Electrical Resistivity of thin metal films-Peter Wibmann, Hans-Ulrich Finzel

Later, however, it became obvious that grainboundary scattering [4, 5] and surface roughness [6, 7] play a decisive role inthe resistivity behaviour of polycrystalline films.. hy-The influ

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of Thin Metal Films

With 110 Figures

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47798 Krefeld, Germany E-mail: HU.Finzel@web.de

Library of Congress Control Number: 2006935051

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ISSN print edition: 0081-3869

ISSN electronic edition: 1615-0430

ISBN-10 3-540-48488-4 Springer Berlin Heidelberg New York

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The technical application of thin metal films in such diverse fields as croelectronics (e.g chips, sensor design, solar cells), optical filters, catalysis

mi-or cmi-orrosion-resistant coatings have led to a large database on the electricalproperties The interpretation of the data has, however, often been contro-versial A remarkable progress in understanding the physical basis of thephenomena was achieved by first depositing monometallic films under well-defined ultra-high-vacuum conditions, and then by studying the influence ofthe residual gas by additional gas adsorption experiments in a second step.Polycrystalline as well as single-crystalline films could be prepared by vary-ing the substrate material and the deposition conditions in a proper manner.Modern structure investigations and high-resolving spectroscopic techniqueshave helped to obtain a more accurate picture of the character and strength

of the metal/gas interaction during the crystal growth So, general approval

is presently given to the idea that gaseous adatoms should display featuressimilar to alloy formation on a pure metal surface, e.g the generation of newscattering centres for the conduction electrons The so-called scattering hy-pothesis holds quantitatively in many cases as has been previously shown forthe dependence of the resistivity on film thickness

Here we will concentrate on the effect of annealing and gas adsorption onfilms of the noble metals silver and gold which have model character withrespect to a weak metal/gas interaction Corresponding phenomena on theresistivity of bulk metal samples are widely unknown for obvious reasons.The experimental data have been accumulated in the last three decades andallow a detailed and independent check regarding whether the scattering hy-pothesis can be used for a theoretical prediction of the film resistivity or not

A sufficient structural characterisation of the films is an important site for such efforts Moreover, possibilities to recalculate the resistivity fromoptical, photoelectric or infrared absorption data will be critically discussed.The conclusions drawn may shed new light on the interpretation of the elec-trical properties of films with more complicated structures, compositions andchemical reactivities These films are usually prepared under worse vacuumconditions but represent the centre of practical interest

The authors are deeply obliged to all coworkers of the Institute of Physicaland Theoretical Chemistry of the University Erlangen-N¨urnberg With theirengaged scientific efforts, they have enabled us to present this survey Also

we are obliged Frau B Eichel for typing the script and to Springer Verlag forthe excellent cooperation

1 Introduction 1

References 2

2 The Scattering Hypothesis 3

References 7

3 The Effect of Annealing on the Electrical Resistivity of Thin Silver Films 9

References 32

4 The Effect of Annealing on the Electrical Resistivity of Thin Gold Films 35

References 51

5 The Interaction of Oxygen and Ethylene with Silver and Gold Films 53

References 78

6 Other Adsorbates on Silver and Gold Films 81

6.1 Xenon on Silver and Gold Films 81

6.2 CO on Silver and Gold Films 86

6.3 Hydrogen on Gold Films 91

6.4 Palladium on Gold Films 93

References 95

7 Further Selected Adsorption Systems 97

7.1 Adsorption of CO and O2 on Palladium Films 97

7.2 The Fe/O System 103

7.3 The Ge/CO System 115

References 120

8 Conclusions and Outlook 123

References 124

Index 125

Thin metal films have received widespread attention for technical applicationslike conducting connections in microelectronics, optical elements tailored withdesired spectral properties or supported adsorbents in heterogeneous cataly-sis The electrical resistivity is an easily accessible and informative quantity

to characterise the material

K Fuchs [1] has predicted in a famous theoretical paper published in 1938that the electrical resistivity of thin metal films increases with decreasingthickness The scattering of conduction electrons at the film surfaces wasconsidered to be responsible for this phenomenon Since good agreement wasfound with early experimental data [2, 3], the interpretation was not called

in question for a long period Later, however, it became obvious that grainboundary scattering [4, 5] and surface roughness [6, 7] play a decisive role inthe resistivity behaviour of polycrystalline films Moreover, ultra-thin filmsmay crack and form an island structure [8, 9] Thus, the measured thicknessdependence of the electrical resistivity differs from Fuchs’ results in such away that the resistivity increase with decreasing thickness is more pronouncedthan the theoretical prediction

The corresponding extension of Fuchs’ theory leads to the scattering pothesis [10,11], where not only surface scattering, but also crystallic bound-ary scattering, surface roughness and adsorption phenomena are includedinto description A brief survey of this hypothesis is presented in Chap 2 Inspecial cases, however, more complicated mechanisms must be included intodiscussion in order to explain the results The aim of the present booklet is toreport on selected examples of such complications and to show possibilitiesfor solving the problems

hy-The influence of annealing on the resistivity of silver and gold films istreated in Chaps 3 and 4 It is shown that the pure grain boundary scattering

is not sufficient to explain the resistivity properties of polycrystalline films.Obviously, the healing of lattice defects in the interior of the grains must beadditionally taken into consideration On the other hand, both effects can beneglected in the case of single-crystal films

Then, we elucidate in Chap 5 the problems of a theoretical calculation ofthe scattering cross section for the example of oxygen and ethylene adsorption

on silver and gold films The application of the model described in Chap 2 is

P Wissmann and H.-U Finzel: Electrical Resistivity of Thin Metal Films, STMP 223, 1–2

