Quantum size effects on the work functionof metallic thin film nanostructuresJungdae Kima, Shengyong Qina, Wang Yaob, Qian Niua, M. Y. Chouc, and Chih-Kang Shiha,1

aDepartment of Physics, University of Texas, Austin, TX 78712; bDepartment of Physics and Center for Theoretical and Computational Physics, Universityof Hong Kong, Hong Kong, China; and cSchool of Physics, Georgia Institute of Technology, Atlanta, GA 30332

Edited by Steven G. Louie, University of California, Berkeley, CA, and approved June 16, 2010 (received for review December 31, 2009)

In this paper, we present the direct observation of quantum sizeeffects (QSE) on the work function in ultrathin Pb films. By usingscanning tunneling microscopy and spectroscopy, we show thatthe very existence of quantum well states (QWS) in these ultrathinfilms profoundly affects the measured tunneling decay constant κ,resulting in a very rich phenomenon of “quantum oscillations” in κas a function of thickness, L, and bias voltage, Vs. More specifically,we find that the phase of the quantum oscillations in κ vs. Ldepends sensitively upon the bias voltage, which often resultsin a total phase reversal at different biases. On the other hand,at very low sample bias (jVsj < 0.03 V) the measurement of κ vs.L accurately reflects the quantum size effect on the work function.In particular, the minima in the quantum oscillations of κ vs. L occurat the locations where QWS cross the Fermi energy, thus directlyunraveling the QSE on the work function in ultrathin films, whichwas predicted more than three decades ago. This further clarifiesseveral contradictions regarding the relationship between theQWS locations and the work function.

STM ∣ quantum well states ∣ Pb

The work function, the minimum energy required to move anelectron from a solid into the vacuum, is the most fundamen-

tal material parameter in surface science. It plays a key role, forexample, in the photo-electric effect, one of the first phenomenathrough which quantum mechanics unveiled itself. The workfunction is the result of a complex interplay between quantummechanics and forces on the atomic scale. Recent studies suggestthe exciting possibility of controlling the work function throughquantum engineering of electronic structures at the nanoscale(1–5). Such efforts are, however, still in their infancy, and thereare many puzzles and contradictions in the observations so farthat hinder further progress (1–5). In this report, we show directevidences for quantum size effects on the work function in ultra-thin Pb films. We further establish the direct correlation of thisquantity with the behavior of quantum well states. With the abilityto control the growth of metallic thin films with atomic layerprecision, one can anticipate the possibility of tuning the workfunction of metallic thin films through the quantum size effects,thus influencing chemical processes on surfaces.

It should be noted that theoretical investigations of quantumsize effects (QSE) on the work function (Φ) were carried out sev-eral decades ago (6). By using a jellium model and continuouslyvarying the film thickness, Schulte showed that the QSE leads towork function oscillations as a function of layer thickness. More-over, it was found that theΦ oscillations are directly related to theenergy locations of the quantum well states (QWS). In particular,whenever a QWS channel crosses the Fermi level (EF), there is anegative cusp in Φ as a function of film thickness (L), whichresults from the sudden increase of the surface dipole due tothe additional electronic density outside the surface contributedby the new QWS (6). Subsequent theoretical investigations usingmore sophisticated methods, while allowing more realistic elec-tronic structure calculations, did not call into question this basictenet regarding the correlation of Φ and QWS (7, 8).

