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Electronic properties and dielectric response of surfaces and nanowires of silicon from ab-initio approaches
Superlattices and Microstructures 46 (2009) 234–239
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Electronic properties and dielectric response of surfaces and nanowires of silicon from ab-initio approaches M. Palummo a,∗ , F. Iori b , R. Del Sole a , S. Ossicini c a
European Theoretical Spectroscopy Facility (ETSF), NAST, Dipartimento di Fisica - Università di Roma, ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133 Roma, Italy b
European Theoretical Spectroscopy Facility (ETSF), Laboratoire des Solides Irradiés, École Polytechnique, F-91128 Palaiseau, France
c
INFM-S3 ‘‘nanoStructures and bioSystems at Surfaces’’, Dipartimento di Scienze e Metodi dell’Ingegneria, via G. Amendola 2, Universitá di Modena e Reggio Emilia, Italy
article
info
Article history: Available online 21 January 2009 Keywords: ab-initio DFT GW BSE Surfaces Nanowires
abstract We present here an ab-initio study, within the Density Functional Theory (DFT), of the formation energy of doped Silicon Nanowires (Si-NWs). While this theoretical approach is appropriate to calculate the ground-state properties of materials, other methods, like Many-Body Perturbation Theory (MPBT) or Time Dependent Density Functional Theory (TDDFT), formally provide a correct description of the electronic excited states. Then, in the second part of this paper, we show how the many-body effects, introduced using the MBPT, modify the optical properties of the Si(100) surface. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Due to the widespread use of Si in microelectronics, silicon based materials are still intensively studied at the level of fundamental research, both from experimental and theoretical point of view. Among its surfaces, the (100) is certainly the most important termination because it is involved in all the electronic devices and it is the template for the the formation of well-ordered organic thin films in hybrid devices. Furthermore, in the last years, thanks to their possible efficient integration in conventional Si-based microelectronics, Si-NWs are becoming especially attractive for their potential use in future nanoelectronics applications. Some results about the ground-state and excited states properties of these systems will be presented here.
∗
Corresponding author. Tel.: +39 06 72594894. E-mail address: [email protected] (M. Palummo).
0749-6036/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2008.12.026
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2. Theoretical methods DFT [1], is based on the idea that the ground state energy of a system of N interacting electrons in an external potential Vext (r) can be written as a functional of the ground state electronic density. In the Kohn and Sham (KS) formulation it consists of a self-consistent solution of one-particle effective Schrödinger-like equations where, beyond the kinetic one-particle term and the external potential, there appear the classical Hartree potential and the exchange–correlation potential which, in principle, contains all the many-body effects.1 DFT total energy calculations are used here,2 to evaluate what amount of energy (i.e the formation energy (FE)) is required to sustain the doping process (with acceptor and donor impurities, such as Boron (B) and Phosphorus (P)) in Si-NWs. Starting from the undoped Sin Hm -NW, the FE of the neutral B or/and P impurities can be defined as the energy needed to insert one B and/or one P atom within the Si-NW after removing one (or two) Si atoms (transferred to the chemical reservoir, assumed to be bulk Si) [4–10]
1Ef = E (Sin−l−k Bk Pl Hm ) − E (Sin Hm ) + (k + l)µSi − kµB − lµP ,
(1)
where E is the total energy of the systems involved, µSi the total energy per atom of bulk Si, µB(P) the total energy per atom of the impurity. k and l can be 0 or 1, thus one has undoped, single B- or P-doped, B and P codoped Si-NWs. The B or/and P impurity atoms have always been located substitutionally in the Si wire. Despite the enormous successes of DFT to calculate ground-state properties, the KS eigenvalues do not correspond to the true band energies. These can be obtained within the MBPT with the use of the Green’s functions approach, by means of the so-called GW method to calculate the nonlocal and dynamic self-energy operator Σ [11]. The poles of the Green’s function yield the quasi-particle (QP) energies, which are the band energies measured, for instance, in direct and inverse photoemission. In addition, a realistic many-body description of two-particle neutral excitations, which are involved in the dielectric response, and hence in optical and energy loss spectroscopies, is today feasible thanks to the solution of the so-called Bethe–Salpeter equation (BSE) which correctly describes the electron–hole interaction [12]. Since the simplicity of the