First-principles study on the electronic structure of n-type magnetic semiconductor Ba(Zn 1−x Co x )2As2

We perform systematic first-principles calculations on the electronic structure of n-type magnetic semiconductor Ba(Zn 1−x Co x )2As2 with the facilitation of HSE06 hybrid functional. Supercells are used to consider the doping of Co atoms, and the first-principles band structures are unfolded for clarity. Based on the calculation results, magnetic states are preferred by individual Co atoms doped in Ba(Zn 1−x Co x )2As2 at diluted limit, and carriers are originated mainly from situations where only one Co atom exists in the nearest neighbor Zn sites out of certain doped Co atoms. The origination of carriers can be explained by the density of states and the unfolded band structure, where it is found that the scattering effects from single Co atom is small but quite large when more Co atoms are located at adjacent Zn sites. The large scattering effects of two adjacent Co atoms will alter the band structures near the Fermi-level. Carriers in Ba(Zn 1−x Co x )2As2 mainly originate from the As-4p orbitals, with partial contributions from the Co-3d orbitals. Our work provides new insights into the origin of the n-type carriers in magnetic semiconductors and will inspire the development of new magnetic semiconducting systems.


Introduction
Spintronics, also known as spin electronics, is the study that involves active control and manipulation of both charge and spin degrees of freedom in solid-state materials [1,2].It attracts growing attentions since its potential application in future quantum computing devices [3][4][5] and the growing demand for next generation computers in industry.The functionality of many spintronics devices relies on the properties of magnetic semiconductors (MSs), which are a type of materials combined with both semiconducting and ferromagnetic behaviors [6,7].For example, the use of MSs in Spin field-effect transistor (spinFET) may provide a promising solution to minimize the charge current flow, leading to nonvolatile transpinor [4,8,9].The practical application of many spintronics devices need the generation of spin-polarized electrons in non-magnetic semiconducting materials at room temperature, one most promising method is spin injection from MSs [10,11].Thus, many researches have been done recent decades for finding new type of MSs that are suitable for these usages.

Results
The space group of BaZn 2 As 2 is I4/mmm (No. 139).The conventional and the primitive unit cell of undoped BaZn 2 As 2 are illustrated in figures 1(a) and (b), respectively.Doping of Co atoms is considered by employing supercells contains 18 Ba atoms, figures 1(c)-(f) show the supercells where 1 to 3 Co atoms are doped at adjacent Zn sites.The k-path for band structure calculation are illustrated in figures 1(g) and (h).

Undoped BaZn 2 As 2
Firstly, we calculate the electronic structure of undoped BaZn 2 As 2 with the primitive unit cell.As shown in figure 2(a), the band structure calculated with PBE functional shows no energy gap, while BaZn 2 As 2 is a semiconductor with an indirect gap of ∼0.2 eV as determined by Xiao et al [32].The failure can be understood by the incorrect estimation of the exchange energy of electrons with PBE functional, which will under-estimate the energy gap of insulators and semiconductors.For BaZn 2 As 2 , the energy gap is too small to be preserved when using the PBE functional.This result is consistent with the calculations performed on Cr doped BaZn 2 As 2 [33].Therefore, HSE06 hybrid functional is needed to calculate the exchange and correlation energy accurately.With HSE06 functional, the calculated band structure of undoped BaZn 2 As 2 exhibits typical indirect-gap semiconductor characteristics, as shown in figure 2(b).The energy gap between the valance band maximum (VBM) at Γ point and the conduction band minimum (CBM) at Z point is ∼0.3 eV, which is close to the experimental result.Wannier functions are obtained through projection using an initial guess composed of the atomic orbitals of Ba-d x 2 −y 2 , Zn-d, and As-p.Comparing the band structures shown in figures 2(a) and (b), we can see that, apart from the band gap, the band structures are very similar in nature.The band structure near the Fermi-level can be reconstructed well with these Wannier functions, as shown by the blue dashed line in figures 2(a) and (b).These results show that the bands of BaZn 2 As 2 near VBM and CBM are mainly contributed by As-p and Ba-d x 2 −y 2 , respectively, and HSE06 hybrid functional can restore the energy gap of BaZn 2 As 2 correctly.For accuracy, the DFT calculation of all Co doped cases provided in this work are performed with the use of HSE06 hybrid functional.

