1

SECTION 2: NONMAGNETIC EXCITATIONS IN BULK MATERIALS © M.G. Cottam, 2005

We continue considering bulk (effectively infinite) materials and introduce several examples of the excitations (waves) in the case of nonmagnetic materials. The approach follows that in Cottam and Tilley (CT), parts of sections 2.1, 2.2, 2.4 and 2.5. 2.1. One-dimensional lattice dynamics In a full discussion of lattice dynamics it is necessary to include the 3D character of the crystal structure in question. As a start, however, it is sufficient to review the properties of the simplest 1D models, described in terms of masses connected by springs. 2.1.1 Monatomic lattice The simplest case is the monatomic lattice shown below, where an infinite number of identical masses m are joined by identical springs of spring constant C. m m

C z

If un is the longitudinal displacement of mass n from its equilibrium position, the equation of motion is

2

1 1 1 12 ( ) ( ) ( 2nn n n n n n n

um C u u C u u C u ut + − + −

∂= − + − = + −

∂)u

i t

(2.1)

For the normal mode solutions, all the masses vibrate at the same angular frequency ω, so we take exp( )nu ω∝ − . Then Eq. (2.1) becomes (2.2) 2

1 1(n n nm u C u u uω + −− = + − 2 )n

2

This equation is the typical member of an infinite set of coupled difference equations. The set is solved by use of Bloch’s theorem (see Section 1.4), according to which (2.3) 0 exp( )nu u inqa=where a is the equilibrium distance between the masses. The solution corresponds to a wave travelling along the 1D array. Substitution of Eq. (2.3) into (2.2) gives (2.4) 2 ( / )( 2) (4 / )sin ( / 2)iqa iqaC m e e C m qaω −= − + − = ω

2 /C m

0 / aπ q

2

We note the limiting expressions for qa2 2( / )Ca m qω = 2 1 << when qa2 4 /C mω = π= (2.5) The first of these shows that the mode is a sound wave, vqω = with , in the long-wavelength limit, and the second gives the frequency at the Brillouin-zone boundary

2 2 /v Ca m=/q aπ= in

1D. Although the above expression for ω was derived classically, the same result is obtained used quantum field theoretical methods. The quantum of energy associated with the lattice vibration is E ω= and is called a phonon. Another consequence of Eq. (2.1) may be noted: we can pass directly from Eq. (2.1) to a continuum limit which is applicable at long wavelengths, where qa 1<< . Assuming slow spatial variation, we write

2

21 2

12

nn n

uu u a az z±

∂ ∂= ± +

∂ ∂nu (2.6)

and (2.1) becomes

2 2

2

m u uCaa t z

∂=

∂ ∂ 2

∂ (2.7)

This has the form of the 1D wave equation with density /m aρ = and elastic modulus K Ca= . It will be recalled that the velocity of sound is 1/ 2( / )v K ρ= , so Eq. (2.7) is consistent with the first part of Eq. (2.5). 2.1.2. Diatomic lattice We now discuss the 1D diatomic lattice depicted below. We shall see later, when we introduce a surface, that it is the simplest model in which a localized surface mode arises. m1 m2

C C z

We take the equilibrium positions as (2 1)n a+ for masses m1 and 2 for masses mna 2 (n = integer). The equations of motion of adjacent masses are

2

2 11 2 2 22 ( 2n

n n nd um C u u

dt+

+ += + − 2 1)u (2.8)

2

22 2 1 2 12 ( 2n

n nd um C u udt + −= + − 2 )nu (2.9)

From Bloch’s theorem, we write solutions as 2 1 1 exp{ [(2 1) ]}nu U i n qa tω+ = + − (2.10)

3

2 2 exp[ (2 )]nu U i nqa tω= − (2.12) Substitution into Eqs. (2.9) and (2.9) gives two alternative expressions for U1/U2 and equating these we find a quadratic equation for ω2. The solution leads to the dispersion relation which can be written as:

