Experimental evidence for quantum size effects in ultrathin metallic films

Progressin SurfaceScience,Vol. 48, Nos. 14, pp. 287-297.1995 Copyright 0 1995Elsevier ScienceLtd Printed in the USA. All rights reserved 0079-6816195$29.00+ .OO

Pergamon

OW9-6816(95)EOOO29-1

Experimental

Evidence in Ultrathin

for Quantum Size Effects Metallic Films

M. Jafochowski

Institute of Physics,Maria Curie-SklodowskaUniversity, Pl. M.Curie-Sklodowskiej1, PL-20031 Lublin. Poland Abstract

In recent years it has become clear that size quantization in ultrathin metallic films can influence many physical properties but most effects are rather weak. Due to short de Broglie wavelength of electrons at Fermi level most effects can be Recent observed in extremely thin and smooth films. achievements in technology of ultrathin metallic films allow to verify early theoretical predictions. Among them the resistivity and electron of specific measurements photoemission can be used as useful and sensitive tools. Clear QSE effects showed ultrathin layers of Pb, Pb-In alloy and In deposited on Si(lll)-(Gx6)Au substrate. Using Reflection High Energy Electron Diffraction (RHEED) technique the layered structure of ultrathin films was examined and correlated with QSE effects. Abbreviations

FWHM JCDOS) MPF ML

QSE

RHEED UHV UPS

Full Width at Half Maximum Joint (Density of States) Mean Free Path Mono-Layer Quantum Size Effect Reflection High Energy Electron Diffraction Ultra-High Vacuum Ultra-Violet Photoemission Spectroscopy 287

M. Jatochowski

1. Introduction Through continuing evolution of the ultrathin metallic film technology many predictions of Quantum Size Effect (QSE) theory con be nowadays experimentally verified. These include firstly quantum size effects in properties of metallic films. A large amount of theoretical work has been done on this subject (for references see Refs 1I2I3I4 ,5,6I 7I8). Early theories [91 considered only scattering by randomly distributed centres with delta-potential and surface scattering was not included. Using Boltzman density, electron

equation Sandomirskii [91 have obtained formulas for carrier conductivity, Hall constant and magnetoresistance of semimetal

(Bi) film. He obtained an oscillatory dependence of the electrical resistivity of semimetalic films upon thickness. The period of oscillation was of one-half of the Fermi wavelength hi . Bi was chosen as favourable candidate for observation of QSE because of relatively large de Broglie wavelength. On other hand complicated band structure of this material makes difficult interpretation of experimental data and little convincing experimental evidence of QSE is available for Bi. Using fully quantum-mechanical calculations Govindaraj and Devanathan 121 have calculated the resistivity of thin aluminium and copper films. Their result was in good agreement with the result of Sandomirski. The influence of surface roughness on the conductivity of size-quantized thin metal films was studied by Leung 111. He introduced the autocorrelation function of the rough surface profile and pointed out the importance of surface scattering in the description of the electron transport in ultrathin metal films. Tesanovic et al. [4] discussed the effect of surface scattering on quantum transport in thin film with uncorrelated atomically rough surfaces. They found that the full quantummechanical treatment of very pure systems leads to results qualitatively different from those of quasiclassical theory. More recently Trivedi and Ashcroft [3] thoroughly discussed the influence of random surface roughness on the QSE. In their work surface roughness is incorporated as a boundary condition on the Hamiltonian, and for sufficiently small variation of the thickness the problem is handled perturbatively. They also included large-scale thickness fluctuations on a scale larger than the mean free path (MFP) by breaking up the film into units with slightly different thicknesses. Most recently the influence of surface roughness correlations on the surface conductivity of thin films was studied theoretically by Fishman and Calecki [5,7,81. All recent theoretical works stress importance of microscopic surface morphology on electron transport properties of ultrathin metallic films showing QSE effects. So far little convicting experimental evidence of QSE in resistivity of metallic films is available. We have succeeded in growing Ag, Au 1101, Pb 1111, and Pb-In

289

Quantum Size Effects in Films

alloy[l2]

single-crystal

films

on a semi-insulating

substrate,

Siflll),

in quasi-

monolayer-by-monolayer fashion and have measured the electrical resistivity of the growing films. In Pb and Pb-In clear oscillations originating from QSE effects were observed. In contrary to resistivity experiments where only electrons near Fermi level are involved photoemission, in particular the angle-resolved version provides information about the electronic structure in the bulk and at the surface of solid. This technique is particularly well suited for the determination of quantum size energy levels in thin films.

