SECTION 2: NONMAGNETIC EXCITATIONS IN BULK MATERIALS

Transcription

1 1 ECON : NONMAGNEC EXCAON N BUK MAERA M.G. Cottam, 005 We continue considering bulk (effectively infinite) materials and introduce several examles of the excitations (waves) in the case of nonmagnetic materials. he aroach follows that in Cottam and illey (C), arts of sections.1,.,.4 and One-dimensional lattice dynamics n 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 roerties of the simlest 1D models, described in terms of masses connected by srings..1.1 Monatomic lattice he simlest case is the monatomic lattice shown below, where an infinite number of identical masses m are joined by identical srings of sring constant C. m m C f u n is the longitudinal dislacement of mass n from its equilibrium osition, the equation of motion is un m = C( u n+ 1 un) + C( un 1 un) = C( un+ 1+ un 1 un) (.1) t For the normal mode solutions, all the masses vibrate at the same angular frequency ω, so we take u ex( iωt). hen Eq. (.1) becomes n mω un = C( un+ 1+ un 1 un) (.) his equation is the tyical member of an infinite set of couled difference equations. he set is solved by use of Bloch s theorem (see ection 1.4), according to which u ex( ) n = u0 inqa (.3) where a is the equilibrium distance between the masses. he solution corresonds to a wave travelling along the 1D array. ubstitution of Eq. (.3) into (.) gives ( / )( iqa iqa ω = C m e + e ) = (4 C/ m)sin ( qa/) (.4) ω z C/ m 0 π / a q

2 We note the limiting exressions ω = ( Ca / m) q for qa << 1 ω = 4 C/ m when qa = π (.5) he first of these shows that the mode is a sound wave, ω = vq with v = Ca / m, in the longwavelength limit, and the second gives the frequency at the Brillouin-zone boundary q= π / a in 1D. Although the above exression for ω was derived classically, the same result is obtained used quantum field theoretical methods. he quantum of energy associated with the lattice vibration is E = ω and is called a honon. Another consequence of Eq. (.1) may be noted: we can ass directly from Eq. (.1) to a continuum limit which is alicable at long wavelengths, where qa << 1. Assuming slow satial variation, we write un 1 u un± 1 = un ± a + a n (.6) z z and (.1) becomes m u u = Ca (.7) a t z his has the form of the 1D wave equation with density ρ = m/ aand elastic modulus K = Ca. t will be recalled that the velocity of sound is v art of Eq. (.5). 1/ = ( K / ρ), so Eq. (.7) is consistent with the first.1.. Diatomic lattice We now discuss the 1D diatomic lattice deicted below. We shall see later, when we introduce a surface, that it is the simlest model in which a localized surface mode arises. m 1 m C C z We take the equilibrium ositions as (n+ 1) a for masses m 1 and na for masses m (n = integer). he equations of motion of adjacent masses are dun+ 1 m1 = C( u n+ + un n+ 1 dt (.8) dun m = C( u n+ 1+ un 1 u dt n (.9) From Bloch s theorem, we write solutions as u n 1 U1ex{ i[(n 1) qa + ωt]} (.10)

3 3 u = U ex[ i( nqa ωt)] (.1) n ubstitution into Eqs. (.9) and (.9) gives two alternative exressions for U 1 /U and equating these we find a quadratic equation for ω. he solution leads to the disersion relation which can be written as: sin ( qa) ω = C + ± C + (.13) m1 m m1 m mm 1 A tyical grah relating ω to q is shown schematically (assuming m 1 > m ). ω 1/ Cm ( + m ) C/ m C/ m 1 0 π /a q he acoustic (or lower) branch occuies the frequency interval 0 ω ( C/ m ) and the otic 1/ 1/ (or uer) branch occuies ( C/ m) ω [ C( m1+ m)/ mm 1 ]. he interval 1/ 1/ ( C/ m ) < ω < ( C/ m ) is a sto band for bulk modes. 1 We shall see later, when we consider surface effects, that this is the region in which a localized surface mode can occur. 1/ 1.. 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 dislacements go over into the elastic wave equation. n a similar way, we might exect that for long wavelengths the difference equations of motion of masses in a 3D crystal become the elastic equations of motion. Here we simlify the 3D roblem by starting from the standard equations of an elastic medium. t should be noted that, like (.7), the equations of elasticity aly only to the small q art of the disersion curve in the acousticmode branch.

