PHYSICAL REVIEW B 15 NOVEMBER 2000-IVOLUME 62, NUMBER 19 Periodicity and thickness effects in the cross section of quantum well states A. Mugarza,1 J. E. Ortega,1 A. Mascaraque,2 E. G. Michel,2 K. N. Altmann,3 and F. J. Himpsel3 1Donostia International Physics Center and Centro Mixto de Materiales CSIC-UPV, Departamento de Fı´sica Aplicada I, Universidad del Paı́s Vasco, Plaza de On˜ate 2, 20018-San Sebastia´n, Spain 2Instituto ‘‘Nicolás Cabrera’’, Departamento de Fı´sica de la Materia Condensada, C-III Universidad Auto´noma de Madrid, Cantoblanco, E-28049 Madrid, Spain 3Physics Department, University of Madison Wisconsin, 1150 University Avenue, Madison, Wisconsin 53706-1390 ~Received 31 March 2000! The photoemission cross section of quantum well states in very thin Cu films has been analyzed as a function of photon energy within a wide energy range. We show that this cross section is periodical ink space, peaking around vertical transitions of the respective Cu bulk band. The cross section peak width is analyzed in terms of final- and initial-state wave vector broadening. The latter can only be detected at low kinetic energies due to reduction of final state broadening. The initial statek' broadening increases when the Cu film gets thinner, as simply expected from the uncertainty principle. ec u o n th s iti io u ia h te n en ho - n is in u ak at s l- th th t s of al ro o ere o- p po- nd - d ery s. d 2 a 6 - o- re- d ith of till ls. to he gy nal y- the W dif- eed the In a thin metal film deposited on a solid substrate el trons are confined in the perpendicular direction by the s face and interface potentials, leading to the formation quantum well~QW! states. These are known to be respo sible of intriguing phenomena in layered systems, like oscillatory magnetic coupling of two magnetic layers acro a nonmagnetic spacer1 or the giant magnetoresistance.2 Their existence has been frequently probed in noble and trans metals by means of angle-resolved photoemission3 ~ARPES! where they display strong energy-dependent cross sect In general the QW cross section is expected to be maxim near vertical transitions of the bulk crystal of the mater comprising the thin film. This reflects the conservation of t perpendicular wave vector in transitions from thin film sta to final states in the continuum.4 Such behavior has bee qualitatively observed in Cu/Co~100! ~Ref. 5! and Ag/ Cu~111! ~Ref. 6!. Away from this vertical transition region one can also obtain periodic modulations of the QW int sity. They have been attributed to discretization of the p toemission final-state band,6 but also to surface-interface co herent photoemission effects.7,8 Furthermore, it has bee claimed that at low energies and very thin films such em sion dominates over the regular QW photoemission from side the thin film.8 Here we show that this is not true for C films on Co~100!, since the cross section displays clear pe around vertical transitions to the lowest three bulk final-st bands. The cross-section peak width is analyzed in term perpendicular wave vector broadening (Dk') for both final and initial ~i.e., QW! states. The results show that the fina state Dk' dominates at hn;80 eV, whereas athn ;14 eV initial-state effects are necessary to explain width of the transition peak. The thickness dependence of initial stateDk' and its magnitude appears to be related confinement within the QW via the uncertainty principle. High-resolution photoemission experiments were done the Synchrotron Radiation Center in Stoughton, Wiscon (hn,16 eV) and at the VUV photoemission beam line the synchrotron radiation laboratory Elettra at Trieste, It (hn.40 eV). In both cases the polarization of the synch tron light was set top-like in order to enhance sensitivity t PRB 620163-1829/2000/62~19!/12672~4!