Frequency shift and relaxation of localized vibrations at surfaces mediated by substrate anharmonicity

~ ) Solid State Communications, Vol. 87, No. 4, pp. 273-276, 1993. Printed in Great Britain.

0038-1098/93 $6.00+.00 Pergamon Press Ltd

FREQUENCY SHIFT AND RELAXATION OF LOCALIZED VIBRATIONS AT SURFACES MEDIATED BY SUBSTRATE ANHARMONICITY

Zhen Ye* and P. Piercy Department of Physics, University of Ottawa, Ottawa, Canada KIN 6N5

(Received 9 November 1992 by A.A. Maradudin) (accepted for publication 17 May 1993)

A vibrational mode localized at an adsorbed atom or molecule experiences a temperature-dependent frequency shift and line broadening~ due in part to its interaction with substrate phonons. When the bihnear coupling be-- tween the adatom and surface is treated nonperturbativeiy, the aslharmonic interactions between neighbouring substrate atoms couples to the local mode. The effects of the anharmonicity of the substrate on the vibration spectra are demonstrated in a simple model calculation applied t~.~ the adsorption systems C O / P t ( l l l ) and CO/Ni(100).

This letter is devoted to the study of the temperature dependent vibrational line shape of a molecule adsorbed on a crystal surface. Over the last decade, a central edm of the extensive work on infrared absorption line shapes of surface systems has been to define the mechanisms causing the line shapes.[1,2] Theories of phonon-induced line shape effects have mainly con- centrated on the dephasing and decay mechanisms that arise from the anharmonic terms in the molecule-surface adsorption potential. After Persson and Ryberg[3] considered a surface adiabatic effect which was missing in previous works, the temperature-dependence of the line shape of an admolecule was often attributed to its coupling to thermally excited, low-frequency localized vibrational modes such as frustrated lateral translational and rotational modes[2]. However, an exhaustive theoretical description of all the accessible relaxation and dephasing channels is not yet available.

In this letter, we propose a new mechanism contributing to the temperature dependent, vibrational line shape of an adsorbed molecule, mediated by the anharmonicity of the sub3tra~.e. A vibrational mode localized at an adsorbed molecule involves coherent motion of the atoms of the substrate near it. (This is analo- gous to the properties of a localized mass defect in a solid.[4]) Anharmonic interactions between these substrate atoms can therefore couple the localized mode to the delocalized phonons of the substrate, leading to vibrational relaxation and line shift effects. Without parameter fitting, our results suggest that the substrate anharmonicity gives important contributions to the observed line shapes in some systems.

The theoretical description of this phenomena develops in two steps. The vibrational field is expressed in terms of free local and delocalized components in the harmonic approximation, including the bilinear interaction of the adsorbate and substrate non-perturbatively. The substrate anharmonicity is then treated in pertur- bation theory[5,6], to predict its effects on the vibrational line shape of an adsorbed molecule, as a function

* Present address: Ocean Physics, Institute of Ocean Sciences, P.O. Box 6000, Sidney, British Columbia, Canada V8L 4B2

273

of temperature. This is a typical many-body problem, which is readily formulated using a finite temperature field theory[7,8].

We consider here the perpendicular vibration of an adsorbed atom or molecule, corresponding to the stretching of its bond to the surface atoms. (In the case of diatomic admoleeule such as CO, only its centre- of-mass motion, with frequency typically around a few hendred wave numbers, is described here. The effects of the higher frequency, internal C-O vibrational mode on the molecule-surface motion are negligible here.) For simplicity we use a model similar to that described in Refs.[3,6,9], which could be easily extended to treat other local vibrational modes as well. The masses of the molecule and a substrate atom are denoted by m and Ms respectively. The frequency of the vibrational mode is denoted by w0, which is greater than the maximum phonon frequency wo of the substrate. The case in which the frequency w0 is within the continuum phonon band requires a different approach, as addressed elsewhere.[10]

In this paper, without losing generality, the anharmonic interactions between only nearest neighbour atoms, due to their motion perpendicular to the surface, are taken into account. The interaction between a molecule adsorbed at an on-top site and the nearest neighbour surface atom beneath it is described by an adsorption potential V(u - v). Here, v and u are the displacement coordinates for the surface atom and the adsorbed molecule, respectively. The interaction between a pair of nearest neighbour substrate atoms i and j is described by the potential ¢(rlj) depending only on their distance of separation rij.