1/ 22 22

1 2 1 2 1 2

1 1 1 1 4sin ( )qaC Cm m m m m m

ω⎡ ⎤⎛ ⎞ ⎛ ⎞⎢= + ± + −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎥ (2.13)

A typical graph relating ω to q is shown schematically (assuming m1 > m2). ω

1 1

1 22 ( )C m m− −+

22 /C m

12 /C m 0 / 2aπ q

The acoustic (or lower) branch occupies the frequency interval and the optic (or upper) branch occupies . The interval

is a “stop band” for bulk modes.

1/ 210 (2 / )C mω≤ ≤

1/ 2 1/ 22 1 2(2 / ) [2 ( ) / ]C m C m m m mω≤ ≤ + 1 2

21/ 2 1/ 2

1(2 / ) (2 / )C m C mω< < We shall see later, when we consider surface effects, that this is the region in which a localized surface mode can occur. 2.2. Bulk elasticity theory Previously we showed for a 1D monatomic crystal that in the long-wavelength (small q) limit the difference equations for the atomic displacements go over into the elastic wave equation. In a similar way, we might expect that for long wavelengths the difference equations of motion of masses in a 3D crystal become the elastic equations of motion. Here we simplify the 3D problem by starting from the standard equations of an elastic medium. It should be noted that, like (2.7), the equations of elasticity apply only to the small q part of the dispersion curve in the acoustic-mode branch.

4

The state of an elastic medium is characterised by two symmetric second-rank tensors, namely the strain iju and the stress ijσ . With u(r) denoting the displacement from equilibrium at position r, the strain is defined by

12

jiij

j i

uuux x

⎛ ⎞∂∂= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

(2.14)

The definition of the stress is that on a surface of area df whose normal is the unit vector , the component i of the force is

n̂

(2.15) ˆ ˆi ik k ik kk

F n df nσ σ= =∑ df

where the tensor ijσ is symmetric ( ij jiσ σ= ). The basic assumption of linear elasticity theory is the generalised Hooke’s Law that stress and strain are linearly related, that is,

,ij ijkl kl ijkl kl

k lu uσ λ λ= =∑ (2.16)

This introduces the elasticity tensor ijklλ . In the second parts of Eqs. (2.15) and (2.16) we have introduced the summation convention, which will be used later whenever convenient. The convention is that where an index is repeated, summation over that index is implied. Thus in

ˆik kn dfσ summation is over k, and in ijkl kluλ summation is over k and l. Equation (2.16) shows that ijklλ is of fourth rank, and since ijσ and iju are symmetric, ijklλ is symmetric on interchange of i and j and of k and l. It also satisfies ijkl klijλ λ= . Given these symmetries, it can be seen that ijklλ may have up to 21 independent components. However, in a crystal the point-group symmetry generally reduces this number. We shall consider only isotropic media here, for which there are just two independent components. These are conventionally taken as Young’s modulus E and Poisson’s ratio σ. The stress-strain relations are

[(1 ) ( )](1 )(1 2 )xx xx yy zz

E u u uσ σ σσ σ

= − ++ −

+ (2.17)

1xy xy

E uσσ

=+

(2.18)

with the analogous relations for yyσ , yzσ , etc., being found by appropriate permutation of suffixes. Poisson’s ratio satisfies the inequality 1

21 σ− ≤ ≤ for reasons of hydrostatic stability, and in practice the stronger inequality 1

20 σ≤ ≤ is satisfied. The form of propagating waves in a bulk elastic medium is found from the equation of motion for a mass element, namely

2

2ik

k kt xσρ ∂∂

=∂ ∂∑u (2.19)

With the use of Eqs. (2.17) and (2.18) this becomes

5

2

2 ( )2(1 ) 2(1 )(1 2 )2

E Et

ρσ σ σ

∂= ∇ + ∇ ∇

∂ + + −u u u⋅ (2.20)

