First-principles calculations of Cu(001) thin films: quantum size effect in surface energetics and surface chemical reactivities

0.58 MB·English

by  Bo Sun

First-principles calculations of Cu(001) thin films: quantum size effect in surface
energetics and surface chemical reactivities
Bo Sun,1 Ping Zhang,1,∗ Suqing Duan,1 Xian-Geng Zhao,1 Junren Shi,2 and Qi-Kun Xue2
1Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China
7 2Institute of Physics, Chinese Academy of Sciences, Beijing 100080, P.R. China
0
First-principlescalculationsofCu(001)free-standingthinfilmshavebeenperformedtoinvestigate
0
the oscillatory quantum size effects exhibited in surface energy, work function, atomic relaxation,
2
and adsorption energy of the cesium adsorbate. The quantum well states have been shown and

n clarified at particular k-points corresponding to the stationary extrema in bulk Brillouin zone,
a whichareingoodagreementwithexperimentalobservations. Thecalculatedsurfaceenergeticsand
J
geometryrelaxationsareclearlyfeaturedbyquantumoscillations asafunctionofthefilmthickness

3 ofthefilmwith oscillation periodscharacterized byasuperposition oflongandshort length scales.
1 Furthermore, we have investigated Cs adsorption onto Cu(001) thin films as a function of the film
thickness. Oursystematic calculated results clearly show the large-amplitude quantum oscillations
] inadsorptionenergetics,whichmaybeusedtotailorcatalysis,chemicalreactionsandothersurface
l
l processes in nanostructured materials.
a
h
- PACSnumbers: 73.61.-r,73.20.At,73.21.Ac,
s
e
m I. INTRODUCTION cumulation model (PAM)20, have been successfully used
. to interpret the energies of QW states. More sophisti-
t
a cated methods have also been used, such as the tight-
m Whenthe thicknessofthinmetalfilms approachesthe binding approach21 and layer-Korringa-Kohn-Rostoker
- nanoscale, the oscillatory quantum size effects (QSEs) approach22,23. In a few systems the QW states have
d associated with electronic confinement and interference
been investigated by self-consistent density-functional
n will occur1,2,3,4 due to the splitting of the energy-level calculations24,25,26,27,28,29,30,31. There are strong reasons
o
spectrumintosubbandsnormaltotheplaneofthefilms, to use the ab initio methods. First, there are no ad-
c
i.e., the quantum well (QW) states. These QW states
[ justableparameters,andawiderangeofcalculatedstruc-
lead to strongly modified physical properties and thus
tural and electronic properties offer the possibility of a
1 have been the subject of numerous experimental investi-
detailed comparison with experiments. Also, quantities
v gations in recent years5,6. For example, the QW states
suchasthe expectedSTMprofilesandthe amplitudesof
4
are found to be responsible for an unusual metallic film
9 the wave functions of the QW states, which cannot be
growth pattern7,8,9,10, and for the thickness-dependent
2 obtained in simple approaches,can be calculated.
1 stability11 observed in the experiment. The QW states
In this paper we report our first-principles calcula-
0 are directly connected to the oscillation in the exchange
tions of the QSE in a specific QW system, i.e., the
7 coupling between two magnetic materials across a non-
Cu(001) freestanding thin films. Previous QSE studies
0 magneticspacerlayerofvariousthickness12,andtogiant
t/ magnetoresistance13,14,15,16. Moreover, the QW states concerning Cu(001) are mainly focused on the oscilla-
a tions in magnetic interaction between the two ferromag-
also give rise to an oscillatory phonon-electron coupling
m netic layers in fcc M/Cu/fcc M (001) sandwich struc-
asthefilmthicknessvaries,andthusaffectthetransition
tureswhereM denotestheferromagneticmaterial. Ithas
- temperature of the superconductivity17,18.
d been demonstrated that the Cu(001) films have a long
n Experimentally,the characterizationof the QWstates length scale of 5.6 monolayer (ML) and a short length
o
arecommonlymeasuredusingangle-resolvedphotoemis- scale of 2.6 ML of oscillation periods for magnetic cou-
c
sion spectroscopy (ARPES) and the scanning tunneling pling in the [001] direction, corresponding to spanning
:
v microscopy (STM). ARPES can be used to study the vectorsatthebellyandneckofthebulkCuFermisurface
Xi band structure along any direction of the surface Bril- respectively32,33,34,35. Recent experimental efforts have
louinzone(BZ),whileSTMoffersthepossibilitytostudy been focused on the new band structure properties of
r
a local structures, such as islands, chains, dots, etc. Using QW states in Cu(001) system36,37,38,39,40,41,42,43,44. On
the scanning tunneling spectroscopy(STS) technique, in the other side, the QSE of Cu(001) associated with its
which the differential conductance (dI/dV) is measured, energetics was scarcely considered up to now. In partic-
one can reliably determine the energy of quantized elec- ular,there is no clear experimental or theoretic evidence
tronic states in the range of approximately 1 eV below of the interplay between the different oscillation periods
andabovetheFermilevel. Theoretically,anumberofap- ofCu(001)filmsbythebellyandneckextremainbulkCu
proaches have been used in the past in order to describe Fermi surface. In this paper we present a detailed first-
the electronic properties, in particular the QW states, principles study of the surface energetics of the Cu(001)
in ultrathin metallic films. Quasi-one-dimensional mod- freestandingthinfilms. TheQWstatescorrespondingto
els, such as a square well potential19 or the phase ac- the stationary extrema in bulk BZ are studied in detail.
2
Theoscillationsoftheenergeticsversusthe Cu(001)film sorptionpropertiesofalkali-metalCsonCu(001)surface
thicknessareidentifiedandthe correspondingoscillation is discussed in detail by presenting the sensitivity of the
periods are explained. We find that the quantum inter- adsorption energy to the thickness of the Cu(001) films.
ference of the QW states with different in-plane wave Finally,Sec. IVcontainsasummaryoftheworkandour
vectors result in a superposion between long- and short- conclusion.
length oscillating periods.