In a Taylor expansion of the interatomic poten- tials to second order, we get the free Hamiltonian

Ho = hwobtb -b ~ hwkbtbk - mwguv (1) k

where the second derivative of the adsorption potential is V2 = mwg, and {wk} is the phonon spectrum. H0 describes a localized mode coupled to the free phonon continuum,[8] in which the latter is approximated using a bulk Debye model. (A more detailed treatment of the substrate phonon motion, including surface modes,

274

would be expected to give greater accuracy, but not qualitative differences.)

Anharmonie coupling is contained in the interaction Hamiltonian which reads, to fourth order in the displacements,f9]

H, = V3(u- v) 3 + ~V4(u- v : + . . .

1 1 + ~ ¢ , ( v - vl)~ + ~ ¢ ~ ( v - v,)4 + . . . (2)

We will ignore the first two terms in this equation, aris- ing from the anharmonicity of the adsorption potential V(u - v), because contributions from acoustic phonons in these terms are severely reduced by a "reduction factor" described later. The last two terms in Eq. (2) describe anharmonic coupling of the surface atom (of the on-top site) to its nearest neighbour in the second layer of the substrate, which has perpendicular vibrational displacement vl. This picture applies directly to a (100) surface of a simple cubic crystal. Although the lattice structure at other crystal surfaces should be considered, a detailed description of the solid is unnecessary in view of the approximate treatment of the motion of the admolecule. If we ignore lateral motion in the anharmonic coupling terms, and also assume that the nearest neigh- bours of the surface atom vibrate in phase (a long wave length limit), the interaction Hamiltonian for other lattice types simplifies to the form given in eq.(2), with modified coefficients.

The values of the derivatives of the substrate pair potential ¢ are fixed by comparison with the thermal and mechanical properties of the substrate[ll,13]. Specifically, the harmonic and cubic coefficients ¢2 and ¢3 are determined by the bulk modulus and linear thermal expansion coefficient of the solid, as described in Refs. [12,13]. Assuming a Morse potential form for ¢ defines the fourth derivative coefficient as ¢4 = (7¢])/(9¢2).

We now show how the bilinear term mw2oUV gives rise to an interpolation formula[7] relating u and v to free fields u ° and v ° defined below. There are different ways of treating the harmonic Hamiltonian in Eq.(1). Using standard Green's function methods, the Heisenberg equations of motion yield Yang-Feldman- type equations[14,9], which may be solved for the fields u and v to give

= c ~ / ~ ° + c ~ / ~ z ( - o ? ) : (3) v = C~/2v °-I- C~/2A(-O?)u ° (4)

Here, u ° is a harmonic oscillator with frequency w~ near w0,[8] and v ° is the free harmonic displacement of the surface atom, with phonon spectrum {wk}. The oper- ators in the above equations are defined as Z(-O~) = w~/(w~ + 0t a) and A(-0~) = ~k(mw~/M~N)[O~ + w~] -~. Renormalization factors Ca and C2 are determined by the commutation relations of the fields u and v.

Equations (3) and (4) show that the vibrational motion of both the adsorbed molecule, and the surface atom beneath it, contain both local and delocalized components. Of primary importance here, the surface atom displacement v has a local component pro-

t " ~ l / 2 A { , . , : 2 " ~ portional to u °, with relative amplitude d '~1 " ~ 0 z in Eq. (4).[4] The last two interaction terms in Eq. (2), due to the anharmonicity of the substrate, depend on the difference of coordinates v - vl, which may be rewritten, using Eq. (4), as

FREQUENCY SHIFT AND RELAXATION Vol. 87, No. 4

v - - v , = d u ° + q ° (5)

in terms of the modified phonon displacement q0 =

C~/2v ° - V°l . The substrate anharmonicity in H1 therefore couples the local mode displacement u ° to the phonons of the solid, and influences the vibrational line shape as discussed below. (The local component of the second layer atom displacement vl is negligible for the systems considered here[9], and is therefore ignored in the equations above.)