It can be shown from this that longitudinal and transverse waves propagate with different velocities. A longitudinal wave satisfies 0∇× =Lu , since for a plane wave this implies that

, i.e., the displacement is parallel to the propagation vector q. Hence , so that and (2.20) reduces to the ordinary wave equation

0× =Lq u Lu( )∇× ∇× =Lu 0 Lu2( )∇ ∇ ⋅ = ∇Lu

2

2 22 Lv

t∂

= ∇∂

LL

u u (2.21)

where the longitudinal velocity is given by Lv

2 (1 )(1 )(1 2 )L

Ev σρ σ σ

−=

+ − (2.22)

For a transverse wave , since for a plane wave this gives 0∇ ⋅ =Tu 0⋅ =Tq u . The last term in (2.20) then vanishes, so that the transverse velocity is given by

2

2 (1 )TEv

ρ σ=

+ (2.23)

Equations (2.22) and (2.23), together with the inequality satisfied by σ, show that 2Lv ≥ Tv (2.24) To summarize, in the 3D case there will be 1 longitudinal (L) phonon and 2 transverse (T) branches for an elastic solid. In the case of an isotropic elastic material the 2 transverse branches are degenerate with each other, but this will not always be true in general. Also in a lattice dynamics treatment in 3D for ionic solids (such as those having the NaCl structure with 2 atoms per primitive unit cell) there will be both acoustic (A) and optic (O) branches − so there will be 6 phonon modes labeled LA, TA (2 of these), LO and TO (2 of these). 2.3. Some experimental techniques for bulk phonons The following will be mentioned: inelastic light scattering (Raman and Brillouin scattering); inelastic particle scattering (neutron and electron scattering). They are used also for a wide range of other excitations apart from phonons. Phonon frequencies f might often lie in the approximate range from about 1010 to 1013 Hz (i.e. spanning the microwave to infrared region of the electromagnetic spectrum), depending on the material and type of phonon branch. For the different experimental techniques other energy or frequency related units are often used.

6

Conversion factors between frequency- and energy-related units

Frequency (GHz)

Wave number (cm−1)

Electron volts (meV

Temperature (K)

1 GHz 1 0.0334 0.00414 0.0480

1 cm−1 30.0 1 0.124 1.44

1 meV 242 8.07 1 11.6

1 K 20.8 0.695 0.0862 1 The conversion factor is obtained by looking along a row to the column giving the required unit.

These are deduced from: / BE hf hc eV k Tλ= = = = 2.3.1. Inelastic light scattering Raman and Brillouin scattering of light by dense media were first demonstrated in the 1920s and 1930s, but it was not until the advent of the laser, together with other technical developments, that these methods were widely applied to bulk excitations in solids and liquids. The essential distinction between the techniques of Raman and Brillouin scattering lies in the frequency analysis employed for the scattered light. In Raman scattering this is achieved by use of a grating spectrometer, typically with shifts in the wave number of the light in the range 5 – 4000 cm-1. In Brillouin scattering a Fabry-Pérot interferometer is used and the wave number shifts are typically in a range up to about 5 cm-1. The instrumental resolutions obtainable are generally of order 1 cm-1 in the case of conventional Raman scattering and several orders of magnitude smaller for Brillouin scattering. Recently (since the early 1990s) it has become possible to achieve higher-resolution Raman scattering and to extend the range of wave-number shifts measurable. As an example, we show in the figure below a schematic arrangement for Brillouin scattering off the surface of an opaque sample.

(PM = photomultiplier; Stab = stabiliser; Discr = discriminator; MCA = multi-channel analyser).

7

We consider first the kinematics of light scattering in a bulk (effectively infinite) transparent medium. The simplest processes involve the incident light of frequency Iω and wave vector creating or absorbing a single excitation of frequency

Ikω and wave vector q, thereby scattering into

light of frequency Sω and wave vector . These are represented as below and are known as the Stokes and anti-Stokes processes respectively.