dependence on the sole electronic density is lost, and replaced by an explicit dependence on one- or two-particles Green functions, the MBPT approaches are computationally very demanding. In principle, the TDDFT is an alternative route to calculate the neutral electronic excitations, but using standard local or semilocal xc functionals and kernels (where one of the most used approximations is the Time Dependent Local Density Approximation (TDLDA)), it has well known failures in extended systems, while it works reasonably well in predicting the optical spectra of small molecules and clusters [12] and the electron energy loss spectra (EELS) of bulk materials [13]. As already discussed in the literature [12,14,15], formally the response function equations in TDDFT and BSE framework have the same structure, such as: S = S0 + S0 KS. The kernel K of this Dyson-like screening equation, that links the response function S0 of a system of independent Kohn–Sham particles (in TDDFT) or quasi-particles (in MBPT) to the full response S, is composed of two terms: the bare Coulomb interaction v , which is the same in both approaches, and the exchange–correlation functional fxc or electron–hole screened Coulomb interaction W , respectively in TDDFT and BSE approaches [12]. In view of the failure of TDLDA in extended systems, we will not show, here, examples of this kind of calculation. In the next section (Section 3), we will discuss the application of the DFT to the study of the formation energy (FE) of doped and co-doped Si-NWs. In Section 4, we will show how the
1 This latter potential is not known exactly and it is usually calculated in local density (LDA) or semilocal (Generalized gradient Approximation GGA) approximations [1]. 2 All the calculations presented in this paper are performed using codes [2,3] that employ plane wave basis set, within the LDA, or GGA for the exchange–correlation energy. Ultra-soft pseudopotentials are used to obtain the relaxed geometries, while norm-conserving pseudopotentials are used for the electronic and optical properties. The repeated cell approach, with a particular attention to avoid the spurious effects due to the infinite reproduction of the unit cell, allows the simulation of materials with reduced dimensionality, as those studied here.
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Fig. 1. (Color online) Formation energy for a codoped Si-NW (atomic geometry shown in the inset) as a function of the relative position between the two dopants. In this simulation, the B impurity is frozen in a subsurface site, while the P atom occupies different substitutional sites √ labelled as 1, 2, and 3. The line is a guide for the eyes. Here, the size of the simulation cell along the growth direction is abulk / 2. This is the size of the unit cell needed to simulate a corresponding undoped Si-NW.
electron–hole interaction strongly modifies the optical response of the Si(100) surface, with respect to the one particle approach. The conclusions are outlined in Section 5. 3. Doping in Si-NWs The nanowires, considered here, are isolated Si wires, oriented along the [110] direction, passivated at the surface with H and with a linear cross sections l of 8 Å [16]. Single doped and codoped nanowires have been studied, moving the impurities (B and P) in several positions in the unit cell. In order to vary the dopant concentration (from 12.5% to 3.1%) we also performed some simulations, increasing the size of the cell along the periodicity axis of the wire. Fig. 1 shows how the formation energy for the B and P codoped nanowires changes as a function of the position of the dopants within the NW. From this figure it is clear that, when the B atom is fixed in a substitutional site below the surface, the minimum is reached when the P atom moves to a surface position. In the case (not shown in the figure) where the P impurity is located in a subsurface position and the B atom is moved in different sites, from the center to the surface, we found that the most favored configuration is when B is at a surface site. In this case, we obtained that the formation energy becomes negative with the value of −0.28 eV. Each of the two cases demonstrates the tendency of both impurities to move close to the surface of the NW, in order to minimize the total energy and then increase the stability of the codoped system. Table 1 reports, for different doped and codoped nanowires, the number and the positions of the impurities (first column), their mutual distance in the case of codoping (second column), the formation energies (third column) and the electronic energy gaps, calculated at the DFT-GGAs level (fourth column). In the single doped cases (first two rows), the impurities were located in a subsurface position. In this table, all the results are √ obtained using a size of the simulation cell, along the periodicity axis of the NW, equal to abulk / 2, which is the minimum length needed to simulate a corresponding undoped