of nanostructures. Description
of the LF effect within classical electrodynamics is presented in many monographs
and research papers (see Chapter 3 and references therein). The classical approach,
however, cannot include optical excitations on the atomic scale in the whole picture.
Electronic excitations are increasingly important in the optics of nanomaterials with
reduction of the dimensions of nanostructures, thus requiring microscopic modeling.
This section presents the evaluation of the optical dielectric function, including the
LF effect within the perturbation theory (see section B).
The formula for the polarization function (see Eq. (B.30)) can be presented as
2 k ,k
∗k,k
Bn ,n (q + G)Bn,n
PG,G (ω) =
(q + G )
Ω k ,k n ,n
f [El (k + q)] − f [El (k)]
.
El (k + q) − El (k) − (ω + iη)
The Bloch integrals in Eq. (C.1) are defined as
×
(C.1)
,k
Bnk ,n
(q + G) = n , k |ei(q+G)r |k, n
1
(C.2)
=
ψn∗ ,k (r)ei(q+G)r ψn,k d3 r.
Ω
In plane wave representation (2.68) neglecting the umklapp processes (the nonconserving crystal momentum electron–electron scattering) [Bechstedt (2003)] and in
the limit of q → 0 the Bloch integrals have an extremely simple form, given by
k ,k
(G) =
d∗c,k (G1 )dv,k (G1 − G).
(C.3)
Bc,v
G1
Indexes c and v in Eq. (C.3) denote empty antibonding (conducting) and filled
bonding (valence) electron states, respectively, at zero temperature. By derivation
of Eq. (C.3) we used the following properties of direct and reciprocal lattice vectors:
Gi Rj = 2πδij
eiGj Rj = 1
ei(k −k+q)Ri =
δk −k+q,Gi
Ri
Gi
291
(C.4)
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Appendix C
292
Equation (C.1) at zero temperature takes the following form:
k ,k
∗k,k
2 Bn ,n (q + G)Bn,n (q + G )
.
PG,G (ω) =
Ω El (k + q) − El (k) − (ω + iη)
(C.5)
k ,k n ,n
From the orthonormality of the wave functions follow the properties of Bloch integrals:
k ,k
Bc,v
(0) =
d∗c,k (G1 )dv,k (G1 ) = 0,
G1
N
2
k ,k
Bn ,n (q + G) = 1.
(C.6)
n =1 k
For G = 0 and in the limit q → 0 the Bloch integrals have the following properties:
3
k ,k
(q) = δk,k lim i
lim Bc,v
q→0
q→0
=
qα c, k|rα |k, v
α=1
3
δk,k
lim
qα c, k|vα |k, v,
Ec (k) − Ev (k) q→0 α=1
(C.7)
k ,k
Bc,v
(qα )
1
lim
=
c, k|vα |k, v.
q→0
|qα |
Ec (k) − Ev (k)
Here we used the general definition of velocity (or momentum) [Adolph et al. (1996)]
given by
1
H, eiqr .
(C.8)
v = lim
q→0 q
At the limit the velocity is given by
v = i [H, r] .
(C.9)
After projecting on the full set of eigen functions of the Hamiltonian it follows that
nk| [H, rα ] |n k =
nk|H|m, lm, l|rα |n k m,l
−
nk|rα |m , l m , l |H|n k (C.10)
m ,l
= (Enk − En k )nk|rα |n k .
Equation (C.11) represents the relationship between matrix elements of the velocity
and of the induced dipole momentum, which could be used to obtain the relationship
between optical functions calculated in velocity and length gauges [Gavrilenko and
Bechstedt (1997)].
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Appendix D
Optical Field Hamiltonian in Second
Quantization Representation
If there are resonances of electromagnetic field within a cavity the entire field can
be presented as a superposition of single modes in the following form [Jaynes and
Cummins (1963)]:
√ pj (t)E j (r),
(D.1)
E(r, t) = − 4π
j
√ H(r, t) = 4π
ωj qj (t)H j (r).
