Phys. Status Solidi B 247, No. 8, 1965–1968 (2010) / DOI 10.1002/pssb.200983927 p s sb

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basic solid state physics

Simulating the temperature

dependence of surface state contributions toreflection anisotropy spectral features

G. E. Isted*, P. D. Lane, and R. J. Cole

School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UK

Received 5 May 2009, revised 13 January 2010, accepted 22 February 2010

Published online 14 June 2010

Keywords metal surfaces, reflection anisotropy spectroscopy, surface states, temperature effects

*Corresponding author: e-mail gisted@staffmail.ed.ac.uk, Phone: þ44 131 650 5229, Fax: þ44 131 650 5902

By simulating the effect of temperature on the absorption

spectra ofAg andCu (110),we show the differing importance of

the various processes influencing the temperature dependence

of the width and intensity of the 1.7 eV Ag (110) reflection

anisotropy spectroscopy (RAS) peak and the surface state

contribution to the Cu (110) 2.1 eVRAS peak. Our results show

that the significance of these processes in each case is governed

by the energy gap between the associated occupied surface state

involved and the Fermi energy.

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1 Introduction Reflection anisotropy spectroscopy(RAS) [1, 2] has in the last decade or so proved to be ahighly versatile probe of metal surfaces due to its sensitivityto a diverse range of electronic and topographical surfacephenomena. Reflection anisotropy (RA) spectral featuresderive from a variety of processes; in the case of the (110)noble metal surfaces these predominantly consist ofelectronic transitions involving surface modified bulk bandsand surface states [2]. The surface state related features in theRA response of these surfaces are well known to be highlyresponsive to surface temperature, adsorbate coverage andion bombardment, in particular the effect of these variableson the 2.1 eV peak in the Cu (110) RA spectrum has beenextensively studied; for a review see ref. [2]. In recent workfocusing on Ag (110) [3], we have simulated the effect ofthermally created surface defects on the intensity of thesurface state-derived 1.7 eV RAS peak and have shown thatindividual adatoms have the ability to quench the contri-bution to the intensity of this feature over a significantsurrounding area. Similarly, Sun et al. [4, 5] have revealedthat molecular adsorbates induce an equivalent effect on the2.1 eV Cu (110) RAS peak. These studies demonstrate thatsurface state related RAS features have the potential to beused for accurately monitoring a variety of surface kineticprocesses such as the creation of thermal ad atoms from stepedges and the ordering of molecular adsorbates. However, to

enable a highly rigorous quantitative analysis of suchphenomena using RAS, significant advances in the modeldescribed in our recent study [3] are required, in particular anincorporation of the effect that complex defect or adsorbateinteractions, such as the formation of islands, have on theintensity of surface state peaks, which are likely to becomemore pronounced with increasing temperature. In this studyhowever, we will set aside the effect that thermally createddefects have on the intensities of surface state RAS peaks andfocus solely on simulating the effect of temperature on theintrinsic properties of the surface state transitions involved inthe formation of two different RAS peaks.

The 1.7 and 2.1 eV features found in the RA response ofclean Ag and Cu (110), respectively, are often described asanalogous as both derive from electronic transitions betweentwo surface states at Y [2]. However, whereas the peak in theAg spectrum is thought to arise exclusively from this process[6] there is evidence to suggest that the Cu feature has twofurther contributions: transitions between two surface-modified bulk bands at the X point of the Brillouin zoneand anisotropic intraband transitions (referred to as theDrude term) [7]. When discussing Cu (110) in this study wewill only consider the surface state contribution to the 2.1 eVRAS peak; it is the dominant contributor to the intensity ofthe feature and the only one of the three with a knowntemperature dependence.

