Thomas–Fermi Approximation and Basics of the Density Functional

December 22, 2010 13:5 World Scientific Book - 9.75in x 6.5in Gavrilenko˙NanoOptics

Appendix A

Thomas–Fermi Approximation and Basicsof the Density Functional Theory

As stated at the beginning of section 2.7 the total energy is a key function describ-ing the basic physical and chemical properties of materials: the ground state. Itconsists of both kinetic (describing motion) and potential energy parts. To makea theoretical model realistic it is very important to incorporate all important con-tributions to both the parts of the total energy. In view of the big number of theparticles involved into the model, this is very challenging for the first-principlestheory. Different approximations are applied in order to achieve a trade-off betweencomplexity and accuracy. Very successful in realistic modeling of the ground stateis the density functional theory, DFT. In this chapter we present the basic ideasof the DFT and demonstrate both advantages and problems for optics within thismethod.

Initially Thomas and Fermi (TF) in the 1920s suggested describing atoms asuniformly distributed electrons (negatively charged clouds) around nuclei in a six-dimensional phase space (momentum and coordinates). This is enormous simplifi-cation of the actual many-body problem. It is instructive to consider the basic ideasof 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 providedhere.

Following the TF approach the total energy of the system could be presentedas a function (functional) of electron density [McQuarrie (1976); Parr and Yang(1989)]. Each h3 of the momentum space volume (h is the Planck constant) isoccupied by two electrons and the electrons are moving in an effective potentialfield that is determined by nuclear charge and by assumed uniform distribution ofelectrons. The density of ΔN electrons in real space within a cube (nanoparticle)with a side l is given by

ρ(r) =ΔN

v=

ΔN

l3. (A.1)

The electron energy levels in this three-dimensional infinite well are given by:

E =h2

8ml2(n2

x + n2y + n2

z) =h2

8ml2R2, nx, ny, nz = 1, 2, 3, . . . (A.2)

279


280 Appendix A

Radius R = Rmax of the sphere in the space (nx, ny, nz) covering all occupiedstates determines the maximum energy of electrons: the Fermi energy F . Thenumber of energy levels within this maximum value at zero temperature is given by

NF =123

4πR3

3=

π

6

(8ml2F

h2

)3/2

. (A.3)

The density of states is defined as

g(E)dE = NF (E + dE) − NF (E) =π

4

(8ml2

h2

)3/2

E1/2dE. (A.4)

At zero temperature all energy levels below the Fermi energy are occupied:

f(E) ={

1 E ≤ F

0 E > F

}. (A.5)

Consequently the total energy of the electrons in one cell will be given by

E =∫ F

0

Ef(E)g(E)dE =4π

h3(2m)3/2l3

∫ F

0

E3/2dE

=π

5

(2l

h

)3

(2m)3/2F 5/2. (A.6)

The Fermi energy F can be obtained from the total number of electrons ΔN in acell:

ΔN = 2∫ F

0

f(E)g(E)dE =π

h3

(2l

h

)3

(2m)3/2F 3/2. (A.7)

Combining Eqs. (A.6) and (A.7) the energy of the electrons in one cell is given by

E =310

(3π2)2/3 l3

(2π)2ρ5/3 = CF

l3

(2π)5/3,

CF =310

(3π2

)2/3= 2.871. (A.8)

In Eq. (A.8) we reverted to atomic units e = h = m0 = 1. The electron density isa smooth function in a real space. For systems without translational symmetry itis different for different cells. However, for spatially periodic systems only consider-ation within the unit cell is required since all unit cells are equivalent. Now addingthe contributions from all cells with energies within F , we obtain

TTF[ρ] = CF

∫ρ5/3(r)d3r. (A.9)

Equation Eq. (A.9) represents the well-known Thomas–Fermi kinetic energy func-tional, 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 themodern DFT, the local density approximation (LDA) [Martin (2004)]. Adding to


Appendix A 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 theoryof atoms:

ETF[ρ(r)] = CF

∫ρ5/3(r)d3r − Z

∫ρ(r)r

d3r

+12

∫ρ(r1)ρ(r2)|r1 − r2|

d3r1d3r2. (A.10)

Note that the nucleus charge Z is measured in atomic units. The energy of theground state and electron density can be found by minimizing the functional (A.10)with the constrain condition

N =∫

ρ(r)d3r. (A.11)

The electron density in Eq. (A.10) has to be calculated in conjunction withEq. (A.11) from the following equation for chemical potential, defined as the varia-tional derivative according to

μTF =δETF[ρ]δρ(r)

=53CTFρ5/3(r) − Z

r+

∫ρ(r2)

|r1 − r2|d3r2. (A.12)

The Thomas–Fermi model provides reasonably good predictions for atoms. It hasbeen used before to study potential fields and charge density in metals and theequation of states of elements [Feynman et al. (1949)]. However, this method isconsidered rather crude for more complex systems because it does not incorpo-rate the actual orbital structure of electrons. In view of the modern DFT theorythe Thomas–Fermi method could be considered as an approximation to the moreaccurate theory.

