Magnetic properties of Mn-doped InSb nanowires from first principles

Room-temperature ferromagnetism (RTFM) has been achieved in Mn-doped InSb nanowires (NWs) through experiment. However, the underlying cause of RTFM remains unclear. In this paper, using first-principles calculations, the distribution of Mn ions and magnetic properties of Mn-doped wurtzite and zinc blend InSb NWs have been investigated. Our results indicate that wurtzite (In,Mn)Sb NWs can exhibit superior ferromagnetic behavior compared to zinc blend (In,Mn)Sb NWs. The distribution of Mn ions and magnetic properties in the (In,Mn)Sb NWs is influenced by their size, surface passivation and crystal structure. Furthermore, the ferromagnetic coupling is short-range in passivated (In,Mn)Sb NWs, and as the size of the NW decreases, the Mn-3d level becomes a deep acceptor in the band gap, resulting in an enhancement of ferromagnetism.


Introduction
Diluted magnetic semiconductors (DMSs) have attracted significant attention as a cornerstone in the design of spin-based devices due to their unique combination of semiconductor and magnetic properties.They are widely utilized in various applications, such as spin field effect transistor [1], spin valves [2], spin qubits [3] and nonvolatile Memory.In the DMS area, the intense research activity has largely been focused on Mn-doped III-V semiconductors because of the related protoype devices demonstration in both theory and experiment [4][5][6][7][8][9][10][11][12][13].InSb stands out among III-V semiconductor candidates due to its exceptional room-temperature electron mobility of 7800 cm 2 V −1 s −1 , narrowest band gap of 0.18 eV, and largest spin-orbit interaction strength [14][15][16].Furthermore, Mn-doped InSb has been shown to exhibit significant negative magnetoresistance [17], making it suitable for infrared spin photonic applications [18].Previously, significant progress has been made in the experimental synthesis of (In,Mn)Sb DMSs.For instance, at lower concentrations, Mn doped InSb homogeneous films can lead to a carrier-mediated ferromagnetism at temperatures below 20 K [19,20], but their Curie temperature (T C ) is much lower than 300K.At higher concentrations, the ferromagnetism with a T C exceeding 300K was observed in bulk InSb:Mn [21,22], this magnetic behavior is attributed to non-interacting MnSb precipitates rather than the hole-mediated ferromagnetism.With the advancements in nano lithographic fabrication and numerical computation methods, it has become feasible to develop DMS nanostructures.Since an experimental reports of T C reaching 200K in (Ga,Mn)As by patterning a heavily Mn-doped (Ga,Mn)As films into NWs [23], this success has inspired researchers to be interested in DMS NWs that takes advantage of the quantum size effect and surface effect of the NW.At the experimental level, the room temperature DMS NWs were developed using Mn-doped GaN NWs [24], Co-doped GaN NWs [25,26], Co-doped ZnO NWs [27], and (Cu,N)-codoped In 2 O 3 NWs [28].Theoretically, Zhang et al suggested that the magnetic moment and the ferromagnetic states in (Ga,Mn)As NWs can be effectively regulated by the surface dangling bonds [29].Arora et al reported that the room temperature ferromagnetism can be achieved in Mn doped H-SiNWs only in the [100] and [111] direction [30].Recently, an experimental study by Hnida et al reported that a robust  ferromagnetism of Mn-doped InSb NW was synthesized by template-assisted pulse electrode position, exhibiting a higher T C with 500K [31].The result suggests that the room temperature ferromagnetism can be achieved in Mn-doped InSb through nanoengineering techniques.However, the distribution of Mn ion and the mechanism of magnetic coupling in Mn-doped InSb NWs remain unclear, including the effects of NW size, surface properties and crystal structure.In this work, our focus is on the theoretical understanding of the formation energies and magnetic properties of Mn-doped InSb NWs.Our studies will provide important theoretical reference for the synthesis of the room temperature InSb-based DMS in nano spintronic devices.

Models and computational methods
In the present work, all structural optimizations and magnetic property calculations are performed using density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP) [32].The  Perdew-Burke-Ernzerhof (GGA-PBE) functional is employed within the generalized-gradient approximation [33].The projector augmented wave method (PAW) is chosen to represent the ionic potentials [34].The energy cutoff for the plane-wave expansion is set to 350 eV.The geometries are fully relaxed until the energy difference between two consecutive ionic steps is less than 10 −4 eV, the energy convergence criterion is set to 10 −6 eV in magnetic calculations, and the force acting on each atom is below 0.05 eV Å, respectively.For Brillouin zone integration, a Monkhorst-Pack 1 × 1 × 5 and 1 × 1 × 7 k-point meshs are chosen for the geometric relaxations and electronic structure calculations [35], respectively.According to the mean field theory, the relationship between T c and energy difference between FM and AFM states is mathematically expressed as [36,37]: where, k B and n represent Boltzmann constant and the number of the Mn atoms, respectively.Here n = 2.For bulk InSb, the optimized lattice constant is 6.46 Å, which is consistent with the experimental values (6.479Å) [38].In the ideal wurtzite (WZ) crystal, the lattice constant is given by / = a a 2

