SECTION 5: SURFACE EXCITATIONS IN THIN FILMS M.G. Cottam, 2005

Transcription

1 SECTION 5: SURFACE EXCITATIONS IN THIN FIMS M.G. Cottam, 005 We now consider the dynamical properties of surfaces by studying the surface ecitations in films, as well as modifications to the bulk ecitations that occurred in infinite materials. This will include the surface counterparts of the ecitations considered beforehand, e.g., phonons, plasmons, polaritons, spin waves, etc. Assuming a film geometry, with flat smooth surfaces parallel to each other, provides a simple case. By letting the film thickness tend to infinity we include the case of a semi-infinite material, where only one surface needs to be taken into account. The approach follows that in Cottam and Tilley (CT), parts of chapters 3 to 6. We start the study of surfaces with the vibrational waves (phonons) that can propagate parallel to the surface of an elastic solid and along an elastic film. These can be described either by the discrete model of lattice dynamics (masses and springs) or by the continuum model of elasticity theory. The corresponding results for a bulk (infinite) medium were given in sections. and attice dynamics It turns out that the simplest eample of a mode that is localised at a surface occurs for the D diatomic lattice. The arrangement of interest is shown below, which is like the case in subsection.., ecept that the crystal terminates at one end on a mass m. m m m 0 C C C 3 4 z The bulk modes in the infinite case were discussed beforehand and the dispersion graph was found to be like: ω Cm ( + m ) C/ m C/ m 0 π /a q

2 / / We note in particular that there was a stop band ( C / m ) < ω < (C / m), assuming m > m, in which there were no real solutions for ω. We now show that this is where a surface mode can occur. All the masses, ecept the end one, have the equation of motion (.8) or (.9): dun+ m = C( u n+ + un n+ dt (.8) dun m = C( u n+ + un u dt n (.9) For the end mass (labeled 0) the equivalent is d u0 m ( ) = C u u0 dt (5.) Suppose we now substitute the epressions (.0) and (.) as trial solutions into the equations of motion (.8), (.9) and (5.): u n Uep{ i[(n ) qa + ωt]} (.0) un = U ep[ i( nqa ωt)] (.) Previously these came from Bloch s theorem and we knew that q had to be a real wave vector for the bulk ecitation. Now Bloch s theorem does not apply, and we have the possibility of finding (decaying) surface ecitations if there are solutions with q comple. It is easy to see that all the equations ecept (5.) are satisfied provided the ratio has the same form as previously in the bulk system. For eample, using (.8) and (.9) we get U Ccos( qa) C mω = = U C mω Ccos( qa) (5.) The etra condition for (5.) to hold is mω U = C[ U ep( iqa) U ] (5.3) which gives another epression for. Upon eliminating this becomes ( C mω )cos( qa) = (C mω )ep( iqa) The last two parts of (5.) can be rearranged as (5.4) 4C sin ( qa) = 4 C( m + m ) ω mm ω (5.5) This last equation can be solved for ω in terms of q to get the previous bulk dispersion relation: 4sin ( qa) ω = C + ± C + (.3) m m m m mm However, we have to remember now that q is not necessarily real (because of the surface). In fact, it is obvious that (5.3) cannot be satisfied for real q (if ω is real). There might still be a solution for comple q. It is easy to show from (5.5) that in the stop band we have sin (qa) > and above the optic modes we have sin (qa) < 0, both of which are impossible for real q. This gives us the following behaviour in the different ranges of frequency: /

3 3 ω < Frequency range C/ m ( C/ m ) < ω < ( C/ m ) / / / / Form of q q real ( C/ m ) < ω < [ C( m + m ) / mm ] q real ω > [ Cm ( + m) / mm] q = iy On eliminating the / ep(iqa) q = (π/a) + iy terms, by using for eample (5.4) and (5.5), we get ω ( m ) mω Cm Cm = 0 (5.6) The first solution gives ω = 0, which corresponds to q = 0 with all the equal (so it is just a uniform translation of the lattice). The second solution can be written as ω = ±ω s, where ω S = Cm ( + m ) (5.7) / / This frequency lies in the stop band, i.e. it satisfies ( C / m ) < ω < (C / m). To prove that it is a surface mode, we need to show that it decays with distance from the surface. In this frequency range we have ep( iqa) ep{ i[( π a) + iy] a} i ep( ya) so we need y > 0 for a decaying ( surface ) solution. Substituting this and (5.6) into (5.5) leads to sinh ( ya) = ( m m ) / 4mm (5.8) which does have a solution with y > 0 for the decay factor. The surface mode we have found occurs when the lighter mass is the end atom and a repetition of the calculation shows that there is no surface mode when the system terminates at the heavier mass m. It can be seen also that there is no surface mode in a monatomic crystal. In fact, the diatomic crystal becomes monatomic if the masses are put equal, m = m. In that case, it is seen from (5.8) that y = 0, so that the wave vector q is purely real and the mode at frequency ω s is not localised at the surface. In terms of the bulk dispersion curve, there is no stop band at q = π / a for m = m, and the surface mode degenerates into one of the bulk modes. m u n Etensions of the theory: One possible etension in D is to consider the corresponding system which also terminates at the right-hand end, and consists of a chain of N masses in all. Some conclusions are: There must be N normal modes in total. If N is an odd integer we can have the situation where the lighter mass m is at each end. Then there are surface modes in general, and the remaining N modes are bulk modes (which in fact become quantized or discrete, because only certain real values of q are allowed as a consequence of the etra boundary condition). If N is an even integer, the lighter mass m will be at one end only, so there is surface mode along with N discrete bulk modes.

