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Scattering of the S1 -surface state electron by an isolated adatom at Si(1 1 1)√3×√3 -Ag surface
ARTICLE IN PRESS
Physica E 40 (2007) 324–327 www.elsevier.com/locate/physe
Scattering of the S 1-surfacepstate p electron by an isolated adatom at Si(1 1 1) 3 3-Ag surface Satoshi Minamoto, Yurika Ogawa, Yoshinori Sano, Hiroyuki Hirayama Department of Materials Science and Engineering, Tokyo Institute of Technology, J1-3, 4259 Nagatsuda, Midori-ku, Yokohama 226-8502, Japan Available online 16 June 2007
Abstract We studied the scattering of the S 1 surface state electrons of two-dimensional electron gas (2DEG) by an isolated adatom at the p p Si(1 1 1) 3 3-Ag surface. The interference between the incident and scattered electrons caused a standing wave pattern around the adatom. It was successfully observed in scanning tunneling microscope (STM) based dI=dV images at room temperature. In the analysis of the standing wave pattern, we numerically confirmed that the S-wave approximation and usage of the asymptotic form of the Bessel function are valid. This enabled us to deduce an analytical equation to fit the experimentally observed standing wave pattern to estimate the scattering potential. The fitting was satisfactory and gave a scattering potential and a surface electron coherent length of 6:5 eV and 15 nm at room temperature. r 2007 Elsevier B.V. All rights reserved. PACS: 73.20.r; 73.20.Ak Keywords: Surface state; Electron standing wave; STM; Adatom; Scattering potential
1. Introduction p p The Si(1 1 1) 3 3-Ag surface is characteristic in its free-electron like surface state (S1 state) [1]. This surface state electron has been reported to be scattered by step p edges and 3-Ag out-of-phase domain boundaries (OPBs) to produce electron standing wave patterns at low temperatures [2,3]. Recently, pwe also drew a line of nanoscale width, where the 3-Ag reconstruction was destroyed locally by a scanning tunneling microscope (STM) tip, and found that the S1 surface state electrons were scattered by the lithographed line. Since local defects like adatoms, steps and lithographed lines behave as potential barriers, the S 1 surface state electron is confined and produces electron standing wave patterns in nano-structures surrounded by these local defects [2–4]. This provides a new way to artificially design of the surface 2D quantum structures by placing these local defects at the place of our demands [4]. However, it is crucial to know the scattering potential of these local Corresponding author. Tel.: +81 45 924 5637; fax: +81 45 924 5685.
E-mail address: [email protected] (H. Hirayama). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.06.019
defects for the surface electrons for engineering the quantized electronic states. Several analyses have reported on the scattering of the surface state electron by an adatom at metal surfaces [5,6]. In these previous studies, the scattering was treated in the framework of partial wave theory, and S-wave approximation was adapted a priori without examining its quantitative validity. Although the phase shift d of the scattered S-wave was deduced by fitting the standing wave pattern with the asymptotic form of the Bessel function, the d was not related to the scattering potential height. In this study, we quantitatively examined the scale for which the S-wave approximation and the usage of the Bessel function’s asymptotic form are valid. Confirming their validity for the scattering from an isolated adatom, we then deduced an analytical equation to fit the standing wave pattern, and estimated the scattering potential.