Sk

,S Sω k ,S Sω k ,I Iω k ,I Iω k ,ω q ,ω q Stokes anti-Stokes The conservation of energy and momentum imply that ωωω ±= SI qkk ±= SI (2.25) where the upper and lower signs refer to Stokes ( IS ωω < ) and anti-Stokes ( IS ωω > ) scattering respectively. In many simple cases involving bulk media, the ratio of intensities for anti-Stokes and Stokes scattering, SAS II , is given by the thermal factor exp( )AS S BI I k Tω= − (2.26) Hence anti-Stokes scattering is usually less intense than Stokes scattering from the same excitation. The conservation conditions in Eq. (2.25) impose limitations on the wave vector q, so that only excitations near the centre of the Brillouin zone are detectable by light scattering. This follows from noting that the optical wave vectors are related to their frequencies by cIII ωη=k cSSS ωη=k (2.27) where Iη and Sη are the refractive indices corresponding to frequencies Iω and Sω . Typically

Iωω << and S Iη η , from which it is a simple exercise to prove that | | 2 sin( / 2) /I I cη ω θq where θ is the angle between the incident and scattered light beams. The connection between theory and experiment is provided by the scattering cross section σ , which is defined as the rate at which energy is removed from the incident light beam by the scattering, divided by the power flow in the incident beam. A calculation of σ for any scattering medium must take account of the mechanism by which light interacts with the crystal excitation (e.g. a phonon) and the transmission of the incident and scattered light beams through the boundaries of the medium. Roughly, the mechanism for interaction with the light is that the excitation modulates the dielectric function of the medium in a wave-like fashion, and the light then scatters from these fluctuations.

8

2.3.2. Inelastic particle scattering Although inelastic light scattering offers very high sensitivity and resolution for studying excitations, it is limited to those excitations with wave vectors very close to the Brillouin zone centre. Under appropriate conditions the inelastic scattering of particles by crystal excitations can involve larger momentum changes, and hence this type of scattering may provide a probe of excitations at wave vectors extending throughout the Brillouin zone (although generally with less favourable resolution than in light scattering). Here we shall be concerned mainly with the scattering of electrons and neutrons. Electron energy loss spectroscopy (EELS) has proved to be successful for studying the excitations. For the early work in the 1960s, which was applied principally to electronic excitations, the energy losses in the inelastic scattering process were typically many eV and the resolution was of order 250 meV. The development of high-resolution EELS (or HREELS) in the 1980s was motivated by studies of phonons and led to much improved sensitivity, corresponding to an energy resolution of ~7 meV (or ~55-60 cm-1). This improved technique can be used to measure dispersion relations of excitations for wave vectors throughout the entire Brillouin zone. A much more successful technique (dating from around the 1960s) for studying phonons is through inelastic scattering of a beam of monoenergetic neutrons with energies of a few meV or more. The neutrons interact chiefly with the nuclei in the solid and the technique is sensitive to displacements of the nuclei occurring through vibrations, i.e. they interact with phonons. An important step was the development of a suitable spectrometer (the so-called triple-axis spectrometer). As in light scattering, the conservation of energy and momentum in a bulk material are again used. In this case they give

22 2( )

2 I Sk kM

ω− = ± ( )I S− = ± +k k q Q (2.28)

Here M is the mass of a neutron and the two sets of signs refer to the two possibilities of the absorption or creation of a phonon, as before. A new feature is the appearance of Q in the momentum equation (where Q denotes any reciprocal lattice vector). The value of Q would always be chosen such that the phonon wave vector q lies in the first Brillouin zone. [The occurrence of a nonzero Q is called an Umklapp process: it is unimportant in light scattering]. 2.4. Single-electron excitations For electronic excitations in solids, there are two distinct effects that are of interest:

• The single-electron states. These are the quantum states of individual electrons in a solid, e.g. as described in QM by the solution of Schrödinger’s equation. A proper treatment will lead to a description of band structure in a solid. It will be discussed in this subsection

• The plasma wave (or plasmon). This is a collective excitation of an electron gas in a solid (e.g. a metal or semiconductor). It involves all of the electrons interacting with one another. It will be discussed later.

