MEASUREMENT OF ELECTRON WAVE FUNCTIONS AND CONFINING POTENTIALS VIA PHOTOEMISSION.pdf

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Measurement of electron wave functions and confining potentials via photoemission

A. Mugarza,1,2 J. E. Ortega,1,2 F. J. Himpsel,3 and F. J. Garca de Abajo1,4,*1Donostia International Physics Center (DIPC), Aptdo. 1072, 20080 San Sebastian, Spain

2Departamento de Fsica Aplicada I, Universidad del Pas Vasco, Plaza Onati 2, 20018 San Sebastian, Spain3Department of Physics, University of Wisconsin, Madison, Wisconsin 53706

4Centro Mixto de Materiales CSIC/UPV, Paseo Manuel Lardizabal 3, E-20018 San Sebastian, Spain

Received 12 December 2002; published 28 February 2003

Wave functions and electron potentials of laterally confined surface states are determined experimentally bymeans of photoemission from stepped Au111 surfaces. The confined nature of the wave function makes itpossible to use an iterative formalism borrowed from x-ray diffraction where real-space wave functions areretrieved from the absolute values of their Fourier transforms. The latter are measured by angle-resolvedphotoemission. The effective confining potential is then obtained by introducing the wave functions intoSchrodingers equation.

DOI: 10.1103/PhysRevB.67.081404 PACS numbers: 79.60.Bm, 68.35.Bs, 73.21.b

The electron wave functions and potentials in low-dimensional systems are of primary importance for tailoringelectronic properties of nanostructures. The electron energy

levels and the probability density are the physical observ-ables. These can be obtained in real space from local con-ductance maps via scanning tunneling microscopy/spectroscopySTM/STS,1,2 or in reciprocal space by angle-resolved photoemission.3,4 The latter has been used to studythin films3 and two-dimensional2Darrays of nanoobjects,4

whereas STM/STS resolves individual nanostructures on asurface.

The question arises whether it is possible to directly de-rive wave functions and/or electron potentials from experi-mental data. The standard procedure consists of assuming amodel potential, whose parameters are obtained by fitting theexperiment. For instance, in core-level photoelectron diffrac-

tion, muffin-tin potentials are assumed for retrieving atomicpositions that fit the experiment.5,6 The electron density ofsurface states confined between two steps has been measuredby STMRefs. 1 and 2and has been modeled by semitrans-parent mirrors located at the step edges.2 It has been arguedthat the position of the mirror has to be moved slightly awayfrom the steps to account for electron spillover.7

We introduce a direct method to determine the wave func-tion and the effective electron potential from angle-resolvedphotoemission data, without any a priori knowledge abouttheir nature. All the necessary information is contained in themomentum distribution of the photoemission intensity, aslong as the wave function is confined to finite areaa terrace

in our case. This allows us to unambiguously determine thepotential and wave functions of one-dimensional quantum-

well states on stepped Au111. The real-space wave func-tions are derived from the Fourier transform of theirmomentum-space representations, the square of which isproportional to the photoelectron intensity under the condi-tions discussed below. However, the phase in momentumspace is not measured and this leads to the well-known phaseproblem in optics, shared with many other techniques, suchas x-ray and electron diffraction.8,9 Various iteration methodshave been devised to retrieve real-space objects from themodulus of their momentum-space representation, in particu-

lar, the oversampling method.10 These methods are valid fora confined wave function because the phase is obtained byrepeatedly diminishing the amplitude of the wave function

outside the confinement region. The one-electron potentialcan be obtained after dividing the Schrodinger equation bythe wave function.

As a test system we use vicinal noble-metal surfaces char-

FIG. 1. Vicinal Au111surfaces with quantum-well states con-fined by step edges. The energy levels are directly measured byangle-resolved photoemission. The confining potential V and thewave functions of statesN1,2,3 are obtained from the momentumdistribution of the photoemission intensity.

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acterized by equally spaced, linear step arrays and one-dimensional surface states confined within individual ter-races of nominal size L. This is the case of Au111vicinal

surfaces with relatively wide terraces, such as Au887 (L38 ) and Au23,23,21(L56 ) shown in Fig. 1Refs.4 and 11. Au887is characterized by two flat quantum lev-els (N12) below the Fermi energy, whereas the widerterraces in Au23,23,21allow for three occupied quantum-well levels (N13). Data points in Figs. 2a and 2bshow the angular photoemission intensity scan for eachquantum level as obtained from the areas of the peaks inindividual photoemission spectra.4,11 Data for Au887weretaken at the SU8 beam line at the synchrotron LURE in ParisFrance using a VSW electron analyzer, whereasAu23,23,21 data were taken at the PGM beam line in theSynchrotron Radiation Center SRC of the UW-Madison,

using a hemispherical Scienta SES200 spectrometer. The ex-periment details have been published elsewhere.4,11 The in-tensity is properly normalized to account for the photon fluxand the transmission function in the Scienta analyzer. Thelines represent a smooth splinefit to data points that go tozero, away from the displayed wave-vector region in Fig. 2,where the photoemission spectra are completely featureless.

