Strain-Spintronics: Modulating Electronic and Magnetic Properties of

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C: Physical Processes in Nanomaterials and Nanostructures

Strain-Spintronics: Modulating Electronic and Magnetic Properties of HfMnCO MXene by Uniaxial Strain 2

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Edirisuriya M. D. Siriwardane, Pragalv Karki, Yen Lee Loh, and Deniz Cakir J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b00594 • Publication Date (Web): 26 Apr 2019 Downloaded from http://pubs.acs.org on April 26, 2019

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The Journal of Physical Chemistry

Strain-Spintronics: Modulating Electronic and Magnetic Properties of Hf2MnC2O2 MXene by Uniaxial Strain Edirisuriya M. D. Siriwardane, Pragalv Karki, Yen Lee Loh, and Deniz Çakir∗ Department of Physics and Astrophysics, University of North Dakota, Grand Forks, North Dakota 58202, USA E-mail: [email protected] Abstract Next-generation spintronic nanoscale devices require two-dimensional (2D) materials with robust ferromagnetism. Among 2D materials, MXenes are favorable for spintronic applications due to their high electron conductivity and mobility. A recently reported MXene, Hf2 MnC2 O2 , possesses a high Curie temperature (greater than 800 K) and a high magnetic moment per formula unit (3 µB ). Since 2D materials have greater elastic strain limits than their bulk counterparts, their properties can be tuned effectively using strain engineering. Here, we investigate modifications in the structural, electronic, and magnetic properties produced by uniaxial strain on a Hf2 MnC2 O2 monolayer. The strain-free Hf2 MnC2 O2 nanosheet is an indirect-band-gap semiconductor. Our calculations predict that an indirect-to-direct band gap transition occurs at about 1%-3% tensile strain applied in the armchair direction. At 7% strain applied in the zigzag direction and 9% strain applied in the armchair direction, this semiconductor material becomes a half-metal which is favorable for spintronic applications. Under 4% compressive strain in either direction, a semiconductor-to-metal transition is predicted.

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More importantly, the Curie temperature of the material can be enhanced significantly by applying tensile strain. For instance, under 8% strain, the Curie temperature becomes greater than 1200 K.

Introduction Spin-polarized transport occurs in ferromagnetic materials, where the density of states of spin-up and spin-down electrons are shifted relative to each other near the Fermi level. 1 Spintronics, which utilizes spin-polarized transport to manipulate spin degrees of freedom, is being investigated intensively in order to develop current- and next-generation technologies such as computer memory units and quantum computing. 1,2 In ferromagnetic materials, a Curie temperature higher than the room temperature is essential to maintain its intrinsic magnetization above the room temperature. 3 High carrier mobility along with enhanced spin-orbit coupling in 2D materials make them potential candidates for spintronic applications. 4 MXenes are 2D transition metal carbides, carbonitrides, or nitrides with the general formula Mn+1 Xn Tx , where M represents an early transition metal, X stands for carbon and/or nitrogen, and T is surface termination group. 5 MXenes are made by etching out the A layers of MAX phases where A is a group-A element such as Al. Since Ti3 C2 Tx was introduced as the first ever MXene, 6 around 20 different MXenes have been synthesized including Ti2 CTx and Hf3 C2 O2 . 7,8 It has been reported that the double transition metal MXenes Ti2 MnC2 (OH)2 , Hf2 MnC2 O2 , and Hf2 VC2 O2 , and also the nitride MXenes Mn2 NF2 , Mn2 NO2 , Mn2 N(OH)2 , and Cr2 NO2 , are ferromagnetic materials exhibiting Curie temperatures well above room temperature. 5,9 Introducing approaches to control the magnetic states in those robust ferromagnetic materials is crucial in quantum information devices. 9,10 Strain engineering can be effectively applied to 2D materials due to the high elastic strain limits of nanosheets as compared to their bulk counterparts. 10–12 Strain engineering has been exploited to tune numerous properties of 2D materials like graphene, transition metal dichalcogenides (TMDs), MXenes, 2