Experimental investigations of this effect remained dormantfor almost 30 years. The situation, however, changed dramaticallywith recent advancements in the quantum control of thin metalfilm growth (9–18), by which epitaxial metallic thin films can begrown on semiconductor substrates with atomic layer precision.One example is shown in Fig. 1A, where an epitaxial Pb film wasgrown on a Si(111) substrate. The scanning tunneling microscopy(STM) image reveals the smoothness and the thickness unifor-mity of the Pb film where the steps on the overlayer directly re-flect the substrate steps. This sample system, epitaxial Pb thin filmon Si(111), is the most widely studied system for the investigationof QSE, and many of its physical quantities have been shown toexhibit so-called “quantum oscillations” as a function of layerthickness (18–20). While such atomically smooth, uniform thick-ness metal films can be grown with near perfection, it is moreconvenient for direct investigation of QSE to use a differentgrowth process that leads to the formation of large 2D flat-topmesas on the stepped region. These mesas contain regions ofdifferent thicknesses in one mesa (15). An example is shownin Fig. 1B, spanning thicknesses ranging from 9 to 19 monolayers(ML). The lateral extent for each film thickness ranges from 50 to100 nm, about two orders of magnitude larger than the thickness.Moreover, the experimental measurements using STM are per-formed at ∼1 nm above the surface. One thus can safely assumethat the measurements performed in the local region can be usedto represent an extended film of the same thickness. On such flat-top mesas, the variation of work function as a function of thick-ness can be probed by measuring the tunneling decay constant (κ).

In an STM configuration the tunneling current between thetwo metallic electrodes (tip and sample) decays exponentiallywith respect to the tip-to-sample distance, z, and can be expressedas I ∝ e−2κz. From this general expression, the tunneling decayconstant κ can be defined operationally as κ ≡ −d ln I∕2dz whichis an experimentally measured quantity. One can further definean effective barrier height simply as Φeff ¼ ðℏ2κ2Þ∕2 m. Often,the tunnel junction is approximated with a one dimensionaltrapezoidal tunneling barrier (see Fig. 1C). Within this approxi-mation, if the density of states (DOS) is smoothly varying inenergy, then the effective barrier can be expressed as

Φeff ¼ ðΦs þ Φt − ejVsjÞ∕2 [1]

where Φs and Φt are the work functions of the sample and tiprespectively and jVsj is the bias applied across the tunnel junction.Shown in Fig. 1C is the configuration with a negative sample biasVs (relative to the tip electrode), where the tunneling currentwould be dominated by the tunneling channel between the high-est occupied states of the sample, namely EF;s, and the unoccu-

Author contributions: C.-K.S. designed research; J.K. and S.Q. performed research; J.K.,S.Q., W.Y., Q.N., M.-Y.C., and C.-K.S. analyzed data; and J.K., W.Y., Q.N., M.-Y.C., andC.-K.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed: E-mail: shih@physics.utexas.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.0915171107/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.0915171107 PNAS ∣ July 20, 2010 ∣ vol. 107 ∣ no. 29 ∣ 12761–12765

APP

LIED

PHYS

ICAL

SCIENCE

S

Dow

nloa

ded

by g

uest

on

Feb

ruar

y 28

, 202

1

pied states of the tip electrode at EF;t þ ejVsj (labeled by the lar-gest horizontal arrows). Note that the same expression (1) appliesfor the case with a positive sample bias, where the dominatingtunneling channel occurs between the highest occupied statesof the tip electrode, namely EF;t and the unoccupied states ofthe sample electrode at EF;s þ eVs. Thus, when the approxima-tion (1) is valid, the sample work function can be extracted ifthe tip work function is known. This is the basic principle behindwork function measurements using STM.

While this approximation ignores the effect of the imagepotential, which can lower the effective barrier (as shown bythe dashed curve in Fig. 1C), it is expected that the correctionshould be relatively minor. The method just described has beenroutinely applied ever since the invention of STM and has largelyyielded good information about the sample work function,although it was also recognized early that the effective barrierheight is often influenced by detailed band structures (21).Furthermore, if one focuses primarily on the work functiondifference on different surfaces, the correction is canceled.

