Variation of exchange energy in δ-(Ga,Mn)As films under tensile strain: PBE and LDA+U calculations

Abstract

PBE and LDA+U calculations were both used to clarify the variation of exchange energy in three kinds of δ-(Ga,Mn)As films under tensile strain. All configurations were strong FM types regardless of which of the two methods was used under strain. Tensile strain could affect the exchange energy and the thickness of three kinds of films. In some configurations, the exchange energy could be enhanced by shape deformation. The thickness of δ-(Ga,Mn)As from LDA+U calculations was greater than from PBE with the same lattice constant in each structure. PBE showed strong double exchange, while LDA+U resulted in a p–d exchange between Mn atoms in the DOS structure. All such films could be grown layer-by-layer based on the formation energy thereof.

---

1 Introduction

As a diluted magnetic semiconductor (DMS),^1–3 the doping of Mn in GaAs, when it was prepared by molecular beam epitaxy in 1996,⁴ has generated much research.^5–8 Films made from (Ga,Mn)As remain worthy of research and in the last two years this had covered topics such as: the spin–orbit torque effective fields,⁹ the vertical magnetisation gradients,¹⁰ electronic excitations of a magnetic impurity state,¹¹ electrical transport mechanisms,¹² and magnetic anisotropy fields.^13–17 To overcome the low solubility of Mn in earlier years,¹⁸ a GaAs/MnAs superlattice (called δ-(Ga,Mn)As) has been constructed by molecular-beam epitaxy (MBE).¹⁹ A Curie temperature (T_c) of up to 250 K has been reported for δ-(Ga,Mn)As with Mn concentrations of up to 14.38%:²⁰ this is much higher than the 170 K of alloy Ga_1−xMn_xAs thin films with high Mn concentrations (12.2% ≤ x ≤ 21.3%) grown by MBE²¹ and 154 K with Mn concentrations ranging from 6 to 15%.²² Sanvito and Hill also investigated the half-metallic properties of δ-GaMnAs from first principles.²³ The exchange interaction was much stronger than that of a random alloy with the same Mn concentration for the confined hole carriers of Mn within a few monolayers around the two-dimensional (2-d) MnAs plane.²³ Based on this model, the shape deformation induced enhancement of ferromagnetism in δ-(Ga,Mn)As was explored from first principles: this could be useful when looking to increase the T_c of δ-(Ga,Mn)As.²⁴

This theoretical research into δ-(Ga,Mn)As used a bulk model.^23,24 Nowadays, the size of electronic devices is near-nanometric. The properties of traditional materials are no longer suitable for current semiconductor research, in which the nano-effect, surface effects, etc. must be considered. Reflected in the DMS, magnetic materials will take the form of thin, 2-d layers in the material.² For example, a film thickness of 10 nm ensures no precipitation of hexagonal MnAs clusters.²¹ Combining our research,²⁴ shape deformation induced magnetism in the δ-(Ga,Mn)As film is an important topic: film lattices should be changed to match the lattice of the substrate. Research into (001) films is rendered necessary and urgent by the large T_c in δ-(Ga,Mn)As.²⁰ Reliable theoretical guidance thus obtained can be applied to δ-DMS research for example to the benefit of materials such as: Cr-doped zincblende BeS,²⁵ Cr-doped rock-salt, SrS, SrSe, and SrTe,²⁶ Li co-doped δ-(Zn,Mn)Se,²⁷ substitutional δ-SiMn,²⁸ and even free-standing single-atom-thick iron in graphene pores.²⁹ Furthermore, local-density approximation plus U (LDA+U) should be also considered³⁰ to describe the localised d states of the Mn atom, and make the peak Mn d states collocate with experimental values thereof.³¹ Although in principle, magnetism can be attributed to the confinement of hole carriers and the surface effect, the details and fundamental mechanism behind the contributions toward the magnetic ordering of δ-DMS film remain unclear.

Here, the shape deformation-induced variation of ferromagnetism was explored by means of systematic density functional calculations (DFT) with, and without, LDA+U in δ-(Ga,Mn)As with three different types of thin layers. All configurations were strong FM ones (in both methods) under strain. Tensile strain affected the exchange energy and the thickness of these films. Perdew–Burke–Ernzerhof (PBE) showed a strong double exchange (DE), while LDA+U resulted in a p–d exchange (PE) between Mn atoms in densities of states (DOS) structures. All of the films could be grown layer-by-layer based on the formation energy thereof. Only Str. A was stable with regards its bulk chemical potentials in PBE calculations.