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and phosphorene. 10,12–18 Enhanced magnetic moments have been observed in T-VS2 15 and WTe2 19 TMDs and also in Ti2 C and V2 N MXenes 20 under biaxial tensile strain. It has been reported that ferromagnetism in Li-doped TiO2 nanosheets can be tuned via uniaxial strain. 21 The ferromagnetic half metallic Ti2 NO2 2D materials can be converted into an antiferromagnetic semiconductor by applying tensile strain greater than 6%. 22 It is also reported that the magnetic properties of 2D CrX2 (X=S, Se, Te) and p-doped PtSe2 can be modified by using strain engineering. 23,24 The band gap engineering of 2D materials using uniaxial strain has been extensively studied in recent years. HSE06 studies show that Hf3 C2 O2 is a semiconductor with a 0.155 eV band gap. At around −2.5 % compressive strain, this monolayer shows metallic properties, whereas a band gap of 0.56 eV can be realized at a strain value of 5.0%. 25 Ti2 CO2 monolayer undergoes an indirect-to-direct band gap transition under biaxial strain of 4% and the uniaxial strain of 6%. 26 Similarly, the band gap of black phosphorus, PtSe2 and CrX (X=S, Se, Te) monolayers can be significantly modified by exploiting uniaxial strain. 23,27,28 Ferromagnetic semiconductors are useful as sources of spin-polarized carriers 29 for spintronic applications. Among magnetic monolayer materials with high Curie temperatures, Hf2 MnC2 O2 is a semiconductor, where it has been already proven that biaxial strain can be employed to tune the electronic and magnetic properties significantly. 30 In this research work, we applied uniaxial compressive and tensile strains on a Hf2 MnC2 O2 monolayer in two different directions, namely zigzag and armchair, to manipulate the electronic, and magnetic properties. Our Density Functional Theory (DFT) calculation show that the conduction bands are highly sensitive to strain. Due to the conduction band shifting under uniaxial strain, semiconductor-to-metal phase transition occurs at 4% compression in both directions, at 7% tension in zigzag direction, and at 9% tension in the armchair direction. The dispersion of bands at the lower part of conduction band also changes as a result of band shifting, which alters the effective mass of the electrons. Moreover, we carried out Monte Carlo simulations to investigate the variation of Curie temperature under uniaxial strain.

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We found that uniaxial tensile strain can be utilized to enhance the Curie temperature in Hf2 MnC2 O2 . For instance, the Curie temperature increases from 851 K for the strain-free material to greater than 1200 K at 8% tensile strain.

Computational Method Our first-principles calculations were based on spin-polarized density functional theory as implemented in the Vienna ab initio Simulation Package (VASP) 31–34 with Perdew-BurkeErnzerhof (PBE) 35,36 pseudopotentials in the Generalized Gradient Approximation (GGA) method. A plane wave basis set was considered with Projected Augmented Wave (PAW) method 37,38 with energy threshold value of 400 eV in the calculations. The criteria for energy and force convergence of the ionic steps were set to 10−6 eV and to 10−2 eV/Å, respectively. A vacuum space of around 20 Å was used in the c-axis direction in order to avoid interaction between two neighboring images. The Brillouin zone was sampled on a Gamma-centered 12×6× 1 Monkhorst-Pack k-mesh. The Density of States (DOS) calculations were performed using a 24 ×12×1 k-mesh. The GGA with on-site Coulomb potential method (GGA+U) 39 was used to treat the strong on-site Coulomb interaction of localized d electrons of the transition metal atoms. The U potentials are 2 eV for Hf and 4 eV for Mn which were taken from a previous. 5 In order to infer the strength of exchange couplings between Mn atoms, different super-cell sizes were considered as described in Section . HSE06 40 calculations for several strain values were done to verify the validity of the GGA+U calculations. The same magnetic ground state can be obtained from both GGA+U and HSE06 methods. Density Functional Perturbation Theory (DFPT) as implemented in VASP 41 was employed for elastic constant (Cij ) calculations. The dynamical stability was investigated by calculating phonon dispersions using PHONOPY code. 42

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Results and Discussion Strain and Structure

Figure 1: (a)Top and (b)side views of Hf2 MnC2 O2 monolayer. Table 1: The elastic constants C11 , C22 , C12 , monolayer thickness (t) and 3D Young’s modulus (Y3D ) values under -4%, 0%, 4% and 8% uniaxial strains. Strain (%) C11 (GPa) C22 (GPa) C12 (GPa) t (nm) Y3Dx (GPa) Y3Dy (GPa)