The other purpose in this paper is to investigate the
QSEcharacterinsurfaceadsorptionenergeticsforarep- II. COMPUTATIONAL METHOD
resentative system in order to shed light on the effect
of the QW states on the surface reactivities. Since the The calculations were carriedout using the Vienna ab
adsorptionpropertyiscloselycharacterizedbythechem- initio simulation package55 based on density-functional
icalbondingbetweentheadsorbateandthesurfaceofthe theory with PAW pseudopotentials56 and plane waves.
substrate, thus when the substrate is ultrathin, the QSE In the present film calculations, free-standing Cu(001)
in the substrate will also influence the behavior of the films by the so-called repeated slab geometries were em-
surface adsorption. Here we address a particular adsor- ployed. This scheme consists in in the construction of a
bate system, i.e., Cs/Cu(001), as a case study to man- unit cell of an arbitrarily fixed number of atomic layers
ifest the QSE in surface adsorption properties. Due to identical to that of the bulk in the plane of the surface
its simple electronic structure and active chemical prop- (defining the two dimensional cell), but symmetrically
erties which is intrinsic for alkali metals, Cs is a unique terminated by an arbitrarily fixed number of empty lay-
adsorbate on metal surfaces, and has been extensively ers (the “vacuum”) along the direction perpendicular to
studied. Concerning Cs adsorption on Cu metal sur- thesurface. Inthepresentstudywehavefixedthewhole
faces,therecentlypublishedarticles45,46,47,48,49,50mainly vacuum region equal to 20 ˚A, which is found to be suffi-
focus on the investigation of electronic, dynamic, and ciently convergent. The two dimensional unit cell of the
geometry properties of Cs layers on Cu(111). The na- fcc Cu(100) surface is a square of edge a/√2 with basis
ture of interaction between the Cs adatom and Cu(001) vectors a =a/2(1,¯1,0) and a =a/2(1,1,0) where a is
1 2
film has been studied both via first-principles pseudopo- the Cu bulk lattice constant. The corresponding surface
tential calculations51 or through and a jellium model BZ is a square with two high-symmetry directions Γ¯-M¯
approach52. andΓ¯-X¯. DuringourslabcalculationstheBZintegration
Compared to the previous Cs/Cu(001) work, as men- wasperformedusingthe Monkhorse-Packscheme57 with
tioned above, our present emphasis is on QSE in sur- a11 11 1k-pointgrid,andtheplane-waveenergycut-
× ×
face adsorption energetics, instead of giving a general offwasset270eV.Furthermore,thegeneralizedgradient
overview of the nature of alkali-metal-atom adsorption approximation(GGA) with PW-91exchange-correlation
on metal surfaces. In particular, although it has been potential has been employed with all atomic configura-
wellknownthatduetochargetransferfromtheadsorbed tions fully relaxed. First the total energy of the bulk fcc
alkali atom to the substrate the alkali adatoms become Cu was calculated to obtain the bulk lattice constants.
partially charged53, the dependence of this chargetrans- The calculated lattice constant is a = 3.639 ˚A, compa-
fer process on the thickness of the ultrathin substrate rable with experimental58 values of 3.61 ˚A, respectively.
film is yet to be understood with high interest. We no- Theuseoflargerk-pointmeshesdidnotalterthesevalues
tice there are emerging some high-quality experimental significantly. A Fermi broadening of 0.1 eV was chosen
data of QSE in charge transfer, via observation of the to smear the occupation of the bands around E by a
F
absorbedworkfunction andphotoemissionas a function finite-T Fermi function and extrapolating to T =0 K.
of the thickness of the ultrathin substrate film54. Fur-
ther measurements related with the QSE effect of the
adsorptionareexpectedtobesystematicallyreportedaf- III. BAND STRUCTURE AND QUANTUM
terwardsduetotheavailabilityofthehigh-qualityquan- WELL STATES
tum metal thin films. From this aspect a thorough the-
oretical investigation is necessary and will be helpful for We first studied the properties of electronic struc-
experimental references in near future. Our results show tures for bulk Cu and Cu(001) films. For this we have
that in the ultrathin Cu(001) films, the surface adsorp- used two kinds unit cells of crystal lattice. Namely,
tion properties display well-defined QSEs. unit cell I is defined via introducing the usual fcc ba-
This paper is organized as follows: In Sec. II, the ab sis vectors a = (a/2)(j+ k), a = (a/2)(i + k), and
1 2
initio based method and computational details are out- a = (a/2)(i+j), while unit cell II is defined by choos-
3
lined. In Sec. III, the band structure and the proper- ing Cu(001) as the basal plane, i.e., a = (a/2)(i j),
1
−
ties of QW states at the belly and neck points in bulk a = (a/2)(i + j), and a = ak. The volume of sec-
2 3
BZ are presented. In Sec. IV, the surface properties of ond unit cell is twice as that of the first one and it’s
the Cu(001)films, surfaceenergy,workfunction, andin- convenient to extend to slab calculation since a is nor-
3
terlayer relaxation, as a function of the thickness of the mal to Cu(001) surface. Correspondingly, the recipro-
films, are presented and discussed. In Sec. V, the ad- cal lattice basis vectors are b = (2π/a)( i + j+k),
1
−
3
20 (a) (b) turns to be at ΓX line. In this paper the neck point will
Energy (eV)100 bttoieonunnagimitveecdseltPlhIeaI)npdroesPsitpifeoocnrtib[viunellykc.oaoOnrdduirnsubaratfenasdcoesfBt(rkZuc(,twkuirt,ehkcra)el]scpoueflcaPt-
x y z
to be (2π/a)(0.4,0.4,0.0).
