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Positional dependence of optical absorption in silicon nanostructure
Journal of Non-Crystalline Solids 293±295 (2001) 705±708
www.elsevier.com/locate/jnoncrysol
Positional dependence of optical absorption in silicon nanostructure Kengo Nishio a,*, Junichiro K oga a, Hiroaki Ohtani b, Toshio Yamaguchi c, Fumiko Yonezawa a b
a Department of Physics, Keio University, Yokohama 223-8522, Japan Theory and Computer Simulation Center, National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan c Department of Physics, Tokyo Women's Medical University, Tokyo 162-8666, Japan
Abstract We present a theoretical study of the optical absorption process of a 9 9 Si quantum wire. We calculate the imaginary part of the dielectric constant e2 and the contribution to e2 due to three Si atoms located in dierent positions using the non-orthogonal tight-binding method. From these calculations, we clearly ®nd for the ®rst time that the optical absorption below 3.4 eV tends to occur in the inner region of the 9 9 Si quantum wire. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 71.24.+q
1. Introduction The observation of ecient and visible photoluminescence (PL) from the Si nanostructures, such as porous Si [1] and Si nanocrystals [2], has attracted much interest in the origin of the PL from the viewpoint of fundamental physics and the potential application to optical devices. Various models [3±6] were proposed to explain the PL origin, but it is still under debate. On the other hand, it is widely accepted that the absorption properties are governed by the crystalline silicon core, because the observed optical band gap energy shows a blue shift with decreasing size [7] and the imaginary part of the dielectric constant e2 maintains some of the original properties of bulk Si [8]. Theoretical studies of the
band gap energy [9] and imaginary part of the dielectric constant [10] are consistent with experimental results. For a further understanding of the absorption process, we investigate where, in the crystalline Si core, the optical absorption occurs. For the purpose of this study, we calculate e2 and the contribution to e2 of the inner 25 Si atoms for a 9 9 Si quantum wire, as a model of Si nanostructure, using the non-orthogonal tight-binding method. As far as we know, there is no study which investigates where, in the crystalline Si core, the absorption process occurs. It is important to understand the optical absorption process for further discussion and understanding of the PL origin.
2. Method * Corresponding author. Tel.: +81-45 563 1141; fax: +81-45 566 1672. E-mail address: [email protected] (K. Nishio).
The model we investigate is a 9 9 Si quantum wire cut from the perfect diamond structure of bulk
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 8 4 9 - 3
706
K. Nishio et al. / Journal of Non-Crystalline Solids 293±295 (2001) 705±708
Si. The atomic con®guration of this model is shown in Fig. 1. We assign periodic boundary condition for the [0 0 1]-direction. In order to eliminate the surface states and provide a smooth termination of the orbitals, the surface dangling bonds are terminated by terminators. The terminator denoted by T has an s orbital which plays the role of only removing surface states from the fundamental band gap [11]. It is a model for a wider band gap insulator enclosing the Si nanostructure [14]. We think that this model is valid for studying the optical absorption process. In the tight-binding method, the Bloch function W lk r is expanded by the atomic orbitals /n r of angular momentum n as follows:
W lk r 1 X X lk ik Rd i Ain e /n r p N in R
R d i ; 1
where l is the band index, k is the wave vector, N is the number of atoms in the Si nanostructure, i is lk the atomic number in the unit cell, Ain is the expansion coecient, d i is the position vector of atom i, and R is the lattice vector. The energy ei lk genvalue El k and eigenvector Ain are determined by the secular equation, X k lk X k lk Him;jn Ajn El k Sim;jn Ajn 2 jn
jn
with the orthonormal condition X X lk k mk Aim Sim;jn Ajn dl;m : im
3
jn
k
The Hamiltonian matrix Him;jn and the overlap k matrix Sim;jn are de®ned as Z X k e ik Rd i d j /m r R d i d j Him;jn R
H /n r dr; k
Sim;jn
X
e
ik Rd i d j
R
Z
4 /m r
R d i
/n r dr;
d j 5
k
k
respectively. In order to build up Him;jn and Sim;jn , we use the empirical parameters proposed by Mattheiss [15] and set the parameters between Si and T to take the dangling bond away from the fundamental band gap [12]. The imaginary part of the dielectric constant e2 and its contribution from the Si atom located at site i, which we denote as e2i , are de®ned as e2 Fig. 1. The atomic con®guration of a 9 9 Si quantum wire unit cell. Each circle represents a Si atom and ellipse represents the terminator denoted by T in the text. The size of the circle represents the component of the [0 0 1]-direction; 3=4a; 2=4a; 1=4a and 0 where a is the lattice constant of diamond bulk Si. The Si atoms represented by partially ®lled circles (25 Si atoms) and denoted by capital letters (A, B and C) are discussed in this study.
