Disorder induced local density of states oscillations on narrow Ag(111) terraces

0.41 MB·English

by  Karina Morgenstern

Disorder induced local density of states oscillations on narrow Ag(111) terraces
Karina Morgenstern and Karl-Heinz Rieder
Institut fu¨r Experimentalphysik, FB Physik, Freie Universit¨at Berlin, Arnimallee 14, D-14195 Berlin, Germany
Gregory A. Fiete
5 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
0 (Dated: February 2, 2008)
0
Thelocal densityofstatesofAg(111)hasbeenprobedindetailondisorderedterracesofvarying
2
width by dI/dV-mapping with a scanning tunneling microscope at low temperatures. Apparent

n shifts of thebottom of the surface-state band edge from terrace induced confinement are observed.
a DisorderedterracesshowinterestingcontrastreversalsinthedI/dVmapsasafunctionoftip-sample
J
voltagepolaritywithdetailsthatdependontheaveragewidthoftheterraceandtheparticularedge

8 profile. In contrast to perfect terraces with straight edges, standing wave patterns are observed
1 parallel to the step edges, i.e. in the non-confined direction. Scattering calculations based on the
Ag(111) surface states reproduce these spatial oscillations and all the qualitative features of the

] standingwave patterns, including thepolarity-dependent contrast reversals.
i
c
s PACSnumbers: 73.20.At,73.20.-r,68.37.Ef
-
l
r
mt I. INTRODUCTION structure of the surface,21,22,23 also occur on surfaces
where disorderis present. Inparticular,the LDOSmod-
.
t ulations on irregular step arrays24 can have a dramatic
a The fcc(111) surfaces of noble metals exhibit a large
m sp-band gap in the projected bulk band structure along impact on these processes. Therefore, it is important
- the Γ-L line.1,2 These gaps reach down below the Fermi to understand and to be able to predict the electronic
d structureondisorderedterracesofvaryingwidthlimited
energy at the center of the surface Brillouin zone and
n support an occupied free electron surface-state.3,4 Elec- by steps with irregular edges. A study of the electronic
o structure of such terraces is the focus of this paper.
trons occupying these Shockley-type surface-states form
c
[ a two-dimensionalnearlyfreeelectrongasparallelto the In a recent publication, we have investigated the de-
surface.5,6,7 Electrons are confined to the vicinity of the pendenceoftheLDOSneartheFermi-energy,EF,onthe
1 top layer by the vacuum barrier on one side and a band width of terraces on Ag(111) at 7 K.25 We showed that
v
gap in the bulk states on the other side. the apparent surface-state band minimum shifts mono-
5
tonically towards the bulk E with decreasing terrace
3 Scanning tunneling microscopy (STM) and spec- F
4 troscopy (STS) are particularly sensitive tools to study widthleadingto adepopulationatameanterracewidth
1 these surface-states that dominate in the zone be- L = 3.2 nm, in quantitative agreement with a Fabry-
0 tween the tip and the surface.8 These techniques P´erot model with energy dependent finite asymmetric
5 map the local electronic density of states (LDOS) in reflectivity of the step edges. In addition, a switch
0
from confinement by terrace modulation to step mod-
real space. STM has provided real space observa-
/
at tion of electrons in surface-states and of their interac- ulation was observed at λF/2 (L=3.7 nm), half of the
m tions with adsorbates,9,10 steps,9,10,11,12,13,14, and other Fermi wavelength on an infinite terrace. This switch
structures9,10,15,16 intheformofstandingwavepatterns. occurs on Cu(111) surfaces also at λF/2 as proven by
- photoemission.14
d These spatial oscillations are quantum mechanical inter-
n ference patterns caused by the scattering of electrons In this paper, we present STS data revealing spatial
o in the two-dimensional electron gas from defects. The oscillations of the local density of states on Ag(111) for
c
standing wave patterns contain information about the narrow disordered terraces of varying width up to 8.5
:
v surface-statedispersionandtheygiveinsightintothein- nm. Apparent band edge shifts of the surface-state on-
Xi teraction between surface-state electrons and scattering set (as reported in Ref. [25]) lead to conductance varia-
sites on the surface.9,10 tionsonterracesofdifferentwidthatthesameenergy. A
r
a STS images of the local differential tip-surface con- standing wave pattern also evolves parallel to the steps,
ductance (called dI/dV maps) have been used to in- i.e. in the non-confined direction. Its wavelength follows
vestigate how the surface-state interacts with a single the dispersion relation of an electron in the clean, infi-
straight atomic step17,18 and with a pair of adjacent nite surface-state. In order to understand the wave pat-
straight steps.11,18 From the measurement of spatial os- tern, theoretical simulations based on scattering theory
cillation periods normal to the step it was possible to areperformedfordisorderedterraces. Thesecalculations
extractFermiwavevectors19andcompleteenergydisper- successfully reproduce the observed oscillations parallel
sion curves.20 Thus, the surface-state on clean, disorder- to the terraces as well as interesting tip-sample voltage
free systems is well characterized. However, growth and polarity dependent effects.