4 4 As N, the spacing in frequency between the quantized modes becomes very small, and in the limit they become the bulk modes. Another possible etension is to 3D, but this is complicated algebraically. However, it would allow us to consider cases for which the wave vector in the surface plane is nonzero (whereas the D models obviously has no variation in the surface plane, and it applies for q = 0 ). Instead, it is easier to use elasticity theory to etend the results (for small wave vector) to 3D. We consider this first before considering eperimental studies of surface phonons. 5.. Surface elastic waves We first recall some results from Section. for bulk elastic waves in an isotropic material. We defined the strain u ij by u u i j uij = + j i and the stress σ by ij F = σ nˆ df = σ nˆ df i ik k ik k k q (.4) (.5) for the component i of the force on a surface of area df whose normal is the unit vector ˆn. The generalised Hooke s aw is σij = λijklukl = λijklukl (.6) kl, which introduces the elasticity tensor λ ijkl. For isotropic media, there are just two independent components of λ ijkl, conventionally written in terms of Young s modulus E and Poisson s ratio σ. The stress-strain relations are E σ = [( σ) u + σ( uyy + uzz )] (.7) ( + σ)( σ) E σ y = u y (.8) + σ with the analogous relations for σ yy, σ yz, etc. A longitudinal wave satisfies u =0, since for a plane wave this implies that the displacement is parallel to the propagation vector q. The wave equation is u u = v u t (.) where velocity v is given by eq. (.). Similarly, a transverse wave satisfies u T =0, which implies that ut is perpendicular to q. There is a similar wave equation but with a different velocity v T given by eq (.3). The ratio of velocities is v ( σ ) = v ( σ ) (5.9) T

5 Reflection of acoustic waves at a free surface As a preliminary to surface waves, we first consider the reflection of bulk acoustic waves at a free surface. We take the elastic medium to be in the half-space z < 0, with the surface as the plane z = 0. The surface plane is assumed to be stress-free, which from the definition of the stress tensor means σ σ = σ = 0. With the use of the relations (.7) and (.8) these may be written in z = yz zz terms of the strain as u = 0 (5.0) z u = 0 (5.) yz σ ( u u ) + ( σ ) u = 0 (5.) + yy zz Consider now a wave incident on the surface at an angle θ to the normal. The propagation vector q and the normal to the plane define the plane of incidence, which we take as the z plane (see the figure). Reflection in s-polarisation (left) and p-polarisation (right) To begin with, we assume that the wave is transverse, with displacement u in the y direction; this is sometimes referred to as s-polarisation. The incident wave is u = u 0 ep[ i( q + qz z ωt)] (5.3) where q / q z = tanθ, q + qz = ω / vt (5.4) with = (0, a, 0). The reflected wave is u 0 u = u ep[ i( q q z z ωt)] (5.5) with u = (0, b, 0). The same q appears in (5.3) and (5.5) because the boundary condition (5.) must be satisfied for all. The fact that the values are the same then follows from (5.4), implying that the angles of incidence and reflection are the same, as indicated in the figure. Finally, application of the boundary condition (5.) shows that b = a for the amplitudes. It becomes more complicated if we consider an incident transverse wave polarised in the z plane, known as p-polarisation. Equations (5.3) and (5.4) still describe the incident wave, but now u = ( a,0, atan θ ). We might attempt to describe the reflection by means of a reflected 0 transverse wave given by (5.5) with u = ( b, 0, btan θ ). However, this introduces only one amplitude b, whereas there are two boundary conditions, (5.0) and (5.), to satisfy. This problem is resolved by observing that, as sketched on the right of the figure, a reflected q z

6 6 longitudinal wave is generated in addition to the reflected transverse wave. For the boundary conditions to be satisfied for all, this wave must have the same q value as the incident wave. It is therefore described by u = u ep[ i( q q z ωt)] (5.6) with q / = tanθ, q + q = ω v (5.7) q / and u = ( c, 0, ccot θ ). Since v > v T, the above equations imply that θ > θt, as drawn in the right figure. In fact sin θ v = (5.8) sin θ vt With the introduction of the longitudinal wave with independent amplitude c, it becomes possible to satisfy the two boundary conditions and solve for the reflection coefficients b/a and c/a. One consequence of (5.8) should be noted. If the angle of incidence θ is larger than a value θ c, defined by sin θ = v / v, then (5.8) would imply that sin >. The meaning of this can be c T θ c seen from (5.7). For θ >, q > ω v, so (5.7) can be satisfied only if is negative. We / put q = iκ and the longitudinal wave in (5.6) becomes u = u ep[ i( q ωt)]ep( κ z) (5.9) This is a wave travelling along the surface, with amplitude decaying eponentially with distance into the medium. It is accompanied by the reflected transverse wave, and the combination is called a pseudo-surface wave. We have discussed incident transverse waves of either polarisation. A similar calculation can be done for longitudinal waves. In particular, an incident longitudinal wave generates reflected longitudinal and p-transverse waves. θ q 5.. Rayleigh waves We now show that the free surface of an isotropic elastic medium supports a surface mode known as a Rayleigh wave. It results from a miture of waves with p-transverse and longitudinal polarizations and its properties follow from etension of results given in Subsections 5... We look for a solution of the and T wave equation, as in (.), together with the boundary conditions (5.0) and (5.). The total displacement is the sum of longitudinal and transverse parts, u = u + u T, each part satisfying the wave equation with the appropriate velocity. To find a surface mode, we draw on the form of the pseudo-surface wave and assume both parts are localized at the surface: u ep[ iq ( ωt)] ep( κ z) (5.0) T,, T Since the sample occupies the half space z < 0, the real eponential factors with κ 0 describe, T > decay of the amplitude away from the surface. The comple eponential term is in accordance with the D form of Bloch s theorem and corresponds to propagation parallel to the surface plane