4 4 he state of an elastic medium is characterised by two symmetric second-rank tensors, namely the strain u ij and the stress σ ij. With u(r) denoting the dislacement from equilibrium at osition r, the strain is defined by 1 u u i j uij = + (.14) x j x i he definition of the stress is that on a surface of area df whose normal is the unit vector ˆn, the comonent i of the force is F = σ nˆ df = σ nˆ df (.15) i ik k ik k k σ ij is symmetric ( σ ij σ ji where the tensor = ). he basic assumtion of linear elasticity theory is the generalised Hooke s aw that stress and strain are linearly related, that is, σij = λijklukl = λijklukl (.16) kl, his introduces the elasticity tensor λ ijkl. n the second arts of Eqs. (.15) and (.16) we have introduced the summation convention, which will be used later whenever convenient. he convention is that where an index is reeated, summation over that index is imlied. hus in σ ˆ ikndf k summation is over k, and in λ ijkl ukl summation is over k and l. Equation (.16) shows that λ ijkl is of fourth rank, and since σ ij and u ij are symmetric, λ ijkl is symmetric on interchange of i and j and of k and l. t also satisfies λ ijkl = λ klij. Given these symmetries, it can be seen that λ ijkl may have u to 1 indeendent comonents. However, in a crystal the oint-grou symmetry generally reduces this number. We shall consider only isotroic media here, for which there are just two indeendent comonents. hese are conventionally taken as Young s modulus E and Poisson s ratio σ. he stress-strain relations are E σxx = [(1 σ) uxx + σ( uyy + uzz )] (.17) (1 + σ)(1 σ) E σ xy = u xy (.18) 1 + σ with the analogous relations for σ yy, σ yz, etc., being found by aroriate ermutation of suffixes. Poisson s ratio satisfies the inequality 1 σ 1 for reasons of hydrostatic stability, and in ractice the stronger inequality 0 σ 1 is satisfied. he form of roagating waves in a bulk elastic medium is found from the equation of motion for a mass element, namely u σ ik ρ = (.19) t x k k With the use of Eqs. (.17) and (.18) this becomes

5 5 u E E ρ u t (1 + σ) (1 + σ)(1 σ) = + ( u) (.0) t can be shown from this that longitudinal and transverse waves roagate with different velocities. A longitudinal wave satisfies u = 0, since for a lane wave this imlies that q u =0, i.e., the dislacement u is arallel to the roagation vector q. Hence ( u ) = 0, so that ( u ) = u and (.0) reduces to the ordinary wave equation u = v u t (.1) where the longitudinal velocity is given by E(1 σ ) v = ρ(1 + σ)(1 σ) v (.) For a transverse wave u =0, since for a lane wave this gives qu = 0. he last term in (.0) then vanishes, so that the transverse velocity is given by E v = (.3) ρ(1 + σ ) Equations (.) and (.3), together with the inequality satisfied by σ, show that v v (.4) o summarize, in the 3D case there will be 1 longitudinal () honon and transverse () branches for an elastic solid. n the case of an isotroic elastic material the 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 atoms er rimitive unit cell) there will be both acoustic (A) and otic (O) branches so there will be 6 honon modes labeled A, A ( of these), O and O ( of these)..3. ome exerimental techniques for bulk honons he following will be mentioned: inelastic light scattering (Raman and Brillouin scattering); inelastic article scattering (neutron and electron scattering). hey are used also for a wide range of other excitations aart from honons. Phonon frequencies f might often lie in the aroximate range from about to Hz (i.e. sanning the microwave to infrared region of the electromagnetic sectrum), deending on the material and tye of honon branch. For the different exerimental techniques other energy or frequency related units are often used.