Si-NW. A dependence of the FE from the impurity distance is visible. It is worthwhile to underline that in all cases of single-doped Si-NW (upper two rows) the FE shows higher positive value with respect to all the co-doped cases. This confirms the stabilizing role of compensated doping already found in Si nanocrystals [6,10]. Concerning the electronic properties, the band structures show a direct energy gap behavior at Γ , whose values depend on the impurity position. For different BP distances, (labelled 1, 2 and 3 in Fig. 1, these values are 0.63 eV, 0.08 eV and 0.97 eV respectively. It is interesting to note that a switch between the positions of the impurities (see the last two rows of Table 1) results in a change of the energy gap from 0.97 to 1.24 eV. In the case of
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Table 1 BP distance (second column), formation energies (third column) and energy gaps (fourth column) for the single doped and the √ codoped nanowires. The size of the simulation cell along the growth NW direction is abulk / 2. This is the size of the unit cell needed to simulate a corresponding undoped Si-NW. Nanowire (l = 0.8 nm) Si15 BH12 Si15 PH12 Si14 BPH12 B and P subsurface P center, B subsurface P surface, B subsurface B surface, P subsurface
DBP (Å)
FE (eV)
EG (eV)
1.13 0.66 4.3 1.95 2.17 2.17
0.41 0.10 −0.05 −0.28
0.08 0.63 0.97 1.24
Table 2 Formation energies and energy gaps of codoped nanowires (l = 0.8 nm), simulated using unit cells of different size along the growth NW direction (see text). B and P are both located in substitutional sites in the first silicon layer below the surface. Nanowire (l = 0.8 nm)
DBP (Å)
FE (eV)
EG (eV)
Si14 BPH12 Si30 BPH24 Si46 BPH36 Si62 BPH48
4.3 4.35 4.2 4.35
0.41 −0.15 −0.6 −0.64
0.08 1.1 1.43 1.51
1 unit cell 2 unit cell 3 unit cell 4 unit cell
single doping, the impurity levels, appearing in the energy gap of the Si-NW, present a non negligible energy dispersion, due to the interaction of the impurities in neighbors’ cells. Moreover, differently from the Si bulk case, they cannot be considered shallow, being located a few tenths of eV from the band edges. Regarding the electronic states of the codoped Si-NWs, it is important to mention here, that the quasi-particle corrections, obtained from extensive GW calculations, show a weak dependence on doping. A complete analysis of this finding will be published elsewhere [17]. In Table 2 we report the formation energies and the electronic gaps, obtained in different simulations where the size of the unit cell along the growth √ NW axis has been changed to be two, three and four times bigger than the single unit cell (abulk / 2). For each situation, the B and P impurities are introduced substitutionally at the same distance in the first subsurface silicon layer. Increasing the dimension of the unit cell means increasing the number of Silicons while keeping fixed the number of impurities. In this way, we note that lowering the dopant concentration, the FE lowers, resulting in a gain of stability for the nanowire. Moreover, as the dopant concentration decreases, the energy gap slowly increases towards the value of the corresponding undoped NW, EG = 1.66 eV. 4. Optical properties of Si(100) surface We move now to the discussion of the excited state properties of the Si(100) surface through a MBPT calculation. The geometrical reconstruction considered here is the c(4 × 2), which is the equilibrium structure of this surface, formed by rows of dimers, alternately buckled along the dimer rows and along the direction perpendicular to the dimer rows. Actually, first we perform DFTLDA calculations, modelling the surface using a repeated slab containing 16 atomic layers and a vacuum region of more than 1 nm. Then Kohn–Sham eigenvalues and eigenvectors are used to calculate the full dielectric matrix within the Random Phase Approximation (RPA) and the quasiparticle energies within the GW approach. Finally these quantities are used as input in the calculation of the two particle Bethe–Salpeter equation (BSE), that describes the electron–hole pair dynamics. It has already been shown elsewhere [18] how the inclusion of the many-body effects, namely self-energy and electron–hole interaction, influences the reflectance anisotropy spectrum of this surface, leading to a very good agreement with the experimental data. Here we show the half-slab polarizabilities [19] (which are intermediate physical quantities which enter in the calculation of the reflectance anisotropy spectrum (RAS)), obtained at the independent quasi-particle level (GW)
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Fig. 2. (Color online) Half-slab polarizabilities of the Si(100)c4 × 2, obtained at the GW (or independent quasi-particle) level (red dashed), BSE (black solid), Left panel: for light polarized along the dimers. Right panel: for light polarized along the dimers rows. The plotted quantities have dimension of length [19].