4p2 e2 X 2 2jhWc;k je pjWv;k ij dEc k m2 x2 v;c;k Ev k
e2i
hx;
6
4p2 e2 X 2 2jhWc;k je pjWv;k ij m2 x2 v;c;k Wi
vk
dEc k
Ev k
hx;
7
K. Nishio et al. / Journal of Non-Crystalline Solids 293±295 (2001) 705±708
707
respectively, where e is the polarization vector, p is lk the momentum operator and Wi is a weighting function de®ned as X X 1 h lk k lk lk A Wi S A 2 im im;jn jn m jn i lk k lk Ajn Sjn;im Aim X X h lk k lk i R Aim Sim;jn Ajn : 8 m
jn
2
The transition probability 2p= hjhWc;k je pjWv;k ij is evaluated in the two manners in which (a) we regard it as constant and (b) we calculate it using the parameters [16] used in [12]. We calculate e2 and e2i polarized in [1 1 0]-direction, which is most in¯uenced by the quantum con®nement. 3. Result Figs. 2(a) and (b) show the imaginary part of the dielectric constant e2 ([1 1 0]-polarization) for the 9 9 Si quantum wire and the contribution to e2 of the 25 Si atoms shown in Fig. 1. Fig. 3 shows the contribution to the imaginary part of the dielectric constant e2i ([1 1 0]-polarization) for the three Si atoms denoted by capital letters (A, B and C) in Fig. 1, for the 9 9 Si quantum wire.
Fig. 2. The imaginary part of the dielectric constant e2 ([1 1 0]polarization) for the 9 9 Si quantum wire (solid line) and its contribution from the 25 Si atoms is shown in Fig. 1 (dashed line). The transition probability is evaluated in the manner in which (a) we regard it as constant and (b) we calculate it using the parameters [16] used in [12].
4. Discussion We have veri®ed, previous to this study, that the onset of the imaginary part of the dielectric constant e2 for bulk Si is 3.4 eV and the band structure of 9 9 Si quantum wire shows 2.0 eV direct band gap at C-point. Therefore we discuss properties of e2 in the energy range between 2.0 and 3.4 eV which re¯ects the intrinsic properties of Si nanostructures. Fig. 2 shows that a large part of the imaginary part of the dielectric constant e2 below 3.4 eV is contributed by the inner 25 Si atoms. This result indicates that the optical absorption below 3.4 eV tends to occur at the inner 25 Si atoms. For a further understanding of the optical absorption process, we have calculated the contribution to e2 for the three Si atoms denoted by
Fig. 3. The contribution to the imaginary part of the dielectric constant e2i ([1 1 0]-polarization) for three dierent Si atoms A (solid line), B (dashed line) and C (dash±dot line) is shown in Fig. 1, for the 9 9 Si quantum wire. The transition probability is evaluated using the parameters [16] used in [12].