For systems like molecules and solids, much better predictions are provided bythe DFT. Search for the ground state within the DFT follows the rule that theelectron density is a basic variable in the electronic problem (the first theorem ofHohenberg and Kohn [Hohenberg and Kohn (1964)]) and another rule that theground state can be found from the energy variational principle for the density (thesecond theorem of Hohenberg and Kohn [Kohn (1999)]). According to the DFT thetotal 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 isthe electrostatic energy due to Coulomb interactions. The most important partin the DFT is EXC, the exchange and correlation (XC) energy that includes allmany-body contributions to the total energy. The charge density is determined bythe wave functions, which for practical computations could be constructed fromsingle orbitals, φj (e.g. antisymmetrized product—the Slater determinant, atomicor Gaussian orbitals, linear combinations of plane waves). The charge density isgiven by

ρ(r) =∑

j

|φj(r)|2 , (A.14)


282 Appendix A

where the sum is taken over all occupied j orbitals. In the spin-resolved case therewill be orbitals occupied with spin-up and spin-down electrons. Their sum givesthe total charge density and their difference gives the spin density. In terms of theelectron orbitals the energy components are given in atomic units as

T = −12

∫ ∑j

φ∗j (r)|∇2|φj(r)d3r, (A.15)

U = −n∑j

N∑α

∫φ∗

j (r)∣∣∣∣ Zα

(Rα − r)

∣∣∣∣ φj(r)d3r

+12

∑i,j

∫φ∗

i (r1)φ∗j (r2)

1(r1 − r2)

φi(r1)φj(r2)d3r1d3r2

+N∑α

N∑β<α

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 rep-resents nucleus–nucleus repulsion. In Eq. (A.16) Zα refers to the charge on nucleusα 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 localdensity approximation, LDA, which assumes that the charge density varies slowlyon the atomic scale, i.e., the effect of other electrons on the given (local) electrondensity is described as a uniform electron gas. The XC energy can be obtained byintegrating with the uniform gas model (see e.g. [Ceperley and Adler (1980)]):

EXC∼=

∫ρ(r)EXC[ρ(r)]d3(r), (A.17)

where EXC[ρ(r)] is XC energy per particle in a uniform electron gas. For manysystems a good approximation provides analytic expression for EXC[ρ(r)] suggestedby Perdew and Wang (1992). In practical calculations through minimization of thetotal energy Eq. (A.13) one determines self-consistently the electron density andthe actual XC part. A variational minimization procedure leads to a set of coupledequations proposed by Kohn and Sham [Kohn and Sham (1965)]:[

−12∇2 − VN + Ve + μXC(ρ)

]φj = Ejφj , (A.18)

with

μXC =∂

∂ρ(ρEXC) , (A.19)

The solution of the Kohn–Sham equation provides the equilibrium geometryand the ground-state energy of the system. However, eigen functions and eigenenergies of the Kohn–Sham equation cannot be interpreted as the quasiparticle


Appendix A 283

quantities needed for optics. The term quasiparticle refers to a particle-like entityarising in certain systems of interacting particles. If a single particle moves throughthe system, surrounded by a cloud of other interactiong particles, the entire entitymoves along somewhat like a free particle (but slightly different). The quasiparticleconcept is one of the most important in materials science, because it is one of the fewknown ways of simplifying the quantum mechanical many-body problem describingexcitation 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 resultautomatically in correct prediction of excitation energies required for optics. Forexample, in nonmetallic systems the predicted value of the energy difference (energygap) between the highest occupied molecular orbital (HOMO) and the lowest un-occupied molecular orbital (LUMO) in most cases is underestimated (gap problem).Special corrections (quasiparticle, QP corrections) are required to get more accu-rate excitation energies [Onida et al. (2002)]. Without corrections in semiconductorsand insulators the local density approximation, LDA, substantially underestimatesforbidden gap values. In this chapter we present LDA results for optics with dif-ferent QP corrections avoiding, however, detailed analysis of theoretical methods.For advanced reading of the DFT one can recommend original papers [Hohenbergand 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 withinthe Perturbation Theory

In this section we describe evaluation of the light-filled induced charge within per-turbation theory using plane wave representation, which is used in this chapter forcalculations of optical functions (see section 2.7). Equilibrium electron charge den-sity is defined through the density operator (using definition of Trace, Tr, as a sumof the diagonal elements):

neq(r) = eTr[ρ0, δ(r − r0)]. (B.1)