(D.2)
j
The total energy of the field is given by
1
1 2
H=
(|E|2 + |H|2 )d3 r =
(p + ωj2 qj2 )
8π
2 j j
(D.3)
The Hamiltonian equation of motion is given by
∂H
= pj ,
∂pj
∂H
p˙j = −
= −ωj2 qj .
∂qj
q˙j =
(D.4)
Mathematically, quantization of the field is represented by commutations rules for
the canonically conjugated coordinates and momenta:
[qi , qj ] = 0,
[pi , pj ] = 0,
[qi , pj ] = iδij .
(D.5)
The Hamiltonian of an optical field is convenient to present in terms of second
quantization operators, the Bosonic operators of creation (â†j ) and annihilation (âj )
of photons using the definitions [Davydov (1976)]
ωj (D.6)
âj + â†j ,
pj =
2
ωj ωj qj = i
(D.7)
âj − â†j ,
2
with commutation rule
[âi , â†j ] = δij .
293
(D.8)
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Appendix D
The properties of âi operators are described through their action on state vector
φ(n1 , n2 , . . . ni . . .) according to
√
âi φ(n1 , . . . ni−1 , ni , ni+1 . . .) = ni φ(n1 , . . . ni−1 , ni−1 , ni+1 . . .),
√
â†i φ(n1 , . . . ni−1 , ni , ni+1 . . .) = ni + 1φ(n1 , . . . ni−1 , ni+1 , ni+1 . . .).
The Hamiltonian of the quantized optical field is now given by
1
†
ωi âi âi +
H=
.
2
i
(D.9)
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Appendix E
Surface Plasmons and Surface Plasmon
Polaritons
Collective electronic excitations (also known as plasma excitations) at metal surfaces and/or metal/dielectric interfaces play a key role in the optics of nanomaterials, ranging from physics and materials science to biology. A unified theoretical
description of these phenomena is based on the many-body dynamical electronic
response of solids (see section B of the Appendix), which underlines the existence of
various collective electronic excitations at metal surfaces, such as the conventional
surface plasmon, multipole plasmons, and the acoustic surface plasmon. A detailed
description of the surface plasmon polariton (SPP) phenomena and its applications
in modern optical spectroscopy is out of the scope of the present book. Several specialized monographs and reviews can be recommended for advanced reading [Tudos
and Schasfoort (2008); Pitarke et al. (2007); Liebsch (1997); Ritchie (1973); Venger
et al. (1999); Raether (1988)]. Here we present the basic conditions and properties
of SPP that follow from classical electrodynamics.
Consider a model consisting of two semi-infinite nonmagnetic media with local (frequency-dependent) dielectric functions εd (ω) (dielectric) and εm (ω) (metal)
separated by a planar interface at z = 0 [Pitarke et al. (2007)]. The full set of
Maxwell’s equations in the absence of external sources can be expressed as follows
[Jackson (1975)]:
1 ∂
En,
c ∂t
1 ∂
H n,
∇ × En = −
c ∂t
∇ · (εn E n ) = 0,
∇ × Hn = ε
∇ · H n = 0,
(E.1)
(E.2)
(E.3)
(E.4)
where the index n = d in dielectric (at z < 0) and n = m in metal (at z ≥ 0).
Within the classical picture, the metal can be treated as a semi-infinite electron
gas with an abruptly terminated profile of the electron density function. The surface
charge density (see Eq. (B.1)) on the metal–dielectric interface can be presented as
Liebsch (1997); Raether (1988)
n(r, ω) ≈ ejq r δ(z).
295
(E.5)
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Appendix E
The electric field associated with the density Eq. (E.5) is given by the Gauss
law, which follows from Eq. (E.3) in the presence of charge:
∇ · E(r, ω) = −4πn(r, ω).