2. Experiments Experiments were carried out in an ultra-high vacuum (UHV) apparatus equipped with an Ag Knudsen cell and
ARTICLE IN PRESS S. Minamoto et al. / Physica E 40 (2007) 324–327
an STM unit [7]. The base pressure of the apparatus was below 1 108 Pa. In the UHV apparatus, Si samples were degassed overnight at 600 C, flashed at 1200 C for 10 s. and then slowly cooled to room temperature to p expose p the clean 7 7 reconstructed surface. The Si(1 1 1) 3 3-Ag reconstructed surface was prepared by depositing one monolayer (ML) Ag atoms on the Si(1 1 1)7 7 surface at 600 C. After the deposition, the surface was p cooled to room temperature. The surface showed the 3-Ag reconstruction, but local defects of isolated adatoms, p vacancies, and OPBs eventually appeared on the 3-Ag reconstruction. The surface structure and electron standing wave pattern around the local defects were observed by STM and STM-based dI=dV images. The dI=dV image (i.e. conductance map) was taken simultaneously with STM using a lock-in technique by applying a small sinusoidal modulation (100 meV pp , 1.7 kHz) to the sample bias voltage. The tunneling current was set at 0.2–0.3 nA. The scale in the STM image was calibrated p p by referring to the lattice distance of the Si(1 1 1) 3 3-Ag reconstruction. Both the STM and dI=dV images were taken at room temperature. 3. Results Fig. 1(a) shows a typical STM image of the p p Si(1 1 1) 3 3-Ag reconstructed surface with local defects. In the figure, an isolated adatom exists on the p 3-Ag surface at the upper right. A vacancy complex and a vacancy complex with an adatom at its center are observed at the upper left and bottom in the figure. The horizontal line at the center of the image was not the defect but was due to dragging of the STM tip during the scan. Fig. 1(b) shows the dI=dV image taken simultaneously with the STM image. In the dI=dV image, electron standing wave patterns with concentric fringes were observed around the local defects. In particular, we paid attention to the standing wave pattern around the isolated adatom in this study for the reason described in the next section. Ag adatoms are highly p mobile and do not rest at any sites on the 3-Ag surface at
325
Fig. 2. A cross-sectional line profile of the quantum interference pattern from the isolated adatom (the upper right one in Fig. 1) at the p p Si(1 1 1) 3 3-Ag surface. The sample bias voltage was þ1:0 V. The tunneling current was 0.3 nA.
room temperature [8]. Thus we think that the isolated adatom was probably Si. Fig. 2 shows an example of the cross-sectional line profile of the standing wave pattern around the isolated adatom. An oscillatory pattern was observed in the figure (the dashed line). Peaks in this figure were separated by 2 nm in any line profiles taken at the bias voltage of 1.0 eV. Although the oscillation decayed with distance from the adatom, two or three peaks were observed in cross-sectional line profiles taken at room temperature. Surface electrons lose coherency rapidly at room temperature, so it was very difficult to acquire the standing wave pattern at room temperature. Thus, we did not succeed to take a perfect series of dI=dV images with sequentially changed bias voltages at the same place. However, we observed similar damping oscillations in which the fringe pattern tended to become smaller with increasing the sample bias voltage, as expected from the free-electron like dispersion for the S1 surface state electrons. This demonstrated that the concentric fringe pattern was due to the quantum interference of the incoming and scattered surface electron waves around the defects. 4. Discussion The quantum interference around the isolated adatom is described by 2D Schro¨dinger equation of cylindrical symmetry. In polar coordinate r ¼ ðr; yÞ, the radial part of the wave function cðr; yÞ ¼ uðrÞ expði‘yÞ should satisfy the following equation [9]: 2 2 2 _2 d 1d _ ‘ þ þ V ðrÞ uðrÞ ¼ EuðrÞ, uðrÞ þ 2me dr2 r dr 2me r2 (1)
Fig. 1. Simultaneously taken STM (a) and dI=dV image (b) around p p isolated defects at the Si(1 1 1) 3 3-Ag surface. The sample bias voltage was þ1:0 V. The tunneling current was 0.3 nA.
where _ is the reduced Planck constant, me the electron effective mass, ‘ the angular momentum quantum number ð‘ ¼ 0; 1; 2; . . .Þ, V ðrÞ the scattering potential of the isolated adatom, and E is the energy.