7 7 along the direction. We must assume that both the and T components have the same q so that the boundary conditions can be satisfied for all values of. Since the components separately satisfy the corresponding wave equation, the constants κ and κ T are defined by / κ = q ω / v (5.), T (, T ) The and z components of u and ut are related through the equations u = 0 and u T =0, which imply iqu z κ u = 0 and iqut + κ T utz = 0 These can be satisfied by introducing two amplitude factors a and b: u = aq (, 0, iκ ) ep[ iq ( ωt)] ep( κ z) u T = b( κt, 0, iq) ep[ i( q ωt)] ep( κ Tz) With these forms substituted, the first boundary condition, (5.0), gives κ q a + ( q + κ T ) b = 0 The other boundary condition, (5.), yields v κ q b + [ v ( κ q ) + v q ] a = 0 T T T where we have used (5.9) to rewrite σ in terms of the velocity ratio. The eliminated in favour of κ T T κ T q b + ( q + κ ) a = 0 with the help of (5.) to give Finally, (5.) and (5.3) give two independent epressions for the ratio obtain the dispersion equation for the Rayleigh surface wave: κ κ q = ( q + κ ) 4 T T κ κ T 4 4 6q ( q ω / v )( q ω / vt ) = (q ω / vt ) q Substituting for and we find the more eplicit form This looks like a complicated result connecting ω and and denote dimensionless ξ = ω vq, we have 8 q T 4 6( ξ )[ ( v vt) ξ ] = ( ξ ) This just gives a number when solved, and so ω is proportional to κ (5.) term can be (5.3) a / b. Equating these, we (5.4). However, if we divide throughout by q for the wave: ω = ξv q (5.5) T The number ξ is determined by the ratio of velocities, so in view of (5.9) it depends on Poisson s ratio σ. The dependence on σ (which can range from 0 to 0.5) is actually very weak. In fact, ξ is always close to the value 0.9, to within a few percent. Equation (5.5) shows that the Rayleigh wave is like an ordinary acoustic wave, but with a velocity ξ vt that is slower than either of the bulk velocities v and vt. It is sometimes convenient to represent the dispersion relation as in the figure below.

8 8 Dispersion relation for Rayleigh wave This answers the question: What modes propagate for a given value of the in-plane wave vector? The lowest frequency mode is the Rayleigh wave. At a somewhat higher frequency T bulk q modes given by ω = v q + q ) / can occur. However, on the diagram only q is given and T ( z can take any value. Thus the transverse modes occupy a bulk continuum defined by ω v q. At T higher frequencies still is the longitudinal bulk continuum, ω v q. These bulk continuum regions are shaded regions in the figure Other surface elastic waves The Rayleigh wave is actually the easiest eample of a surface elastic wave to calculation, and it is also the easiest to detect eperimentally (see later). It occurs in p-polarisation, and there are no s- polarised surface modes for a semi-infinite medium. However, for a finite-thickness film, other possibilities arise. The Rayleigh wave calculation can be generalized, leading to a surface wave at each surface. Also it is found that s-polarised waves can occur: these are usually called amb waves. When there is an interface between two different isotropic media, other possibilities arise because the boundary is usually no longer stress free. The simplest waves in this kind of geometry are called Stoneley waves and ove waves for p-polarisation and s-polarisation, respectively Eperiments for surface phonons and surface elastic waves The most relevant eperimental techniques are: inelastic light scattering (usually Brillouin scattering); inelastic helium atom scattering; and inelastic electron scattering. A) Brillouin scattering qz

9 9 This is a sensitive method for detecting surface waves under conditions where the scattering medium is relatively opaque to the light (so that it penetrates the medium only in the surface region). This is usually the case in metals and semiconductors. When there is a surface, the conservation conditions discussed in Section, eq. (.5), for the bulk case become modified to ω ω ± ω k = k ± q (5.6) I = S I, S, where the upper and lower signs again refer to the Stokes (phonon creation with ω < ω ) and anti-stokes (phonon absorption with ω S > ω I ) scattering respectively. The energy conservation result is the same as before. However, in the momentum result, the conservation property only applies to the components parallel to the surface. A surface mode is characterized only by a D wave vector (and a decay factor), so it appears as a sharp peak in the spectrum. A bulk mode has 3 components of wave vector and the 3 rd component (perpendicular to the surface is not fied). Therefore a bulk mode should appear broadened in the spectrum. This is illustrated in the eperimental data below (R = Rayleigh, T = transverse acoustic, = longitudinal acoustic). S I Brillouin scattering from phonons at a Si (00) surface (Sandercock 978). B) Helium atom scattering Inelastic neutron scattering is not useful in general for surface studies, because neutrons are highly penetrative. By contrast, if neutral atoms of He are used instead, they scatter from just the one or two first layers at the surface. The technique was pioneered in the 980 s using low energy beams (~ 0 mev). As with neutron scattering, it is applicable to all wave vectors in the (D) Brillouin zone. Studies can be made of both acoustic and optic phonon branches. An eample of some measurements is shown below, along with some theory curves, for scattering from the surface of a metal.

10 0 He atom scattering from the surface and bulk phonons for a Ag () surface (Doak et al 983). C) Electron scattering This was described in Section for the bulk case. Using electron energies of a few 00 ev, it has been developed as a surface sensitive technique (since the 980 s). An eample is given below for TaC and compared with theory (a 3D generalization of the diatomic chain calculation). Electron scattering from a surface phonon in the stop band of TaC (00) (Oshima et al 984) Surface polaritons Polaritons in the bulk case (for a nonmagnetic material) were dealt with in Section.6 with phonon-polaritons and plasmon-polaritons as the main eamples. The main result is that the bulk dispersion relation is given by