Supercell doped with 1 Co atom (uniformly doped case)
To figure out the influence of Co doping, we construct a 3 × 3 × 1 supercell from the conventional unit cell (figure 1(a)) of BaZn 2 As 2 .The supercell contains 18 Ba atoms, 36 Zn atoms and 36 As atoms.We then consider the diluted limit of Co doping by substituting one Co atom for one of 36 Zn atoms in supercell, which is equivalent to a compound Ba(Zn 0.972 Co 0.028 ) 2 As 2 where the Zn site is uniformly doped with ∼2.8% of Co, as shown in figure 1(c).Based on the calculation results, the magnetic configuration energy calculated by assuming a ferromagnetic order of magnetic moments on the doped Co atom is 1.73 eV lower than that calculated by assuming a non-magnetic order.This difference indicates that Co atoms favor magnetic state in the diluted limit of Ba(Zn 1−x Co x ) 2 As 2 .We construct Wannier functions with atomic orbitals of Ba-d x 2 −y 2 , Zn-d, Co-d and As-p, and the Wannier interpolated band structure for the majority spin and the minority spin are shown in figures 3(a) and (b), respectively.The energy gap is ∼0.2 eV, suggesting no charge carriers introduced by uniformly doped Co atoms.
Although no carriers are obtained, careful examination of these results is not only crucial for a comprehensive understanding of the electronic structures, but also significant for characterizing the role of individual Co atoms in the doped systems.Due to the significantly larger number of bands in the supercell compared to the primitive unit cells, the supercell band structure becomes considerably complex.This complexity makes it difficult to extract valuable information or compare it with the bands of primitive unit cells.To make the discussion more straightforward, we use the unfolding method [34] to derive a band structure that is similar to the bands calculated with the primitive unit cell of BaZn 2 As 2 .In the process of unfolding, Co-d and Zn-d are regarded as the same orbitals to keep the translational symmetry.The unfolded band structures are illustrated in figures 3(c) and (d) for the majority spin and the minority spin, respectively.As we can see from the results, with 1/36 uniformly Co doping, the VBM and CBM are located at Γ and Z points, respectively, which is the same situation as the undoped case.Different from the undoped case, the unfolded bands in figures 3(c) and (d) appear blurry with Co doping, which can be understood by the scattering effect introduced by Co atoms.In some region, e.g.energy ranging from −6 eV to −5 eV, the scattering effect of dopants is large; while in some other region, e.g.energy ranging from −3 eV to −2 eV between Γ and Z points, the scattering effect is small.In the energy ranges of [−6.The results obtained with supercell doped with 1 Co atom indicate that for individual Co atoms doped in Ba(Zn 1−x Co x ) 2 As 2 at diluted limit, magnetic state is preferred, the scattering effect introduced by the Co-d orbitals is significant at deep energy levels but very small near the Fermi-level, and no carriers are introduced.

Supercell doped with 2 Co atoms at adjacent Zn sites
In realistic materials, generally, the doped atoms are not uniformly distributed in ionic sites.To trace the origination of n-type carriers in Co doped BaZn 2 As 2 , we consider the cases that Co atoms are randomly distributed on Zn ionic sites.Each substituted Co atom has 4 nearest neighbored (NN) Zn sites, the 4 sites as well as the substituted Co atom are located in the same ab-plane.As is shown in figure 4, taking NN Zn sites into account, there are five situations for a Co atom substituted for a Zn atom.Assuming the doping level of Co is 2.8% (∼ 1  36 ), the possibilities of the five situations are 89.3%, 10.2%, 0.44%, 0.008% and 0.00 006%, respectively.The contribution to charge carriers of the last two situations can be safely ignored for their low possibilities.The first situation can be considered by the uniformly doped supercell with doping level of 1  36 , as discussed above.We will consider the second and the third situations.
Then, let us pay attention to the second situation in figure 4. We substitute 2 Co atoms for 2 of 36 Zn atoms at adjacent Zn sites in the supercell containing 18 Ba atoms, as shown in figure 1(d).In our calculations, we considered non-magnetic (NM) order and two type of magnetic orders, in which the magnetic moments on the two doped Co atoms are parallel (mag F ) or anti-parallel (mag A ). Results show that magnetic states are energetically preferred and the magnetic configuration energies of mag F and mag A   majority spin, the band structures exhibit with a gap of ∼0.4 eV and introduces no carriers.These results give us a possible origination of carriers in Co doped BaZn 2 As 2 .