In the line shape calculations to follow, the effects of the anharmonic terms in the adsorption potential V(u - v) are ignored. This is justified by considering the so-called "reduction factor" for the phonon field, defined as the coefficient in front of v ° in u - v, which is

given by/i~p = C~12w~/(w2 o - w~). If, for typical system parameters, Rp << 1, then acoustic phonon coupling via the adsorption potential may be neglected. In the particular case where m << M, and WD << w0, we find C2 ---+ 1, and Rp agrees with the work of Persson and Ryberg[3].

Now we give general formulae for the line shape of the local vibrational mode, as mediated by substrate anharmonicity alone. The line shape is calculated to have an approximately Lorentzian form, described by a peak frequency shift and linewidth, as derived from the real and imaginary parts of the vibrational self energy.f8] Considering the interaction terms given by eq.(2), it is easy to check that the main contributions to the temperature-dependent frequency shift of the local mode come from the two diagrams depicted in fig.l(a, b), due to the fluctuating thermal phonon motion described by the phonon propagator. Using real time field theory at finite temperature,f7] the total frequency shift Aw = Aw~ + Awb is found to be composed of a red shift contribution

Aw~ - 2row0 k 2M, Nw~ jfkl~ h

× ~ IA, l~p + 2,,(,.,.,k,)] (6) 2MflVwk "K

due to the cubic interaction term ¢3, while the quartic term gives a blue shift portion

, (7)

In these equations, n(wk) is the bosonic thermal distri-

bution function, and [fk[ 2 = 1 + C2 - 2C~/2 sin kro/kro, where r0 is the lattice constant.

For an adsorption system such as CO/Ni(100), for which the vibrational frequency is less than twice the maximum phonon frequency, the main line width contribution due to substrate anharmonicity is given by the two-phonon term in Fig. l(c) as

~rhd2 ¢~ Ifkl21f~,l 2 rc = 16mwoM~N 2 Y~ [1 + n(wk) + n(wk,)]

k k I taJkiaJk t

x , ~ ( ~ o - ~ k - ~ k , ) ( 8 )

Noting that the dashed line in the figure represents the modified phonon field q0, this llne width term is inter- preted as due to vibrational relaxation via a weighted combination of two-phonon absorption and emission processes.

Vol. 87, No. 4

f ~ - .-,,.% / \ [ \

1

"-r" "" I

I I I

Fig.l(a)

/ \ /

1 \ /

FREQUENCY SHIFT AND RELAXATION 275

-1

-2

-3

Fig.

i I I I I

0 2 0 0 4 0 0

Temperature (K)

2. Temperature dependent line shift versus temperature for CO adsorbed at on-top sites of P t ( l l l ) and Ni(100). The anhaxmonic coefficients of these solids axe derived from bulk elastic properties to be[9,13,15] ¢3 = -1.78 x 1013J/m 3 and ¢4 = 2.7 x 1024J/m 4 for PL, and ¢3 = -0.994 × 1013J/m 3 and ¢4 = 8.89 x 102ZJ/m 4 for Ni.

Fig.l(b)

f

( ) ,.... J

,... J

Fig.l(c)

Fig. 1. Feynman diagrams contributing to the self- energy. The solid line represents the propagator for the T-mode and the dash line stands for the Green's function iO°(t) = (Tq°(t)q°(O)) for the modified phonon coordinate q0 defined in the text. Here we only draw the diagrams which have significant contributions to the temperature- dependence of either line shift or width.

The results above can now be applied to study the effects of substrate anhavmonicity on the vibrational line shapes in specific adsorption systems. We first consider the adsorption of CO at on-top sites on the P t ( l l l ) surface, for which the molecule-surface vibrational frequency is known from experiments[16] to be 470cm -1. Using a Debye model (with Debye frequency WD = 167cm-Z[15]) to describe the free phonon modes of the substrate, we calculate the coefficients of the delocalized and localized components of the surface atom

displacement to be C~/2 = 0.93 and d = -0.14, respectively. The two lowest order temperature-dependent terms in the frequency shift, Aw~ and Awb, axe re- lated by Eq. (7) giving Aw, ~ --3.6Awb. As plotted versus temperature in Fig. 3, we calculate a net red shift which increases by 1.8cm -1 over the temperature range from 100K to 290K. This is close to the experimentally measured shift of about 2cm -1 by Ryberg[16], showing that the anhaxmonicity in the substrate crystal plays a rather important part in defining the frequency shift. However, in this case, substrate anharmonicity does not give a significant contribution to the linewidth, since the vibrational decay channel would re- quire an inefficient, three-phonon process. Furthermore~ vibrational dephasing due to elastic phonon scattering at the admolecule - mediated by ¢4 - is higher order in the small mass ratio re~Ms ~ 0.14 of admolecule to substrate atom. In this work, we have neglected the anharmonic coupling of the adsorption bond to lower frequency quasilocalized modes, which may also con- tribute significantly to the line width and shift in this system. [2,16]