The UPS studies of Ag films on Si(ll1)

1131,Na and Ba

on Cu(ll1) [141, Ag on Au(ll1) [151 and Ag on Cu(ll1) [16] have significantly advanced our understanding of QSE’s in thin metallic films. Energy band structure above Fermi level in quantum wells of Fe films embedded in Au(100) was studied using inverse photoemission 1171. Both photoemission and inverse photoemission methods were used to analyse band structure of quantum-well states and oscillatory magnetic coupling in thin films composed of ferromagnets and noble metals [18]. Recently the photoemission from small Ag islands formed on GaAs(ll0) was observed and the oscillating features were analysed in term of commensurate states introduced in fundamental electron tunneling experiments of Jaklevic et al. [19]. All experiments

stressed importance

of thickness fluctuation

smearing out the QSE

features. Therefore we choose the well-defined system Pb/Si(lll)-(bx6)Au because of approximate layer-by-layer growth*of Pb and Pb-In on cooled substrate. Using simple model of finite quantum well and taking into account the mode of growth we were able to calculate photoemission intensity as a function of Pb film thickness and qualitative agreement between calculated and experimental data was achieved

m. Less attention was paid other experimental techniques although theoretical works clearly show on possible observation of other size-dependent effects. Schulte [21] has calculated

electron density,

potentials

and work function

of thin metal films.

All

three parameters showed oscillations with a period of one-half Fermi wavelength. Fibelman [22] has calculated work function for Al(111) and Mg(0001) films composed of monoatomic slabs and concluded that QSE should be observable as oscillations versus number of slabs. More detailed discussion of quantum-size-effects in Al(111) layers was presented by Batra et al. [23]. Variations of work function measured by means two independent methods, e.g. a version Kelvin method and diode method were observed experimentally in polycrystalline indium deposited on polycrystalline, texturized gold substrate by Marliere [24]. The period of oscillations was about Inm. Using photoelectric method we observed variations of joint density of states (JDOS) during evaporation of ultrathin Pb films.

M. Jdochowski

2. Experimental

Details

Specific resistivity and photoemission experiments were performed in UHV molecular-beam epitaxy system equipped with various electron-and photon induced electron spectroscopies. The system was pumped by a titanium sublimation pump and by an LN2 - baffled diffusion pump which produced a base pressure of 4x10-9 Pa and maintained a pressure below 1x10-9 Pa during deposition. Pb and In were deposited from MO crucibles and Au from a W crucible by electron-bombardment heating. The substrates were Sit1111 single crystals with 1000 Rem resistivity at room temperature and typical dimensions 16x4x0.8 mm3. They were polished, etched in 19:l HN03 + HF solution, rinsed in distilled water and methanol, and mounted in MO holder. The final surface cleaning consisted in flashing for a few seconds to about 1500 K. This treatment produced a sharp (7x7) superstructure RHEED pattern. Direct resistive heating of Si crystal was used. The (6x6)Au superstructure was produced using the method described in Ref. [12]. The electrical resistivity was measured as follows. The signal from an ac generator was multiplied with the dc signal from quartz-crystal monitor which is proportional

to the mass of the deposited film. The ac-dc product voltage was applied to the Si substrate. A signal was obtained from potential contacts consisting of electrochemically etched W wires pressed against the Si crystal. For photoelectron spectroscopy we used the He I line (21.22 eV1 from a Leybold cold cathode capillary discharge lamp. The spectra were recorded with an energy resolution of better than 100 meV. An aperture of the entrance of the analyser input lens reduced the acceptance to a cone with 1 half angle. The angle between incident light and detected electrons was always 36’. Integrated

photoemission

and optical reflectivity

experiments

were performed

in

other UHV system with a base pressure of 4x 10-g Pa .The system was equipped with RHEED, set of evaporators, precise quartz-crystal monitor, liquid helium cryostat and sapphire and/or glass windows. 20 W deuterium discharge lamp with monochromator was used to excite photoelectrons. Emission of photoelectrons as a function

of continuously evaporated Pb film or as a function of photon energy was measured directly by means of electrometer. The sample was biased with -18 V and no angular selection of emitted photoelectrons was performed. To improve signalto-noise ratio the electrometer was operating in the charge mode. To eliminate possible spurious currents the light beam was chopped 10 times per minute and background signal was numerically subtracted.