-130(Γc ) W L Γ Χ DOS 3(d) Now we turn to study the electronic structure of
2 2 Cu(001) thin film with focus on QW states. Since the
Energy (eV)-101 -101 opsrceivlliaotuisngstpuedriieosd1s2ihnaivnetesrhlaoywenrmthaagtnetthiec eloxnchgaanngdeisnhtoerrt-
actionoriginatefrombanddispersionsatΓandPpoints
-20 3 6 9 12 15 18 21 24 Γ Χ -20 3 6 9 12 15 18 21 2 4 P P’
Thickness N (ML) (0,0,0) (2π/a)(0,0,0.5) Thickness N (ML) (2π/a)(0.4,0.4,0 ) (2π/a)(0.4,0.4,0.5) respectively,weexpecttheotherintrinsicfilmproperties
suchasfilmenergetics,atomicstructurerelaxations,and
even surface chemistry are closely related with the QW
FIG. 1: (Color online) (a) GGA energy bandsand densityof states at these two kinds of stationary k-points. Fig-
electronstates(rightpanel)offccbulkCu;(b)TheCuFermi ure 1(c) shows the energies of the QW states at Γ point
surface showing the belly and neck regions. The bulk and
as a function of the film thickness without interlayer re-
surface BZ are also depicted; (c) Calculated (GGA) energies
laxation. The energy zero is set at the Fermi level of
at Γ in Cu(001) thin films as a function of thickness, with
each film. Interlayer relaxation effect is also studied and
the energy set to zero at the Fermi level. The right panel
it is found that the overall thickness dependence of the
plots the bulk energy dispersion in the [001] direction; (d)
Calculated (GGA) energies at P in Cu(001) thin films as a energiesaresimilarto thatwithoutrelaxation. Forcom-
functionofthickness,withtheenergysettozeroattheFermi parison, also plotted in Fig. 1(c) (right panel) is the
level. Theright panelplotsthebulkenergy dispersion in the energy dispersionin the bulk along [001](Γ X) direc-
→
[001] direction. The dotted line denotesFermi level. tion, which corresponds to the center Γ of (001) surface
BZ and determines the energy range for the QW states
shown in the left panel. One can see from Fig. 1(c)
b = (2π/a)(i j+k), and b = (2π/a)(i+j k) for that as the thickness of the film increases, the energy
2 3
unit cell I and−b = (2π/a)(i j), b = (2π/a−)(i+j), of a given QW state also increases. When the thick-
1 2
and b =(2π/a)k for unit cell−II. Figure 1(a) shows the ness of the film is increased to be about 5.0 monolayers,
3
band structure and the DOS of bulk fcc Cu (unit cell I). then a quantum well state, with the energy crossing the
The highly dispersive s-p band, typical of noble metals, Fermi level, occurs. The next energy crossing with the
is present. The more intense d band region is located Fermi level occurs at the film thickness of 11 mono-
∼
between 2.0 and 4.5 eV below Fermi surface and can be layers. Our calculated result of quantum well states is
measured by a large rise of the intensity in photoemis- in goodagreementwith the recentexperimentalARPES
sion experiments12. The s-d hybrid band region with measurement12, in which the photoemission intensity as
two less intense dispersive states can also be seen in Fig. afunctionofenergyforfixedCuthicknessandasafunc-
1(a). TheFermisurfaceofbulkCuandtheBrillouinzone tion of Cu thickness for fixed energy clearly show the
for unit cell I are schematically shown in Fig. 1(b). Of existence of the QW states of the sp electrons in the Cu
particular interest for Cu(001) thin film studies are the layer. Fig. 1(d)shows(leftpanel)theenergiesoftheQW
two extremal points, i.e., the belly and neck points at states at P point as a function of film thickness without
the bulk Fermi surface. Previous extensive studies have interlayerrelaxation. Theenergyzeroissetatthe Fermi
shown that the stationary spanning vectors connecting levelofeachfilm. Againthebulkenergydispersionalong
the belly and neck points respectively play a key role in [001]direction(startingfromP point). One cansee that
determining the oscillatory behavior of interlayer mag- for every incremental increase in the film thickness of
netic exchange interaction J in fcc M/Cu/M (001) about 2.6 ML, then a new quantum well state crosses
M/M
sandwich structures where M denotes the ferromagnetic the Fermi level. A comparison between the left panel
material. The belly extrema was found to correspond to and right panel in Fig. 1(c) [or (d)] shows that the bulk
long oscillating period in J while the neck extrama electron band of Cu works very well for QW states as
M/M
corresponds to short period. the Cuthickness is greaterthan5 ML,whichagreeswell
with the experimental results59.
One inconvenience in discussing properties of Cu(001)
film via the bulk Fermi surface calculated from unit cell The quantitative understanding of the QW states
I lies in the fact that none of three basis vectors of unit showed in Figs. 1(c)-(d) is usually analyzed using the
cellIisnormalto(001)surface,whichleadstoanoblique so-called phase accumulation model (PAM)60,61. Here
projectionofbulkBZonto(001)surfaceBZ.Onthecon- the free standing Cu(001) film is considered as a quan-
trary, by use of unit cell II the discussions, particularly tum well confining electrons between the two vacuums
for QW states of (001) thin films, become very conve- in the slab. Since the system is invariant on transla-
nient. Beginning with this unit cell construction, the tion parallel to the (001) plane, the in-plane wavevector
belly point in bulk BZ is still projected to Γ point, like k (k ,k ) is a good quantum number. Thus, for a
k x y
≡
withunitcellI.Whereastheneckpoint,afterprojection, givenk ,thequantizationconditionforanelectronstate
k
4

in such a well is given by (a)
0.4
2k⊥Nd+2φ=2πn, (1)
n 0.2
)
where N is the number of atomic layers in the film, πof 0.0
dth=eeale/c2trtohnewianvteerfluanyecrtiospnaucpinogn,rφeBflecistiothneatphthaesethgeaifinlmo-f unit -0.2
vacuum interface, n is the number of half-wavelengths n
confined inside the QW and k⊥ describes the Cu elec- φ (i -0.4
tron wavevector component alonng [001] direction for the hift (b)
nth QW state. Equation (1) has been successfully used S -0.2
e
to model QW states in metal thin films, and its valid- s
a -0.4
ityhasbeenrigorouslytestedbyfull-scalefirst-principles h
P
calculations for some special systems29. It should be
-0.6
addressed that QW states will not only form at the Γ
pointbutalsoforminalargepartofthesurfaceBZ,and -0.8
the quantization condition (1) may be applied through-
out the entire zone. Using Eq. (1) one can calculate -1.0 -0.5 0.0 0.5 1.0
the periodicity for the QW states crossing the Fermi Energ y (eV)
level, ∆N = π/[k⊥(k )d], where k⊥(k ) is the bulk
F k F k
Fermi wavevector along [001] direction for a given in-
plane wavevector kk. For the Γ point (kk = 0), from FIG.2: (Color online) The phaseshift of theelectronic wave
the right panel in Fig. 1(c), one can see that the upper function at the film-vacuuminterface as a function of energy
branch of the bulk sp band runs through 33% of the of QW states at (a) Γ and (b) P points.