Presumably, to unravel the QSE on the work function of ultra-thin films, one can simply measure the tunneling decay constant,κ, as a function of the layer thickness. Earlier works by Qi et al. (3)andMa et al. (4) used precisely this approach and claimed to haveobserved QSE on the work function of ultrathin films. However,as we show in this paper, the very existence of QWS in theseultrathin films has a profound effect on the measurements ofκ itself, resulting in a very rich phenomenon of quantum oscilla-tions in κ as a function of thickness, L, and also as a function ofsample bias voltage, Vs. The phenomenon is rooted in the sharpresonances of QWS in these thin films, which renders approxi-mation (1) inapplicable for most of the bias range except forthe limit where Vs approaches zero. As a result, κðVs;LÞ showsmarkedly different behaviors in three different regimes: (i) emptystates (Vs > 0.2 V), (ii) filled states (Vs < −0.2 V), and (iii) thevery small bias regime (jVsj < 0.03 V). We further show that

indeed in regime iii the measurements yield correct informationabout the work function, thus allowing us to unravel the QSE onthe work function of this prototypical materials system.

Results and Discussion(i) Empty-State (Vs > 0.2 V) Regime. The inapplicability of approxi-mation (1) in the presence of sharp QWS resonances can be un-derstood by looking at the schematic diagram shown in Fig. 2A.Considering first a positive sample bias that lines up the tip Fermilevel (EF;t;1) with the lowest unoccupied QWS (LUQWS) abovethe sample Fermi level (EF;s), the effective tunneling barrierheight is marked by a vertical arrow and labeled as Φeff;1. Whenthe bias is raised so that EF;t;2 is above the first QWS, fromapproximation (1), one would expect a lowering of the effectivebarrier height. However, the effective barrier height actually in-creases (as shown by the dashed vertical arrow, labeled Φeff;2),because the tunneling is still dominated by the LUQWS. Thistrend would continue until the tip Fermi level reaches the nextunoccupied QWS. The tunneling will then be dominated bythe second QWS, and one expects a sudden drop in effectivebarrier height. As a result, the decay constant (κ) should showoscillatory behavior as a function of bias, and the valleys in κoscillation should coincide with the QWS peaks. Shown inFig. 2 B and C are spectra of dI∕dV and κ vs. Vs respectivelyfor 9 and 10 ML films. Indeed, the oscillations in κ vs. Vs arecorrelated with the QWS positions with the valleys in κ vs. Vscoinciding with the peaks in dI∕dV (labeled by the arrows withdifferent colors for different thicknesses).

This correlation leads to very interesting behaviors of bias-de-pendent mapping of the decay constant κ. Shown in Fig. 2 E and Fare images of κ of an island, containing local regions with thick-nesses ranging from 9 to 19 MLs, at two different bias voltages,Vs ¼ þ1.2 V and þ0.7 V, respectively. One obvious feature inthe κ-image is the contrast between different thicknesses. Forexample, in Fig. 2E all odd layers have higher κ values thanthe neighboring even layers. On the other hand, in Fig. 2F, thecontrast is exactly reversed: All even layers have higher κ valuesthan the odd layers. The odd-even bilayer oscillations, often re-ferred to as quantum oscillations in the literature (10–17), are theconsequence of nearly half integer phase matching between theFermi wavelength and the interlayer spacing for Pb (111) ultra-thin films (7, 8). The contrast reversal in κ, on the other hand, is aconsequence of the onset of different QWS at the imaging bias:At Vs ¼ þ0.7 V, the bias is close to the peak positions of theQWS for the odd layers (thus depressions in κ for odd layers),while at Vs ¼ þ1.2 V it is close to the QWS peaks for even layers.Although this phenomenon itself is interesting, such a bias-dependent behavior of κ vs. L and its correlation with theQWS locations (see also the top panel of Fig. 4A) raise seriousquestions about using measurements of effective barrier heightsto extract the work function because the contrast in workfunctions at different thicknesses should be a fixed quantity. Thiseffect is explored further using theoretical simulations.