291

Quantum Size Effects in Films

3. Results and Discussion A. Film structure and growth model For our studies we need atomically smooth samples with well-defined thickness and crystalline structure. Common method for the determination of growth mode of ultrathin films is the analysis of RHEED data. The structure of thin Pb and Pb-In films was determined from RHEED pattern. The full widths at half maximum (FWHM’s) of RHEED streaks, after subtraction of the instrumental broadening were about 0.04 A NO%), corresponding to an average terrace size of about 25 A. Fig 1

35-

305 s25L cii zzo5n15E iti IO-

5-

0 0 2

4 6 8 1012141618202224262'3

THICKNESS (ML)

Fig.1. RHEED specular beam intensity oscillations during growth of Pb, Pb-lO.l%In, Pb-46.9%In and In films on Si(lll)-(6xb)Au surface at 1OOK [112] azimuth, glancing angle is 0.3’ . shows the intensity of the specular beam as a function of the thickness of Pb, Pb-In alloy and In layers. For the Pb- In alloy more than 200 periods of oscillations could be observed easily. The damping of oscillations changes with amount of In and at around 30 at % of In the oscillations

were best developed.

The amplitude

of RHEED

intensity oscillations is commonly regarded as a measure of the growth mode. For perfect monolayer-by-monolayer growth 2 ML of thin film are involved in the formation of the growth front. For less perfect growth 3 or more ML are growing simultaneously. Our comparative calculations of the thickness-dependent coverage performed with “distributed growth mode” have shown that Pb and Pb - In films

M. Jabchowski

292

grew in almost perfect monolayer-by-monolayer mode 1201. Fig 2. shows calculated thickness-dependent coverage 0, the specular beam intensity and rms surface roughness 6d/d for two adjustable parameters A [25] describing the growth perfection. The main feature of these results important for our further discussion is oscillation both RHEED intensity and rms surface roughness with period of 1 ML.

“0

A-=o.%s

$ 0.5 V

O-O m5 THICKNESS

(ML OF Pb)

Fig. 2. The coverage 0 (solid), the specular beam intensity I (dashed) and rms surface roughness (6d/d12 [in (MLJ2 units1 calculated according to the “distributed growth model” [251. B. Fine structures in the resistivity

Fig. 3a shows the specific conductivity data for Pb, Pb - In and In films measured during evaporation on cooled to 90 K Si(llll46x6)Au substrate. Similar to our earlier investigations 1111the experimental data could be fitted by the Sondheimer approximation to the Fuchs formula for d>I with d the thickness d and I mean free path as follows: pW=p,[l+3N1-/71/84

(3.1)

The specific resistivity of the bulk material pm and the parameter I(l-p) were obtained from a last-squares fit within the thickness range in which the fine structure in the resistivity is week. This range began typically at about 15 A. Fig. 3b shows the data obtained after subtraction of l/p(d) calculated according to Eq. (3.1)

293

Quantum Size Effects in Films

from

the measured

data.

Three

distinct

oscillations

oscillations

in all samples, 2 ML period of oscillations

oscillations

in In.

are seen: 1 ML

period

in Pb and 3 ML period of

6

Pb+GOwln

15 THICKNESS

20 (ML)

25

30

-21r 0

I”‘~,~~I~/‘~~~I~III,~ 10 15 20 25 THICKNESS (ML)

30

Fig.3. Specific conductivity vs thickness of film of Pb, Pb with 60% In and pure In measured at 1OOK (a), and specific conductivity differences between experimental data and calculated according Eq.(3.1) for the same samples (b). Detailed discussion of these effects is given in the Ref. [12] where quantum theory of Trivedi and Ashcroft [3] was applied. The 2 ML periodicity in Pb is caused by the size quantization of the energy band structure while the 1 ML periodicity results from the periodic variation of the surface roughness during monolayer-by-monolayer growth. In lead and in indium the de Broglie wavelength at the Fermi level ELF are in general incommensurate and and the thickness of the separate islands nd, mhF =nd, (QSE condition) is fulfilled only over limited the matching condition thickness ranges for which mhF = nd,. For heat, =3.9118 A [121 and Xl+= 6.16 A 1261 this occurs for Pb with every 2 ML with condition 3(3L,1+/2)=2ML Pb(lll) and for In with every 3 ML with condition 4(h~l,,/2)=3 ML In (10~) This is consistent with the data of Fig. 3 where these conditions are clearly marked by 1 ML periodicity This periodicity originated from periodic variation of the surface roughness. corresponds also the oscillations of electron specular beam intensity recorded during RHEED experiments presented in Fig. 1.

294

C. Modification

M. Jaiochowski

of the photoemission

by quantum size fffects

Thorough analysis and discussion of QSE during photoemission from Pb and Pb-In quantum wells were recently presented in Ref. 1201. Here we describe briefly the main results of these investigations. Examples of photoemission intensities of Pb and Pb - 30%In films as a function of thickness are shown in Fig. 4. In contrary tothe resistivity measurements where only energy levels at the Fermi level are involved, the photoemission samples the band structure below EF..