BZ, kf = 0.33π/c. One gets ∆N = 6. Th∼erefore a new
⊥
QW state of k =(0,0) occurs every 5.6 ML increase of
k often used in the literature and has the form62
the film thickness, which is verified in the left panel in
Fig. 1(c) that an energy branch moves down, crossing
3.4 (eV)
the Fermi level for every incrementalincrease in the film φ(E)=π π. (2)
4.5 (eV) (E ~2k2/2m) −
thickness of 6 layers. Similarly for P point correspond- s − − k
ing to k = (2π/a)(0.4,0.4) in bulk BZ, one can derive
k
Here4.5eVistheCuworkfunction(inthethinthickness
from the right panel in Fig. 1(c) kf = 0.75π/c. In this
⊥ limit), E is theelectronenergymeasuredfromthe Fermi
case, Eq. (1) gives the periodicity for the QW states to
level and k is the the in-plane wavevector of QW. The
be ∆N = 2.6. Therefore, Figs. 1(c)-(d) reveal that the k
imagepotentialduetotheelectron–holeattractionislike
periodicbehavioroftheCu(001)QWstatesisessentially
aone-dimensionalhydrogenatom. Thatis wherethe 3.4
governedby the bulk Cu electronic structure properties.
eV is from which actually equals one quarter of the hy-
In particular, the extremal points of the Fermi surface
drogen ionization energy. For the belly of the Cu Fermi
andthe extremalpoints ofthe QWbranchesclosetothe
surface k = 0 while for neck point we obtain k = 0.98
Fermi level occur at the same k , i.e., at k = (0,0) in k k
k k ˚A−1. Equation(2)isplottedasadashedcurveinFig. 2.
the bulk BZ corresponding to Γ in the (001) surface BZ
Note that the data points within a given energy range
and at the neck extrema k = (2π/a)(0.4,0.4) both in
k can come from films of very different thickness. The
the bulk BZ andin the surfaceBZ.This relationshipbe-
datasets at both Γ and P vary smoothly as a function
tweentheFermisurfaceandthequantum-welldispersion
of energy with little scattering, indicating that Eq. (1)
turns outto be veryfruitful forthe understandingofthe
does determine the energies of the QW states quantita-
connectionbetweentheperiods inenergeticsandsurface
tively, provided the phase shifts are known.
reactivity discussed below.
The energiesofQWstates nearthe Fermilevelcanbe
Usingthe energydataforQWstates inFigs. 1(c)-(d),
measuredbySTSexperiments,inwhichthelocaldensity
and with the help of Eq. (1) and the bulk band struc-
of states (LDOS) is probed through the dI/dV curve.
ture shown in the right panels in Figs. 1(c)-(d), one can
The distinctive sharp peaks in dI/dV curve are charac-
deduce the phase shifts φ for the QW states at Γ and P
teristic of the QW states at different quantum numbers.
points. They are plotted as filled circles in Fig. 2(a)-(b)
Themosteasy-to-seequantityinSTSmeasurementisthe
respectively. Since the energy range of interest is quite
energy gap ∆E between the highest occupied QW state
farfromthecalculatedvacuumlevelof4.5eV(relativeto
and lowest unoccupied QW state. To see the thickness
the Fermi level), this phase shift curve is nearly feature-
dependence of ∆E, taking the derivative of Eq. (1) with
less for most of the energy range, except for the feature
respectto energyand evaluatingit at the Fermi levelfor
near the top of the band. In general the phase shift de-
a given N, one has29
pends on the incident (with respect to the film/vauum
interface)energyand momentumof the electron. A sim- 1 2d 1
N + φ′(E ) (3)
ple expressionobtainedfromthe WKBapproximationis ∆E ≈ hv 2π F
F
5

2.8
2.4
) 2.0 u.)
V
.
e a
1/ 1.6 E(F
E ( 1.2 at

∆ S
1/ 0.8 O
D
0.4
8 10 12 14 16 18 20 22
0.0
3 6 9 12 15 18 21 24 Thickness N (ML)
Thickness N (ML)
FIG. 4: Calculated electronic density of states at the Fermi
energy as a function of Cu(001) thin film thickness.
FIG. 3: The inverse of the energy gap between the highest
occupiedQWstateandthelowestunoccupiedQWstateasa
function of Cu(001) film thickness. The squares in thefigure
aretheresultsfor ΓpointinsurfaceBZ,whilethecircles are ) -5 (a)
V
for P point (projection of bulk Fermi surface’s neck onto the e
(001) plane). N ( -6
) /
N
(
E -7
wherev istheFermivelocityobtainedfromtheslopeof V) (b)
F e 0.62
the bulk band at the Fermi level and φ′(EF) the energy y (
derivative of the interface electronic phase shift at the g
r
e
Fermi level. Therefore, the measured 1/∆E curve is a En 0.60
linearfunctionofN,withaslopeconnectedtovF. Figure e
c
3 shows the calculated 1/∆E at Γ and P points using a
rf
the QW energies in Figs. 1(c)-(d). One can see that Su 0.58
thetwocurvesfollowastraightlinewithdifferentslopes. 0 3 6 9 12 15 18 21 24 27
Also it can be seen that whence a new branch of QW Thickness N (ML)
statescrossestheFermilevel,thenakinkoccurs. Dueto
the different periods at Γ and P, the kinks in these two
curvesarealsolocatedatdifferentvaluesofN. Notethat FIG. 5: (a) Monolayer energy E(N)/N and (b) correspond-
the intersection of the linear curves with the horizontal ing surface energy for fully relaxed Cu(001) 1×1 slabs as a
axis is not necessarily at N = 0, due to the nonzero function of thickness. The red curve is a least-squares fit to
thesurface energy.
energy derivative of the interfacial phase shifts at the
Fermi level30.
IV. FILM ENERGETICS AND INTERLAYER
Another central quantity closely related with above
RELAXATION
discussed QW electronic structure of Cu(001) thin film
is the film electronic DOS at the Fermi level, D(E ).
F
As shown in Fig. 4, the film D(E ) with respect to the Figure 3(a) shows the total energy per monolayer
F
film thickness displays well-defined oscillations with os- E(N)/N as a function of the number N of layers in the
cillationperiods characterizedby asuperpositionoflong slab. Theatomsintheslabshavebeenfullyrelaxeddur-
and short length. Note that in obtaining Fig. 4 we have ingthecalculations. OnecanseefromFig. 3(a)thatwith
increased the k-grid for integration to 35 35 1 for increasing the thickness, E(N)/N gradually approaches
× ×
each value of N, which ensures the precision of the re- aconstantvaluewhichinthe limitisequaltotheenergy
sult. The oscillations in D(E ) give an periodic band per atom in bulk Cu.