/$15.00 - r- f - e s on ns. m l e s - - - - s e of e e o at in y - D1-symmetry initial states. The photoemission spectra w normalized to the photon flux. The Cu~100! surface was electrochemically polished prior to thein situ sputter- annealing cycles.1 After substrate preparation, a 10 mon layer~ML ! thick Co film was grown, and immediately on to of this film Cu was deposited. Both Co and Cu were eva rated from electron-beam-heated sources onto the Cu~100! substrate held at 300 K, with a deposition rate of;1 Å/min determined with a quartz microbalance. Epitaxial growth a film quality was controlled by low-energy electron diffrac tion ~LEED!, which always displayed very low backgroun and sharp spots. The characteristic QW features for ev integral layer allowed further calibration of the Cu thicknes The valence band photoemission spectra in Figs. 1 an show the QW-state valence band spectra for a 3 ML and ML Cu film, respectively, at different photon energies~the numbering follows Ref. 3!. In contrast to other QW reso nances nearEF ,9 both n52 for 3 ML ~at E2EF5 20.9 eV) andn53 for 6 ML ~at E2EF521.2 eV) states display no appreciable dispersion. They are truly tw dimensional spin-down states, totally confined by Bragg flection at the Co minority gap.3 The peaks appear broadene due to emission from smaller patches of the surface w 61 ML thickness. This is more evident in the left panel Fig. 2, i.e. at lower energies. Although the main feature s corresponds to then52 QW state of 3 ML, the high-energy resolution allows to distinguish from up to 4 different leve The lack of lateral uniformity in the growing film appears be characteristic of the room-temperature~RT! evaporated films, since it persists for nominally complete layers. T QW intensity varies strongly in the wide photon ener range studied, i.e., the range of vertical transitions to fi states in the second, third, and fourth branches of theD1 bulk band. At the lower energy transition the strong energ dependent cross section washes away QW features in range of just a few eV. This effect, and the presence of Q peaks from patches with different thicknesses, makes it ficult to provide an accurate coverage determination. Ind the relative peak intensities are dramatically affected by 12 672 ©2000 The American Physical Society a g L e ce ho rg io re fte in m th te d ne 1 d n ng ses nto o- ng, ate. nly r of Cu of - erly ith ith ns of . It nsity e er an k- r a 3 e n- k. PRB 62 12 673BRIEF REPORTS cross section change within 1 eV photon energy. In this c we separate the specific layer contribution by performin Gaussian fitting. The resulting peak for 3.0 ML and 6.0 M is marked in Figs. 1 and 2 with a thick line. On the oth hand, this sharpk' resonance in the cross section enhan the QW peak intensity. This fact, together with a better p ton and electron energy resolution, makes the low-ene regime the optimum for QW state studies. In Fig. 3 we have plotted the peak intensity as a funct of the photon energy. The intensity is defined as the a under the QW peak in the spectra of Figs. 1 and 2, a subtraction of a smooth background. As an example, we clude in Figs. 1 and 2 the background lines used for so spectra. For the low-energy range, the area is that of Gaussian fit to the 3.0 ML and the 6.0 ML peaks indica with thicker lines. The error in the data is plotted jointly an is mostly determined by the election of the background li The cross section maxima appear at 14.6 eV, 81 eV, and eV for 6 ML, whereas for 3 ML intensity maxima are foun at 13.8 eV and 82 eV. The higher-energy transition has been studied in the latter case. Away from the energy ra FIG. 1. Photoemission spectra as a function of the photon ergy for a 6 ML Cu film showing then53 minority spin QW state. A smaller 5.0 ML QW peak is better distinguished in the low energy spectra. In that case the thick line represents a Gaussi to the n53 peak for 6.0 ML. The dotted lines are typical bac grounds used to determine the area under the peak~see Fig. 3!. se a r s - y n a r - e e d . 