(E.6)
Neglecting the retardation effects only the longitudinal plasma oscillations are
considered here. In terms of the scalar potential φ the electric field is given by
E(r, ω) = ∇ · φ(r, ω).
(E.7)
For the choosen model system the potential φ is given by
φ(r, ω) = ejq r φ(z, ω)
(E.8)
The z components of the field decay evanescently into both media. This follows
from the electroneutrality since ∇2 φ = 0 must be valid everywhere except at z = 0
[Liebsch (1997)]. Consequently the potential in Eq. (E.8) must be taken in the
form
φ(r, ω) = φ0 ejq r e−q|z| ,
(E.9)
where q ≡ qz . In this case the field determined by Eqs. (E.7) and (E.9) varies
continuously within the interface, however, the normal component is discontinuous.
The components above and below the interface are given by
Ez (z + 0) = q φ0 ejq r ,
Ez (z − 0) = −q φ0 ejq r .
(E.10)
The field and charge distribution for such a surface mode is represented in
Fig. E.1.
In the long-wavelength limit the boundary condition can be written as
εm (ω)Ez (0− ) = εd Ez (0+ ).
(E.11)
For the metal–vacuum interface (i.e., by εd = 1) the condition for the existence
of the surface plasmons followed from Eqs. (E.10) and (E.11) now reads
εm (ω) = −1.
(E.12)
Fig. E.1 Schematic of charge and field distribution for a surface plasmon described by Eq. (E.9).
Adapted from [Hofmann (2008)]
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Appendix E
Gavrilenko˙NanoOptics
297
Equation (E.12) defines the frequency of the surface plasmon in the q = 0
limit [Liebsch (1997)]. Dielectric dispersion in metal can be described by the Drude
model (see Eq. (2.53) in section 2.5), which can be written in the form
ωp2
(E.13)
ω(ω + jΓ)
Pluging the Eq. (E.12) into Eq. (E.13) results
ωp
ωs = √
(E.14)
2
The plasma frequency ωp is given by Eq. (2.54) (see section 2.5). The existence
of the electronic excitations on the interfaces was predicted by Ritchie (1957). If
the overlayer has the dielectric constant εd > 1 the condition Eq. (E.12) is given by
εm (ω) = 1 −
εm (ω) = −εd
(E.15)
Consequently, the surface plasmon frequency is redshifted according to
ωp
ωs = √
.
(E.16)
εd + 1
This redshift has been observed on different metal–dielectric interfaces (see e.g.
[Ritchie (1973); Pitarke et al. (2007); Raether (1988)] and references therein). It is
a key point of surface plasmon resonance (SPR)-based sensing spectroscopic tools
widely used for different fundamental and applied studies (see section 10.5)
Consider now the dispersion of surface plasmons. The solution of the system Eqs.
(E.1) to (E.4) will be separated into s- (E vector perpendicular) and p-polarized (E
vector parallel to the plane of incidence) EM modes. If there is a wave propagating
along the interface, it should contain the electric field E component perpendicular
to the interface (the p-polarized mode) and thus the s mode is not relevant. Consequently, the problem is now formulated as the search for the conditions of the
propagation of the p-polarized EM wave along the interface. Choosing the wave
propagation direction along the x-axis the solution should be taken in the form
[Pitarke et al. (2007)]
0
0
, 0, En,z
)ej(qn x−ωt) e−kn |z| ,
E n = (En,x
Hn =
0
(0, Hn,y
, 0)ej(qn x−ωt) e−kn |z| ,
(E.17)
(E.18)
where qn denotes a two-dimensional wave vector q of the wave propagating along
the interface.
Substituting Eqs. (E.17) and (E.18) into Eqs. (E.1) to (E.4) results in the
following set of equations:
ω
(E.19)
kd Hd,y = εd Ed,x ,
c
ω
km Hm,y = − εm Em,x ,
(E.20)
c
and
ω 2
.