ARTICLE IN PRESS 326
S. Minamoto et al. / Physica E 40 (2007) 324–327
The centrifugal potential V c ðrÞ ¼ _2 ‘2 =2me r2 pushes the electron with non-zero quantum number ‘ away from the origin. We plotted the V c for me ¼ 0:13 m0 as a function of the distance from the origin (i.e. the center of the adatom) in Fig. 3. (In the present system, me was estimated to be 0:13m0 as described below.) For the electron energy of 1.0 eV, the state of ‘X1 cannot enter the region rp0:54 nm, while the size of the adatom was estimated to be R ¼ 0:40 nm in the STM image. Thus, only the S-wave state (i.e. ‘ ¼ 0) is subjected to the scattering potential of the isolated adatom V ðrÞ. Thus, the S-wave approximation is valid in scattering by an isolated adatom. The S-wave function contains the zeroth Bessel function J 0 ðzÞ. Here, z ¼ k== r, and k== is the surface parallel wave vector of the electron in the S 1 surface state. For quantitative analysis of the scattering potential, it is crucial to use the asymptotic form of the Bessel function rffiffiffiffiffi 2 p J 0 ðzÞ cos z . (2) pz 4 We plotted the r-dependence of the strict and asymptotic forms of the Bessel function J 0 ðzÞ in Fig. 4. As shown in the figure, the asymptotic form is useful for zX0:7. Since k== ¼ 1:81 109 m1 for the S 1 surface state electron of 1.0 eV, the asymptotic form of the Bessel function is valid only for rX0:39 nm in our case. This means the adatom sizeð¼ 0:40 nmÞ is at the very limit of what is permissible to apply the asymptotic form of the Bessel function.
Fig. 3. Centrifugal potential for ‘ ¼ 1 and 2 as a function of the distance from the origin r. The electron mass was taken to be 0:13 m0 . The horizontal line indicates the electron energy of 1.0 eV.
Since both the S-wave approximation and the use of the asymptotic form of the Bessel function are valid, the radial wave function uðrÞ of the electron outside the adatom is safely described as sffiffiffiffiffiffiffiffiffiffiffi 2 p (3) cos k== r þ d . uðrÞA pk== r 4 Here, d is the scattering induced phase shift. In this case, the standing wave pattern around the isolated adatom is given [6,10] by 2A 2 cos k== r p þ d cos2 k== r p . DDOS / pk== r 4 4 (4) The equation indicates that the standing wave decays as 1=r. However, it decayed more rapidly due to the short coherent length ‘c in the experiment. Thus, we added a term expðr=‘c Þ to the above equation, and fitted the standing wave pattern in Fig. 2. The fitting was satisfactory, and we obtained free parameters d, k== and ‘c as 62 , 1:81 109 m1 , and 15 nm for the S 1 surface state electron of E ¼ 1:0 eV at room temperature. The S1 surface band has its bottom very close to the Fermi level [4]. Thus, we regarded the band dispersion as Eðk== Þ ¼ _2 k2== =2me in the fitting. Substituting k== ¼ 1:81 109 m1 for E ¼ 1:0 eV, we obtained the surface electron mass me of 0:13m0 . This me is consistent with the me of 0:13m0 reported in recent ultraviolet photoelectron spectroscopic (UPS) studies [11] and is reasonable. At room temperature, the surface electron life time is dominated by electron–phonon scattering. A recent UPS study reported the electron–phonon coupling constant p p l of 0.4 for the S 1 state electron at the Si(1 1 1) 3 3-Ag surface [12]. Since the electron life time t ¼ _=2plkB T and ‘c ¼ t v where v is on the order of the Fermi velocity 106 m=s, ‘c is estimated to be 10 nm. In this respect, the ‘c of 15 nm in the fitting is reasonable. Finally, we related the d to the scattering potential of the isolated adatom. Here, we assumed the isolated adatom as a cylindrical scatterer of a potential height of V and radius of R ¼ 0:40 nm. Inside the adatom region ðrpRÞ, the radial wave function is described aspuðrÞaI 0 ðkrÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi where a the normalized factor, k ¼ ð1=_Þ 2me ðV EÞ, and I 0 the zeroth modified Bessel function. For numerical analysis, we fitted I 0 ðkr) by a third polynomial equation. We obtained an analytical equation relating d to V by connecting smoothly this wave function to that of Eq. (4) for rXR at r ¼ R. For d ¼ 62 , R ¼ 0:40 nm, E ¼ 1:0 eV and me ¼ 0:13m0 , the equation gave us a scattering potential V of 6.5 eV. 5. Summary
Fig. 4. Zeroth Bessel function J 0 ð0Þ and its asymptotic form.