Supercell doped with 3 Co atoms at adjacent Zn sites
To figure out whether there are carriers introduced by the third situation in figure 4, we substitute 3 Co atoms for 3 of 36 Zn atoms at adjacent Zn sites in the supercell containing 18 Ba atoms.For this situation, the angle of Co-Co-Co can be 90 • and 180 • , corresponding to 2 possible situations for the 3 Co atoms, as shown by the supercells illustrated in figures 1(e) and (f).For each situation, we consider 3 different magnetic orders, as shown in the insets of figures 6(a)-(f), and calculate the DOS for each magnetic order with HSE06 hybrid functional.The resulted DOSs are shown in figures 6(a)-(f).We find that the magnetic structure significantly influences the DOS near the Fermi-level.Specifically, when the magnetic moments of the 3 Co are parallel, there is a notable DOS at the Fermi-level for the majority spin channel, as depicted in figures 6(a) and (d).However, when one of the magnetic moments is anti-parallel to the other two, the DOS for both spin channel is significant for magnetic order of type II and V (figures 6(b) and (e)) , while it vanish for magnetic order of type III (figure 6(c)) and diminish to a very small value (∼0.03 eV −1 per formula unit, which is an order of magnitude smaller than that for magnetic order I, II, IV and V) for magnetic order of type VI (figure 6(f)).The magnetic configuration energies, the net magnetic moments per Co atom and the energy gaps of these calculations are summarized in table 1.The magnetic order energetically favored is III for a Co-Co-Co angle of 90 • and VI for 180 • .Although a few carriers will emerge for the two energetically preferred magnetic orders (III and VI), the carriers introduced by these two cases are neglected in our following discussions due to the negligible DOS at the Fermi-level and the small probability of the situation where 3 Co atoms locate at adjacent Zn sites.
The above findings imply that in Co doped BaZn 2 As 2 , carriers predominantly emerge when two or more Co atoms substitute for Zn atoms at adjacent sites and all their magnetic moments align in parallel.If any of magnetic moment is anti-parallel to the other(s), the DOS at the Fermi-level may remain significant (as in case II and V for supercell with 3 Co atoms), decrease notably (as in case VI for supercell with 3 Co atoms) or even vanish (for supercell with 2 Co atoms, or case III for supercell with 3 Co atoms).
In BaZn 2 As 2 , there may be other types of defects that may affect the formation of carrier, such as a Co atom substituted for a Ba atom and As vacancy.So, we also calculated the formation energies (E f ) of these defects.Results show that, E f of a Co atom substituted for a Ba atom is 5.1 eV and E f of an As vacancy is 1.9 eV.As a comparison, E f of a Co atom substituted for a Zn atom is −0.4 eV.Thus, the contribution to carrier density of defects such as a Co atom substituted for a Ba atom and As vacancy can be safely neglected.
Table 1.The magnetic configuration energy (Em), net magnetic moment per Co atom (m) and energy gap calculated with HSE06 functional.Em of each non-magnetic (NM) state is set to zero as a reference.

Supercell with
a For magF, all magnetic moments are parallel to each other.b For magA, the two magnetic moments in supercells with 2 Co atoms are anti-parallel to each other.c The magnetic structures of type I, II, III, IV, V and VI are shown with the insets of figures 6(a)-(f), respectively.