2 Method and model

The calculations were as implemented in the VASP package.³² The pseudopotentials were generated by projector-augmented wave (PAW) method at the level of semi-local PBE.³³ The strong correlation effect was described by the LDA+U approach.³⁴ For Mn atoms, U was chosen as 4.0 eV in Dudarev's approach.³⁰ An energy cut-off of 300 eV was chosen for the expansion of the wave functions. Testing the calculations with E_cut values of 350 eV and 400 eV indicated that the strength of the Mn–Mn exchange interaction (exchange energy), measured as the difference between the total energy of the AFM and FM states, ΔE_AFM–FM = E_AFM − E_FM, converged to less than 2 meV. For static calculations, 16 × 16 × 1 Monkhorst–Pack K-points were used. For geometric relaxation, the same concentrated K-points were used under the condition that the Hellmann–Feynman forces were less than 10⁻² eV Å⁻¹. Gauss smearing was used for the partial occupancies. A smearing width of 0.02 eV was used to improve the accuracy of the calculations.

The crystal constant of bulk GaAs was chosen as 5.74 Å in the PBE calculations. Three types of thin layers were chosen as shown in Fig. 1 (labelled A to C, respectively). All of these thin layers were constructed in the model such that, three GaAs cubic cells (eight atoms in a diamond crystal cell) were aligned along the z-direction. A vacuum region of 21.2 Å was set between the slabs for structure (Str.) A. One of the Ga planes was substituted by Mn in the middle of the body and periodic boundary conditions were applied and referred to as δ-(Ga,Mn)As below. This led to the development of a superlattice (GaAs)₃/Mn/(GaAs)₃ for Str. A as shown in Fig. 1. It meant that the outermost layers were both Ga in Str. A. In Str. B, one of the outermost Ga layers was deleted compared with Str. A. Str. B stands for the type in which the outermost layers were Ga and As atoms. In Str. C, another Ga layer was also deleted, compared to Str. B. Str. C represents the type in which the outermost layers were both As. Str. A, B, and C contained 14, 12, and 10 cations, respectively: the substitution of two Mn atoms corresponded to an Mn concentration of 14.29%, 16.67%, and 20.00%, respectively, which was close to the experimental value of 14.38% for δ-type (Ga,Mn)As.²⁰ The δ-(Ga,Mn)As thin films changed with a, as shown in Fig. 1. The height of the film changed as it relaxed because of the vacuum region.

Fig. 1 Three kinds of (Ga,Mn)As film configurations: A, B and C are shown in ball format. Small blue, big green and middle purple balls represent As, Ga and Mn atoms respectively. x, y and z are axes of the film; a is lattice parameter of film in the x- and y- directions, named as lattice parameter. [100] and [001] is the direction of x and z, respectively.

3 Results and discussion

To see how tensile strain affected the exchange energy, a was varied around the equilibrium lattice constant in each film, by increasing and decreasing its values in 0.02 Å increments. The PBE and LDA+U results for three kinds of films were shown in Fig. 2. All configurations in Fig. 2 were strong FM types in both methods, and regardless of strain. However, the value, and variation, of exchange energy differed with different structures and methods. In Str. A, the exchange energy was much larger in PBE calculations at each lattice parameter, because of the strong double exchange (DE) in the PBE results^24,27,30 and weak PE in the LDA+U results.³⁰ The same phenomenon was also obvious in Str. B and C as shown in Fig. 2. It was found that the exchange energy increased when a decreased in the PBE results, but the opposite trend emerged from LDA+U calculations in Str. A. Comparing the exchange energy at the lattice constant (the value at 5.74 Å for PBE, and 5.76 Å for LDA+U), strain could enhance the exchange energy, but the trend indicated the reverse. Why was there a change of direction between approach PBE and LDA+U? It was noted that the FM energy curve in PBE did not keep up with that of LDA+U. At small a, the FM structure was more stable in PBE, and at large a, the LDA+U FM structure was more stable. This was due to the short-ranged DE, and long-ranged PE, effects in PBE and LDA+U methods, respectively.³⁰ While the opposite trend emerged in the AFM structure. The fewer electrons around the Fermi energy (E_F) in LDA+U made the super-exchange (SE) more suitable at small a and less suitable at large a, compared to the PBE results.³⁰ This caused the different trend in exchange interaction in both methods. In Str. B, the largest exchange energy was at the lattice constant of 5.70 Å in PBE, while it had the same trend as Str. A in LDA+U. The differences between PBE and LDA+U were similar to those in Str. A, but without one napped Ga layer, the lattice constant was a little less than in its bulk state, and this changed the crystal field. The trend of FM energy in PBE and LDA+U was almost the same, although there remained a small difference. In Str. C, the exchange energy became larger with increasing a in PBE, which differed from Str. A and B: it was vibrational with the largest mode at the lattice constant of 5.52 Å in LDA+U. Also, it was seen that, the lattice constant of the film decreased from Str. A to C with reducing layer thickness, and the lattice constant of Str. C deviated from its bulk state value. This meant that Str. C retained little of the characteristics of the ZB structure, and its magnetism became more complex. This explained the exchange energy direction change between Str. A and B, and Str. C.