(E.21)
kn = qn2 − εn
c
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Appendix E
The standard boundary conditions require that the components of both electric
and magnetic fields must be continuous [Jackson (1975)]. Consequently, Eqs. (E.19)
and (E.20) result in
kd
km
Hd,y +
Hm,y = 0,
εd
εm
(E.22)
Hd,y = Hm,y
(E.23)
and
The system of Eqs. (E.22) and (E.23) has a solution if the determinant is equal
to zero:
εm
εd
+
=0
(E.24)
kd
km
Equation (E.24) represents the surface plasmon condition [Pitarke et al. (2007)].
The boundary conditions also require continuity of the two-dimensional wave vector
q in Eq. (E.21), i.e., qd = qm = q. Based on this condition and combining Eq.
(E.24) and Eq. (E.21) one arrives at
ω2
ω2
2
2
2
2
(E.25)
εd q − ε m 2 = εm q − ε d 2 .
c
c
Equation (E.25) leads to another widely used form of the surface plasmon condition [Ritchie and Eldridge (1962); Ritchie (1973)]:
εd εm
ω
.
(E.26)
q(ω) =
c εd + εm
For a metal–dielectric interface with the dielectric constant εd , the solution ω(q)
√
of Eq. (E.26) has a slope equal to c/ εd at the point q = 0 and is a monotonic
√
increasing function of q, which is always smaller than cq/ εd and for a large q is
asymptotic to the value given by the solution of
εd + εm = 0.
(E.27)
This is the nonretarded surface plasmon condition that follows from Eq. (E.24)
at kd = km = q. This is valid as long as the phase velocity is much smaller than
the speed of light, i.e., ω/q c.
It is instructive now to analyze the dispersion of the SPP propagation on the
interface between metal and dielectric. The q0 = ω/c equation represents the magnitude of the light wave vector. Assume that for the dielectric εd = 1. In this case
Eq. (E.26) yields
ω 2 − ωp2
ω
q(ω) =
.
(E.28)
c 2ω 2 − ωp2
The dispersion relation described by Eq. (E.28) is represented in Fig. E.2
[Hofmann (2008)].
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Appendix E
Gavrilenko˙NanoOptics
299
Fig. E.2 Bold solid lines represent the dispersion of light in the retarded (upper line) and the
nonretarded surface plasmon polariton regions (lower line). By the thin line the dispersion of light
striking the interface at different angles is shown. The
√ thin horizontal lines indicate the values of
bulk ωp and surface plasmon frequencies ωs = ωp / 2. (Adapted from [Hofmann (2008)]).
The upper solid line in Fig. E.2 represents the dispersion of light in solid. The
lower solid line is the surface plasmon polariton dispersion curve, which is given by
ω 2 (q) = ωs2 + c2 q 2 − ωs4 + c4 q 4
(E.29)
√
where ωs = ωp / 2 represents the classical nondispersive surface plasmon frequency.
In the retarded region (q < ωs /c), the surface plasmon polariton dispersion curve
approaches the light line (ω = cq , see the thin line in Fig. E.2). At short wavelengths where q ωs /c the surface plasmon polariton
√ approaches asymptotically
the nonretarded surface plasmon frequency ωs = ωp / 2 (see the horizontal dashed
line in Fig. E.2).
Important conclusions can be made regarding the excitation of the surface plasmon polaritons corresponding to the lower branch in Fig. E.2. The wave vector of
the SPPs has the value of the two-dimensional vector within the interface plane,
q . Depending on the angle of incidence it varies from q = 0 (normal incidence) to
|q | = q (for grazing incidence, qz = 0). The light dispersion will change from the
vertical line to that given by ω = cq (see Fig. E.2). For any other angle the light
dispersion is given by ω = c q2 + qz2 . Consequently, the light dispersion line and
the surface plasmon polariton dispersion curve never cross, and hence there cannot
be any excitation of SPP on an ideal interface considered above.
