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Formation of lithium clusters and their effects on conductivity in diamond: A density functional theory study
Diamond & Related Materials 16 (2007) 840 – 844 www.elsevier.com/locate/diamond
Formation of lithium clusters and their effects on conductivity in diamond: A density functional theory study Hulusi Yilmaz a,⁎, Brad R. Weiner b , Gerardo Morell a a b
Department of Physics, University of Puerto Rico, San Juan, PR 00931, USA Department of Chemistry, University of Puerto Rico, San Juan, PR 00931, USA Available online 24 January 2007
Abstract We have applied Density Functional Theory (DFT) within the Generalized Gradient Approximation to study the behavior of lithium impurities in the diamond lattice. It was found that, although isolated interstitial lithium atoms in tetrahedral sites have a donor state relatively close to the conduction band at 0.35 eV, their behavior is critically affected by adjacent interstitials. Lithium atoms that occupy adjacent interstitial tetrahedral sites cluster spontaneously and induce carbon–carbon bond breaking in their neighborhood. The corresponding charge distribution and electronic density of states analyses show that charge localization takes place at the carbon atoms around the lithium clusters and deeper mid-gap states are introduced in the diamond band gap. The formation energy was found to be 1.88 eV and 3.09 eV, for clusters of two and three lithium atoms, respectively, considered in this study. The calculations also indicate that Li clustering would not take place if Li ions instead of atoms were incorporated interstitially. © 2007 Elsevier B.V. All rights reserved. Keywords: Lithium; Diamond; n-type; DFT
1. Introduction Besides substitutional dopants such as nitrogen [1,2], phosphorus [3,4] and sulfur [5], interstitial lithium has been considered as a candidate to achieve shallow n-type doping in diamond. Many attempts have been made to incorporate lithium in diamond, including implantation [6–8], addition during CVD growth [9–11] and diffusion [12–16]. Car–Parrinello calculations by Kajihara et al. [17] indicated the possibility of lithium diffusion and its shallow n-type activity in diamond. However, diffusion experiments on lithium-implanted diamond showed that lithium does not diffuse in single-crystal diamond even though it does in polycrystalline diamond through grain boundaries [8]. According to Job et al. [12], the presence of defects in the diamond lattice reduces the diffusion of lithium and hinders its n-type conductivity, especially at high temperatures. On the other hand, Popovici et al. [13] performed Li incorporation in diamond by forcing Li with electric fields together with Cl and O, and reported a high concentration of ⁎ Corresponding author. Tel.: +1 787 529 2941; fax: +1 787 756 7717. E-mail address: [email protected] (H. Yilmaz). 0925-9635/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2007.01.008
lithium in high quality diamond. They subsequently reported that Li-diffused diamond is n-type and attributed this behavior to interstitial lithium [18]. They also built a p–n junction using boron and lithium-doped diamond, but could not establish whether the n-type component was due to shallow lithium doping or defects [19]. Moreover, te Nijenhuis et al. [14] studied lithium diffusion into an initially defect-free single-crystal diamond and reported lithium concentrations of 1.0 × 1019 cm− 3 at a depth of 0.5 μm below the crystal surface and a much higher concentration at the surface. However, they could not unambiguously conclude the occurrence of n-type conductivity and further analysis of their samples showed that lithium atoms clustered below the diamond surface. Uzan-Saguy et al. [15] arrived at a similar conclusion after diffusing Li in diamond. Almeida et al. [16] reported a direct relationship between lithium incorporation and boron concentration in diamond, pointing at the possibility of boron co-doping and inactivation of lithium. Therefore, despite the various methods employed and some encouraging reports summarized above, there is no reproducible and definite report on n-type conductivity due to interstitial lithium in diamond, nor a satisfactory detailed theoretical explanation to this situation.
