- Email: [email protected]
Density of states of a two-dimensional electron gas measured by high-resolution photoelectron spectroscopy
PERGAMON
Solid State Communications 110 (1999) 661–666
Density of states of a two-dimensional electron gas measured by high-resolution photoelectron spectroscopy M.G. Betti a,*, V. Corradini a, V. De Renzi a, C. Mariani a, P. Casarini a, A. Abramo b a
Istituto Nazionale per la Fisica della Materia, Dipartimento di Fisica, Universita` di Modena e Reggio Emilia, Via G. Campi 213/A, I-41100 Modena, Italy b DIEGM, Universita` di Udine, Udine, Italy Received 12 March 1999; accepted 16 March 1999 by E. Molinari
Abstract We present high energy-resolution photoemission measurements of the spectral density at the discrete quantized electronic levels of a two-dimensional (2D) electron gas. The dynamical 2D electron gas has been obtained by generating a strong accumulation layer at the (110) surface of narrow-gap III–V semiconductors. Exploitation of a number of cases generating band bending (metallic chains or clusters, atomic structure, defects) demonstrates the generality of 2D electron gas formation at charge-accumulated semiconductor surfaces. A self-consistent solution of the Poisson and Schro¨dinger equations gives the potential well shape, the sub-band energy level position and the accumulated charge density, in excellent agreement with the present experimental data. q 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: A. Semiconductors; D. Electronic states (localized); E. Photoelectron spectroscopies
Space charge layer formation at semiconductor surfaces and at metal–semiconductor interfaces provides an ideal opportunity to investigate a degenerate two-dimensional (2D) electron gas. Charge confinement in the near-surface region can be achieved generating a strong accumulation layer in the conduction band of a semiconductor. The electrons confined in the quantum well present quantized energy levels in the direction perpendicular to the surface, while they are nearly free in the surface layer. The knowledge of the spectral density of the accumulated layer of a semiconductor is important both to investigate the fundamental physics of a degenerate 2D electron gas and to enlighten the driving forces responsible for the Fermi level pinning. * Corresponding author. E-mail address: [email protected] (M.G. Betti)
Since the 1960s, transport and optical response of a 2D electron gas have been attracting active theoretical and experimental interest [1]. Few experimental studies have been devoted to the investigation of the electronic properties and of the spectral density of a confined 2D electron gas at surfaces [2–6]. The quasi2D behaviour of the accumulated carriers has been experimentally deduced from the dispersion of the free-carrier plasmon of narrow gap III–V surfaces [3–5]. Recently, an angular resolved photoelectron spectroscopy study [7] of clean InAs surfaces showed an emission from the conduction band, due to the charge accumulated in the near-surface region. Native point defects in these molecular-beam-epitaxy (MBE) grown InAs surfaces induce a narrow space charge layer, where the 2D electron gas is confined. The asymmetric lineshape of the spectral density has been attributed to the presence of discrete quantum
0038-1098/99/$ - see front matter q 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00156-8
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M.G. Betti et al. / Solid State Communications 110 (1999) 661–666
Fig. 1. Angle-integrated HRUPS spectra in the valence band (VB, on the left) and conduction band (CB, on the right) energy region for the clean n-InAs(110) surface and at different Cs exposures. The continuous lines superimposed to the spectra in the CB are the result of a fitting to the experimental data (see text). Photon energy of 21.218 eV (HeI). The band bending potential as a function of Cs exposure is plotted in the inset.