There are two basic approaches to generate SPP. It can be generated on a grating. Additional periodic profile on the surface causes modifications of the wave
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Appendix E
vector selection rules (like additional Bragg reflection in superlattices). According to the superperiodicity the dispersion curve will get folded, crossing the light
dispersion line and thus allowing excitation of the SPPs. This has been observed
experimentally by Wood at the beginning of the last century [Wood (1902)], and he
described it as an “anomalous diffraction gratings” effect [Wood (1935)]. The same
effect can be achieved by a rough surface that can be viewed as a superposition
of many gratings with different periodicities [Ritchie (1973); Venger et al. (1999);
Raether (1988)]. The excitation of the SPPs via surface roughness is thought to
play a role in surface-enhanced Raman scattering (see Chapter 6).
The other way to achieve the coupling is to use an optical system where the value
of the photon wave vector will increase, thus reducing the slope of the curve. Optical
systems with a total light reflection inside a prism mounted in a short distance over
the surface are widely used. In this case, an evanescent electric field penetrates the
gap between prism and surface. The field decays exponentially because the wave
vector contains an imaginary q value in the z direction (see the dashed line in Fig.
E.2). The complex value of the light wave vector causes slope decrease of the light
dispersion curve in Fig. E.2 that results in the situation when the light dispersion
line and the surface plasmon polariton dispersion curve cross, thus allowing the
excitation of the SPPs. Examples of the prism systems generating an evanescent
light field are shown in Figs. 3.20 and 10.3a. This design is widely used in SPRbased optical spectroscopic tools for materials characterization, biosensing, etc. (see
section 10.5).
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Index
additional boundary conditions (ABC),
123
Autler–Townes splitting, 194
dielectric function, 52–55, 289
differential scattering cross section, 176
Dirac point, 38
discrete dipole approximation, 93, 175
DNA-based nanotechnology, 29
dressed state, 186
Drude model, 48, 62
band folding, 216
Bethe–Salpeter equation, 122, 148
bioconjugates, 263
biolabels, 19
biological nanomaterials, 29
biosensors, 272, 275
Bloch functions, 285, 286
Bloch integrals, 291, 292
block conjugated polymers, 255
Bosonic operators, 293
effective mass approximation (EMA), 34,
126
effective medium approximation, 67
electric field induced SHG (EFISH), 211,
215
electro-optical spectroscopy, 215
electromagnetic field enhancements, 68
electromagnetic wave equation, 44
electron charge density, 285
electron–phonon coupling, 151
electroreflectance, 215
entanglement, 182–185, 189
exciton, 119
biexcitons, 144
Bohr radius, 123
Frenkel, 119
singlet exciton, 122
triplet exciton, 122
Wannier–Mott, 119, 120
exciton Raman scattering, 155
C-dot, 19, 263
carbon fibers, 9
carbon nanotubes, 5, 7
charge conservation, 44
chemisorption, 3, 172
chromoionophore, 257
Clausius–Mossotti equation, 67
COIN (composite organic-inorganic
nanoparticles), 269
colloidal crystals, 26
conjugated polymers, 29, 243
constitutive relations, 43
continuity equation, 44
Coulomb interaction, 62
fabrication, 1
GaN nanowires, 108
Ag nanoparticles, 26
carbon nanoparticles, 7
CdSe nanocrystals, 18
CdSe-ZnS core/shell nanocrystals, 19
CVD technique, 2
Davydov splitting, 249
deformation potential interaction, 151
delta function, 288
density functional theory (DFT), 50, 279
density of states, 38, 280
density operator, 285, 286