19 ot e shown in Fig. 3, the intensity of the QW peak decrea strongly. The photon energy scale is converted on top of Fig. 3 i a final-state~reduced! wave vector scale assuming phot emission final-state bands as in bulk Cu. Strictly speaki the final state should be defined as a mixed Co/Cu st However, due to the finite escape depth this fact is o important for thin films and low energies~around 14 eV!, where the electron inelastic mean free path is of the orde 8 atomic layers. Even in this case, since both Co and share a very similar band structure for energies aboveEF 110 eV,9 the deviation from the the electronic structure bulk Cu is expected to be small.10 Thus we can assume Cu like final-state bands. At higher energies these are prop represented by a free-electron-like parabola calculated w an inner potentialV02EF527.2 eV.11 The lower-energy band is taken from the experimental data of Ref. 12. W respect tok' , the cross section is periodic, i.e., all transitio occur at the same valuek';0.75(2p/a) for 3.0 ML and k';0.74(2p/a) for 6.0 ML.13 Such a periodic variation in the bulk Brillouin zone is directly related to the properties the QW-state wave function in the perpendicular direction is analogous to the case of surface states, where the inte n- fit FIG. 2. Photoemission spectra at several photon energies fo ML Cu film. Although the 3 ML QW state peak is dominant, w detect again emission from locally different thickness at low e ergy. The thick lines represent Gaussian fits to the 3.0 ML pea ru e p st e s le a he g n . er h s- io i m um re- can idth ML in er e at L h te are e t te nal- in in the pth, tic d- ibu- if- w ulk 12 674 PRB 62BRIEF REPORTS peaks at the fundamental frequency of the Fourier spect in the vertical direction, i.e.,kedge in the bulk Brillouin zone.14 In QW statesk' can be deduced from the envelop function model.15 In this model the wave function is made u with a Bloch-like rapid oscillation derived from the close bulk band edge (kedge52p/a), modulated by an envelop function (kenv52p/d) that allows the boundary condition to be met at both the surface and the interface. If we neg changes in the bonding near the interfaces, the total w vectork'(E)5kedge6kenv follows the dispersion of the bulk (s,p) band. This allows us to locate the QW state within t bulk Brillouin zone in the perpendicular direction. Assumin the lower initial-state (s,p) band given in Ref. 12, we obtai k';0.77(2p/a) for 3.0 ML and k';0.75(2p/a) for 6.0 ML, in fair agreement with the results obtained in Fig. 316 Note that the envelope function model accounts for the p odic variation of the potential within the film via the Bloc oscillation kedge. Assuming only surface and interface di continuities in the potential to calculate the photoemiss matrix element one cannot explain the periodicity shown FIG. 3. Photoemission intensity variation of the 6 ML~thin line, dots! and the 3 ML~thick line, circles! QW states of Cu~100! films. The lines are Lorentzian fits to the data points. At low energy include the Lorentzian fit to a constant-initial-state curve for b Cu ~dotted line!. m ct ve i- n n Fig. 3.8 Thus in our case the inner corrugation of the fil cannot be neglected, even for the thinnest one. Optical transitions can be described in the moment space using complex perpendicular wave vectorsk'5k' R 1 ik' I for the initial state~i! and the final state (f ).17 In this way, the cross section can be thought of as thek' convolu- tion of two Lorentzians, where the imaginary part ofk' is the Lorentzian broadening. The curves in Fig. 3 can be garded as intensity scans in the constant-initial-state~CIS! photoemission mode. Since final states are extended we define the group velocity\v f5]E/]k' and then write17 Gm \ 1 v f 5Dk',i1Dk', f , ~1! whereGm stands for the measured cross section peak w and Dk'52k' I . In Table I the experimental widths@full width at half maximum~FWHM!