- Email: [email protected]
Instability of one-dimensional dangling-bond wires on H-passivated C(001), Si(001), and Ge(001) surfaces
Surface Science 605 (2011) L13–L15
Contents lists available at ScienceDirect
Surface Science j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s u s c
Instability of one-dimensional dangling-bond wires on H-passivated C(001), Si(001), and Ge(001) surfaces Jun-Ho Lee, Jun-Hyung Cho ⁎ Department of Physics and Research Institute for Natural Sciences, Hanyang University, 17 Haengdang-Dong, Seongdong-Ku, Seoul 133-791, Republic of Korea
a r t i c l e
i n f o
Article history: Received 20 November 2010 Accepted 6 January 2011 Available online 14 January 2011 Keywords: Density functional calculations Surface electronic phenomena
a b s t r a c t We investigate the instability of one-dimensional dangling-bond (DB) wires fabricated on the H-terminated C (001), Si(001), and Ge(001) surfaces by using density-functional theory calculations. The three DB wires are found to show drastically different couplings between charge, spin, and lattice degrees of freedom, resulting in an insulating ground state. The C DB wire has an antiferromagnetic spin coupling between unpaired DB electrons, caused by strong electron–electron interactions, whereas the Ge DB wire has a strong charge-lattice coupling, yielding a Peierls-like lattice distortion. For the Si DB wire, the antiferromagnetic spin ordering and the Peierls instability are highly competing with each other. The physical origin of such disparate features in the three DB wires can be traced to the different degree of localization of 2p, 3p, and 4p DB orbitals. © 2011 Elsevier B.V. All rights reserved.
Scanning tunneling microscope (STM) has been a powerful tool not only for investigating the physical, chemical, and electronic properties of surfaces, but also to create atomic-scale structures that play an important role in the development of a future nanotechnology.1 Atom manipulation can be achieved by a precise control of interactions between the STM tip and the adsorbed atom on surfaces, thereby fabricating various nanostructures such as quantum dots and quantum wires.2 It is of crucial importance to understand the underlying physics of such quantized low-dimensional systems for the application to novel nanoelectronic devices. Especially, the confinement of electrons in onedimensional (1D) systems provides many exotic physical phenomena such as Peierls instability,3 Jahn–Teller distortion,4 spin polarization5 or the formation of non-Fermi-liquid ground states.6 Recently, a variant of hydrogen resist STM nanolithography technique, termed feedback controlled lithography,7,8,9 was used to 1 C. Bai, Scanning tunneling microscopy and its applications (Springer Verlag, New York, 2000), and references therein. 2 P. Rodgers, Nanoscience and Technology (World Scientific, Singapore, 2009). 3 R. E. Peierls, Quantum Theory of Solids, Oxford Classics Series (Oxford University Press, Oxford, 2001). 4 H. A. Jahn and E. Teller, Proc. R. Soc. London, Ser. A 161, 220 (1937). 5 R. Arita, Y. Suwa, K. Kuroki, and H. Aoki, Phys. Rev. Lett. 88, 127202 (2002). 6 S. Tomonaga, Prog. Theor. Phys. 5, 544 (1950); J. M. Luttinger, J. Math. Phys. 4, 1154 (1963). 7 J. W. Lyding, T.-C. Shen, J. S. Hubacek, J. R. Tucker, G. C. Abeln, Appl. Phys. Lett. 64, 2010 (1994). 8 T.-C. Shen, C. Wang, G. C. Abeln, J. R. Tucker, J. W. Lyding, Ph. Avouris and R. E. Walkup, Science 268, 1590 (1995). 9 M. C. Hersam, N. P. Guisinger and J. W. Lyding, J. Vac. Sci. Technol. A18, 1349 (2000).