We observed concentric electron standing wave patterns p p around isolated defects on the Si(1 1 1) 3 3-Ag surface at room temperature. We especially paid attention to the
ARTICLE IN PRESS S. Minamoto et al. / Physica E 40 (2007) 324–327
scattering around the isolated adatom, and verified numerically that the S-wave approximation and the usage of the asymptotic form of the Bessel function are valid for describing the radial wave function for rXR. Then, we deduced a correct analytical form of DDOS and fitted the experimentally observed standing wave pattern by it to obtain d ¼ 62 , k== ¼ 1:81 109 m1 , and ‘c ¼ 15 nm. By smoothly connecting uðrÞ for rXR to that for rpR, we obtained the scattering potential of the isolated adatom of 6:5 eV. References [1] H. Aizawa, M. Tsukada, N. Sato, S. Hasegawa, Surf. Sci. 429 (1999) L509.
327
[2] M. Ono, Y. Nishigata, T. NIshio, T. Eguchi, Y. Hasegawa, Phys. Rev. Lett. 96 (2006) 016801. [3] N. Sato, S. Takeda, T. Nagao, S. Hasegawa, Phys. Rev. B 59 (1999) 2035. [4] S. Minamoto, T. Ishiduka, H. Hirayama, Phys. Rev. B., submitted for publication. [5] D. Eigler, E.K. Schweizer, Nature 363 (1993) 524. [6] F. Crommie, C. Lutz, D. Eigler, Science 262 (1993) 218. [7] M. Watai, H. Hirayama, Phys. Rev. B 72 (2005) 085435. [8] S. Hasegawa, J. Condens. Matter 12 (2000). [9] J. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press, Cambridge, 1998. [10] J. Ziman, Theory of Solids, Cambridge University Press, Cambridge, 1972. [11] T. Hirahara, I. Matsuda, M. Ueno, S. Hasegawa, Surf. Sci. 563 (2004) 191. [12] C. Liu, I. Matsuda, S. Hasegawa, private communication.
Physica E 40 (2007) 324–327 www.elsevier.com/locate/physe
Scattering of the S 1-surfacepstate p electron by an isolated adatom at Si(1 1 1) 3 3-Ag surface Satoshi Minamoto, Yurika Ogawa, Yoshinori Sano, Hiroyuki Hirayama Department of Materials Science and Engineering, Tokyo Institute of Technology, J1-3, 4259 Nagatsuda, Midori-ku, Yokohama 226-8502, Japan Available online 16 June 2007
Abstract We studied the scattering of the S 1 surface state electrons of two-dimensional electron gas (2DEG) by an isolated adatom at the p p Si(1 1 1) 3 3-Ag surface. The interference between the incident and scattered electrons caused a standing wave pattern around the adatom. It was successfully observed in scanning tunneling microscope (STM) based dI=dV images at room temperature. In the analysis of the standing wave pattern, we numerically confirmed that the S-wave approximation and usage of the asymptotic form of the Bessel function are valid. This enabled us to deduce an analytical equation to fit the experimentally observed standing wave pattern to estimate the scattering potential. The fitting was satisfactory and gave a scattering potential and a surface electron coherent length of 6:5 eV and 15 nm at room temperature. r 2007 Elsevier B.V. All rights reserved. PACS: 73.20.r; 73.20.Ak Keywords: Surface state; Electron standing wave; STM; Adatom; Scattering potential
1. Introduction p p The Si(1 1 1) 3 3-Ag surface is characteristic in its free-electron like surface state (S1 state) [1]. This surface state electron has been reported to be scattered by step p edges and 3-Ag out-of-phase domain boundaries (OPBs) to produce electron standing wave patterns at low temperatures [2,3]. Recently, pwe also drew a line of nanoscale width, where the 3-Ag reconstruction was destroyed locally by a scanning tunneling microscope (STM) tip, and found that the S1 surface state electrons were scattered by the lithographed line. Since local defects like adatoms, steps and lithographed lines behave as potential barriers, the S 1 surface state electron is confined and produces electron standing wave patterns in nano-structures surrounded by these local defects [2–4]. This provides a new way to artificially design of the surface 2D quantum structures by placing these local defects at the place of our demands [4]. However, it is crucial to know the scattering potential of these local Corresponding author. Tel.: +81 45 924 5637; fax: +81 45 924 5685.