(2007)

DOI 10.1007/3-540-48490-6 1
Springer-Verlag Berlin Heidelberg 2007c

the interior of the film Oxygen on iron films at higher temperatures showssuch a behaviour Finally, we discuss the adsorption of carbon monoxide

on semiconducting Ge films, where the scattering hypothesis totally fails toexplain the results Doping phenomena cannot be excluded in this case, even

if the thickness dependence of the resistivity seems to be in agreement with

a scattering mechanism

References

1 K Fuchs Proc Cambridge Phil Soc 94, 100 (1938)

Verlagsge-sellschaft Stuttgart, p 178 (1955)

3 K.L Chopra Thin Film Phenomena, Mc Graw-Hill, New York (1969)

4 A.F Mayadas and M Shatzkes Phys Rev B1, 1382 (1970)

5 P Wissmann Thin Solid Films 5, 329 (1970)

6 Y Namba Japan J Appl Phys 9, 1326 (1970)

7 H.-U Finzel and P Wissmann Ann Phys 43, 5 (1986)

8 T.J Coutts Electrical Conduction in Thin Metal Films, Elsevier, Amsterdam

p 205 (1974)

9 H.-U Finzel and P Wissmann Z Naturforsch 40a, 161 (1985)

10 P Wissmann The Electrical Resistivity of Pure and Gas Covered Metal Films

Springer Tracts Mod Phys., 77, 1 (Springer-Verlag, Berlin, 1975)

11 P Wissmann Thin metal films and gas chemisorption in: Studies in Surface

Science and Catalysis, 32, 53 (Elsevier, Amsterdam, 1987)

The scattering hypothesis is based on the assumption that Matthiessen’srule can be applied, i.e all scattering contributions compose additively [1]according to

density as the films; K and C are scattering constants and hence proportional

to the mean free path l0of the electrons, and B is a measure of the asperity

where p is the fraction of electrons specularly reflected at the film surfaces.

It is easily recognised that an upper limit Cmax = (3/16)l0 is implied in

Eq (2.2) since the quantity p is defined to vary between 0 and 1 at the outer

centres in the grain boundary.

While discussing Eq (2.1), we assume that the free electron density neand hence ρ0l0[1] is independent of the film thickness d Since the justification

of this assumption is still disputed in the literature [6–9], we may insert someremarks on this topic in the following

Hall effect measurements are a suitable method to determine the free

elec-tron density ne in thin films The evaluation of literature data shows that

the Hall constant RH of thin gold films of medium thickness yield electrondensities equal to values obtained for bulk gold [10] Only for ultra-thin films

a certain enhancement of RHis reported This enhancement should not, ever, be attributed to size effects as predicted theoretically by Sondheimer [3];more likely, an influence of surface roughness is effective [5]

how-A further powerful method for the determination of free electron density

in thin films is the analysis of the real portion ε1of the dielectric function in

the Drude range According to [11], a proportionality between ε1 and ne isexpected in the infrared wavelength region, where the constant of proportion-

ality depends only on the light wavelength λ Hence one would expect that the ε1 values do not differ much while comparing thin film data with bulkgold data Typical results are reported in [12], where 30-nm thick gold filmsare deposited on glass substrates and subsequently annealed at 423 K for 1 h.Indeed, a good agreement is found between film and bulk data These filmsare known to be rather flat with a homogeneous structure [13] One should

be more cautious, however, while analysing inhomogeneous films with roughsurfaces Since the optical data represent an average of metal and vacuum,

the ε1 values come out to be too small for rough films, which often leads tomisinterpretations [7]

Further information is obtained from the measured thickness dependence

of the resistivity of single-crystal films These films are widely free of angle grain boundaries and roughnesses, at least in the higher thicknessrange [14, 15] The film resistivity is then nearly equal to the well-knownbulk resitivity; deviations are observed only for lower thicknesses, where thesurface roughness cannot be neglected

large-In conclusion, we will state that the electron density in the films can beconsidered as a material constant and hence does not depend on film thickness

or annealing temperature

Now we return to Eq (2.1b), where the scattering centres can be easilyincreased by adsorbing gas on the film surface Quantitatively,

C = nAl0 (2.5)

is valid where n is the gas coverage in molecules/cm2 of the geometrical

film surface area and A is the corresponding scattering cross section From Eqs (2.1) and (2.5), we obtain by differentiating for D = d and B = 0 [1],

d(∆ρ) dn



n→0

= A ρ0l0

Since ρ0l0 is known from literature and d is measured independently,

Eq (2.6) can be used to determine A from the initial slope of the curves

On the other hand, it is desirable to calculate A theoretically in order

to compare with the experimental values The first attempt to realise thisconcept was published by Persson [16, 17] on the basis of the analysis offrustrated infrared spectroscopy vibrations However, the results obtainedfor the Pd/CO system were very disappointing [18]

Therefore, we decided to follow the predictions of the Ziman–Mott formula[19]:

rs = 3.02 and EF= 5.48 eV [20], and f = 0.68 Hence, the main problem is

to obtain an estimation on the excess charge z in Eq (2.7) Recently, Ricart

et al [21] have performed cluster calculations on the basis of self-consistingfield approximation