11 q = ε ( ω) ω / c (.34) This was obtained from Mawell s equations (leading to a wave equation for E) and the transversality condition that q E =0. The dielectric function ε ( ω ) contains the characteristic frequency dependence for the material, usually in the approimation that spatial dispersion can be ignored (i.e. there is no q dependence in the dielectric function) Dispersion relation for a single-interface polariton As the simplest eample, we consider the plane interface between two semi-infinite dielectric media and, as shown in the figure. We take the z ais normal to the interface and the ais as the direction of propagation of the mode. z Medium ε y Medium ε Calculations show that there is no surface mode when E is in the y direction (i.e. in s-polarization), so we take E in the z plane (p-polarization): E = ( E, 0, E )ep( iq iωt)ep( iq z) (5.7) z z in medium, with a similar form in medium. We take the frequency as ω and, since boundary conditions will be applied on the whole plane z = 0, the wave vector component must be the same in both media. Our aim is to find the dispersion equation ω ( q ). For the mode to be localized, E must decrease in magnitude with distance from the interface in both directions. Thus we have Im( q z ) > 0 and Im( q z ) < 0. In order for (5.7) to satisfy the electromagnetic wave equation (Mawell s equations) in both media, the wave vectors must satisfy q + q = ε ω / c i =, (5.8) iz i Thus, when ε and ε are real, the localization requirement is q > εiω / c. The equation D = 0 gives the ratios of the field amplitudes in (5.7): qei + qizeiz = 0 i =, (5.9) The amplitudes in the two media are related by the usual electromagnetic boundary conditions, i.e. continuity of the tangential component of E and the normal component of D: E (5.30) E = q

12 ε Ez = ε Ez (5.3) Equations (5.9) (5.3) are four homogeneous equations in the four field amplitudes, and the solvability condition is the required dispersion relation. Equations (5.9) and (5.30) give in terms of E, and then substitution in (5.3) gives qz ε = qz ε (5.3) Substitution from (6.) leads to the eplicit result for the surface polariton dispersion equation: ω εε q = (5.33) c ε+ ε We can also deduce some etra necessary conditions. Provided damping is neglected, ε and ε are both real. From (5.8) and the localisation requirement it then follows that q z and q z are both pure imaginary, and so we write q z = iκ and q z = iκ, where the signs incorporate the localisation condition. Equation (5.3) now implies that for a surface polariton to eist, ε and ε must have opposite signs: ε ε < 0 (5.34) Since the right-hand side of (5.33) must be positive, we have the further condition that ε + ε 0 (5.35) < Equations (5.34) and (5.35) can be used to determine the frequency intervals in which the surface polaritons may occur. In many eperimental situations, one of the dielectric functions is frequency-independent and positive, as is the case if medium, say, is air or vacuum. In that case, (5.34) and (5.35) show that the other dielectric constant, ε say, has to be negative and satisfies ε <. Medium is then called the surface-active medium. ε E iz The electrostatic limit In addition to the general result given in (5.33) that follows from the full form of Mawell s equations (with retardation), it is interesting to consider the electrostatic limit. Just as in the magnetostatic limit mentioned in Sec. 3, this corresponding to ignoring retardation by formally taking the limit c. Equation (5.33) can be rearranged as ε + ε q ω = εε c The right side vanishes in the above limit, and so this implies as the limiting result of the dispersion relation ε+ ε = 0 (5.36) The solution of this equation gives some fied frequency (or frequencies), independent of q. The above result can also be derived directly using the electrostatic scalar potential V, which is defined by E = V. We also have E = 0 and so, if media and are described by potentials and V they must satisfy V

13 3 V i = 0 i =, (5.37) A plane wave localised at the interface and travelling in the direction is described by V = V0 ep( κ z)ep[ i( q ωt)] (5.38) with a similar epression for. When these are substituted into (6.7), we obtain the result that κ = i q V for i =,. Thus the potentials are Vi = Vi0 ep( q z)ep[ i( q iωt)] (5.39) with the upper (negative) sign for medium and the lower sign for medium (to give localization in both cases). The corresponding fields are E i = ( iq,0, ± q) Vi (5.40) with the upper sign again corresponding to medium. Equations (5.39) and (5.40) have the property that the decay constant normal to the interface is equal in magnitude to the wavevector along the interface. It is seen from (5.40) that the field components E and E in either medium are equal in magnitude but 0 90 out of phase. In order to derive the frequency of the surface wave, we apply standard boundary conditions to the fields given by (5.40). Continuity of gives V 0 = V 0 and then continuity of D z gives εqv0 = ε qv0 Taken together, these two equations lead to the previously derived (5.36). E z Eample. Medium is vacuum and medium is an electron plasma (semiconductor or metal). ω p ε = and ε( ω) = ε ω Substituting into (5.36) and solving for the frequency gives ω = ω where ωs = + ωp = ωp if ε = (5.4) ε This surface plasmon mode occurs in the stop band below the plasma frequency: ω S ω p

14 4 0 cq Eample. Medium is vacuum and medium is a polar medium (with optic phonons) ω ω T ε = and ε( ω) = ε + ωt ω In this case the solution (representing a surface optic phonon) corresponds to ε ω + ωt ωs = (5.4) ε + Again this represents a frequency in the stop band, which occurs for ωt < ω < ω: ω ω ω T 0 cq Surface plasmon-polaritons We take the choice for a vacuum/plasma system as in eample and substitute the dielectric functions into the general dispersion relation (5.33). We find a surface plasmon-polariton branch that etends from ω = 0 to ω = ωs as a function of q. The result is as shown below (taking the case of ε = ).

15 5 The bulk continuum is shown shaded Surface phonon-polaritons We take the choice for a vacuum/polar-dielectric system as in eample and substitute the appropriate dielectric functions into the general dispersion relation (5.33). We find a surface phonon-polariton branch that etends from ω = ωt to ω = ωs as a function of q. The result is as shown below (taking the parameters appropriate to GaAs): Notice in this case that the surface branch occurs only to the right of the light line (it starts at some nonzero value of q ) Eperimental methods for surface polaritons The two most important general methods involve (A) inelastic light scattering (usually Raman scattering) and (B) attenuated total reflection. A) Raman scattering

16 6 The method and the modified form of the conservation laws were described previously. The technique has been successfully used to verify the predicted results for both the surface plasmonpolaritons and surface phonon polaritons. An eample is given below for a GaP sample in a backscattering geometry (where incident angle θ controls the in-plane wave vector). Raman scattering from GaP (Denisov et al 987) B) Attenuated total reflection A basic property of the surface polariton discussed earlier is that it is required to satisfy q > εω / c where ε is the dielectric constant of (eternal) medium, occupying the half space z > 0. This means that the surface mode cannot be ecited by a light beam incident in the normal way in medium, since such a beam satisfies q + qz = εω / c (5.43) with q z real. For this reason the surface polariton is called nonradiative. The restriction to real q z is removed in the method of attenuated total reflection (ATR), which was introduced in the mid-970 s. Three different ways of coupling an incident beam in medium to nonradiative modes are illustrated in the figure below.