E-mail address: [email protected] (H. Hirayama). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.06.019
defects for the surface electrons for engineering the quantized electronic states. Several analyses have reported on the scattering of the surface state electron by an adatom at metal surfaces [5,6]. In these previous studies, the scattering was treated in the framework of partial wave theory, and S-wave approximation was adapted a priori without examining its quantitative validity. Although the phase shift d of the scattered S-wave was deduced by fitting the standing wave pattern with the asymptotic form of the Bessel function, the d was not related to the scattering potential height. In this study, we quantitatively examined the scale for which the S-wave approximation and the usage of the Bessel function’s asymptotic form are valid. Confirming their validity for the scattering from an isolated adatom, we then deduced an analytical equation to fit the standing wave pattern, and estimated the scattering potential.
2. Experiments Experiments were carried out in an ultra-high vacuum (UHV) apparatus equipped with an Ag Knudsen cell and
ARTICLE IN PRESS S. Minamoto et al. / Physica E 40 (2007) 324–327
an STM unit [7]. The base pressure of the apparatus was below 1 108 Pa. In the UHV apparatus, Si samples were degassed overnight at 600 C, flashed at 1200 C for 10 s. and then slowly cooled to room temperature to p expose p the clean 7 7 reconstructed surface. The Si(1 1 1) 3 3-Ag reconstructed surface was prepared by depositing one monolayer (ML) Ag atoms on the Si(1 1 1)7 7 surface at 600 C. After the deposition, the surface was p cooled to room temperature. The surface showed the 3-Ag reconstruction, but local defects of isolated adatoms, p vacancies, and OPBs eventually appeared on the 3-Ag reconstruction. The surface structure and electron standing wave pattern around the local defects were observed by STM and STM-based dI=dV images. The dI=dV image (i.e. conductance map) was taken simultaneously with STM using a lock-in technique by applying a small sinusoidal modulation (100 meV pp , 1.7 kHz) to the sample bias voltage. The tunneling current was set at 0.2–0.3 nA. The scale in the STM image was calibrated p p by referring to the lattice distance of the Si(1 1 1) 3 3-Ag reconstruction. Both the STM and dI=dV images were taken at room temperature. 3. Results Fig. 1(a) shows a typical STM image of the p p Si(1 1 1) 3 3-Ag reconstructed surface with local defects. In the figure, an isolated adatom exists on the p 3-Ag surface at the upper right. A vacancy complex and a vacancy complex with an adatom at its center are observed at the upper left and bottom in the figure. The horizontal line at the center of the image was not the defect but was due to dragging of the STM tip during the scan. Fig. 1(b) shows the dI=dV image taken simultaneously with the STM image. In the dI=dV image, electron standing wave patterns with concentric fringes were observed around the local defects. In particular, we paid attention to the standing wave pattern around the isolated adatom in this study for the reason described in the next section. Ag adatoms are highly p mobile and do not rest at any sites on the 3-Ag surface at
325
Fig. 2. A cross-sectional line profile of the quantum interference pattern from the isolated adatom (the upper right one in Fig. 1) at the p p Si(1 1 1) 3 3-Ag surface. The sample bias voltage was þ1:0 V. The tunneling current was 0.3 nA.