Discussion
With these results, we find that the carriers in Co doped samples are mainly contributed by the case where 2 Co atoms are doped at adjacent Zn sites.The difference between the densities of n-type and p-type carriers is small (<0.01/Co).Assuming the Co atoms are randomly distributed in the doped system, for Co doped BaZn 2 As 2 with a doping level x, i.e.Ba(Zn 1−x Co x ) 2 As 2 , the probability for 2 Co atoms occupying two adjacent Zn sites are 4x(1 − x) 3 , and the density of n-type (and p-type) carriers can be estimated by where V cell is the volume of the supercell and N is the number of Zn sites in the supercell.For x = 4%, the derived carrier densities by these equations are ρ n = 5.6 × 10 18 cm −3 and ρ p = 4.1 × 10 19 cm −3 .In experiments [26] of Hall effect and Seebeck effect, the majority carrier for 4% doped Ba(Zn 0.96 Co 0.04 ) 2 As 2 is n-type, and the carrier density is 10 17 − 10 18 cm −3 , which is 3 − 4 orders of magnitude smaller than that of doped conventional semiconductors.This can be explained by the generally larger mobility of electron-type carriers than that of hole-type carriers.In our calculations, we derived the value of carrier density for n-type and p-type separately from the band structure, while in experiments of Hall effect and Seebeck effect, the obtained carrier density and carrier type is not only affected by the difference in carrier density between the two type of carriers, but also affected by the difference in mobility of the two type of carriers.Therefore, our current hypothesis is that the n-type carriers of Co doped BaZn 2 As 2 in experiments arise from the difference in carrier mobility between n-type and p-type carriers.
With the small difference in the density of n-type and p-type carriers (<0.01/Co), it can be reasonably speculated that controlling the type and density of carrier in this material will be relatively easy.Actually, this conjecture has been confirmed by recent experiments conducted by our group [35], where the type and density of the majority carrier can be regulated by doping different ions together with Co into BaZn 2 As 2 .
In conclusion, we performed systematic DFT based first-principles calculations on Co doped BaZn 2 As 2 .Our calculations with primitive unit cell of undoped BaZn 2 As 2 show that PBE functional can not restore the energy gap of BaZn 2 As 2 correctly and the use of HSE06 hybrid functional is necessary for exploring the electronic properties near Fermi-level.In our calculations, the doped systems are considered by the substitution of 1-3 Co atom(s) for adjacent Zn atoms in a supercell with 18 Ba atoms.Our results show that individual Co atoms doped in Ba(Zn 1−x Co x ) 2 As 2 at diluted limit do not introduce any carriers but favor magnetic state.For supercells with 2 Co atoms at adjacent Zn sites, magnetic states of the Co atoms are also energetically preferred.n-type and p-type carriers are introduced at the same time by the 2 Co atoms substituted for adjacent Zn atoms.For supercells with 3 Co atoms at adjacent Zn sites, the magnetic orders with all magnetic moments parallel to each other are not energetically preferred, and the contribution to carrier density is small.In all these calculations, the density of states introduced by the doping of Co atoms are mostly more than ∼1 eV away from Fermi-level and the carriers in Co doped BaZn 2 As 2 are provided by As-4p orbitals (mostly) and Co-3d orbitals (in small part).The origination of the n-type carriers can be explained as follows.It is found that the scattering effect on electronic states introduced by individual Co atoms doped in Ba(Zn 1−x Co x ) 2 As 2 at diluted limit is significant at deep energy levels but very small near the Fermi-level; while, when there exists only one Co atom in the nearest Zn sites of a certain doped Co atom, the scattering effect on electronic states of the dopants becomes quite large and substantially alters the band structures near the Fermi-level.The calculated carrier density of n-type and p-type carriers from the calculated electronic structures are non-zero at the same time.We hypothesize that the n-type carriers of Co doped BaZn 2 As 2 in experiments of Hall effect and Seebeck effect is related to the difference in carrier mobility of n-type and p-type carriers.Considering the huge computational cost for estimating the mobility of this system in DFT framework, further study is needed in the future.