-4 429.7 394.3 91.8 0.911 408.3 374.6

zigzag (x) 0 4 8 385.2 328.1 258.5 384.9 373.4 347.8 88.2 82.4 82.5 0.901 0.894 0.888 364.9 309.9 238.9 364.7 352.7 321.4

-4 407.9 386.5 104.7 0.910 379.6 359.6

armchair (y) 0 4 8 385.2 312.9 214.7 384.9 381.7 382.1 88.2 82.4 83.6 0.901 0.894 0.887 364.9 295.1 196.4 364.7 359.9 349.5

In order to apply uniaxial strain, the rectangular cell structure for Hf2 MnC2 O2 shown in Fig. 1 was considered. The GGA+U calculations give the optimized monolayer lattice parameters as a0 = 3.253 Å and b0 = 5.634 Å. As can be seen in Fig. 1 (b), Mn is sandwiched 5



between two O-Hf-C layers where O is the surface termination. Thus, there are four Hf, four O, four C, and two Mn atoms in the rectangular simulation cell where Hf makes bonds with both C and O atoms while Mn makes bonds only with C atoms. The computed Hf-C, Mn-C and Hf-O bond lengths are 2.319, 2.201, and 2.115 Å, respectively. 2.6

(a)

2.6

Hf1-C1 Hf2-C1 Mn1-C1 Mn2-C1 Hf1-O1 Hf2-O1

2.5

Bond Length (Å)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2.5

2.4

2.4

2.3

2.3

2.2

2.2

2.1

2.1

2 -4 -2

0

2

4

6

Strain (%)

8

(b)

10

2 -4 -2

0

2

4

6

Strain (%)

8

10

Figure 2: Variation of the bond lengths under uniaxial strain in the (a) x and (b) y directions. Strain is applied in the zigzag (x ) or armchair (y ) directions as x =

a−a0 a0

and y =

b−b0 b0

where a and b are the lattice constants of the strained monolayer in the x and y directions, respectively. Due to the symmetry, studying Hf1-O1, Hf2-O1, Hf1-C1, Hf2-C2, Mn1-C1 and Mn2-C1 is sufficient for investigating the variation of the bond lengths due to the applied uniaxial strain. As can be seen in Fig. 2(a), transition metal (TM)-C bond lengths change differently according to the direction of uniaxial strain. For instance, Hf1-C1 and Mn1C1 increase under tensile strain in the y direction whereas those quantities decrease under tensile strain in the x direction. On the contrary, Hf2-C1 and Mn2-C1 bond lengths rise (decline) as tensile strain is applied in the x (y) direction. These distinct behaviors exist due to the Poisson effect where the structure contracts (expands) in one direction when a tension (compression) is applied in the perpendicular direction. Our calculations show that the Poisson ratio is around 0.22, which indicates that the bond length change in the opposite direction due to the strain is very small. Despite the fact that TM-C bond lengths 6


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are behaving differently as above, both Hf-O bond lengths increase (decrease) under uniaxial tensile (compressive) strain applied in the zigzag and armchair directions. The strain energy (Es ) under uniaxial strain is calculated using Es = E − E0 where E is the total energy of the strained material and E0 is the total energy of the strain-free material. The stress is calculated as a function of strain using the equation σ =

1 ∂E V0 ∂i

where

V0 is the volume of the strain-free monolayer and i is the strain in the direction i (x or y). Here, the thickness (6.961 Å) was considered as the distance between two oxygen layers. As was illustrated by Fig. 3, the monolayer is able to handle higher stress values when the uniaxial strain is applied in the zigzag direction. This is due to the fact that TM-C bond lengths do not increase much at higher tensile strains in the x direction compared to that in the y direction as illustrate in Fig. 2. Thus, at higher strains in the y direction, bond lengths become weaker. The ultimate stress in the x direction is 46.3 GPa while that in the y direction is 35.7 GPa. Moreover, the corresponding ultimate strain is 18% in both directions. 50

(b)

(a) 40

εx εy

3

Stress (GPa)

4

Strain Energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

1

0 -4

30

20

10

0

4

8

12

0

16

Strain (%)

0

5

10

15

20

Strain (%)

Figure 3: (a) Strain energy per formula unit and (b) stress as a function of uniaxial strain in the zigzag and armchair directions.