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We studied the behavior of interstitial lithium atoms in diamond with the supercell method, placing lithium atoms in adjacent and in distant tetrahedral sites, and fully relaxing the diamond lattice to observe the lithium interactions with each other and with the lattice. The results provide a mechanistic understanding for the absence of n-type behavior in lithiumdoped diamond. 2. Computational methods Lattice relaxation and total energy calculations were performed using a plane wave-based density functional theory (DFT) [20,21] code — Vienna ab-initio simulation package (VASP) [22–24]. Calculations were performed using DFT within the generalized gradient approximation (GGA), with the exchange-correlation functional of Perdew and Wang [25]. We used gradient-corrected ultrasoft pseudopotentials [26] and expanded the valence orbitals on a plane wave basis set. The kinetic energy cutoff for plane wave basis set was set to 320 eV. For the Brilliouin zone integrations we used 2 × 2 × 2 Monkhorst–Pact grid [27] together with a Gaussian smearing method with σ = 0.2 eV. A finer, 3 × 3 × 3 Monkhorst–Pact grid with tetrahedron method [28] was used to calculate the electronic density of states (DOS) on the initially optimized structure of atomic coordinates. The fast-Fourier-transformation grid used for these calculations was 64 × 64 × 64. The electronic minimization algorithm for total energy calculations was based on a residual minimization scheme–direct inversion in the iterative subspace (RMM–DIIS) [29,30] and the geometry optimizations were performed via the conjugate-gradient minimization method. The optimization of atomic coordinates was continued until force on each atom in the supercell was reduced below 0.01 eV/Å. Lithium–lithium interactions in diamond crystal were studied in a 216-carbon atom diamond supercell. The lattice constant and the bulk modulus of diamond calculated with the above parameters were 3.567 Å and 4.37 Mbar, respectively, in very good agreement with the experimental values of 3.5668 Å and 4.43 Mbar. The cell shape and volume were kept constant for all calculations and the symmetries were turned off. In this supercell, the distance between a lithium atom and its closest images is 10.7 Å.
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as a reference, the formation energy of an isolated interstitial lithium atom in diamond in the neutral charge state was calculated as 8.2 eV. The diffusion of lithium in diamond is restricted by the energy barriers that are located at the hexagonal sites between tetrahedral sites. This energy barrier was found to be 1.09 eV. The formation energy and the calculated diffusion barrier energies are larger than those calculated by Ref. [17], which were 5.5 eV and 0.85 eV, respectively. Although the activation energy for lithium diffusion calculated here is in agreement with the experimental value of 0.9 ± 0.3 eV [14], another experimental study [15] reported a much smaller, 0.26 eV, activation energy for lithium diffusion in diamond. We have calculated the donor state of an isolated interstitial lithium atom using the so-called empirical marker method (EMM) [31–33]. This method provides the donor state of an impurity by comparing its ionization energy with that of a reference impurity with known donor state using the following expression: X ð0=þÞ ¼ Rð0=þÞ þ f½EðX 0 Þ−EðX þ Þ−½EðR0 Þ−EðRþ Þg:
ð1Þ
Here X (0/+) and R(0/+) stand for the donor state to be calculated and the known reference state, respectively. The terms in
3. Results and discussion 3.1. Single lithium atom in diamond Lithium atoms energetically prefer the interstitial tetrahedral sites in diamond [17]. In the tetrahedral site, the first neighbor C–C bond lengths around Li atoms are stretched out by 4.5% from their initial position after relaxation. The C–C bond lengths between the first and second neighbor carbon atoms are shorter than 1.54 Å, and depending on their direction, they vary from 1.49 to 1.52 Å. Although a minor increase is observed in the C–C bond lengths between the second and third neighbor carbon atoms, the distortions beyond that point are negligible, based on the error bar of the calculations. Taking lithium metal
Fig. 1. Structural diagrams of lithium clusters formed with (a) two and (b) three lithium atoms. Nearby carbon atoms around the lithium atoms are dark colored, lithium atoms are shown in dot–circles and dashed lines represent broken C–C bonds.
