Comment on “Quantum Confinement and Electronic Properties of Silicon Nanowires”
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Comment on ‘‘Quantum Confinement and Electronic Properties of Silicon Nanowires’’ In a recent Letter, Zhao et al. [1] investigated the elec- tronic properties of isolated silicon nanowires by means of ab initio calculations based on, and going beyond, density functional theory. One important conclusion of their work concerns the anisotropy of the dielectric response of these systems that is claimed to vanish for wires with diameters as small as 2 nm. We believe that this conclusion has only a very limited meaning, as the authors completely neglect depolarization effects, which dramatically change the an- isotropy. In the following, we will illustrate this fact, both by ab initio calculations and by using the effective medium theory (EMT). Zhao et al. calculated absorption spectra using Fermi’s golden rule in the independent-particle transition picture. This scheme leads to a macroscopic dielectric function in the form of the spatial average of the dielectric matrix, hence neglecting the effects due to the nonhomogeneity of space. This approximation is not sufficient to predict or even to interpret anisotropies in the spectra of nanostruc- tured materials [2]. In fact, in these systems it is important to take into account the microscopic components arising in the response to a perturbing macroscopic field. In particu- lar, the induced microscopic variations of the Hartree potential, the so-called depolarization or local-field effects (LFE), suppress absorption at lower energies when the polarization of the light is perpendicular to a surface. Although further contributions arise from quasiparticle and excitonic effects, the LFE are the key ingredients for the discussion of strong anisotropies in nanostructured systems [3]. In order to assess the importance of LFE for Si nano- wires, we performed response calculations [4] within the random phase approximation (RPA) for the largest wire considered in Ref. [1], the d 2:2 nm 110 wire, in the same supercell geometry as in Ref. [1], but using a larger interwire distance ( 2:5 nm). This is necessary as LFE (and the calculation of any excited states quantity, like also GW corrections) depend much more on the distance be- tween wires than a ground state calculation. By neglecting LFE we reproduce the spectra of Ref. [1] for both the component of the imaginary part of the dielectric function along the wire axis " k 2 and the one perpendicular to the wire axis " ? 2 (Fig. 1, solid and dotted lines). Our spectra are rigidly redshifted with respect to Ref. [1], as we do not include GW corrections (this does not influence the con- clusions). When LFE are included, no significant changes in " k 2 (dashed line) are observed. However, it turns out that " ? 2 (dash-dotted line) is extremely sensitive to LFE: the absorption is shifted to higher energies and the nanowires become transparent up to 6–7 eV. Hence, we prove that the anisotropic contribution to the spectra due to LFE does not disappear for d> 2 nm. This result can be better under- stood if we use the fact that, for large wires, the ab initio calculations tend to reproduce the classical limit given by the EMT [5]. The gray lines in Fig. 1 show the results of the EMT, using the RPA bulk silicon dielectric function. The agreement between the model and the ab initio calculations is extremely good for both polarization directions. This shows that the observed anisotropy can be explained al- most entirely by classical effects (that are contained in the ab initio calculations when LFE are included), and that one can use the EMT to estimate anisotropies for larger di- ameters. In view of our conclusions, the results of Zhao et al. imply that only the contribution to the anisotropy due to band structure vanishes for d> 2 nm. Their results do not allow one to estimate, even approximately, the anisot- ropy that might be actually measured. Computer time was granted by IDRIS (Project No. 544). This work has been supported by the EU NoE NANOQUANTA. S.B. acknowledges financial support by the Marie Curie Actions of the EU. Fabien Bruneval, Silvana Botti, and Lucia Reining Laboratoire des Solides Irradie ´s CNRS-CEA-E ´ cole Polytechnique F–91128 Palaiseau, France Received 6 July 2004; published 2 June 2005 DOI: 10.1103/PhysRevLett.94.219701 PACS numbers: 73.21.Hb, 73.22.Dj, 78.67.Lt [1] X. Zhao, C.M. Wei, L. Yang, and M.Y. Chou, Phys. Rev. Lett. 92, 236805 (2004). [2] See, e.g., A. G. Marinopoulos et al., Phys. Rev. Lett. 91, 046402 (2003); S. Botti et al., Phys. Rev. Lett. 89, 216803 (2002). [3] See, e.g., E. Chang et al., Phys. Rev. Lett. 92, 196401 (2004). [4] X. Gonze et al., Comput. Mater. Sci. 25, 478 (2002); http://www.abinit.org; http://theory.lsi.polytechnique.fr/ codes/. [5] S. Datta et al., Phys. Rev. B 48, 14936 (1993). 0 2 4 6 8 10 Energy [eV] 10 20 30 ε 2 ε || without LFE ε ⊥ without LFE ε || with LFE ε ⊥ with LFE ε || EMT ε ⊥ EMT FIG. 1. Imaginary part of the dielectric functions for 110 d 2:2 nm nanowires, calculated within the RPA. PRL 94, 219701 (2005) PHYSICAL REVIEW LETTERS week ending 3 JUNE 2005 0031-9007= 05=94(21)=219701(1)$23.00 219701-1 2005 The American Physical Society
Transcript of Comment on “Quantum Confinement and Electronic Properties of Silicon Nanowires”
0 2 4 6 8 10Energy [eV]
10
20
30
ε 2
ε|| without LFE
ε⊥ without LFEε|| with LFE
ε⊥ with LFEε|| EMT
ε⊥ EMT
FIG. 1. Imaginary part of the dielectric functions for �110� d �2:2 nm nanowires, calculated within the RPA.