catalysis, which are both influenced by the electronic Thispaperisorganizedinthefollowingway. InSec.II
2
we discuss the details of the experiment, including sam- waves parallel to the steps. This leads to a wave pat-
ple preparation, the types of measurements made, and tern perpendicular to the steps. Fig. 1a shows these well
the main features observed in the data. In Sec. III we known spatial oscillation of the surface-state electrons
discuss the theoretical model used to interpret the ex- perpendiculartopre-existingstepedgesonalargeterrace
periments and its assumptions. We also present the re- of 51 nm in width. In addition, surface impurities act as
sults of model calculations and discuss their connection point defects and lead to radial wave patterns centered
with experiment. Finally, in Sec. IV we give the main ontheimpurities. At32mVthewaveshavetheexpected
conclusions of our work. wavelength of λ/2= 3.5 nm (See Ref. [20]). At 7 K five
maxima are discernable already in the topographic im-
age. Notetheirregularitiesoftheplanewavesintroduced
by the naturaldeviationofthe stepedge positionfroma
II. EXPERIMENT
straight line.
The LDOS (as recorded in dI/dV spectra) displays a
A. Sample preparation and data acquisition
step-like function marking the sharp increase of differ-
ential conductivity around the surface-state band onset
Theexperimentshavebeenperformedinultrahighvac-
(Fig. 1b). Following Ref. [16], we measure the onset of
uum with a low temperature scanning tunneling micro-
the surface-state, U = (U +U )/2, and a width,
on top bot
scope that operates at temperatures between 6 and 300
∆=U −U , of the onset by continuing the slope at
top bot
K. The single crystalline Ag(111) surface is cleaned by
the middle of the rise to the bottom and the top of the
sputtering and annealing cycles. The surface separates
riseasindicatedinFig.1b. ThevaluesofU =−(68±4)
on
into largeflat terraces andstep bunches with anaverage
meV and ∆=8 meV are in goodagreementwith earlier
terracewidthof4nm.24 Inthispaper,weinvestigatethe measurements.18
spatialoscillationsoftheLDOSinthesesteppedregions.
Confinement on narrow terraces leads to both a shift
Measurements are performed at 7 K.
in energy of the surface-state related peak and a broad-
The topographic images are taken in constant current eningofit.25 Moreover,additionalmaximadevelop. The
mode. dI/dV spectra are recordedwith the lock-intech-
example shown in Fig. 1c, taken on a terrace of 6.1 nm
nique; the ac tunnel current is driven by a 4 mV signal
width,displaysonsetsat-53meVand32meVwithonset
added to the junction bias with a frequency of 327.9 Hz,
widths of ∆ = 23 meV and ∆ = 32 meV,
−53meV 32meV
381.8Hz,or738.1Hz. Voltagesareappliedtothesample
respectively. Thesetwopeakscorrespondtothefirsttwo
with respect to the tip. Thus, for a negative voltage the
transversemodesofthesurfacestateconfinedbytheter-
occupiedsideofthespectrumisprobed. dI/dVmapsare
race edges.
recorded simultaneously with the topographic images at
The spectra vary for terraces of different width, but
the same voltage by recording the lock-in signal at each
also for different positions on the same terrace (Fig. 2)
pixel. ItisnotpossibletoruntheSTMinconstantheight
with slight variations in apparentsurface state onset en-
modeinimagingasteparrayofmorethan1nmchangein
ergyandlargevariationsindI/dVintensity(Figs.2band
height. Thus,theconstant-currentimagerecordedsimul-
c). In particular, close to the step edges, U shifts to
taneouslywith the dI/dVmapis usedfor anadjustment bot
lower values influencing the onset value U (Fig. 2b).
in tip-sample separation. This leads to a superposition on
We have shown before that these variations close to the
of the dI/dV signal with a variance in height at the step
step edges are in accordance with a Fabry-P´erot model
edges.
