On NaCl(100)/Cu(111) an interface state band is observed that descends from
the surface-state band of the clean copper surface. This band exhibits a
Moire-pattern-induced one-dimensional band gap, which is accompanied by strong
standing-wave patterns, as revealed in low-temperature scanning tunneling
microscopy images. At NaCl island step edges, one can directly see the
refraction of these standing waves, which obey Snell's refraction law.Comment: 4 pages, 4 figure

F. Forstmann

Gerhard Meyer

J. Kliewer

Jascha Repp

Karl-Heinz Rieder

L. Esaki

N. W. Ahscroft

English

arXiv

Crossref

Snell’s Law for Surface Electrons: Refraction of an Electron Gas Imaged in Real Space

Repp, Jascha

Meyer, Gerhard

Rieder, Karl-Heinz

University of Regensburg Publication Server

P H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3Snell’s Law for Surface Electrons: Refraction of an Electron Gas Imaged in Real SpaceJascha Repp,1,2 Gerhard Meyer,1,2 and Karl-Heinz Rieder21IBM Research, Zurich Research Laboratory, CH-8803 Ru¨schlikon, Switzerland2Institut fu¨r Experimentalphysik, Freie Universita¨t Berlin, Arnimallee 14, D-14195 Berlin, Germany(Received 15 August 2003; published 22 January 2004)036803-1On NaCl100=Cu111 an interface state band is observed that descends from the surface-state bandof the clean copper surface. This band exhibits a Moire´-pattern-induced one-dimensional band gap,which is accompanied by strong standing-wave patterns, as revealed in low-temperature scanningtunneling microscopy images. At NaCl island step edges, one can directly see the refraction of thesestanding waves, which obey Snell’s refraction law.DOI: 10.1103/PhysRevLett.92.036803 PACS numbers: 73.20.At, 68.37.Ef, 73.61.Ngtion of surface electronic waves at island step edges bymeans of LT-STM. Moreover, we show that natural Moire´FIG. 1 (color online). STM image of NaCl on Cu(111). Thenumbers indicate the thicknesses of the NaCl islands in layers.Ever since the mapping of standing waves of a two-dimensional electron gas (2DEG) on the close-packedCu(111) surface by means of low-temperature scanningtunneling microscopy (LT-STM) [1,2], the 2DEG turnedout to be an ideal playground for a variety of LT-STMexperiments [3–12]. In the STM images the surface elec-trons reveal their wavelike behavior, often discussed inanalogy to light. This analogy has been addressed di-rectly in experiments such as the confinement of a2DEG between two parallel step edges, considered as acounterpart of the optical Fabry-Perot resonator [3], thestriking experiment of an electronic Mach-Zehnder in-terferometer [4], and the spectacular quantum mirageexperiment [5]. However, despite this strong analogy,the counterpart of optical refraction has not yet beenobserved for surface electrons.Such an experiment has to comply with several re-quirements. First of all, two different media are neces-sary. For surface electronic states, two regions havingdifferent dispersions may act as the media and be realizedby a partial coverage of the surface with an adlayer thatmodifies the dispersion, as was observed for various thinmetal films on Cu(111) [6–9] as well as for Xe=Cu111[10]. Second—in terms of optics—the interface betweenthe two media has to be transparent. It was reported thatstep edges of metals exhibit a transmission close to zerofor surface electrons, i.e., that the wave patterns on thetwo sides of a step edge are not related to each other [3].However, in the case of an insulator adsorbed on a metalsurface, this may be different: An insulator does notcontribute to the electronic states, therefore the electronsare still confined to the surface of the metal underneath,and thus may better overlap with the electrons of the cleansurface. Third, in addition to being inherently transpar-ent, the interface between the two media also has to bevery smooth. This means that a perfect step edge isneeded to observe refraction of surface electrons.As NaCl=Cu111 meets the above requirements, weare able to report here the first observation of the refrac-0031-9007=04=92(3)=036803(4)$22.50 patterns, inherent in incommensurate growth, generate aband gap within a two-dimensional (2D) interface stateband. Thereby the textbook band-structure model ofnearly free electrons (NFE) is exploited and visualizedin real space.Our experiments were performed with a LT-STM [13]operated at 9 K. The sample was cleaned by severalsputtering and annealing cycles. NaCl was evaporatedthermally, while the sample temperature was kept at320 K. Bias voltages refer to the sample voltage withrespect to the tip, and we used electrochemically etchedtungsten wire as STM tips.NaCl forms (100)-terminated islands that are up toseveral microns in diameter [14] (see Fig. 1). The NaClislands start with a double-layer thickness and perfectnonpolar step edges, in which the anions and the cationsalternate. On top of the initial double layer, islands ofadditional layers are formed. Substrate defect steps aresmoothly overgrown in the so-called carpetlike growth[15]. Because of the different symmetries of NaCl(100)and