Without illumination if the system is periodic (at least in one dimension) the den-sity operator could be defined in energy representation on a set of Bloch functionsaccording to [Davydov (1980)]

ρ|s〉 = ρ0|k, l〉 = f(Ek,l)|k, l〉, (B.2)

where the equilibrium Fermi distribution function is given by

f(Es) =(e

F−EskT − 1

)−1

, (B.3)

The Bloch functions

|s〉 = |k, l〉 = uk,l(k)eikr, (B.4)

are solutions of an undisturbed Schrodinger equation with periodic potential:

H0|k, l〉 =[−1

2∇2 + V0(r)

]|k, l〉 = Ek,l|k, l〉. (B.5)

In an external optical field when light quanta strike electrically neutral atoms theequilibrium is broken through the deformation of electron clouds. Time-dependentchanges of the electron charge density can be represented as Taylor expansion. Thenumber of the terms to be included into the Taylor sum for the induced part of thecharge depends on the excitation intensity:

n(r, t) = neq(r) + nind(r, t) = eTr[ρ, δ]

= eTr[ρ0, δ] + eTr[ρ(1), δ] + eTr[ρ(2), δ] + . . . (B.6)

The first- and higher-order corrections to the density operator are determined fromthe standard perturbation theory:

i�dρ

dt= [H, ρ] = (Hρ − ρH), (B.7)

285


286 Appendix B

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 couldbe included through the relaxation time. Plugging (B.8) in (B.7) and choosing termsof the same order on both the left and right side of the equation of motion for thedensity operator, Eq. (B.7) splits into a series of equations for zero, first, second,etc., orders of perturbations, respectively:

i�ρ0 = [H0, ρ0],

i�ρ(1) =[H0, ρ

(1)]

+[V (1), ρ0

](B.9)

i�ρ(2) =[H0, ρ

(2)]

+[V (1), ρ(1)

]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,

i�ρ(ω) = −�ωρ(ω). (B.10)

It is convenient now to switch to the matrix representation in Eq. (B.10) by pro-jecting the relevant quantities on a set of Bloch functions in Eq. (2.68). To thisend 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 ofthe relevant equation. Through integration over the entire space and by taking intoaccount orthonormality conditions for Bloch functions, this leads to the followingexpression for the first-order terms:

−�ωρ(1)ss′ = (Es − Es′)ρ(1)

ss′ +∑

t

V(1)st ρ0

ts′ −∑

p

ρ0spV

(1)ps′ , (B.11)

The density operator defined as Eq. (B.2) in matrix presentation has the form

ρ(0)ss′ = f(Es)δss′ . (B.12)

Equation (B.11) is now transformed into

−�ωρ(1)ss′ = (Es − Es′)ρ(1)

ss′ + [f(Es′) − f(Es)] Vss′ . (B.13)

At zero temperature, optical excitations occur between completely filled and emptystates with Fermi functions equal to either 1 or zero, respectively. Consequently,

ρ(1)ss′(ω) =

f(Es′) − f(Es)Es − Es′ − �ω

Vss′ = (Es − Es′ − �ω)−1Vss′ |T=0. (B.14)

For the second-order perturbation one needs to use the first-order solutionEq. (B.14). Plugging it into Eq. (B.10) after some algebra leads to the follow-ing expressions at T = 0:

−�ωρ(2)ss′ = (Es − Es′)ρ(2)

ss′ +∑s′′

V(1)ss′′ρ

(1)s′′s′ −

∑s′′

ρ(1)ss′′V

(1)s′′s′ , (B.15)


Appendix B 287

or

ρ(2)ss′(ω) =

1Es − Es′ − �ω

∑s′′

[V (1)ss′′V

(1)s′′s′(

1Es′′ − Es′ − �ω

− 1Es − Es′′ − �ω

)]. (B.16)

Equations (B.14) and (B.16) can be used now to obtain induced charge density fromEq. (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 theperturbation potential is given by

V (r, t) =∑qG

∫ ∞

−∞V (q + G, ω)ei(q+G)rdω, (B.17)

where G is the reciprocal lattice vector. For Fourier transform of the potential

V (r, t) =∫ ∞

−∞V (r, ω)eiωt, (B.18)

the expansion of the potential is given by

V (r, ω) =∑qG

V (q + G, ω)ei(q+G)r. (B.19)

In a periodic system with all equivalent atoms separated by Ri one has for thecharge:

neq(r0 + Ri) = neq(r0), (B.20)

with

n(r) =∑

q

eiqrn(q) =∑

q

eiqr∑G

δqGn(G). (B.21)

