Density-functional theory study of half-metallic heterostructures: interstitial Mn in Si

Using density-functional theory within the generalized gradient approximation, we show that Si-based heterostructures with 1/4 layer $\delta$-doping of {\em interstitial} Mn (Mn$_{\mathrm int}$) are half-metallic. For Mn$_{\mathrm int}$ concentrations of 1/2 or 1 layer, the states induced in the band gap of $\delta$-doped heterostructures still display high spin polarization, about 85% and 60%, respectively. The proposed heterostructures are more stable than previously assumed $\delta$-layers of {\em substitutional} Mn. Contrary to wide-spread belief, the present study demonstrates that {\em interstitial} Mn can be utilized to tune the magnetic properties of Si, and thus provides a new clue for Si-based spintronics materials.

Using density-functional theory within the generalized gradient approximation, we show that Sibased heterostructures with 1/4 layer δ-doping of interstitial Mn (Mnint) are half-metallic. For Mnint concentrations of 1/2 or 1 layer, the states induced in the band gap of δ-doped heterostructures still display high spin polarization, about 85% and 60%, respectively. The proposed heterostructures are more stable than previously assumed δ-layers of substitutional Mn. Contrary to wide-spread belief, the present study demonstrates that interstitial Mn can be utilized to tune the magnetic properties of Si, and thus provides a new clue for Si-based spintronics materials. Doping of semiconductors with magnetic 3d transition metal impurities has attracted much interest: recent advances in spintronics have been largely due to magnetic semiconductors produced in this way. Their spin-dependent electric transport properties [1] could be exploited in novel magneto-electronic devices, e.g. spin filters. Since it has long been known that manganese impurities can have a high magnetic moment [2,3], research has focused Mn-doped magnetic semiconductors based on GaAs and Ge [4,5,6]. Moreover, doping of GaAs by a δ-layer of Mn has been proposed theoretically [7], and an experimental realization has been reported recently [8]. The calculations of spin-polarized transport [7] employing density-functional theory (DFT) within the local-spin-density approximation (LSDA) and pseudopotentials predicted that GaAs doped with a δ-layer of substitutional Mn (Mn sub ) on Ga sites is a two-dimensional (2D) ferromagnetic half-metal with large exchange coupling. Indeed, a Curie temperature of up to 250 K has been measured on Mn δ-doped GaAs samples [8]. Adding magnetic functionality to the most common semiconductor, Si, is still in its infancy, despite recent reports about ferromagnetism in Mn-doped Si created by ion implantation [9]. Recently, Qian et al. [10] proposed, on the basis of pseudopotential calculations within the generalized gradient approximation (GGA), that a Si heterostructure with Mn sub δ-doping should be a 2D ferromagnetic half-metal. So far, research on Mn-doped Si has concentrated on substitutional Mn [11,12], probably motivated by the physics of dilute magnetic semiconductors, where it has been argued that Mn sub is crucial for ferromagnetism [5,6,13,14]. However, these substitutional Mn impurities in Si are energetically less stable than interstitial ones [15,16,17]. This has been considered a serious obstacle in creating magnetic Mn:Si calling for growth techniques far from equilibrium [15].
In this Letter, we investigate the role of interstitial Mn (Mn int ) impurities for ferromagnetism in Si by means of all-electron full-potential DFT calculations. We propose a novel type of heterostructures with Mn int δ-doping.
For occupation of each forth interstitial site in a single layer by Mn, we find 2D half-metallic behavior with a Kohn-Sham gap of 0.5 eV in the electronic density of states in the majority spin channel. Even for higher doping densities, the spin-polarization is still large, >60%, judged from the density of states at the Fermi level. This is different from the previously proposed Mn sub δlayers [10], where a (smaller) gap in the minority spin channel between Mn states of e and t 2 -type symmetry was responsible for the half-metallicity. Moreover, our proposed Mn int δ-layer is calculated to be 0.5 eV per Mn atom lower in energy than the heterostructure proposed by Qian et al. [10], in line with the known higher stability of isolated Mn int impurities compared to Mn sub impurities in Si [16,17]. Consequently, Si self-interstitials can destroy the half-metallicity of the heterostructure assumed by Qian et al. by energetically favorable site exchange with Mn sub .