withthestepsactingasfinite,energy-dependentbarriers,
AdI/dVmapof256x256pixelsisrecordedinabout80
so this is no indication of a shift in energy.25 As a conse-
minutes. Series of dI/dV maps therefore requires a high
quence of strong scattering of the surface-state electrons
stability STM, which our system provides. Typically,
by the step potential, the amplitude of the surface-state
the drift is less than 0.3 nm during a series of 10 dI/dV
issmallerclosetothestepedges.28 Thus,thepeakinten-
maps, i.e. 13 hours. Major drift problems arise from the
sityofdI/dVvariesforthesameterracewidthdepending
increasingtemperatureattheendofheliumevaporation.
onthedistancefromthestepedges(Fig.2c),beinghigh-
Therefore, measurement series on the same spot of the
estinthemiddlebetweenthestepedgesforthefirstpeak
crystalarelimitedbytheheliumamountavailableinthe
in dI/dV and showing two maxima for the second peak
bath cryostat.
indI/dVonthe largestterrace. Onlythe widestterrace,
the second one in Fig. 2a, shows two peaks in dI/dV.
The second peak arising in dI/dV due to local con-
B. Experimental results
finement on a single terrace should show a quadratic de-
pendence of energy with terrace width according to a
For a defect-free terrace of ideally parallel steps, the particle-in-a-box model for the simple case of infinitely
surface-state electrons on each terrace are confined per- highpotentialwalls:29E =h¯2π2/2m L2,withm
gnd eff eff
pendicular to steps but are free parallel to them. Thus, theeffectivemassandLtheterrace(confinement)width.
the wave functions of surface-state electrons are separa- The plot of the energy of the second peak relative to
ble, with standing waves normal to the steps and plane apparent surface-state onset vs. 1/L2 (Fig. 3) shows
3
indeed a linear dependence for terrace widths > λ /2, alongtheperpendicularwavesonthelefthalfandonthe
F
i.e. for those terraces where the Fabry-P´erot model is right half of the terrace are independent of each other.
applicable.25 From the prefactor of (6.371± 1.794) eV This leads to an interference pattern at the middle of
nm2, we determine mterr = (0.59±0.17) m , which de- the terrace, which changes wavelengthwith energy (Fig.
eff e
viates fromthe effective mass ofelectronsin the surface- 6c to d). Note the differences of terrace “3” from the
state band where m =0.4m . slightly narrower terrace “5” at + 15 mV and +30 mV.
eff e
Toinvestigatespatialvariationsofthedifferentialcon- Terrace “5” shows lower conductivity at its center than
ductance in more detail, we have recorded dI/dV maps at its sides, while terrace “2” shows higher conductivity
onthesesteparraysatdifferentenergies. Someexamples atitscentercomparedtoitssides. Alsoterraces“3”and
are displayed in Figs. 4 to 7. “4” show wave patterns of opposite contrast.
Below the onset of the surface-state, dI/dV maps dis- The wavelength dependence on energy of the parallel
play bright lines and spots on a continuous background wave pattern is investigated in more detail on the unoc-
(Fig.4btod). Thebrightlinescorrespondtothesurface cupied side of the LDOS (Fig. 7). The standing wave
stepsandareanartifactoftheconstantcurrentscanning pattern parallel to the surface steps on terrace “3” at
process (see above). The brightest spot in the dI/dV positive polarity is contrast-reversedwith respect to the
maps corresponds to an impurity at the step edge. It is standing wave pattern shown in Fig. 4. The dI/dV map
visible in the topography as a protrusion imaged 20 pm at -30 mV (Fig. 7b) shows that this is an effect of the
higherthanthesurroundingstepedge(Fig.4a). Thisim- opposite polarity of the tip-sample voltage bias and not
puritycanbeusedasareferencepointtocompensatefor of the terrace width. Due to a subsurface impurity (see
thermaldrift. Thetwolessbrightspotsarenotvisiblein below) the wave pattern in the lower part of this image
thetopographicSTMimageinthecontrastshown. Con- is less regular. Thus, we concentrate on the upper part
trast enhancement shows them as protrusions of only 4 of the image.
pm in height. We therefore attribute them to subsurface
The average wavelength changes with bias voltage
impurities.