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parentheses represent the ionization energy of the two systems. We used phosphorous with Ec − 0.6 eV as a reference, following the recommendation of Goss et al. [31]. The donor state of lithium atom calculated with this method was Ec − 0.35 eV, which is deeper than the activation energy obtained from experimental studies [7,12,34], namely 0.23, 0.16 and 0.22 eV, respectively. While the theoretical study of Kajihara et al. [17] reports the activation energy of lithium to be 0.1 eV below the conduction band, that of Anderson and Mehandru [35] predicts lithium to be readily ionized in diamond. Notice that, although the reported activation energies differ among themselves, all of them indicate a relatively shallow donor doping activity for interstitial lithium in diamond. The results discussed below further indicate that this shallow donor characteristic is readily lost if lithium atoms form clusters in diamond. 3.2. Formation of lithium atom clusters in diamond Lithium atoms that are initially placed in non-adjacent tetrahedral sites remain stationary and show n-type behavior, similar to the single lithium atom case discussed above. However, if two or more lithium atoms are initially placed in adjacent tetrahedral sites, they tend to cluster by breaking the C–C bonds near them. Fig. 1a and b shows the final structure of the lithium clusters formed by two and three lithium atoms, respectively. For illustration purposes, nearby carbon atoms around the lithium atoms are dark colored, lithium atoms are shown in dot–circles, and dashed lines represent the broken C–C bonds, in Fig. 1. Lithium atoms were initially placed in consecutive tetrahedral sites along a diffusion chan-
nel and all the atomic coordinates in the supercell were fully relaxed. This configuration results in the spontaneous formation of lithium clusters, i.e. there is no barrier for this reaction. In the case of two Li clusters (Fig. 1a), two C–C bonds are broken and one sp2 -type C bond is formed. The bond length of this sp2 -type C bond is 1.41 Å, which is slightly larger than a typical sp2 C bond. The distance between two lithium atoms is 1.53 Å and the two broken C–C bonds are nearly equal, 2.14 Å and 2.15 Å. Similarly, two C–C bonds break when three lithium atoms are initially placed in consecutive tetrahedral sites in a diffusion channel (Fig. 1b). The distances between the three consecutive lithium atoms are equal, i.e. 1.64 Å, and the distances between C atoms after bond breaking are also equal, i.e. 2.25 Å. The carbon atom that lost two of its four sp3 -type C bonds formed two shortened sp2 -type C bonds, i.e. 1.36 Å, with its remaining neighbor carbon atoms. In both cases, the charge density analyses show that lithium atoms are almost fully ionized and the charge is localized over the carbon atoms around the Li clusters. The formation energy of these two and three Li atoms clusters is 1.88 eV and 3.09 eV, respectively. We calculated the formation energy of lithium clusters assuming that the lithium atoms were initially located at non-interacting distances in tetrahedral sites in the diamond lattice. Since the energy gain per lithium atom is higher in the three Li cluster than in the two Li cluster, these results seem to indicate that larger lithium clusters would be favored in diamond. Nevertheless, studying the formation of lithium clusters with larger number of lithium atoms is required in order to be able to derive a general conclusion in this respect.
Fig. 2. Electronic density of state diagrams of diamond with: (a) one isolated interstitial Li, and the defect structure formed by two (b) and three (c) lithium atoms.
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To verify whether the lithium clustering observed here is linked to the supercell size, we performed a test calculation using a 360-atom diamond supercell with two lithium atoms at adjacent tetrahedral sites. C–C bond breaking and lithium clustering were also observed in this supercell after relaxation. (This calculation was done with 2 × 2 × 1 k-mesh and energy cutoff of 300 eV.) Lithium clustering was also observed in a test calculation performed with PAW-PBE [36,37] potentials. Although the electronic levels obtained by DFT are known not to be accurate in absolute numbers, the DOS corresponding to the diamond supercell with a single lithium atom (Fig. 2a) and that with two and three lithium atom clusters (Fig. 2b–c) show distinct effects due to lithium clustering. There are no mid-gap defect states in the diamond band gap in the case of an isolated interstitial lithium atom. However, new electronic states are introduced in the diamond band gap when two and three lithium atoms form clusters by inducing C–C bond breaking. The energy levels in the DOS plots are shown relative to the Fermi level, at zero energy, which separates the occupied and unoccupied electronic states. Compared to the shallow donor level of an isolated interstitial lithium atom calculated in the previous section, the activation energy of these localized electrons in the deep mid-gap states should be much higher, thus leading to electrically inactive Li doping. These results help to explain the experimental observations of te Nijenhuis [14] and the conclusion arrived by Uzan-Saguy et al. [15]. In both of these studies, it is pointed out that lithium could be clustering below the surface of the diamond samples, and the absence of n-type conductivity was attributed to this behavior. 