9

For the single-electron states we need (in principle) to solve Schrödinger’s equation to determine the energy states for an electron moving in a periodic potential (whose form depends on the arrangement of the atoms in the crystal lattice). Schematically we have something like Potential energy U(x) a x where the dots denote the location of the ion cores. Hence we have a periodic variation (in 3D) of the potential energy, giving an array of quantum wells. One way to analyze the essential features of this in 1D is to use the Kronig-Penney model. This replaces the potential energy function by the rectangular form below: U(x) U0 ε x b a The Schrödinger equation in 1D for the wave function ( )xψ of an electron is

2 2

2 ( )2

d U xm dx

ψ ψ εψ− + =

where U(x) is the periodic potential energy and ε is the energy eigenvalue. We take 0 < ε < U0 appropriate to bound states of the electron. In a “well” region (such as 0 < x < a) where U = 0, the wave function is a combination of traveling waves:

( ) iKx iKxx Ae Beψ −= + where K has to satisfy 2 2 2K mε = . In the barrier region (such as −b < x < 0) where U = U0, the wave function is a combination of real exponentials:

( ) Qx Qxx Ce Deψ −= +

10

where Q has to satisfy 2 20 2U Qε− = m .

Because of the translational symmetry through any multiple of ( )a b+ , we can use Bloch’s theorem in 1D. This tells us that, for example,

( )( ) ( 0) ik a ba x a b b x eψ ψ +< < + = − < < (2.29) which defines a wave vector k used to describe the electronic state. Relationships between the constants A, B, C and D can be written down by applying the usual QM boundary conditions at x = 0 and x = a (i.e. ψ and d dxψ must both be continuous there). At x = 0 these give

A B C D+ = + ( ) (iK A B Q C D− = − )

)+

+

At x = a, with the use of Bloch’s theorem, this gives

(( )iKa iKa Qb Qb ik a bAe Be Ce De e− −+ = + , ( )( ) ( )iKa iKa Qb Qb ik a biK Ae Be Q Ce De e− −− = −

The four equations have a nontrivial solution only if the determinant of the coefficients of A, B, C and D vanishes. After some algebra, this gives the following dispersion relation:

2 2[( ) 2 ]sinh( )sin( ) cosh( )cos( ) cos[ ( )]Q K QK Qb Ka Qb Ka k a b− + = + (2.30) This can be studied as it is, but usually it is simplified by taking a limit in which the potential barriers become delta functions. This is the limit of taking and in such a way that

remains finite. This is equivalent to putting 0b → 0U → ∞

0bU 2 2Q ba P= , where P is a finite quantity related to the strength of each delta function. The dispersion relation can then be shown to take the slightly simpler form:

( )sin( ) cos( ) cos( )P Ka Ka Ka ka+ = (2.31) The solutions can be studied graphically and numerical solutions deduced for any chosen value of P (e.g. see the book by Kittel). Plotting the left-hand side of eq (2.31) as a function of Ka gives the figure below. If this is to be equal to the cosine factor on the right-hand side, then only certain ranges of values of Ka lead to a physical solution. This means energy bands (in ε depending on the value of wave vector k).