In the ease of CO adsorbed on a Ni(100) surface, the molecule-surface vibrational frequency is w0 = 476cm -1 [17,18], which is less than twice the value of the maximum phonon frequency wo = 271cm -1. The coefficients of the interpolation formula (Eq. 4.) be-

come C~ 12 = 0.77 and d = -0.45, indicating a greater coupling of the local mode with the phonon continuum, than in the CO/Pt case. Then from eq.(8), we estimate a full linewidth at room temperature 2F = 8.4cm -1, which makes up a significant part of the experimentally observed width of 15cm -1 by Chiang et all18]. By including next-nearest neighbour interactions, lateral mo- tions of the admolecule and substrate atoms, and the detailed structure of the crystal and the surface phonons, a more precise comparison with experiment could be made. Although we are not aware of experimental data for the frequency shift in this system, the calculated line shift for CO/Ni(100) is shown in Fig. 2 to be smaller than that predicted for C O / P t ( l l l ) .

276 FREQUENCY SHIFT AND RELAXATION Vol 87, No 4

In this letter, we described a new channel for energy dissipation and exchange between the adsorbate and substrate, mediated by the anhaxmonicity of the solid. It was demonstrated specifically, from first princi- ples, how the vibrational line shape of an adsorbed atom or molecule is influenced by this interaction channel The local vibrational mode of the admolecule includes coherent motion of the surface atom(s) near it in the harmonic approximation, which allows the anharmonio- ~ty of the substrate to couple directly to the local mode

Without parameter fitting, the calculated line shape parameters are found to make major contributions to the infrared absorption spectra for the adsorption systems CO/Ni(100) and CO/Pt(111)

Acknowledgement This research is funded througi~ the Networks of Centres of Excellence program and the Natural Sciences aad Engineering Research Councii of Canada,

R e f e r e n c e s

[1'~ ~. J. Chabal, Surface Science Rep. 8, 211- 357(1988)

[2) D. C. Langreth and M. Persson, Phys. Rev. B43. 1353(1991)

[3} B. N. J. Persson and R. Ryberg, Phys Rev. B40, 10273(1989)

[4] O. Madelung, Introduction to Solid State Theory, Springer-Verlag, Berlin, 1978; see section 9.1.4.

[5] A. A. Maradudin and A. E. Fein, Phys. Rev. 128, 2589(1962)

[6] A. A. Maradudin, P. A. Flinn and R. A. Coldwell- Horsfall, Ann. of Phys.I5, 360(1961)

[71 H. Umezawa, H. Matsumoto, and M. Tachiki, Thermo Field Dynamics and Condensed States. North-Holland(1982)

[8] G. D. Mahan, Many-Particle Physics, Plenum Press, New York(1981)

[9] Zhen Ye and P. Piercy, Phys. Rev. B47, 9797 (1992).

[10] Zhen ¥% Renfa~g Tu, m:d P. P:ercy~ te be pub lished,

[11] A. A. Maradudm and F A Fhnn, Phys, Rev 129, 2529(1963.)

[12] W. Ludwig, J Phys. Chem Solid 4, 2831(1958)

[13] G. Leibfrled and W Ludwig, in Sol~d State Physzc,. 12, 275(1961)

[14] see for example, D. Lurle, Particle, and Fields, In- terscience Publisher(1968)

[15] C. Kittel, Introduction to Solid State Physics, John Wiley and Sons, New York(1976)

[16] a. Ryberg, Phys. R~v. B40, 8567(1989)

[17] J. C. hriyasu et al, Phys. Rev. B28, 6123(1983)

[18] S. Chiang, R. G. Tobin, P. L. Richard and P. A. Thiel, Phys. Rev. Lett. 52, 648(1984)

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Frequency shift and relaxation of localized vibrations at surfaces mediated by substrate anharmonicity