Calculation parameters
The DFT calculations are performed with Vienna Ab initio Simulation Package (VASP) [36,37].The PBE exchange correlation functional [38] and the HSE06 hybrid functional [39][40][41] were employed.The energy cutoff of the plane-wave basis are up to 450 eV.Wannier functions and Wannier Hamiltonian are derived with Wannier90 package [42], and further symmetrized with WannSymm package [43].The initial guess of the Wannier functions involves 17 (primitive unit cells) or 306 (supercells) atomic orbitals, which includes Ba-d x 2 −y 2 , Zn-d, Co-d and As-p.The unfolding of first-principles band structure are performed based on the Wannier functions with the methods proposed by Ku et al [34].The total energy converges to 0.1 meV/supercell.A 4 × 4 × 4 k-mesh is used for supercells in HSE06 calculations to derive Wannier functions.In the calculation of density of states with HSE06 functional, a 5 × 5 × 5 k-mesh was employed for further accuracy.

Structural parameters
The structure of primitive unit cell of BaZn 2 As 2 is relaxed from the experimental crystal structure [44] until the atomic forces are less than 0.01 eV Å −1 with PBE functional.The differences between the lattice constants before and after relaxation are smaller than 1%.The supercells are constructed based on the relaxed crystal structure.For the doped supercells with 1 or 2 Co atoms, the structures are further relaxed until the atomic forces are less than 0.04 eV Å −1 with PBE functional.

Figure 1 .
Figure 1.(a) Conventional unit cell of BaZn2As2; (b) primitive unit cell of BaZn2As2; (c)-(f) supercell of Ba(Zn 1−x Cox)2As2 consists of (c) one Co at a Zn site; (d) two Co at two adjacent Zn sites; (e) three Co at three adjacent Zn sites; (f) three Co at three adjacent Zn sites; (a)-(f) with the help of VESTA 3 package [31].First Brillouin zone for primitive unit cell (g) and supercells (h).

Figure 2 .
Figure 2. VASP bands (red solid lines) and Wannier interpolated bands (blue dashed lines) calculated with primitive unit cell of BaZn2As2.(a) Using PBE functional; (b) using HSE06 hybrid functional.

Figure 3 .
Figure 3. Calculation results of supercell with only one Co atom.(a) and (b) Wannier interpolated band structure and density of states; (c) and (d) unfolded band structure; (a), (c) the majority spin; (b), (d) the minority spin.
5, −4] eV for the majority spin, and [−4, −3] eV and [0.8,2] eV for the minority spin, the scattering effect is extremely large, and the unfolded band structure in these areas differs significantly from that of the undoped case.These differences are originated from the bands contributed by the d orbitals of Co atoms as well as the structure distortion around the dopants.These can be demonstrated by the projected density of states (DOS) illustrated in the right panel of figures 3(a) and (b): in the energy range of [−6, 0] eV, DOS is contributed mainly by As-p with limited contribution from Co-d; the contribution of Ba-d x 2 −y 2 are mainly located on conduction bands.For the majority spin, Co-d mainly affects DOS in the range of [−6.5, −4] eV; for the minority spin, Co-d mainly affects DOS in the range of [−4, −3] eV and [0.8, 2] eV.

Figure 4 .
Figure 4.All 5 possible situations of a certain doped Co atom and its 4 nearest Zn sites.Numbers after each situation are the probability of that situation when doping level of Co is 1/36 assuming the Co is randomly distributed at Zn sites.There are two possible situations in (c), the angle of Co-Co-Co is 180 • in one and 90 • in the other one.The probability of (d)-(e) is extremely lower than (a)-(c) and can be omitted.

Figure 5 .
Figure 5. Calculation results of supercell with two Co atoms located at two adjacent Zn sites.(a) and (b) Wannier interpolated band structure and density of states; (c) and (d) unfolded band structure; (a), (c) the majority spin; (b), (d) the minority spin.

Figure 6 .
Figure 6.Density of states of each magnetic structure for supercells with 3 Co atoms located at adjacent Zn sites.The magnetic structures of type I, II, III, IV, V and VI are shown with insets, where the blue balls represent Co atoms, the white balls represent Zn atoms, and the arrows represent the direction of magnetic moments.The insets to the left of (a)-(f) are the zoomed-in figures near the Fermi-level.The angle of Co-Co-Co for the three Co atoms are 90 • for (a)-(c) and 180 • for (d)-(f).