For induced charge density in (B.6) we have

nind(r, ω) =∑qG

nind(q + G, ω)ei(q+G)r. (B.22)

Where Fourier transform of the induced charge is given by Fourier integral by

nind(q + G, ω) =∫

nind(r, ω)e−i(q+G)rd3r, (B.23)

Linear part of the induced charge in (B.23) follows from (B.6)

nind(q + G, ω) = e

∫Tr

[ρ(1)(ω), δ(r′ − r)

]e−i(q+G)r′

d3r′

= eTr[ρ(1)(ω), e−i(q+G)r

], (B.24)

The trace of the operator product is calculated according to

Tr(AB

)=

∑m

〈m|AB|m〉 =∑m

∑n

AmnBnm. (B.25)


288 Appendix B

Eq. (B.24) projected on the plane wave basis 2.68 can be represented as

nind = nind(q + G, ω)

= e∑k+q

∑k,l

〈l′,k + q|ρ(1)ω |k, l〉〈l,k|δ(r′ − r)|k + q, l′〉e−i(q+G)r′

= e∑k+q

∑k,l

〈l′,k + q|ρ(1)ω |k, l〉〈l,k|

∑G′

ei(r−r′)G′e−i(q+G)r′ |k + q, l′〉

= e∑k+q

∑k,l

〈l′,k + q|ρ(1)ω |k, l〉〈l,k|e−i(q+G)r|k + q, l′〉. (B.26)

In (B.26) the bracket notation means space integration. We also used the definitionof the δ function:

δ (r − r′) =∑G

ei(r−r′)G. (B.27)

Now Eq. (B.14) can be written as

〈l′,k + q|ρ(1)ω |k, l〉 =

f [El′(k + q)] − f [El(k)]El′(k + q) − El(k) − ω − iη

〈l′,k + q|V (r, ω)|k, l〉

=f [El′(k + q)] − f [El(k)]

El′(k + q) − El(k) − ω − iη

×∑q,G

V (q + G, ω)〈l′,k + q|ei(q+G)r|k, l〉. (B.28)

The complex part of the energy in the denominator of Eq. (B.28) is introducedto 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∑G′

PG,G′V (q + G, ω). (B.29)

Using the notation (k′ = k + q) the polarization function is defined as

PG,G′(ω) =∑k′,k

∑l′,l

〈l′,k′|ei(q+G)r|k, l〉〈l,k|ei(q+G′)r|k′, 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 potentialin 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 potentialthrough the induced charge in materials. This can be presented in terms of dielectricfunction:

V (q + G, ω) =∑G′

ε−1G,G′Vext(q + G′, ω), (B.32)


Appendix B 289

or

Vext(q + G′, ω) =∑G

εG,G′V (q + G, ω), (B.33)

Equations (B.32) and (B.33) can be considered as the microscopic definition of thedielectric function. The described computation of ε presents a transformation fromthe microscopic (atom-related) quantities to the macroscopic values used in classicalelectrodynamics theory. For advanced reading related to the definition of the opticalfunction within the first-principles theory, one can recommend [Martin (2004); Yuand Cardona (2010)] or original papers (see e.g. [Onida et al. (2002); Gavrilenkoand Bechstedt (1997)] and references therein).The induced potential satisfies the Poisson equation:

Vind(q + G, ω) =4π

|q + G|2 nind(q + G, ω). (B.34)

From Eqs. (B.29), (B.31), (B.33), and (B.34) we obtain

V (q + G, ω) =∑G′

[εG,G′(ω) +

4π

|q + G|2 PG,G′

]V (q + G, ω)

=∑G′

δG,G′V (q + G, ω). (B.35)

The dielectric function can now be expressed in terms of the polarization function:

εG,G′(ω) = δG,G′ − 4π

|q + G|2 PG,G′ . (B.36)

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Appendix C

Local Field Effect in Optics of Solids fromthe First Principles

Local field (LF) effect plays a key role in the optics of nanostructures. Descriptionof the LF effect within classical electrodynamics is presented in many monographsand 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 withreduction of the dimensions of nanostructures, thus requiring microscopic modeling.This section presents the evaluation of the optical dielectric function, including theLF effect within the perturbation theory (see section B).