The photoemission intensity in reciprocal space qcan beunderstood in the framework of the standard photoemissiontheory, assuming a one-electron description. We start withthe photoemission intensity, which is proportional to the ma-trix element

IN qqeikrA N

2. 1

Nis the initial surface-state wave function, q is the finalelectron state of momentumq(q,qz), qis the momentumcomponent parallel to the surface, A is the light polarizationvector, k is the momentum of the photon, and Nrefers to thequantum number of the wave function. To proceed further, itwill be assumed that the surface-state wave functions can be

factorized into components that are either parallel or perpen-dicular to the surface, i.e.,12

N rN Rz ,

where R denotes the coordinates along the surface. If, inaddition to the surface-state confinement in the perpendiculardirectionz, the electron is also bound to a 2D region of thesurface, the parallel component of the wave function N(R)must adopt a form that depends on the detailed shape of theconfining region and on the boundary conditions at its bor-der. Thus, INcan be written as

IN q CqN2,

where

C qze ikzzAz z

2

depends very weakly onq, so that it can be absorbed into anormalization factor that we set to 1. Therefore, there is adirect relation between photoemission intensity IN(q) andthe Fourier transform of the wave function N(q):

N qqN dReiqRN R. 2

Thereby, we neglect multiple scattering in the final state,which at the atomic level is partly due to the lateral atomiccorrugation, small compared to that in the bulk. Also, theprobability of the surface-state electron to propagate elasti-cally towards the bulk region and back to the vacuum issmall due to inelastic attenuation. Multiple scattering involv-ing different terraces can also play a role resulting in Fouriercomponents qg with a reciprocal terrace-lattice vectorg . However, these affect the intensity distribution betweendifferent Brillouin zones but not the momentum distributionwithin one Brillouin zone compare this to the spot profilesin low-energy electron diffraction, which are not affected byelastic scattering.13 This allows one to obtain the wave func-tion in real space N(R) from the intensity in reciprocalspace as

N R dq22

e iqRIN qeiN(q). 3

However, this equation still contains an unknown phaseN(q). We have explored two methods to obtain this phase:1an iterative procedure using oversampling810 and2anexpansion of the wave function into a Fourier series com-bined with a least-squares fit. In both cases, the strategyminimizes outside the confinement region.

FIG. 2. Angular distribution of the photoemission intensity fromone-dimensional quantum-well states at vicinal aAu887and bAu23,23,21surfaces. The lines are smooth splinefits to the datapoints and are used as input intensity functions in Eq. 3 see thetext.

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For the iterative method, we begin with a constant phaseN(q)0 and the smooth fitting curves of Fig. 2 as inputdata to start the iteration with Eq.3. The resulting N(R) iscorrected outside the confining region, and transformed backintoqspace using Eq. 2. The phase ofN(q) is extractedand inserted again into Eq. 3to start a new iteration. Theconfinement length is obtained directly from the photoemis-sion data see below, and the noted correction consists ofsubtracting from the newly calculated real-space wave func-tion the wave function at the previous step multiplied by afactor 0.1 outside the confinement region. This ensures con-vergence to a wave function that vanishes outside the con-finement region. As a strong test of convergence of this it-erative technique, we have used the phase of the N2 stateto start the iteration of the N1 state. The first iteration stepleads to a wave function that resembles that of the N2state, a clear indication of the importance of the phase,8 andconvergence to the true N1 state is achieved after severalhundred iterations.

This iterative method leads to results that are in agree-ment with those derived from an expansion of the wave

function into sine functions that vanish at the edges of theconfinement region, and therefore, we only present the re-sults derived from the iterative method, which can be easilygeneralized to more complicated systems. In either case, thelength of the region where the wave function takes non-negligible values is not sensitive to the input value for thelength of the confinement region.

The size of the confining region is obtained from the Fou-rier transform of the experimental intensity distribution,which can be written as the self-convolution of the real-spacewave function,

dq

22e iqRI

N q

dR

N R

N

* RR. 4

This convolution takes nonzero values in an area twice thesize of the confining region. For the data of Fig. 2, in par-ticular, one obtains a function which is basically zero outsidea region of 80 and 120 in diameter not shown, inagreement with terrace-width confinement in Au887 andAu23,23,21, respectively.