PRL 94, 219701 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending3 JUNE 2005
Comment on ‘‘Quantum Confinement and ElectronicProperties of Silicon Nanowires’’
In a recent Letter, Zhao et al. [1] investigated the elec-tronic properties of isolated silicon nanowires by means ofab initio calculations based on, and going beyond, densityfunctional theory. One important conclusion of their workconcerns the anisotropy of the dielectric response of thesesystems that is claimed to vanish for wires with diametersas small as 2 nm. We believe that this conclusion has only avery limited meaning, as the authors completely neglectdepolarization effects, which dramatically change the an-isotropy. In the following, we will illustrate this fact, bothby ab initio calculations and by using the effective mediumtheory (EMT).
Zhao et al. calculated absorption spectra using Fermi’sgolden rule in the independent-particle transition picture.This scheme leads to a macroscopic dielectric function inthe form of the spatial average of the dielectric matrix,hence neglecting the effects due to the nonhomogeneity ofspace. This approximation is not sufficient to predict oreven to interpret anisotropies in the spectra of nanostruc-tured materials [2]. In fact, in these systems it is importantto take into account the microscopic components arising inthe response to a perturbing macroscopic field. In particu-lar, the induced microscopic variations of the Hartreepotential, the so-called depolarization or local-field effects(LFE), suppress absorption at lower energies when thepolarization of the light is perpendicular to a surface.Although further contributions arise from quasiparticleand excitonic effects, the LFE are the key ingredients forthe discussion of strong anisotropies in nanostructuredsystems [3].
In order to assess the importance of LFE for Si nano-wires, we performed response calculations [4] within therandom phase approximation (RPA) for the largest wireconsidered in Ref. [1], the d � 2:2 nm �110� wire, in thesame supercell geometry as in Ref. [1], but using a largerinterwire distance (�2:5 nm). This is necessary as LFE(and the calculation of any excited states quantity, like alsoGW corrections) depend much more on the distance be-tween wires than a ground state calculation. By neglectingLFE we reproduce the spectra of Ref. [1] for both thecomponent of the imaginary part of the dielectric functionalong the wire axis "k2 and the one perpendicular to the wireaxis "?2 (Fig. 1, solid and dotted lines). Our spectra arerigidly redshifted with respect to Ref. [1], as we do notinclude GW corrections (this does not influence the con-clusions). When LFE are included, no significant changesin "k2 (dashed line) are observed. However, it turns out that"?2 (dash-dotted line) is extremely sensitive to LFE: theabsorption is shifted to higher energies and the nanowiresbecome transparent up to 6–7 eV. Hence, we prove that theanisotropic contribution to the spectra due to LFE does not
0031-9007=05=94(21)=219701(1)$23.00 21970
disappear for d > 2 nm. This result can be better under-stood if we use the fact that, for large wires, the ab initiocalculations tend to reproduce the classical limit given bythe EMT [5]. The gray lines in Fig. 1 show the results of theEMT, using the RPA bulk silicon dielectric function. Theagreement between the model and the ab initio calculationsis extremely good for both polarization directions. Thisshows that the observed anisotropy can be explained al-most entirely by classical effects (that are contained in theab initio calculations when LFE are included), and that onecan use the EMT to estimate anisotropies for larger di-ameters. In view of our conclusions, the results of Zhaoet al. imply that only the contribution to the anisotropy dueto band structure vanishes for d > 2 nm. Their results donot allow one to estimate, even approximately, the anisot-ropy that might be actually measured.
Computer time was granted by IDRIS (Project No. 544).This work has been supported by the EU NoENANOQUANTA. S. B. acknowledges financial supportby the Marie Curie Actions of the EU.
Fabien Bruneval, Silvana Botti, and Lucia ReiningLaboratoire des Solides IrradiesCNRS-CEA-Ecole PolytechniqueF–91128 Palaiseau, France
Received 6 July 2004; published 2 June 2005DOI: 10.1103/PhysRevLett.94.219701PACS numbers: 73.21.Hb, 73.22.Dj, 78.67.Lt
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[1] X. Zhao, C. M. Wei, L. Yang, and M. Y. Chou, Phys. Rev.Lett. 92, 236805 (2004).
[2] See, e.g., A. G. Marinopoulos et al., Phys. Rev. Lett. 91,046402 (2003); S. Botti et al., Phys. Rev. Lett. 89, 216803(2002).
[3] See, e.g., E. Chang et al., Phys. Rev. Lett. 92, 196401(2004).
[4] X. Gonze et al., Comput. Mater. Sci. 25, 478 (2002);http://www.abinit.org; http://theory.lsi.polytechnique.fr/codes/.
[5] S. Datta et al., Phys. Rev. B 48, 14936 (1993).
2005 The American Physical Society
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