The surface Young’s modulus Y2D in the x and y directions for a rectangular structure can be calculated using Eq. 1 and Eq. 2 where the elastic constants C11 and C22 are the axial compression moduli in the x and y directions, respectively, and C12 is the transverse

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expansion modulus: 11 Y2Dx =

C11 C22 − C212 C22

(1)

Y2Dy =

C11 C22 − C212 C11

(2)

Since the thickness changes under uniaxial strain (see TABLE 1), we normalized Y2D as Y3D = Y2D /t and the elastic constants as Cij = Cij(2D) /t where i and j are 1 and 2, respectively. The thicknesses are calculated by considering multilayer configurations with van der Waals interactions as defined by DFT-D3 with Becke-Jonson damping. 43,44 Thus, TABLE 1 provides the Young’s modulus values and the elastic constants in GPa. At 0% strain, C11 = C22 = 385 GPa and Y3Dx = Y3Dy = 365 GPa due to the hexagonal symmetry of the strain-free Hf2 MnC2 O2 monolayer. When uniaxial strain is applied, those quantities are no longer equal (C11 6= C22 and Y3Dx 6= Y3Dy ), since the symmetry is broken. At 8% strain in either direction, there is a significant difference between x and y components of Y3D , where the latter is higher than the former. Thus, stiffness will be low in the zigzag direction, when a high uniaxial tension is applied (Y3Dx = 238.9 GPa at x = 8% and Y3Dx = 196.4 GPa at y = 8% ). Even though stiffness declines in both directions under tension due to bond weakening, there is still a remarkably high stiffness (Y3Dy > 310 GPa) in the armchair direction at 8% strains compared to that in the zigzag direction. As shown in Fig. S1 ( see the Supplementary Material), we calculated the ab-initio phonon calculations along the high symmetry directions of Brillouin zone for large strain values (10 and 18%) in order to elucidate the dynamical stability Hf2 MnC2 O2 under strain. It is clear that the Hf2 MnC2 O2 monolayer is stable at strain values as large as 10%. Imaginary vibrational frequencies in x = 18% indicates that the nanosheet cannot retain its stability at much higher strains like 18%, even though Fig. 3 indicates that the material’s ultimate strain in both directions is 18%.

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Uniaxial Strain Effects on Electronic Properties The band structures were obtained by considering the high-symmetry points (see Fig. 4 for an unstrained case) of the rectangular supercells, i.e. Γ (0,0,0), X (0.5,0,0), S (0.5,0.5,0) and Y (0,0.5,0), under uniaxial strains applied in the zigzag (x ) and armchair (y ) directions as shown in Fig. 5. Strain-free Hf2 MnC2 O2 is a semiconductor with an indirect band gap of 0.282 eV. The conduction band minimum (CBM) resides at the S point, while the valence band minimum (VBM) resides at the Γ point. As strain is applied, the lower three bands (see Fig. 4), namely CB1, CB2, and CB3, shift considerably as illustrated in Fig. 5. DOS calculations reveal that TM-d and O-p states hybridize to form CB1, while TM-d orbitals together with p orbitals of Hf, C, and O hybridize to construct CB2 and CB3. CB1 band is occupied by the spin-up electrons while CB2 and CB3 bands are occupied by the spin-down electrons, as shown in Fig. 4.

Figure 4: Spin-resolved electronic band structure and partial density of states (DOS) for strain free rectangular super-cell of Hf2 MnC2 O2 monolayer. Spin-up and spin-down bands are shown with red and blue colors, respectively. The Brillouin Zone for the rectangular super-cell is included with high symmetric points. Fermi energy marks zero energy. TABLE 2 reveals that Hf2 MnC2 O2 monolayer remains an indirect-band-gap semiconductor under tensile strain in the zigzag direction. While tensile strain rises in this direction, CB1 does not move noticeably with respect to the top of the valence band until strain of 4% 9



Figure 5: The calculated spin-resolved band structure of Hf2 MnC2 O2 for different uniaxial strain values in (a) zigzag and (b) armchair directions. Spin-up and spin-down bands are in red and blue, respectively. Fermi energy marks zero energy.