F
energy contribution to the total energy, thus producing An energetic quantity more suitably tailored to QSE
oscillatorysurfaceenergeticsandreactivitywhichwillbe is the surface energy which is defined as one-half of the
shown below. energydifference between the film and the bulk with the
6

0.2 4.8 BAfetfeorr aeb asbosroprtpiotinon eV) 4
0.1 n (eV)4.6 ential ( 0 EF
) (eV) 0.0 ork functio24..44 ostatic pot-4
∆(N -0.1 W 2.2 Electr-8

-0.2 0 3 6 9 12 15 18 21 24 27 -12-15 -10 -5 0 5 10 15
Thickness N (ML) Z (Å)
-0.3
0 3 6 9 12 15 18 21 24 27
Thickness N (ML) FIG.7: (a)WorkfunctionforcleanandCs-adsorbedCu(001)
thin films as a function of thickness. (b) Planar-averaged
electrostatic potential of clean (solid curve) andCs-adsorbed
(dotted curve) Cu(001) thin film (ten-layerCu slab).
FIG.6: Theseconddifferenceofthetotalenergyasafunction
of thickness for freestanding Cu(001) thin films.
three dimensions, the stability of the film could also be
same number of atoms, including the proper subtraction affectedduetothevariationintheelectronicenergy64. In
ofatermlinearinN63. Thethicknessdependenceofsur- the presentcase,since the abovediscussionshaveclearly
face energy is calculated and shown in Fig. 3(b) (curve shown that the neck point P in the bulk Fermi surface
withsquares). Itrevealsthatthesurfaceenergyapproxi- plays the same important role as the bell point (corre-
matelyfollowsaperiodicoscillatoryform,inaccordwith spondingto Γ insurfaceBZ)doesindetermining the os-
the oscillation properties of film D(E ). This can be cillatory energetics properties of Cu(001) thin film, thus
F
simply reasoned by the fact that the total band energy one can reasonably expect that the stability of the film
with respect to the Fermi level is related to the DOS by alsoexhibitsasuperposedoscillatorybehaviorasafunc-
the equation E = EF(E E )D(E)dE. Therefore, tion of the film thickness, with the oscillation periods
band F
− consisting of long and short length. To illustrate this,
eachtimethetotalDOScrossestheFermilevel,theband
R followingRef.30,weusetheseconddifferenceofthetotal
energywillalsoasaresponsehaveanoscillatorychange,
energyasameasureoftherelativestabilityofanN-layer
whichconsequentlyresultsinperiodicoscillationsinsur-
film with respect to the films of N +1 and N 1 layers,
face energy. In the same manner as discussed above,
−
which is defined as
onecanexpectthattheoscillationsinthesurfaceenergy
consistofasuperpositionofoscillationswith the periods
∆(N)=E(N +1)+E(N 1) 2E(N) (5)
corresponding to the extremal points of the bulk Fermi − −
surface. To illustrate this, also plotted in Fig. 3(b) (red
whereE(N)isthecalculatedtotalenergyofthefullyre-
curve)isaleast-squaresfittothesurfaceenergywiththe
laxedN-layerfilmwiththein-planelatticeconstantfixed
following expression
at the theoretical value. The result is shown in Fig. 5.
Apeakinthe figureindicatesahighrelativestabilityfor
2
2πN
thefilm. Itisnotsurprisingthatthestabilityisfeatured
E (N)= A sin +φ , (4)
surf i i
i=1 (cid:18) Λi (cid:19) by a superposition of long- and short-period oscillations
X asisevidentfromthediscussionabove. Remarkably,one
where N is the number of monolayers (ML), Λ = 5.6 can see the N =8 film is mostly stable while the N =9
1
andΛ =2.7aretheperiodicities(inunitofatomicML) film is particularly unstable.
2
corresponding to the bell and neck extrema in bulk Cu In addition to the surface energy, we have also calcu-
Fermi surface, A and φ are the fitting parameters. It lated the work function W of freestanding Cu(001) thin
i i
canbe seenthatthe fitting curvewellreproducesthe os- film. The work function, defined as the minimum en-
cillatingbehaviorinthesurfaceenergy. Thustheperiods ergyrequiredto emit anelectronfromthe surfaceto the
extractedfromthecalculatedCu(001)surfaceenergyare vacuum,isonefundamentalphysicalquantityforsurface
well consistent with the periods extracted from the bulk reactivity. Anelementarypictureoftheworkfunctionin-
Fermisurface,andagainthestationaryextremainFermi volvesasurfacedipole layerthatavalenceelectronmust
surface play a key role in determining the oscillatorybe- overcomein order to escape. Since the workfunction, as
havior in surface energetics. withmanyotherproperties,isalsoafunctionoftheelec-
Some authors have used the energies of QW states at tronic density, thus the changes in the electronic density
Γ to discuss the stability of the film. See Ref.30 for an by the crossings between the QW states and Fermi level
example. The key point employed there is that even will influence the work function in an oscillatory way.
though the confinement takes place only in one of the In addition to its relevance as an important element in
7

our understanding of surface science, a modified, or tun- 1
able,workfunctioncanbeusefulforapplicationssuchas
catalysis, because a slight change in the energy scale is
0
exponentially amplified for chemical reactions65. Recent
in situ experimentshavemeasuredatomic-layer-resolved ∆d
work function and shown clear QSE in Ag/Fe(100)66 -1 1,2
and Pb/Si(111)67 systems. Here we have carried out %) ∆d2,3
ficlresatn-pCrinuc(0ip0l1e)sscuarlfcauclea.tiTonhseorfestuhlet iwsosrhkowfunncintioFnigo.f7t(hae) (1 -2 ∆d3,4
+ ∆d
(lafixlleeddastqoumaircesg)eoams eatrfyu.nOctnioencaonf sfielemthtahtictkhneeswsowrkitfhunrce-- ∆di,i -3 4,5
tion is featured by an oscillatory behavior. As with the
surface energy, the oscillations in the work function of -4
Cu(001)thinfilmsconsistofasuperpositionoflong-and
short-length periods.