Fig. 2 FM (black square line), AFM (red circle line) and exchange energy (blue angle line) of Str. A (A), Str. B (B) and C (C) as a function of the lattice parameter a. The thick solid lines are for the results with PBE, while the thin dashed lines are for LDA+U. Short vertical lines give out the lattice constant in FM state in each film.

Under tensile strain, the thickness of each film also changed as shown in Fig. 3. All the films varied linearly with a and were thicker at smaller values of a. There was only one case in which the exchange energy strengthened with smaller a (PBE calculations in Str. A, see Fig. 2). This case was compatible with the bulk state case, where lattice parameter change in the z-direction (perpendicular to the Mn layer) strengthened the exchange energy significantly.²⁴ Indeed Str. A had the same constant 5.74 Å as the bulk state in PBE calculations. The overall thickness from LDA+U calculations was thicker than in PBE with the same a. The thickness differences between FM and AFM structures in LDA+U in all structures were nearly stable with a, but were large at large a, and became zero at small a in PBE. Compared to FM results, the thickness of AFM structures was more sensitive to calculation method in Str. A and B with a larger lattice parameter, while it was almost the same in both FM and AFM for Str. C with its much smaller lattice constant.

Fig. 3 Variation of the thickness as a function lattice parameter a for Str. A, Str. B and Str. C, respectively. Black square and red circle lines represent FM and AFM states respectively. Thick solid and thin dashed lines represent PBE and LDA+U calculations respectively.

For further research, the partial DOS of two types of structures in Str. 1 were chosen, as shown in Fig. 4. The differences between FM and AFM structures were significant. In the FM panel, the wide DOS distribution across the E_F in the spin-up channel meant that there were only a few states in the spin-down channel; however, in the AFM panel, a high peak was seen near E_F in both spin channels.

Fig. 4 AFM DOS (left panel) and FM DOS (right panel) of Mn atom of Str. A with two types of cases: 5.64 Å and 5.74 Å. Thin red and thick blue lines represent PBE and LDA+U calculations respectively. Zero energy indicates the position of the E_F. Positive and negative values of DOS are for the spin-up and spin-down channels, respectively.

In PBE calculations for δ-(Ga,Mn)As film, the main peak of d electrons of Mn was at −2.7 eV in the FM structure, which matched published data.³⁰ A mass of d electrons remained around E_F. In the FM panel, with the magnetic moments of two Mn atoms which were parallel to each other, a strong DE was shown in the wide DOS distribution in the gap. With the hole, the E_F lay in the spreading states, which meant that the high energy zone remained unoccupied. The energy gain would be caused by the DE with a hole, leading to a lowering in energy of the FM state. If the energy gain were larger than that induced by SE in the AFM state, the transition from Str. A to C would then be shown as an FM state (see Fig. 2). Conversely, the Mn majority level was shifted to a lower energy (−4.6 eV) in the FM panel when U = 4 eV was used in LDA+U calculations (see Fig. 4). Obviously, −4.6 eV was close to the resonant synchrotron radiation results (−4.5 eV (ref. 31)). With LDA+U calculations, the FM DOS showed a typical PE mechanism, where Mn d states and As states near the Mn atom pushed to each other in each spin-up and spin-down channel, and the hole ensured that the higher energy level remained unoccupied. At the same time, the SE was much weaker due to the departure of the main Mn d states in LDA+U calculations, and these systems were also shown in the FM state as shown in Fig. 2.

The hole from the Mn atoms concentrated into itself: this made the DE very strong. Although the long-range effect of PE was prominent due to the 2-d effect of the Mn layer, the exchange energy in DE was still much larger than PE in δ-(Ga,Mn)As film as shown in Fig. 2.