Band Gap (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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εx εy

0.2

0.1

0 -4

-2

0

2

4

6

8

10

Strain (%)

Figure 6: Variation of band gap as a function of strain. (see Fig. 5). Thus, band gap remains between 0.28 eV and 0.24 within x =0% to x =4% range, as shown in Fig. 6. After x = 4%, the bottom of CB2 becomes CBM, which appears at Y point. Since CB2 moves towards the valence band as tensile strain rises further in the zigzag direction, there is a sudden band gap drop after x = 4%. The CB2 and valence band start crossing the Fermi level at x = 7%. At x = 7%, only the bands with spin-up electrons cross the Fermi level. As a consequence, Hf2 MnC2 O2 becomes a half-metal, where spin-down

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(a)

εx = -4%

2

εx = 0%

εx = 4%

εx = 8%

PDOS

1 0 -1 -2

(b)

Hf-p Mn-p

Hf-d Mn-d -2

0

2

-2

εy = -4%

2

0

C-p O-p 2

-2

εy = 0%

0

2

-2

εy = 4%

0

2

εy = 8%

1

PDOS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 -1 -2

-2

0

2

-2

0

2

-2

0

2

-2

0

2

Energy (eV)

Figure 7: The calculated Partial Density of States (PDOS) of Hf2 MnC2 O2 for different uniaxial strain values in (a) zigzag and (b) armchair directions. Fermi energy marks zero energy. electron bands make a band gap of 0.61 eV. The partial density of states (PDOS) studies show that even though the main contributor in conduction bands of spin-up electrons near the Fermi level is Mn-d orbitals at 0% strain, Mn-d, Hf-d and C-p contribute under tensile strains. The contributions from each orbital for the valence bands near the Fermi level barely changes as strain is applied (see Fig. 7). Even though the band gap of Hf2 MnC2 O2 changes in a similar way under the strain in the armchair direction (see Fig. 6), the CBM appears at the different k-points as illustrated in TABLE 2. For the strains greater than 3% in this direction, the CBM resides between Γ and X points since CB3 reaches the Fermi level. Similar to the case of zigzag direction, the spinup electron bands are populated near the Fermi level at higher strains. Thus, Hf2 MnC2 O2 turns into a half-metal at y = 9% with a band gap of 0.82 eV made by the bands with spin-down electrons. It is clear that there is a sharp drop in band gap when compressive strain is applied on

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Table 2: The position of the conduction band minimum (CBM) for each 2D material under uniaxial strain in the zigzag (x) and armchair (y) directions. VBM resides always at the Γ point. Strain (%) -3 -2 -1 0 1 2 3 4 5 6 7 8

CBMx CBMy Γ S Γ S Γ S S S S Γ S Γ S Γ S Γ-X Y Γ-X Y Γ-X Γ-X Γ-X

Table 3: The average Bader charge transfer ∆q for each atom are mentioned for strained and unstrained Hf2 MnC2 O2 . Property ∆qHf (e) ∆qO (e) ∆qC (e) ∆qM n (e)

-4% 3.937 -2.001 -2.687 1.504

-2% 3.937 -2.007 -2.674 1.489

zigzag (x) 0% 2% 4% 3.939 3.938 3.943 -2.011 -2.011 -2.023 -2.669 -2.660 -2.652 1.480 1.466 1.464

8% 3.861 -2.058 -2.635 1.426

-4% 3.939 -2.004 -2.690 1.511

-2% 3.939 -2.007 -2.680 1.496

armchair (y) 0% 2% 3.939 3.936 -2.011 -2.009 -2.669 -2.659 1.480 1.463

4% 3.815 -2.014 -2.521 1.439

8% 3.859 -2.093 -2.474 1.417

the monolayer in either direction. CB1 forms the CBM which appears at the Γ point from x = y = -1% to -3%. CB1 moves downward and crosses the Fermi-level at x = y = -4%. Since a very small portion of the bottom of the conduction band and the top of the valence bands crosses the Fermi level, Hf2 MnC2 O2 is a semi-metal at x = y = -4%. Bottom of CB1 and VBM move towards each other to reduce the separation between the anti-bonding and bonding states. Since Hf-O bond lengths become shorter at negative strains, TM-d and O-p bonding enhances. Materials like Hf2 MnC2 O2 monolayers require special attention due to the presence of partially filled d orbitals. 45 The Hartree electron-electron repulsion potential in DFT implies how an electron interacts with the electron-density of the system. Since the electron-density contains the charge of that electron as well, an unphysical self interaction energy is present 12