Consider that the sample DOS, DðEÞ, is largely reflected inthe dI∕dV spectra through the relation dI

dV s∝ DðeVsÞ

exp½−2zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mℏ2 ðΦs−eVsþΦt

2Þ

q� (see SI Text). This relationship allows

us to extract DðEÞ from the experimental dI∕dV spectra inFig. 2B. Using the extracted DðEÞ, the tunneling current versusa small change in barrier width z is calculated to obtain κ. Shownin Fig. 2D are theoretical simulation curves of κ vs.Vs for 9 and10 MLs, respectively. Note that all calculations are done for in-finite flat surfaces. The solid lines are the theoretical results witha fixed work function Φ≡ ΦsþΦt

2¼ 4 eV for both 9 and 10 MLs.

One can see that the oscillatory behavior of κ vs. Vs is wellreproduced and the phases of oscillations for 9 and 10 MLsare complementary, as we have observed, except at very low sam-ple bias. We notice that if the value of Φ is lowered by 0.08 eV for

Fig. 1. (A and B) STM topography images of a globally flat 2ML Pb film and aflat-top Pb mesa on Si(111), respectively. The images were taken with samplebiasVs ¼ 2 V,andtunnelingsetcurrent I0 ¼ 100 pA.(C)Anillustrationofaone-dimensional tunneling junction for the STM configuration. The effect of theimage potential is shown as a dashed curve between the sample and tip. Φs

and Φt are the work functions for the sample and tip, respectively. Φeff is theeffective tunneling barrier height with an applied bias Vs across the junction.

12762 ∣ www.pnas.org/cgi/doi/10.1073/pnas.0915171107 Kim et al.

Dow

nloa

ded

by g

uest

on

Feb

ruar

y 28

, 202

1

10 ML in the simulation, then the low bias behavior is also wellreproduced (dashed line), suggesting the existence of a true workfunction difference between 9 and 10 MLs. Nevertheless, such aninference is not a direct experimental measurement.

(ii) Filled-State Regime (Vs < −0.2 V). The behavior of κ vs. Vs and κvs. L in the filled states (i.e., negative sample biases), on the otherhand, is very different from the behavior in the empty states. Asillustrated by the schematic shown in Fig. 3A, at a negative sample

Fig. 2. (A) A schematic illustration of the STM tun-neling process when probing the empty states ofthe sample (i.e., when Vs is positive). The Fermi levelsof the sample and tip are labeled by EF;s and EF;t , re-spectively. EF;t;1 and EF;t;2 show the energy level of EF;t

at two different applied biases Vs;1 and Vs;2. The un-occupied QWS are represented by the curved lines inthe sample side. The vertical arrows represent theeffective barrier height, Φeff, at different bias condi-tions. (B) Differential conductance spectra (dI∕dV) for9 and 10 MLs. The peak positions in the dI∕dV spectracorrespond to the locations of QWS in the emptystates for each thickness. (C) Measured κ vs. Vs for9 and 10 MLs. All κ images used to extract κ vs. Vs

were taken under same tunneling set current I0 ¼100 pA without changing the tip or sample. (D) The-oretical simulation of the bias dependence of κ for 9and 10MLwith an assumption ofΦ≡

ΦsþΦt2 ¼ 4 eV for

the solid lines and 3.92 eV for the dashed line of10 ML. (E and F) κ images taken at Vs ¼ 1.2 V and0.7 V respectively for the same island shown in Fig. 1B.Odd numbers labeled on the images indicate thenumber of underlying ML. The apparent contrast ofeach layer clearly exhibits a bilayer oscillation beha-vior and the contrast of E is completely reversed in F.

Fig. 3. (A) A schematic illustration of the STM tunnelingprocess when probing the sample filled states. EF;s and EF;t

represent Fermi levels of the sample and tip, respectively.EF;t;1 and EF;t;2 show different energy levels of EF;t at twodifferent applied biases. (B) Differential conductance spec-tra (red) and κ (blue) for 9 ML. (C–E) Special mappings of κtaken at large negative biases: Vs ¼ −1.0 V, −1.5 V, and−2.0 V, respectively, for the same island shown in Fig. 1B.