⁎ Corresponding author. Tel.: +82 2 2220 0915; fax: +82 2 2295 6868. E-mail address: [email protected] (J.-H. Cho). 0039-6028/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2011.01.011
generate 1D arrays of individual dangling bonds (DBs) by the selective removal of H atoms from an H-passivated Si(001) surface along one side of an Si dimer row.10 This technique can be extended to fabricate the same 1D arrays on the H-passivated C(001) and Ge(001) surfaces. Such fabricated DB wires have a single DB per atom, offering quasi-1D metallic systems with a half-filled DB state, crossing the Fermi level. As stated by Peierls in the 1950s,3 those quasi-1D metals may be unstable against metal-insulator transition, where electrons and holes near the Fermi level often couple strongly with a lattice vibration, thereby resulting in formations of a charge density wave (CDW) and an electron band gap at the Fermi level.11 However, it was recently proposed12,13 that the Si DB wire exhibits the preference of the antiferromagnetic (AF) ordering rather than the Peierls instability. In this work, using first-principles density-functional calculations, we demonstrate that the three DB wires fabricated on the Hpassivated C(001), Si(001), and Ge(001) surfaces undergo a metalinsulator transition, driven by drastically different couplings between charge, spin, and lattice degrees of freedom. We find that in the C DB wire [see Fig. 1(a)], the strong electron–electron interactions give rise to the preference of the AF spin coupling between unpaired DB electrons, while in the Si DB wire, the stability of the AF spin ordering is weakened but still favored over the Peierls model. However, the Ge DB wire is found to show a significant preference for the Peierls instability [see Fig. 1(b)] that exhibits alternating up and down
10 T. Hitosugi, S. Heike, T. Onogi, T. Hashizume, S. Watanabe, Z.-Q. Li, K. Ohno, Y. Kawazoe, T. Hasegawa, and K. Kitazawa, Phys. Rev. Lett. 82, 4034 (1999). 11 S. Watanabe, Y. A. Ono, T. Hashizume, and Y. Wada, Phys. Rev. B 54, R17308 (1996). 12 C. F. Bird and D. R. Bowler, Surf. Sci. 531, L351 (2003). 13 J. Y. Lee, J.-H. Cho and Z. Zhang, Phys. Rev. B 80, 155329 (2009).
SU RF A LE CE TT S ER CIE S NC E
a
J.-H. Lee, J.-H. Cho / Surface Science 605 (2011) L13–L15
b
Fig. 1. Optimized structure of the DB wires on H-terminated (a) C(001) and (b) Ge(001) surfaces. For distinction, the atoms composing the DB wires are drawn with dark colors. The side view of the Ge DB wire is also given in the inset of (b).
vertical displacements of the Ge atoms composing the DB wire, accompanying a charge transfer from the down to the up Ge atom. Our analysis shows that the on-site electron−electron interactions decrease in strength as the sequence of C N Si N Ge DB wires due to the different degree of localization of 2p, 3p, and 4p DB orbitals, while the strength of electron-lattice coupling decreases as the reversed sequence. The total-energy and force calculations were performed using non spin-polarized or spin-polarized density functional theory14 within the generalized-gradient approximation.15 The C (Si, Ge, and H) atom is described by ultrasoft16 (norm-conserving17) pseudopotentials. The surface was modeled by a periodic slab geometry. The slab for the C DB wire consists of twelve C atomic layers, where the upper and bottom sides are passivated by one H atomic layer to have an inversion symmetry. However, the slab for the Si or Ge DB wire contains six atomic layers plus one passivating H atomic layer, where the bottom side is terminated by two H atoms per Si or Ge atom. We simulated the DB wires of infinite length by using a 4 × 1 or 4 × 2 unit cell that includes two dimer rows perpendicular to the wire direction. The electronic wave functions were expanded in a plane-wave basis set with a cutoff of 25 Ry. The k-space integration was done with sixteen (eight) points in the surface Brillouin zone of the 4 × 1 (4 × 2) unit cell. The present calculational scheme has been successfully applied for the adsorption of various molecules on the C(001), Si(001), and Ge(001)surfaces.18 We first determine the atomic structures of the C, Si, and Ge DB wires within the nonmagnetic (NM), ferromagnetic (FM), and AF configurations. Here, the NM configuration is based on the Peierls model where the surface atoms composing the DB wire are displaced up and down alternatively.11 To obtain the energy gains caused by Peierls distortion and magnetic orderings, we calculate the energies of the NM, FM, and AF configurations relative to the non-distorted NM configuration (designated as NM0) where each atom composing the DB wire is constrained to have an identical height. The results for the three DB wires are plotted in Fig. 2. We find that for the C and Si DB
14 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964); W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 15 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 16 D. Vanderbilt, Phys. Rev. B 41, 7892 (1990); K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt, Phys. Rev. B 47, 10142 (1993). 17 N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 (1991). 18 J.-H. Cho, L. Kleinman, K.-J. Jin and K.S. Kim, Phys. Rev. B (2002) 66, 113306, J.-H. Cho and L. Kleinman, Phys. Rev. B (2003) 68, 195413, J.-H. Choi and J.-H. Cho, Phys. Rev. Lett. (2007) 98, 246101; 102, 166102 (2009).