6 9 12 15 18 21 24 27
Thickness N (ML)
TABLE I: Interlayerrelaxations given in percent,∆di,i+1,of
Cu(001) as a function of thethickness of the film.
N ∆d12 ∆d23 ∆d34 ∆d45 ∆d56 ∆d67 FIG.8: InterlayerrelaxationsofCu(001)thinfilmsasafunc-
tion of thefilm thickness.
2 -5.186
3 -3.351 -3.350
4 -3.131 +0.616 -3.144
that we considered, the topmost interlayer relaxation is
5 -3.425 +0.081 -0.099 -3.431 always inward(∆d <0), with ∆d starting from the
1,2 1,2
6 -2.798 +0.477 -0.213 +0.487 -2.800 largest value of -5% for a slab with only two monolay-
7 -4.076 +0.126 +0.398 +0.394 -0.115 -4.066 ers, and approaches a final value of -3% with increasing
8 -2.693 +0.189 -0.320 -0.066 -0.317 +0.185 the thickness of Cu(001) films. Whereas, the second in-
9 -2.917 +0.369 +0.133 +0.484 +0.492 +0.134 terlayer relaxation is always outward (∆d2,3 > 0). Note
that the first interlayer separation on most metal sur-
10 -3.358 +0.343 -0.364 +0.039 -0.335 -0.042
faces is contracted, Cu(001) is one of the typical exam-
11 -2.805 +0.415 +0.256 +0.415 -0.029 -0.025
ples. (ii)Theinterlayerspacingsoscillateasafunctionof
12 -3.279 +0.296 -0.274 +0.382 -0.219 -0.498
the thicknessofthe filmwiththeperiodagainconsisting
13 -3.699 +0.237 -0.322 -0.134 -0.407 -0.938
of long- and short-length scales. After 26 monolayers,
14 -2.968 +0.469 +0.019 +0.177 -0.041 +0.037 which is the maximal layers considered here, the oscil-
15 -3.286 +0.127 -0.315 -0.245 -0.721 -1.015 lations are invisible which suggest that the semi-infinite
surface limit is now reached.
In addition to the above discussed film energetics, the
relaxed atomic structures of the Cu(001) thin film will
alsobe influenced by the occurrenceofQWstatesatthe V. ADSORPTION OF CESIUM: QSE IN
SURFACE CESIATION
Fermi level. Since the 1 1 supercells are employed and
×
k is a good quantum number, thus only atomic inter-
k
layer relaxation along [001] direction is allowed during In the above discussions we have extensively studied
our calculations. Here the interlayer relaxation, ∆d , theQWstatesinCu(001)thinfilmsandthecorrespond-
i,i+1
is given in percent with respect to the unrelaxed inter- ingQSEinvariousphysicalquantitiessuchasthesurface
layer spacings, d , i.e., ∆d = 100(d d )/d . energy, work function, and interlayer relaxation. To fur-
0 i,i+1 i,i+1 0 0
−
d istheinterlayerdistancebetweentwoadjacentlay- therillustratethephysicalpropertiesinfluencedbyfinite
i,i+1
ersparalleltothesurfacecalculatedbytotalenergymin- sizeofthethinfilms,inthissectionwefocusourattention
imization. d =a/2 is the bulk interlayer distance along to the adsorptionofCs onCu(001)thin films. Note that
0
[001] direction. Obviously, the signs + and of ∆d ourstudyinthissectionisnotjustforanalternativever-
i,i+1
−
indicateexpansionandcontractionoftheinterlayerspac- ification of QSE in solid thin film. On the contrast, the
ings, respectively. The interlayer relaxations of Cu(001) present study of surface adsorption as a function of the
films asa functionofthe film thicknessis summarizedin film thickness has its own cause to be emphasized. It is
Table I. Furthermore, the interlayer relaxations are also wellknownthatmetalsurfacesareaprototypeheteroge-
plottedinFig. 8as afunctionofN forclearillustration. neous catalyst,andhavebeen widely studiedin terms of
One can see: (i) The two outmost layers relax signifi- the dissociative chemisorptionof the reactantmolecules.
cantly from the bulk value, in agreement with the result Foragivensolid, exceptfor surfaceirregularitiessuchas
fromFLAPWcalculation68. Inthe wholerangeoflayers steps and defects, its surface reactivity is almost solely
8
(a) (b)
3
0.04
a 2 0.02
2 -0.002
-0.02
1 3 4 1 -0.04
-0.07
2 0 Cs
Å
a
1 -1
-2
Cu
-3
-1 0 1 2 3
FIG. 9: The four different adsorption sites for Cs adatom on Å
Cu(001) surface.
FIG. 10: (Color online) (a) The contour plot of charge den-
determined by the crystallographic orientation. At re- sity difference, ρ(Cs/Cu(001))−ρ(Cu(001))−ρ(Cs), in the
(100) plane, for the cesiated Cu(001) surface. (b) (Left
duced size and/or dimensionality, particularly when the
panel) band structure of cesiated Cu(001) thin film (5-layer
characterized size enters nanometer scale, however, the
Cu slab) with coverage Θ=0.5; The right panel depicts the
situation could be dramatically different from that of
decomposition of local DOS for Cs adatom (dotted curves)
the bulk. Infact, size-dependentsurface chemicalactivi-
and surface Cu atom (solid curves) into states with s (cyan
ties in metal films with the thickness of nanometer scale
curves),p(redcurves),andd(blackcurves)characterforthe
havebeenobservedinpreviousexperimentalreports69,70. Cs/Cu(001)system. Theinset in (b) pedicts thebandcharge
Remarkably, in a very recent oxygen adsorption experi- densities around theFermi level (lower panel) and above the
ment on high-quality Pb(111) thin films, Ma et al have Fermi energy (upperpanel) of typical valueof 3.5 eV.
clearly observed an oscillatory dependence of the chemi-
cal reactivities on the film thickness67, thus providing a
most convincing proof of the key role played by the QW numbertothesubstrate71,72,73,74,75,76,77. Thusinthefol-
states in changing surface reactivity. It is expected that lowing the atomic Cs is always put on site-4 during the
more experiments will address the issue of QSE in sur- simulation. This adsorption site is independent on the
face chemical reactivities. From this aspect, the present coverageΘ of Cs atoms. In this paper, we only consider
detailed theoretical analysis for the dependence of be- the case of Θ=0.5.