Kim et al. PNAS ∣ July 20, 2010 ∣ vol. 107 ∣ no. 29 ∣ 12763

APP

LIED

PHYS

ICAL

SCIENCE

S

Dow

nloa

ded

by g

uest

on

Feb

ruar

y 28

, 202

1

bias, whenever the tip Fermi level is below the highest occupiedQWS (HOQWS), the tunneling process is dominated by thisHOQWS (labeled by the black arrow), regardless of the presenceof other occupied QWS. Thus, the effective barrier height will beprimarily determined by this tunneling channel. Indeed, this isobserved experimentally. Shown in Fig. 3B are dI∕dV and κvs. Vs for 9 ML film probed at negative sample biases. While thedI∕dV spectrum reveals the clear signature of QWS at −0.45 V(albeit with smaller contrast), the oscillation in κ vs. Vs is nowabsent. Instead, a monotonic decrease of κ is observed as thenegative sample bias varies from low to high. Nevertheless, abovethe location of the highest occupied QWS (HOQWS), one ob-serves a tapering of slope in the κ vs. Vs curve.

The thickness-dependence of the tunneling decay constant(κ vs. L) continues to exhibit quantum oscillations. This can beseen in the spatial imaging of the tunneling decay constant, κ,shown in Fig. 3 C–E. Most interestingly, here the contrast ofspatial imaging of κ remains identical, as long as Vs < −0.5 V(namely, the tip Fermi level is below the HOQWS). This can beseen in the middle panel of Fig. 4a, in contrast to empty-state κ vs.L behavior where the phase of the oscillation changes drasticallyat different biases (upper panel of Fig. 4A). In fact, for Vs <−0.5 V, κ vs. L is anticorrelated with the location of the HOQWS(red dots in Fig. 4B): That is to say, the closer the HOQWS is toEF;s, the lower the measured value of κ (with an exception at10 ML). This can be easily understood as we discussed above:At a negative sample bias, whenever EF;t is below the HOQWS,the tunneling process is dominated by this HOQWS, regardlessof the presence of other occupied QWS, thus determining theeffective barrier height measurements. Such an antiphase rela-tionship between κ and HOQWS for Pb (111) thin films indeedwas reported previously but was wrongly understood as direct evi-dence for the QSE on the work function (3, 4).

At the bias between 0 and −0.5 V, κ vs. Vs deviates from theabovebehavior.Whenthenegativesamplebias isdecreasedtowardzero, the κ vs.L oscillation continues to undergo a profound phasechange (lower panel of Fig. 4A), especially in the crossover region(shaded region in Fig. 4A). The question that arises is whetherany of these measurements tells us about the work function.

(iii) Very Low Bias Regime (−0.03 V < Vs < 0.03 V). Within the one-particle picture, the work function of the metal can be definedas the energy difference between the Fermi level and the vacuumlevel. This definition suggests that one should use the measure-ments performed at the bias voltage close to zero because onlythen are the states near EF being probed. Moreover, at such verylow bias, the results from filled-state and empty-state measure-ments should be consistent, and this is indeed what we observed(top panel of Fig. 4C). This result would suggest that the mea-surements of the effective barrier height at very low bias rangereflect the true QSE on the work function in these ultrathin films.

According to theoretical calculation using a jellium model (6)with thickness varied continuously, the Fermi level with respect tothe vacuum level varies as a function of film thickness. This is dueto the variation of the depth of the “effective potential,” which isin turn determined by the number of subbands occupied by thequantized electrons in the film. As the thickness of the film in-creases, new subbands get occupied one by one. Whenever anew subband becomes occupied at a particular thickness, the the-oretical work function, being a function of a continuous thicknessvariable, exhibits a cusp right at the thickness where a new sub-band crosses the Fermi level (6, 7). In reality, the film thickness isnot continuous but quantized, and the corresponding workfunction should take the discretized value. So far all theoreticalcalculations of the L-dependent work function of the Pb (111)surface are carried out only for free standing film and thus cannotbe compared directly with our measurements. However, the un-derlying principle that the QSE on work function is due to the