6 6 Conversion factors between frequency- and energy-related units Frequency Wave number Electron volts emerature (GHz) (cm 1 ) (mev (K) 1 GHz cm mev K he conversion factor is obtained by looking along a row to the column giving the required unit. hese are deduced from: E = hf = hc/ λ = ev = kb.3.1. nelastic light scattering Raman and Brillouin scattering of light by dense media were first demonstrated in the 190s and 1930s, but it was not until the advent of the laser, together with other technical develoments, that these methods were widely alied to bulk excitations in solids and liquids. he essential distinction between the techniques of Raman and Brillouin scattering lies in the frequency analysis emloyed for the scattered light. n Raman scattering this is achieved by use of a grating sectrometer, tyically with shifts in the wave number of the light in the range cm -1. n Brillouin scattering a Fabry-Pérot interferometer is used and the wave number shifts are tyically in a range u to about 5 cm -1. he 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 ossible to achieve higherresolution Raman scattering and to extend the range of wave-number shifts measurable. As an examle, we show in the figure below a schematic arrangement for Brillouin scattering off the surface of an oaque samle. (PM = hotomultilier; tab = stabiliser; Discr = discriminator; MCA = multi-channel analyser).

7 7 We consider first the kinematics of light scattering in a bulk (effectively infinite) transarent medium. he simlest rocesses involve the incident light of frequency ω and wave vector k creating or absorbing a single excitation of frequency ω and wave vector q, thereby scattering into light of frequency ω and wave vector k. hese are reresented as below and are known as the tokes and anti-tokes rocesses resectively. ω, k ω, k ω, k ω, k ω, q ω, q tokes anti-tokes he conservation of energy and momentum imly that ω = ω ± ω k = k ± q (.5) where the uer and lower signs refer to tokes ( ω < ω ) and anti-tokes ( ω > ω ) scattering resectively. n many simle cases involving bulk media, the ratio of intensities for anti-tokes and tokes scattering,, is given by the thermal factor A A = ex( ω kb) (.6) Hence anti-tokes scattering is usually less intense than tokes scattering from the same excitation. he conservation conditions in Eq. (.5) imose limitations on the wave vector q, so that only excitations near the centre of the Brillouin zone are detectable by light scattering. his follows from noting that the otical wave vectors are related to their frequencies by k = η ω c k = η ω c (.7) where η and η are the refractive indices corresonding to frequencies ω and ω. yically ω << ω and η η, from which it is a simle exercise to rove that q η ω sin( θ / ) / c where θ is the angle between the incident and scattered light beams. he connection between theory and exeriment is rovided 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 ower 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 honon) 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 8.3.. nelastic article 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 aroriate conditions the inelastic scattering of articles by crystal excitations can involve larger momentum changes, and hence this tye of scattering may rovide a robe 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 sectroscoy (EE) has roved to be successful for studying the excitations. For the early work in the 1960s, which was alied rincially to electronic excitations, the energy losses in the inelastic scattering rocess were tyically many ev and the resolution was of order 50 mev. he develoment of high-resolution EE (or HREE) in the 1980s was motivated by studies of honons and led to much imroved sensitivity, corresonding to an energy resolution of ~7 mev (or ~55-60 cm -1 ). his imroved technique can be used to measure disersion relations of excitations for wave vectors throughout the entire Brillouin zone. A much more successful technique (dating from around the 1960s) for studying honons is through inelastic scattering of a beam of monoenergetic neutrons with energies of a few mev or more. he neutrons interact chiefly with the nuclei in the solid and the technique is sensitive to dislacements of the nuclei occurring through vibrations, i.e. they interact with honons. An imortant ste was the develoment of a suitable sectrometer (the so-called trile-axis sectrometer). As in light scattering, the conservation of energy and momentum in a bulk material are again used. n this case they give ( k k) = ± ω ( k k) = ± q+ Q (.8) M Here M is the mass of a neutron and the two sets of signs refer to the two ossibilities of the absortion or creation of a honon, as before. A new feature is the aearance of Q in the momentum equation (where Q denotes any recirocal lattice vector). he value of Q would always be chosen such that the honon wave vector q lies in the first Brillouin zone. [he occurrence of a nonzero Q is called an Umkla rocess: it is unimortant in light scattering]..4. ingle-electron excitations For electronic excitations in solids, there are two distinct effects that are of interest: he single-electron states. hese are the quantum states of individual electrons in a solid, e.g. as described in QM by the solution of chrödinger s equation. A roer treatment will lead to a descrition of band structure in a solid. t will be discussed in this subsection he lasma wave (or lasmon). his is a collective excitation of an electron gas in a solid (e.g. a metal or semiconductor). t involves all of the electrons interacting with one another. t will be discussed later.