708
K. Nishio et al. / Journal of Non-Crystalline Solids 293±295 (2001) 705±708
capital letters (A, B and C) in Fig. 1. From the result shown in Fig. 3, we ®nd that the optical absorption below 3.4 eV increases, as the site of the Si atom changes from the surface (C) to the center (A). This is because the surface boundary condition causes wave functions of the levels near the valence band maximum to be localized in the central core of Si atoms. It is a property in which the 9 9 Si quantum wire diers from bulk Si; the optical absorption depends on the site of Si atom we examine. It is worth discussing the dierence between Figs. 2(a) and (b). The imaginary part of the dielectric constant near the band gap (2.0 eV) in Fig. 2(a) increases as the hx increases, although that in Fig. 2(b) is small. The small e2 in Fig. 2(b) is attributed to a small transition probability near the band gap. This corresponds to the experimental results [8] and is similar to those reported by other workers [10,13]. The small transition probability indicates that the Brillouin zone folding makes it possible for the direct transition to take place at low energy, but nevertheless the k-selection rule of bulk Si remains. 5. Conclusion We have calculated the imaginary part of the dielectric constant e2 and the contribution to e2 for the three Si atoms, for a 9 9 Si Quantum wire, using the non-orthogonal tight-binding method. We have shown that a large part of e2 below 3.4 eV is contributed by the inner 25 Si atoms. We have also shown that the contribution to e2 below 3.4 eV increases, as the site of the Si atom changes from the surface Si atom to the center Si atom. From these results, we have clearly found for the ®rst time that the optical absorption below 3.4 eV tends to occur in the inner 25 Si atoms and that the frequency of the optical absorption depends on the site of Si atom we examine. This is because the surface boundary condition causes wave functions of the levels near the valence band maximum to be
localized in the central core of Si atoms. These are the properties in which the 9 9 Si quantum wire diers from bulk Si. Finally, we think that the ®ndings of this study, the positional dependence of optical absorption, true of all Si nanostructures. Acknowledgements The authors are grateful to Mr Okumura for valuable discussions. This work has been supported by `Research for the Future Project' on `Physics of Nanocrystalline Semiconductors and Their Application to New Functional Devices' at Ritsumei University (JSPS-RFTF96I00102). References [1] L.T. Canham, Appl. Phys. Lett. 57 (1990) 1046. [2] H. Takagi, H. Ogawa, Y. Yamazaki, A. Ishizaki, T. Nakagiri, Appl. Phys. Lett. 56 (1990) 2379. [3] P.D.J. Calcott, K.J. Nash, L.T. Canham, M.J. Kane, D. Brumhead, J. Phys. 5 (1993) L91. [4] F. Koch, V. Perova-Koch, D. Muschik, J. Lumin. 57 (1993) 271. [5] M.B. Robinson, A.C. Dillon, D.R. Haynes, S.M. George, Appl. Phys. Lett. 61 (1992) 1414. [6] Y. Kanemitsu, S. Okamoto, M. Otobe, S. Oda, Phys. Rev. B. 55 (1997) 7375. [7] Y. Kanemitsu, H. Uto, Y. Masumoto, Phys. Rev. B 48 (1993) 2827. [8] N. Koshida, H. Koyama, Y. Suda, Y. Yamamoto, M. Araki, T. Saito, K. Sato, N. Sata, S. Shin, Appl. Phys. Lett. 63 (1993) 2774. [9] D.J. Lockwood, Solid. State. Commun. 92 (1994) 101. [10] M. Cruz, M.R. Beltran, C. Wang, J. Taguena-Martinez, Mater. Res. Soc. Symp. Proc. (USA) 452 (1997) 69. [11] The terminator T has the same role as the hydrogen atom in Ref. [12,13]. In order to distinguish a real hydrogen atom and the hydrogen atom introduced to terminate the surface dangling bond [12,13], we call the latter terminator. [12] J.P. Proot, C. Delerue, G. Allan, Appl. Phys. Lett. 61 (1992) 1948. [13] G.D. Sanders, Y.-C. Chang, Phys. Rev. B 45 (1992) 9202. [14] B. Delley, E.F. Steigmeier, Appl. Phys. Lett. 67 (1995) 2370. [15] L.F. Mattheiss, J.R. Patel, Phys. Rev. B 23 (1981) 5384. [16] J. Petit, G. Allan, M. Lannoo, Phys. Rev. B 33 (1986) 8595.