wires, the AF configuration is more stable than the NM and FM ones, whereas for the Ge DB wire, the NM configuration with a Peierls distortion is the most stable. Note that the AF configuration has an undistorted structure [see Fig. 1(a)], indicating the absence of spinPeierls instability. As shown in Fig. 2, the energy gain arising from a Peierls distortion is 46 and 116 meV/DB for the Si and Ge DB wires, respectively. However, the C DB wire does not show a Peierls distortion, indicating that the electronic energy gain obtained by the charge redistribution cannot prevail over the larger elastic energy cost of a lattice distortion. On the other hand, the energy gain caused by the AF spin ordering is 132, 53, and 35 meV/DB for the C, Si, and Ge DB wires, respectively. We note that the FM configuration is less stable than the AF one by 31, 58, and 70 meV/DB for the C, Si, and Ge DB wires, respectively (see Fig. 2). Thus, we can say that the C DB wire significantly favors the AF spin coupling between DB electrons, while the Ge DB wire exhibits the Peierls instability driven by a strong electron-lattice coupling. However, for the Si DB wire, the AF spin ordering is only 7 meV more favored over the Peierls instability, indicating that the electron– electron and electron–lattice interactions are highly competing with each other. Fig. 3 shows the electronic band structures for the NM (or NM0) and AF configurations of the C, Si, and Ge DB wires. In the NM0 configuration, a surface band due to the DB electrons is found to cross the Fermi level at an almost midpoint of the symmetry line ΓJ, yielding a half-filled band [see the S state in Fig. 3(a)]. This half-filled DB state has band widths of 1.04, 0.75, and 0.63 eV for the C, Si, and Ge DB wires, respectively. It is noticeable that the AF spin ordering splits the half-filled DB band into two subbands (S1 and S2) where the indirect band gap (Egap) is as large as 1.37, 0.55, and 0.43 eV for the C, Si, and Ge DB wires, respectively. On the other hand, the Peierls instability opens Egap = 0.32 and 0.62 eV for the Si and Ge DB wires, respectively. Thus, for the C and Si DB wires, the electronic energy gain due to the AF energy gap is greater than that of the Peierls gap (zero in the C DB wire), whereas for the Ge DB wire, the former is smaller than the latter. These different features of the electronic band structures between the AF and NM configurations give rise to an AF ground state for the C and Si DB wires and a NM ground state for the Ge DB wire. It is noteworthy that the different degree of localization of 2p, 3p, and 4p DB orbitals should influence the strength of electron–electron and electron–lattice interactions in the three DB wires. In the case of the C DB wire, the highly localized 2p orbital enhances more effectively the electron–electron interactions, giving rise to the preference of the AF spin ordering. However, in the Ge DB wire, the rather delocalized 4p orbital is likely to facilitate the electron-lattice coupling, producing the Peierls instability. The different degree of localization of DB orbitals can be seen from their charge and spin
50 NM AF FM 0
ΔE(meV/DB)
L14
-50
-100
-150
C
Si
Ge
Fig. 2. Calculated energies (per dangling bond) of the NM, FM, and AF configurations relative to the NM0 configuration for the C, Si, and Ge DB wires.