For the case of the Ag/O system, the excess charge z was determined by

the Mulliken population analysis

b) Rough silver surfaces tend to reconstruct during chemisorption of oxygen[24] Details depend on the surface structure which is especially complex

in the case of polycrystalline films

c) The cluster calculation has clearly shown that the excess charge cannot

be considered as a point charge, it is rather distributed over several metalatoms [21, 25]

d) Born’s approximation was assumed to hold while deriving Eq (2.7) It

is known from alloy physics, however, that this approximation can betroublesome in the case of heavy metals [26] Then, resistivity values areobtained which are too high as compared to experimental data, leading

to a common preference of the partial wave method for the calculation

of resistivity of alloys [19] It remains to be checked whether Born’s proximation is better applicable for adsorption systems where the surfacecharge seems to be smeared over a larger area of surface atoms as com-pared to bulk properties

ap-Fortunately, the various restrictions underlying the calculation seem tomutually compensate to some extent so that the results for the Pd/CO andCu/O systems appear to be reasonable

In principle, the excess charge z can also be calculated from the change

in work function ∆φ according to [27]

where the quantity on the left-hand-side of Eq (2.8a) can be taken from the

linear slope of the measured curves ∆φ versus n (examples are presented in Chaps 5 to 7) and M is the dipole moment originating from charge transfer

in the adsorption bond, which can be described to a first approximation by

a linear dipole model [28],

where l is the distance between the centre of charges forming the dipole.

Here, one should keep in mind, however, that in addition to the restrictionsmentioned above (i.e reconstruction, extended charge distribution etc.) an

overlapping of σ- and π-bonds (‘backdonation’ [29]) or a permanent dipole

moment [30] of the adsorbed molecule can revoke the justification for theapplication of Eq (2.8b) Consequently, the discrepancy between calculated

(0.11 eV [31]) and experimental ∆φ values (1.1 eV [18]) is extremely large

for the example of the Pd/CO system More details on this topic are given

in Sect 7.1

References

1 P Wissmann Thin metal films and gas chemisorption, In: Studies in Surface

Science and Catalysis, 32, 53 (Elsevier, Amsterdam, 1987)

2 H.-U Finzel and P Wissmann Ann Phys 43, 5 (1986)

3 E.H Sondheimer Adv Phys 1, 1 (1952)

4 A.F Mayadas and M Shatzkes Phys Rev B1, 1382 (1970)

5 P Wissmann Springer Tracts Mod Phys 77, 1 (Springer, Berlin, 1975)

Verlagsge-sellschaft, Stuttgart, p 178 (1955)

7 H Bispinck Z Naturforsch 25a, 70 (1970)

Braunschweig, p 265 (1982)

9 G Fahsold, M Sinther, A Priebe, S Diez and A Pucci Phys Rev B65,

235408 (2002)

10 K.L Chopra and S.K Bahl J Appl Phys 38, 3607 (1967)

11 R.E Hummel Optische Eigenschaften von Metallen und Legierungen, Springer,Berlin (1971)

12 P Wissmann and E Wittmann Thin Solid Films 138, L67 (1986)

135 (1985)

14 W Fischer, H Geiger, P Rudolf and P Wissmann Appl Phys 13, 245 (1977)

15 D Dayal, P Rudolf and P Wissmann Thin Solid Films 79, 193 (1981)

16 B.N.J Persson, Phys Rev B44, 3277 (1991)

17 B.N.J Persson, D Schumacher and A Otto Chem Phys Lett 178, 204 (1991)

18 M Rauh, B Heping and P Wissmann Appl Phys A 61, 587 (1995)

19 J.H Ziman Electrons and Phonons, Clarendon Press, Oxford, p 342 (1960)

p 185 (1980)

21 J.M Ricart, J Torras, A Clotet and J.E Sheiras Surface Sci 301, 89 (1994)

22 P Wissmann In: L Eckertova and T Ruzicka (eds.) Growth and Applications

of Thin Films, Prometheus, Prague, p 25 (1994)

23 K Christmann Surface Physical Chemistry, Springer Verlag, New York (1991)

24 G.A Somorjai Surface Chemistry and Catalysis, Wiley, New York (1993)

25 J Torras, J.M Ricart, F Illas and J Rubio Surface Sci 297, 57 (1993)

26 N.F Mott and H Jones The Theory of the Properties of Metals and Alloys,Dover, New York, p 268 (1958)

Mod Phys 85, 1 (Springer, Berlin, 1979)

(1962)

29 D Dayal and P Wissmann Vakuum – Technik 38, 121 (1989)

30 E Schmiedl, M Watanabe, P Wissmann and E Wittmann Appl Phys A

35, 13 (1984)

31 C Mijoule, Y Bouteiller and D.R Salahub Surface Sci 253, 375 (1991)

The electrical resistivity of polycrystalline metal films usually decreases ing an annealing treatment Figure 3.1 shows a typical example measured forsilver films deposited at 77 K on a glass substrate and subsequently annealed

dur-for 1 h at the temperature TA (filled circles [1]) Measuring temperature wasalways 77 K

Fig 3.1 Resistivity ρ of a 50-nm-thick silver film in dependence on the annealing

and Fig 3.7a

A spherical glass cell has been used for the measurements as shown in

Fig 3.2 Details are described elsewhere [2] Two platinum foils F were molten

on the inner wall of the cell It was assured that the tungsten helix H bearingthe silver pearl did not cast any shadow on the deposited film area near the

platinum foils [4] The geometric factor F , necessary to calculate the absolute resistivity ρ from the measured resistance R according to

ρ = Rd

was determined by calibration measurements with copper at 77 K [5] The

film thickness d was estimated during deposition from the resistivity data and

was finally determined after completion of the experiments and chemicallydissolving the silver by atomic absorption spectroscopy The accuracy was

P Wissmann and H.-U Finzel: Electrical Resistivity of Thin Metal Films, STMP 223, 9–34

