Surface Reactivity and Quantum-Size effects on the Electronic Density Decay Length of ultrathin Metal Films

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by  N. Binggeli

Surface Reactivity and Quantum-Size effects on the Electronic Density Decay Length
of ultrathin Metal Films
N. Binggeli1,2 and M. Altarelli1,3
1 The Abdus Salam International Center for Theoretical Physics, Trieste 34014 , Italy
2INFM-CNR DEMOCRITOS National Simulation Center, Trieste 34014 , Italy
6 3 European XFEL Project Team, DESY, Notkestraße 85, 22607, Hamburg, Germany
0
0 The origin of the correlation between surface reactivity and quantum-size effects, observed in
2 recentexperimentsontheoxidationofultrathinmagnesiumfilms,isaddressedbymeansofabinitio
calculationsandmodelpredictions. Weshowthatthedecaylengthinvacuumoftheelectroniclocal
n
densityof statesat theFermienergy exhibitssystematic oscillations with film thickness,with local
a
J maxima induced when a quantum well state at kk = 0 crosses the Fermi energy. The predicted
changes in the decay length are expected to have a major impact on the electron transfer rate by
7
1 tunneling,which has been proposed to control the initial sticking of O2 in theoxidation process.

PACSnumbers: 73.20.At,73.21.Fg,82.65.+r,68.43.-h
]
i
c
s
- There is considerable interest in identifying methods is closest to the Fermi energy. The predicted changes in
l
r totailorthechemicalreactivityofsurfaces,acrucialfac- the decay length are significant. They are expected to
t
m tor in many technologically relevant surface phenomena, have a direct, exponential impact on the electron trans-
including oxidation and catalysis. The recent observa- ferratebytunneling,whichhasbeenproposedtocontrol
.
t tion of a correlation between quantum-size effects and the initial sticking of O on the metal surface [3]. The
a 2
m the surface reactivity of ultrathin metal films [1] is an changesin λ shouldbe observableby scanning tunneling
exciting development in this area, both because of its microscopy (STM), near terrace steps, on surfaces char-
-
d fundamental interest for a quantitative understanding of acterized by regions of different film thicknesses [1].
n the structure-size dependence of the reactivity and in Model predictions for λ.—Withinanindependentelec-
o
viewoftheimportanceofnano-scaleandlow-dimensional trondescriptionandconsidering,forsimplicity,asquare-
c
[ structures in modern technology. Experimentally, oscil- well potential along the z direction, normal to the thin
lations were observed in the oxidation rate of ultrathin film, with a constant depth V and a variable width L
1
Mg(0001) films on W(110), as a function of film thick- (film thickness), the electronic states are solutions of a
v
ness, with the largest oxidation rate occurring when a separableprobleminthez and(x,y)variables,andread:
4
6 quantum-wellstate was found to cross the Fermi energy,
3 in photoemission (PE) spectra [1] taken near normal in- ψnE,kx,ky(x,y,z)∼χn(z)·ei(kx·x+ky·y), (1)
1 cidence. Inparticular,thechangesobservedintheinitial
0 oxidation rate—when most of the film was still metallic, with energy E = En + ~2(kx2 + ky2)/2m∗, where m∗
6
0 were found to be dramatic. The origin of these changes stands for the electron effective mass. En and χn(z) are
/ has not been established yet. The changes in reactivity the eigenvalues and eigenstates of the one-dimensional
at were suggested to be due to a change in the density of square-well potential problem, with the following prop-
m states (DOS) at, or near the Fermi energy E [1], as the erties:
F
- oscillationsinthe reactivitywerefoundtocorrelatewith
d χ (z) e−αn·z (2)
n the oscillations in the PE intensity at EF. However,un- n ∼
like the normal-incidence PE intensity, which measures
o for z L/2, assuming the z origin at the center of the
c the partial DOS at kk = 0, the total DOS of the films film, a≥nd:
: at E is bound to exhibit a monotonic, staircase-likein-
v F
i crease with film thickness [2], and cannot thus simply α =(√2m∗/~) p E , (3)
X account for the oscillations in the reactivity. n · − n
r
Here we propose a theoretical interpretation of these where the zero of energy is taken at the vacuum level.
a
observations,inwhichthe decaylengthinthevacuumof In the three-dimensionalproblem, E coincides with the
n
theelectroniclocaldensityofstatesaroundtheFermien- energy E of the subband state n at k = 0, measured
k
ergyisidentifiedasthekeyparameterresponsibleforthe relative to the vacuum level.