levels [7]. Recently, Le Lay and coworkers [8] have performed a photoemission experiment on the InAs(110) surface where a giant accumulation layer was induced by depositing alkali metals. A spectral density of electronic states from the conduction band levels localized at the G point of the Brillouin zone has been measured on III–V(110) surfaces, for the maximum band bending achieved [8–12]. Photoelectron spectroscopy is the most direct method for a detailed characterization of occupied electronic states. In this letter, we present an investigation of the spectral density at the discrete quantized electronic levels populated in the conduction band of clean and alkali-covered narrow gap III–V(110) surfaces, by means of high energy-resolution and high luminosity UV photoemission spectroscopy. Our purpose is to explore a wide range of physical phenomena which can engender the formation of an accumulation layer at semiconductor surfaces. A set of different potential wells has been exploited, by
changing either the substrate or the band bending sources (defects, metal chains or clusters, etc.). High-resolution spectroscopy allows to clearly resolve the discrete quantized levels of the 2D electron gas for different potential quantum wells. A self-consistent solution of the Poisson and Schro¨dinger equations gives the potential well shape, the subband energy levels and the accumulated charge density at the semiconductor surface for the 2D electron gas, in excellent agreement with the whole set of experimental data. The comparison with the theoretical results clearly shows how the quantum well shape is independent from the driving forces giving rise to the band bending (defects, metal adatoms, surface disorder, etc.) and the corresponding density of states mirrors a 2D free electron behaviour. The experiments, carried out at the surface physics laboratory LOTUS (Dipartimento di Fisica, Universita` di Modena) were performed in a ultra-highvacuum (UHV) chamber containing a High Resolution Ultraviolet Photoelectron Spectroscopy (HRUPS) apparatus and other ancillary facilities for sample preparation. All photoelectron spectra were excited with a high intensity He discharge lamp (HeI and HeII photons, hn 21.218 and 40.814 eV, respectively). The photoemitted electrons were analyzed in the plane of incidence, with a Scienta SES-200 hemispherical analyzer; the integration angle was about ^88 with respect to the normal emission direction. The instrumental energy resolution was better than 15 meV, as determined on the Fermi level (EF) of freshly evaporated Au in good electrical contact with the semiconductor samples. The (110) clean surfaces were obtained by cleaving in situ n-InAs and n-InSb single crystals (n 4 × 10 17 and n 8 × 10 15 atoms/cm 3, respectively). Pure Cs was evaporated from well outgassed SAES Getters alkali dispensers at a pressure better than 7 × 10 211 mbar on the semiconductor substrate at 270 K. Thickness was calibrated following the work function evolution and the Cs 5p core level intensity. The energy distribution curves (EDCs) of the clean InAs(110) surface and those obtained after a tiny deposition of Cs, in the energy region of the upper valence band and the lowest conduction band, are shown in Fig. 1. Though the InAs valence band shape is not affected by the Cs deposition, a low spectral density of states with a step-like lineshape is
M.G. Betti et al. / Solid State Communications 110 (1999) 661–666
663
Table 1 Binding energy of the potential well eigenstates deduced from the solution of the Poisson–Schro¨dinger equations (Th.) compared with the results of a fitting procedure to the experimental data (Exp.). Energy values given in meV EF-CBM
InAs clean InAs-Cs
InSb-Cs Sb/InAs-Cs
230 364 384 442 510 570 600 247 273 400 565
E0 Th.
Exp.
E1 Th.
Exp.
E2 Th.
Exp.
118 169 177 204 237 264 279 101 111 186 262
135 ^ 10 170 ^ 5 173 202 240 262 275 100 ^ 5 117 190 ^ 10 245
84 99 102 112 125 137 144 35 40 105 136
90 ^ 10 100 ^ 5 102 118 130 143 145 35 ^ 5 44 100 ^ 10 135
83 84 84 85 88 92 94 4 6 84 91
85 ^ 10 85 ^ 5 85 85 90 90 95 6^5 8 85 ^ 10 90
observed within the conduction band. The metallicity of the semiconductor surface is settled by the high energy cutoff of the density of states in the conduction band at the Fermi energy. Moreover, the energy difference between the valence band maximum and the onset of the spectral density in the conduction band corresponds to the InAs band gap, and there is not any detectable structure in the gap. Thus, we can exclude the presence of gap states coming from Cs atoms which should be barely detectable at these low coverages. The evolution of the band bending potential Vbb, as deduced from the In-4d bulk component energy shift, is displayed in the inset of Fig. 1. At extremely low Cs coverage, the Fermi level pins well into the conduction band with a maximum band bending at 0.05 ML of Cs. This maximum value of the downward Vbb occurs at the same Cs coverage for which the highest density of states is emitted from the conduction band. These findings suggest that the spectral density can be associated with conduction band states within the potential well formed at the surface by the downward pinned band. The step-like density of states reflects the charge distribution of the quantized electronic levels generated within the potential well. The density of states of an electron gas confined in the direction perpendicular to the surface (z) and free on the surface plane (xy) is constant, independent from the energy, and shows discontinuities at each eigenvalue En [13]. In Table 1 we report the energy