QWS crossing the Fermi level should remain valid. Here theQWS locations as a function of thickness are directly measured.By using these data points, we can interpolate the locations in L(as a continuous variable) where the QWS channels cross theFermi level (see the crossings of the black curves with the bluedashed horizontal line in Fig. 4B). By using these crossing points,the period and phase of the oscillating work function as a functionof continuous variable L are determined. The calculated contin-uous curve is shown in the lower panel of Fig. 4C, while bluesquare points on the calculated curve indicate discrete thicknessvalues. It is evident in Fig. 4C that the behavior of the measured κat low biases of �0.01 V indeed agrees with theoretical simula-tion based on the principle of Fermi level crossing of QWS chan-nels. 17 ML is expected to show the lowest work function value,because one of its QWS is located exactly at the point where thesubband crosses the Fermi energy level, as shown in Fig. 4B.

The consistent result of experimentally measured κ vs. L atVs → 0 with the expected work function oscillations predictedbased on the Fermi level crossing of QWS (Fig. 4C) further

Fig. 4. (A) Measured κ as a function of Pb thickness with various samplebiases for 9–23 ML. (B) The locations of QWS obtained from the peak posi-tions in dI∕dV spectra. The horizontal dashed line at 0 eV is a guided line forthe Fermi level. The red dots represent the highest occupied QWS for eachthickness and the black dashed curves indicate a series of energy subbands.(C) Comparison between measured κ at �0.01 V (upper) and the work func-tion simulation (lower). The thickness-dependence of the work function iswell reflected on measurements of κ vs. L at �0.01 V.

12764 ∣ www.pnas.org/cgi/doi/10.1073/pnas.0915171107 Kim et al.

Dow

nloa

ded

by g

uest

on

Feb

ruar

y 28

, 202

1

confirms that such measurements indeed truthfully reflect theQSE of work function in ultrathin films.

It should be emphasized that the underlying physics of thebilayer oscillations in κ at �0.01 V and κ at large negative or po-sitive biases are due to totally different origins. The oscillation ofκ vs. Lmeasured at a large negative bias and its antiphase relationto the HOQWS are direct consequences of how QWS influencethe tunneling process and do not involve work function oscilla-tion. Similarly the rich behavior of κ measured at a positive biasas a function of L and Vs is also a direct manifestation of QWS inthe tunneling process. Only when κ is probed at very low samplebias (jVsj < 0.03 V) does the result truthfully reflect quantumoscillations of work function vs. layer thickness.

MethodsThe experiments were conducted in a home-built low temperature STMsystem with in situ sample preparation chamber. Pb was deposited ontoatomically clean Si(111) 7 × 7 surfaces at room temperature with a flux

rate of 0.5 ML∕min by using a thermal evaporation technique. The roomtemperature growth of Pb leads to the formation of (111)-oriented 2Dflat-top mesas on the stepped region, which is convenient for STM investiga-tion because one mesa contains regions of different thicknesses. Moreover,the local thickness referenced to the wetting layer can be unambiguouslydetermined. The base pressure of the system was maintained at better than1.0 × 10−10 torr during the sample preparation. All STM measurements weredone at 78 K with liquid nitrogen cooling.

The derivative of the tunneling current with respect to the tip-to-sampledistance, dI∕dz, is acquired by using the lock-in amplifier with a z-modulationamplitudeof0.01nmandamodulationfrequencyof2kHz,which is faster thanthe feedback time constant. This results in a small error current superimposedon the feedback set current Io. The tunnelingdecay constant, κ ≡ −d ln I∕2dz, isdirectly related to the measured quantity dI∕dz as κ ¼ −ðdI∕dzÞ∕2Io.