17 7 Techniques for coupling light to surface polaritons In the grating method (figure a), a diffraction grating is laid down on the surface of interest. The / incident light has surface wavevector ε ( ω / c) sinθ I. By Bloch s theorem, this couples to modes with / q = ε ( ω / c)sinθ I + mπ / d (5.44) where d is the grating periodic distance and m is an integer. A disadvantage is that the application of a grating perturbs the modes being studied. This problem is overcome using attenuated total reflection (ATR). Two different versions are sketched in figures b and c. Both methods involve a three-layer geometry and in both the light is incident in a prism of dielectric constant ε. In the Otto configuration (figure b), a spacing medium, typically vacuum, is adjacent to the prism, with the surface-active medium as medium 3. The prism is chosen with a relatively large refractive inde, ε > ε, and the angle of incidence θ I is larger than the critical angle θ for total internal reflection at the - interface, / / C θ I > θ C = sin ( ε / ε ). Thus if medium 3 were absent, the incident light would be totally reflected. However, total reflection involves the presence of an evanescent mode with decreasing amplitude (imaginary q z ) in medium. The tail of this evanescent mode can ecite a surface mode on the -3 interface; this mode removes energy and thus gives a reduction (attenuation) of the total reflection. More concisely, the in-plane wavevector component / q = ε ( ω / c) sinθ (5.45) I is the same in all three media, and with sufficiently large ε and θ I it can be made to satisfy the condition / q > ε ω / c (5.46)

18 8 necessary for ecitation of a non-radiative surface mode at the -3 interface. Then when the incident values of ω and q lie on the dispersion curve of a surface mode the reflectivity is reduced below unity. The main technical problem is the control and uniformity of the spacer thickness, i.e. the thickness of medium. In the alternative Raether-Kretschmann configuration (figure c), the surface-active medium is deposited direct on the surface of the prism. Thus the method is most readily applicable when the material of medium is easily evaporated, and the configuration has been mostly used to study surface plasmon-polaritons on metals such as Al and Ag. In this case ε is negative. As in the Otto configuration, the reflectivity is reduced when ω and lie on the dispersion curve. q ATR for GaP (Marschall and Fischer 97) In the ATR spectra above, the sample is GaP, and the Otto configuration is used with a Si prism and air as the gap. k0 = ω c Surface magnetic modes The bulk magnetic modes (spin waves) were discussed previously in Section 3. Recall that different cases occurred depending on whether the echange interactions or the dipolar interactions are dominant. In turn, this depends on the wave vector regime of interest (see the table in Sec. 3.). Here, for simplicity, we will consider surface spin waves in just one case: the magnetostatic case, where echange effects are negligible compared with the dipolar effects. We need to recall some results obtained when considering the infinite medium in Section 3. The relevant Mawell s equations for the fluctuating fields were: h ( r) = 0, [ hr ( ) + mr ( )] = 0 (3.7) The magnetostatic scalar potential ψ was introduced with the definition that h = ψ (3.8) From the second of Mawell s equations above, this led to

19 9 ( ψ ψ ψ + χ a ) + + = 0 (3.9) y z where χ a is one of the susceptibility components obtained from χ a = ω m ω0 /( ω0 ω ), χ b = ω m ω /( ω 0 ω ) (3.3) We have defined the following quantities that have the dimension of angular frequencies: ω 0 = γb 0, ω m = γµ M 0 0 (3.4) All of this, together with the use of Bloch s theorem in 3D gave a bulk dispersion relation / ω( q) = [ ω ( ω + ω sin θ)] (3.3) 0 0 m where θ denotes the angle between q and the magnetization direction. The frequencies eist in a continuous band corresponding to / ω ω( q) [ ω ( ω + ω m )] For the surface case, we start with a thick material (so there is just one surface to consider) and then generalize to a film of finite thickness Magnetostatic modes in a semi-infinite ferromagnet The results depend on the direction of the magnetization vector relative to the surface. The usual configuration for surface magnetostatic modes is to have the static magnetisation M 0 parallel to the surfaces. We assume a geometry as in the figure below (with thickness infinite at present). The z ais is chosen to be along and the ais is perpendicular to the surface. The D inplane wave vector q y z M 0 has components given by q = ( q, q ) = q (sin φ,cos φ) (5.47) where q = q + q ) and φ is the angle between and M. The theory for this geometry was ( y z q 0 first presented by Damon and Eshbach (96). We need to solve for the magnetostatic potential ψ for the regions outside ( > 0) and inside ( < 0) the ferromagnet. Eq. (3.9) applies inside, while for the region outside it simplifies to ψ ψ ψ + + = 0 (5.48) y z