position of the discrete energy levels for the InAs(110) surface, obtained by fitting the experimental spectra with a step-like 2D electron density of states, multiplied by the Fermi–Dirac distribution function, and convoluted with a gaussian curve representing the experimental broadening. The width of each step takes also into account the hole-lifetime of the corresponding energy level. The results of the experimental fitting procedure are reported in Fig. 1, superimposed to the experimental photoelectron emission from the conduction band. Within the parabolic approximation, we can determine the eigenvalues, the eigenfunctions, the potential well and the charge distribution, via the solution of the Poisson–Schro¨dinger equations. The Schro¨dinger and Poisson equations for a jellium-like 2D electron gas are self-consistently solved using an iterative procedure. The simulation starts assuming a guess for the electrostatic potential and introducing the nominal doping concentration. Based on this, the envelope function equation, i.e. the Schro¨dinger equation in the effective mass approximation, is solved. Spherical symmetry is assumed for InAs, with parabolic mass of 0.023 me for electrons and 0.4 me for holes, a dielectric constant of 14.60 and the energy gap of 0.354 eV at room temperature (RT). The eigenvectors are determined using the inverse iteration technique, while the eigenvalues are obtained via Rayleigh quotient. Once the eigenstates are computed, the classical charge density for holes and the quantum one for electrons
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M.G. Betti et al. / Solid State Communications 110 (1999) 661–666
Fig. 2. Result of a self-consistent solution of the Poisson–Schro¨dinger equations: (a) potential well shape and sub-band energy levels; (b) density of states in the parabolic approximation multiplied by the Fermi–Dirac distribution function at T 0 K; (c) photoemission spectrum from the CB for 0.05 ML-Cs/InAs(110) (white dots), and theoretical results convoluted with the experimental broadening at RT (continuous line).
are calculated. Poisson equation is then solved in its non-linear form to obtain the new electrostatic potential. On the right panel (a) of Fig. 2, the potential well as a function of distance from the surface and the first three quantized discrete energy levels, as obtained after the solution of the Poisson–Schro¨dinger equations, are displayed. The calculation is performed for a potential well of 570 meV with respect to the Fermi level, corresponding to the experimentally measured band bending. The three lowest energy levels are localized at E0 2 264 meV, E1 2 137 meV and E2 2 92 meV. The density of states at T 0 K is reported in Fig. 2(b), where the 2D spectral density is constant for each eigenvalue En in the potential well, and a 3D free-electron gas simulates the density of states from the bulk conduction band minimum (CBM) to the surface Fermi level. The density of states, calculated at T 300 K and convoluted with a gaussian curve, is shown in Fig. 2(c), superimposed to the experimental data. The fitting procedure of the experimental spectrum for Vbb 570 meV gives eigenvalues at E0 2 262, E1 2 143 and E2 2 90 meV (see also Table 1), in excellent agreement with the
theoretical predictions. The charge density responsible of the downward band bending is about 7.8 × 10 11 e/cm 2 at the maximum Fermi level pinning achieved. The eigenvalues obtained via the self-consistent solution of the Poisson–Schro¨dinger equations as a function of the surface Fermi level position with respect to the conduction band minimum (EF-CBM), are collected in Fig. 3 and reported in Table 1, along with the results of the fitting procedure to the experimental data. The excellent quantitative agreement between the experimental data and the theoretical results for different potential wells in a wide range of band bending clearly confirms how the density of states can be described as a 2D jellium, and how it represents a general property of charge accumulated semiconductor surfaces. A further step is to understand if the spectral density of a confined 2D electron gas generated by an accumulated charge is independent from the band bending sources, from the semiconductor surface, and from the structure of the metal adatoms deposited on the surface. We have performed further measurements on the n-type doped InSb(110) surface, where Cs
M.G. Betti et al. / Solid State Communications 110 (1999) 661–666
Fig. 3. Calculated energy eigenvalues (black dots) of the potential well for InAs(110) as a function of Vbb potential deduced from the solution of Poisson–Schro¨dinger equations. Experimental energy eigenvalues (open squares). The energy values are also reported in Table 1.