9 9 For the single-electron states we need (in rincile) to solve chrödinger s equation to determine the energy states for an electron moving in a eriodic otential (whose form deends on the arrangement of the atoms in the crystal lattice). chematically we have something like Potential energy U(x) a x where the dots denote the location of the ion cores. Hence we have a eriodic variation (in 3D) of the otential 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. his relaces the otential energy function by the rectangular form below: U(x) U 0 ε x b a he chrödinger equation in 1D for the wave function ψ ( x) of an electron is d ψ + U( x) ψ = εψ mdx where U(x) is the eriodic otential energy and ε is the energy eigenvalue. We take 0 < ε < U 0 aroriate to bound states of the electron. n a well region (such as 0 < x < a) where U = 0, the wave function is a combination of traveling waves: ikx ikx ψ ( x) = Ae + Be where K has to satisfy ε = K m. n the barrier region (such as b < x < 0) where U = U 0, the wave function is a combination of real exonentials: Qx Qx ψ ( x) = Ce + De

10 10 where Q has to satisfy U 0 ε Q = m. Because of the translational symmetry through any multile of ( a+ b), we can use Bloch s theorem in 1D. his tells us that, for examle, ik ( a+ b) ψ ( a< x< a+ b) = ψ ( b< x< 0) e (.9) which defines a wave vector k used to describe the electronic state. Relationshis between the constants A, B, C and D can be written down by alying the usual QM boundary conditions at x = 0 and x = a (i.e. ψ and dψ dx 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+ b) Ae + Be = ( Ce + De ) e, ika ika Qb Qb ik ( a+ b) ik( Ae Be ) = Q( Ce De ) e he 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 disersion relation: [( Q K ) QK]sinh( Qb)sin( Ka) + cosh( Qb)cos( Ka) = cos[ k( a + b)] (.30) his can be studied as it is, but usually it is simlified by taking a limit in which the otential barriers become delta functions. his is the limit of taking b 0 and U0 in such a way that bu 0 remains finite. his is equivalent to utting Qba = P, where P is a finite quantity related to the strength of each delta function. he disersion relation can then be shown to take the slightly simler form: ( PKa)sin( Ka) + cos( Ka) = cos( ka) (.31) he solutions can be studied grahically and numerical solutions deduced for any chosen value of P (e.g. see the book by Kittel). Plotting the left-hand side of eq (.31) as a function of Ka gives the figure below. f 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 hysical solution. his means energy bands (in ε deending on the value of wave vector k).

11 11 (lotted for P = 3π/: adated from Kittel). Recalling that has the form (for P = 3π/):- = ( ) 1/ Ka mε a, the lot of ε versus ka (adated from Kittel) here are energy gas (or sto bands) at integer multiles of π/a.5. Plasmons in metals and semiconductors We now consider the natural oscillation frequency in an electron-gas lasma. he lasma is assumed to contain equal concentrations of ositive and negative charges, and at least one charge tye is mobile. For examle, we might have a gas of mobile electrons in a background of heavier (therefore essentially static) ositive ions, as in a metal or n-tye semiconductor. he electrons can oscillate subject to a restoring force rovided by the electric field, and they give rise to a collective excitation because the electrons interact with one another (e.g. through Coulomb interactions). he lasma frequency can be found from the following simle argument. uose the electron gas is instantaneously dislaced satially by a small amount r. he corresonding olarization P (electric diole moment er unit volume) is P = ner, where n is the number of electrons er unit volume. n a slab of material this will give rise to an deolarization electric field E d where E d = P/ε 0 = (ne/ε 0 )r he equation of motion for an electron is then md r dt = ee = ( ne / ε ) r his describes HM at an angular frequency ω = ω where ω = ε is called the lasma frequency of oscillation. ne 0m he corresonding quantum of energy ω is called a lasmon. For a metal it might tyically be a few ev in energy, and the EE technique rovides a good exerimental method for study. For d 0