20 0 From the property of translational invariance in the y and z directions it follows that ψ (r) must be of the form ψ ( )ep( i q r ), where r = ( yz, ). For (5.48) to be satisfied we have ψ q ψ = 0 and the solution that vanishes at = is ψ ( ) = a ep( q ) > 0 (5.49) For (3.9) for < 0 to be satisfied we have ψ ( + χa) q yψ qzψ = 0 Its general solution is of the form ψ ( ) = a ep( iq ) + a ep( iq ) < 0 (5.50) 3 q where the quantity ( + χa)( q + q ) χaqz, which can be real or imaginary, must satisfy = 0 (5.5) The amplitude coefficients a j (j =,, 3) could be determined from standard electromagnetic boundary conditions at = 0. In terms of the scalar potential these boundary conditions are (i) that ψ must be continuous across a boundary, and (ii) that ( h + m ) inside the ferromagnet at a boundary must be equal to h outside. First, if q is real we have a bulk wave in the ferromagnet, and rearranging (5.5) gives q y χa q q = with χ a = ( ω0 ω )/ ωmω0 ( + χa ) The solution is ω =± ωb( q, q ) where / q + q y ωb( q, q ) = ω0 + ω0ωm q + q As epected, this is the same as the result in Eq. (3.3) for an infinite ferromagnet. The solutions 0 for the mode frequencies are particularly simple when q z = 0, i.e., angle φ = 90 for the propagation direction. This is referred to as the Voigt configuration. The above result simplifies to ω B ( q, q ) = [ ω0( ω0 + ωm)] (5.5) In the Voigt configuration, (5.5) simplifies to ( + χ )( q + q ) = 0 (5.53 a This has two solutions: if the first term vanishes, we have χ a = which gives the bulk solution above, whereas if the second vanishes we have imaginary q (a surface mode) with q =±q i. If we (arbitrarily) choose the sign and substitute into the solution for the magnetostatic potential we get ψ ( ) = a ep( q ) < 0

21 together with a 3 = 0. We now apply the boundary conditions at = 0, it is easy to show that this leads to a surface-mode frequency ωs ( q ) given by ω ( q ) = ω + ω (5.54) S 0 m In this special case of the Voigt configuration, both ωb( q, q ) and ωs ( q ) are independent of wave vector and they satisfy the inequality ω ( q, q ) < ω ( q ) B S Choosing the other sign for q does not lead to any surface mode Magnetostatic modes in a ferromagnetic film We now consider the same geometry as before but we assume a finite film thickness. There are two surfaces corresponding to = 0 and =. The general solution for the scalar potential becomes a ep( q ) > 0 ψ ( ) = a ep( iq) + a3 ep( iq) 0 > > (5.55) a4 ep( q ) < where q again satisfies (5.5). The 4 amplitude coefficients a j can now be determined by applying the standard electromagnetic boundary conditions, as stated before. There are boundary conditions at each of surfaces, so they lead to four homogeneous linear equations for the four coefficients, and the condition for a solution is found to be q q q ( + χ )cot( q ) q ( + χ ) q χ 0 (5.56) When modes. q + a a y b = is substituted from (5.5), the above equation determines the dispersion relations of the 0 We once more consider the Voigt geometry ( q z = 0 or φ = 90 ) to simplify the calculation. This means that q satisfies (5.53). As before, one term gives the bulk modes with the same dispersion relation as in (5.5). The other term gives q = ±iq, and both of these possibilities now correspond to surface modes in the film. The surface solutions have a common frequency, which is found by substituting for q in (5.56). On using the epression for the susceptibility component, the result can be rearranged as ω ( q ) = [( ω + ω ) ω ep( q (5.57) S 0 m 4 m )] It is important to note that the mode solutions, as given in terms of the amplitude coefficients in (5.55), are different for the two surface states. If it can be shown that the two cases of q = iq and q = iq correspond to surface states localized near the lower surface ( = ) and the upper surface ( = 0 ), respectively, and vice versa if q < 0. This property is an aspect of non-reciprocal propagation, which we shall discuss later. It follows from the dispersion relations q y > 0 y a j

22 that ω S is always greater than ω B (just as in the semi-infinite case), and the limiting surface mode frequencies are [ ω0( ω0 + ωm)] ( q << ) ω S ( q ) = (5.58) ( ω0 + ωm) ( q >> ) The latter case corresponds to the surface mode for a semi-infinite ferromagnet. The behaviour of ω and ω as functions of q is sketched in the figure (taking ω ω 0 = 6 ). S B m In general, the term non-reciprocal propagation means that the mode frequency is changed when the wave vector is reversed. For a surface mode it implies that ω q ) ω ( ) S ( S q and we have found that this is the case for the surface magnetostatic modes. Indeed no surface mode is possible, localised at that same surface, when is reversed. The behaviour on reversing q is illustrated in the figure below for the case of the Voigt configuration. q We suppose that for one direction of q there is a surface mode localised at the lower surface of the slab, as in (a). Then if is reversed in direction, keeping and M fied, the new surface q B0 0 mode is localised near the upper surface, as in (b). The two modes are degenerate in frequency and have the same attenuation length q.

23 3 The above results for the eistence of surface waves and their dispersion relation applied for the special case of the Voigt geometry (with q z = 0 or φ = 90 0 ). The plot below shows what happens if q z 0. In particular, it can be shown that the surface waves eist only for φ in a certain range: φc < φ < π φ c where sinφ c = ω / 0 ω0 + ωm (5.59) Eperimental studies of surface magnetostatic modes The first eperiments were with microwaves in thin films (thickness ~ 0.5 mm) and other dimensions of several mm, as illustrated. A surface wave is launched from one end of the film. It travels on one surface (say, the top) a distance d to the other end and then back along the lower surface. The path length is approimately d (assuming << d). In the main eperiment the conditions were varied until there was constructive interference with the signal being emitted at the source. This occurs when the wavelength λ is such that d = nλ, where n is an integer. Thus there is a series of discrete wave vector values corresponding to ( n) q = nπ d (5.60) /

24 4 By varying the source frequency ω to match with ω S at the constructive resonances, it was possible to verify the dispersion relation for the surface modes. In practice, it is easier to keep ω fied but vary the field B 0. The accuracy of this type of eperiment was improved later by using thinner samples. Brundle and Freedman (968). Much more convincing (and accurate) eperiments were later performed using Brillouin light scattering. These allowed the property of non-reciprocal propagation to be clearly demonstrated. A Brillouin spectrum on a thick (semi-infinite) sample of the ferromagnet EuO is shown in the net figure. Surface and bulk spin wave peaks are labeled by S and B respectively. The surface peak appears on one side of the spectrum only, because the light cannot penetrate to the lower surface of the optically absorptive medium. However, this can be switched by reversing the direction of q. The measured frequencies for the bulk and surface modes agree very well with the theoretical results.