3.3. Li ions in diamond The results discussed above indicate that there is a relation between Li clustering and their valence electrons lowering their energy by moving into mid-gap states. In order to further explore this relation, using the same initial configurations, we repeated the calculations with +2 and +3 charged states to represent two and three lithium ions, respectively. In both of these cases no C–C bond breaking took place and the lithium ions remained stationary in their initial tetrahedral sites. Moreover, when we remove one electron per Li atom from the relaxed cell discussed in Section 3.2 and allow the system to relax again, the lithium ions move back to their initial positions and the broken C–C bonds repair themselves. In the charged states hereby considered, there is repulsion energy between lithium atoms, which is 0.86 eV and 1.87 eV for two and three lithium atoms, respectively. This repulsion energy may actually prevent occupation of neighboring tetrahedral sites in diamond when lithium is incorporated in ionic form. However the strength of the electric field applied to incorporate lithium ions in diamond might suffice to overcome this repulsion. 4. Conclusions A DFT study of lithium atoms and ions in diamond was performed. Although isolated interstitial lithium atoms in tetrahedral sites have a donor state that is 0.35 eV below the
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conduction band, their behavior is critically affected by the presence of adjacent interstitial atoms. We found that interstitial lithium atoms that are located in adjacent tetrahedral sites spontaneously induce C–C bond breaking and form lithium clusters. The charge distribution analysis shows that charge localization takes place at the defect sites over the carbon atoms around the lithium clusters. Correspondingly, the electronic DOS analysis shows that mid-gap states are introduced in the diamond band gap, thus providing lower energy states to the otherwise near-conduction electrons. Since the defect states are much deeper in the diamond band gap compared to those of the isolated interstitial lithium, these electrons require much higher energy to be activated. We also found that when lithium ions instead of atoms are considered in the calculations, none of the previously-described Li clustering or C–C bond breaking takes place. Collectively, these results imply that n-type doping of diamond with Li is in principle feasible, provided that a low doping concentration is employed and precautions are taken to ensure that the Li atoms remain mono-dispersed during doping. Note added in proof A recent study of Goss et al, to be published in PRB, in issue 7 of volume 74, confirms some of the findings presented here.
Acknowledgements We gratefully thank Dr. Marius J. Vilkas and Dr. Jonathan Goss for their invaluable contribution to this work. This research was supported by the Department of Energy (Grant DE-FG02-01ER45868) and the National Aeronautics Space Administration (NASA research grant NCC3-1034, NASA Training Grant NNG05GG78H, NASA Cooperative Agreement NCC5-595). We gratefully acknowledge the use of UPR's High Performance Computing Facility. References [1] W.J.P. van Enkevort, E.H. Verteegen, J. Phys., Condens. Matter. 4 (1992) 2361. [2] K. Jackson, M.R. Pederson, J.G. Harrison, Phys. Rev., B 41 (1990) 12641. [3] S.N. Schauer, J.R. Flemish, R. Wittstruck, M.I. Landstrass, M.A. Plano, Appl. Phys. Lett. 64 (1994) 2094. [4] J.R. Flemish, S.N. Schauer, R. Wittstruck, M.I. Landstrass, M.A. Plano, Diamond Relat. Mater. 3 (1994) 672. [5] R. Kalish, A. Reznik, C. Uzan-Saguy, C. Cytermann, Appl. Phys. Lett. 76 (2000) 757. [6] S. Prawer, C. Uzan-Saguy, G. Braustin, R. Kalish, Appl. Phys. Lett. 63 (1993) 2502. [7] I.M. Buckley-Golder, R. Bullugh, M.R. Hayns, J.R. Willis, R.C. Piller, N.G. Blamires, G. Gard, J. Stephen, Diamond Relat. Mater. 1 (1991) 43. [8] C. Cytermann, R. Brener, R. Kalish, Diamond Relat. Mater. 3 (1994) 677. [9] K. Okumura, J. Mort, M. Machonkin, Appl. Phys. Lett. 57 (1990) 1907. [10] T.H. Borst, O. Weis, Diamond Relat. Mater. 4 (1995) 948. [11] H. Sternschulte, M. Schreck, B. Stritzker, A. Bergmair, G. Dollinger, Diamond Relat. Mater. 9 (2000) 1046. [12] R. Job, M. Werner, A. Denisenko, A. Zaitsev, W.R. Fahrner, Diamond Relat. Mater. 5 (1996) 757. [13] G. Popovici, T. Sung, M.A. Prelas, S. Khansawinah, A.A. Melnikov, R.G. Wilson, Mater. Res. Soc. Symp. Proc 339 (1994) 601.