(Fig. 7m). For an ideal terrace with impenetrable edges,
With increasing voltage a difference in contrast devel-
the isotropic free surface dispersion relation E(~k) =
opsbetweenterracesofdifferentwidth. Forthe broadest
terrace“3”,thereisacontrastswitchfromdarktobright E0+ 2h¯m2ekf2f, with E0 = −0.065 eV, is modified by “con-
between -50 mV and -40 mV (Fig. 4g to h). Narrower finement” due to the terrace structure,
terraces switch contrast at voltages closer to E . The
F
dependence of the switch in intensity on terrace width
results from the different apparent surface-state onsets ¯h2π2n2 ¯h2k2
k
E (k )=E + + , (1)
(see Fig. 2a). On the largest terrace “3” a wave pattern n k 0 2m L2 2m
eff eff
evolvesparallel tothe stepedges. Thewavelengthofthis
wave pattern increases as the energy is lowered with re-
spect to EF. While a faint wave pattern discernable at where kk ≡ 2π/λk and n is the number of transverse
-50 mV on terrace “3” (Fig. 4g) seems to originate from modes. This equation predicts that for a fixed energy,
the subsurface impurity, the standing wave pattern at narrowerterraceswithsmallerwidthLwillhavealarger
smaller energies is only perturbed by this impurity. The parallel wavelength, λk. Examples of this effect can be
wave pattern does not originate from it. seen in Figs. 4 to 7, although the oscillations in the nar-
Thereisnoobviouscorrespondencebetweenwavepat- rower terraces are less regular for the narrower terrace.
ternsonadjacentterraces. Forexample,at-30meV(Fig. We attribute this to larger relative variations in width
4i)thedistancebetweentheuppermostmaximaisdiffer- for fixed edge variations in the narrowerterraces.
ent on terrace “3” and on terrace “4”. At -10 mV (Fig. Furthermore,a quadraticvariationofλ with E is ex-
k
4k) the wave patterns on terraces “3” and “4” are phase pectedforafreeelectrondispersion(seeEq.(1))provided
shifted in the upper half of the image. noadditionaltransversemodesareoccupiedovertheen-
Fig. 5 compares the parallel wave pattern (-10 mV, ergy range probed. Indeed, the data can be well fitted
Fig. 4k) to a pattern at opposite polarity, i.e. in the by a(E −E )−1/2. We find E = −(67±5) meV, the
0 0
unoccupied region (+90mV). At +90 mV the dominant same value as the bottom of the surface-state band de-
wave pattern on terrace “3” runs perpendicular to the termined from the surface-state onset on large terraces
stepedges,however,withcontrastvariations. Thus,per- (see above) and from perpendicular wave patterns on
pendicular andparallelwavepatterns are superimposed. large terraces.17 Also the wavelengths determined from
Note that even at +90 meV the narrower terraces, e.g. the negative dI/dV maps of Fig. 4 fit a quadratic dis-
“5”, show the parallel wave pattern only. persion with E =−(68±19) meV. Fig. 7n displays the
0
This superposition of parallel and perpendicular wave dispersion relation deduced from this measurement that
patterns becomes more pronouncedon even broaderter- reproduces the dispersion relation determined from per-
races(Fig.6). Fortheterrace“2”of8.5nminwidth,the pendicularwavepatterns.20 Thus,parallelwavepatterns
perpendicular wave pattern dominates at -38 mV (Fig. originate also from the scattering of surface-state elec-
6b). At +15 mV and +30 mV the contrast variation trons.
4
III. THEORY B. Theoretical calculations
1. Model calculation for qualitative effects
A. Model and approach
To understand the qualitative effects we expect from
a scattering theory approach to the LDOS modulations
Inthispaperwearefocusedonunderstandingtherole fromdisorderedterraces,wefirstcarriedoutcalculations
of disorder in the terrace edge profile on the observed ona single terracewith perfectly straightwallsand then
modulations in the local density of states. Disorder in added in increasing amounts of disorder. The terraces
the edge profile breakstranslationalsymmetry alongthe were modeled by placing 21 point (s-wave) scatterers at
direction parallelto the terrace edges. This brokensym- a distance of λF/4 along the terrace edge. See Fig. 8.
metrypreventsasimpleseparationoftheenergy(andthe The dI/dV maps were computed using the surface-
wavefunctions)ofthe electronsinto a parallelandtrans- statedispersionE(~k)=E + h¯2k2 . (Confinementeffects
verse component, as was done in Eq. (1). In order to that do not cause an app0aren2mtesfhfift in the surface-state
conveniently handle the broken translational symmetry band minimum are captured automatically by electron
another approach is needed. We chose to use a scatter- scattering from the terrace “edges”.) An approximation
ing theory26 that has been quite successful in predicting to the “constant current” condition used in the exper-
the standing wave patterns in the LDOS observed with iment (while simultaneously aquiring topographic and
the STM around impurities and step edges on the sur- dI/dV maps) is achieved by calculating
faces of the noble metals that support surface-states.