11

2 1/ 2(2 )Ka m aε=(plotted for P = 3π/2: adapted from Kittel). Recalling that , the plot of ε versus ka has the form (for P = 3π/2):-

(adapted from Kittel) There are energy gaps (or stop bands) at integer multiples of π/a 2.5. Plasmons in metals and semiconductors We now consider the natural oscillation frequency in an electron-gas plasma. The plasma is assumed to contain equal concentrations of positive and negative charges, and at least one charge type is mobile. For example, we might have a gas of mobile electrons in a background of heavier (therefore essentially static) positive ions, as in a metal or n-type semiconductor. The electrons can oscillate subject to a restoring force provided by the electric field, and they give rise to a collective excitation because the electrons interact with one another (e.g. through Coulomb interactions). The plasma frequency can be found from the following simple argument. Suppose the electron gas is instantaneously displaced spatially by a small amount r. The corresponding polarization P (electric dipole moment per unit volume) is P = −ner, where n is the number of electrons per unit volume. In a slab of material this will give rise to an depolarization electric field Ed where Ed = −P/ε0 = (ne/ε0)r The equation of motion for an electron is then 2 2 2

0( / )dm d dt e ne ε= − = −r E r This describes SHM at an angular frequency ω = ωp where

20p ne mω ε= is called the plasma frequency of oscillation.

The corresponding quantum of energy pω is called a plasmon. For a metal it might typically be a few eV in energy, and the EELS technique provides a good experimental method for study. For

12

a doped semiconductor the excitation might occur in the meV range, and Raman scattering is a convenient technique. We can modify the above argument to obtain the dielectric function associated with a plasma oscillation. If an electric field of frequency ω is applied parallel to a slab of the material, there is no depolarization field, and then

2 2m d dt e= −r E implies 2e mω=r E , which gives 2 2ne ne mω= − = −P r E Next we use 0 0( )ε ε ω ε= =D E +E P , which gives

22

2 20

( ) 1 1 pnem

ωε ω

ε ω ω= − = −

If the positive-ion background has a dielectric constant ε∞ (independent of ω) the final result is

2

2( ) 1 pωε ω ε

ω∞

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠ , 2

0p ne mω ε ε∞= (2.32)

Notice that this changes sign at the plasma frequency, which has consequences for the propagation of an electromagnetic wave in a plasma. Recall that the wave dependence is like exp[ ( ) ]i cε ω x in the direction x.

For the plasma wave that we have just calculated, all the electrons oscillate in phase, i.e. it corresponds to a wave with infinite wavelength or equivalently wave vector q = 0. Physically, this is because we have ignored interactions (e.g. Coulomb repulsion) between the electrons. If these are included, we get a modified dispersion relation like 2 2( ) p qω ω= +q 2β

for an isotropic medium (where β is a constant). In most cases the effect of q (giving the so-called spatial dispersion) can be neglected for the plasmons. 2.6. Polaritons We have just seen, in the case of an electron plasma, how the electromagnetic wave propagation is influenced by the frequency dependence of the dielectric function ( )ε ω . We can now take the

13

study of the interaction of light with a solid one step further by considering polaritons. These are “mixed” excitations that arise when the light (photon) couples to an excitation of the crystal (such as a phonon or plasmon). 2.6.1 The polariton dispersion relation The propagation of light through a nonmagnetic bulk medium characterized by a frequency-dependent dielectric function is governed by Maxwell’s equations. For a plane wave in an isotropic or cubic medium the equation div D = 0, or equivalently ( ) 0ε ω ∇ ⋅ =E , leads to

( ) 0ε ω ⋅ =q E where all fields are assumed proportional to exp( )i i tω⋅ −q r in the bulk medium, so . This equation has the solutions that

i∇ → q

0)( =ωε or (2.33) 0⋅ =q EThe first of these gives, for example, pωω = for a plasma with the previous dielectric function. It describes longitudinal modes, which are relatively uninteresting and will not be discussed further. The second solution in Eq. (2.33) is a transversality condition (E transverse to q) and we study its consequences. Two of Maxwell’s equations (assuming a nonmagnetic material) are

0 tµ∇× = − ∂ ∂E H and 0 ( ) tε ε ω∇× = ∂ ∂H E With the assumption of plane waves as above, they become 0µ ω× =q E H and 0 ( )ε ε ω ω× = −q H E On eliminating one of the field vectors (say H), we have 2