and with the inclusion of the excitonic effects (BSE level). The aim is to show how the electron–hole interaction modifies the optical response with respect to the corresponding curves calculated at the quasi-particle level of approximation, and then induces an important modification of the reflectance anisotropy spectrum. From Fig. 2 it is clear that the BSE curves show a clear red-shift and a redistribution of the peaks’ intensity with respect to the curves obtained within the independent quasi-particles approach (GW). An anisotropic behavior in the optical response, at the origin of the RAS signal, is evident at the two levels of theoretical approximation, not only for the surface to surface transitions below 3 eV, but also in the bulk-modified transitions region (see Ref. [18] for more details). Excitonic effects – that is the difference between BSE and GW curves – are important, especially below and in the region of the E1 bulk like transitions. Calculations usually report RA spectra or SDR (Surface Differential Reflectivity) spectra (where the reflectance of the clean surface is subtracted of the reflectance of the saturated surface). Some information is of course lost in this process. Here we show, instead, the optical properties of the clean surfaces for two light polarizations, parallel or perpendicular to the dimers. Transitions across surface states are clearly visible below 1 eV and close to 1.5 eV. The latter have been clearly seen in RA spectra. Fig. 2 shows that this structure is stronger for light polarized perpendicular to the dimers (contradicting, in this way, the general wisdom based on chemical intuition), in good agreement with the experiments. On the other hand, the structure below 1 eV is only weakly anisotropic, such that it is hardly visible in the calculated RA spectra (not shown here). RA experiments have not been carried out so far in the infrared energy range, while a structure is clearly visible in unpolarized SDR close to 0.9 eV [20]. 5. Conclusion In the present paper, we have shown some results of ab-initio ground-state and excited state calculations, regarding the formation energy of Si nanowires and the optical properties of the (100) surface of Si. It results in the codoped nanowires tending to increase their stability when the impurities are placed substitutionally at and below the surface, and also when the concentration of the compensated dopants tends to decrease with respect to the overall number of silicon atoms in the cell. Moreover, our analysis shows how the codoping with B and P substitutional impurities can be seen as a possible way to modulate and tune the electronic properties of Si nanowires. Regarding the Si(100) surface, the present study points out how the inclusion of the excitonic effects strongly influences the surface optical response of Si(100), determining an important redistribution of the optical peaks and a consequent visible change in the surface optical properties with respect to the independent quasi-particle calculation. We have shown here the optical properties of the Si(100) surface, for light parallel and perpendicular to the surface dimers, from which RA and SDR spectra can be calculated. We obtain a better understanding of, and a good agreement with, the available experimental data.
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Acknowledgments This work was funded in part by the EU’s Sixth Framework Programme through the Nanoquanta Network of Excellence (NMP4-CT-2004-500198) and by MIUR-PRIN2007. We acknowledge the CINECA CPU time granted by INFM. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