The formula for the polarization function (see Eq. (B.30)) can be presented as

PG,G′(ω) =2Ω

∑k′,k

∑n′,n

Bk′,kn′,n(q + G)B∗k,k′

n,n′ (q + G′)

× f [El′(k + q)] − f [El(k)]El′(k + q) − El(k) − (ω + iη)

. (C.1)

The Bloch integrals in Eq. (C.1) are defined as

Bk′,kn′,n(q + G) = 〈n′,k′|ei(q+G)r|k, n〉

=1Ω

∫ψ∗

n′,k′(r)ei(q+G)rψn,kd3r. (C.2)

In plane wave representation (2.68) neglecting the umklapp processes (the noncon-serving crystal momentum electron–electron scattering) [Bechstedt (2003)] and inthe limit of q → 0 the Bloch integrals have an extremely simple form, given by

Bk′,kc,v (G) =

∑G1

d∗c,k′(G1)dv,k(G1 − G). (C.3)

Indexes c and v in Eq. (C.3) denote empty antibonding (conducting) and filledbonding (valence) electron states, respectively, at zero temperature. By derivationof Eq. (C.3) we used the following properties of direct and reciprocal lattice vectors:

GiRj = 2πδij

eiGjRj = 1 (C.4)∑Ri

ei(k′−k+q)Ri =∑Gi

δk′−k+q,Gi

291


292 Appendix C

Equation (C.1) at zero temperature takes the following form:

PG,G′(ω) =2Ω

∑k′,k

∑n′,n

Bk′,kn′,n(q + G)B∗k,k′

n,n′ (q + G′)El′(k + q) − El(k) − (ω + iη)

. (C.5)

From the orthonormality of the wave functions follow the properties of Bloch inte-grals:

Bk′,kc,v (0) =

∑G1

d∗c,k′(G1)dv,k(G1) = 0,

N∑n′=1

∑k′

∣∣∣Bk′,kn′,n(q + G)

∣∣∣2 = 1. (C.6)

For G = 0 and in the limit q → 0 the Bloch integrals have the following properties:

limq→0

Bk′,kc,v (q) = δk,k′ lim

q→0i

3∑α=1

qα〈c,k|rα|k, v〉

=δk,k′

Ec(k) − Ev(k)limq→0

3∑α=1

qα〈c,k|vα|k, v〉, (C.7)

limq→0

Bk′,kc,v (qα)|qα|

=1

Ec(k) − Ev(k)〈c,k|vα|k, v〉.

Here we used the general definition of velocity (or momentum) [Adolph et al. (1996)]given by

v = limq→0

1q

[H, eiqr

]. (C.8)

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′〉 =∑m,l

〈nk|H|m, l〉〈m, l|rα|n′k′〉

−∑m′,l′

〈nk|rα|m′, l′〉〈m′, l′|H|n′k′〉 (C.10)

= (Enk − En′k′)〈nk|rα|n′k′〉.

Equation (C.11) represents the relationship between matrix elements of the velocityand of the induced dipole momentum, which could be used to obtain the relationshipbetween optical functions calculated in velocity and length gauges [Gavrilenko andBechstedt (1997)].


Appendix D

Optical Field Hamiltonian in SecondQuantization Representation

If there are resonances of electromagnetic field within a cavity the entire field canbe presented as a superposition of single modes in the following form [Jaynes andCummins (1963)]:

E(r, t) = −√

4π∑

j

pj(t)Ej(r), (D.1)

H(r, t) =√

4π∑

j

ωjqj(t)Hj(r). (D.2)

The total energy of the field is given by

H =18π

∫(|E|2 + |H|2)d3r =

12

∑j

(p2j + ω2

j q2j ) (D.3)

The Hamiltonian equation of motion is given by

qj =∂H

∂pj= pj ,

pj = −∂H

∂qj= −ω2

j qj . (D.4)

Mathematically, quantization of the field is represented by commutations rules forthe canonically conjugated coordinates and momenta:

[qi, qj ] = 0, [pi, pj ] = 0, [qi, pj ] = i�δij . (D.5)

The Hamiltonian of an optical field is convenient to present in terms of secondquantization operators, the Bosonic operators of creation (a†

j) and annihilation (aj)of photons using the definitions [Davydov (1976)]

pj =

√�ωj

2

(aj + a†

j

), (D.6)

ωjqj = i

√�ωj

2

(aj − a†

j

), (D.7)

with commutation rule

[ai, a†j ] = δij . (D.8)

293


294 Appendix D

The properties of ai operators are described through their action on state vectorφ(n1, n2, . . . ni . . .) according to

aiφ(n1, . . . ni−1, ni, ni+1 . . .) =√

niφ(n1, . . . ni−1, ni−1, ni+1 . . .),

a†iφ(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

H =∑

i

�ωi

(aia

†i +

12

). (D.9)