induces an analogous though less intense accumulation layer. Also in this case the intensity evolution of the spectral density in the InSb conduction band reflects the rise of the band bending potential. The electrons photoemitted from the conduction band show a spectral density with discontinuities corresponding to the eigenvalues of the 2D confined electron gas. Assuming a free electron gas in the surface plane, with parabolic mass of 0.0145 me for electrons and 0.4 me for holes, a dielectric constant of 15.68 and the energy gap of 0.176 eV at RT, we have obtained from the self-consistent solution of the Poisson and Schro¨dinger equations the eigenvalues reported in Table 1, along with the results of the experimental fitting. Also for this substrate, changing only the substrate parameters in the model, we obtain the theoretical energy eigenvalues in very good agreement with the experimental results. A further question is to enlighten the role of the metal adatoms atomic geometry. Cesium deposited on InAs(110) and InSb(110), at extremely low ˚ ) sepacoverages, forms long 1D chains (,1000 A rated hundreds of angstroms from each other
665
[14,15]. The Cs adatoms in the chains create adatom-induced surface states of donor-type, responsible for the charge accumulation. Are these 1D structures responsible for the spectral density, and do we expect a different density of states if the metal adatoms are disposed with a different atomic geometry? Charge accumulation is also achieved by depositing Cs on an Sb-precovered InAs(110) surface. In this case, Cs does not forms 1D chains but disordered 2D clusters, as can be observed by scanning-tunneling-microscopy (STM) images [15]. Even for this system we measure a step-like density of states in the conduction band, in which measured energy levels are compared with the calculated eigenvalues and reported in Table 1. The good agreement confirms that the spectral density is only determined by the depth of the potential well, and does not depend on the structural atomic geometry of the metal adatoms. Band bending potential can also be induced by the presence of defect states created in the surface preparation [7]. Our samples, prepared by cleaving the semiconductor InAs(110) surface, show an initial band bending due to a small density of cleavageinduced defects. Thus, at the clean n-type doped InAs(110) surface, the Fermi level position is located at 0.230 eV above the conduction band minimum, while in the flat band condition we would have expected a value of 0.083 eV above the CBM for the nominal doping n 4 × 10 17 atoms/cm 3, and effective mass m* 0.028 me. A careful observation of the spectral density in the conduction band of the clean surface reveals the presence of a low but measurable density of states, with two subbands positioned at 2130 meV and 290 meV, again in excellent agreement with the theoretical results (Fig. 3), reported in Table 1. In conclusion, we presented a collection of photoemission data showing step-like spectral density formed at the bottom of the conduction band at charge-accumulated semiconductor surfaces. These results are in excellent agreement with a self-consistent theoretical approach. This collection of experimental results clearly shows how the properties of the 2D electron gas is only determined by the semiconductor substrate and independent from the sources of the band bending potential (defects, metal adatoms, structural ordering at the surface, etc.). Metal adatoms and/or defect states generate the potential well, they
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are responsible for the energy position of the donor- or defect-induced states in the conduction band, and settle the position of the surface Fermi level. Despite the different origin of charge accumulation, the spectral density of states of the 2D electron gas only depends on the amount of charge accumulated in these states. Acknowledgements Work financed under the LOTUS project of INFM, and by the Ministero per l’Universita` e la Ricerca Scientifica e Tecnologica (MURST) under ex-40% and 60% funds. References [1] T. Ando, A.B. Fowler, F. Stern, Rev. Mod. Phys. 54 (1982) 437 and references therein. [2] D.C. Tsui, Phys. Rev. Lett. 24 (1970) 303. [3] Y. Chen, J.C. Hermanson, G.J. Lapeyre, Phys. Rev. B 39 (1989) 12682.