(2007)

DOI 10.1007/3-540-48490-6 3
Springer-Verlag Berlin Heidelberg 2007c

Fig 3.2 Schematic representation of the glass cell used for the resistivity

measure-ments [2] F are two platinum foils, H is a tungsten helix with a pearl of specpuresilver molten on The platinum wire K serves as a cathode for measurements withthe help of the space-charge limited diode (SLD) [3] G, gas shower; E, electricalfeedthroughs; P, connection to the pumping line

about 5% Occasionally, the films were co-deposited on small glass plates

at the bottom of the glass cell After removing from the cell, the plateswere additionally analysed by transmission electron microscopy (TEM) Theanalysis was possible because silver adheres relatively weakly on the glasssubstrates Hence small portions of silver can be removed from the substratewith the help of a sticking paste and can be transferred to a transmissionelectron microscope A strong asymmetric size distribution is obtained by aquantitative size analysis where a second satellite maximum can be detected

in the high size regime, which is typical for a coalescence stage during crystal

growth [6] We will identify the mean crystallite size in the film plane D ||

with the marked main maximum in the size distribution curve

Many reasons like an enhanced growth of crystallites with a reduction

of crystallite boundary scattering or healing of lattice distortions inside thecrystallites may be discussed in order to explain the dependence shown inFig 3.1 Moreover, a smoothening of the film surface and (111) alignment ofthe crystallites must be taken into account during the heating procedure.The healing of lattice distortions has been extensively discussed in a pa-per of Schumacher and Stark [7] These authors applied Vand’s theory [8]with good success They neglected totally, however, crystallite boundaryscattering On the other hand, the effect of crystallite boundary scatteringcan be quantitatively predicted by the scattering hypothesis on the basis of

Eq (2.1a) In order to quantify the phenomenon, we have determined the

The silver film with a thickness of 50 nm has been deposited on a glasssubstrate cooled with liquid nitrogen [11] The deposition rate was 1 nm/min.

A tungsten wire of 0.3-mm diameter with a pearl of specpure silver molten on

it served as the evaporation source After completion of deposition, the filmswere heated step by step to 480 K in the UHV diffraction chamber while mon-itoring the exact temperature with a Ni/CrNi thermocouple and analysing insitu the (111) peak after each heating step Selected (111) reflections detected

in the XRD chamber are shown in Fig 3.3

The diffracted intensity is plotted versus the double diffraction angle 2θ; parameter is the annealing temperature TA in the range from 87 to 400 K.Only irreversible changes of the structure are discussed; the reversible changes

do not influence the peak shape remarkably The signal/noise ratio is rathersmall because of the absorption of the glass window used in the vacuumchamber, but the intensities of the (111) reflections (Fig 3.3) as well asthe (200) reflections (Fig 3.4) are clearly detectable and sufficient for aquantitative analysis On the other hand, the (220) and (222) reflections

I (a.u.)

I (a.u.) I (a.u.)

Fig 3.4 Selected (200) reflections of the film of Fig 3.3

partially disappear in the background noise and hence cannot be evaluatedwith sufficient accuracy Since the intensity of the reflections is proportional

to the amount of net planes parallel to the film surface [12], we concludethat the crystallite orientation is preferentially (111), and to a smaller degree(200) In order to get more quantitative information on the dependence of filmorientation on annealing, we have evaluated the peak intensity of Figs 3.3and 3.4 with the help of a planimeter

The result is shown in Figs 3.5a and 3.5b where the intensity I is plotted versus the annealing temperature TA In the case of (111) oriented crystals,the curve increases particularly steeply around room temperature At thesame time, the curve for the (200) oriented crystals decreases Obviously, the(111) growth is strongly preferred; one reason for this phenomenon may beseen in the fact that thermally induced strains favour the growth of net planesdensely packed parallel to the film surface [12] Nevertheless, the amount of(200) oriented crystals remains remarkable even at the highest annealingtemperature applied here

The film under investigation has been deposited at nitrogen temperatureand subsequently annealed at higher temperatures The degree of order is

higher, however, when the films are directly deposited at the temperature TA

So a portion of (200) oriented crystals of only 13% was reported for silver filmsdeposited at room temperature and that of less than 1% for films deposited

at 473 K [13] Schlemminger and Stark [14] observed a marked crystal growth

at a deposition temperature of 225 K and traced back their results to a stronginfluence of the Debye temperature on the growth mechanism Discrepancies

in the results may arise from differences in the evaporation rate which is a veryimportant production parameter with respect to the structure properties [15]

In our case, the rate was rather small (0.5 nm/min [11])

A further insight into the crystallite orientation is provided by a polefigure shown in Fig 3.6 [16] Details of the texture analyser used for theseexperiments have been described elsewhere [13] For reasons of clarity, onlythe (111) peaks have been included in Fig 3.6

The half width of full maximum βH is a measure of the scattering of thetexture; the values obtained are listed in Table 3.1

Table 3.1 Scattering of the (111) texture βH (refer to Fig 3.6) for 50-nm-thicksilver films on glass or Si(111) substrates [13]

I (a.u.)