25 5 Brillouin scattering (Grunberg and Metawe 977) Another eample of a Brillouin scattering spectrum, due to Sandercock and Wettling (979), is shown below for Fe. The broad peaks at about 7 GHz are the epected spin waves (and are very well described by the theory of magnetostatic modes). The sharper peak at about 40 GHz is a surface magnetic mode. Frequency shift (GHz) The frequencies of the bulk (b) and surface (s) spin waves as a function of applied field B 0 (epressed here in koe units 0. T) are shown below.

26 6 Applied field (koe) Some more results for Fe, but for thin films of varying thickness, are shown below. When the film thickness is reduced to the same order as the optical penetration depth ~ 5 nm in Fe), a surface peak can be seen on both sides of the spectrum. Brillouin scattering (Grunberg et al 98) 5.5. Nonlinear electromagnetic surface waves All of the eamples of waves up to this point have been linear waves, i.e. the equation of motion (usually a finite-difference equation or a differential equation) has been linear in the amplitude factor, such as the displacement vector (for a phonon), a spin component (for a spin wave) or an electric field component (for an electromagnetic wave or a polariton).

27 7 As an eample here, we consider briefly nonlinear optics and discuss how a nonlinear surface mode can eist at an interface under these conditions Brief introduction to nonlinear optics The development of a theoretical framework for nonlinear optics (NO) dates from the invention of the laser in the early 960s. The nonlinear effects are often more pronounced in thin films, superlattices and other nanostructures. The starting point for linear optics is the relation between the field vectors P and E. In terms of wavevector and frequency its usual form is P ( q, ω) = ε 0 γ ( q, ω) E ( q, ω) (5.6) i ij j in which γ ij ( q, ω) is the linear susceptibility tensor. Equation (5.6) allows for spatial dispersion, that is the q dependence, in addition to the frequency dependence of γ ( q, ω). As mentioned before, spatial dispersion is unimportant n most cases and the q = 0 form of the above is adequate. The standard approach to nonlinear optics is to regard (5.6) as the first term in a Taylor epansion, the subsequent terms of which lead to the various nonlinear effects. In making this epansion we should be careful to use the real-variable forms of all the field quantities since, for eample, the product of the real parts of two comple numbers is not equal to the real part of the product. In a typical eperiment, one or more intense laser beams pass through a nonlinear-optic crystal, and in addition a dc field may be applied. Since the laser beams are usually monochromatic, to a good approimation, the electric field is written as E ( t) = E0 + [ E ep( iωt) + E ep( iωt)] (5.6) ω > 0 ω ω Here the condition for E(t) to be real is E ω = E ω, E0 is the dc field, and the sum is over the frequencies of the laser beams and of any output beams that are produced. The induced polarization can written as a sum of terms proportional to ij E, E, E 3 and so on. In the term of n th n order a variety of frequencies appears since for each factor E in the product E any one of the dc term E, the positive-frequency amplitude E and the negative-frequency amplitude E ω ( n) ( t) = 0 + ( ω )ep( ω i t) σ σ n, ωσ 0 can be selected. The full epression for the Taylor epansion is of the form P P P (5.63) where ( n) ( n) ( Pω ) 0 K... ( ;,..., ) ( ) ( )...( ) σ µ = ε χµα α α ω n σ ω ωn Eω α E ω α E ωn α (5.64) n and ω is defined by σ ω σ = ω ω + ω n (5.65) Here each of ω, ω,... ω n can be regarded as positive or negative (or zero) depending on which term in (5.6) is selected, so that ω σ is the frequency of the output beam. The summation convention for tensor suffices applies and K is a combinatorial factor. ω

28 8 The first-order effect in the above epansion is just a description of linear optics. Here we consider the additional effects of the second-order and third-order nonlinear terms. (A) Second-order nonlinear effects The χ () effects arising from taking n = in (5.64) and (5.65) are () () Pω + ω = ε K χµα α ( ( ω + ω ); ω, ω ) Eω E ω (5.66) ( ) 0 ( ) ( ) µ α Various physical effects can arise depending on the combination of the frequency factors. For two incident optical beams of different frequencies the possible effects are sum-frequency generation, E E with ω ω ω σ = ω + ω and difference-frequency corresponding to the product ( ) ( ) α α generation, corresponding to the product ( ω ) ( ω ) α α α E E and ω σ = ω ω. For a single input optical beam of frequency ω, the corresponding effects are second-harmonic generation (or SHG), ω σ = ω + ω = ω, and optical rectification, = ω ω = 0. The second of these describes the () appearance of a static polarization P 0. ω σ SHG can be seen very clearly in eperiments on thin films using intense light beams to increase the nonlinear behavior. The intensity of the SHG fluctuates strongly with respect to the thickness of the nonlinear film and is strongly influenced by the choice of (linear) capping material. Data (eperiment and theory) are shown below for AlP films with GaP capping layers SHG measurements in a film (Hashizume et al 995).