WZ ZB
and the lattice constant along the c axis is related to the in-plane lattice constant by / = c a 8 3 WZ [39].In our study, we have extracted [0001] WZ and [111] zinc blende (ZB) NWs of varying sizes from an optimized bulk material.These NWs are named WZNW1, WZNW2, ZBNW1 and ZBNW2 with the corresponding diameters of 2.3 nm, 1.4 nm, 1.6 nm and 1.0 nm respectively, as illustrated in figure 1.The WZ elementary cell has been doubled along the z direction to eliminate the interaction of the Mn ion and its periodic images.The lateral supercell size is used so that NWs are separated by 15 Å vacuum layers, ensuring no inter-image interactions occur.Based on previous calculations of III-V NWs [40][41][42], it has been determined that [0001] WZ and [111] ZB NWs with small diameters are the most stable configurations.Therefore, these two types of NWs are selected, where a Mn atom is introduced to replace the In atom at various positions (refer to figure 1).

Results and discussion
To understand the distribution of Mn ions in InSb NW and identify stable dopant position, we calculated the formation energies with a Mn atom replacing various nonequivalent In atom inside and at the surface of the NW [43] where E InSb Mn : ( )is the total energy of the doped NW, E InSb ( )denotes the total energy of undoped NW, m In ( )and m Mn ( )are the chemical potentials of bulk In and Mn, respectively.The calculated results are presented in figure 2. For the bare WZNW, we note that the formation energy of the Mn ion in the NW inside is larger than that at the surface.Furthermore, the formation energy of the Mn ions at non-equivalent positions in the NW inside is approximately equal (see figure 2(a)), indicating that the Mn ion is nearly homogeneous distribution in bare WZNWs independent of the NW diameter, this is agreement with the experimental observations [33].In contrast, in bare ZBNW1 that the Mn ion prefer to occupy the lateral surface position 5 with an surface dangling bond (SDB), the self-healing effect is present.However, in bare ZBNW2 the lowest formation energy is obtained when the Mn ion occupy the subsurface position 2.This result indicates that the distribution of Mn ion in ZBNW is related to the NW size.
To study the effect of surface passivation on the distribution of Mn ion, we take the pseudohydrogen (PH) atoms Z H with both fractional nuclear charge Z and fractional shell electron to terminate the SDBs of the NW.In bulk InSb, the number of In valence electron is three, an In gives 3/4 electrons to one of the nearest anions.Based on the fact that an H needs two electrons to form the 1s shell, Z = 2-3/4 = 5/4 pseudohydrogen ( 5/4 H) has to be chose to passivate the In SDBs.In the same way, the number of Sb valence electron is five, to form an eightelectron stable closed shell, we choose Z = (8-5)/4 = 3/4 pseudohydrogen ( 3/4 H) to passivate the Sb SDBs.In our earlier studies [44][45][46], this method is applied to passivate the SDBs of the NW and quantum dots, obtaining some good results.For the WZNW, after surface passivation, the formation energy of Mn atom at the surface position can be significantly increased, but slightly higher than that in the interior as shown in figure 2(c).This result is consistent with the Mn-doped AlN NWs by Zhang et al [47].In figure 2(d), for passivated ZBNW, we can clearly see that the formation energy of Mn ions at different positions is approximately equal, suggesting that the distribution of the Mn ion tends to uniform.The above results indicate that surface passivation on the distribution of Mn ion is sensitively dependent on the crystal structure and the NW size.We also find that in all passivated configurations, the total and the Mn local magnetic moment is about 3.5 m B and 3.45 m , B respectively, which is smaller than the theoretical value (4 m B ), this is attributed to the strong Mn 3d and Sb p states hybridization.Here, to verify the localization of the Mn magnetic moment, in figure 3, we plot the DOS and the spin density distribution for a Mn atom in four configurations: figure 3(a), a Mn doped WZNW1; figure 3(b), a Mn doped WZNW2, figure 3(c); a Mn doped ZBNW1; figure 3(d), a Mn doped ZBNW2.For all configurations, there is a strong localized magnetic moment at the Mn atom position.The Mn local orbital has a d character with spin up, however, in the Mn-Sb bond direction, the orbital changes to p character with spin down.
To investigate the interaction between Mn dopants, two In atoms with two Mn atoms in InSb NWs were substituted, the models are shown in figure 4. We chose six pair configurations with different positions and distances, and the calculated results are listed in table 1.For the bare NWs, we find that the lowest energy is corresponding to Mn atoms located away from each other and inside the NW.In contrast, for the passivated WZNW2, two Mn atoms perfer to located close to each other and inside the NW.For the passivated ZBNW2, the lowest energy is that one Mn atom resides inside the NW and the other on the surface.
To research the effect of surface passivation on the magnetism of Mn doped InSb NWs, we calculated the energy difference between the total energy of the NW with Mn dopants at the ferromagnetic (AFM) and antiferromagnetic (FM) states, as shown in figure 5(a), where the positive and negative values correspond to FM and AFM states.We found that, for the WZNW2, after surface passivation, the FM state is obviously enhanced.For the bare ZBNW2, the lower energy corresponds to the AFM state of the Mn spins.However, for the PHpassivated ZBNW2, a strong FM coupling has been obtained.This shows that the surface passivation has a significant effect on the magnetic coupling strength and enhance the FM coupling in Mn-doped InSb NWs.In figure 5(b), the T c of NW2 is estimated, we observe that the T c can be raised above room temperature after NW passivated by PH.To probe the causes of the different magnetic coupling behavior in those cases, we calculated the DOS in FM states NW near the Fermi level, as shown in figure 6.From figure 6(c) we can see that for the case of the bare ZBNW2, the surface states stemming from 2-fold coordinated atoms with additional dangling bonds are present in the band gap, a distinct metallic property is present.However, when the ZBNW2 is passivated by PH, there is a band gap in the spin down channel and the half-metallicity in the spin up channel, the surface states are removed, displayed in figure 6(d).This explains why the transition of the ZBNWs from the AFM to the FM state is occurred by the surface passivation.
To compare the different magnetic behavior of Mn doping in InSb NWs, we calculated the energy difference (dE) between AFM and FM states of the NW with different Mn-Mn distance, as shown in figure 7. We find that, when the two Mn atoms in the NW are separated by more than one bridging Sb atom, the configuration is less stable, the strength of the FM coupling rapidly decreases as a function of the Mn-Mn distance.This result suggests the spin exchange interaction between two Mn atoms is a short-range one.This may be a reason for the transition from paramagnetic states to ferromagnetic states of the Mn doped InSb NWs by the experimental observations [33].The reason is that the four nearest Sb atoms of Mn form an tetrahedral structure, under the tetrahedral crystal field, the Mn-3d orbitals are split into the high-lying 3 fold-degenerate t g 2 orbitals (d , xy d yz and d xz ) and low-lying 2 fold-degenerate e g orbitals (d z2 and - d x y 2 2 ) .When the distance between the two Mn atoms increases, the bonding orbitals formed by the two spin-down t orbitals is smaller than the antibonding orbitals, so the spin exchange interaction strength gradually decreases.Importantly, when the two Mn atoms bonding to the same Sb atom, the spin exchange interaction in WZNW1 (ZBNW1) is larger than that in WZNW2 (ZBNW2).The reason is that the NWs with smaller size have larger relaxation effect, which leads to the decrease of the distance between the two Mn atoms.For instance, the Mn-Mn distance (4.50 Å) in the WZNW1 is less than that in WZNW2 (4.53 Å).The results show that the quantum size effect can enhance the ferromagnetic stability in Mn doped passivated InSb NW.In addition, the T c increases as the distance between Mn atoms decreases and the NW diameter reduces.
To probe the origin of the different FM coupling behavior of the NW with various sizes, the bandstructures of the systems with an isolated Mn atom are illustrated in figure 8.In all the considered configurations, we can see that the Mn-3d orbitals are more localized than those of the host.With the decreasing NW size, the lowering of the VBM is most dramatic, the difference between  E t 2 and E VBM is significantly increased.