a AF
NM0
Energy (eV)
1
S2
S
0
S1
-1 Γ
J
Γ
J
b NM
AF
Energy (eV)
1
S2
S2
S1
S1
0
-1 Γ
J
Γ
J
c AF
NM
Energy (eV)
1
S2
S2
0
S1
S1 -1 Γ
J
Γ
J
Fig. 3. Calculated band structures for the (a) C, (b) Si, and (c) Ge DB wires within the NM and AF configurations. The energy zero represents the Fermi level. The direction of ΓJ line is along the DB wire. The solid lines represent the subbands due to the DB electrons. In the AF configuration, the subbands of the majority and minority spins are equal to each other. The charge or spin characters of subbands at the J point are also given. The plots are drawn in the vertical plane containing the maximum magnitude of charge or spin density. In the AF configuration, the solid (dotted) line represents the majority (minority) spin density. The first contour line and the line spacings are the same as 0.005 electrons/bohr3.
CE N IE SC S R CE E A TT RF LE SU
J.-H. Lee, J.-H. Cho / Surface Science 605 (2011) L13–L15
L15
To examine how the on-site electron–electron interactions vary among the three DB wires, we evaluate the Hubbard correlation energy U by using the constrained DFT calculations.19 Here, we simulate one minority-electron transfer from the S1 state to the S2 state in the AF configuration and calculate the change in the total energy. We obtain U = 3.45, 1.71, and 1.36 eV for the C, Si, and Ge DB wires, respectively. This sequence of U corresponds to that of the localization degree of 2p N 3p N 4p DB orbitals. We note that the values of U are significantly larger than the electron hopping parameter t = 1.04, 0.75, and 0.63 eV, which are estimated from the bandwidth of the corresponding NM0 configuration. Thus, it is most likely that the relatively larger ratio of U/t = 3.32 (2.28) in the C (Si) DB wire gives rise to the preference of the AF spin ordering rather than a CDW formation. The physics of the C, Si, and Ge DB wires could be described by the so-called Peierls–Hubbard (PH)20 Hamiltonian. In the U → ∞ limit, the PH model can be converted to the Heisenberg spin-Peierls21,22 model, which becomes the Heisenberg Hamiltonian if the spin-Peierls effects were ignored. Noting that the C DB wire has a larger value of U without the spin-Peierls effects, its AF spin ordering may be well described by the Heisenberg Hamiltonian. On the other hand, if the effect of U were ignored, the PH model becomes the Peierls model which describes properly the Peierls-like distortion in the Ge DB wire. The detailed model analysis will be done in our future work. In summary, we demonstrated drastically different features of electron–electron and electron–lattice interactions in 1D DB wires fabricated on the H-terminated C(001), Si(001), and Ge(001) surfaces, caused by the different degree of localization of 2p, 3p, and 4p DB orbitals. We found that the C DB wire prefers the AF spin ordering due to strong electron–electron interactions, the Ge DB wire shows a Peierls-like lattice distortion with a strong charge-lattice coupling, whereas the Si DB wire shows a high competition between the electron–electron and electron–lattice interactions. These disparate electronic properties of the C, Si, and Ge DB wires might be a promising perspective for the design of nanoelectronic devices for storages and processing of quantum information on the three typical semiconductor surfaces.
Acknowledgement This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (KRF-3142008-1-C00095 and KRF-2009-0073123).
characters in Fig. 3. It is notable that the NM configuration of the Si and Ge DB wires shows a charge transfer from the down to the up atoms, representing a CDW formation. On the other hand, the AF configuration of the three DB wires involves an opposite spin orientation between adjacent dangling bonds, as shown in Fig. 3.
19 V. I. Anisimov, I. V. Solovyev, M. A. Korotin, M. T. Czyżyk and G. A. Sawatzky, Phys. Rev. B 48, 16929 (1993). 20 D. K. Campbell, J. T. Gammel, and E. Y. Loh, Jr., Phys. Rev. B 42, 475 (1990). 21 E. Pytte, Phys. Rev. B 10, 4637 (1974). 22 W. Barford and R. J. Bursill, Phys. Rev. Lett. 95, 137207 (2005).