changesinthe reactivity. Onthe basisofab initio calcu- All states ψE belonging to subband n, with en-
n,kx,ky
lations, inspired by model predictions for quantum well ergy E E , are thus characterized by the same decay
n
≥
states,weshowthatthedecaylength,λ,ofthelocalden- length 1/α 1/√ E , for a given film thickness L. If
n n
∼ −
sity of states at the Fermi energy exhibits an oscillatory E is locatedbetweenthe levels E andE , the dom-
F n n+1
behavior with film thickness, with local maxima present inantdecaylengthoftheelectronicstatesatE is1/α ,
F n
when the highest occupied quantum-well state at k =0 namely the decay length of the highest occupied band
k
2
state atk =0. With increasingwidth L ofthe well,the Mg(0001) (a = 3.21 ˚A, b = √3a ) and W(110)
k Mg Mg Mg
energiesofthequantum-wellstatesdecreasewithrespect (a = 3.16 ˚A, b = √2a ) are 1.6% and 20%
W W W
∼
to E ; hence, the decay length λ of the electronic den- [4]. To model such epitaxial systems, a commensurate
F
sity atE firstdecreasesas 1/2α 1/√ E , until the interface atomic structure is needed in the calculations.
F n n
∼ −
nextquantum-wellstateat Γ crossesE ,atwhichpoint The inclusion of dislocations would require prohibitively
F
λincreasestothenewvalue1/2αn+1 1/p En+1. The large lateral dimensions of the supercell, and the details
∼ −
decaylengththendecreasesagainwithincreasingL,dis- of the atomic structure at the interface are unknown in
playing systematic oscillations with L. Considering next any case. In order to simulate an unstrained Mg film on
adiscretenumber ofatomic layers,the periodicityofthe W(110), we thus elected to laterally strain the W(110)
crossing of the Fermi energy may be derived from the slabto the in-planelattice parametersofMg(0001). The
Bohr-Sommerfeldrule,whichforMg(0001)yieldsa peri- epitaxial alignment was made by positioning the atoms
odicity of 7.7 monolayers (ML) [4]. ofthe firstW(110)layer,adjacenttothe Mg,inthe con-
Fromthemodeldescription,weexpectthusoscillations tinuation of the Mg (0001) hcp lattice. The three outer-
inλ,withlocalmaximaoccurringwhenthehighestoccu- most layers of the Mg film were relaxed, while the other
pied quantum-well states at Γ is closest to E . A more interlayer spacings were kept frozen at their bulk posi-
F
realistic description of the system, however, is clearly tion (using the theoretical values of the bulk lattice con-
needed to confirm the robustness of this behavior, and stants: aMg =3.19 ˚A, cMg =5.18 ˚A, and aW =3.21 ˚A).
to quantifythe effect. Inparticular,electroninteraction, It should be noted, however, that the Mg(0001) surface
atomic orbitals and bonds, surface states, and the pres- relaxations are small [11, 12], and were found to have a
ence of the substrate, are all factors which are expected negligible impact on λ. The self-consistent calculations
toinfluencethedecaylengthλ. Wehavethereforeexam- werecarriedoutusinga(20,20,1)MPgrid. Forthe local
inedthe effectofquantumconfinementonλbymeansof density of states, we used a (48,48,1)grid centered at Γ,
ab initio calculations performed for epitaxial Mg(0001) with a Gaussian level smearing of 0.005 Ry. The decay
films on W(110) and also, for comparison, for the corre- length was derived from a fit, assuming an exponential
sponding unsupported Mg(0001) films in vacuum. decay of the local density of states at distances beyond
First-principles results.—The calculations were per- 2.15 ˚A from the outermost atomic plane.