0 0 0( ) ( )µ ω ε µ ε ω ω× × = × = −q q E q H EUsing 2

0 0 1 cε µ = and a standard vector identity, this can be rewritten as 2 2( ) ( ) [ ( ) / ]cε ω ω⋅ − ⋅ = −q E q q q E EThe first term vanishes because of the transversality condition, leaving (2.34) 222 /)( cq ωωε=This is the bulk polariton dispersion relation for the transverse mode. 2.6.2. Plasmon-polaritons In this case all that we need to do is substitute the plasmon expression for )(ωε given in Eq. (2.32) into (2.34):

2

2 2 2 221 ( / ) ( )(p

pq c cω 2 2 )ε ω ε ωω∞ ∞

⎛ ⎞= − = −⎜ ⎟⎜ ⎟

⎝ ⎠ω

This gives: 2 2 2( )p cω ω ε∞= + 2q (2.35) Its limiting cases are pω ω for small q (i.e. it is plasmon-like), and

cqω ε∞ for large q (i.e. it is photon-like). The overall behavior looks like:

14

ω There is a frequency gap (or stop band) for 0 pω ω< < . The horizontal dashed line is pω the plasmon line (ignoring

spatial dispersion); the sloping dashed line is the so-called “light line” for an uncoupled

photon: cqω ε∞= 0 cq 2.6.2. Phonon-polaritons In this case we need to find the form of the dielectric function )(ωε that corresponds to phonons. We will work this out for optic phonons in an ionic solid (such as those having the NaCl structure), ignoring spatial dispersion as in the above case (i.e. we need consider only the phonons at q = 0). For simplicity, we restrict attention to an isotropic or cubic medium, in which vector quantities, such as P and E, are parallel. We may then use the lattice dynamics of a 1D diatomic lattice as our starting point. As in Sec. 2.1.2, we consider an infinite diatomic lattice in which masses and

alternate. The crystal is assumed to be ionic, so that the two masses are associated with opposite electric charges. The mode that couples to electromagnetic radiation is the long-wavelength ( ) optic phonon at frequency

1m

2m

0=q Tω . This follows by considering the amplitude ratio that can be derived from earlier results. This ratio is easily shown to be negative for

so that the two masses move in antiphase. Since the masses carry opposite charges, the optic phonon therefore carries an oscillating dipole moment.

21 /UU0=q

We denote by u the relative displacement of the two types of masses. The polarization P will contain a term proportional to u, as well as a term due to the electrical susceptibility:

)(0 EuP χαε += (2.36) For an isotropic medium P, u and E will be parallel. In Eq. (2.36) E is the macroscopic electric field, the value found by averaging the local field over many unit cells. Values of the constants of proportionality α and χ depends upon details of the lattice dynamics and electronic structure.

locE

With the electric field E and all other field variables assumed proportional to exp( )i tω− , the equation of motion (ignoring damping) for u is 2 2

T lω ω β− = − +u u E oc

The first term on the right-hand side is a restoring force that ensures that in the absence of coupling to the electric field the mode frequency is Tω , and the second is the driving force due to the electric field (with β constant). In order to derive an expression for the dielectric function from

15

the above it is necessary to have a relation between and E. The usual assumption is that they must be simply proportional to one another, and so the last equation can therefore be replaced by an equation involving E rather than (only the constant is different):

locE

locE 2 2

Tω ω γ− = − +u u E (2.37) Equations (2.36) and (2.37) are readily solved for P:

0 2T

αγε χω ω

⎛ ⎞= +⎜ −⎝ ⎠

2

EP ⎟E (2.38)

The defining relations for the dielectric function give EPED )(00 ωεεε =+= Combining this with Eq. (2.38) leads to