R.M. Drezler, E.K.U Gross, Density Functional Theory, Springer Verlag, 1990 and references therein. S. Baroni, et al. http://www.pwscf.org, 2001. A. Marini, et al. http://www.yambo-code.org, 2008. S.B. Zhang, J.E. Northrup, Phys. Rev. Lett. 67 (1991) 2339. G. Cantele, E. Degoli, E. Luppi, R. Magri, D. Ninno, G. Iadonisi, S. Ossicini, Phys. Rev. B 72 (2005) 133303. S. Ossicini, E. Degoli, F. Iori, E Luppi, R Magri, G. Cantele, F. Trani, D. Ninno, Appl. Phys. Lett. 87 (2005) 173120. G.M. Dalpian, J.R. Chelikowsky, Phys. Rev. Lett. 96 (2006) 226802. H. Peelaers, et al., Nano Lett. 6 (2006) 2781. Q. Xu, J.-W. Luo, S.-S. Li, J.-B. Xia, J. Li, S.-H. Wei, Phys. Rev. B 75 (2007) 235304. F. Iori, E. Degoli, R. Magri, I. Marri, G. Cantele, D. Ninno, F. Trani, O. Pulci, S. Ossicini, Phys. Rev. B 76 (2007) 085302. F. Aryasetiawan, O. Gunnarsson, Rep. Prog. Phys. 61 (1998) 237–312 and references therein. G. Onida, L. Reining, A. Rubio, Rev. Modern Phys. 74 (2002) 601 and references therein. V. Olevano, L. Reining, Phys. Rev. Lett. 86 (2001) 5962. L. Reining, V. Olevano, A. Rubio, G. Onida, Phys. Rev. Lett. 88 (2002) 066404. S. Botti, et al., Phys. Rev. 69 (2004) 155112. M. Bruno, M. Palummo, A. Marini, R.D. Sole, S. Ossicini, Phys. Rev. Lett. 98 (2007) 036807. M. Palummo, F. Iori, S. Ossicini, preprint. M. Palummo, N. Witkowski, O. Pluchery, Rodolfo Del Sole, Y. Borensztein, Phys. Rev. B (2009) (in press). R. Del Sole, in: P. Halevi (Ed.), Photonic Probes of Surfaces, Elsevier, 1995, p. 131. Y.J. Chabal, S.B. Christman, E.E. Chaban, M.T. Yin, J. Vac. Sci. Technol. A 1 (1983) 1241.
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Electronic properties and dielectric response of surfaces and nanowires of silicon from ab-initio approaches M. Palummo a,∗ , F. Iori b , R. Del Sole a , S. Ossicini c a
European Theoretical Spectroscopy Facility (ETSF), NAST, Dipartimento di Fisica - Università di Roma, ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133 Roma, Italy b
European Theoretical Spectroscopy Facility (ETSF), Laboratoire des Solides Irradiés, École Polytechnique, F-91128 Palaiseau, France
c
INFM-S3 ‘‘nanoStructures and bioSystems at Surfaces’’, Dipartimento di Scienze e Metodi dell’Ingegneria, via G. Amendola 2, Universitá di Modena e Reggio Emilia, Italy
article
info
Article history: Available online 21 January 2009 Keywords: ab-initio DFT GW BSE Surfaces Nanowires
abstract We present here an ab-initio study, within the Density Functional Theory (DFT), of the formation energy of doped Silicon Nanowires (Si-NWs). While this theoretical approach is appropriate to calculate the ground-state properties of materials, other methods, like Many-Body Perturbation Theory (MPBT) or Time Dependent Density Functional Theory (TDDFT), formally provide a correct description of the electronic excited states. Then, in the second part of this paper, we show how the many-body effects, introduced using the MBPT, modify the optical properties of the Si(100) surface. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Due to the widespread use of Si in microelectronics, silicon based materials are still intensively studied at the level of fundamental research, both from experimental and theoretical point of view. Among its surfaces, the (100) is certainly the most important termination because it is involved in all the electronic devices and it is the template for the the formation of well-ordered organic thin films in hybrid devices. Furthermore, in the last years, thanks to their possible efficient integration in conventional Si-based microelectronics, Si-NWs are becoming especially attractive for their potential use in future nanoelectronics applications. Some results about the ground-state and excited states properties of these systems will be presented here.
∗
Corresponding author. Tel.: +39 06 72594894. E-mail address: [email protected] (M. Palummo).