[4] M.G. Betti, R. Biagi, U. del Pennino, C. Mariani, Europhys. Lett. 32 (1995) 235. [5] M. Noguchi, K. Hirakawa, T. Ikoma, Phys. Rev. Lett. 66 (1991) 2243. [6] S. Kawaji, H.C. Gatos, Surf. Sci. 7 (1967) 215. ¨ . Olsson, C.B.M. Andersson, M.C. Ha¨kansson, J. Kanski, [7] L.O L. Ilver, U.O. Karlsson, Phys. Rev. Lett. 76 (1996) 3626. [8] V.Yu. Aristov, G. Le Lay, P. Soukiassian, K. Hricovini, J.E. Bonnet, J. Osvald, O. Olsson, J. Vac. Sci. Technol. B 12 (1994) 2709. [9] V.Yu. Aristov, G. Le Lay, M. Grehk, V.M. Zhilin, A. TalebIbrahimi, G. Indlekofer, P. Soukiassian, Surf. Rev. Lett. 2 (1995) 723. [10] Y. Enta, T. Kinoshita, S. Suzuki, S. Kono, Phys. Rev. B 36 (1987) 9801. [11] V.Yu. Aristov, M. Grehk, V.M. Zhilin, A. Taleb-Ibrahimi, G. Indlekofer, Z. Hurych, G. Le Lay, P. Soukiassian, Appl. Surf. Sci. 104 (1996) 73. [12] Y. Enta, T. Kinoshita, S. Suzuki, S. Kono, Phys. Rev. B 39 (1989) 1125. [13] C. Weisbuch, B. Vinter, Quantum semiconductor structures, Academic Press, New York, 1991. [14] L.J. Whitman, J.A. Stroscio, R.A. Dragoset, R.J. Celotta, Phys. Rev. B 44 (1991) 5951. [15] S. Modesti et al. (unpublished).
Solid State Communications 110 (1999) 661–666
Density of states of a two-dimensional electron gas measured by high-resolution photoelectron spectroscopy M.G. Betti a,*, V. Corradini a, V. De Renzi a, C. Mariani a, P. Casarini a, A. Abramo b a
Istituto Nazionale per la Fisica della Materia, Dipartimento di Fisica, Universita` di Modena e Reggio Emilia, Via G. Campi 213/A, I-41100 Modena, Italy b DIEGM, Universita` di Udine, Udine, Italy Received 12 March 1999; accepted 16 March 1999 by E. Molinari
Abstract We present high energy-resolution photoemission measurements of the spectral density at the discrete quantized electronic levels of a two-dimensional (2D) electron gas. The dynamical 2D electron gas has been obtained by generating a strong accumulation layer at the (110) surface of narrow-gap III–V semiconductors. Exploitation of a number of cases generating band bending (metallic chains or clusters, atomic structure, defects) demonstrates the generality of 2D electron gas formation at charge-accumulated semiconductor surfaces. A self-consistent solution of the Poisson and Schro¨dinger equations gives the potential well shape, the sub-band energy level position and the accumulated charge density, in excellent agreement with the present experimental data. q 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: A. Semiconductors; D. Electronic states (localized); E. Photoelectron spectroscopies
Space charge layer formation at semiconductor surfaces and at metal–semiconductor interfaces provides an ideal opportunity to investigate a degenerate two-dimensional (2D) electron gas. Charge confinement in the near-surface region can be achieved generating a strong accumulation layer in the conduction band of a semiconductor. The electrons confined in the quantum well present quantized energy levels in the direction perpendicular to the surface, while they are nearly free in the surface layer. The knowledge of the spectral density of the accumulated layer of a semiconductor is important both to investigate the fundamental physics of a degenerate 2D electron gas and to enlighten the driving forces responsible for the Fermi level pinning. * Corresponding author. E-mail address: [email protected] (M.G. Betti)
Since the 1960s, transport and optical response of a 2D electron gas have been attracting active theoretical and experimental interest [1]. Few experimental studies have been devoted to the investigation of the electronic properties and of the spectral density of a confined 2D electron gas at surfaces [2–6]. The quasi2D behaviour of the accumulated carriers has been experimentally deduced from the dispersion of the free-carrier plasmon of narrow gap III–V surfaces [3–5]. Recently, an angular resolved photoelectron spectroscopy study [7] of clean InAs surfaces showed an emission from the conduction band, due to the charge accumulated in the near-surface region. Native point defects in these molecular-beam-epitaxy (MBE) grown InAs surfaces induce a narrow space charge layer, where the 2D electron gas is confined. The asymmetric lineshape of the spectral density has been attributed to the presence of discrete quantum
0038-1098/99/$ - see front matter q 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00156-8
662
M.G. Betti et al. / Solid State Communications 110 (1999) 661–666
Fig. 1. Angle-integrated HRUPS spectra in the valence band (VB, on the left) and conduction band (CB, on the right) energy region for the clean n-InAs(110) surface and at different Cs exposures. The continuous lines superimposed to the spectra in the CB are the result of a fitting to the experimental data (see text). Photon energy of 21.218 eV (HeI). The band bending potential as a function of Cs exposure is plotted in the inset.