12 1 a doed 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 lasma oscillation. f an electric field of frequency ω is alied arallel to a slab of the material, there is no deolarization field, and then md r dt = ee imlies r = ee mω, which gives P= ner = ne E mω Next we use D = ε 0εω ( ) E = ε0e+ P, which gives ne ω εω ( ) = 1 = 1 ε mω ω 0 f the ositive-ion background has a dielectric constant ε (indeendent of ω) the final result is ω εω ( ) ε 1 =, ω ω = ne ε ε0m (.3) Notice that this changes sign at the lasma frequency, which has consequences for the roagation of an electromagnetic wave in a lasma. Recall that the wave deendence is like ex[ i ε( ω c) x ] in the direction x. For the lasma wave that we have just calculated, all the electrons oscillate in hase, i.e. it corresonds to a wave with infinite wavelength or equivalently wave vector q = 0. Physically, this is because we have ignored interactions (e.g. Coulomb reulsion) between the electrons. f these are included, we get a modified disersion relation like ω q ( ) = ω + β q for an isotroic medium (where β is a constant). n most cases the effect of q (giving the so-called satial disersion) can be neglected for the lasmons..6. Polaritons We have just seen, in the case of an electron lasma, how the electromagnetic wave roagation is influenced by the frequency deendence of the dielectric function ε ( ω ). We can now take the

13 13 study of the interaction of light with a solid one ste further by considering olaritons. hese are mixed excitations that arise when the light (hoton) coules to an excitation of the crystal (such as a honon or lasmon)..6.1 he olariton disersion relation he roagation of light through a nonmagnetic bulk medium characterized by a frequencydeendent dielectric function is governed by Maxwell s equations. For a lane wave in an isotroic or cubic medium the equation div D = 0, or equivalently ε ( ω) E = 0, leads to ε ( ω) qe = 0 where all fields are assumed roortional to ex( iqr iωt) in the bulk medium, so iq. his equation has the solutions that ε ( ω) = 0 or qe =0 (.33) he first of these gives, for examle, ω = ω for a lasma with the revious dielectric function. t describes longitudinal modes, which are relatively uninteresting and will not be discussed further. he second solution in Eq. (.33) is a transversality condition (E transverse to q) and we study its consequences. wo of Maxwell s equations (assuming a nonmagnetic material) are E= µ 0 H t and H = ε 0 εω ( ) E t With the assumtion of lane waves as above, they become q E=µ 0ω H and q H = ε0 εωω ( ) E On eliminating one of the field vectors (say H), we have q ( q E) = µωq H = ε µε( ωω ) E Using εµ 0 0 = 1 c and a standard vector identity, this can be rewritten as ( qeq ) ( qqe ) = [ εωω ( ) / c ] E he first term vanishes because of the transversality condition, leaving q = ε ( ω) ω / c his is the bulk olariton disersion relation for the transverse mode. (.34).6.. Plasmon-olaritons n this case all that we need to do is substitute the lasmon exression for ε (ω ) given in Eq. (.3) into (.34): ω q = ε 1 ( ω / c ) ( ε c )( ω = ω ) ω his gives: ω = ω + ( c ε ) q (.35) ts limiting cases are ω ω for small q (i.e. it is lasmon-like), and ω cq for large q (i.e. it is hoton-like). ε he overall behavior looks like:

14 14 ω ω 0 cq here is a frequency ga (or sto band) for 0< ω < ω. he horizontal dashed line is the lasmon line (ignoring satial disersion); the sloing dashed line is the so-called light line for an uncouled hoton: ω = cq ε.6.. Phonon-olaritons n this case we need to find the form of the dielectric function ε (ω ) that corresonds to honons. We will work this out for otic honons in an ionic solid (such as those having the NaCl structure), ignoring satial disersion as in the above case (i.e. we need consider only the honons at q = 0). For simlicity, we restrict attention to an isotroic or cubic medium, in which vector quantities, such as P and E, are arallel. We may then use the lattice dynamics of a 1D diatomic lattice as our starting oint. As in ec..1., we consider an infinite diatomic lattice in which masses m 1 and m alternate. he crystal is assumed to be ionic, so that the two masses are associated with oosite electric charges. he mode that coules to electromagnetic radiation is the longwavelength ( q = 0 ) otic honon at frequency ω. his follows by considering the amlitude ratio U 1 /U that can be derived from earlier results. his ratio is easily shown to be negative for q = 0 so that the two masses move in antihase. ince the masses carry oosite charges, the otic honon therefore carries an oscillating diole moment. We denote by u the relative dislacement of the two tyes of masses. he olarization P will contain a term roortional to u, as well as a term due to the electrical suscetibility: P = ε 0( αu + χe) (.36) For an isotroic medium P, u and E will be arallel. n Eq. (.36) E is the macroscoic electric field, the value found by averaging the local field over many unit cells. Values of the constants of roortionality α and χ deends uon details of the lattice dynamics and electronic structure. With the electric field E and all other field variables assumed roortional to ex( iω t), the equation of motion (ignoring daming) for u is ω u= ωu+ βel oc he first term on the right-hand side is a restoring force that ensures that in the absence of couling to the electric field the mode frequency is ω, and the second is the driving force due to the electric field (with β constant). n order to derive an exression for the dielectric function from E loc