0749-6036/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2008.12.026
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235
2. Theoretical methods DFT [1], is based on the idea that the ground state energy of a system of N interacting electrons in an external potential Vext (r) can be written as a functional of the ground state electronic density. In the Kohn and Sham (KS) formulation it consists of a self-consistent solution of one-particle effective Schrödinger-like equations where, beyond the kinetic one-particle term and the external potential, there appear the classical Hartree potential and the exchange–correlation potential which, in principle, contains all the many-body effects.1 DFT total energy calculations are used here,2 to evaluate what amount of energy (i.e the formation energy (FE)) is required to sustain the doping process (with acceptor and donor impurities, such as Boron (B) and Phosphorus (P)) in Si-NWs. Starting from the undoped Sin Hm -NW, the FE of the neutral B or/and P impurities can be defined as the energy needed to insert one B and/or one P atom within the Si-NW after removing one (or two) Si atoms (transferred to the chemical reservoir, assumed to be bulk Si) [4–10]
1Ef = E (Sin−l−k Bk Pl Hm ) − E (Sin Hm ) + (k + l)µSi − kµB − lµP ,
(1)
where E is the total energy of the systems involved, µSi the total energy per atom of bulk Si, µB(P) the total energy per atom of the impurity. k and l can be 0 or 1, thus one has undoped, single B- or P-doped, B and P codoped Si-NWs. The B or/and P impurity atoms have always been located substitutionally in the Si wire. Despite the enormous successes of DFT to calculate ground-state properties, the KS eigenvalues do not correspond to the true band energies. These can be obtained within the MBPT with the use of the Green’s functions approach, by means of the so-called GW method to calculate the nonlocal and dynamic self-energy operator Σ [11]. The poles of the Green’s function yield the quasi-particle (QP) energies, which are the band energies measured, for instance, in direct and inverse photoemission. In addition, a realistic many-body description of two-particle neutral excitations, which are involved in the dielectric response, and hence in optical and energy loss spectroscopies, is today feasible thanks to the solution of the so-called Bethe–Salpeter equation (BSE) which correctly describes the electron–hole interaction [12]. Since the simplicity of the dependence on the sole electronic density is lost, and replaced by an explicit dependence on one- or two-particles Green functions, the MBPT approaches are computationally very demanding. In principle, the TDDFT is an alternative route to calculate the neutral electronic excitations, but using standard local or semilocal xc functionals and kernels (where one of the most used approximations is the Time Dependent Local Density Approximation (TDLDA)), it has well known failures in extended systems, while it works reasonably well in predicting the optical spectra of small molecules and clusters [12] and the electron energy loss spectra (EELS) of bulk materials [13]. As already discussed in the literature [12,14,15], formally the response function equations in TDDFT and BSE framework have the same structure, such as: S = S0 + S0 KS. The kernel K of this Dyson-like screening equation, that links the response function S0 of a system of independent Kohn–Sham particles (in TDDFT) or quasi-particles (in MBPT) to the full response S, is composed of two terms: the bare Coulomb interaction v , which is the same in both approaches, and the exchange–correlation functional fxc or electron–hole screened Coulomb interaction W , respectively in TDDFT and BSE approaches [12]. In view of the failure of TDLDA in extended systems, we will not show, here, examples of this kind of calculation. In the next section (Section 3), we will discuss the application of the DFT to the study of the formation energy (FE) of doped and co-doped Si-NWs. In Section 4, we will show how the
1 This latter potential is not known exactly and it is usually calculated in local density (LDA) or semilocal (Generalized gradient Approximation GGA) approximations [1]. 2 All the calculations presented in this paper are performed using codes [2,3] that employ plane wave basis set, within the LDA, or GGA for the exchange–correlation energy. Ultra-soft pseudopotentials are used to obtain the relaxed geometries, while norm-conserving pseudopotentials are used for the electronic and optical properties. The repeated cell approach, with a particular attention to avoid the spurious effects due to the infinite reproduction of the unit cell, allows the simulation of materials with reduced dimensionality, as those studied here.
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Fig. 1. (Color online) Formation energy for a codoped Si-NW (atomic geometry shown in the inset) as a function of the relative position between the two dopants. In this simulation, the B impurity is frozen in a subsurface site, while the P atom occupies different substitutional sites √ labelled as 1, 2, and 3. The line is a guide for the eyes. Here, the size of the simulation cell along the growth direction is abulk / 2. This is the size of the unit cell needed to simulate a corresponding undoped Si-NW.