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in the total energy. 46,47 This self interaction error is significant in the transition metal atoms where the unpaired electrons are delocalized to minimize the self-interaction. Our GGA calculations in Fig. S2 show that all the strained Hf2 MnC2 O2 monolayers are metallic systems (see Supplementary Material) as a result of delocalization of unpaired d-electrons. To overcome this issue, GGA+U method utilizes the on-site Coulomb potential (U), which localizes the transition metal electrons in the partially filled energy states. Furthermore, HSE06 calculations constitute the nonlocal Hartree-Fock exchange potential to minimize the effects of so called self interaction. 48,49 Thus, both GGA+U and HSE06 methods prove that the strain-free Hf2 MnC2 O2 nanosheets are semiconductors, and it retains the semiconductor properties at 4% strains in both zigzag and armchair directions, as illustrated by Fig. S3 (see Supplementary Material). As mentioned in the TABLE S1 and S2, hybrid functional calculations provide higher band gaps. For instance, 0.484 eV and 0.324 eV band gaps are obtained for x = 0% and x = 4%, respectively. On the contrary, GGA+U provides smaller band gap values of 0.282 eV and 0.239 eV at x = 0% and x = 4%, respectively. HSE06 method confirms that bands are shifting in the same way as in GGA+U method. Thus, zero band gaps exist under x = y =-4% and x =8% and also a small band gap (≈ 0.103 eV) exists when y =8% due to bands shifting. Band structure variations are directly related to the charge transfer in each atomic species. Thus, we studied the Bader charge analysis for several strain values in the zigzag and armchair directions, as mentioned in TABLE 3. 50 It is clear from TABLE 3 that the structural variations under compression in either direction considerably affect the charge transfer of Mn and C atoms as compared to those of Hf and O atoms. Small compressive strains alter ∆qMn and ∆qC noticeably as compared to ∆qHf and ∆qO . This is mainly because of considerable shortening of Mn1-C1 and Mn2-C1 bond lengths, while uniaxial compression is applied in the x and y directions respectively (see Fig. 2). However, Hf-O bond length change within 0% to -4% is relatively small. At large tensile strains, less charge transfer for Hf, Mn and C atoms can be seen due to the fact that charges are more localized when the bond strengths

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become weaker. (a) 2.5

(b) m h (m0)

1.5 1

*

*

m e(m0)

2

0

1.2

Γ-X Γ-Y

1 0.8 0.6

0.5 -2

0

2

4

0.4

6

-2

0

εx (%)

4

6

4

6

(d) 1 0.8

*

m h (m0)

*

2.5 2 1.5 1 0.5 0

2

εx (%)

(c) 3 m e (m0)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-2

0

2

4

0.6 0.4

6

-2

0

2

εy (%)

εy (%)

Figure 8: Variation of the effective mass (m∗ ) of (a) electrons and (b) holes under uniaxial strain in the zigzag direction (x ) and m∗ of (c) electrons and (d) holes under uniaxial strain in the armchair direction (y ). Here, m0 is the rest mass of electron.

The effective mass is inversely proportional to the curvature of the band dispersion. We computed the effective masses of electrons and holes as a function of uniaxial strain. The effective mass (m∗ ) can be obtained by using the equation

∗

2

m ~ =h ¯

∂ 2E ∂k 2

−1 ~e~k

(3)

where E and k are the energy and reciprocal lattice vector in a high symmetric direction ~k = k~e~k . 17 The high symmetric directions Γ−X and Γ−Y in reciprocal lattice correspond to the armchair and zigzag directions in real space, respectively.Thus, by calculating the effective mass of electrons and holes in Γ−X and Γ−Y directions, we are able to find a direction with low effective mass, which enhances the transport and thermoelectric properties. 27,51,52 Regardless of the strain direction, Hf2 MnC2 O2 monolayer is a semiconductor in the strain range of [-2%, 6%]. Thus, this strain region was considered for m∗ calculations of this 2D material. For unstrained Hf2 MnC2 O2 , the effective masses of electrons and holes are larger along the Γ−Y direction, since the bottom of CB1 in the Γ−Y direction is flatter than that in the Γ−X direction. When strain is applied in the zigzag direction, m∗e remains 14


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around 0.3m0 -0.4m0 in the Γ−X direction, while m∗e is between 2m0 and 2.5m0 in the other direction. Same kind of behavior of m∗e can be seen for y case, except at 6% strain, where m∗e is approximately same in both directions as a result of change in energy order of CB1 and CB2. Therefore, as evidenced by Fig. 8, the armchair direction contains low effective mass electrons.