We perform DFT-GGA [20] calculations using the full-potential augmented plane-wave plus local-orbital method [21], which was shown to be appropriate for description of the Mn:Si system [18]. The lattice con- stant of 5.48Å for bulk Si calculated in GGA is used to construct the heterostructures we study. The muffintin sphere radii are chosen to be 1.11Å for both the Mn and Si atoms. Converged results are obtained at a cut-off energy of 13.8 Ryd for the interstitial plane-wave expansion, and with a set of 12×12×1 special k-points for integrations over the Brillouin zone of the (1 × 1) heterostructure (and an equivalent density of k-points for larger supercells, e.g., 6×5×1 for the c(4 × 2) cell). All Si and Mn atoms except for the central Si layer are relaxed until the calculated atomic force for each of them is smaller than 0.05 eV/Å. The c(4 × 2) heterostructure (see Figs. 1(a) and 1(b)) displays a half-metallic electronic band structure as shown by the density of states (DOS) derived from the Kohn-Sham energy levels (see Fig. 2(a)). The spin-up channel (dashed blue lines) is insulating with a Kohn-Sham gap of 0.5 eV. The spin-down channel (solid red lines) is metallic, and the Fermi level lies 0.2 eV below the bottom of the spin-up conduction bands. While the Mn 3d spin-up orbitals are fully occupied, the pla-nar spin-down x 2 − y 2 orbitals form a conduction band with bandwidth of 0.7 eV, contributing to the DOS in the metallic channel. The narrower xz-and yz-derived conduction bands are also partially occupied and add to the DOS at the Fermi energy [22]. The coplanar Si (Si 0 ), the first-neighbor-layer Si (Si 1 ) and the second-neighborlayer Si (Si 2 ) (cf. Fig. 1(b)) also contribute considerably to the half-metallic DOS through Mn 3d-Si 3s3p hybridizations. Such contribution decreases strongly for the third-neighbor-layer Si (Si 3 ) and eventually vanishes for the central-layer Si (Si 8 ). The Mn atom has a local spin magnetic moment (within the muffin-tin sphere) of 2.56 µ B , and its four Si neighbors Si 1 (to which it is tetrahedrally coordinated) get spin-polarized with a spin moment of about 0.024 µ B each. Any other Si atom has a spin moment less than 0.01 µ B . Taking also into account the spin moment of 0.25 µ B outside the muffin-tin spheres, stemming from the Mn 3d-Si 3s3p hybridized bands, the total spin moment is an integer of 3 µ B per supercell [23]. Note that due to the long distance of 8.7Å between the center and corner Mn atom(s) in Fig. 1(a) the magnetic coupling turns out to be weak, but ferromagnetic (FM): Our calculations show that the FM state is slightly more stable than the antiferromagnetic (AF) one by 5 meV/Mn, consistent with the short range of FM interactions in zincblende MnSi and MnGe [24,25].
The 1/4 ML p(2 × 2) heterostructure (not shown, but refer to Fig. 1(c) and remove the central Mn in the p(2 × 2) cell) also displays half-metallicity, similar to the above discussed 1/4 ML c(4 × 2) heterostructure. The spindown x 2 − y 2 orbitals hybridized with Si sp and the spindown xz and yz orbitals determine the half-metallic DOS (not shown). The p(2 × 2) superstructure is only sightly less stable than the above c(4 × 2) one by 20 meV per supercell, which shows that the c(4 × 2) superstructure gains a little more by lattice relaxation.
For the 1/2 ML p(2×2) heterostructure (see Fig. 1(c)), the DOS is displayed in Fig. 2(b), indicating a high spinpolarization at the Fermi level of 85%. The Mn spin moment is 2.72 µ B and the total one is 6.34 µ B per supercell (3.17 µ B per Mn). Compared with the c(4 × 2) supercell, the conduction bands become broader due to decreased Mn-Mn distance and enhanced Mn d-Si sp hybridization. The FM exchange coupling of Mn to its four first neighbors at 5.5Å is quite large, J = 13 meV. This estimate is based on a Heisenberg model with local Mn spin of S = 3/2, and the calculated stability of the FM state over the AF state 4JS 2 = 120 meV per Mn atom. Thus the 1/2 ML p(2 × 2) heterostructure combines the advantage of a stronger in-plane FM coupling with a still high spin-polarization. Now we turn to the 1 ML (1 × 1) heterostructure (see Figs. 1(d) and 1(e)). Even in this case, spin-polarization is still > 60%, as shown by our calculations (Fig. 2(c)). The Mn and Si atoms form broad conduction bands. The Mn spin moment is 2.68 µ B and the total one is 3.12 µ B per (1 × 1) supercell. According to our calculations, the Mn-Mn in-plane FM coupling turns out to be further increased: the FM state is more stable than the AF state by 570 meV/Mn. Since the direct magnetic coupling between two neighboring Mn atoms (with distance of 3.9Å) is AF, we attribute the increased FM coupling with shorter Mn-Mn distance to an enhanced interaction of double-exchange type. It operates in the impurity band formed by the spin-down 3d states of Mn with t 2 character, as seen in Fig. 2, and is mediated by spin-down Si sp states. As a result, we find the Si atoms to have small spin moments parallel to Mn in all the above calculations.