electron–hole interaction strongly modifies the optical response of the Si(100) surface, with respect to the one particle approach. The conclusions are outlined in Section 5. 3. Doping in Si-NWs The nanowires, considered here, are isolated Si wires, oriented along the [110] direction, passivated at the surface with H and with a linear cross sections l of 8 Å [16]. Single doped and codoped nanowires have been studied, moving the impurities (B and P) in several positions in the unit cell. In order to vary the dopant concentration (from 12.5% to 3.1%) we also performed some simulations, increasing the size of the cell along the periodicity axis of the wire. Fig. 1 shows how the formation energy for the B and P codoped nanowires changes as a function of the position of the dopants within the NW. From this figure it is clear that, when the B atom is fixed in a substitutional site below the surface, the minimum is reached when the P atom moves to a surface position. In the case (not shown in the figure) where the P impurity is located in a subsurface position and the B atom is moved in different sites, from the center to the surface, we found that the most favored configuration is when B is at a surface site. In this case, we obtained that the formation energy becomes negative with the value of −0.28 eV. Each of the two cases demonstrates the tendency of both impurities to move close to the surface of the NW, in order to minimize the total energy and then increase the stability of the codoped system. Table 1 reports, for different doped and codoped nanowires, the number and the positions of the impurities (first column), their mutual distance in the case of codoping (second column), the formation energies (third column) and the electronic energy gaps, calculated at the DFT-GGAs level (fourth column). In the single doped cases (first two rows), the impurities were located in a subsurface position. In this table, all the results are √ obtained using a size of the simulation cell, along the periodicity axis of the NW, equal to abulk / 2, which is the minimum length needed to simulate a corresponding undoped Si-NW. A dependence of the FE from the impurity distance is visible. It is worthwhile to underline that in all cases of single-doped Si-NW (upper two rows) the FE shows higher positive value with respect to all the co-doped cases. This confirms the stabilizing role of compensated doping already found in Si nanocrystals [6,10]. Concerning the electronic properties, the band structures show a direct energy gap behavior at Γ , whose values depend on the impurity position. For different BP distances, (labelled 1, 2 and 3 in Fig. 1, these values are 0.63 eV, 0.08 eV and 0.97 eV respectively. It is interesting to note that a switch between the positions of the impurities (see the last two rows of Table 1) results in a change of the energy gap from 0.97 to 1.24 eV. In the case of
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Table 1 BP distance (second column), formation energies (third column) and energy gaps (fourth column) for the single doped and the √ codoped nanowires. The size of the simulation cell along the growth NW direction is abulk / 2. This is the size of the unit cell needed to simulate a corresponding undoped Si-NW. Nanowire (l = 0.8 nm) Si15 BH12 Si15 PH12 Si14 BPH12 B and P subsurface P center, B subsurface P surface, B subsurface B surface, P subsurface
DBP (Å)
FE (eV)
EG (eV)
1.13 0.66 4.3 1.95 2.17 2.17
0.41 0.10 −0.05 −0.28
0.08 0.63 0.97 1.24
Table 2 Formation energies and energy gaps of codoped nanowires (l = 0.8 nm), simulated using unit cells of different size along the growth NW direction (see text). B and P are both located in substitutional sites in the first silicon layer below the surface. Nanowire (l = 0.8 nm)
DBP (Å)
FE (eV)
EG (eV)
Si14 BPH12 Si30 BPH24 Si46 BPH36 Si62 BPH48
4.3 4.35 4.2 4.35
0.41 −0.15 −0.6 −0.64
0.08 1.1 1.43 1.51
1 unit cell 2 unit cell 3 unit cell 4 unit cell
single doping, the impurity levels, appearing in the energy gap of the Si-NW, present a non negligible energy dispersion, due to the interaction of the impurities in neighbors’ cells. Moreover, differently from the Si bulk case, they cannot be considered shallow, being located a few tenths of eV from the band edges. Regarding the electronic states of the codoped Si-NWs, it is important to mention here, that the quasi-particle corrections, obtained from extensive GW calculations, show a weak dependence on doping. A complete analysis of this finding will be published elsewhere [17]. In Table 2 we report the formation energies and the electronic gaps, obtained in different simulations where the size of the unit cell along the growth √ NW axis has been changed to be two, three and four times bigger than the single unit cell (abulk / 2). For each situation, the B and P impurities are introduced substitutionally at the same distance in the first subsurface silicon layer. Increasing the dimension of the unit cell means increasing the number of Silicons while keeping fixed the number of impurities. In this way, we note that lowering the dopant concentration, the FE lowers, resulting in a gain of stability for the nanowire. Moreover, as the dopant concentration decreases, the energy gap slowly increases towards the value of the corresponding undoped NW, EG = 1.66 eV. 4. Optical properties of Si(100) surface We move