# Gm obtained for the dif- ferent peaks are summarized. We include data for 4.0 and 5.0 ML determined from the spectra on the left panels Figs. 1 and 2, as well as a fit obtained for bulk Cu at low energy~dotted line in Fig. 3!. We note that peak widths ar very similar for 3 ML and 6 ML in the second resonance ;80 eV, butGm decreases as we go from 3.0 ML to 6.0 M and to bulk Cu at the;14 eV resonance. Based in Eq.~1! we can interpret the experimental widt of the cross section curve in terms of initial- and final-sta wave vector broadening. To this end in Table I we comp Gm /v f with Dk', f . At high energiesv f can be assumed to b constant in the energy rangeGm defined by the peak width. I is readily obtained from the free-electron-like final-sta band mentioned before. For the lower resonance, the fi state band is close to theX1 point, wherev f changes much faster as a function of the energy. In order to fit the data this case,v f(E) is obtained from the parabolic band given Ref. 12 and it is already introduced as a correction to experimental Lorentzian fit in Fig. 3.Dk', f in Table I is defined as the inverse of the photoelectron escape de which in turn is estimated from the experimental inelas mean free path~IMFP! for bulk Cu.12 At the higher-energy transition around;80 eV Dk', f is similar to Gm /(\v f). Thus in this case the final-state broa ening appears to shade the initial state broadening contr tion. This is also supported by the fact that there is no d ference inGm /(\v f) from 3.0 ML to 6.0 ML. In contrast, e a and TABLE I. Experimental cross section linewidths and the corresponding initial- and final-stateDk' for the 6 ML and the 3 ML QW states at two different energy ranges~see the text!. For the low-energy regime dat for 4 ML (n52 state! and 5 ML (n53 state! have been included. IMFP data for final-state broadening v f values have been taken from Ref. 12. Thickness~ML ! Gm (ev) Gm /\v f (Å 21) Dkf (Å 21) Dki (Å 21) hn;82 eV 3 8.760.9 0.2360.05 0.250 6 961 0.2460.03 0.250 hn;14 eV 3 2.760.2 0.1560.01 0.07 0.0860.01 4 3.160.6 0.1860.03 0.06 0.1260.03 5 2.260.1 0.1260.02 0.09 0.0460.02 6 2.160.3 0.1160.02 0.07 0.0360.02 Bulk Cu 1.760.5 0.0960.03 0.07 0.0260.03 e u i t e s e he er en - nt a th t c in du n uch oss rve ith n ate ter- ble for ial- , e- er C o. V .M. PRB 62 12 675BRIEF REPORTS Gm /(\v f) increases by 35% from 6 ML to 3 ML at th lower-energy resonance. At around 14 eV the IMFP goes to 15 Å in Cu,12 i.e., Dkf is reduced to 0.07 Å21, about half of the value ofGm /(\v f). Therefore Dki must be brought in at low energy. The resulting values are listed Table I. The error bars are estimated from the experimen well as from the fitting procedure. In the thinnest layersDki is far from the hole lifetime contribution, which should b lower than 0.02 Å21.18 On the other hand, the thicknes dependence suggests the influence of confinement insid quantum well.19 This leads to wave vector broadening via t uncertainty principle. In our case 1/d ranges from 0.185 Å21 to 0.093 Å21 for 3.0 to 6.0 ML. The experi- mentalDki values listed in Table I lie within the same ord of magnitude and roughly reproduce the decreasing tr from 3 ML to 6 ML. However they clearly fall short com pared with 1/d. A possible explanation could be a significa penetration of the QW wave function inside the Co band g such that the effective QW thickness is actually larger.20 In- deed our QW states lie very close to the edge of minority-spin band gap~nominally3 at 20.7 eV belowEF); thus a large penetration is expected. At the same time, strong wave function tailing inside Co can result in an effe tive smoothening of the interface step barrier. The smooth of the interface and the surface potential barriers, either to electronic effects or to roughness, can explain the abse R ns ur s lm ns p n as the d p, e he - g e ce of large final-state surface-interface interference effects, s as those observed in other thin films.7,8 