havior of Cs adsorptionupon the Cu(001) film thickness To illustrate the surface bond formed between Cs
is highly interesting and will be helpful for experimental adatom and Cu surface atoms, we depict in Fig. 10(a)
reference in near future. the contour map of the electron-density difference for
BeforewestudytheCsadsorptionpropertiesasafunc- Cs/Cu(001). Here the plane depicted is normal to the
tionofthethicknessofCu(001)thinfilms,weneedtode- surfaceandcontainsaCsandtwoCu’s. Thedensitydif-
terminetheenergeticallyfavorableadsorptionsite. Since ferencewasobtainedbysubtractingthe densitiesofnon-
the preference of adsorption site is not sensitive to the interactingcomponentsystems,ρ(Cu(001))+ρ(Cs),from
thickness of the substrate, thus to look for this pref- the density of the Cs/Cu(001) system, ρ(Cs/Cu(001)),
erence of the adsorption site, it is sufficient to give a while retaining the positions of the component systems
study on the slabs with fixed thickness of the Cu(001) at the same location as Cs/Cu(001). The solid and bro-
substrate, which at present is chosen to be 5 ML. We ken lines denote an increase and decrease, respectively,
choosefourmostprobableadsorptionsites,whichisenu- in electron density upon Cs adsorption onto the surface.
merated in Fig. 9. The binding energy is calculated Covalentbondingisevidentfromtheaccumulationofthe
using the following equation: Binding energy [atomic charge between the Cs adsorbate and the Cu(001) sub-
Cs]= (E[Cs/Cu(001)] E[Cu(001)] 2E[Cs])/2 where strate. This charge is drawn principally from the atomic
− − −
E[Cs/Cu(001)]isthetotalenergyofaslabwhichincludes 6s state of Cs adlayer. On the vacuum side of the over-
two Cs atoms inside with a symmetric configuration, layerthechargeisdepleted. ThustheCs6sstatecausesa
E[Cu(001)]is total energyof the slabwithout Cs atoms, polarization toward the Cu(001) surface. As a response,
andE[Cs]istotalenergyofafreeCsatomwhichisputin the work function will be decreased by this occurrence
a16˚A 16˚A 16˚Asupercell. ThecalculatedCs/Cu(001) of surface polarization. On the other side, a significant
× ×
bindingenergiesare0.9189eV,0.9176eV,0.9174eV,and counter-polarization of the Cs 5p semicore charge is also
0.9327eVforsite-1,site-2,site-3,andsite-4respectively. quite evident from Fig. 10(a). This effect opposes the
Thus site-4 is most stable for adsorption. This is in ac- reduction of the work function and the ultimate value
cord with recent experiments that for substrate surfaces of work function represents a self-consistent compromise
withsquareorrectangularsymmetrythealkaliatomsoc- between 6s and 5p polarization. The net result of these
cupy adsorption sites which maximize the coordination multiple surface dipoles, as shown in Fig. 7(b), is a low-
9

ering of the work function upon cesiation from 4.473 eV V)
e (a)
[clean ten-layer Cu(001) slab] to 2.259 eV for the cover- y ( 1.00
age Θ = 0.5, corresponding to the relaxed height of the erg
Cs atom above the Cu film of 2.988 ˚A. Also due to this En 0.95
opposite orientationbetween 6s and 5p polarization,the g
n
work function of Cs/Cu(001) displays a non-monotonic ndi 0.90
variationwith increasingthe coverageofCs: Atlowcov- Bi
ecroavgeera,gteh,egowionrgkthfurnouctgihonadmecinreimasuems w(aitbhouintcarteaΘsin=g0t.h5e) Åht () 3.05 (b)
andincreasesalittle bitfromthereonto the highcover- g 3.00
ei
age value78. The similar picture of cesiation process has H
largely discussed in Cs/W(001) system79. A well-know ate 2.95
conclusion in studying alkali-metal-atom chemisorption orb 2.90
onto a metal surface is that although the region around s 2.85
d
A
thealkali-metaladatomiselectricallyneutralandnonet 0 2 4 6 8 10 12 14 16 18 20 22 24 26
charge transfer towards the metal surface is exhibited, Thickness N (ML)
theelectronsofthealkali-metaladatomundergoastrong
mixing with the substrate electron orbitals during the
adsorption process. Such a process is local around the FIG.11: Calculated(a)bindingenergyofCsadatomand(b)
adatom and accompanied by a screening process, which adsorbate height as a function thicknessof Cu(001) films.
is responsible for work function change. Furthermore,
Figure 10(b) shows the band structure (left panel) and
that the calculated data in Fig. 11 have been carefully
the orbital-resolved local DOS (right panel) for the Cs
checked. This ensures that the oscillating behavior of
adatomandsurfaceCu atom,respectively. The complex
theadsorptionenergyhasnothingtodowiththeconver-
Cs-induced charge rearrangement is more obvious. The
gence problem, which may be encountered in the cluster
Cs adatom experiences during adsorption a repulsive in-
calculations. The supercell approachretains 2Dperiodic
teraction of the valence electron with the induced image
boundary conditions during calculation, and can avoid
charge of the ionic nuclear core. As a result, the valence
this problem. Therefore, as with other quantities, the
electronsshiftupwardinenergyandhybridizewiththose
ofthesubstrateintobondingandantibondingstates80,81. oscillations in adsorption energy is physically caused by
the periodic change in the DOS at E . In fact, from
ThebondingstatesaremainlyaroundtheFermileveland F
what we have shown in Figs. 10, or from the simplest
are dominated by hybridization between adatom sp and
Anderson-Grimsley-Newns adsoptin model82,83, one re-
surface Cu sp orbitals, as shown in the lower panel in
calls that as a Cs atom approaches to the Cu(001) sur-
the inset in Fig. 10(b). Whereas, the antibonding states
face (which can be approximated by a free-electron-like
shiftupawayfromtheFermilevelwithatypicalvalueof
metal), the valence s-p states of Cs adatom are broaden
3.8 eV, and largely consists of hybridization between Cu
intoresonantstatesdue toasaresultofinteractionwith
s and Cs d orbitals, which can be seen from the upper
themetalbands. Surfaceelectrostaticpotentialwillshift
panel in the inset in Fig. 10(b). A thorough descrip-
this hybridization down in energy to below E for sus-
tion of the bonding properties in Cs/Cu(001) system is F
taining the whole neutralization. A lower DOS at E
beyond our intention in this paper. F
implies that the Cu(001) film has fewer electronic states
After finding the preferred atomic Cs adsorption site to respond, whereas a higher D(E ) means the film has
F
[site-4inFig. 9]andgettingfamiliarwiththeCs-Cu(001) more electronic states to respond, to the presence of the
bonding properties, we turn now to our central focus on Cs adsorbate. Therefore, higher D(E ) implies higher
F
theQSEinsurfacechemisorptionandreactivity. Forthis probability in the above hybridization process, causing
purpose, we have given a series of calculations for the the sp resonance to move to lower energies and to be
binding energy (the reverse of the adsorption energy) of occupied. This leads to a higher surface reactivity and
theCsadsorbateasafunctionofthethicknessofCu(001) higher adsorption energy. This physical picture can be
thin films. Here as mentioned above we only consider formulated by expressing the adsorption energy as
the case of Θ = 0.5. The results are depicted in Fig.