2 2( ) (1 )T

αγε ω χω ω

= + +−

It is more conventionally written in the form

2 2

2 2( ) 1 L T

T

ω ωε ω εω ω∞

⎛ ⎞−= +⎜ −⎝ ⎠

⎟ (2.39)

where we define χε +=∞ 1 , )1/(22 χαγωω ++= TL

We can identify ∞ε as the high-frequency dielectric constant and Lω as the LO phonon frequency. Notice that the definition makes )(ωε vanish as required at the frequency Lω . Also

L Tω ω> . The behavior of )(ωε is plotted below, taking 1ε∞ = and 2L Tω ω = . It is negative for

T Lω ω ω< < , implying that this is a stop band for optical propagation.

The dispersion relation for phonon-polaritons is obtained by substituting Eq. (2.39) into (2.34):

2 2 2

22 2 21 L T

T

qc

ω ω ωεω ω∞

⎛ ⎞−= +⎜ ⎟−⎝ ⎠

or 2 2 2

22 2 2

L

T

qc

ε ω ω ωω ω

∞ ⎛ ⎞−= ⎜ −⎝ ⎠

⎟ (2.40)

The second form above can be rewritten as a quadratic expression in ω2, which can then be solved for ω as a function of q. The quadratic equation gives rise to two branches for the dispersion relation. Some asymptotic forms of Eq. (2.40) are

16

2 2 2

2T

2

L2 2 2

~ (0) / for

for

0 (i.e. no solution) for 0 for

~ / for

T

T L

q c

q

qq

q c

ε ω ω ω

ω ω

ω ω ωω ω

ε ω ω∞

<<

→ ∞ →

< <= =

→ ∞

< (2.41)

The overall behavior is illustrated below, showing the stop band for LT ωωω << . ω Lω Tω 0 cq The dispersion curve may be seen as resulting from the crossing of the phonon line Tωω = and the photon curve , where ε is imagined to change slowly with increasing frequency from

cq /2/1 ωε=)0(ε to ∞ε . These modes interact strongly, and the crossover is therefore eliminated with

repulsion of the curves. Thus the full dispersion curve describes a mode of mixed phonon-photon character.

2.6.3 Excitons and exciton-polaritons Excitons are bound electron-hole pairs occurring in semiconductors. An example occurs in a semiconductors such as GaAs or ZnSe, where an exciton can be formed from an electron near the bottom of the conduction band and a hole near the top of the valence band. Schematically:- Energy Conduction band e Band gap Eg h Valence band

17

If the electron (e) and hole (h) are weakly bound (by the screened Coulomb interaction), they may be several atomic spacings apart, forming a “large” exciton (like a H-atom but the masses are different).

We restrict attention to the simplest model, in which an electron of effective mass is bound by the Coulomb interaction to a hole of effective mass . If the exciton moves as a wave with wavevector q its energy may be represented as

em

hm

(2.42) 2 2( ) ( / 2 )eq qω ω= + Mwhere ( )e hM m m= + is the total mass entering into the kinetic energy term, and 0EEge −=ω is the energy of the exciton at rest (the band-gap energy minus the exciton binding energy denoted by ). The exciton can couple strongly to light and its contribution to the dielectric function is found to be

gE

0E

2 2( , )e

SqDq

ε ω ε 2ω ω∞= ++ −

(2.43)

where constant , S is a dipole strength of the exciton resonance, and 1/D ∝ M ∞ε is the background dielectric constant. The exciton-polariton is the most important case in which spatial dispersion of the dielectric function must be taken into account, i.e. we cannot ignore the q-dependence as in previous examples. The exciton-polariton dispersion relation is then found from Eq. (2.43) as before. It has two main branches, like in the phonon-polariton case, but the shapes are different because of the extra q terms. An important practical difference is that exciton-polaritons occur near the semiconductor gap frequency, typically in the visible or near infrared region, whereas phonon-polaritons occur in the far infrared.

© M.G. Cottam, 2005