levels [7]. Recently, Le Lay and coworkers [8] have performed a photoemission experiment on the InAs(110) surface where a giant accumulation layer was induced by depositing alkali metals. A spectral density of electronic states from the conduction band levels localized at the G point of the Brillouin zone has been measured on III–V(110) surfaces, for the maximum band bending achieved [8–12]. Photoelectron spectroscopy is the most direct method for a detailed characterization of occupied electronic states. In this letter, we present an investigation of the spectral density at the discrete quantized electronic levels populated in the conduction band of clean and alkali-covered narrow gap III–V(110) surfaces, by means of high energy-resolution and high luminosity UV photoemission spectroscopy. Our purpose is to explore a wide range of physical phenomena which can engender the formation of an accumulation layer at semiconductor surfaces. A set of different potential wells has been exploited, by
changing either the substrate or the band bending sources (defects, metal chains or clusters, etc.). High-resolution spectroscopy allows to clearly resolve the discrete quantized levels of the 2D electron gas for different potential quantum wells. A self-consistent solution of the Poisson and Schro¨dinger equations gives the potential well shape, the subband energy levels and the accumulated charge density at the semiconductor surface for the 2D electron gas, in excellent agreement with the whole set of experimental data. The comparison with the theoretical results clearly shows how the quantum well shape is independent from the driving forces giving rise to the band bending (defects, metal adatoms, surface disorder, etc.) and the corresponding density of states mirrors a 2D free electron behaviour. The experiments, carried out at the surface physics laboratory LOTUS (Dipartimento di Fisica, Universita` di Modena) were performed in a ultra-highvacuum (UHV) chamber containing a High Resolution Ultraviolet Photoelectron Spectroscopy (HRUPS) apparatus and other ancillary facilities for sample preparation. All photoelectron spectra were excited with a high intensity He discharge lamp (HeI and HeII photons, hn 21.218 and 40.814 eV, respectively). The photoemitted electrons were analyzed in the plane of incidence, with a Scienta SES-200 hemispherical analyzer; the integration angle was about ^88 with respect to the normal emission direction. The instrumental energy resolution was better than 15 meV, as determined on the Fermi level (EF) of freshly evaporated Au in good electrical contact with the semiconductor samples. The (110) clean surfaces were obtained by cleaving in situ n-InAs and n-InSb single crystals (n 4 × 10 17 and n 8 × 10 15 atoms/cm 3, respectively). Pure Cs was evaporated from well outgassed SAES Getters alkali dispensers at a pressure better than 7 × 10 211 mbar on the semiconductor substrate at 270 K. Thickness was calibrated following the work function evolution and the Cs 5p core level intensity. The energy distribution curves (EDCs) of the clean InAs(110) surface and those obtained after a tiny deposition of Cs, in the energy region of the upper valence band and the lowest conduction band, are shown in Fig. 1. Though the InAs valence band shape is not affected by the Cs deposition, a low spectral density of states with a step-like lineshape is
M.G. Betti et al. / Solid State Communications 110 (1999) 661–666
663
Table 1 Binding energy of the potential well eigenstates deduced from the solution of the Poisson–Schro¨dinger equations (Th.) compared with the results of a fitting procedure to the experimental data (Exp.). Energy values given in meV EF-CBM
InAs clean InAs-Cs
InSb-Cs Sb/InAs-Cs
230 364 384 442 510 570 600 247 273 400 565
E0 Th.
Exp.
E1 Th.
Exp.
E2 Th.
Exp.