Cu(111), the growth is incommensurate. In the case ofonly a few layers of NaCl, the nonvanishing tail of chargedensity from the metal that extends through the insulatoris sufficient to take STM images without crashing the tip.In atomically resolved STM images, only the Cl ions areimaged as protrusions [16–18].2004 The American Physical Society 036803-1P H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3Upon adsorption of NaCl, the Cu(111) surface-stateband survives and forms an interface state band that isconfined to the insulator/metal interface and has a largerFermi wavelength of F  3:8 nm (clean Cu: F 3:0 nm). To measure the dispersion of this modified 2Dinterface state band, differential conductance (dI=dV)images were taken at various bias voltages, as shown inFigs. 2(a)–2(d). By extracting the wavelength of the wavepatterns seen in these images [19], the dispersion relationcan be established [see Fig. 2(e)], as was done for theclean Cu surface [1,2,20]. The band minimum E0 wasobtained by recording the dI=dV signal at a fixed positionas a function of energy [20,21] [crosses in Fig. 2(e)] and isshifted upward in energy by 230 30 meV in the caseof the adsorption of an insulating NaCl overlayer. Theresulting parabolic dispersion E  E0   hk2=2m isslightly wider; i.e., the effective mass m has increasedfrom 0:40 0:02me to 0:46 0:04me, where me de-notes the free electron mass.The energy shift can be qualitatively understoodwithin the one-dimensional (1D) phase-accumulationmodel [22,23]. In this analysis, the wave-function  ofsurface electrons is found by matching the wave-functionphase of the analytical solution inside the copper crystalFIG. 2 (color online). (a)–(d) dI=dV images of the same re-gion near a NaCl step edge, taken at various voltages. In allcases the wavelength of the NaCl=Cu111 interface state islarger than that of the Cu(111) surface state. (e) Dispersion ofthe 2DEG for the clean Cu(111) surface and the NaCl100=Cu111 interface state band (square data points and fit curves).(f) Using the phase-accumulation model, the probability dis-tributions jj2 perpendicular to the surface are determined,yielding only small changes upon the adsorption of NaCl. Theorigin (z  0) was chosen to be the Cu(111) substrate surface.Circles denote the position of Cu atoms.036803-2[24] to the one of the numerical solution for the outsideregion, obtained by integrating the Schro¨dinger equation[22]. For the latter an electrostatic potential had to beassumed: For the clean Cu surface, this is given by theimage potential [25]. For NaCl=Cu111 we consideredthe modified image potential [23,25] as well as the low-ering of the work function upon NaCl adsorption [14].This model yields an upward shift of the dispersionof E ’ 300 meV, in qualitative agreement with ourexperiment. Even more relevant, it reveals that the wavefunctions of the surface electrons are barely modifiedupon adsorption of NaCl [see Fig. 2(f)] and thus rendera high transmission through NaCl step edges promising.Moreover, it justifies an even simpler understanding ofthe upward shift in the dispersion within first-orderperturbation theory, E  hjVji. Owing to theexponentially decaying wave function  outside the Cusubstrate, the perturbation potential V, i.e., the poten-tial difference with and without NaCl, is relevant for theenergy shift E only in immediate proximity of thecopper surface. The upward shift thus results fromthe positive perturbation potential at the interface givenby the weaker image potential [25] due to the polarizationof the adsorbed dielectric. The work-function change [14]as well as the image potential at the insulator-to-vacuuminterface are negligible for E. Consequently, the dis-persion for three layers of NaCl does not differ from thatfor two layers, which was verified in the experiment.The above results show that for interface states of aninsulator/metal system, as described here, the main con-tribution of the electronic wave functions resides withinthe substrate [see Fig. 2(f)]. Therefore, the effective massof the shifted interface state will mainly be determinedby the metal. For copper an upward shift of the surfacestate is expected to be correlated with a slight increase ofthe effective mass as discussed in Ref. [22], in goodagreement with the present experiment. The energeticshift of the band for other insulator/metal systems willdepend critically on the structure of the interface.However, the adsorption of an insulator that does notstrongly affect the metal surface but weakens its imagepotential will, in general, result in an upward shift of thesurface-state band. These findings explain that in a pre-vious experiment on Xe=Cu111 [10] a similar upwardshift and effective mass increase were observed.Another condition for the observation of the electronrefraction at island edges is the presence of standingwaves. Continuous waves lead to a uniform probabilitydistribution and thus prevent the observation of refractionin the STM images, even if the refraction took place. Inthe case of NaCl=Cu111, strong electronic plane