� 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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1966 G. E. Isted et al.: Simulating the temperature dependence of surface state contributionsp

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The most noteworthy difference between the surfacestate transitions involved in the RA response of these twosurfaces is the fact that the binding energy of the involvedoccupied state at Y is considerably closer to the Fermi energy(Ef) in the case of Ag (110) than it is for Cu (110). In thiswork, we simulate the temperature dependence of theimaginary part of the complex surface dielectric function,e00s , and hence the absorption spectra ofAg andCu (110) in the1.7 and 2.1 eV photon energy regions, respectively. By doingthis we demonstrate the fundamental importance of the sizeof the energy gap between Ef and the associated occupiedstate in determining the role of the processes contributing tothe width and the intensity of e00s and thus the consequent RAresponse of the two surfaces.

2 Simulating the temperature dependence ofthe surface state transitions To simulate the effect oftemperature on e00s in the energy regions around 1.7 and2.1 eV forAg andCu (110), respectively, we adopt themodeldeveloped by Gerlach et al. [8] to determine the temperaturedependence of the 1.7 eV SHG signal from Ag (110), andused in recent RAS studies of noblemetal surfaces [3, 7]. Theparabolic shape of the occupied and unoccupied states aredescribed using the expression

Tabunoasso

Cu

Ag

� 20

EðkÞ ¼ E0 þ�h2

2mjk � k0j2; (1)

where k is a surface state electron wave vector, k0 theposition in reciprocal space of the Y point, E0 the surfacestate energy at Y and m is the effective mass of the surfacestate electrons (assumed to be temperature independent).Photoemission studies have shown that the occupied stateinvolved in both the Ag (110) 1.7 eV and the Cu (110)2.1 eV transitions exhibit a monotonous shift to lowerbinding energies with increasing temperature [8, 9]. Wemake the assumption that the binding energy of theunoccupied state in each case is independent of temperature.The rate of change of E0 with temperature along with otherdetails used in our simulations related to the occupied andunoccupied states are shown in Table 1.

The model allows vertical electronic transitions tobe induced from any point on the occupied state. Byintegrating over the 2D Brillouin zone around k0 we obtainthe shape of the absorption spectrum and hence e00s . Atemperature dependent Gaussian line L of widths¼ 50meVþ (0.1meV/K)T to account for hole lifetime

le 1 Values of E0, effective mass, m, in terms of free electronccupied surface states used in the Ag and Cu (110) e00s simulations.ciated references.

surface state E0 (eV) Ref. effective mass (m0) R

(110) occupied �0.51 [9] 0.27 [unoccupied 1.8 [11] 0.8 [

(110) occupied �0.106 [8] 0.26 [unoccupied 1.66 [8] 0.9 [

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broadening and a Fermi-Dirac distribution function F todetermine the electronic occupation of the state near Ef

for any given temperature are also incorporated into themodel. The relationship between e00s and the surface statetransition is:

maValu

ef.

10]11]8]8]

; TÞ / 1

ð�hvÞ2Z1

�1

L½�hv� EfðkÞþEiðkÞ; sðTÞ�

� F½EiðkÞ; T �kdk;

(2)

where Ei and Ef are the initial and final energies of anelectronic transition induced by a photon of frequency v. Atelevated temperatures, the F function allows transitions tobe induced from the occupied state at points above Ef thusadding intensity to the lower photon energy side of asimulated peak in e00s . The L function however, redistributesthe intensity of a peak in e00s by effectively spreading theintensity associated with individual transitions over atemperature dependent energy range. The effect oftemperature on the simulated e00s spectra for Cu and Ag(110), along with the effect of removing L and F togetherand separately from Eq. (2), is shown in Fig. 1.

Although not performed in this study, the RA responseresulting from these surface state transitions can becalculated from e00s using the approach described by Coleet al. [12] which requires the derivation of the real part of thecomplex surface dielectric function, e0s, from e00s usingKramers–Kronig relations.