Contents lists available at ScienceDirect
Surface Science j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s u s c
Instability of one-dimensional dangling-bond wires on H-passivated C(001), Si(001), and Ge(001) surfaces Jun-Ho Lee, Jun-Hyung Cho ⁎ Department of Physics and Research Institute for Natural Sciences, Hanyang University, 17 Haengdang-Dong, Seongdong-Ku, Seoul 133-791, Republic of Korea
a r t i c l e
i n f o
Article history: Received 20 November 2010 Accepted 6 January 2011 Available online 14 January 2011 Keywords: Density functional calculations Surface electronic phenomena
a b s t r a c t We investigate the instability of one-dimensional dangling-bond (DB) wires fabricated on the H-terminated C (001), Si(001), and Ge(001) surfaces by using density-functional theory calculations. The three DB wires are found to show drastically different couplings between charge, spin, and lattice degrees of freedom, resulting in an insulating ground state. The C DB wire has an antiferromagnetic spin coupling between unpaired DB electrons, caused by strong electron–electron interactions, whereas the Ge DB wire has a strong charge-lattice coupling, yielding a Peierls-like lattice distortion. For the Si DB wire, the antiferromagnetic spin ordering and the Peierls instability are highly competing with each other. The physical origin of such disparate features in the three DB wires can be traced to the different degree of localization of 2p, 3p, and 4p DB orbitals. © 2011 Elsevier B.V. All rights reserved.
Scanning tunneling microscope (STM) has been a powerful tool not only for investigating the physical, chemical, and electronic properties of surfaces, but also to create atomic-scale structures that play an important role in the development of a future nanotechnology.1 Atom manipulation can be achieved by a precise control of interactions between the STM tip and the adsorbed atom on surfaces, thereby fabricating various nanostructures such as quantum dots and quantum wires.2 It is of crucial importance to understand the underlying physics of such quantized low-dimensional systems for the application to novel nanoelectronic devices. Especially, the confinement of electrons in onedimensional (1D) systems provides many exotic physical phenomena such as Peierls instability,3 Jahn–Teller distortion,4 spin polarization5 or the formation of non-Fermi-liquid ground states.6 Recently, a variant of hydrogen resist STM nanolithography technique, termed feedback controlled lithography,7,8,9 was used to 1 C. Bai, Scanning tunneling microscopy and its applications (Springer Verlag, New York, 2000), and references therein. 2 P. Rodgers, Nanoscience and Technology (World Scientific, Singapore, 2009). 3 R. E. Peierls, Quantum Theory of Solids, Oxford Classics Series (Oxford University Press, Oxford, 2001). 4 H. A. Jahn and E. Teller, Proc. R. Soc. London, Ser. A 161, 220 (1937). 5 R. Arita, Y. Suwa, K. Kuroki, and H. Aoki, Phys. Rev. Lett. 88, 127202 (2002). 6 S. Tomonaga, Prog. Theor. Phys. 5, 544 (1950); J. M. Luttinger, J. Math. Phys. 4, 1154 (1963). 7 J. W. Lyding, T.-C. Shen, J. S. Hubacek, J. R. Tucker, G. C. Abeln, Appl. Phys. Lett. 64, 2010 (1994). 8 T.-C. Shen, C. Wang, G. C. Abeln, J. R. Tucker, J. W. Lyding, Ph. Avouris and R. E. Walkup, Science 268, 1590 (1995). 9 M. C. Hersam, N. P. Guisinger and J. W. Lyding, J. Vac. Sci. Technol. A18, 1349 (2000).