dI dI dI
(eV) ≡ (eV) −h i (2)
The scattering theory given in Ref. [26] most natu- (cid:20)dV (cid:21) (cid:20)dV (cid:21) (cid:20)dV (cid:21)
meas calc calc
rally describes the scattering of electrons in the surface-
state from point-like impurities (adatoms) where an s- where h dI i = eV dǫ dI(ǫ) (eV)−1. This
dV calc 0 dV calc
wave scattering approximation can be made; scattering procedur(cid:2)e co(cid:3)rrectly repRroduce(cid:2)s brig(cid:3)htness variations in
from an extended impurity, such as a terrace edge, re- dI/dV maps simultaneously taken with topographic
quires some modification of the theory, which otherwise maps.27
cannotdirectly describe scatteringfromanextendedob- Intheseinitialmodelcalculations,eachscattererisas-
ject. Fortunately, since the standing wave patterns we sumedto havea phaseshift ofπ/2. This value turns out
are interested in (those parallelto the terrace edges) are to be almost “universal” for adatoms on the surfaces of
observed most clearly in the central portion of the ter- noble metals26 so we take it as a starting point; later we
race, we may make a simple approximation in the scat- willfindadifferentvaluematchestheexperimentsbetter
tering theory for the terrace edge: We treat the terrace whenweattempt tomakeadirect, quantitativecompar-
edge as a dense “row” of point (s-wave) scatterers. Far ison between theory and experiment. The initial choice
from the terrace edges (near the middle of the terrace, ofπ/2isarbitraryanddoesnotaffectanyofthequalita-
for example) the results will be indistinguishable from tive conclusions reached from the calculations presented
a terrace edge that is continuous. A “row” of atoms in Figs. 8 and 9. While this value works well for single
is expected to be a good approximation to the terrace adatoms, the physics of scattering from a terrace edge
edge provided the atoms that make up the wall are well is very different from the physics of scattering from an
within a wavelength(the wavelengthof the surface-state isolated atom, and this is why the best phase shift for
electron) of each other and the distances probed are of the terrace edge atom differs from π/2.
the orderofthe separationofthe atomsorlarger. Inthe Fig. 8a shows the calculated [dI/dV] map for a
calc
calculationsdescribedbelowourobjectivewillbe firstto troughof5λ in lengthandλ /4inwidthlimited by21
F F
understand the qualitative features we can expect from point scatterers on each side. While the perpendicular
scattering theory and second to compare as directly as standing wave pattern outside the trough is clearly visi-
possible with experiment. The latter requires determin- ble,itsnarrownessinhibitsasimilarwavepatternwithin
ing the phase shift of a single atom (one of many that the trough. It is too small to support even half of a
makeupthe terraceedge)sothatthe correctphaseshift Fermi-wavelength. The introduction of an arbitrary de-
results for the edge as a whole. Since the input to the fect in one of the terrace walls leads to standing waves
scatteringtheoryis the wallprofile,the dispersionofthe parallel to the rows of atoms with the same wavelength
surface-state electrons, and the s-wave scattering phase λ /2 as the perpendicular waveoutside the trough(Fig.
F
shift δ, the only free parameter we have in the theory is 8b). Note the faint signs of oscillations in the “perfect”
δ. We tune δ for a fixed wall profile and surface-state terrace as well. These originate from the finite length
dispersion to reproduce the experimental dI/dV vs. V of the trough, where its ends act as defects. These first
spectra at the center of the terrace. Once δ is optimized calculations demonstrate that a parallel wave pattern is
in this way, we computed the dI/dV image maps (at a indeed a result of terrace disorder.