wavesare already inherent in the system: The incommensurategrowth gives rise to various Moire´ patterns [26], whichcan be seen in atomically resolved STM images[Fig. 3(a)]. The Moire´ patterns are associated with dis-placements of the charged ions and are thus accompaniedby a modulation of the electrostatic potential seen by the036803-2P H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3surface electrons. One general conclusion of the NFEmodel [27] is that any periodic potential modulationwill give rise to a band gap. Just below and just abovethe band gap, the electronic wave functions are expectedto be standing waves, and the probability distributionsjj2 have a phase shift of  with respect to each other, asillustrated in Fig. 3(c). For the case discussed here, thisphysics textbook statement can be imaged in real space.As a first indication, STM images such as that in Fig. 3(b)show strong standing plane waves that correspond to thisparticular Moire´ pattern in terms of wavelength anddirection [dotted lines in Figs. 3(a) and 3(b)].For a more precise analysis, we recorded a dI=dVimage in constant-height mode, but whereas we scannedwithin one scan line perpendicular to the plane waves [asindicated by the solid line in Fig. 3(b)], we did not movethe tip in the other direction. Instead, we changed the biasvoltage after each scan line. The resulting STM image inFig. 3(d) thus reflects a map of the probability distributionFIG. 3 (color online). (a) An atomically resolved STM imageshows one of the various possible Moire´ patterns that resultfrom incommensurate growth [26]. (b) This pattern gives riseto a strong plane-wave pattern. (c) Illustration of a band gapin the NFE model. Just below and just above the band gap(marked by circles), the electronic wave functions are expectedto be standing waves, and the probability distributions jj2will have a phase shift of  with respect to each other. (d) ThisdI=dV image (acquired in constant-height mode) maps theprobability distribution jj2 in real space and energy. It showsthe behavior expected from the NFE model.036803-3jj2 in space and energy. As expected, the image shows aregion of reduced intensity, which represents the band-gap region. Just above and just below this region, there arestrong standing-wave patterns shifted by  to each other.The standing-wave pattern caused by a periodic scat-tering potential contains only the wavelength compo-nents of this potential modulation. Thus in Fig. 3(d), thestanding-wave pattern is given by the Moire´ pattern’speriodicity and does not change with energy. The positionin energy of the band gap can vary between 20 and400 meV above the Fermi level from one rotational do-main to the next. In all cases, this energy position can belinked to the Moire´ patterns in such a way that the wavevector at the band gap corresponds to half of the wavevector of the potential modulation, in agreement with theNFE model. In the particular case shown in Fig. 3, theMoire´ pattern has a periodicity of ’ 1:62 nm correspond-ing to k  2=16:2 A1, kgap  0:194 A1, and thusEgap ’ 81 meV. As this condition is met only in onedirection, we have a 1D band gap in a 2D electron gas,i.e., a pseudogap. For some rotational domains, the con-dition was not met in the experimentally accessible en-ergy range, and no band gap was observed, as in the caseof the dispersion shown in Fig. 2(e).The standing-wave pattern persists far above the bandgap because of the one dimensionality of the band gap.This is also one of the reasons for the remaining lowintensity within the band gap. In addition, the finite life-time of the interface states [11] gives rise to an energeticbroadening of the states, which further ‘‘smears out’’ theband gap and may also lead to interference effects.To observe the refraction, we took advantage of thestrong standing plane waves just below the band gap byapplying a corresponding bias voltage of V  55 meV.Figure 4 shows the refraction of the surface electronsin a differential conductance (dI=dV) image of an islandedge of NaCl=Cu111. The region on the NaCl island(to the left) shows a strong standing plane-wave patternthat is due to the formation of a band gap. Reflectioninduces an additional modulation of the pattern parallelto the step edge. More importantly, the image clearlyreveals standing plane waves on the clean copper surface(to the right). These patterns at the island step edge obeySnell’s refraction law, sin= sin  1=2. Note that and 2 are given by the geometry of this particularMoire´ pattern, 1 is the electron wavelength on the cleanCu surface corresponding to E  EF  55 meV, and  isgiven by Snell’s law. This law follows directly from theconservation of the component of the momentum (andwavelength) parallel to the step. In other words, thepatterns on both sides have a fixed phase correlation atthe step edge, which is directly evident in the images andfollows from the postulation of continuous wave functionsin quantum mechanics.In everyday life, optical refraction manifests itself inthe bending of light rays, and we cannot observe the