⁎ Corresponding author. Tel.: +82 2 2220 0915; fax: +82 2 2295 6868. E-mail address: [email protected] (J.-H. Cho). 0039-6028/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2011.01.011
generate 1D arrays of individual dangling bonds (DBs) by the selective removal of H atoms from an H-passivated Si(001) surface along one side of an Si dimer row.10 This technique can be extended to fabricate the same 1D arrays on the H-passivated C(001) and Ge(001) surfaces. Such fabricated DB wires have a single DB per atom, offering quasi-1D metallic systems with a half-filled DB state, crossing the Fermi level. As stated by Peierls in the 1950s,3 those quasi-1D metals may be unstable against metal-insulator transition, where electrons and holes near the Fermi level often couple strongly with a lattice vibration, thereby resulting in formations of a charge density wave (CDW) and an electron band gap at the Fermi level.11 However, it was recently proposed12,13 that the Si DB wire exhibits the preference of the antiferromagnetic (AF) ordering rather than the Peierls instability. In this work, using first-principles density-functional calculations, we demonstrate that the three DB wires fabricated on the Hpassivated C(001), Si(001), and Ge(001) surfaces undergo a metalinsulator transition, driven by drastically different couplings between charge, spin, and lattice degrees of freedom. We find that in the C DB wire [see Fig. 1(a)], the strong electron–electron interactions give rise to the preference of the AF spin coupling between unpaired DB electrons, while in the Si DB wire, the stability of the AF spin ordering is weakened but still favored over the Peierls model. However, the Ge DB wire is found to show a significant preference for the Peierls instability [see Fig. 1(b)] that exhibits alternating up and down
10 T. Hitosugi, S. Heike, T. Onogi, T. Hashizume, S. Watanabe, Z.-Q. Li, K. Ohno, Y. Kawazoe, T. Hasegawa, and K. Kitazawa, Phys. Rev. Lett. 82, 4034 (1999). 11 S. Watanabe, Y. A. Ono, T. Hashizume, and Y. Wada, Phys. Rev. B 54, R17308 (1996). 12 C. F. Bird and D. R. Bowler, Surf. Sci. 531, L351 (2003). 13 J. Y. Lee, J.-H. Cho and Z. Zhang, Phys. Rev. B 80, 155329 (2009).
SU RF A LE CE TT S ER CIE S NC E
a
J.-H. Lee, J.-H. Cho / Surface Science 605 (2011) L13–L15
b
Fig. 1. Optimized structure of the DB wires on H-terminated (a) C(001) and (b) Ge(001) surfaces. For distinction, the atoms composing the DB wires are drawn with dark colors. The side view of the Ge DB wire is also given in the inset of (b).
vertical displacements of the Ge atoms composing the DB wire, accompanying a charge transfer from the down to the up Ge atom. Our analysis shows that the on-site electron−electron interactions decrease in strength as the sequence of C N Si N Ge DB wires due to the different degree of localization of 2p, 3p, and 4p DB orbitals, while the strength of electron-lattice coupling decreases as the reversed sequence. The total-energy and force calculations were performed using non spin-polarized or spin-polarized density functional theory14 within the generalized-gradient approximation.15 The C (Si, Ge, and H) atom is described by ultrasoft16 (norm-conserving17) pseudopotentials. The surface was modeled by a periodic slab geometry. The slab for the C DB wire consists of twelve C atomic layers, where the upper and bottom sides are passivated by one H atomic layer to have an inversion symmetry. However, the slab for the Si or Ge DB wire contains six atomic layers plus one passivating H atomic layer, where the bottom side is terminated by two H atoms per Si or Ge atom. We simulated the DB wires of infinite length by using a 4 × 1 or 4 × 2 unit cell that includes two dimer rows perpendicular to the wire direction. The electronic wave functions were expanded in a plane-wave basis set with a cutoff of 25 Ry. The k-space integration was done with sixteen (eight) points in the surface Brillouin zone of the 4 × 1 (4 × 2) unit cell. The present calculational scheme has been successfully applied for the adsorption of various molecules on the C(001), Si(001), and Ge(001)surfaces.18 We first determine the atomic structures of the C, Si, and Ge DB wires within the nonmagnetic (NM), ferromagnetic (FM), and AF configurations. Here, the NM configuration is based on the Peierls model where the surface atoms composing the DB wire are displaced up and down alternatively.11 To obtain the energy gains caused by Peierls distortion and magnetic orderings, we calculate the energies of the NM, FM, and AF configurations relative to the non-distorted NM configuration (designated as NM0) where each atom composing the DB wire is constrained to have an identical height. The results for the three DB wires are plotted in Fig. 2. We find that for the C and Si DB
14 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964); W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 15 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 16 D. Vanderbilt, Phys. Rev. B 41, 7892 (1990); K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt, Phys. Rev. B 47, 10142 (1993). 17 N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 (1991). 18 J.-H. Cho, L. Kleinman, K.-J. Jin and K.S. Kim, Phys. Rev. B (2002) 66, 113306, J.-H. Cho and L. Kleinman, Phys. Rev. B (2003) 68, 195413, J.-H. Choi and J.-H. Cho, Phys. Rev. Lett. (2007) 98, 246101; 102, 166102 (2009).