15 15 the above it is necessary to have a relation between and E. he usual assumtion is that they must be simly roortional to one another, and so the last equation can therefore be relaced by an equation involving E rather than (only the constant is different): E loc ω u= ωu+ γe (.37) Equations (.36) and (.37) are readily solved for P: αγ E P= ε 0 + χe (.38) ω ω he defining relations for the dielectric function give D = ε 0E + P = ε 0ε ( ω) E Combining this with Eq. (.38) leads to αγ εω ( ) = (1 + χ) + ω ω t is more conventionally written in the form ω ω εω ( ) = ε 1+ ω ω (.39) where we define ε = 1 + χ, ω = ω + αγ /(1 + χ ) We can identify ε as the high-frequency dielectric constant and ω as the O honon frequency. Notice that the definition makes ε (ω ) vanish as required at the frequency ω. Also ω > ω. he behavior of ε (ω ) is lotted below, taking ε = 1 and ω ω =. t is negative for E loc ω < ω < ω, imlying that this is a sto band for otical roagation. he disersion relation for honon-olaritons is obtained by substituting Eq. (.39) into (.34): ω ω ω ε ω ω ω q = ε 1+ or q = ω ω c c ω ω (.40) he second form above can be rewritten as a quadratic exression in ω, which can then be solved for ω as a function of q. he quadratic equation gives rise to two branches for the disersion relation. ome asymtotic forms of Eq. (.40) are

16 16 q q q ~ ε(0) ω / c for ω << ω for ω ω < 0 (i.e. no solution) for ω < ω < ω (.41) q = 0 for ω = ω q ~ εω / c for ω he overall behavior is illustrated below, showing the sto band for ω < ω < ω. ω ω ω 0 cq he disersion curve may be seen as resulting from the crossing of the honon line ω = ω and the hoton curve q = ε 1 / ω / c, where ε is imagined to change slowly with increasing frequency from ε (0) to ε. hese modes interact strongly, and the crossover is therefore eliminated with reulsion of the curves. hus the full disersion curve describes a mode of mixed honon-hoton character..6.3 Excitons and exciton-olaritons Excitons are bound electron-hole airs occurring in semiconductors. An examle occurs in a semiconductors such as GaAs or Zne, where an exciton can be formed from an electron near the bottom of the conduction band and a hole near the to of the valence band. chematically:- Energy e Conduction band Band ga E g h Valence band

17 17 f the electron (e) and hole (h) are weakly bound (by the screened Coulomb interaction), they may be several atomic sacings aart, forming a large exciton (like a H-atom but the masses are different). We restrict attention to the simlest model, in which an electron of effective mass the Coulomb interaction to a hole of effective mass wavevector q its energy may be reresented as ω( q) = ω + ( q / M) e m h m e is bound by. f the exciton moves as a wave with (.4) where M = ( me + mh) is the total mass entering into the kinetic energy term, and ω e = E g E0 is the energy of the exciton at rest (the band-ga energy minus the exciton binding energy denoted by E 0 ). he exciton can coule strongly to light and its contribution to the dielectric function is found to be ε( q, ω) = ε + (.43) ω + Dq ω e where constant D 1/ M, is a diole strength of the exciton resonance, and ε is the background dielectric constant. he exciton-olariton is the most imortant case in which satial disersion of the dielectric function must be taken into account, i.e. we cannot ignore the q- deendence as in revious examles. he exciton-olariton disersion relation is then found from Eq. (.43) as before. t has two main branches, like in the honon-olariton case, but the shaes are different because of the extra q terms. An imortant ractical difference is that exciton-olaritons occur near the semiconductor ga frequency, tyically in the visible or near infrared region, whereas honon-olaritons occur in the far infrared. E g M.G. Cottam, 005