wavefronts directly. In Fig. 4 we do not observe the bending of036803-3FIG. 4 (color online). (a) STM dI=dV image (constant-current mode) of a NaCl island step edge reveals strongstanding-wave patterns, not only on the NaCl island but alsoon the clean Cu(111) surface (V  55 mV). (b) Adding somelines as guides to the eye shows that the standing-wave patternobeys Snell’s refraction law. Note that the contrast for thearea of clean Cu(111) is increased, compared with the rest ofthe image.P H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3rays, but directly image the wave fronts, visualizing therefraction model. Another difference to optics is evident:In optics the refraction is usually determined by one termonly, namely, the refractive index of the medium. In thecase of electron refraction, the band minimum as well asthe effective mass of the electrons may change and thuscontribute to the refraction in different ways.The observed band-gap formation due to Moire´ pat-terns is similar to the superlattice concept [28,29] basedon the idea of creating artificial, tunable 1D band gaps bygrowing a vertical lattice of alternating semiconductors.Whereas in semiconductor superlattices the band gaps canbe tuned by the layer thicknesses during growth, Moire´-pattern-induced band gaps can be tuned by the choice oflattice constant mismatch. This phenomenon shows a newway to tailor the properties of a 2DEG for future appli-cations. These possibilities can be further extended bycontrolled sequential growth of different dielectric mate-rials. In contrast to surface states, interface states areinherently protected by the dielectric adlayer and caneven be studied under ambient conditions.In summary, we observed the formation of a 1D bandgap in the 2D electron gas of the NaCl=Cu111 inter-face state band. The band gap is because of the Moire´patterns of the incommensurate growth and was observedin an STM image displaying the electron-probabilitydistribution in space and energy. At NaCl island edges,036803-4the refraction of standing waves could be observed, obey-ing Snell’s refraction law.We thank Frank Forstmann and Rolf Allenspachfor fruitful discussions, and acknowledge partial fund-ing by the EU-TMR projects ‘‘AMMIST’’ and ‘‘NANO-SPECTRA’’ and the Deutsche ForschungsgemeinschaftProject No. RI 472/3-2.[1] M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature(London) 363, 524 (1993).[2] Y. Hasegawa and Ph. Avouris, Phys. Rev. Lett. 71, 1071(1993).[3] L. Bu¨rgi et al., Phys. Rev. Lett. 81, 5370 (1998).[4] Y. Ji et al., Nature (London) 422, 415 (2003).[5] H. C. Manoharan, C. P. Lutz, and D. M. Eigler, Nature(London) 403, 512 (2001).[6] J. Kliewer and R. Berndt, Appl. Phys. A 72, S155 (2001).[7] T. Suzuki et al., Phys. Rev. B 64, 081403(R) (2001).[8] A. Bendounan et al., Phys. Rev. B 67, 165412 (2003).[9] L. Diekho¨ner et al., Phys. Rev. Lett. 90, 236801 (2003).[10] J.-Y. Park et al., Phys. Rev. B 62, R16 341 (2000).[11] J. Kliewer et al., Science 288, 1399 (2000).[12] J. Repp et al., Phys. Rev. Lett. 85, 2981 (2000).[13] G. Meyer, Rev. Sci. Instrum. 67, 2960 (1996).[14] R. Bennewitz et al., Surf. Sci. 438, 289 (1999).[15] C. Schwennicke, J. Schimmelpfennig, and H. Pfnu¨r, Surf.Sci. 293, 57 (1993).[16] W. Hebenstreit et al., Surf. Sci. 424, L321 (1999).[17] F. Olsson, M. Persson, J. Repp, and G. Meyer(unpublished).[18] J. Repp, S. Fo¨lsch, G. Meyer, and K.-H. Rieder, Phys. Rev.Lett. 86, 252 (2001).[19] The dI=dV images presented in Figs. 2 and 4 are re-corded in constant-current mode to enable simultaneousimaging on both sides of the step edge. To determine thedispersion, constant-height images were used as well.[20] G. Ho¨rmandinger, Phys. Rev. B 49, 13897 (1994).[21] J. Li et al., Phys. Rev. Lett. 81, 4464 (1998).[22] N.V. Smith, Phys. Rev. B 32, 3549 (1985).[23] A. Hotzel et al., Appl. Phys. B 68, 615 (1999).[24] F. Forstmann, Z. Phys. 235, 69 (1970).[25] As in Ref. [23] we treated the image potential classicallyby applying the static dielectric constant of NaCl. Thenonphysical divergence of the image potential was sup-pressed. The average barrier potential within the NaCllayer is given by the upper edge of the band gap.[26] The pattern shown in Fig. 3(a) corresponds to a NaCllattice constant of a  5:52 A and a rotational domainin which the polar NaCl direction is almost parallel(  0:2) to the close-packed direction of the coppersubstrate. This alignment of the two directions seemsfavorable, although other rotational domains also exist.[27] N.W. Ahscroft and N. D. Mermin, in Solid State Physics,edited by D. G. Crane (Holt, Rinehart, and Winston,Orlando, 1976), Chap. 9.[28] L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61 (1970).[29] R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev.Lett. 34, 1327 (1975).036803-4

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Snell's law for surface electrons: Refraction of an electron gas imaged
  in real space

Snell's law for surface electrons: Refraction of an electron gas imaged in real space

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