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Ab initio calculations of phonons at semiconductor surfaces
ELSEVIER
Physica B 219&220 (1996) 434-435
Ab initio calculations ofphonons at semiconductor surfaces U. Schr6der *, J. Fritsch, P. Pavone Institut fiir Theoretische Physik, Universitgit Reyensbur 9, D-93040 Regensbur 9, Germany
Abstract
In the context of an ab initio linear-response approach we have applied the density-functional theory to the investigations of structural and dynamical properties of semiconductor surfaces. The relaxation geometry was found by minimizing the total energy with the help of the Hellmann- Feynman forces. We present the full phonon dispersion of GaAs (1 1 0) and InP (1 1 0) along high symmetry lines of the surface Brillouin zone and the bond-stretching frequencies of these surfaces covered with one monolayer hydrogen. All calculated frequencies compare very well with all available experimental data from HREELS and He atom scattering.
1. Introduction
In recent years advances in inelastic-surface-scattering techniques have generated substantial interest in the study of surface dynamics of crystals. Detailed comparison between experimental results and theoretical studies of surface structures and of surface dynamics has now become feasible. We report here on ab initio calculations of the phonon dispersion curves of the GaAs (1 1 0) and InP (1 1 0) surfaces, which agree very well with all available experimental data. In addition we have determined the vibrational properties of the hydrogen covered surfaces of these semiconductors.
spacing equal to three removed atomic layers. For the k point sampling we use six special points in the irreducible part of the surface Brillouin zone. The determination of the relaxation geometry was done by minimizing the total energy with the help of the HellmannFeynman forces starting from the ideal surface. The dynamics of the fully relaxed slabs were treated within the framework of the density-functional perturbation scheme [6, 7]. The hydrogen-covered surfaces are calculated using the same formalism with a film of seven substrate layers which is covered on both surfaces with hydrogen atoms placed in the dangling bond direction of the substrate. Two neighboring films are separated by a distance corresponding to six interlayer distances prior to hydrogenation.
2. Theoretical method Our calculations are carried out within the densityfunctional theory in the local-density approximation [1, 2]. For the exchange-correlation potential we use the form of Ceperley and Alder [3] in the parametrization of Perdew and Zunger [4]. The electronic wave functions are expanded in plane waves up to a kinetic energy cutoff of 10 Ry. For the electron-ion interaction we use nonlocal norm-conserving pseudopotentials [5]. The surface properties are calculated using the periodic slab method. Each slab spans nine atomic layers, two neighboring slabs are separated by a vacuum * Corresponding author. 0921-4526/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 092 1-4526(95)00769-5
3. Results
Fig. 1 shows our results for GaAs. The large shaded area indicates the surface-projected bulk band structure. As can be seen, the calculated dispersion curves are in excellent agreement with the He-scattering experiments [8, 9] and the experimental data from HREELS [10]. A full discussion of the theoretical results is given in Ref. [11]. Fig. 2 summarizes the calculated spectra of InP including a comparison with the experimental data of the HREELS analysis of Nienhaus and M6nch [12]. There are two gaps in the surface-projected bulk band structure which is
U. Schr6der et al./ Physica B 219&220 (1996) 434~35 4O
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222meV, H - A s 259meV, H - I n 204meV, and H - P 278 meV, which compare very well with the experimental data of 229, 262, 208, and 281meV, respectively [14]. A more detailed analysis of the structure, the phonon spectrum, and the hydrogen modes is given in Ref. [15].
............................
=o
Acknowledgements
ii!!!iiiii iiiii::::
,,z, l O
: :.:,:.: >: :!:i:i:i:i~:~ ....
O ~ ~' f ..:. 0 5:"7 ' 7
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Fig. 1. Phonon dispersion of the relaxed GaAs (110) surface in comparison with the experimental data from He-atom scattering (triangles, Refs. [8] and [9]) and from HREELS (squares, Ref. [10]). The calculated surface localized phonon modes are indicated by the solid lines, the shaded area represents the surface projected bulk band structure.
We are grateful to S. Baroni and P. Giannozzi for providing numerical support. Our calculations have been accomplished by using the Cray-YMP supercomputer of the HLRZ of the KFA in Jiilich under Contract No. K271000 and the Leibniz Rechenzentrum in Miinchen. This work has been supported by a grant of the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg "KomplexitS.t in Festk6rpern: Phononen, Elektronen und Strukturen".