www.elsevier.com/locate/jnoncrysol
Positional dependence of optical absorption in silicon nanostructure Kengo Nishio a,*, Junichiro K oga a, Hiroaki Ohtani b, Toshio Yamaguchi c, Fumiko Yonezawa a b
a Department of Physics, Keio University, Yokohama 223-8522, Japan Theory and Computer Simulation Center, National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan c Department of Physics, Tokyo Women's Medical University, Tokyo 162-8666, Japan
Abstract We present a theoretical study of the optical absorption process of a 9 9 Si quantum wire. We calculate the imaginary part of the dielectric constant e2 and the contribution to e2 due to three Si atoms located in dierent positions using the non-orthogonal tight-binding method. From these calculations, we clearly ®nd for the ®rst time that the optical absorption below 3.4 eV tends to occur in the inner region of the 9 9 Si quantum wire. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 71.24.+q
1. Introduction The observation of ecient and visible photoluminescence (PL) from the Si nanostructures, such as porous Si [1] and Si nanocrystals [2], has attracted much interest in the origin of the PL from the viewpoint of fundamental physics and the potential application to optical devices. Various models [3±6] were proposed to explain the PL origin, but it is still under debate. On the other hand, it is widely accepted that the absorption properties are governed by the crystalline silicon core, because the observed optical band gap energy shows a blue shift with decreasing size [7] and the imaginary part of the dielectric constant e2 maintains some of the original properties of bulk Si [8]. Theoretical studies of the
band gap energy [9] and imaginary part of the dielectric constant [10] are consistent with experimental results. For a further understanding of the absorption process, we investigate where, in the crystalline Si core, the optical absorption occurs. For the purpose of this study, we calculate e2 and the contribution to e2 of the inner 25 Si atoms for a 9 9 Si quantum wire, as a model of Si nanostructure, using the non-orthogonal tight-binding method. As far as we know, there is no study which investigates where, in the crystalline Si core, the absorption process occurs. It is important to understand the optical absorption process for further discussion and understanding of the PL origin.
2. Method * Corresponding author. Tel.: +81-45 563 1141; fax: +81-45 566 1672. E-mail address: [email protected] (K. Nishio).
The model we investigate is a 9 9 Si quantum wire cut from the perfect diamond structure of bulk
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 8 4 9 - 3
706
K. Nishio et al. / Journal of Non-Crystalline Solids 293±295 (2001) 705±708
Si. The atomic con®guration of this model is shown in Fig. 1. We assign periodic boundary condition for the [0 0 1]-direction. In order to eliminate the surface states and provide a smooth termination of the orbitals, the surface dangling bonds are terminated by terminators. The terminator denoted by T has an s orbital which plays the role of only removing surface states from the fundamental band gap [11]. It is a model for a wider band gap insulator enclosing the Si nanostructure [14]. We think that this model is valid for studying the optical absorption process. In the tight-binding method, the Bloch function W lk r is expanded by the atomic orbitals /n r of angular momentum n as follows:
W lk r 1 X X lk ik Rd i Ain e /n r p N in R
R d i ; 1
where l is the band index, k is the wave vector, N is the number of atoms in the Si nanostructure, i is lk the atomic number in the unit cell, Ain is the expansion coecient, d i is the position vector of atom i, and R is the lattice vector. The energy ei lk genvalue El k and eigenvector Ain are determined by the secular equation, X k lk X k lk Him;jn Ajn El k Sim;jn Ajn 2 jn
jn
with the orthonormal condition X X lk k mk Aim Sim;jn Ajn dl;m : im
3
jn
k
The Hamiltonian matrix Him;jn and the overlap k matrix Sim;jn are de®ned as Z X k e ik Rd i d j /m r R d i d j Him;jn R
H /n r dr; k
Sim;jn
X
e
ik Rd i d j
R
Z
4 /m r
R d i
/n r dr;
d j 5
k
k
respectively. In order to build up Him;jn and Sim;jn , we use the empirical parameters proposed by Mattheiss [15] and set the parameters between Si and T to take the dangling bond away from the fundamental band gap [12]. The imaginary part of the dielectric constant e2 and its contribution from the Si atom located at site i, which we denote as e2i , are de®ned as e2 Fig. 1. The atomic con®guration of a 9 9 Si quantum wire unit cell. Each circle represents a Si atom and ellipse represents the terminator denoted by T in the text. The size of the circle represents the component of the [0 0 1]-direction; 3=4a; 2=4a; 1=4a and 0 where a is the lattice constant of diamond bulk Si. The Si atoms represented by partially ®lled circles (25 Si atoms) and denoted by capital letters (A, B and C) are discussed in this study.