Appendix E

Surface Plasmons and Surface PlasmonPolaritons

Collective electronic excitations (also known as plasma excitations) at metal sur-faces and/or metal/dielectric interfaces play a key role in the optics of nanomate-rials, ranging from physics and materials science to biology. A unified theoreticaldescription of these phenomena is based on the many-body dynamical electronicresponse of solids (see section B of the Appendix), which underlines the existence ofvarious collective electronic excitations at metal surfaces, such as the conventionalsurface plasmon, multipole plasmons, and the acoustic surface plasmon. A detaileddescription of the surface plasmon polariton (SPP) phenomena and its applicationsin modern optical spectroscopy is out of the scope of the present book. Several spe-cialized monographs and reviews can be recommended for advanced reading [Tudosand Schasfoort (2008); Pitarke et al. (2007); Liebsch (1997); Ritchie (1973); Vengeret al. (1999); Raether (1988)]. Here we present the basic conditions and propertiesof SPP that follow from classical electrodynamics.

Consider a model consisting of two semi-infinite nonmagnetic media with lo-cal (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 ofMaxwell’s equations in the absence of external sources can be expressed as follows[Jackson (1975)]:

∇ × Hn = ε1c

∂

∂tEn, (E.1)

∇ × En = −1c

∂

∂tHn, (E.2)

∇ · (εnEn) = 0, (E.3)

∇ · Hn = 0, (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 surfacecharge density (see Eq. (B.1)) on the metal–dielectric interface can be presented asLiebsch (1997); Raether (1988)

n(r, ω) ≈ ejq‖r‖δ(z). (E.5)

295


296 Appendix E

The electric field associated with the density Eq. (E.5) is given by the Gausslaw, 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 areconsidered 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 followsfrom the electroneutrality since ∇2φ = 0 must be valid everywhere except at z = 0[Liebsch (1997)]. Consequently the potential in Eq. (E.8) must be taken in theform

φ(r, ω) = φ0ejq‖r‖e−q|z|, (E.9)

where q ≡ qz. In this case the field determined by Eqs. (E.7) and (E.9) variescontinuously within the interface, however, the normal component is discontinuous.The components above and below the interface are given by

Ez(z + 0) = q‖φ0ejq‖r‖ ,

Ez(z − 0) = −q‖φ0ejq‖r‖ . (E.10)

The field and charge distribution for such a surface mode is represented inFig. E.1.

In the long-wavelength limit the boundary condition can be written as

εm(ω)Ez(0−) = εdEz(0+). (E.11)

For the metal–vacuum interface (i.e., by εd = 1) the condition for the existenceof 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)]


Appendix E 297

Equation (E.12) defines the frequency of the surface plasmon in the q‖ = 0limit [Liebsch (1997)]. Dielectric dispersion in metal can be described by the Drudemodel (see Eq. (2.53) in section 2.5), which can be written in the form

εm(ω) = 1 −ω2

p

ω(ω + jΓ)(E.13)

Pluging the Eq. (E.12) into Eq. (E.13) results

ωs =ωp√

2(E.14)

The plasma frequency ωp is given by Eq. (2.54) (see section 2.5). The existenceof the electronic excitations on the interfaces was predicted by Ritchie (1957). Ifthe overlayer has the dielectric constant εd > 1 the condition Eq. (E.12) is given by

εm(ω) = −εd (E.15)

Consequently, the surface plasmon frequency is redshifted according to

ωs =ωp√

εd + 1. (E.16)

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 isa key point of surface plasmon resonance (SPR)-based sensing spectroscopic toolswidely 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 (Evector parallel to the plane of incidence) EM modes. If there is a wave propagatingalong the interface, it should contain the electric field E component perpendicularto the interface (the p-polarized mode) and thus the s mode is not relevant. Con-sequently, the problem is now formulated as the search for the conditions of thepropagation of the p-polarized EM wave along the interface. Choosing the wavepropagation direction along the x-axis the solution should be taken in the form[Pitarke et al. (2007)]

En = (E0n,x, 0, E0

n,z)ej(qnx−ωt)e−kn|z|, (E.17)

Hn = (0,H0n,y, 0)ej(qnx−ωt)e−kn|z|, (E.18)

where qn denotes a two-dimensional wave vector q‖ of the wave propagating alongthe interface.