References 50
40
12
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Fig. 2. Phonon dispersion of the relaxed InP (1 l 0) surface in comparison with the experimental data from HREELS (squares, Ref. [12]). The calculated surface localized phonon modes are indicated by the solid lines, the shaded areas represent the surface projected bulk band structure. represented by the shaded areas. The structural data and the phonon spectrum are discussed in detail in Ref. [13]. For the hydrogen-covered surfaces the most prominent features seen in the HREELS spectra are the stretching modes of the hydrogen atoms vibrating in the bond direction [14]. Our results for these modes are: H~Ga
[1] P. Hohenberg and W. Kohn, Phys. Rev. B 136 (1964) 864. [2] W. Kohn and L.J. Sham, Phys. Rev. A 140 (1965) 1133. [3] D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. [4] J. Perdew and A. Zunger, Phys. Rev. B 23 (1981) 5048, [5] L. Pavesi, P. Giannozzi and K.F. Reinhart, Phys. Rev. B 42 (1990) 1864. [6] S. Baroni, P. Giannozzi and A. Testa. Phys. Rev. Lett. 58 (1987) 1861. [7] P. Giannozzi, S. de Gironcoli, P. Pavone and S. Baroni, Phys. Rev. B 43 (1991) 7231. [8] U. Harten and J.P. Toennies, Europhys. Lett. 4 (1987) 833. [9] R.B. Doak and D.B. Nguyen, J. Electron. Spectrosc. Relat. Phenom. 44 (1987) 205. [10] H. Nienhaus and W. M6nch, Phys. Rev. B 50 (1994) 11750. [ll] J. Fritsch, P. Pavone and U. Schr6der, Phys. Rev. Lett. 71 (1993) 4194. [12] H. Nienhaus and W. M6nch, Surf. Sci. 328 (1995) L 561. [13] J. Fritsch, P. Pavone and U. Sehr6der, Phys. Rev. 52 (1995) 11326. [14] U. del Pennino, C. Mariani, A. Amoddeo, R. Biagi, F. Proix and C. S6benne, J. Phys.: Condens. Matter 5 (1993) 6613; U. del Pennino, C. Mariani, A. Amoddeo, F. Proix and C.A. S~benne, J. Electron. Spectrosc. Relat. Phenom. 64/65 (1993) 491. [15] J. Fritsch, A. Eckert, P. Pavone and U. Schr6der, J. Phys.: Condens.Matter 7 (1995) 7717.
Physica B 219&220 (1996) 434-435
Ab initio calculations ofphonons at semiconductor surfaces U. Schr6der *, J. Fritsch, P. Pavone Institut fiir Theoretische Physik, Universitgit Reyensbur 9, D-93040 Regensbur 9, Germany
Abstract
In the context of an ab initio linear-response approach we have applied the density-functional theory to the investigations of structural and dynamical properties of semiconductor surfaces. The relaxation geometry was found by minimizing the total energy with the help of the Hellmann- Feynman forces. We present the full phonon dispersion of GaAs (1 1 0) and InP (1 1 0) along high symmetry lines of the surface Brillouin zone and the bond-stretching frequencies of these surfaces covered with one monolayer hydrogen. All calculated frequencies compare very well with all available experimental data from HREELS and He atom scattering.
1. Introduction
In recent years advances in inelastic-surface-scattering techniques have generated substantial interest in the study of surface dynamics of crystals. Detailed comparison between experimental results and theoretical studies of surface structures and of surface dynamics has now become feasible. We report here on ab initio calculations of the phonon dispersion curves of the GaAs (1 1 0) and InP (1 1 0) surfaces, which agree very well with all available experimental data. In addition we have determined the vibrational properties of the hydrogen covered surfaces of these semiconductors.
spacing equal to three removed atomic layers. For the k point sampling we use six special points in the irreducible part of the surface Brillouin zone. The determination of the relaxation geometry was done by minimizing the total energy with the help of the HellmannFeynman forces starting from the ideal surface. The dynamics of the fully relaxed slabs were treated within the framework of the density-functional perturbation scheme [6, 7]. The hydrogen-covered surfaces are calculated using the same formalism with a film of seven substrate layers which is covered on both surfaces with hydrogen atoms placed in the dangling bond direction of the substrate. Two neighboring films are separated by a distance corresponding to six interlayer distances prior to hydrogenation.