11(a). One can see that the binding energy of Cs onto Ead =Eband(un-adsorbed) Eband(adsorbed) (6)
−
Cu(001) thin films depends on the film thickness in a EF
damped oscillatory way. These oscillations are featured E∆DdE,
≈
byasuperpositionoflong-andshort-lengthperiods,thus Z−∞
indicating a well-definedQSEin the surface chemicalre- where E (un-adsorbed) is the band energyfor a com-
band
actionofCu(001)thinfilmwithrespecttoCsadsorption. bined Cs/Cu(001) system with Cs adatom and Cu(001)
In experiment this QSE of atomic adsorption can be ob- film largely separated such that no chemsorption oc-
servedbyinvestigatingthedependenceofCscoverageon curs, E (adsorbed) is the band energy for the ad-
band
the monolayers of Cu(001) thin films. We address here sorbed Cs/Cu(001) system, ∆D =DOS(un-adsorbed)
−
10
DOS(adsorbed) is the difference in the electronic DOS dence of electronic structure, surface energetics, and in-
between the un-adsorbed and adsorbed Cs/Cu(001)sys- terlayerrelaxationuponthethicknessofthefilmhasbeen
tems. As the film thickness changes, QW state en- fully investigated, clearly showing the metallic QSEs of
ergy levels shift. Whenever a QW state crosses the the film. These QSEs have been shown to be closely
Fermi level from above, it adds energy to the E (un- related with the occurrence of QW states at the Fermi
band
adsorbed) and adds electronic density to un-adsorbed level. UnlikesomeothersimplespmetalssuchasAl,Mg,
D(E ). Whereas E (adsorbed) decreases upon the andPb,the Fermisurfaceof bulk Cuis characterizedby
F band
crossing between the QW states and the Fermi level due multiple stationary extrema such as the belly and the
to the abovementionedcausethat the Cs-Cuorbitalhy- neck points. As a consequence, the Cu(001) QW states
bridization shifts more down in energy by the increase also display the abundant properties at these surface-
of D(E ). Thus the net consequence is an anti-phase projected k-extrema. For example, we have shown the
F
oscillation mode of the adsorption energy with respect different film-thickness oscillation modes by calculating
the oscillation mode of D(E ) as a function of the film the energies of QW states in correspondence with these
F
thickness. stationary extrema. Due to the interference between
The Cs adsorbate height is plotted in Fig. 7(b) as a these two kinds QW states, the energetics and the cor-
function ofthe film thickness,whichalso showsthe peri- responding stability (Fig. 6) of the Cu(001) thin films
odicoscillationsindicativeofQSE.Comparedtothe fea- have been shown in a consistent way to display a quan-
tures in thickness dependence ofbinding energy,one can tum beating behaviorasa function ofthe film thickness,
notice that oscillations in the adsorbate height are much with the oscillation periods consisting of long and short
weak and the periods are somewhat not easily resolved. length. In addition, the calculated energy gap between
The cause may be due to the fully interlayer relaxations the highest occupied QW state and lowest unoccupied
asawholestructureduringcalculations. Ifthesubstrate QWstate,whichcanbedirectlymeasuredviaSTStech-
is not allowed to relax during the adsorption process, nique, have been shown to display different slopes and
then it can be expected that there is a consistent corre- kinks at these extrema in bulk BZ.
spondenceintheoscillationsbetweenthe binding energy Wehavealsoextensivelydiscussedtheotherhighlyin-
and the height of the adsorbate. terested issue, i.e., the oscillatory QSE in surface chemi-
Moreover, we have also calculated the work function cal reactivities, via studying the Cs adsorption onto the
of Cs/Cu(001) as a function of the film thickness. The Cu(001)thin films. Throughsystematic calculations, we
results are shown in Fig. 7 (open squares). Compared have shown that the Cu(001) surface chemical catalysis
to the case of clean Cu(001) films, the work function is for Cs adsorption can be uniquely modulated and con-
decreased by a typical value of 2.2 eV [see Fig. 7(b) for trolledbytheoccurrenceofQWstatesattheFermilevel.
10-layer case]. Also, the adsorbed work function oscil- TheCsadsorptionenergyandtheworkfunctionafterce-
lates with increase of the film thickness. However, one siationarefeaturedbyquantumoscillationsasafunction
can see that the oscillating mode is different from that ofthefilmthickness. Theamplitudesintheseoscillations
without adsorption, and also the oscillating amplitude reach a typical value of 40 meV. From the application
becomes much weak. This implies that the QW states pointofview,this resultisquite intriguingbecause1ML
are different for clean and cesiated Cu(001) thin films. changeinthefilmthicknesscancauseaneffectequivalent
As a result, there no longer exists a simple one-to-one to a temperature change as high as 400 degrees. These
correspondencebetweentheadsorbedQWstatesandthe calculated results thus may be used as a guide to tailor
bulk Cu Fermi surface. More detailed work will be done catalysis, chemical reactions and other surface processes
for this issue. in nanostructured materials.
VI. CONCLUSION
Acknowledgments
Insummary,the cleanandcesiatedCu(001)thinfilms
have been extensively studied by density-functional the- ThisworkwaspartiallysupportedbytheCNSFunder
orypseudopotentialplane-wavecalculations. Thedepen- Grant No. 10544004and 10604010.
∗ Electronic address: zhang [email protected] 4 A. Kawabata, J. Phys.Soc. Jpn.62, 3988 (1993).
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