3 Discussion The results shown in Fig. 1 show theeffect that theL andF functions in Eq. (2) have on the widthand intensity (integrated intensity beneath the spectra) of thesimulated Ag and Cu (110) peaks in e00s . It is clear from thesesimulated spectra that both F and L play a more crucial rolein shaping the profile of the e00s spectrum for Ag than for Cu.For example, removing F from the Ag simulation (opencircles in Fig. 1) at 700K results in a flat spectrum with zerointensity, whereas for Cu the omission of this function hasonly a small effect on the peak intensity at this temperature.Similarly, it is clear from the spectra that the L functiondominates the width (FWHM) of the e00s peak for Ag acrossthe entire temperature range, while its omission from thesimulated Cu spectra has only a notable effect on the FWHMof the peak at higher temperatures (see blue spectra in Fig. 1).A more detailed analysis of the effects of F and L on e00s can

ss, m0, symmetry type, dE0/dT and k0, for the occupied andes of E0 are predicted values at 0K extrapolated from results in

dE0/dT (meV/K) Ref. symmetry type k0 (A�1) Ref.

0.26 [9] py (0, 0.87) [7]0 [11] sþ pz0.17 [8] py (0, 0.73) [8]0 s

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Phys. Status Solidi B 247, No. 8 (2010) 1967

Original

Paper

1.4 1.55 1.7 1.85 1.65 1.8 1.95 2.1 2.25

700 K

100 K

500 K

300 K

700 K

100 K

500 K

300 K

Photon energy (eV)

Inte

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(arb

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its)

Inte

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(arb

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y un

its)

Photon energy (eV)

a) b)

Figure 1 (online colour at: www.pss-b.com)The effect of temperature on e00s for (a)Ag (110)and (b) Cu (110) calculated using Eq. (2) withboth theF andL functionsomitted (black line),justL omitted (blue line), justF omitted (opencircles) and neitherF norL omitted (red line).

Figure 2 Theintensity (squares), intensitywithF functionomitted(triangles) and width with both the F and L functions omitted(diamonds) of the simulated Ag (110) (black) and Cu (110) (grey)peaks in e00s as a function of temperature. Dashed lines show widthinduced by the L and F (2 kT) functions.

be made using Fig. 2, which illustrates the effect oftemperature on the intensity and the width of the simulatedAg and Cu (110) peaks is e00s along with the FWHM of thelifetime broadening effect (equal to 2.35s) and the width, onthe low photon energy side, of the smoothed Fermi edge(equal to 2 kT).

The data points labelled using diamonds in Fig. 2 showthewidth of the peak in e00s for the two surfaces as a function oftemperature when both the L and F functions are removedfrom the simulation and correspond to the spectra shownusing a solid black line in Fig. 1. The width of the e00s peak inthese spectra derives solely from the differing parabolicshapes of the occupied and unoccupied states (defined byEq. 1) and the k points at which the occupied state crosses Ef.The fact that the binding energy of the occupied state for bothsurfaces reduces with increasing temperature causes adecrease in the width in k-space of the occupied state belowEf and hence a decrease in the range of photon energies ableto induce transitions between the two states (we will refer tothis effect on the width of the peak in e00s as the thermal shifteffect). The decrease in binding energy of the occupied statewith increasing temperature is also responsible for the redshift in the position of the peak in e00s observed in the spectra inFig. 1. The diamond labelled data in Fig. 2 illustrates that forboth surfaces the thermal shift effect causes a linear decreasein the range of absorbed photon energies with increasingtemperature (and hence a reduction in the width of thecorresponding e00s peaks labelled with a black line in Fig. 1),until at �620K when the occupied state in the case of Ag

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(110) completely crosses Ef, causing the peak width due tothis effect to drop to zero (see black diamonds, Fig. 2). Thesignificantly higher binding energy of the occupied state forCu (110) however, ensures that at this temperature thecorresponding e00s peak has only lost approximately a third ofits width (see grey diamonds), indicating that the thermalshift effect plays a significantly more important role indefining the width of e00s peak for this surface across theinvestigated temperature range than for Ag (110).