Fig 3.6 Schematical (111) pole figure and texture profile I = f (β)

For a 50-nm-thick polycrystalline silver film, we find βH= 11.3◦, which is

a rather high scattering of the preferred (111) orientation of the crystallitesaround the fibre axis Such a behaviour is typical for a so-called mosaic struc-

ture [16] For single-crystal silver films, βH = 1.5◦ was recorded and hence

a value much smaller than that for polycrystalline films but still larger thanthe experimental broadening of the main peak estimated to be less than 0.3◦from the broadening of the (111) peak of the underlying silicon substrate [17].Next topic in our discussion is the line width of the peaks in Figs 3.3 and3.4 The half width of full maximum is a measure of the mean crystallite size

D ⊥ in the direction perpendicular to the film plane It can be determined onthe basis of Scherrer’s formula [18],

where λ is the wavelength of the copper tube used (λ = 0.15418 nm) and

θB is Bragg’s angle for the (111) or (200) reflections βp is the physical linebroadening which can be approximated in the present case by the difference

of the measured broadening and the instrumental broadening determined arately by a calibration measurement with a highly annealed silver standardsheet [14]

sep-The D ⊥ values obtained from Figs 3.3 and 3.4 are plotted in Figs 3.7a

and 3.7b versus the annealing temperature T

Fig 3.7 D ⊥ depending on annealing temperature for (111) orientation (a) and

An increase is observed for the (111) as well as (200) orientation The

slope of the curves D ⊥ versus TA is again particularly steep around roomtemperature The (200) oriented crystallites are distinctly smaller than those

of (111) orientation For the following discussion, therefore, we will trate on the (111) oriented crystallites, the growth of which can be illustrated

concen-by the qualitative picture of Fig 3.8

No X-ray reflection of an intensity sufficient for the evaluation can bedetected immediately after deposition at 77 K After annealing at 150 K, the

crystal exhibits a more or less spherical shape where D ⊥ is equal to D || but

is much smaller than the film thickness [19, 20] Hence, many crystaIlites arestapled one above the other Crystal growth proceeds particularly drastically

near room temperature At 300 K, D ⊥ = D || still holds as deduced from thecomparison of the TEM value in Fig 3.7 with the X-ray values Now, however,

Fig 3.8 Schematical crystal growth in thin silver films

Fig 3.9 Temperature dependence of the resistivity of a 40-nm-thick silver film [21]

D || ≈ d leads to the qualitative picture of Fig 3.8b It should be emphasised that the assumption D || ≈ d strictly holds only at this temperature, and

hence the application of the equation

ρ = ρ0

1 + K l0d



(3.3)

commonly used for the interpretation of resistivity data is restricted to thetemperature range around room temperature In all other cases, Eq (2.1a)

with D = D || should be preferred [5] This statement is also valid for still

higher annealing temperatures, where D ⊥ reaches a saturation value for

ob-vious reasons, but D ||can further increase because of a stronger extension ofthe crystallites in the film plane (Fig 3.8c)

Equation (3.3) fails completely in explaining the resistivity data if a

crit-ical temperature TCis reached Then the films begin to crack and coagulate,which results in an abrupt increase of the resistivity while at the same time

D ⊥ as well as D || remain more or less constant Figure 3.9 shows an

exam-ple [21] that is untypical in the beginning of the curve because ρ increases with TAcontrary to the usual behaviour reported in [6] (here a compensation

of the influence of crystal growth by the positive temperature coefficient ofresistivity becomes effective)

The TC value depends in a complicated manner on film thickness andthe exact preparation conditions [6] For ultra–thin silver films, an islandstructure can be formed even at room temperature [22]

all the details.

Now we will deal with the roughness of the surface and the preferred [111]orientation An excellent qualitative picture of the surface is provided by thescanning tunnelling micrographs published by Schumacher [23] More quan-titative information is obtained by work function measurements Both theincreasing smoothness and the alignment towards (111) lead to an enhance-ment of the work function [3] because the surface atoms become packed moredensely Consequently, the work function should approach the correspondingsingle-crystal value for flat and well-oriented films The influence of surfaceroughness is predicted by the so-called uniform background model of Smolu-chowski [24] According to this model, the roughness produces an additionaldipole in the surface, which partially compensates the dipole of the well-ordered surface Hence, the net work function is smaller than that for thesmooth films Photoelectric measurements are a very suitable method to de-termine the work function Then one should keep in mind, however, that thelowering is amplified by the fact that rough portions of the surface enter the

average of the photoelectric yield Y with a higher weight as compared to

measured at different measuring temperatures and no dependence of the workfunction can be detected Moreover, any influence of film thickness is missing

in Fig 3.11

The data of two independent authors confirm the excellent reproducibility

of the photoelectric results The mean value of 4.38 eV differs slightly from theproperties of (111) oriented silver single crystals (4.5 eV [31], reconfirmed bythe values of Dayal [28] for (111) oriented single-crystal films) For unannealed

Fig 3.10 Fowler-plot according to Eq (3.4) for a 45-nm-thick silver film deposited

at room temperature; parameter is the measuring temperature [29]

Fig 3.11 Work function of thin silver films on glass substrates in dependence on

films at 77 K, a value of 4.2 eV [6] was found, which is again only 0.3 eVlower than the (111) single-crystal value It is noteworthy that ultra-thin filmswith an inhomogeneous structure cannot be investigated by photoelectricmethods because of instabilities in the photocurrent Figure 3.12 shows a

typical example for a 15-nm-thick silver film where the photocurrent I is

plotted versus the time The arrows indicate the switch-on and switch-off ofthe light One easily recognises that charging of isolated silver islands has alarge influence on the photoelectric yield; the kinetics show time constants inthe minute range