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[14] J. te Nijenhuis, G.Z. Cao, P.C.H.J. Smits, W.J.P. van Enckevort, L.J. Giling, P.F.A. Alkemade, M. Nesladek, Z. Remes, Diamond Relat. Mater. 6 (1997) 1726. [15] C. Uzan-Saguy, C. Cytermann, B. Fizgeer, V. Richter, R. Brener, R. Kalish, Phys. Status Solidi, A 193 (2002) 508. [16] E.C. Almeida, V.J. Trava-airoldi, N.G. Ferreira, J.M. Rosolen, Diamond Relat. Mater. 14 (2005) 1673. [17] S.A. Kajihara, A. Antonelli, J. Bernholc, R. Car, Phys. Rev. Lett. 66 (1991) 2010. [18] G. Popovici, M.A. Prelas, T. Sung, S. Khansawinah, A.A. Melnikov, V.S. Varichenko, A.M. Zaitsev, A.V. Denisenko, W.R. Fahrner, Diamond Relat. Mater. 4 (1995) 877. [19] G. Popovici, A. Melnikov, V.V. Varichenko, T. Sung, M.A. Prelas, R.G. Wilson, S.K. Loyalka, J. Appl. Phys. 81 (1997) 2429. [20] W. Kohn, L. Sham, Phys. Rev., A 140 (1965) 1133. [21] R.O. Jonas, O. Gunnarsson, Rev. Mod. Phys. 61 (1989) 698. [22] G. Kresse, J. Hafner, Phys. Rev., B 48 (1993) 13115. [23] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15.
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Formation of lithium clusters and their effects on conductivity in diamond: A density functional theory study Hulusi Yilmaz a,⁎, Brad R. Weiner b , Gerardo Morell a a b
Department of Physics, University of Puerto Rico, San Juan, PR 00931, USA Department of Chemistry, University of Puerto Rico, San Juan, PR 00931, USA Available online 24 January 2007
Abstract We have applied Density Functional Theory (DFT) within the Generalized Gradient Approximation to study the behavior of lithium impurities in the diamond lattice. It was found that, although isolated interstitial lithium atoms in tetrahedral sites have a donor state relatively close to the conduction band at 0.35 eV, their behavior is critically affected by adjacent interstitials. Lithium atoms that occupy adjacent interstitial tetrahedral sites cluster spontaneously and induce carbon–carbon bond breaking in their neighborhood. The corresponding charge distribution and electronic density of states analyses show that charge localization takes place at the carbon atoms around the lithium clusters and deeper mid-gap states are introduced in the diamond band gap. The formation energy was found to be 1.88 eV and 3.09 eV, for clusters of two and three lithium atoms, respectively, considered in this study. The calculations also indicate that Li clustering would not take place if Li ions instead of atoms were incorporated interstitially. © 2007 Elsevier B.V. All rights reserved. Keywords: Lithium; Diamond; n-type; DFT
1. Introduction Besides substitutional dopants such as nitrogen [1,2], phosphorus [3,4] and sulfur [5], interstitial lithium has been considered as a candidate to achieve shallow n-type doping in diamond. Many attempts have been made to incorporate lithium in diamond, including implantation [6–8], addition during CVD growth [9–11] and diffusion [12–16]. Car–Parrinello calculations by Kajihara et al. [17] indicated the possibility of lithium diffusion and its shallow n-type activity in diamond. However, diffusion experiments on lithium-implanted diamond showed that lithium does not diffuse in single-crystal diamond even though it does in polycrystalline diamond through grain boundaries [8]. According to Job et al. [12], the presence of defects in the diamond lattice reduces the diffusion of lithium and hinders its n-type conductivity, especially at high temperatures. On the other hand, Popovici et al. [13] performed Li incorporation in diamond by forcing Li with electric fields together with Cl and O, and reported a high concentration of ⁎ Corresponding author. Tel.: +1 787 529 2941; fax: +1 787 756 7717. E-mail address: [email protected] (H. Yilmaz). 0925-9635/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2007.01.008