Uniaxial Strain Effects on Magnetic Properties According to GGA+U calculations, the total magnetic moment (µ) per formula unit is 3.00µB (µB is the Bohr magneton). The main contribution to µ is from the Mn atom, which is in the +4 oxidation state. Our spin-polarized calculations show that contributions from other ions are negligible. The electronic configuration for Mn4+ is [Ar] 3d3 . The uncoupled 3d electrons occupy the t2g states (dxy , dyz , and dxz ), while the eg states (dx2 − dy2 and dz2 ) are empty. 5 The magnetic moment behaves approximately in the same way under uniaxial strain in both directions [see Fig.9(a)]. While tension does not affect µ, compressive strain decreases µ starting at around −2%. This can be attributed to the increased charge transfer from Mn atom to C atom under compression as mentioned in the TABLE 3. Thus, Mn atoms lose electrons, while C atoms gain them under compressive strain. µ becomes 2.986 µB at x = y = −4%. Since the magnetic moment change is negligible, we can assume that magnetic moment is approximately constant within this range of strain. Our HSE06 calculations also show a constant magnetic moment throughout this range of strain, as stated in TABLE S1 and S2 (see Supplementary Materials). In order to find the magnetic ground state of the monolayer at each strain applied, we calculated the total energies for various ferromagnetic (FM), antiferromagnetic (AFM), and non-magnetic (NM) configurations as illustrated in Fig. 10. In the AFM1 configuration [Fig. 10(b)], spin-down Mn atoms form straight lines, whereas in AFM2 [Fig. 10(c)] they form zigzag lines. AFM3 [Fig. 10(d)] has spin-down stripes that are two sites thick. AFM4 [Fig. 10(e)] is the same as AFM1 rotated by 120◦ . As shown in Fig. 11 and S4 (see Supple15



(a)

3

(b)

εx εy

εx εy

1200

TC (K)

2.995

μ (μΒ)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 27

2.99

1000

2.985 800

2.98 -4

-2

0

2

4

6

8

10

-4

Strain (%)

-2

0

2

4

6

8

Strain (%)

Figure 9: (a) The total magnetic moment and (b) Curie temperature (TC ) of Hf2 MnC2 O2 as functions of strain.

Figure 10: (a) FM, (b) AFM1, (c) AFM2, (d) AFM3 and (e) AFM4 spin configurations with rectangular super-cells indicated. Only Mn atoms are considered, since contribution for magnetic moment from other atoms are negligible. mentary Materials), we find that EFM has the lowest than energy compared to any of the other configurations. Thus the ground state is ferromagnetic under uniaxial strain within the range −4% to 10%. ENM − EFM is higher than 2.3 eV whereas EAFM − EFM is always very low (less than 0.2 eV). We approximated the magnetic behavior of Hf2 MnC2 O2 by a classical spin-half Ising model on a triangular lattice, with strain-induced anisotropy. The Hamiltonian is 53

H = E0 − J1

X

σi σj − J2

hijix

X

σi σ j ,

(4)

hijidiag

where σi = ±1 are Ising spins, hijix indicates that the sum is over nearest neighbors in the horizontal x direction, hijidiag indicates nearest neighbors in other directions, J1 and J2 are

16


Page 17 of 27

0.2

(a)

(b)

EAFM1-EFM EAFM2-EFM EAFM3-EFM EAFM4-EFM

0.15

EAFM - EFM (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.1

0.05

0 -4 -2

0

2

4

6

8

10 -4 -2

Strain (%)

0

2

4

6

8

10

Strain (%)

Figure 11: The energy differences per formula unit between each AFM and FM phases (EAFM − EFM ) in the (a) zigzag and (b) armchair directions.