DE and PE depended on the position of the d states of the magnetic impurities. To match experimental results, the LDA+U method was usually used to calculate these zincblende DMSs. However, too many Mn d states concentrate at lower energies in LDA+U, which differs from the experimental results.³¹ Mn d states around E_F were also characteristic of the DE, but the far fewer states induced by LDA+U calculations made the DE much weaker compared to the experimental results. DE and PE are extreme cases of a more general type of interaction.³⁰ They should both be considered in the calculations of zincblende DMS. Considering both the FM exchange, and the results in Fig. 2, these films that grew on the substrate around the lattice constant of GaAs may have had the largest T_c.

To investigate the stability of different δ-(Ga,Mn)As films, their formation energies, defined as:

E_F = E_film − mu_Mn − nu_Ga − su_As

were plotted using the chemical potentials of Mn, Ga, and As in both bulk, or one layer, phases. Such a definition covers two cases, using either bulk, or individual, atom layers as reservoirs, it was found that it was rather difficult to form films from the bulk phase in both methods, except for Str. A in PBE. Fig. 5 showed that the formation energies all remained positive side in LDA+U calculations. In PBE calculations, Str. C and Str. A took the positive and negative sides, respectively, and Str. B stayed at zero. It was difficult to form such thin films using the melting method in these experiments. However, the formation energy of such films was always below zero from its single layer phases in both calculation methods. This meant that these films could be grown layer-by-layer. To obtain the perfect film, the height of the film should be larger than Str. A.

Fig. 5 Formation energies for different films as a function of chemical potentials from μ_b (bulk) and μ_l (single layer).

4 Conclusions

Ab initio density functional calculations showing the variation of exchange energy in three types of δ-(Ga,Mn)As films, under tensile strain, were presented. Tensile strain could change the exchange energy, and the thickness, of the three types of films analysed. In some configurations, the exchange energy could be enhanced by shape deformation. PBE showed the strong DE, while LDA+U resulted in the PE between Mn atoms in DOS structure. All of the films could be grown layer-by-layer based on the formation energy thereof. Only Str. A was stable with regard to its bulk chemical potentials in the PBE calculations.

Acknowledgements

This work was funded by the National Natural Science Foundation of China (NSFC), Grant no. 11204131, 11374159 and 11447211. This work was also funded by State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, CHINA, Grant no. KF2014_02. Y. Pan also thank the Foundation for Graduate Innovation Centre in NUAA (Grant no. kfjj201449) and the “Fundamental Research Funds for the Central Universities” for their financial support as well as Fudan University High-end Computing Center for providing us with a calculation platform.