lithium in high quality diamond. They subsequently reported that Li-diffused diamond is n-type and attributed this behavior to interstitial lithium [18]. They also built a p–n junction using boron and lithium-doped diamond, but could not establish whether the n-type component was due to shallow lithium doping or defects [19]. Moreover, te Nijenhuis et al. [14] studied lithium diffusion into an initially defect-free single-crystal diamond and reported lithium concentrations of 1.0 × 1019 cm− 3 at a depth of 0.5 μm below the crystal surface and a much higher concentration at the surface. However, they could not unambiguously conclude the occurrence of n-type conductivity and further analysis of their samples showed that lithium atoms clustered below the diamond surface. Uzan-Saguy et al. [15] arrived at a similar conclusion after diffusing Li in diamond. Almeida et al. [16] reported a direct relationship between lithium incorporation and boron concentration in diamond, pointing at the possibility of boron co-doping and inactivation of lithium. Therefore, despite the various methods employed and some encouraging reports summarized above, there is no reproducible and definite report on n-type conductivity due to interstitial lithium in diamond, nor a satisfactory detailed theoretical explanation to this situation.
H. Yilmaz et al. / Diamond & Related Materials 16 (2007) 840–844
We studied the behavior of interstitial lithium atoms in diamond with the supercell method, placing lithium atoms in adjacent and in distant tetrahedral sites, and fully relaxing the diamond lattice to observe the lithium interactions with each other and with the lattice. The results provide a mechanistic understanding for the absence of n-type behavior in lithiumdoped diamond. 2. Computational methods Lattice relaxation and total energy calculations were performed using a plane wave-based density functional theory (DFT) [20,21] code — Vienna ab-initio simulation package (VASP) [22–24]. Calculations were performed using DFT within the generalized gradient approximation (GGA), with the exchange-correlation functional of Perdew and Wang [25]. We used gradient-corrected ultrasoft pseudopotentials [26] and expanded the valence orbitals on a plane wave basis set. The kinetic energy cutoff for plane wave basis set was set to 320 eV. For the Brilliouin zone integrations we used 2 × 2 × 2 Monkhorst–Pact grid [27] together with a Gaussian smearing method with σ = 0.2 eV. A finer, 3 × 3 × 3 Monkhorst–Pact grid with tetrahedron method [28] was used to calculate the electronic density of states (DOS) on the initially optimized structure of atomic coordinates. The fast-Fourier-transformation grid used for these calculations was 64 × 64 × 64. The electronic minimization algorithm for total energy calculations was based on a residual minimization scheme–direct inversion in the iterative subspace (RMM–DIIS) [29,30] and the geometry optimizations were performed via the conjugate-gradient minimization method. The optimization of atomic coordinates was continued until force on each atom in the supercell was reduced below 0.01 eV/Å. Lithium–lithium interactions in diamond crystal were studied in a 216-carbon atom diamond supercell. The lattice constant and the bulk modulus of diamond calculated with the above parameters were 3.567 Å and 4.37 Mbar, respectively, in very good agreement with the experimental values of 3.5668 Å and 4.43 Mbar. The cell shape and volume were kept constant for all calculations and the symmetries were turned off. In this supercell, the distance between a lithium atom and its closest images is 10.7 Å.