Fig. 4. Photoemission intensities of Pb, and Pb30%In thickness. The parameter is the binding energy Eg in eV.

films

as a function

of

-1.0 s-o.5 al - 0.0 L u 0.5 iii w 1.0 0 f, 1.5 0 f, 2.0

m 2.5

0

5

10 15 THICKNESS

20 25 (ML)

30

Fig. 5. Theoretical (dots) and experimental (open squares) data of the QSE levels in thin Pb(ll1) films as a function of the film thickness in steps of 1 ML [20]. Simple calculations for the finite-potential quantum well give the sets of energy levels shown in Fig. 5 and are compared in this figure with the Eg values of the observed photoemission

intensity

maxima.

The dots denote the calculated

values

295

Quantum Size Effects in Films

whereas the open squares are the experimental

data.

Several features influence

emission of photoelectrons from quantum well. conservation rule for the normal component photoelectron inelastic mean free path was taken The photoelectron intensity presented in Fig. growth mode. This was confirmed by calculations

In Ref. [20] the relaxation of the of the momentum and finite into account. 4 was strongly modified by the presented in Fig. 6.

60

0 0

4

8 12 16 20 THICKNESS (ML)

24

28

0

4

E 12 16 20 THICKNESS (ML)

24

28

Fig. 6. Calculated photoemission intensity as a function of the average Pb film thickness. The parameter is the binding energy En in eV [20]. D. Joint density of states Schulte [21] discussed electron density and work function of thin metal film. Numerical results for their dependence on film thickness showed oscillations with a period of one-half the Fermi wavelength. Similar results were obtained by Rogers varied

et al. [27] . Both results were

obtained

for the films with continuously

thickness.

Obviously this is not realistic assumption. The thickness of a crystalline film cannot be varied infinitesimally. QSE for crystalline film was studied theoretically by Fibelman [22]. As in the case of resistivity and photoemission effects variation of the thickness with increment of 1 ML which in general is incommensurate with de Broglie wavelength modifies strongly oscillating behaviour of QSE. Near-ideal condition occurs in Pb(ll1) thin films. On the basis of Hartree - Fock band calculations Saalfrank [28] has found oscillations for Fermi energy, electron distribution, and the electronic densities of states at the Fermi level. Variation of last quantity can be detected by means of electron photoemission. Angular integrated photoemission simultaneously probes various positions in the Brillouin zone and the spectra obtained correspond to joint density

296 of states (JDOS) a. written as:

M. Jabchowski

The measured integrated photoemission intensity can be

I(kw) 0c.1PwJ T(E)N&J

(3.2)

with T(E) being the function related to the electron escape depth h(E) [291. The

0.0 4

]

(

(

(

,

(

,

0123456$ NUMBER OF MONOLAYERS

Fig.7. (a) Integrated photoemission intensity of Pb(llI) film as a function of thickness . The parameter is photon energy ho in eV. (b) The slope of photoemission intensity vs photon energy dependence (solid) and theoretical DOS at the Fermi level [28] (dashed 1. escape function D(E) accounts for the potential step at the surface caused by the work function , With +f~ close to work function it will certainly sample the DOS at the

.

Fermi level. Fig. 7a shows thickness dependent integrated over angle photoelectron intensity for the growing sample Pb on Si(lll)-(6xG)Au held at 70 K irradiated with photons with tiw = 4.2 up to 5.6 eV with increment of 0.1 eV. Fig. 7b presents the slope of I@$ within energy range from 4.5 to 5.0 eV together with theoretical data for DOS at the Fermi level calculated by Saalfrank [281. We stress also the difference between two plots: the theoretical data are the DOS at the Fermi level whereas experimental data represent rather JDOS for which DOS at Fermi level is only one component. Another important feature is the onset of /fhw) which according to the expression (3.2) corresponds to the work function.

297

Quantum Size Effects in Films

Within

experimental

error this quantity

remains constant.

This is in contradiction

to the results of the work of 1281 where clear 2 ML periodicity was predicted. On other hand the used experimental method may not be applicable in the system with discrete energy levels and this question remains open. Acknowledgements This work was supported in part by Grant No.2-0382-91-01 and in part by the Deutsche Forschungsgemeinschaft. The author thanks E.Bauer for stimulating discussion, and H.Knoppe, G.Lilienkamp and M.Strbjak for their help.

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