Substituting Eqs. (E.17) and (E.18) into Eqs. (E.1) to (E.4) results in thefollowing set of equations:

kdHd,y =ω

cεdEd,x, (E.19)

kmHm,y = −ω

cεmEm,x, (E.20)

and

kn =

√q2n − εn

(ω

c

)2

. (E.21)


298 Appendix E

The standard boundary conditions require that the components of both electricand magnetic fields must be continuous [Jackson (1975)]. Consequently, Eqs. (E.19)and (E.20) result in

kd

εdHd,y +

km

εmHm,y = 0, (E.22)

and

Hd,y = Hm,y (E.23)

The system of Eqs. (E.22) and (E.23) has a solution if the determinant is equalto zero:

εd

kd+

εm

km= 0 (E.24)

Equation (E.24) represents the surface plasmon condition [Pitarke et al. (2007)].The boundary conditions also require continuity of the two-dimensional wave vectorq‖ 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

ε2d

(q2 − εm

ω2

c2

)= ε2

m

(q2 − εd

ω2

c2

). (E.25)

Equation (E.25) leads to another widely used form of the surface plasmon con-dition [Ritchie and Eldridge (1962); Ritchie (1973)]:

q(ω) =ω

c

√εdεm

εd + εm. (E.26)

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 isasymptotic 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 thanthe speed of light, i.e., ω/q � c.

It is instructive now to analyze the dispersion of the SPP propagation on theinterface between metal and dielectric. The q0 = ω/c equation represents the mag-nitude of the light wave vector. Assume that for the dielectric εd = 1. In this caseEq. (E.26) yields

q(ω) =ω

c

√ω2 − ω2

p

2ω2 − ω2p

. (E.28)

The dispersion relation described by Eq. (E.28) is represented in Fig. E.2[Hofmann (2008)].


Appendix E 299

Fig. E.2 Bold solid lines represent the dispersion of light in the retarded (upper line) and thenonretarded surface plasmon polariton regions (lower line). By the thin line the dispersion of lightstriking the interface at different angles is shown. The thin horizontal lines indicate the values ofbulk 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. Thelower solid line is the surface plasmon polariton dispersion curve, which is given by

ω2(q) = ω2s + c2q2 −

√ω4

s + c4q4 (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 wave-lengths where q‖ ωs/c the surface plasmon polariton approaches asymptoticallythe 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 plas-

mon polaritons corresponding to the lower branch in Fig. E.2. The wave vector ofthe 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 thevertical line to that given by ω = cq‖ (see Fig. E.2). For any other angle the light

dispersion is given by ω = c√

q2‖ + q2

z . Consequently, the light dispersion line andthe surface plasmon polariton dispersion curve never cross, and hence there cannotbe any excitation of SPP on an ideal interface considered above.

There are two basic approaches to generate SPP. It can be generated on a grat-ing. Additional periodic profile on the surface causes modifications of the wave


300 Appendix E

vector selection rules (like additional Bragg reflection in superlattices). Accord-ing to the superperiodicity the dispersion curve will get folded, crossing the lightdispersion line and thus allowing excitation of the SPPs. This has been observedexperimentally by Wood at the beginning of the last century [Wood (1902)], and hedescribed it as an “anomalous diffraction gratings” effect [Wood (1935)]. The sameeffect can be achieved by a rough surface that can be viewed as a superpositionof many gratings with different periodicities [Ritchie (1973); Venger et al. (1999);Raether (1988)]. The excitation of the SPPs via surface roughness is thought toplay 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 valueof the photon wave vector will increase, thus reducing the slope of the curve. Opticalsystems with a total light reflection inside a prism mounted in a short distance overthe surface are widely used. In this case, an evanescent electric field penetrates thegap between prism and surface. The field decays exponentially because the wavevector 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 lightdispersion curve in Fig. E.2 that results in the situation when the light dispersionline and the surface plasmon polariton dispersion curve cross, thus allowing theexcitation of the SPPs. Examples of the prism systems generating an evanescentlight field are shown in Figs. 3.20 and 10.3a. This design is widely used in SPR-based optical spectroscopic tools for materials characterization, biosensing, etc. (seesection 10.5).


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Index

additional boundary conditions (ABC),123

Autler–Townes splitting, 194

band folding, 216Bethe–Salpeter equation, 122, 148bioconjugates, 263biolabels, 19biological nanomaterials, 29biosensors, 272, 275Bloch functions, 285, 286Bloch integrals, 291, 292

block conjugated polymers, 255Bosonic operators, 293

C-dot, 19, 263carbon fibers, 9carbon nanotubes, 5, 7charge conservation, 44chemisorption, 3, 172chromoionophore, 257Clausius–Mossotti equation, 67COIN (composite organic-inorganic

nanoparticles), 269colloidal crystals, 26conjugated polymers, 29, 243constitutive relations, 43continuity equation, 44Coulomb interaction, 62