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as a reference, the formation energy of an isolated interstitial lithium atom in diamond in the neutral charge state was calculated as 8.2 eV. The diffusion of lithium in diamond is restricted by the energy barriers that are located at the hexagonal sites between tetrahedral sites. This energy barrier was found to be 1.09 eV. The formation energy and the calculated diffusion barrier energies are larger than those calculated by Ref. [17], which were 5.5 eV and 0.85 eV, respectively. Although the activation energy for lithium diffusion calculated here is in agreement with the experimental value of 0.9 ± 0.3 eV [14], another experimental study [15] reported a much smaller, 0.26 eV, activation energy for lithium diffusion in diamond. We have calculated the donor state of an isolated interstitial lithium atom using the so-called empirical marker method (EMM) [31–33]. This method provides the donor state of an impurity by comparing its ionization energy with that of a reference impurity with known donor state using the following expression: X ð0=þÞ ¼ Rð0=þÞ þ f½EðX 0 Þ−EðX þ Þ−½EðR0 Þ−EðRþ Þg:
ð1Þ
Here X (0/+) and R(0/+) stand for the donor state to be calculated and the known reference state, respectively. The terms in
3. Results and discussion 3.1. Single lithium atom in diamond Lithium atoms energetically prefer the interstitial tetrahedral sites in diamond [17]. In the tetrahedral site, the first neighbor C–C bond lengths around Li atoms are stretched out by 4.5% from their initial position after relaxation. The C–C bond lengths between the first and second neighbor carbon atoms are shorter than 1.54 Å, and depending on their direction, they vary from 1.49 to 1.52 Å. Although a minor increase is observed in the C–C bond lengths between the second and third neighbor carbon atoms, the distortions beyond that point are negligible, based on the error bar of the calculations. Taking lithium metal
Fig. 1. Structural diagrams of lithium clusters formed with (a) two and (b) three lithium atoms. Nearby carbon atoms around the lithium atoms are dark colored, lithium atoms are shown in dot–circles and dashed lines represent broken C–C bonds.
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parentheses represent the ionization energy of the two systems. We used phosphorous with Ec − 0.6 eV as a reference, following the recommendation of Goss et al. [31]. The donor state of lithium atom calculated with this method was Ec − 0.35 eV, which is deeper than the activation energy obtained from experimental studies [7,12,34], namely 0.23, 0.16 and 0.22 eV, respectively. While the theoretical study of Kajihara et al. [17] reports the activation energy of lithium to be 0.1 eV below the conduction band, that of Anderson and Mehandru [35] predicts lithium to be readily ionized in diamond. Notice that, although the reported activation energies differ among themselves, all of them indicate a relatively shallow donor doping activity for interstitial lithium in diamond. The results discussed below further indicate that this shallow donor characteristic is readily lost if lithium atoms form clusters in diamond. 3.2. Formation of lithium atom clusters in diamond Lithium atoms that are initially placed in non-adjacent tetrahedral sites remain stationary and show n-type behavior, similar to the single lithium atom case discussed above. However, if two or more lithium atoms are initially placed in adjacent tetrahedral sites, they tend to cluster by breaking the C–C bonds near them. Fig. 1a and b shows the final structure of the lithium clusters formed by two and three lithium atoms, respectively. For illustration purposes, nearby carbon atoms around the lithium atoms are dark colored, lithium atoms are shown in dot–circles, and dashed lines represent the broken C–C bonds, in Fig. 1. Lithium atoms were initially placed in consecutive tetrahedral sites along a diffusion chan-
nel and all the atomic coordinates in the supercell were fully relaxed. This configuration results in the spontaneous formation of lithium clusters, i.e. there is no barrier for this reaction. In the case of two Li clusters (Fig. 1a), two C–C bonds are broken and one sp2 -type C bond is formed. The bond length of this sp2 -type C bond is 1.41 Å, which is slightly larger than a typical sp2 C bond. The distance between two lithium atoms is 1.53 Å and the two broken C–C bonds are nearly equal, 2.14 Å and 2.15 Å. Similarly, two C–C bonds break when three lithium atoms are initially placed in consecutive tetrahedral sites in a diffusion channel (Fig. 1b). The distances between the three consecutive lithium atoms are equal, i.e. 1.64 Å, and the distances between C atoms after bond breaking are also equal, i.e. 2.25 Å. The carbon atom that lost two of its four sp3 -type C bonds formed two shortened sp2 -type C bonds, i.e. 1.36 Å, with its remaining neighbor carbon atoms. In both cases, the charge density analyses show that lithium atoms are almost fully ionized and the charge is localized over the carbon atoms around the Li clusters. The formation energy of these two and three Li atoms clusters is 1.88 eV and 3.09 eV, respectively. We calculated the formation energy of lithium clusters assuming that the lithium atoms were initially located at non-interacting distances in tetrahedral sites in the diamond lattice. Since the energy gain per lithium atom is higher in the three Li cluster than in the two Li cluster, these results seem to indicate that larger lithium clusters would be favored in diamond. Nevertheless, studying the formation of lithium clusters with larger number of lithium atoms is required in order to be able to derive a general conclusion in this respect.