Measurements with the space-charge limited diode (SLD method [3], anexample of the experimental arrangement was still shown in Fig 3.2) are not

a realistic alternative procedure because only relative changes in the work

function can be deduced from the shift in the IV characteristics [3] A more

Fig 3.12 Kinetics of the photocurrent I for a 15-nm-thick silver film on glass [32]

quantitative correlation of the changes in the work function with the asperity

height B in Eq (2.1c) seems difficult for the time being Unfortunately, the

definition of the surface roughness is not unique in the literature; a discussion

of the problems involved was presented in [6] Summarising, we may statethat the reduction of crystal boundary scattering is the dominant processduring annealing of thin silver films It covers about 70% of the observedresistivity change The residual 30% must be attributed to an ordering ofthe lattice inside the crystallites as well as to an increasing smoothening and

an enhanced (111) orientation of the crystallites Similar results have beenobtained previously for polycrystalline nickel films [5]

The highly annealed films are astonishingly stable against oxidation Wehave floated the vacuum cell with 15 mbar pure oxygen and have studied thestructural changes of the film during heating for another hour at 478 K Thehalf width of full maximum value remained stable after this treatment, andthe colour was still metallically glancing instead of milky white characteristicfor the formation of Ag2O Obviously, the interaction with oxygen is restricted

to the uppermost silver layers even at these high temperatures [33] A moredrastic reactivity, however, is expected for thinner or rougher films

Another prediction of scattering hypothesis is related to the dependence ofthe film resistivity on the measuring temperature The absolute temperature

coefficient of resistivity dρ/dT (ATCR) should be equal to the ing bulk value dρ0/dT , independent of the film thickness and the annealing

Fig 3.13 ∆ρ/∆T in dependence on the film thickness d [21]; the straight line

corresponds to the bulk value [15]

One easily recognises that the prediction of Eq (3.5) is fulfilled in thethickness range under investigation; i.e the ATCR of the films is equal tothe bulk value For thinner films, however, an increase of ATCR values isexpected due to enhanced structural inhomogeneities [34,35] A typical resultfor about 10-nm-thick silver films was published by Wedler et al [22] whereannealing even leads to negative ATRC values

The resistivity behaviour of the silver films is totally changed if singlecrystal films are included into consideration Such films can be prepared by

an epitaxial growth of silver on Si(111) single crystals The crystals werechemically polished according to the usual etching procedure of Chang [36]and were then put into the sample holder of a separate vacuum apparatus

in order to deposit about 1-µm-thick contact layers With the help of a able mask, the tantalum layers were produced and connected with electricalfeedthroughs by tantalum springs The contact layers were found to be nec-essary for the resistivity measurements in order to bridge the highly ohmictransition resistivities between the tantalum springs and the silicon crystalscovered with the silver films under investigation [6] On the other hand, nocontact layers were used in the case of the X-ray diffraction as well as theoptical studies in order to avoid an occasional shadowing of parts of the film

suit-by the electrical connections during deposition

source described in a proper manner [37].

Immediately after insertion of the crystal into the substrate holder, rities containing carbon and/or oxygen were detected in the Auger spectra,but never any traces of tantalum from the contact layers All contaminantsquantitatively disappeared after flashing the sample at 1500 K, and finallythe well-known 7× 7 superstructure of silicon developed in the LEED pat-

impu-tern With this superstructure characteristic for very clean silicon surfaces

in (111) orientation [38, 39], we have started to deposit the silver The filmgrows epitaxially in the Stranski–Krastanov mode [40] Since the silver atomsare weakly bound to the silicon surface, the mobility of the silver atoms israther high As a consequence, the film formation resembles a layer-by-layergrowth at least at higher coverages [41]

The orientation is (111), but the films exhibit small irregularities in thesurface as can be derived from the broadening of the LEED pattern, whichhas been published in [6] They clearly show that large angle grain boundariesare absent in the films except at very low thicknesses

The electrical resistivity of the films increases with increasing annealingtemperature A chemical reaction between silicon and silver can be widelyexcluded [42] Nevertheless, the resistivity versus temperature curve runsthrough a maximum at about 473 K as can be seen in two typical exam-ples in Fig 3.14

Fig 3.14 Resistivity–temperature curves for two thin silver films of different

thick-ness on Si(111) [43] The ordinate scaling is logarithmic

of holes and silver islands Consequently, the resistivity increase becomessteeper The thinner the films are, the lower are the temperatures where thecoagulation starts, similar to the case of glass substrates Hence, the poly-crstalline and the single-crystal silver films show a similar response to theannealing treatment in spite of the fact that the large–angle grain boundariesare missing in the case of single-crystal substrates, and Eq (2.1a) cannot beused to explain the experimental evidence Moreover, an upper limit of theresistivity increase is reached when the bridging of the silver islands by thesemiconducting silicon substrate becomes effective After a transition region,the decrease follows an experimental law as expected for pure silicon [39].Some samples were transferred into a scanning electron microscope (SEM)after completion of the electrical measurements in order to get information

on the shape and the size of the silver islands Figure 3.15 shows an examplefor a silver film annealed at 623 K [44]

The island density varies when different regions of the substrate are tigated The boundaries of the islands exhibit sixfold symmetry; the lateral

inves-extension D || scatters around a mean value as shown in Fig 3.16

Fig 3.16 Size distribution for the crystallites of the silver film of Fig 3.15 [44]

The distribution curve has at least two maxima, which is typical for a

coalescence stage during crystal growth [6] We identify D || with the mainmaximum at 2.5µm is much larger than the initial film thickness d = 11 nm

[44]

A more detailed evaluation of scanning electron micrographs for theAg/Si(111) system was published by Venables et al [41] These authorsshowed that the sidewalls of the islands are inclined in various orientations,preferably in (111) orientation Sometimes, the walls exhibit a curvature or afaceting The well-known√

3 × √3 superstructure was detected in-betweenthe silver island [6] One should be very cautious, however, with the quanti-tative transfer of these findings to the present investigation We obtained a