ACKNOWLEDGMENTS. This work was supported by NSF Grant DMR-0906025,CMMI-0928664, Welch Foundation F-1672, and Texas Advanced ResearchProgram 003658-0037-2007. M.-Y.C. acknowledges support by DOE GrantDE-FG02-97ER45632.

1. Paggel JJ, et al. (2002) Atomic-layer-resolved quantum oscillations in the workfunction: Theory and experiment for Ag/Fe(100). Phys Rev B 66:233403.

2. Liu X, et al. (2008) Wavevector-dependent quantum-size effect in electron decaylength at Pb thin film surfaces. Appl Phys Lett 93:093105.

3. Qi Y, et al. (2007) Atomic-layer-resolved local work functions of Pb thin films and theirdependence on quantum well states. Appl Phys Lett 90:013109.

4. Ma XC, et al. (2007) Experimental observation of quantum oscillation of surfacechemical reactivities. Proc Natl Acad Sci USA 104:9204–9208.

5. Miller T, Chou MY, Chiang TC (2009) Phase relations associated with one-dimensionalshell effects in thin metal films. Phys Rev Lett 102:236803.

6. Schulte FK (1976) Theory of Thin Metal-Films—Electron-density, Potentials and WorkFunction. Surf Sci 55:427.

7. Wei CM, ChouMY (2002) Theory of quantum size effects in thin Pb(111) films. Phys RevB 66:233408.

8. Jia Y, Wu B, Weitering HH, Zhang ZY (2006) Quantum size effects in Pb films from firstprinciples: The role of the substrate. Phys Rev B 74:035433.

9. Smith AR, Chao KJ, Niu Q, Shih CK (1996) Formation of atomically flat silver films onGaAs with a “silver mean” quasi periodicity. Science 273:226–228.

10. Czoschke P, Hong HW, Basile L, Chiang TC (2004) Quantum beating patterns observedin the energetics of Pb film nanostructures. Phys Rev Lett 93:036103.

11. Luh DA, et al. (2001) Quantum electronic stability of atomically uniform films. Science292:1131–1133.

12. Su WB, et al. (2001) Correlation between quantized electronic states and oscillatorythickness relaxations of 2D Pb islands on Si(111)-(7 × 7) surfaces. Phys Rev Lett 86:5116.

13. Yeh V, et al. (2000) Role of the metal/semiconductor interface in quantum size effects:

Pb/Si(111). Phys Rev Lett 85:5158.14. Chang SH, et al. (2002) Electronic growth of Pb islands on Si(111) at low temperature.

Phys Rev B 66:245401.15. Eom D, Qin S, Chou MY, Shih CK (2006) Persistent superconductivity in ultrathin Pb

films: A scanning tunneling spectroscopy study. Phys Rev Lett 96:027005.

16. Hupalo M, et al. (2002) Atomic models, domain-wall arrangement, and electronicstructure of the dense Pb/Si(111)-root 3x root 3 phase. Phys Rev B 66:161410.

17. Zhang ZY, Niu Q, Shih CK (1998) “Electronic growth” of metallic overlayers on

semiconductor substrates. Phys Rev Lett 80:5381.18. Qin SY, Kim J, Niu Q, Shih CK (2009) Superconductivity at the two-dimensional limit.

Science 324:1314–1317.19. OzerMM, et al. (2007) Tuning the quantum stability and superconductivity of ultrathin

metal alloys. Science 316:1594–1597.

20. Ozer MM, Thompson JR, Weitering HH (2006) Hard superconductivity of a softmetal in the quantum regime. Nat Phys 2:173–176.

21. Stroscio JA, Feenstra RM, Fein AP (1986) Electronic-structure of the Si(111)2X1 surfaceby scanning-tunneling microscopy. Phys Rev Lett 57:2579.

Kim et al. PNAS ∣ July 20, 2010 ∣ vol. 107 ∣ no. 29 ∣ 12765

APP

LIED

PHYS

ICAL

SCIENCE

S

Dow

nloa

ded

by g

uest

on

Feb

ruar

y 28

, 202

1