References

1. T. Dietl, H. Ohno, F. Matsukura, J. Cibert and D. Ferrand, Science, 2000, 287, 1019–1022.
2. A. Majid, RSC Adv., 2015, 5, 72592–72600.
3. W. Liu, L. He, Y. Xu, K. Murata, M. C. Onbasli, M. Lang, N. J. Maltby, S. Li, X. Wang, C. A. Ross, P. Bencok, G. van der Laan, R. Zhang and K. L. Wang, Nano Lett., 2015, 15, 764–769.
4. H. Ohno, A. Shen, F. Matsukura, A. Oiwa, A. Endo, S. Katsumoto and Y. Iye, Appl. Phys. Lett., 1996, 69, 363–365.
5. I. Ahmada and B. Amin, Comput. Mater. Sci., 2013, 68, 55–60.
6. M. Bombeck, J. V. Jäger, A. V. Scherbakov, T. Linnik, D. R. Yakovlev, X. Liu, J. K. Furdyna, A. V. Akimov and M. Bayer, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 87, 060302.
7. T. Jungwirth, J. Wunderlich, V. Novák, K. Olejník, B. L. Gallagher, R. P. Campion, K. W. Edmonds, A. W. Rushforth, A. J. Ferguson and P. Němec, Rev. Mod. Phys., 2014, 86, 855–896.
8. K. Pappert, S. Humpfner, C. Gould, J. Wenisch, K. Brunner, G. Schmidt and L. W. Molenkamp, Nat. Phys., 2007, 3, 573–578.
9. B. Howells, K. W. Edmonds, R. P. Campion and B. L. Gallagher, Appl. Phys. Lett., 2014, 105, 012402.
10. J. Leiner, B. J. Kirby, M. R. Fitzsimmons, K. Tivakornsasithorn and X. Liu, J. Magn. Magn. Mater., 2014, 350, 135–140.
11. M. Kobayashi, H. Niwa, Y. Takeda, A. Fujimori, Y. Senba, H. Ohashi, A. Tanaka, S. Ohya, P. N. Hai, M. Tanaka, Y. Harada and M. Oshima, Phys. Rev. Lett., 2014, 112, 107203.
12. J. C. Angelico, A. L. J. Pereira, L. B. de Arrudaa and J. H. Dias da Silva, J. Alloys Compd., 2015, 630, 78–83.
13. N. Tesařová, D. Butkovičová, R. P. Campion, A. W. Rushforth, K. W. Edmonds, P. Wadley, B. L. Gallagher, E. Schmoranzerová, F. Trojánek, P. Malý, P. Motloch, V. Novák, T. Jungwirth and P. Němec, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 90, 155203.
14. E. de Ranieri, P. E. Roy, D. Fang, E. K. Vehsthedt, A. C. Irvine, D. Heiss, A. Casiraghi, R. P. Campion, B. L. Gallagher, T. Jungwirth and J. Wunderlich, Nat. Mater., 2013, 12, 808–814.
15. M. Fhokrul Islam and C. M. Canali, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 85, 155306.
16. M. Bozkurt, M. R. Mahani, P. Studer, J. M. Tang, S. R. Schofield, N. J. Curson, M. E. Flatté, A. Y. Silov, C. F. Hirjibehedin, C. M. Canali and P. M. Koenraad, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 88, 205203.
17. M. R. Mahani, A. Pertsova and C. M. Canali, J. Phys.: Condens. Matter, 2014, 26, 394006.
18. S. Sanvito and N. A. Hill, Phys. Rev. B: Condens. Matter Mater. Phys., 2000, 62, 15553–15560.
19. R. K. Kawakami, E. Johnston-Halperin, L. F. Chen, M. Hanson, N. Guébels, J. S. Speck, A. C. Gossard and D. D. Awschalom, Appl. Phys. Lett., 2000, 77, 2379–2381.
20. A. M. Nazmul, T. Amemiya, Y. Shuto, S. Sugahara and M. Tanaka, Phys. Rev. Lett., 2005, 95, 017201.
21. S. Ohya, K. Ohno and M. Tanaka, Appl. Phys. Lett., 2007, 90, 112503.
22. S. Ohya, K. Takata and M. Tanaka, Nat. Phys., 2011, 7, 342–347.
23. S. Sanvito and N. A. Hill, Phys. Rev. Lett., 2001, 87, 267202.
24. Y. Zhu, C. L. Ma, D. N. Shi and K. C. Zhang, Phys. Lett. A, 2014, 378, 2234–2238.
25. A. Mokaddem, B. Doumi, A. Sayede, D. Bensaid, A. Tadjer and M. Boutaleb, J. Supercond. Novel Magn., 2015, 28, 157–164.
26. B. Doumi1, A. Mokaddem, L. Temimi, N. Beldjoudi, M. Elkeurti, F. Dahmane, A. Sayede, A. Tadjer and M. Ishak-Boushaki, Eur. Phys. J. B, 2015, 88, 93.
27. Y. Pan, Y. Zhu, D. N. Shi, X. Y. Wei, C. L. Ma and K. C. Zhang, J. Alloys Compd., 2015, 644, 341–345.
28. S. Kahwaji, R. A. Gordon, E. D. Crozier, S. Roorda, M. D. Robertson, J. Zhu and T. L. Monchesky, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 88, 174419.
29. J. Zhao, Q. M. Deng, A. Bachmatiuk, G. Sandeep, A. Popov, J. Eckert and M. H. Rümmeli, Science, 2014, 343, 1228–1232.
30. K. Sato, L. Bergqvist, J. Kudrnovský, P. H. Dederichs, O. Eriksson, I. Turek, B. Sanyal, G. Bouzerar, H. Katayama-Yoshida, V. A. Dinh, T. Fukushima, H. Kizaki and R. Zeller, Rev. Mod. Phys., 2010, 82, 1633–1690.
31. J. Okabayashi, A. Kimura, T. Mizokawa, A. Fujimori, T. Hayashi and M. Tanaka, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, R2486.
32. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758–1775.
33. J. P. Perdew, et al., Phys. Rev. Lett., 1996, 77, 3865–3868.
34. V. Anisimov, et al., J. Phys.: Condens. Matter, 1997, 9, 767–808.

---

Footnote

† PACS numbers: 71.15.Nc, 73.20.At, 73.61.Ey.

---