D ⊥ /D || ratio (refer to Fig 3.16 and XRD analysis in the following) around0.012, which is remarkably smaller than the values published by Venables

et al [41] This discrepancy points to the fact that not only initial film ness and annealing temperature are important parameters for the islandsstructure, but also deposition rate, heating rate, lattice fault density in thesilicon surface etc

thick-The samples were additionally put into the manipulator of a highly solving X-ray diffraction analyser (Philips PW 1710 in the Bragg–Brentanofocusing mode [45]), where the illuminated area widely corresponds to thetotal area of the samples Only the Ag(111) and Ag(222) peaks could be de-tected, and fortunately there was no overlapping with the relatively strongSi(111) reflection characteristic for the silicon single-crystal substrate.Figure 3.18 shows a typical example of the (111) reflection on a magni-fied scale One easily detects two satellite peaks positioned symmetrically to

re-the main peak and indicated by arrows The angular distance ∆(2θ) of re-the

Fig 3.17 X-ray diffraction pattern of a 11-nm-thick silver film on Si(111) [44]

Fig 3.18 Ag(111) peak of the film of Fig 3.17 For details, see the text

two satellites opens the possibility of an uncomplicated determination of the

crystallite height D according to [46].

where λ and θBhave the same meaning as in Eq (3.2) The evaluation yields

D = 30 nm in good agreement to the value obtained from Eq (3.2) Such

satellites are typical for crystals where the mean extension in the directionperpendicular to the film plane does not scatter much around a mean value

In the present case, the quantitative evaluation yields a scattering width of

13 nm which corresponds to roughly 43% of the island height [47]

Summarising, we state that the islands are higher than the initial film

thickness (d = 11 nm) but much lower than the lateral extension (D =

2.5µm) Hence the islands exhibit a plate-like shape

In this thickness region, the p-conducting silicon is doped by the n-conductingsilver, which causes the sample resistivity to increase Only for a sufficientlylarge amount of deposited silver, a lower ohmic conducting path is established

at the surface parallel to the higher ohmic substrate material, thus reducingthe resistivity drastically

The island structure is still more pronounced if polycrystalline films areinvestigated Such films can be produced using silicon substrates that havenot been flashed prior to deposition but which have only been heated at 673

K for 1 h in the UHV chamber A distinct oxygen peak can be detected in theAuger spectra; the films behave similarly to the case of glass substrates [49].The measured resistivity versus thickness curve is shown in Fig 3.19

A relative increase in resistivity of 15% is observed and about 9 nm ofsilver are necessary to produce the first coherent conducting paths of sil-ver before the characteristic resistivity drop for homogeneous films emerges.Here local pn-transistions between the n-conducting silver islands and thep-conducting silicon seem to be responsible for this shift of the resistivitydrop to higher thicknesses

Fig 3.19 Thickness dependence of the resistance of ultra-thin Ag films deposited

on slightly oxygen-covered Si(111) crystals [48]

Fig 3.20 Schematics of the four-pole contacts for glass (a) and silicon (111)

substrates (b) A indicates the illuminated area and C the re-entrant cavity

Another problem of actual interest is the relationship between resistivityand optical properties of the films as mentioned in Chap 2 In the framework

of Drude’s theory [50, 51], the dielectric function ε = ε1 − iε2 can be culated by

cal-ε1= 1− ωp2

ω2; ε2= ωsω

2 p

Here, ne, e and me are the density, charge, and mass of the free

elec-trons, respectively ε0 is the dielectric constant of the vacuum ωs and ωp

are the collision and plasma frequency Two different experimental

arrange-ments have been used to measure the dielectric function ε and the resistivity

ρ independently on the same sample.

a) The polycrystalline films were prepared in a UHV glass cell that wasadjusted in the beam path of an automatic ellipsometer Four tungstenterminals molten in the glass substrate were polished optically flatly, al-lowing an in situ recording of the electrical resistivity (Fig 3.20a [51]).The re-entrant cavity C of the substrate could be cooled to 77 K by filling

in liquid nitrogen or could be heated up to 500 K by introducing a able heating device The thickness of the silver films was determined byatomic absorption spectroscopy after completion of the experiments andchemically dissolving the films in HNO3

suit-b) In the case of single-crystal films, the measuring cell consists of a stainlesssteel vessel with two glass windows at a fixed angle of incidence of 70◦.Silicon single-crystals of 0.5-mm thickness serving as substrates were cut

Fig 3.21 Schematical two-layer (a) and three-layer model for rough (b) and

The dielectric function ε was calculated on the basis of the two-layer

model shown in Fig 3.21a Applying Fresnel’s formulae leads to [53]

inci-kF which can be determined using an iteration procedure [54] The

transfor-mation into the dielectric function ε = ε1− iε2 is given by

ε1= n2F− k2

In order to check the validity of Eq (3.7), we have evaluated the ε2values

as well as their dependence on temperature analogous to Eq (3.5),

Fig 3.22 ε2 (a) and ∆ε2/∆T (b) in a double-logarithmic plot as a function of

the wavelength λ for a 33-nm-thick silver film The straight lines have the slope 3,

the higher range investigated here So we restrict the further discussion to

λ = 1152 nm in the near infrared where Drude’s theory should represent

a good approximation Complications arise, however, for wavelengths lowerthan 500 nm where Drude’s theory does no longer hold because of the well-known plasmon excitation in silver [50]