wires, the AF configuration is more stable than the NM and FM ones, whereas for the Ge DB wire, the NM configuration with a Peierls distortion is the most stable. Note that the AF configuration has an undistorted structure [see Fig. 1(a)], indicating the absence of spinPeierls instability. As shown in Fig. 2, the energy gain arising from a Peierls distortion is 46 and 116 meV/DB for the Si and Ge DB wires, respectively. However, the C DB wire does not show a Peierls distortion, indicating that the electronic energy gain obtained by the charge redistribution cannot prevail over the larger elastic energy cost of a lattice distortion. On the other hand, the energy gain caused by the AF spin ordering is 132, 53, and 35 meV/DB for the C, Si, and Ge DB wires, respectively. We note that the FM configuration is less stable than the AF one by 31, 58, and 70 meV/DB for the C, Si, and Ge DB wires, respectively (see Fig. 2). Thus, we can say that the C DB wire significantly favors the AF spin coupling between DB electrons, while the Ge DB wire exhibits the Peierls instability driven by a strong electron-lattice coupling. However, for the Si DB wire, the AF spin ordering is only 7 meV more favored over the Peierls instability, indicating that the electron– electron and electron–lattice interactions are highly competing with each other. Fig. 3 shows the electronic band structures for the NM (or NM0) and AF configurations of the C, Si, and Ge DB wires. In the NM0 configuration, a surface band due to the DB electrons is found to cross the Fermi level at an almost midpoint of the symmetry line ΓJ, yielding a half-filled band [see the S state in Fig. 3(a)]. This half-filled DB state has band widths of 1.04, 0.75, and 0.63 eV for the C, Si, and Ge DB wires, respectively. It is noticeable that the AF spin ordering splits the half-filled DB band into two subbands (S1 and S2) where the indirect band gap (Egap) is as large as 1.37, 0.55, and 0.43 eV for the C, Si, and Ge DB wires, respectively. On the other hand, the Peierls instability opens Egap = 0.32 and 0.62 eV for the Si and Ge DB wires, respectively. Thus, for the C and Si DB wires, the electronic energy gain due to the AF energy gap is greater than that of the Peierls gap (zero in the C DB wire), whereas for the Ge DB wire, the former is smaller than the latter. These different features of the electronic band structures between the AF and NM configurations give rise to an AF ground state for the C and Si DB wires and a NM ground state for the Ge DB wire. It is noteworthy that the different degree of localization of 2p, 3p, and 4p DB orbitals should influence the strength of electron–electron and electron–lattice interactions in the three DB wires. In the case of the C DB wire, the highly localized 2p orbital enhances more effectively the electron–electron interactions, giving rise to the preference of the AF spin ordering. However, in the Ge DB wire, the rather delocalized 4p orbital is likely to facilitate the electron-lattice coupling, producing the Peierls instability. The different degree of localization of DB orbitals can be seen from their charge and spin
50 NM AF FM 0
ΔE(meV/DB)
L14
-50
-100
-150
C
Si
Ge
Fig. 2. Calculated energies (per dangling bond) of the NM, FM, and AF configurations relative to the NM0 configuration for the C, Si, and Ge DB wires.