Davydov splitting, 249deformation potential interaction, 151delta function, 288density functional theory (DFT), 50, 279density of states, 38, 280density operator, 285, 286

dielectric function, 52–55, 289differential scattering cross section, 176Dirac point, 38discrete dipole approximation, 93, 175DNA-based nanotechnology, 29dressed state, 186Drude model, 48, 62

effective mass approximation (EMA), 34,126

effective medium approximation, 67electric field induced SHG (EFISH), 211,

215electro-optical spectroscopy, 215electromagnetic field enhancements, 68electromagnetic wave equation, 44electron charge density, 285electron–phonon coupling, 151electroreflectance, 215entanglement, 182–185, 189exciton, 119

biexcitons, 144Bohr radius, 123Frenkel, 119singlet exciton, 122triplet exciton, 122Wannier–Mott, 119, 120

exciton Raman scattering, 155

fabrication, 1GaN nanowires, 108Ag nanoparticles, 26carbon nanoparticles, 7CdSe nanocrystals, 18CdSe-ZnS core/shell nanocrystals, 19CVD technique, 2

327


328 Index

DAE-E DX tile nanotubes, 30FePt nanoparticles, 27GaMnN nanostructures, 21GaN nanowires, 21Stranski–Krastanow growth, 2titania nanoparticles, 15

Fabry–Perot resonator, 185Fermi energy, 280fluorescence emitters, 262fluorophores, 262Fourier transform, 287Frohlich constant, 155Frohlich interaction, 151, 153fullerenes, 5

graphene, 9, 11, 36Green’s function, 170

Huckel model, 245highest occupied molecular orbital

(HOMO), 35, 283Holliday junction, 267hollow nanoparticles, 23, 75hot spot, 69, 168, 177hyper-polarizability, 203hyper-Rayleigh scattering, 203

interchain polymer distance, 253invisibility cloak, 82

Jaynes–Cummings model (JCM), 183jellium approximation, 90

Kirkendall diffusion effect, 23

Laplace equation, 63LC nanoelement circuit, 223left-handed materials, 78lithography, 13local density approximation (LDA), 282local field, 53local field effect, 46, 54, 291localized atomic orbitals (LCAO), 35Lorentz force, 222Lorentz-force field, 224lowest unoccupied molecular orbital

(LUMO), 283

matrix representation, 286Maxwell’s equations, 43

Maxwell–Garnett approximation, 67metallic carbides, 5metamaterials, 77, 221Mie resonance, 224Mie theory, 62, 65, 91MOCVD, 22molecular nanocrystals, 234Mollow triplet, 192Moore’s law, 261

nanocomposites, 238, 240nanoporous carbon, 6near-field optics, 84negative-index materials, 81nonlinear optics, 202normal modes, 150

oligomers, 235optical field Hamiltonian, 293optical functions

dielectric constant, 43dielectric permittivity, 43, 81displacement, 43extinction coefficient, 44index of refraction, 44magnetic permeability, 43permeability, 81polarizability, 46refraction coefficient, 44susceptibility, 43, 48

optical labeling, 262optical loss, 47optical rectification, 45organic nanocrystals, 234organic nanofibers, 235oscillator strength, 42

para-quaterphenylene, 235perturbation theory, 285phase velocity, 45phonon bottleneck, 147phonon confinement, 161, 162phonons, 150photonic crystal, 190physisorption, 3plane wave representation, 291plasma excitations, 59plasma frequency, 48plasmon resonance, 62

electrostatic theory, 63


Index 329

plasmonic density of states, 73plasmonics, 59PMMA, poly(methyl methacrylate), 4Poisson equation, 289polarization function, 45, 46, 52, 288,

291polymer–metallic nanomaterials, 29polymers, 4, 239potential well, 34Purcell effect, 188Purcell factor, 188

quantization of the field, 293quantum confinement, 35, 40quantum dots, 41quantum electrodynamics (QED), 181quantum well, 33quasiparticle, 34, 283quasistatic approximation, 63

Rabi oscillations, 184Raman polarization function, 150Raman spectroscopy, 149, 160

surface effect, 164Raman tensor, 150, 173refractive index, 65, 80, 81Rydberg atoms, 184

Schrodinger equation, 41, 120, 285second harmonic generation, 205second quantization, 293

sensorsbiosensors, 263polymer-based sensors, 257

silanization, 263silicon carbide, 216single-wall carbon nanotubes (SWNTs),

124spherical harmonics, 40split-ring resonator, 223strong coupling, 183, 188surfac-enhanced infrared absorption

(SEIRA), 271surface-enhanced Raman scattering

(SERS), 149, 165, 219surface plasmon polariton (SPP), 295surface plasmon resonance (SPR), 61, 273,

297, 300surface plasmons, 59

Taylor expansion, 285Taylor series, 46Thomas–Fermi approximation, 279trioctylphosphine oxide, 258two-level atomic system, 183

velocity operator, 292vibronic states, 160

Wannier–Mott excitons, 120weak coupling, 188Wigner symbols, 128

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