In summary, we have studied the photoemission cr section in thin Cu films in a wide energy range. We obse a periodic behavior in the reduced wave vector scale, w peaks at the QW statek' . The peaks in the cross sectio curve have been analyzed in terms of initial- and final-st wave vector broadening. Initial-state broadening is de mined only at the lowest-energy range, due to the noticea reduction in final-state broadening. The values obtained Dki as well as its thickness dependence indicate that init state broadening is linked to confinement within the film~via the uncertainty principle! rather than to photohole lifetime which should be the dominating effect in thicker films. J.E.O. and A. Mu. are supported by the Max-Planck R search Award and the Universidad del Paı´s Vasco~Projects UPV057.240-EA197/97 and 026/98!. E.G.M. and A. Ma. are funded by the Spanish DGES~PB-970031! and the Comu- nidad de Madrid~07N/0031/1998!. Experiments at ELET- TRA have been performed within the TMR program und Contract No. ERBFMGECT950022. Experiments at SR were supported by NSF under Grant No. DMR-9704196, N DMR-9815416, and No. DMR-9531009~Synchrotron Radia- tion Center, University of Wisconsin-Madison!. We ac- knowledge the technical support from the staff of the VU beam line at Elettra as well as helpful discussions with P Echenique. d the ly nd ro- 8 n tt. ent - n- , as the d- er 1J.E. Ortega and F.J. Himpsel, Phys. Rev. Lett.69, 844 ~1992!. 2S.S.P. Parkin, N. More, and K.P. Roche, Phys. Rev. Lett.64, 2304 ~1990!. 3For a review see F.J. Himpsel, J.E. Ortega, G.J. Mankey, and Willis, Adv. Phys.47, 511 ~1998!. 4P.D. Loly and J.B. Pendry, J. Phys. C16, 423 ~1983!. 5E.D. Hansen, T. Miller, and T.-C. Chiang, J. Phys.: Conde Matter.9, L435 ~1997!. 6T. Miller, A. Samsavar, and T.-C. Chiang, Phys. Rev. B50, 17 686 ~1994!. 7A. Carlsson, D. Claesson, S.A. Lindgren, and L. Wallden, S Sci. 352, 656 ~1996!. 8M. Milun, P. Pervan, B. Gumhalter, and D.P. Woodruff, Phy Rev. B59, 5170~1999!. 9P. Segovia, A. Mascaraque, E.G. Michel, A. Na¨rmann, and J.E. Ortega, Surf. Sci.433-435, 425 ~1999!. 10Such similarity is not found in other systems, like Ag/V~100!. In this case chosing a correct final-state band for very thin fi could be more conflicting, since Ag~100! and V~100! have dif- ferent crystal lattices and bulk band topologies@see for instance D.P. Woodruff, M. Milun, and P. Pervan, J. Phys.: Conde Matter 11, L105 ~1999!#. 11V.N. Strocov, H.I. Starnberg, and P.O. Nilsson, Phys. Rev. B56, 1717 ~1997!. 12J.A. Knapp, F.J. Himpsel, and D.E. Eastman, Phys. Rev. B19, 4952 ~1979!. .F. . f. . s . 13There is a slight dispersion ofk' from the first to the second an the third maxima in Fig. 3 for the same thickness. However, values for 3.0 ML lie always above 6.0 ML, and this is on consistent with the vertical transition scheme. 14S.G. Louie, P. Thiry, R. Pinchaux, Y. Ptroff, D. Chandesris, a J. Lecante, Phys. Rev. Lett.44, 549 ~1980!. 15G. BastardWave Mechanics Applied to Semiconductor Het structures~Les Editions de Physique, Les Ulis, France, 198!, Chap. III. 16The small differences ink' obtained from the peaks in Fig. 3 ca be explained by the uncertainty in the final-state band~Ref. 10!. 17N.V. Smith, P. Thiry, and Y. Petroff, Phys. Rev. B47, 15 476 ~1993!. 18J.J. Paggel, T. Miller, and T.-C. Chiang, Science283, 1709 ~1999!; J.J. Paggel, T. Miller, and T.-C. Chiang, Phys. Rev. Le 81, 5632~1998!. 19For a two-dimensional system one has to consider two differ contributions toDk' in the initial state, i.e., the photohole life time and electron confinement within the QW. For sharp co finement the latter can be the dominant broadening effect shown in surface states~Ref. 14!. 20A more accurate uncertainty relation can be obtained from analytical expression of the quantum well wave function, inclu ing the part that tails inside the Co crystal. To this aim, furth theoretical work is currently being undertaken.