Indeed, by comparing the data plotted with the greydiamonds in Fig. 2 with the dashed lines (indicating thepeak width attributed to the F and L functions), it is clearthat the thermal shift effect dominates the width of theCu (110) e00s peak at all temperatures up to �480K,beyond which it is overpowered by the lifetime broadeningeffect (the L function). In the case of Ag (110) however,the L function dominates the width of the e00s peak acrossthe entire temperature range investigated (see blackdiamonds).

The effect of temperature on the intensity of thesimulated e00s peaks for Ag and Cu (110) is plotted in Fig. 2using black and grey squares, respectively. The expectedlinear reduction in intensity with increasing temperature [8,9], induced by the depopulation of the occupied state as itcrosses Ef, is demonstrated across the entire temperaturerange for Cu (110) and up to �300K for Ag (110). Attemperatures beyond�300K however, the Ag data begins tolose its linear trend due to the increasing effect that thesmoothing of the Fermi edge has on the number of possibleelectronic transitions with increasing temperature.Removing the F function from the Ag simulation (blacktriangles, Fig. 2) results in a dramatic change in the intensityof the peak in e00s above �300K, causing it to decreaselinearly with increasing temperature beyond this tempera-ture until the occupied state crosses Ef at �620K where itconsequently drops to zero. In the case of the Cu surfacehowever, removing F from the simulation has almost noeffect on the intensity of the corresponding peak in e00s ,indicating that it is the shift in binding energy of the occupiedstate with temperature, and not the smoothing of the Fermiedge that is its dominant contributing factor.

The difference in the importance of theF function on theintensity of the peaks in e00s for the two surfaces arises as aresult of the significant difference in the energy gap betweenthe occupied state and Ef in each case (see Table 1). Theconsiderably closer proximity of the state to Ef in the case ofAg (110) ensures that at any given temperature thecorresponding peak in e00s not only exhibits a significantlylower intensity than its Cu (110) equivalent, but a greaterproportion of its intensity, and hence the resultant RAS peakintensity, arises due to electronic transitions induced fromthe smoothed Fermi edge aboveEf. This effect (controlled bythe F function) therefore plays a decisive role in thetemperature dependence of the 1.7 eV Ag (110) RAS peakintensity, whereas the temperature dependence of theintensity of the surface state contribution to the 2.1 eV Cu

� 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

(110) RAS peak is instead dominated by the thermal shift ofthe occupied state.

4 Conclusion By simulating the effect of temperatureon the absorption spectra (e00s as a function of photon energy)of Ag and Cu (110), we have demonstrated the differingimportance of the various processes influencing the tem-perature dependence of the width and intensity of the 1.7 eVAg (110) RAS peak and the surface state contribution to theCu (110) 2.1 eVRAS peak. Our results show that the thermalsmoothing of the Fermi edge of the occupied state involvedin the formation of the Ag (110) e00s peak is highly influentialon its intensity across the majority of the temperature rangeinvestigated. In the case of Cu (110) however, this effect issignificantly less important, with the intensity of thecorresponding peak in e00s being primarily governed by thethermal shift in binding energy of the occupied state, which isalso the prime factor controlling the temperature dependenceof its width at lower temperatures (<480K). At highertemperatures for Cu (110) however, and for the entiretemperature range investigated for Ag (110), it is the lifetimebroadening effect that dominates the width of the corre-sponding peak in e00s . The results of this study show that thetemperature dependence of the width and intensity of the1.7 eV Ag (110) RAS peak and the surface state contributionto the 2.1 eV Cu (110) RAS peak are controlled by severalprocesses, the significance of which is governed by theenergy gap between the associated occupied surface stateand Ef.

References

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[3] G. E. Isted, P. D. Lane, and R. J. Cole, Phys. Rev. B 79,205424 (2009).

[4] L. D. Sun, R. Denk, M. Hohage, and P. Zeppenfeld, Surf. Sci.602, L1–L4 (2008).

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[10] S. D. Kevan, N. G. Stoffel, and N. V. Smith, Phys. Rev. B31(6), 3348 (1985).

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