room temperature [8]. Thus we think that the isolated adatom was probably Si. Fig. 2 shows an example of the cross-sectional line profile of the standing wave pattern around the isolated adatom. An oscillatory pattern was observed in the figure (the dashed line). Peaks in this figure were separated by 2 nm in any line profiles taken at the bias voltage of 1.0 eV. Although the oscillation decayed with distance from the adatom, two or three peaks were observed in cross-sectional line profiles taken at room temperature. Surface electrons lose coherency rapidly at room temperature, so it was very difficult to acquire the standing wave pattern at room temperature. Thus, we did not succeed to take a perfect series of dI=dV images with sequentially changed bias voltages at the same place. However, we observed similar damping oscillations in which the fringe pattern tended to become smaller with increasing the sample bias voltage, as expected from the free-electron like dispersion for the S1 surface state electrons. This demonstrated that the concentric fringe pattern was due to the quantum interference of the incoming and scattered surface electron waves around the defects. 4. Discussion The quantum interference around the isolated adatom is described by 2D Schro¨dinger equation of cylindrical symmetry. In polar coordinate r ¼ ðr; yÞ, the radial part of the wave function cðr; yÞ ¼ uðrÞ expði‘yÞ should satisfy the following equation [9]: 2 2 2 _2 d 1d _ ‘ þ þ V ðrÞ uðrÞ ¼ EuðrÞ, uðrÞ þ 2me dr2 r dr 2me r2 (1)
Fig. 1. Simultaneously taken STM (a) and dI=dV image (b) around p p isolated defects at the Si(1 1 1) 3 3-Ag surface. The sample bias voltage was þ1:0 V. The tunneling current was 0.3 nA.
where _ is the reduced Planck constant, me the electron effective mass, ‘ the angular momentum quantum number ð‘ ¼ 0; 1; 2; . . .Þ, V ðrÞ the scattering potential of the isolated adatom, and E is the energy.
ARTICLE IN PRESS 326
S. Minamoto et al. / Physica E 40 (2007) 324–327
The centrifugal potential V c ðrÞ ¼ _2 ‘2 =2me r2 pushes the electron with non-zero quantum number ‘ away from the origin. We plotted the V c for me ¼ 0:13 m0 as a function of the distance from the origin (i.e. the center of the adatom) in Fig. 3. (In the present system, me was estimated to be 0:13m0 as described below.) For the electron energy of 1.0 eV, the state of ‘X1 cannot enter the region rp0:54 nm, while the size of the adatom was estimated to be R ¼ 0:40 nm in the STM image. Thus, only the S-wave state (i.e. ‘ ¼ 0) is subjected to the scattering potential of the isolated adatom V ðrÞ. Thus, the S-wave approximation is valid in scattering by an isolated adatom. The S-wave function contains the zeroth Bessel function J 0 ðzÞ. Here, z ¼ k== r, and k== is the surface parallel wave vector of the electron in the S 1 surface state. For quantitative analysis of the scattering potential, it is crucial to use the asymptotic form of the Bessel function rffiffiffiffiffi 2 p J 0 ðzÞ cos z . (2) pz 4 We plotted the r-dependence of the strict and asymptotic forms of the Bessel function J 0 ðzÞ in Fig. 4. As shown in the figure, the asymptotic form is useful for zX0:7. Since k== ¼ 1:81 109 m1 for the S 1 surface state electron of 1.0 eV, the asymptotic form of the Bessel function is valid only for rX0:39 nm in our case. This means the adatom sizeð¼ 0:40 nmÞ is at the very limit of what is permissible to apply the asymptotic form of the Bessel function.
Fig. 3. Centrifugal potential for ‘ ¼ 1 and 2 as a function of the distance from the origin r. The electron mass was taken to be 0:13 m0 . The horizontal line indicates the electron energy of 1.0 eV.