A plot of ε1versus film thickness is given in Fig 3.23 One easily recognisesthat the data scatter strongly around the literature value [56] A systematic

dependence on d cannot be detected in agreement with Eq (3.7) We attribute

the strong scattering to the influence of surface roughness that cannot becontrolled with sufficient accuracy in spite of the fact that all preparationparameters have been kept as constant as possible in Fig 3.23

The roughness should be described by a third layer at the vacuum/filminterface, but the optical properties of this intermediate layer are widelyunknown (refer to Fig 3.21b) Attempts to describe such a rough overlayerhave been published by Fenstermaker and Mc Crackin [57] who made clearthat the assumption of only one overlayer may not be sufficient for a goodagreement with the experimental data

As far as the relation between ε2 and ρ is concerned, it should be

em-phasised that both quantities have been measured simultaneously for eachfilm under investigation Figure 3.24 clearly shows that a recalculation of theresistivity from the optical data is, strictly speaking, not allowed Obviously,the surface roughness has a quite different influence on the optical [51] andelectrical [6] data

In this context, it is very interesting to include the temperature

depen-dence of ε into discussion because ∆ρ/∆T of the same films has been

Fig 3.23 ε1 in dependence on film thickness d (λ = 1152 nm [21]); the solid line

corresponds to the literature value [56]

Fig 3.24 ε2 in dependence on ρ [21]; the solid line corresponds to Eq (3.7)

presented earlier in Fig 3.13 The correlation between ∆ε2/∆T (refer to

Eq (3.10)) and ∆ρ/∆T is given in Fig 3.25.

The scattering of the measured points is again evident; no connection

can be detected in spite of the fact that ωp in Eq (3.7) should not depend

on temperature Solving this discrepancy, some authors introduce an tical mass’ which differs from the well-known electron mass [50] Such anassumption, however, has only a formalistic character and does not specifythe effect of surface roughness A further possible explanation is based onso-called thermally induced strains [12] Their origin is the difference in thethermal expansion coefficients of film and substrate Hence they are particu-larly effective if annealing and measuring temperature are not the same Thestrains can change the plasmon frequency and the curvature of the energybands [58]

‘op-Fig 3.25 ∆ε2/∆T in dependence on ∆ρ/∆T ; the solid line corresponds to

Eq (3.7)

Fig 3.26 ε2for thin single-crystal silver films evaluated on the basis of a two-layer

model (refer to Fig 3.21a [44]) Parameter is the film thickness d

The optical behaviour totally changes if single-crystal silver films are sidered The preparation of these films has been described above The eval-uation is again based on the two-layer model of Fig 3.21a, where one layer

con-is the silicon substrate The evaluation of the wavelength dependence of ε2

leads to an enhancement compared to bulk silver and the enhancement islarger when the films are thinner (Fig 3.26 [44, 59])

Such resonances are well known from the excitation of surface plasmonpolaritons at the transition between planar noble metal films and silicon[60,61] Because of completeness, we should mention that the optical influence

Fig 3.27 Wavelength dependence of the imaginary part of the dielectric function

of the pure silicon substrate has been eliminated by the evaluation in Fig 3.26[44]

The problem arises, however, that such an excitation by light is usuallyforbidden for flat films because of basic considerations of momentum conser-vation [62] Following the arguments of Raether [63], the residual roughness

of the silver films can promote the generation of surface plasmons by light.The most striking example for the latter case is the extraordinary amplifi-cation of Raman signals on rough silver surfaces, usually known as SERS(surface-enhanced raman scattering [64]) effect

If we look at the Ag/Si(111) system, the roughness may be caused by anembedding of single silver atoms in the 7 × 7 superstructure of silicon [38]

by steps and surface defects in the silicon surface or by the formation of a

√

3× √3 R 30◦superstructure [39] In all cases, isolated silver atoms coupled

to the silver film are embedded into the rough neighbouring silicon surface

with the extremely high ε value (εS ≈ 15 [65]) This coupling seems to be

the dominant factor for the strong red shift of the wavelength of the surfaceplasmons

The resonances are still present if the films are annealed at 623 K under

UHV conditions Figure 3.27 shows the spectra of ε2versus λ for six films of

about 30 nm islands’ height [44], the lateral extension of the islands is much

larger (D || ≈ 2.5 µm; refer to Fig 3.15) Since the penetration depth of light is

too small for an excitation of surface plasmons at the silicon/silver interface,

we expect that the excitation mainly occurs at the more transparent fringearea of the islands

One can further try to use classical Mie theory for the interpretation ofthe results Mie plasmon polaritons develop in noble metal clusters even in

the visible spectral range [62] and for substrates with ε2 values much lowerthan those of pure silicon Unsolvable difficulties, however, arise while doingso:

a) The quasi-static approximation as well as the Maxwell–Garnett theory [62]

are not applicable because the condition D ||  λ does not hold.

b) The island size scatters around a mean value, and the shape varies markably from island to island

re-c) The islands are not embedded in a homogeneous medium as assumed inthe Mie theory, but have contact to both the vacuum and the siliconsubstrate

d) The islands show strong deviations from the sphere shape postulated inthe Mie theory; details of the exact specification of the side boundarieslike faceting or inclination angles are unknown

Hence, essential extension of the Mie assumptions are necessary in order

to check the applicability of the theory to the present results in future work.Nevertheless, we mainly attribute the resonances in the near infrared to sur-face plasmon polaritons, for the islands should behave quite similar to planarfilms because of the extremely large lateral extension of the islands

In conclusion, we state that a recalculation of the electrical resistivity fromthe optical data is not possible for the silver films evaluated here Thus, thevalidity of Eq (3.7) is limited due to the different influences of the structural

peculiarities of the films on ρ and ε.

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