The natural normalization for both the two-dimensionalwave functionNand the measured intensity is provided byEq. 4if one sets R0:

dq

22IN qNN1.

Applying this procedure to the angular scans of Figs. 2aand 2b, we have obtained the surface-state wave functionsfor quantum-well levels that are shown in Fig. 1. All wavefunctions are confined to a region whose width matches theterrace width L. Thus, the wave functions exhibit a clearterrace confinement that has not been assumed by our recon-struction procedure, but rather it has emerged from the infor-mation contained in the photoemission data.

The actual effective potential of the terrace can be re-trieved from the Schrodinger equation as

V REN2

2m*N R2N R, 5

wherem *0.26mis the effective mass of the electron in theinitial state of energy EN, taken from the dispersion of thesurface state on flat Au111. The potential becomes indepen-dent of the quantum number N if many-electron effects areabsent, as we assumed initially. The effective potential canalso change due to a different wave-function penetration in-side the bulk, although this effect is expected to be negligiblesince quantum levels are separated by energies much smaller

than the vertical confining barrier crystal gap.For Au23,23,21, the experimental wave functions ofFig. 1 have been introduced into Eq. 5 and the resultingelectron potential has been represented in Fig. 3. In order tocompare the potential derived from the quantum states withvariousN, we add on the left-hand side of Eq. 5the respec-tive experimental values EN

exp shown in Fig. 1 horizontalbars. An excellent mutual agreement is obtained for theshape of the potential derived independently from each of thethree wave functions N13, proving the validity of ourmethod. The potential exhibits a smooth central region andsharp boundaries that force electron confinement. It must bestressed that the validity of Eq. 5 is limited to regionswhere the wave function is not too small, and therefore, theasymptotic limit of the step barrier potential cannot be deter-mined.

The thick lines in Fig. 1 represent the average potentialobtained from Eq. 5 for both Au23,23,21 and Au887using the wave functions of the different quantum levels ineach case, shown in Fig. 1. The width of the potential well inAu887 is clearly narrower than the terrace size, as alsoobserved in thin Pb films.7 Near the step, the potential ex-hibits a steeper increase in Au887 than in Au23,23,21,which could be explained by a sharper terrace-width distri-bution in the former case.14

FIG. 3. Confining potentials Vobtained from the experimentalphotoemission intensities of Fig. 2 for the N1,2,3 states inAu23,23,21using the oversampling method.

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In summary, we have introduced a simple procedure fordirectly obtaining wave functions and effective potentials ofconfined electronic states from the momentum distribution ofthe photoemission intensity. This procedure, which relies onphotoemission theory within the one-electron approach, hasbeen successfully tested in the case of laterally confined sur-face states on Au111 vicinal surfaces. The method is ofgeneral applicability for discrete states because they are spa-

tially confined and allow an iterative determination of thephase by forcing 0 outside the confinement region. It canbe applied to any type of surface nanostructure confined in atleast one dimension, such as arrays of quantum wires andquantum dots. While STM provides direct images of the lowspatial frequencies, photoemission determines the high spa-tial frequencies. It would be interesting to pursue the reverse

transformation from the real space in STM to in qspace using the technique described here. A connection be-tween and the E(q) band dispersion has been madealready.2,15

A.M. and J.E.O. are supported by the Universidad delPas Vasco Grant No. 9/UPV 00057.240-13668/2001 andthe Max Planck Research Institute. J.G.A. gratefully ac-

knowledges support by the Basque Departamento de Educa-cion, Universidades e Investigacion, the University of theBasque Country UPV/EHUContract No. 00206.215-13639/2001, and the Spanish Ministerio de Ciencia y TecnologaContract No. MAT2001-0946. F.J.H. acknowledges supportby the NSF under Grants Nos. DMR-9815416 and DMR-0084402 SRC.

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12 The approximation of a factorized wave function is reasonable forstates that are confined to a single terrace island. It breaks downfor a superlattice with strong overlap between adjacent terraceswhere is delocalized and discrete states become bands.

13 M. Henzler, Appl. Phys. 9, 111976.14 The barrier appears slightly sharper than the terrace width distri-

bution in both surfaces. This could be due to the backgroundremoval in line fitting to individual photoemission spectra,which tends to filter out contributions coming from the smallestand the largest terraces at higher and lower binding energies,

respectively Ref. 11.15 J.I. Pascual, Z. Song, J.J. Jackiw, K. Horn, and H.-P. Rust, Phys.Rev. B 63, 241103 2001.

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