118 169 177 204 237 264 279 101 111 186 262
135 ^ 10 170 ^ 5 173 202 240 262 275 100 ^ 5 117 190 ^ 10 245
84 99 102 112 125 137 144 35 40 105 136
90 ^ 10 100 ^ 5 102 118 130 143 145 35 ^ 5 44 100 ^ 10 135
83 84 84 85 88 92 94 4 6 84 91
85 ^ 10 85 ^ 5 85 85 90 90 95 6^5 8 85 ^ 10 90
observed within the conduction band. The metallicity of the semiconductor surface is settled by the high energy cutoff of the density of states in the conduction band at the Fermi energy. Moreover, the energy difference between the valence band maximum and the onset of the spectral density in the conduction band corresponds to the InAs band gap, and there is not any detectable structure in the gap. Thus, we can exclude the presence of gap states coming from Cs atoms which should be barely detectable at these low coverages. The evolution of the band bending potential Vbb, as deduced from the In-4d bulk component energy shift, is displayed in the inset of Fig. 1. At extremely low Cs coverage, the Fermi level pins well into the conduction band with a maximum band bending at 0.05 ML of Cs. This maximum value of the downward Vbb occurs at the same Cs coverage for which the highest density of states is emitted from the conduction band. These findings suggest that the spectral density can be associated with conduction band states within the potential well formed at the surface by the downward pinned band. The step-like density of states reflects the charge distribution of the quantized electronic levels generated within the potential well. The density of states of an electron gas confined in the direction perpendicular to the surface (z) and free on the surface plane (xy) is constant, independent from the energy, and shows discontinuities at each eigenvalue En [13]. In Table 1 we report the energy
position of the discrete energy levels for the InAs(110) surface, obtained by fitting the experimental spectra with a step-like 2D electron density of states, multiplied by the Fermi–Dirac distribution function, and convoluted with a gaussian curve representing the experimental broadening. The width of each step takes also into account the hole-lifetime of the corresponding energy level. The results of the experimental fitting procedure are reported in Fig. 1, superimposed to the experimental photoelectron emission from the conduction band. Within the parabolic approximation, we can determine the eigenvalues, the eigenfunctions, the potential well and the charge distribution, via the solution of the Poisson–Schro¨dinger equations. The Schro¨dinger and Poisson equations for a jellium-like 2D electron gas are self-consistently solved using an iterative procedure. The simulation starts assuming a guess for the electrostatic potential and introducing the nominal doping concentration. Based on this, the envelope function equation, i.e. the Schro¨dinger equation in the effective mass approximation, is solved. Spherical symmetry is assumed for InAs, with parabolic mass of 0.023 me for electrons and 0.4 me for holes, a dielectric constant of 14.60 and the energy gap of 0.354 eV at room temperature (RT). The eigenvectors are determined using the inverse iteration technique, while the eigenvalues are obtained via Rayleigh quotient. Once the eigenstates are computed, the classical charge density for holes and the quantum one for electrons
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M.G. Betti et al. / Solid State Communications 110 (1999) 661–666
Fig. 2. Result of a self-consistent solution of the Poisson–Schro¨dinger equations: (a) potential well shape and sub-band energy levels; (b) density of states in the parabolic approximation multiplied by the Fermi–Dirac distribution function at T 0 K; (c) photoemission spectrum from the CB for 0.05 ML-Cs/InAs(110) (white dots), and theoretical results convoluted with the experimental broadening at RT (continuous line).
are calculated. Poisson equation is then solved in its non-linear form to obtain the new electrostatic potential. On the right panel (a) of Fig. 2, the potential well as a function of distance from the surface and the first three quantized discrete energy levels, as obtained after the solution of the Poisson–Schro¨dinger equations, are displayed. The calculation is performed for a potential well of 570 meV with respect to the Fermi level, corresponding to the experimentally measured band bending. The three lowest energy levels are localized at E0 2 264 meV, E1 2 137 meV and E2 2 92 meV. The density of states at T 0 K is reported in Fig. 2(b), where the 2D spectral density is constant for each eigenvalue En in the potential well, and a 3D free-electron gas simulates the density of states from the bulk conduction band minimum (CBM) to the surface Fermi level. The density of states, calculated at T 300 K and convoluted with a gaussian curve, is shown in Fig. 2(c), superimposed to the experimental data. The fitting procedure of the experimental spectrum for Vbb 570 meV gives eigenvalues at E0 2 262, E1 2 143 and E2 2 90 meV (see also Table 1), in excellent agreement with the
theoretical predictions. The charge density responsible of the downward band bending is about 7.8 × 10 11 e/cm 2 at the maximum Fermi level pinning achieved. The eigenvalues obtained via the self-consistent solution of the Poisson–Schro¨dinger equations as a function of the surface Fermi level position with respect to the conduction band minimum (EF-CBM), are collected in Fig. 3 and reported in Table 1, along with the results of the fitting procedure to the experimental data. The excellent quantitative agreement between the experimental data and the theoretical results for different potential wells in a wide range of band bending clearly confirms how the density of states can be described as a 2D jellium, and how it represents a general property of charge accumulated semiconductor surfaces. A further step is to understand if the spectral density of a confined 2D electron gas generated by an accumulated charge is independent from the band bending sources, from the semiconductor surface, and from the structure of the metal adatoms deposited on the surface. We have performed further measurements on the n-type doped InSb(110) surface, where Cs
M.G. Betti et al. / Solid State Communications 110 (1999) 661–666
Fig. 3. Calculated energy eigenvalues (black dots) of the potential well for InAs(110) as a function of Vbb potential deduced from the solution of Poisson–Schro¨dinger equations. Experimental energy eigenvalues (open squares). The energy values are also reported in Table 1.