29 9 (B) Third-order nonlinear effects There are two reasons for paying special attention to third-order NO effects. First, it turns out that the second-order effects vanish in centrosymmetric materials, so that third-order effects are the leading terms in such materials. Second, it will be seen that the third-order effect usually known as the nonlinear Kerr effect involves only one frequency and this has important physical consequences. The χ (3) effects arises from taking n = 3 in (5.64) and (5.65). The eplicit form is like that for the second-order NO but with three input frequencies ω, ω and ω 3. Since output frequencies ± ω ± ω ± can all be generated a wide variety of possible effects might be discussed. It will ω 3 be sufficient here to consider only the effects that occur with a single input beam, of frequency ω (3) say. These are third-harmonic generation, arising from the coefficient χ ( 3 ωωωω ;,, ) and (3) the Kerr effect (sometimes called self action), arising from χ ( ω; ω, ω, ω. The first is 3 µα α α 3 µα α α ) similar to second-harmonic generation. As we see below, the Kerr effect can be interpreted in terms of an intensity-dependent contribution to the dielectric function, or equivalently to the refractive inde. In the case of the Kerr effect, the general epression for the nonlinear polarization is (3) (3) * P ε K χ E E E (5.67) i = 0 ijkl j k l This can be added to the linear polarization that the inclusion of function given by eff ε = ε + K χ (3) P i E E (3) * ij ij ijkl k l () i P = ε 0 ε E ij j at the same frequency, so we conclude is equivalent to defining an effective intensity-dependent dielectric (5.68) A useful special case of this is for a linearly polarized wave in an isotropic medium (e.g. glass). If the optical E field lies along it can then be shown that the only non-vanishing tensor component (3) (3) is χ and (5.68) reduces to an intensity-dependent modification of the isotropic (scalar) χ dielectric function ε of the form eff (3) ε = ε + K χ E = ε + η E (5.69) (3) where we introduce η = Kχ to simplify the notation for later use. We will show that the appearance of the nonlinear term (proportional to η) in the above can give rise to a new type of (localized) surface wave that does not eist in a linear material Eample of a surface nonlinear electromagnetic wave It might be epected that, for a nonlinear material with a positive Kerr effect ( η > 0 ), a localized or surface wave will travel along the interface with a linear material of lower dielectric constant. Roughly, this because there will be a surface region in the nonlinear material where the dielectric

30 30 constant (and hence the refractive inde) is larger than on both sides. Therefore a wave traveling almost parallel to the surface will undergo total internal reflection to stay confined within the surface region. This is sometimes called self guiding. We now need to do a proper calculation to find if this is possible and what the conditions are. Since the wave is nonlinear, we epect that the intensity I = E enters the dispersion relation as a parameter, so that more precisely ω = ω( ; I ). We use the aes and notation defined in the figure for the interface between the two media. We assume that the lower medium is nonlinear with dielectric constant ε 0 + η E and that the upper medium is linear with dielectric constant ε. For simplicity, we assume that ε and ε 0 are positive constants. Also we require that η be positive, consistent with self guiding. z q Taking the case of s-polarization, we look for a mode with E = ( 0, E,0) that propagates as a plane wave along the direction. All field quantities are proportional to ep( iq iωt) in accordance with Bloch s theorem. In each medium we know that E satisfies the wave equation εω E+ E c = 0 With these assumptions and with the assumed Kerr nonlinearity in medium, the wave equation in medium is d E ω κ η E E dz c = 0 (5.70) where we define κ = q ε 0( ω c ) (5.7) We restrict attention to solutions with E real so we ignore the modulus signs. If we denote p = de/dz, (5.70) is dp ω κ ηe E dz c = 0 On multiplying by p it becomes

31 3 dp ω de p κ ηe E 0 = dz c dz This can be integrated to give the energy integral de ω 4 κ E 4η E + = k dz c The constant of integration k has to be set equal to zero here because if we assume that the solution is localized at the interface then E 0 and de / dz 0 as z. ω 4 κ E η E 0 de + = dz c (5.7) Equation (5.7) can be integrated in terms of elementary functions, and the result (for positive η) in medium is found to be / c E = κ / sech [ κ( z+ z ) ] (5.73) η ω where is another constant of integration. z In the upper medium, the solution of the wave equation that satisfies the condition E 0 z is simply E = E ep( κ z) (5.74) since medium is linear, where κ = ε ( ω ) (5.75) and E q c is arbitrary. as We now use boundary conditions at z = 0. Since the parallel component of magnetic field H is proportional to de / dz the boundary conditions at z = 0 are that E and de / dz are continuous. Application of these leads to / c κ / sech( κ z ) = E η ω and / c κ sech( κ / z ) tanh( κ z ) = κ E η ω Dividing the above two equations eliminates E to give κ tanh( κ z = κ (5.76) ) We seek the dependence of q on ω. For given ω, enters (5.76) through the definitions for κ q z so it does not provide the and κ. However, (5.76) also contains the second unknown complete solution. The reason is, as anticipated, that the intensity is a nontrivial parameter in this problem.

32 3 We can relate the electric field intensity to the power flow in the system, and therefore we evaluate this quantity, which is found as the time average of the Poynting vector. This should allow us to find a second equation between and z. q Before doing this, however, it is helpful to note some constraints on the form of the solution. First, since the tanh function never eceeds unity, (5.76) implies κ < κ or equivalently ε > (5.77) ε 0 as a necessary condition for a solution. Second, κ and κ are required to be real and positive. It follows from the definitions that when the constant ε > 0 for the linear medium, the condition for this is q > ε / ω / c (5.80) In other words, the dispersion curve lies to the right of the medium- light line. The Poynting vector S = E H has and z components, but the latter contains a factor cos( q ωt) sin( q ωt) so that its time average is zero. The results for the component are found to be c qκ S sech [ ( )] cos = κ z + z ( q t) 3 ω (5.8) µ 0ηω for z < 0 and q S = E ep( κ z)cos ( q ωt) (5.8) µ ω for 0 > 0. The average of the time-dependent terms gives z per unit length in the y direction is given by the integral of S : P 0 S, and the time-averaged power flow = dz (5.83) Here the factor is the time average and (5.8). The integrals are elementary and after eliminating one finds 0 S c qκ κ P = sech ( κ z ) + + tanh( κ z ) 3 µ 0ηω κ denotes the time-independent factors in (5.8) and Using (5.76) together with the identity sech u = tanh u reduces this to cq κ κ κ P = (5.84) µηω 0 κ κ We have now completed the formulation of the problem. For given values of ω and P, (5.76) and (5.84) are simultaneous equations for the unknowns z and q. Hence it is possible to solve for the dispersion relation in the form q methods. E = q ( ω; P). This can be done numerically using graphical