2. Theoretical method Our calculations are carried out within the densityfunctional theory in the local-density approximation [1, 2]. For the exchange-correlation potential we use the form of Ceperley and Alder [3] in the parametrization of Perdew and Zunger [4]. The electronic wave functions are expanded in plane waves up to a kinetic energy cutoff of 10 Ry. For the electron-ion interaction we use nonlocal norm-conserving pseudopotentials [5]. The surface properties are calculated using the periodic slab method. Each slab spans nine atomic layers, two neighboring slabs are separated by a vacuum * Corresponding author. 0921-4526/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 092 1-4526(95)00769-5
3. Results
Fig. 1 shows our results for GaAs. The large shaded area indicates the surface-projected bulk band structure. As can be seen, the calculated dispersion curves are in excellent agreement with the He-scattering experiments [8, 9] and the experimental data from HREELS [10]. A full discussion of the theoretical results is given in Ref. [11]. Fig. 2 summarizes the calculated spectra of InP including a comparison with the experimental data of the HREELS analysis of Nienhaus and M6nch [12]. There are two gaps in the surface-projected bulk band structure which is
U. Schr6der et al./ Physica B 219&220 (1996) 434~35 4O
30 ~ : ~ . : ' 5 . : . :
>
20
:: ~ .
7.5
435
222meV, H - A s 259meV, H - I n 204meV, and H - P 278 meV, which compare very well with the experimental data of 229, 262, 208, and 281meV, respectively [14]. A more detailed analysis of the structure, the phonon spectrum, and the hydrogen modes is given in Ref. [15].
............................
=o
Acknowledgements
ii!!!iiiii iiiii::::
,,z, l O
: :.:,:.: >: :!:i:i:i:i~:~ ....
O ~ ~' f ..:. 0 5:"7 ' 7
~
x=
Fig. 1. Phonon dispersion of the relaxed GaAs (110) surface in comparison with the experimental data from He-atom scattering (triangles, Refs. [8] and [9]) and from HREELS (squares, Ref. [10]). The calculated surface localized phonon modes are indicated by the solid lines, the shaded area represents the surface projected bulk band structure.
We are grateful to S. Baroni and P. Giannozzi for providing numerical support. Our calculations have been accomplished by using the Cray-YMP supercomputer of the HLRZ of the KFA in Jiilich under Contract No. K271000 and the Leibniz Rechenzentrum in Miinchen. This work has been supported by a grant of the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg "KomplexitS.t in Festk6rpern: Phononen, Elektronen und Strukturen".
References 50
40
12
........................
E
-1-
v
>-
~ 20
.
~ ~ :
:: : ::::
: ,- . ....
... -.
>
w ~~i:iiiii;!ii;!iiiiiii;i z
i;iii?i)) )iiiiiii::"
m 10 ' : : : : " : : ° ' ~ ; " " : .:.?i,:.:.:.:~. 0
7
~
0
x'
Fig. 2. Phonon dispersion of the relaxed InP (1 l 0) surface in comparison with the experimental data from HREELS (squares, Ref. [12]). The calculated surface localized phonon modes are indicated by the solid lines, the shaded areas represent the surface projected bulk band structure. represented by the shaded areas. The structural data and the phonon spectrum are discussed in detail in Ref. [13]. For the hydrogen-covered surfaces the most prominent features seen in the HREELS spectra are the stretching modes of the hydrogen atoms vibrating in the bond direction [14]. Our results for these modes are: H~Ga
[1] P. Hohenberg and W. Kohn, Phys. Rev. B 136 (1964) 864. [2] W. Kohn and L.J. Sham, Phys. Rev. A 140 (1965) 1133. [3] D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. [4] J. Perdew and A. Zunger, Phys. Rev. B 23 (1981) 5048, [5] L. Pavesi, P. Giannozzi and K.F. Reinhart, Phys. Rev. B 42 (1990) 1864. [6] S. Baroni, P. Giannozzi and A. Testa. Phys. Rev. Lett. 58 (1987) 1861. [7] P. Giannozzi, S. de Gironcoli, P. Pavone and S. Baroni, Phys. Rev. B 43 (1991) 7231. [8] U. Harten and J.P. Toennies, Europhys. Lett. 4 (1987) 833. [9] R.B. Doak and D.B. Nguyen, J. Electron. Spectrosc. Relat. Phenom. 44 (1987) 205. [10] H. Nienhaus and W. M6nch, Phys. Rev. B 50 (1994) 11750. [ll] J. Fritsch, P. Pavone and U. Schr6der, Phys. Rev. Lett. 71 (1993) 4194. [12] H. Nienhaus and W. M6nch, Surf. Sci. 328 (1995) L 561. [13] J. Fritsch, P. Pavone and U. Sehr6der, Phys. Rev. 52 (1995) 11326. [14] U. del Pennino, C. Mariani, A. Amoddeo, R. Biagi, F. Proix and C. S6benne, J. Phys.: Condens. Matter 5 (1993) 6613; U. del Pennino, C. Mariani, A. Amoddeo, F. Proix and C.A. S~benne, J. Electron. Spectrosc. Relat. Phenom. 64/65 (1993) 491. [15] J. Fritsch, A. Eckert, P. Pavone and U. Schr6der, J. Phys.: Condens.Matter 7 (1995) 7717.