∼
formed within density functional theory, using the InFig.1,wereportthecalculateddecaylengthλofthe
Perdew-Burke-Ernzerhofexchange-correlationfunctional Mg films, as a function of film thickness. The results are
[5], Troullier-Martins pseudopotentials [6], and a plane- shown for both the Mg films on W and the isolated Mg
wave basis set. The Mg pseudopotential was generated films in vacuum. The behaviors are very similar in the
inthe atomicconfiguration: 3s1.83p03d0.2, usingas core- twocases. Thedecaylengthexhibits apronouncedoscil-
radii cutoff: rs,p,d = 2.8 a.u.. For W we treated the 6s, lation,withamaximumat9layersandaminimumat6-7
6p, and 5d orbitals as valence states, using the same pa- layers. The presence of the tungsten substrate tends to
rametersasinRef. [7]. The overlapbetweenvalence and reduce the amplitude of the variation of λ, from 17% to
coreelectronswasaccountedforusingthenon-linear-core 10%,buthasnosignificantimpactonthepositionofthe
correction to the exchange-correlation potential [8]. A extrema. The densities of states at Γ of the Mg (0001)
kinetic-energycutoffof49Rywasusedfortheplane-wave films on W(110)aredisplayedin Fig. 2. With increasing
expansionoftheelectronicorbitalsoftheMg/Wsystems. filmthickness,anunoccupiedquantum-wellstatecrosses
Forthe isolatedMg(0001)slabswe useda kinetic-energy the Fermi energy at 9 layers. This coincides with the
∼
cutoff of 14 Ry. The Brillouin-zone sampling was done maximal decay length λ found in Fig. 1, consistent with
with a Monkhorst-Pack (MP) grid [9], and we used a the model prediction. In Fig. 2, we also reported the
Gaussian smearing of the electronic levels of 0.02 Ry to energy positions of the electronic states at Γ obtained
determine the Fermi energy. The films were modeled us- from the calculations for the isolated Mg slabs in vac-
ingslabgeometriesinsupercells. FortheMgfilmsonW, uum. Also in this case, a quantum-well state is found
weconsideredslabscontaining7W(110)ML,terminated to pass through the Fermi energy at 9 layers, which
∼
by5to12Mg(0001)MLononeside,andby2Mg(0001) correspondsto the largestλ obtained in Fig. 1. We note
ML onthe other side. The Mg bilayerwas introducedin that we have also examined the variation of the work-
order to avoid the presence of an electric field—due to function of the Mg films on W with the number of Mg
the different work functions of W and Mg—in the vac- layers [13], as quantum size effects can be expected on
uum regions separating the periodic images of the slab. this value [14]. For the Mg films on W, however, the
Weusedvacuumregionswithaminimalthicknessof20.1 calculated changes in the work function are found to be
˚A, in order to obtain well-convergedvalues of λ. very small, namely a variation of 0.05 eV in the range
Experimentally, above 3 ML, the Mg films are known 5-12 layers. With increasing thickness, the workfunction
togrowepitaxiallyonW(110)[4],withlatticeparameters first decreases, in the range 5-8 layers, from 3.74 to 3.69
corresponding to their bulk values [10]. The mismatches eV, and then increases, in the range 8-10 layers, from
between the experimental in-plane lattice parameters of 3.69to 3.72eV, and then saturatesat3.72eV for higher
3
SS
0.54
12
0.52
)
( A 11
λ
0.5
) 10
s
t
i
n
0.48 u
b. 9
r
a
0.46 S ( 8
O
5 6 7 8 9 10 11 12 D
7
Number of Mg Layers
6
FIG.1: (Coloronline)Calculateddecaylengthinvacuumof
the electronic local density of states at the Fermi energy of
Mg(0001)filmsofvariousthicknessesonW(110)(disks). The 5
SS
decay length of thecorresponding Mg(0001) films isolated in
-8 -6 -4 -2 0 2
vacuum is also shown for comparison (diamonds).
E (eV)
coverage(ingoodagreementwiththeexperimentalvalue FIG. 2: (Color online) Calculated density of states at Γ
of 3.66 eV for Mg [15]). of the Mg (0001) films on W(110) (solid lines). The film
Inspection of Fig. 2 reveals a striking correspondence thicknessincreasesfrom5to12atomiclayers(bottomtotop
curve). The densities of states have been convoluted with a
between the positions of the peaks in the DOS of the
Gaussian ofwidth 0.2 eV.Thesymbols(diamonds) showthe
Mg films on W and the positions of the levels of the
energiesoftheelectroniclevelsat Γ oftheisolatedMg(0001)
isolated Mg films in vacuum. The largest difference be-
filmsinvacuum. Thestatesindicatedby“SS”originatefrom
tween the two sets of energies does not exceed 0.2 eV in the Shockley surface state of Mg(0001) (see text). The zero
therange[E 8eV,E +2eV].Thestatesindicatedby of energy corresponds to theFermi level.