fixed voltage) over all positions using this value and the Forabetterunderstandingofparticularfeaturesinthe
exact edge profile of the terraces. wave patterns observed in the experiments, we compare
5
[dI/dV] maps of varying terrace width and terrace r =0.48−2.86E/eV17 for the descending and the as-
calc asc
disorder (Fig. 9). Terrace disorder has been modeled in cending step, respectively. Spectra 15, 16, and 17 are
the way described in the figure caption. Fig. 9 a, d, and measured at a distance of 0.6 nm, 1.2 nm, and 1.8 nm
g support the idea that a wave pattern perpendicular to from spectra 14 are compared to calculations performed
the stepscanonlydevelopforlargerterraces. Theinten- at ‘0.7 nm’, ‘1.5 nm’, and ‘2.0 nm’ from the center of
sityoftheparallelwavepatternincreaseswithincreasing the terrace, respectively. In all cases, the peak at ≈ 20
disorder (Figs. 9a to c) and shows the same qualitative meV lies at almost exactly the same energy and its rela-
featuresasinthe experiments. Forexample,someofthe tive intensity is well reproduced. The first peak lies also
maxima in Fig. 9c are brighter than neighboring ones; at approximately the correct energy. However, a sharp
some of them seem to be disconnected from neighboring rise in the theoretical spectra (an artifact of the calcula-
maxima by deep minima, while between other maxima tionprocedure)makesacomparisondifficult forthe first
there are only little contrast variations. Fig. 9 c and i peak. Inboththeoryandexperiment,thesecondpeakis
showthattwobrightprotrudinglines areseeninsteadof broader than the first peak, because the reflectivity and
one for a larger terrace width. For the broadest disor- thusthe lifetime ofthe resonancedecreaseswithincreas-
deredterrace(Fig.9h,i)thetwowavepatternsarethus ing energy.
superimposed. Minimainthetwoperpendicularpatterns In Fig. 10 the peaks are sharper and more intense in
are displaced with respect to each other by up to half a theory than in experiment. Several processes may ex-
wave length (Fig. 9i) as in the experiment. Many of the plain this difference in sharpness and intensity. On the
qualitativefeaturesoftheexperimentscanthusberepro- experimental side, the spectra show thermal and modu-
ducedby the scatteringtheory,showingthey resultfrom lation voltage broadening. We simulate the modulation
terrace edge disorder. voltage averaging and temperature broadening by aver-
aging the theoretical spectra (Fig. 10b). Indeed, this
broadening makes the additional peak in the theory at
2. Direct comparison with experiment ≈ −20 meV undetectable in the experiment and lowers
theintensityofthepeakat+20meV.However,thetheo-
reticalpeaksremainsomewhatsharper,presumablyfrom
The calculations presented so far were meant to illus-
details associated with how the terrace edges are mod-
trate that the main physics is captured by the scatter-
eled. One may think that the “porous” nature of the
ing theory and to emphasize the quality of the quali-
walls my lead to a shorter lifetime, and hence a broader
tative agreement between the scattering theory and ex-
thoerypeakthanwhatisobservedexperimentally. How-
periment. Now, we go beyond these simple qualitative
ever,thisappearsnottobethecase. Changingthenum-
model calculations and use the actual surface-state dis-
ber ofatomsthat makeupthe edgeprofile by 50%has a
persion for Ag(111) listed above and a terrace edge pro-
negligible effect on the widths of the peaks.
file taken from experiment. The surface-state onset on a
large terrace lies at E = −0.065 eV; the effective mass Discrepancies may result from not correctly modeling
0
is m = 0.4m . These values lead to λ = 7.38 nm theenergydependenceofthereflectivitybyreplacingthe
eff e F
for the Fermi-wavelength on an infinite terrace. First, stepsbypointscatters: The peaksindI/dVcanbe asso-
wecalculatepositiondependent[dI/dV] spectrafora ciated with Fabry-P´erot type resonance with an energy
calc
7.9nmwideterracewithvaryingphaseshiftandcompare dependent reflectivity.25 As a narrowerresonance means
themtotheexperimentalspectraforthebroadestterrace a longer lifetime, and thus a greater reflectivity, we con-
in Fig. 2. The phase shift of the scatterers is chosen to clude that theory has an effective reflectivity which is
bestreproducetheFabry-Perottypeoscillationswiththe larger than experiment. The calculation has an energy
tip-samplebias,asshowninFig.10. Thebestagreement dependent reflectivity, but because the walls are built
is reached with a phase shift of δ = (0.9±0.1)π. Those up from point scatterers, it is difficult to determine the
spectraresembletheexperimentaloneswithashiftinthe reflectivity exactly in order to compare it directly with
apparentsurface-stateonsetandasecondpeakintheun- experimentalresults for a “perfect” terrace. With effort,
occupied region. Note that in order to keep our fitting a very close match could be made by making the phase
parameter to only the number δ, we do not include any shift energy dependent and possibly adding imaginary
energy dependence in the phase shift. parts as well, but extremely accurate numerical agree-
There is good agreement between the spectra calcu- ment is not our goal here. Rather, we wish to under-
lated for the right half of the terrace (Fig. 10a), if spec- standwhatis responsibleforthe essentialfeaturesofthe
trum number 14 showing no second maximum is taken data. A more accurate calculation than ours would also
as the central spectra, though it is measured 0.6 nm to include changes to the surface-state band dispersion it-
therightofthegeometricalcenter. Thismightbesimply selfdue to the terraceedges. Recallthatourtheory uses
due to difficulties in determining the geometrical center the surface-state dispersion of the free surface-states.