now to the discussion of the excited state properties of the Si(100) surface through a MBPT calculation. The geometrical reconstruction considered here is the c(4 × 2), which is the equilibrium structure of this surface, formed by rows of dimers, alternately buckled along the dimer rows and along the direction perpendicular to the dimer rows. Actually, first we perform DFTLDA calculations, modelling the surface using a repeated slab containing 16 atomic layers and a vacuum region of more than 1 nm. Then Kohn–Sham eigenvalues and eigenvectors are used to calculate the full dielectric matrix within the Random Phase Approximation (RPA) and the quasiparticle energies within the GW approach. Finally these quantities are used as input in the calculation of the two particle Bethe–Salpeter equation (BSE), that describes the electron–hole pair dynamics. It has already been shown elsewhere [18] how the inclusion of the many-body effects, namely self-energy and electron–hole interaction, influences the reflectance anisotropy spectrum of this surface, leading to a very good agreement with the experimental data. Here we show the half-slab polarizabilities [19] (which are intermediate physical quantities which enter in the calculation of the reflectance anisotropy spectrum (RAS)), obtained at the independent quasi-particle level (GW)
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Fig. 2. (Color online) Half-slab polarizabilities of the Si(100)c4 × 2, obtained at the GW (or independent quasi-particle) level (red dashed), BSE (black solid), Left panel: for light polarized along the dimers. Right panel: for light polarized along the dimers rows. The plotted quantities have dimension of length [19].
and with the inclusion of the excitonic effects (BSE level). The aim is to show how the electron–hole interaction modifies the optical response with respect to the corresponding curves calculated at the quasi-particle level of approximation, and then induces an important modification of the reflectance anisotropy spectrum. From Fig. 2 it is clear that the BSE curves show a clear red-shift and a redistribution of the peaks’ intensity with respect to the curves obtained within the independent quasi-particles approach (GW). An anisotropic behavior in the optical response, at the origin of the RAS signal, is evident at the two levels of theoretical approximation, not only for the surface to surface transitions below 3 eV, but also in the bulk-modified transitions region (see Ref. [18] for more details). Excitonic effects – that is the difference between BSE and GW curves – are important, especially below and in the region of the E1 bulk like transitions. Calculations usually report RA spectra or SDR (Surface Differential Reflectivity) spectra (where the reflectance of the clean surface is subtracted of the reflectance of the saturated surface). Some information is of course lost in this process. Here we show, instead, the optical properties of the clean surfaces for two light polarizations, parallel or perpendicular to the dimers. Transitions across surface states are clearly visible below 1 eV and close to 1.5 eV. The latter have been clearly seen in RA spectra. Fig. 2 shows that this structure is stronger for light polarized perpendicular to the dimers (contradicting, in this way, the general wisdom based on chemical intuition), in good agreement with the experiments. On the other hand, the structure below 1 eV is only weakly anisotropic, such that it is hardly visible in the calculated RA spectra (not shown here). RA experiments have not been carried out so far in the infrared energy range, while a structure is clearly visible in unpolarized SDR close to 0.9 eV [20]. 5. Conclusion In the present paper, we have shown some results of ab-initio ground-state and excited state calculations, regarding the formation energy of Si nanowires and the optical properties of the (100) surface of Si. It results in the codoped nanowires tending to increase their stability when the impurities are placed substitutionally at and below the surface, and also when the concentration of the compensated dopants tends to decrease with respect to the overall number of silicon atoms in the cell. Moreover, our analysis shows how the codoping with B and P substitutional impurities can be seen as a possible way to modulate and tune the electronic properties of Si nanowires. Regarding the Si(100) surface, the present study points out how the inclusion of the excitonic effects strongly influences the surface optical response of Si(100), determining an important redistribution of the optical peaks and a consequent visible change in the surface optical properties with respect to the independent quasi-particle calculation. We have shown here the optical properties of the Si(100) surface, for light parallel and perpendicular to the surface dimers, from which RA and SDR spectra can be calculated. We obtain a better understanding of, and a good agreement with, the available experimental data.
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Acknowledgments This work was funded in part by the EU’s Sixth Framework Programme through the Nanoquanta Network of Excellence (NMP4-CT-2004-500198) and by MIUR-PRIN2007. We acknowledge the CINECA CPU time granted by INFM. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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