Figure 1 .
Figure 1.The geometrical structures of ZB and WZ InSb NWs with different diameters and surfaces grown along [0001] and [111] directions.The top row represents bare NWs, the bottom row shows PH-passivated NWs.Blue, red, yellow and pink spheres denote In, Sb, 1.25H and 0.75H, respectively.Arabic numerals indicate the substitutional position of Mn dopants.

Figure 2 .
Figure 2. The formation energy of ZBNW and WZNW with different diameters and surfaces as a function of different Mn atom positions, as labeled in figure 1.

Figure 3 .
Figure 3. DOS and the spin density distribution for a Mn doped InSb (a) WZNW1, (b) WZNW2, (c) ZBNW1 and (b) ZBNW2, respectively.Where, the yellow is the Mn d 3 orbitals, the green represents the Sb p orbitals.

Figure 5 .
Figure 5. (a) The energy differences between the AFM and FM states for the bare WZNW2, PH-passivated WZNW2, bare ZBNW2 and PH-passivated ZBNW2.(b) The estimated T c for the bare WZNW2, PH-passivated WZNW2 and PH-passivated ZBNW2.

Figure 6 .
Figure 6.DOS of NW with FM coupling.(a) bare WZNW2, (b) passivated WZNW2, (c) bare ZBNW2 and (d ) passivated ZBNW2.The dash lines indicate the position of the Fermi level.

Figure 7 .
Figure 7.The FM coupling strength and the T c on the distance between Mn atoms for (a) WZNW and (b) ZBNW.

Figure 8 .
Figure 8.The band structure of PH-passivated NW with the strongest FM coupling.(a), (b), (c) and (d) corrspond to WZNW1, WZNW2, ZBNW1 and ZBNW2, respectively.The purple lines and red dotted lines correspond to the total band structure and Mn-3d band structure, respectively.The horizontal dash lines indicate the position of the Fermi level.

Table 1 .
Relative ΔE(eV) total energy respective to the most stable configuration.The total energy of the most stable configuration is set to zero.