4p2 e2 X 2 2jhWc;k je pjWv;k ij dEc k m2 x2 v;c;k Ev k
e2i
hx;
6
4p2 e2 X 2 2jhWc;k je pjWv;k ij m2 x2 v;c;k Wi
vk
dEc k
Ev k
hx;
7
K. Nishio et al. / Journal of Non-Crystalline Solids 293±295 (2001) 705±708
707
respectively, where e is the polarization vector, p is lk the momentum operator and Wi is a weighting function de®ned as X X 1 h lk k lk lk A Wi S A 2 im im;jn jn m jn i lk k lk Ajn Sjn;im Aim X X h lk k lk i R Aim Sim;jn Ajn : 8 m
jn
2
The transition probability 2p= hjhWc;k je pjWv;k ij is evaluated in the two manners in which (a) we regard it as constant and (b) we calculate it using the parameters [16] used in [12]. We calculate e2 and e2i polarized in [1 1 0]-direction, which is most in¯uenced by the quantum con®nement. 3. Result Figs. 2(a) and (b) show the imaginary part of the dielectric constant e2 ([1 1 0]-polarization) for the 9 9 Si quantum wire and the contribution to e2 of the 25 Si atoms shown in Fig. 1. Fig. 3 shows the contribution to the imaginary part of the dielectric constant e2i ([1 1 0]-polarization) for the three Si atoms denoted by capital letters (A, B and C) in Fig. 1, for the 9 9 Si quantum wire.
Fig. 2. The imaginary part of the dielectric constant e2 ([1 1 0]polarization) for the 9 9 Si quantum wire (solid line) and its contribution from the 25 Si atoms is shown in Fig. 1 (dashed line). The transition probability is evaluated in the manner in which (a) we regard it as constant and (b) we calculate it using the parameters [16] used in [12].
4. Discussion We have veri®ed, previous to this study, that the onset of the imaginary part of the dielectric constant e2 for bulk Si is 3.4 eV and the band structure of 9 9 Si quantum wire shows 2.0 eV direct band gap at C-point. Therefore we discuss properties of e2 in the energy range between 2.0 and 3.4 eV which re¯ects the intrinsic properties of Si nanostructures. Fig. 2 shows that a large part of the imaginary part of the dielectric constant e2 below 3.4 eV is contributed by the inner 25 Si atoms. This result indicates that the optical absorption below 3.4 eV tends to occur at the inner 25 Si atoms. For a further understanding of the optical absorption process, we have calculated the contribution to e2 for the three Si atoms denoted by
Fig. 3. The contribution to the imaginary part of the dielectric constant e2i ([1 1 0]-polarization) for three dierent Si atoms A (solid line), B (dashed line) and C (dash±dot line) is shown in Fig. 1, for the 9 9 Si quantum wire. The transition probability is evaluated using the parameters [16] used in [12].