Next we show that the heterostructure with Mn sub (Figs. 1(f) and 1(g)) proposed by Qian et al. [10] is energetically unfavorable. Fig. 3(a) shows that our calculations yield half-metallic behavior with a Kohn-Sham gap of 0.25 eV for 1 ML Mn sub (1 × 1), in agreement with Qian et al. [10]. The Mn spin moment is 2.97 µ B , each first-neighbor Si has a spin moment of −0.116µ B , and the total one is again integer, 3 µ B per (1 × 1) supercell. However, the Mn sub heterostructure turns out to be less stable than the Mn int one by 490 meV per (1 × 1) supercell, assuming bulk Si as reservoir for the extra Si. Knowledge about Mn-Si compounds corroborates the metastability of tetrahedrally coordinated Mn sub : Mn is known to prefer high coordination to Si, as exemplified by the seven-fold Mn-Si coordination in the ground-state crystal structure of MnSi, or the eight-fold coordination in the CsCl structure [18]. In contrast, the hypothetic zinc-blende structure of MnSi with tetrahedral coordination of Mn to Si is calculated to be extremely unstable, 2.3 eV higher in energy than the ground state. The heterostructures with Mn int proposed by us, having four second-neighbor Si atoms (2.7Å) in addition to four first-neighbor Si atoms (2.4Å), comply with the rules for the optimal Mn-Si coordination derivable from bulk phase stability.
We note that both types of δ-layers considered are only metastable in the thermodynamic sense, i.e. they could be destroyed by annealing. However, the tendency of Mn to form ordered subsurface layers, as ob- served recently [19], suggests that growth kinetics could be exploited to fabricate the Mn int heterostructure proposed by us. Although the reported Mn-induced c(4 × 2) structure has so far been observed only in islands, extended layers might be obtainable by improved growth techniques, and could be overgrown at lower temperatures with a capping layer. After successful preparation, the heterostructures are still jeopardized by degradation due to highly mobile [26] Si self-interstitials: By studying the 1 ML Mn sub p(2 × 2) heterostructure with an extra 1/4 ML co-planar interstitial Si, we find that interstitial Si is capable of site exchange with one Mn sub , thereby considerably lowering its energy by 1.2 eV. This irreversibly destroys the half-metallicity of the Mn subheterostructure (cf. Ref. 10) by degrading the spinpolarization at the Fermi level to about 40%, as seen in Fig. 3(b). In the Mn int -heterostructure, trapped coplanar Si interstitials have a less drastic effect on the electronic structure, but they could shift the Fermi level out of the half-metallic gap if present in high concentrations.
We stress that the half-metallicity in our proposed Mn int -heterostructure (metallic minority spin channel) and in the Mn sub -heterostructure of Qian et al. (metallic majority spin channel) have different physical origin. This is explained in the level diagram of Fig. 4. The Mn int 3d states are energetically located near the top of the Si-host valence bands ( Fig. 2(a)). The weak crystal field with T d symmetry and strong Hund exchange stabilize a high-spin state for Mn int [2,3]. The 4s →3d electron transfer leads to a 3d 7 (t 3 2↑ e 2 ↑ t 2 2↓ ) configuration ( Fig.  4(a)). Thus the partially filled t 2↓ band is responsible for the metallic spin-down DOS seen in Fig. 2(a). For Mn sub , however, the Mn states of t 2 symmetry mix strongly with Si sp states of the same symmetry, thus splitting the t 2 states into a bonding and an antibonding band, lying in the valence and in the conduction band of the Si host, respectively (c.f. Fig. 3(a)). The Mn sub 3d t 4−2y 2 and 4s 2y , in total four electrons, fill up the bonding states, together with the Si sp electrons (Fig. 4(b)). Note that the 4s → 3d electron transfer in the Mn sub is not as complete as in the Mn int , as our calculations find the Mn sub 3d occupation (within the muffin-tin sphere) to be less than the Mn int one by 0.2 electrons. Since the non-bonding e ↑ states (lying below the t 2↑ states for T d symmetry) are fully occupied by two electrons, the remaining one electron resides in the t 2↑ antibonding band that gives rise to the DOS in the spin-up channel in Fig. 3(a).
To conclude, we proposed a stable Si-based heterostructure using δ-doping by interstitial Mn, and proved it to be half-metallic for 1/4 monolayer of Mn, using DFT calculations. A recent combined STM-DFT study [19] lets us expect that such structures could be prepared by molecular beam epitaxy. In order to achieve ferromagnetic ordering at room temperature, one should aim at higher Mn concentrations. Even for a full δ-layer of Mn int , the spin polarization of the conduction electrons should still be as high as 60%.