(a)

(b)

Figure 12: Magnetization of Hf2 MnC2 O2 as a function of temperature for strain (ε) directions (a) x and (b) y from anisotropic Ising model simulations of 128×128 triangular lattice. couplings along horizontal and diagonal directions as shown in Fig. 10, and E0 represents the non-magnetic part of the energy. According to Eq. 4, the energies of the spin configurations

17



Page 18 of 27

in Fig. 10(a,b,e) are

EFM = E0 − J1 − 2J2

(5)

EAFM1 = E0 − J1 + 2J2

(6)

EAFM4 = E0 + J1 .

(7)

Solving for the couplings yields J1 = 41 (2EAFM4 − EAFM1 − EFM ) and J2 = 14 (EAFM1 − EFM ), which allows us to infer J1 and J2 from the results of DFT energy calculations. Thus, we obtain J1 and J2 as a function of uniaxial strain εx or εy . We used the above Ising model to predict the finite-temperature magnetic behavior of Hf2 MnC2 O2 monolayer. Although Ising models on two-dimensional lattices can be solved using analytic methods, 54 here we used Monte Carlo methods (in particular the Wolff singlecluster algorithm 55 ), which are easily generalized to multilayers or 3D models. Figure 12 shows results of Ising model simulations of 128 × 128 lattices. The magnetization M drops to zero near the Curie temperature Tc , which is strain-dependent. The Binder cumulant crossing-point method 56 was used to determine Tc accurately (see Fig. S4 in Supplementary Materials). Tc increases with increasing strain, as shown in Fig. 9(b). As illustrated by Fig. 9, Tc becomes greater than 1200 K while it converts into a half-metal under the uniaxial tensile strain higher than 8%. Thus, Hf2 MnC2 O2 monolayer can be transformed into an immensely useful material in the spintronic applications.

Conclusion A first principle study based on density functional theory was carried out on Hf2 MnC2 O2 ferromagnetic semiconductors to study the structural, electronic and magnetic property variations. According to the direction of uniaxial strain, different bond lengths are altered differently. It is clear that TM-C bonds are weakened significantly by the tensile strain in the

18


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armchair direction compared to the tensile strain in the zigzag direction. Thus, the maximum stress that the system can handle becomes lower when the tensile strain is applied in the armchair direction. The tensile strain (compressive strain) in both directions lowers (enhances) the Young’s modulus. The phonon dispersions indicate that Hf2 MnC2 O2 is dynamically stable under high strains like 10%. As a result of the conduction bands shifting under uniaxial strain, variation of CBM location can be observed while VBM always resides at the Γ point. Consequently, indirect-to-direct band gap transition occurs in 1% to 3% range of tensile strain in the armchair direction. Hf2 MnC2 O2 monolayer becomes a half-metal at x = 7% and y = 9%. Semiconductor-to-metal transition occurs at x = y = −4%. It is worth noticing that Hf2 MnC2 O2 becomes half-metal within the strain range between x = 7% and x = 10% in which Hf2 MnC2 O2 was found to be dynamically stable. Our effective mass calculations show that the armchair direction is preferable for the electron transport due to small effective masses in the Γ−X direction. Moreover, the magnetic moment remains constant (there is only a negligible change) within -4% to 10% strain range. It is found that Curie temperature is highly tunable under uniaxial strain. For instance, TC is around 706 K at -4% in both directions while it is greater than 1200 K at 8% strain.

Supplementary Material • The Band structure calculations, the band gap and the total magnetic moment values using GGA (without U) and HSE06 • Energy differences between nonmagnetic phase and ferromagnetic phase • Binder plot for finding the Curie temperature of strain free monolayer • Cohesive energy calculations and Phonon dispersions

19



Acknowledgement Computer resources used in this work was provided by Computational Research Center (HPC-Linux cluster) at University of North Dakota. A part of this work was supported by University of North Dakota Early Career Award (Grant number: 20622-4000-02624). We also acknowledge financial support from ND EPSCoR through NSF grant OIA-1355466. This work was performed, in part, at the Center for Nanoscale Materials, a U.S. Department of Energy Office of Science User Facility, and supported by the U.S. Department of Energy, Office of Science, under contract no. DE-AC02-06CH11357.

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