708
K. Nishio et al. / Journal of Non-Crystalline Solids 293±295 (2001) 705±708
capital letters (A, B and C) in Fig. 1. From the result shown in Fig. 3, we ®nd that the optical absorption below 3.4 eV increases, as the site of the Si atom changes from the surface (C) to the center (A). This is because the surface boundary condition causes wave functions of the levels near the valence band maximum to be localized in the central core of Si atoms. It is a property in which the 9 9 Si quantum wire diers from bulk Si; the optical absorption depends on the site of Si atom we examine. It is worth discussing the dierence between Figs. 2(a) and (b). The imaginary part of the dielectric constant near the band gap (2.0 eV) in Fig. 2(a) increases as the hx increases, although that in Fig. 2(b) is small. The small e2 in Fig. 2(b) is attributed to a small transition probability near the band gap. This corresponds to the experimental results [8] and is similar to those reported by other workers [10,13]. The small transition probability indicates that the Brillouin zone folding makes it possible for the direct transition to take place at low energy, but nevertheless the k-selection rule of bulk Si remains. 5. Conclusion We have calculated the imaginary part of the dielectric constant e2 and the contribution to e2 for the three Si atoms, for a 9 9 Si Quantum wire, using the non-orthogonal tight-binding method. We have shown that a large part of e2 below 3.4 eV is contributed by the inner 25 Si atoms. We have also shown that the contribution to e2 below 3.4 eV increases, as the site of the Si atom changes from the surface Si atom to the center Si atom. From these results, we have clearly found for the ®rst time that the optical absorption below 3.4 eV tends to occur in the inner 25 Si atoms and that the frequency of the optical absorption depends on the site of Si atom we examine. This is because the surface boundary condition causes wave functions of the levels near the valence band maximum to be
localized in the central core of Si atoms. These are the properties in which the 9 9 Si quantum wire diers from bulk Si. Finally, we think that the ®ndings of this study, the positional dependence of optical absorption, true of all Si nanostructures. Acknowledgements The authors are grateful to Mr Okumura for valuable discussions. This work has been supported by `Research for the Future Project' on `Physics of Nanocrystalline Semiconductors and Their Application to New Functional Devices' at Ritsumei University (JSPS-RFTF96I00102). References [1] L.T. Canham, Appl. Phys. Lett. 57 (1990) 1046. [2] H. Takagi, H. Ogawa, Y. Yamazaki, A. Ishizaki, T. Nakagiri, Appl. Phys. Lett. 56 (1990) 2379. [3] P.D.J. Calcott, K.J. Nash, L.T. Canham, M.J. Kane, D. Brumhead, J. Phys. 5 (1993) L91. [4] F. Koch, V. Perova-Koch, D. Muschik, J. Lumin. 57 (1993) 271. [5] M.B. Robinson, A.C. Dillon, D.R. Haynes, S.M. George, Appl. Phys. Lett. 61 (1992) 1414. [6] Y. Kanemitsu, S. Okamoto, M. Otobe, S. Oda, Phys. Rev. B. 55 (1997) 7375. [7] Y. Kanemitsu, H. Uto, Y. Masumoto, Phys. Rev. B 48 (1993) 2827. [8] N. Koshida, H. Koyama, Y. Suda, Y. Yamamoto, M. Araki, T. Saito, K. Sato, N. Sata, S. Shin, Appl. Phys. Lett. 63 (1993) 2774. [9] D.J. Lockwood, Solid. State. Commun. 92 (1994) 101. [10] M. Cruz, M.R. Beltran, C. Wang, J. Taguena-Martinez, Mater. Res. Soc. Symp. Proc. (USA) 452 (1997) 69. [11] The terminator T has the same role as the hydrogen atom in Ref. [12,13]. In order to distinguish a real hydrogen atom and the hydrogen atom introduced to terminate the surface dangling bond [12,13], we call the latter terminator. [12] J.P. Proot, C. Delerue, G. Allan, Appl. Phys. Lett. 61 (1992) 1948. [13] G.D. Sanders, Y.-C. Chang, Phys. Rev. B 45 (1992) 9202. [14] B. Delley, E.F. Steigmeier, Appl. Phys. Lett. 67 (1995) 2370. [15] L.F. Mattheiss, J.R. Patel, Phys. Rev. B 23 (1981) 5384. [16] J. Petit, G. Allan, M. Lannoo, Phys. Rev. B 33 (1986) 8595.