F F
−
“SS” in Fig. 2 originate from the Shockley surface state
of Mg(0001). In the isolated Mg(0001) slab in vacuum,
the Shockley states of the two surfaces strongly inter- Theresonantcharacterofthequantum-wellstateswhich
act, giving rise to a pair of split even and odd states— cross the Fermi energy tends to smoothen the jump in λ
with respect to the center-of-slab reflection plane. The observed for thicknesses between 7 and 9 layers.
resulting splitting is as large as 1.4 eV for 5 Mg layers Hence, in spite of the presence ofthe surface/interface
and 0.6 eV for 12 Mg layers. In the presence of the W states, and the resonant character of the quantum-well
substrate,these statespersistasstrongsurface/interface states crossing the Fermi energy, the qualitative behav-
resonances, with maxima in the probability density on iorofλ,inferredfromtheparticle-in-a-boxmodel,isfully
both the outermost (surface) Mg layer and the inner- supported by the ab initio calculations. The variationin
most (interface) Mg layer. The ratio of the probability the decay length we predict from the ab initio calcula-
density on the surface relative to the interface layer of tions is expected to have a direct, exponential impact of
the high- (respectively, low-) energy state SS tend to in- the electron transfer rate by tunneling—from the metal
crease (decrease) with increasing film thickness, and is to the O2 molecule—which has been proposed to con-
1.3 ( 0.85) in the 12-layer case. In the presence of troltheinitialstickingoftheoxygenmoleculesimpinging
∼ ∼
the W substrate, some of the quantum-well states, and on the surface, via the attractive image charge potential
in particular those in the range [E 6 eV, E 3 eV] on the ionized O− molecule [3]. Assuming a transfer
[4,7],remainfullylocalizedwithinFth−eMgfilmsF—−similar rate by tunneling2proportional to e−d/λ, with d the dis-
totheconfinedquantum-wellstatesobtainedfromabini- tance between the metal surface and the center of mass
tiocalculationsinAg films onFe [16],orsharpFano res- of the molecule, and considering, e.g., a distance d 3.5
≈
onances, located mostly within the Mg film. Other Mg ˚A—within the expected physisorption range of the O2
quantum-wellstates,instead,andinparticularthestates molecules[3,17],a10%variationinλproducesa100%
corresponding to the feature which crosses E , and is changeinthe transferrate. Sucha chargeis ofthe order
F
indicated by the arrow in Fig. 2, become broader reso- ofmagnitudeoftheexperimentalchangeintheoxidation
nances, corresponding to quantum-well states displaying rate at low O exposure [1].
2
anincreasedprobability density within the W substrate. The peak positions of the calculated DOS in the en-
4
ergyrange[E 4eV,E ], inFig.2, comparewellwith tum oscillations in the electronic density decay length,
F F
−
the near-normal-incidence PE measurements [1], except which should be observable by STM.
for a systematic shift, by about +2 Mg ML, of the theo-
retical spectra with respect to the measured spectra. In WethankL.Aballe,A.Barinov,M.Kiskinova,N.Sto-
ourcalculations,thefirstquantum-wellstatewithenergy jic, and G. Trimarchi for helpful discussions. The com-
higher than the surface/interface states SS, enters the putations havebeenperformedusingthe PWscfpackage
occupied-state spectrum at a Mg thickness of 8-9 ML. [22]. We acknowledge support by the INFM within the
In the experiment, this state is found, instead, to en- framework of the “Iniziativa Trasversale Calcolo Paral-
ter the valence-band spectrum at a nominal thickness of lelo”.
6-7 ML. The same shift is observed between the calcu-
latedmaximum/minimaofλandthe experimentalmax-
imum/minima of the reactivity. Such a shift is probably
due to the use of a commensurate,dislocation-free inter-
face atomic structure, in the calculations, to describe a [1] L. Aballe, A. Barinov, A. Locatelli, S. Heun, and M.
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enceofstrainedMglayersordislocationsattheinterface
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could result in the entrance of a quantum-well state at a
when the DOS is normalized to the number of atoms in
smaller value of the number of layers of the Mg film. the film.
Thechangesinλwepredictshouldbeobservable,e.g., [3] A. Hellman, B. Razaznejad, Y. Yourdshahyan, H. Ter-
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be in therange +1.3 to+1.5 %.
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F
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2
the experimental tuning of the reactivity is due to quan- nozzi, URL http://www.pwscf.org/.