Since both the S-wave approximation and the use of the asymptotic form of the Bessel function are valid, the radial wave function uðrÞ of the electron outside the adatom is safely described as sffiffiffiffiffiffiffiffiffiffiffi 2 p (3) cos k== r þ d . uðrÞA pk== r 4 Here, d is the scattering induced phase shift. In this case, the standing wave pattern around the isolated adatom is given [6,10] by 2A 2 cos k== r p þ d cos2 k== r p . DDOS / pk== r 4 4 (4) The equation indicates that the standing wave decays as 1=r. However, it decayed more rapidly due to the short coherent length ‘c in the experiment. Thus, we added a term expðr=‘c Þ to the above equation, and fitted the standing wave pattern in Fig. 2. The fitting was satisfactory, and we obtained free parameters d, k== and ‘c as 62 , 1:81 109 m1 , and 15 nm for the S 1 surface state electron of E ¼ 1:0 eV at room temperature. The S1 surface band has its bottom very close to the Fermi level [4]. Thus, we regarded the band dispersion as Eðk== Þ ¼ _2 k2== =2me in the fitting. Substituting k== ¼ 1:81 109 m1 for E ¼ 1:0 eV, we obtained the surface electron mass me of 0:13m0 . This me is consistent with the me of 0:13m0 reported in recent ultraviolet photoelectron spectroscopic (UPS) studies [11] and is reasonable. At room temperature, the surface electron life time is dominated by electron–phonon scattering. A recent UPS study reported the electron–phonon coupling constant p p l of 0.4 for the S 1 state electron at the Si(1 1 1) 3 3-Ag surface [12]. Since the electron life time t ¼ _=2plkB T and ‘c ¼ t v where v is on the order of the Fermi velocity 106 m=s, ‘c is estimated to be 10 nm. In this respect, the ‘c of 15 nm in the fitting is reasonable. Finally, we related the d to the scattering potential of the isolated adatom. Here, we assumed the isolated adatom as a cylindrical scatterer of a potential height of V and radius of R ¼ 0:40 nm. Inside the adatom region ðrpRÞ, the radial wave function is described aspuðrÞaI 0 ðkrÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi where a the normalized factor, k ¼ ð1=_Þ 2me ðV EÞ, and I 0 the zeroth modified Bessel function. For numerical analysis, we fitted I 0 ðkr) by a third polynomial equation. We obtained an analytical equation relating d to V by connecting smoothly this wave function to that of Eq. (4) for rXR at r ¼ R. For d ¼ 62 , R ¼ 0:40 nm, E ¼ 1:0 eV and me ¼ 0:13m0 , the equation gave us a scattering potential V of 6.5 eV. 5. Summary
Fig. 4. Zeroth Bessel function J 0 ð0Þ and its asymptotic form.
We observed concentric electron standing wave patterns p p around isolated defects on the Si(1 1 1) 3 3-Ag surface at room temperature. We especially paid attention to the
ARTICLE IN PRESS S. Minamoto et al. / Physica E 40 (2007) 324–327
scattering around the isolated adatom, and verified numerically that the S-wave approximation and the usage of the asymptotic form of the Bessel function are valid for describing the radial wave function for rXR. Then, we deduced a correct analytical form of DDOS and fitted the experimentally observed standing wave pattern by it to obtain d ¼ 62 , k== ¼ 1:81 109 m1 , and ‘c ¼ 15 nm. By smoothly connecting uðrÞ for rXR to that for rpR, we obtained the scattering potential of the isolated adatom of 6:5 eV. References [1] H. Aizawa, M. Tsukada, N. Sato, S. Hasegawa, Surf. Sci. 429 (1999) L509.
327
[2] M. Ono, Y. Nishigata, T. NIshio, T. Eguchi, Y. Hasegawa, Phys. Rev. Lett. 96 (2006) 016801. [3] N. Sato, S. Takeda, T. Nagao, S. Hasegawa, Phys. Rev. B 59 (1999) 2035. [4] S. Minamoto, T. Ishiduka, H. Hirayama, Phys. Rev. B., submitted for publication. [5] D. Eigler, E.K. Schweizer, Nature 363 (1993) 524. [6] F. Crommie, C. Lutz, D. Eigler, Science 262 (1993) 218. [7] M. Watai, H. Hirayama, Phys. Rev. B 72 (2005) 085435. [8] S. Hasegawa, J. Condens. Matter 12 (2000). [9] J. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press, Cambridge, 1998. [10] J. Ziman, Theory of Solids, Cambridge University Press, Cambridge, 1972. [11] T. Hirahara, I. Matsuda, M. Ueno, S. Hasegawa, Surf. Sci. 563 (2004) 191. [12] C. Liu, I. Matsuda, S. Hasegawa, private communication.