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328
DAE-E DX tile nanotubes, 30
FePt nanoparticles, 27
GaMnN nanostructures, 21
GaN nanowires, 21
Stranski–Krastanow growth, 2
titania nanoparticles, 15
Fabry–Perot resonator, 185
Fermi energy, 280
fluorescence emitters, 262
fluorophores, 262
Fourier transform, 287
Fröhlich constant, 155
Fröhlich interaction, 151, 153
fullerenes, 5
graphene, 9, 11, 36
Green’s function, 170
Hückel model, 245
highest occupied molecular orbital
(HOMO), 35, 283
Holliday junction, 267
hollow nanoparticles, 23, 75
hot spot, 69, 168, 177
hyper-polarizability, 203
hyper-Rayleigh scattering, 203
interchain polymer distance, 253
invisibility cloak, 82
Jaynes–Cummings model (JCM), 183
jellium approximation, 90
Kirkendall diffusion effect, 23
Laplace equation, 63
LC nanoelement circuit, 223
left-handed materials, 78
lithography, 13
local density approximation (LDA), 282
local field, 53
local field effect, 46, 54, 291
localized atomic orbitals (LCAO), 35
Lorentz force, 222
Lorentz-force field, 224
lowest unoccupied molecular orbital
(LUMO), 283
matrix representation, 286
Maxwell’s equations, 43
Index
Maxwell–Garnett approximation, 67
metallic carbides, 5
metamaterials, 77, 221
Mie resonance, 224
Mie theory, 62, 65, 91
MOCVD, 22
molecular nanocrystals, 234
Mollow triplet, 192
Moore’s law, 261
nanocomposites, 238, 240
nanoporous carbon, 6
near-field optics, 84
negative-index materials, 81
nonlinear optics, 202
normal modes, 150
oligomers, 235
optical field Hamiltonian, 293
optical functions
dielectric constant, 43
dielectric permittivity, 43, 81
displacement, 43
extinction coefficient, 44
index of refraction, 44
magnetic permeability, 43
permeability, 81
polarizability, 46
refraction coefficient, 44
susceptibility, 43, 48
optical labeling, 262
optical loss, 47
optical rectification, 45
organic nanocrystals, 234
organic nanofibers, 235
oscillator strength, 42
para-quaterphenylene, 235
perturbation theory, 285
phase velocity, 45
phonon bottleneck, 147
phonon confinement, 161, 162
phonons, 150
photonic crystal, 190
physisorption, 3
plane wave representation, 291
plasma excitations, 59
plasma frequency, 48
plasmon resonance, 62
electrostatic theory, 63
Gavrilenko˙NanoOptics
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Index
plasmonic density of states, 73
plasmonics, 59
PMMA, poly(methyl methacrylate), 4
Poisson equation, 289
polarization function, 45, 46, 52, 288,
291
polymer–metallic nanomaterials, 29
polymers, 4, 239
potential well, 34
Purcell effect, 188
Purcell factor, 188
quantization of the field, 293
quantum confinement, 35, 40
quantum dots, 41
quantum electrodynamics (QED), 181
quantum well, 33
quasiparticle, 34, 283
quasistatic approximation, 63
Rabi oscillations, 184
Raman polarization function, 150
Raman spectroscopy, 149, 160
surface effect, 164
Raman tensor, 150, 173
refractive index, 65, 80, 81
Rydberg atoms, 184
Schrödinger equation, 41, 120, 285
second harmonic generation, 205
second quantization, 293
Gavrilenko˙NanoOptics
329
sensors
biosensors, 263
polymer-based sensors, 257
silanization, 263
silicon carbide, 216
single-wall carbon nanotubes (SWNTs),
124
spherical harmonics, 40
split-ring resonator, 223
strong coupling, 183, 188
surfac-enhanced infrared absorption
(SEIRA), 271
surface-enhanced Raman scattering
(SERS), 149, 165, 219
surface plasmon polariton (SPP), 295
surface plasmon resonance (SPR), 61, 273,
297, 300
surface plasmons, 59
Taylor expansion, 285
Taylor series, 46
Thomas–Fermi approximation, 279
trioctylphosphine oxide, 258
two-level atomic system, 183
velocity operator, 292
vibronic states, 160
Wannier–Mott excitons, 120
weak coupling, 188
Wigner symbols, 128