induces an analogous though less intense accumulation layer. Also in this case the intensity evolution of the spectral density in the InSb conduction band reflects the rise of the band bending potential. The electrons photoemitted from the conduction band show a spectral density with discontinuities corresponding to the eigenvalues of the 2D confined electron gas. Assuming a free electron gas in the surface plane, with parabolic mass of 0.0145 me for electrons and 0.4 me for holes, a dielectric constant of 15.68 and the energy gap of 0.176 eV at RT, we have obtained from the self-consistent solution of the Poisson and Schro¨dinger equations the eigenvalues reported in Table 1, along with the results of the experimental fitting. Also for this substrate, changing only the substrate parameters in the model, we obtain the theoretical energy eigenvalues in very good agreement with the experimental results. A further question is to enlighten the role of the metal adatoms atomic geometry. Cesium deposited on InAs(110) and InSb(110), at extremely low ˚ ) sepacoverages, forms long 1D chains (,1000 A rated hundreds of angstroms from each other
665
[14,15]. The Cs adatoms in the chains create adatom-induced surface states of donor-type, responsible for the charge accumulation. Are these 1D structures responsible for the spectral density, and do we expect a different density of states if the metal adatoms are disposed with a different atomic geometry? Charge accumulation is also achieved by depositing Cs on an Sb-precovered InAs(110) surface. In this case, Cs does not forms 1D chains but disordered 2D clusters, as can be observed by scanning-tunneling-microscopy (STM) images [15]. Even for this system we measure a step-like density of states in the conduction band, in which measured energy levels are compared with the calculated eigenvalues and reported in Table 1. The good agreement confirms that the spectral density is only determined by the depth of the potential well, and does not depend on the structural atomic geometry of the metal adatoms. Band bending potential can also be induced by the presence of defect states created in the surface preparation [7]. Our samples, prepared by cleaving the semiconductor InAs(110) surface, show an initial band bending due to a small density of cleavageinduced defects. Thus, at the clean n-type doped InAs(110) surface, the Fermi level position is located at 0.230 eV above the conduction band minimum, while in the flat band condition we would have expected a value of 0.083 eV above the CBM for the nominal doping n 4 × 10 17 atoms/cm 3, and effective mass m* 0.028 me. A careful observation of the spectral density in the conduction band of the clean surface reveals the presence of a low but measurable density of states, with two subbands positioned at 2130 meV and 290 meV, again in excellent agreement with the theoretical results (Fig. 3), reported in Table 1. In conclusion, we presented a collection of photoemission data showing step-like spectral density formed at the bottom of the conduction band at charge-accumulated semiconductor surfaces. These results are in excellent agreement with a self-consistent theoretical approach. This collection of experimental results clearly shows how the properties of the 2D electron gas is only determined by the semiconductor substrate and independent from the sources of the band bending potential (defects, metal adatoms, structural ordering at the surface, etc.). Metal adatoms and/or defect states generate the potential well, they
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are responsible for the energy position of the donor- or defect-induced states in the conduction band, and settle the position of the surface Fermi level. Despite the different origin of charge accumulation, the spectral density of states of the 2D electron gas only depends on the amount of charge accumulated in these states. Acknowledgements Work financed under the LOTUS project of INFM, and by the Ministero per l’Universita` e la Ricerca Scientifica e Tecnologica (MURST) under ex-40% and 60% funds. References [1] T. Ando, A.B. Fowler, F. Stern, Rev. Mod. Phys. 54 (1982) 437 and references therein. [2] D.C. Tsui, Phys. Rev. Lett. 24 (1970) 303. [3] Y. Chen, J.C. Hermanson, G.J. Lapeyre, Phys. Rev. B 39 (1989) 12682.
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