of the terrace edges or may be due to the asymmetry The dI/dV spectra to the left and to the right of the
of the reflection properties of ascending and descending center differ in both experiment and theory as a result
step edges.17 The reflectivities of steps on this surface of an asymetry in the disorder of the two terrace edges.
have been measured to be r = 0.72−2.91E/eV and However, for the experimental spectra to the left of the
desc
6
center, the second peak shifts with position on the ter- theimage(Fig.12d). Onlyaphaseshiftinthelowerpart
race. This is not recovered in the calculation (Fig. 10c). is not recovered. Thus, againthe major difference lies in
We attribute this difference to the fact that the spectra intensity. Thislastexampleunderlinesthecomplexityof
arecalculatedforaterracethatissome10%smallerthan the evolved wave pattern and that it may depend quite
the experimental terrace and that has a different shape sensitively on various details of the geometry.
in detail, on the one hand, and to the larger difference
in reflectivity properties between a row of atoms and a
descending step than a row of atoms and an ascending
IV. CONCLUSIONS
step, on the other hand. The asymmetry in scattering
properties of the two steps that can not be captured by
replacing them by two rows of equivalent atoms leads We have probed the LDOS of disordered terraces of
thus to the ‘off-center middle’ (noted earlier in this sec- varying width and observed oscillations parallel to the
tion) and to larger discrepancies on the half of the ter- terraceswithawavelengthinaccordancewithafreeelec-
race that is closer to the descending step edge. Despite trondispersion,andwehaveshownthattheseparallelos-
thesedeviationsindetail,the scatteringtheorydoessat- cillationsarisenaturallyinscatteringtheorycalculations
isfactorilyreproducethe experimentalspectra. Thus,we for disordered terraces. The theory also captures impor-
have calculated dI/dV maps for a particular terrace for tantcontrastvariationswithtip-samplevoltagepolarity.
a direct quantitative comparison of experiments to the- Thus, the wave patterns observed experimentally are a
ory. Wehavechosenterrace“3”ofFig.4duetotheclear result of step disorder.
wavepatternobserved. Thisterraceisalsobroadenough The complete pattern is a combinationofFabry-P´erot
thatchangestothesurface-stateonnarrowterraces25are (surface-state) and disorder scattering effects along with
negligible. surface reconstruction effects. The Fabry-P´erot effects
Theterracewallsareapproximatedby54atomsplaced are intrinsic in scattering theory, but the changes to
as close to the actual boundary as possible (Fig. 11a, the underlying surface-state band structure are not (in
white dots). Fig. 11 compares the [dI/dV] maps, the theortical model used here, which assumes a clean
meas
computed from Eq. (2), of Fig. 4j and k at -20 mV, at surface-state dispersion). As the scattering theory re-
-10 mV, and of Fig. 5c at +90 mV to the calculation. covers all the qualitative features of the experiment
The qualitative agreement is excellent: A similar wave andyieldssemi-quantitativeagreement,thesurface-state
pattern is observed in the calculated spectra as in the scattering effects dominate (at least for the wider ter-
experimental one, both in the occupied and the unoccu- races). Remaining differences are attributed to surface
pied region. Also the decrease in wavelength from -20 reconstruction effects. The semi-quantitative agreement
meV to -10 meV is reproduced in the calculation. The in the simulation also shows that the cross-talk between
comparisonofthe[dI/dV] linescansthroughthecen- the large terraces and their neighbors is negligible.