a AF
NM0
Energy (eV)
1
S2
S
0
S1
-1 Γ
J
Γ
J
b NM
AF
Energy (eV)
1
S2
S2
S1
S1
0
-1 Γ
J
Γ
J
c AF
NM
Energy (eV)
1
S2
S2
0
S1
S1 -1 Γ
J
Γ
J
Fig. 3. Calculated band structures for the (a) C, (b) Si, and (c) Ge DB wires within the NM and AF configurations. The energy zero represents the Fermi level. The direction of ΓJ line is along the DB wire. The solid lines represent the subbands due to the DB electrons. In the AF configuration, the subbands of the majority and minority spins are equal to each other. The charge or spin characters of subbands at the J point are also given. The plots are drawn in the vertical plane containing the maximum magnitude of charge or spin density. In the AF configuration, the solid (dotted) line represents the majority (minority) spin density. The first contour line and the line spacings are the same as 0.005 electrons/bohr3.
CE N IE SC S R CE E A TT RF LE SU
J.-H. Lee, J.-H. Cho / Surface Science 605 (2011) L13–L15
L15
To examine how the on-site electron–electron interactions vary among the three DB wires, we evaluate the Hubbard correlation energy U by using the constrained DFT calculations.19 Here, we simulate one minority-electron transfer from the S1 state to the S2 state in the AF configuration and calculate the change in the total energy. We obtain U = 3.45, 1.71, and 1.36 eV for the C, Si, and Ge DB wires, respectively. This sequence of U corresponds to that of the localization degree of 2p N 3p N 4p DB orbitals. We note that the values of U are significantly larger than the electron hopping parameter t = 1.04, 0.75, and 0.63 eV, which are estimated from the bandwidth of the corresponding NM0 configuration. Thus, it is most likely that the relatively larger ratio of U/t = 3.32 (2.28) in the C (Si) DB wire gives rise to the preference of the AF spin ordering rather than a CDW formation. The physics of the C, Si, and Ge DB wires could be described by the so-called Peierls–Hubbard (PH)20 Hamiltonian. In the U → ∞ limit, the PH model can be converted to the Heisenberg spin-Peierls21,22 model, which becomes the Heisenberg Hamiltonian if the spin-Peierls effects were ignored. Noting that the C DB wire has a larger value of U without the spin-Peierls effects, its AF spin ordering may be well described by the Heisenberg Hamiltonian. On the other hand, if the effect of U were ignored, the PH model becomes the Peierls model which describes properly the Peierls-like distortion in the Ge DB wire. The detailed model analysis will be done in our future work. In summary, we demonstrated drastically different features of electron–electron and electron–lattice interactions in 1D DB wires fabricated on the H-terminated C(001), Si(001), and Ge(001) surfaces, caused by the different degree of localization of 2p, 3p, and 4p DB orbitals. We found that the C DB wire prefers the AF spin ordering due to strong electron–electron interactions, the Ge DB wire shows a Peierls-like lattice distortion with a strong charge-lattice coupling, whereas the Si DB wire shows a high competition between the electron–electron and electron–lattice interactions. These disparate electronic properties of the C, Si, and Ge DB wires might be a promising perspective for the design of nanoelectronic devices for storages and processing of quantum information on the three typical semiconductor surfaces.
Acknowledgement This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (KRF-3142008-1-C00095 and KRF-2009-0073123).
characters in Fig. 3. It is notable that the NM configuration of the Si and Ge DB wires shows a charge transfer from the down to the up atoms, representing a CDW formation. On the other hand, the AF configuration of the three DB wires involves an opposite spin orientation between adjacent dangling bonds, as shown in Fig. 3.
19 V. I. Anisimov, I. V. Solovyev, M. A. Korotin, M. T. Czyżyk and G. A. Sawatzky, Phys. Rev. B 48, 16929 (1993). 20 D. K. Campbell, J. T. Gammel, and E. Y. Loh, Jr., Phys. Rev. B 42, 475 (1990). 21 E. Pytte, Phys. Rev. B 10, 4637 (1974). 22 W. Barford and R. J. Bursill, Phys. Rev. Lett. 95, 137207 (2005).