Fig. 2. Electronic density of state diagrams of diamond with: (a) one isolated interstitial Li, and the defect structure formed by two (b) and three (c) lithium atoms.
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To verify whether the lithium clustering observed here is linked to the supercell size, we performed a test calculation using a 360-atom diamond supercell with two lithium atoms at adjacent tetrahedral sites. C–C bond breaking and lithium clustering were also observed in this supercell after relaxation. (This calculation was done with 2 × 2 × 1 k-mesh and energy cutoff of 300 eV.) Lithium clustering was also observed in a test calculation performed with PAW-PBE [36,37] potentials. Although the electronic levels obtained by DFT are known not to be accurate in absolute numbers, the DOS corresponding to the diamond supercell with a single lithium atom (Fig. 2a) and that with two and three lithium atom clusters (Fig. 2b–c) show distinct effects due to lithium clustering. There are no mid-gap defect states in the diamond band gap in the case of an isolated interstitial lithium atom. However, new electronic states are introduced in the diamond band gap when two and three lithium atoms form clusters by inducing C–C bond breaking. The energy levels in the DOS plots are shown relative to the Fermi level, at zero energy, which separates the occupied and unoccupied electronic states. Compared to the shallow donor level of an isolated interstitial lithium atom calculated in the previous section, the activation energy of these localized electrons in the deep mid-gap states should be much higher, thus leading to electrically inactive Li doping. These results help to explain the experimental observations of te Nijenhuis [14] and the conclusion arrived by Uzan-Saguy et al. [15]. In both of these studies, it is pointed out that lithium could be clustering below the surface of the diamond samples, and the absence of n-type conductivity was attributed to this behavior. 3.3. Li ions in diamond The results discussed above indicate that there is a relation between Li clustering and their valence electrons lowering their energy by moving into mid-gap states. In order to further explore this relation, using the same initial configurations, we repeated the calculations with +2 and +3 charged states to represent two and three lithium ions, respectively. In both of these cases no C–C bond breaking took place and the lithium ions remained stationary in their initial tetrahedral sites. Moreover, when we remove one electron per Li atom from the relaxed cell discussed in Section 3.2 and allow the system to relax again, the lithium ions move back to their initial positions and the broken C–C bonds repair themselves. In the charged states hereby considered, there is repulsion energy between lithium atoms, which is 0.86 eV and 1.87 eV for two and three lithium atoms, respectively. This repulsion energy may actually prevent occupation of neighboring tetrahedral sites in diamond when lithium is incorporated in ionic form. However the strength of the electric field applied to incorporate lithium ions in diamond might suffice to overcome this repulsion. 4. Conclusions A DFT study of lithium atoms and ions in diamond was performed. Although isolated interstitial lithium atoms in tetrahedral sites have a donor state that is 0.35 eV below the
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conduction band, their behavior is critically affected by the presence of adjacent interstitial atoms. We found that interstitial lithium atoms that are located in adjacent tetrahedral sites spontaneously induce C–C bond breaking and form lithium clusters. The charge distribution analysis shows that charge localization takes place at the defect sites over the carbon atoms around the lithium clusters. Correspondingly, the electronic DOS analysis shows that mid-gap states are introduced in the diamond band gap, thus providing lower energy states to the otherwise near-conduction electrons. Since the defect states are much deeper in the diamond band gap compared to those of the isolated interstitial lithium, these electrons require much higher energy to be activated. We also found that when lithium ions instead of atoms are considered in the calculations, none of the previously-described Li clustering or C–C bond breaking takes place. Collectively, these results imply that n-type doping of diamond with Li is in principle feasible, provided that a low doping concentration is employed and precautions are taken to ensure that the Li atoms remain mono-dispersed during doping. Note added in proof A recent study of Goss et al, to be published in PRB, in issue 7 of volume 74, confirms some of the findings presented here.
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