meas
ter ofthe terraces(Fig.11c) showsthis goodwavelength The terraces that the dI/dV maps have been calcu-
agreement for both -20 meV and -10 meV. The fourth lated for are wide enough to allow us to use the surface-
maximumat-20mVinFig.11cissimplynotresolvedin statebandstructurevalidforinfinitelybroadterraces. In
the experimental data. Only the relative intensities are principlenarrowerterracescouldbecalculatedwithinthe
partly recovered. sametheory byusing the appropriatedispersionrelation
Wehavealsosimulatedthebroadestterrace“3”inFig. fornarrowerterraces,whichwillingeneralbeanisotropic
7 for a more detailed comparison of theory with exper- and depend on terrace width.30
iment for wavelength changes in the unoccupied region. As a future project, we suggest to locally investigate
Fig. 12 shows the comparison of simulation and exper- the reaction dependence on the vicinal surface and we
iment at 33, 26, and 17 mV. Unfortunately, the direct expect local variation on the nanometer scale due to the
comparison is somewhat obstructed by an impurity, vis- reported variation of the LDOS.
ible by enhancing the contrast in the topographic image
(Fig. 12a) leading to a very particular wave pattern at
-30 meV (Fig. 7b). At first sight, the comparison seems
V. ACKNOWLEDGEMENT
to be less favourable. The simulation shows two bright
protruding lines close to the step edges separated by a
dark stripe with little contrast variations. However, this We acknowledge financial support by the Deutsche
reproducesthevariationofintensityacrosstheterracein Forschungsgemeinschaft, NSF-9907949, NSF DMR-
the experiments (from bright to weaker to bright). De- 0227743 and the Packard Foundation. We are greatful
spite the impurity the lines cans through the middle of for experimental support by K.-F. Braun, Ohio Univer-
the terraceagree well in wavelengthin the upper partof sity, Athens.
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L17 (1977). L157 (2000).
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8
FIG.1: (a)RealspaceSTMimagetakenat7Kshowingspa-
tial oscillations of electrons in the Ag(111) surface-state on
a large terrace 51 nm in width. Circularly symmetric oscil-
lations in the LDOS are seen around point impurities. The
tip-samplevoltagebiasisU =32mVwithatunnelingcur-
bias
rentItun=0.43nA.(b)Thesurfacestateonsetisindicatedin
the dI/dV spectrum on the large terrace shown in (a). Here
Uon =(Utop+Ubot)/2 is taken as the value of the onset. (c)
The dI/dV spectrum on the edge of a narrow terrace of 6.1
nmwidth(indicatedbyawhitedotontheinsetSTMimage)
shows two peaks in intensity. These peaks can be related to
the onset of the lowest transverse mode of the terrace (first
peak≈−50mV)andthefirstexcitedtranversemode(second
peak ≈50 mV).Here Ubias=-137 mV and Itun=0.34 nA.
9
FIG. 2: (Color online.) Position dependent STS at differ-
ent points across the width of different terraces showing the
dependenceofthesurface-stateonsetandofthesecondmax-
imum in dI/dV spectra on average terrace width and on po-
sition on the terrace. (a) The background grayscale image
is a topographic image (Ubias=-47 mV, Itun=1.3 nA) show-
ingapproximately5terraceswithirregular,disorderededges.
The average widths range from 2.7-7.9 nm. Superimposed
on each terrace of the STM image is a set of inset STS data
taken at various positions (indicated in color) across the ter-
race. The positions are numbered from left (1) to right (30).
(b)Positionofbottom(downtriangle),top(uptriangle),and
middle (diamond) of first (filled symbols) and second (open
symbols) peak in dI/dV spectra of (a). The second peak is
only observed on thewidest terrace (of width 7.9 nm) and is
relatedtoexcitingthesecondtransversemode. Darkvertical
lines indicate positions of the step (terrace) edges. (c) Rela-
tiveintensity[i.e. (dI/dVtop−dI/dVbot)/dI/dVbot]ofthefirst
peak(filledsymbols)andthesecondpeak(opensymbols)for
spectra of (a).
10
FIG. 3: Scaling of the relative position of the first and sec-
ond maximum in dI/dV curves on small terraces vs. inverse
terrace width squared. For terraces wider than L ≈ 4.3 nm,
|U1on−U2on|∝L−2 asshownbythelinearfit,implyinganin-
terpretationconsistentwiththepresenceoftransversemodes
of thesurface state. However, theeffective mass determined,
mteefrfr =(0.59±0.17) me,deviates from theeffectivemass of
electrons in the surface-state band where meff =0.4me. On
the other hand, terraces narrower than L ≈ 3.4 nm do not
scalelinearlywithL−2,butinsteadshowbehaviorconsistent
with step modulation.