1. Antônio J. Roque da Silva (IFUSP), Adalberto Fazzio (IFUSP), Raimundo R. dos Santos (UFRJ), and Luiz Eduardo Oliveira (UNICAMP) Parametrization of Mn-Mn interactions in Ga 1-x Mn x As semiconductors Workshop de Nanomagnetismo – 24 e 25/6/2004 Rede Virtual de Nanociência e Nanotecnologia do Estado do Rio de Janeiro/FAPERJ Financiamento: CNPq FAPERJ FAPESP Inst Milênio de Nanociências, Rede Nacional de Materiais Nanoestruturados

2. Motivation Combination of semiconductor technology with magnetism should give rise to new devices:  Spin-polarized electronic transport  long spin-coherence times (~ 100 ns) have been observed in semiconductors  manipulation of quantum states at a nanoscopic level

3. Magnetic semiconductors Early 60’s: EuO and CdCr 2 S 4  very hard to grow Mid-80’s: Diluted Magnetic Semiconductors II-VI (e.g., CdTe and ZnS) II  Mn  difficult to dope  direct Mn-Mn AFM exchange interaction  PM, AFM, or SG (spin glass) behavior 90’s:  Low T MBE  (In,Mn)As  Uniform (Ga,Mn)As films on GaAs substrates: FM; heterostructures- Possibility of useful devices

4. Ga: [Ar] 3d 10 4s 2 4p 1 Mn: [Ar] 3d 5 4s 2 Mn atoms: provide both magnetic moments and holes  hole-mediated ferromagnetism Ga As Ga 1-x Mn x As

5. Resistance measurements on samples with different Mn concentrations: Metal  R  as T  Insulator  R  as T   Reentrant MIT [Ohno, JMMM 200, 110(1999)] Ga 1-x Mn x As

6. Reproducibility?

7. Hole concentration vs Mn concentration 1 hole/Mn atom

8. A simple mean field treatment † yields 1h/Mn Notice maximum of p(x) within the M phase  correlate with MIT Early predictions [Matsukura et al., PRB 57, R2037 (1999)] log! † [RRdS, LE Oliveira, and J d’Albuquerque e Castro, JPCM (2002)]

9. First principles calculations should shed light into these issues Experimental data very sensitive to growth conditions  what are the dominant mechanisms behind the origin of ferromagnetism in DMS? how delocalized are the holes (are effective mass theories meaningful)? what is the effective Mn-Mn interaction? RKKY? what is the role of disorder?

10. Method Ab initio total energy calculations – DFT -VASP Ultra-soft pseudopotential Supercell calculations – 128/250 atoms (fcc) Spin polarized GGA (Perdew, Burke, Ernzerhof) for exchge-correl’n Plane waves basis set – (cutoff of 230eV, k = L) Final forces smaller than 0.02 eV/Å

11. Mn As Ga Single Mn atom

13. 20.3 Å

14. Isosurfaces for the net local magnetization Mn Ga Ground state: quite localized hole interacting antiferromagnetically with S=5/2 of Mn(d 5 ) Green=0.004e/A 3 Blue= -0.004e/A 3

15. We now consider two Mn atoms per unit cell  Assume all possible non-equivalent positions  For a given relative position, we consider FM and AFM relative Mn orientations, and work out the energy difference  Fit this energy difference to a Heisenberg interaction: thus estimates for J (r 1 – r 2 )

16. Ferromagnetic Mn As Mn-Mn 1 st NN Antiferromagnetic

17. Ferromagnetic Mn As Mn-Mn 1 st NN Antiferromagnetic

18. Ferromagnetic Mn As Mn Mn-Mn 1 st NN Mn-Mn 2 nd NN Antiferromagnetic

19. Ferromagnetic Mn As Mn As Mn-Mn 1 st NN Mn-Mn 2 nd NN Antiferromagnetic Again, note quite localized character of the holes

20. 12 6 24 12 24 8

21. The ferro-antiferro total energy differences yield... 12 6 24 12 24 8 the effective coupling between Mn spins (J Mn-Mn S Mn ·S Mn )

23. Therefore: impurity levels are localized  effective-mass picture for holes may be quite inadequate Mn-Mn interaction mediated by AFM coupling Mn-hole J Mn-Mn always ferromagnetic  non-RKKY estimates for anisotropy and direction dependences for effective J Mn-Mn

24. Our current agenda: 1)Effects of disorder? 2)Effects of concentration?  Preliminary results

25. Strategy (in principle): Randomly place Mn atoms in the Ga sublattice and use a look up table for J’s Ga Mn J1J1 J4J4 J2J2

26. We start with 4 Mn in our 128 atoms supercell: Roadmap 1)Randomly place 4 Mn atoms in the Ga sublattice 2)Calculate, using same ab initio scheme, the total energies for: a)(Mn 1,Mn 2,Mn 3,Mn 4 )=(up,up,up,up) – Ferro b)(Mn 1,Mn 2,Mn 3,Mn 4 )=(down,up,up,up) – Flip Mn 1 c)(Mn 1,Mn 2,Mn 3,Mn 4 )=(up,down,up,up) – Flip Mn 2 d)Etc. 3)Calculate energy differences E (Flip-Mn1)-Ferro, etc. 4)Write up same energy differences using an effective Heisenberg Hamiltonian, and extract effective J n 5)Compare with previous results with only two Mn

27. 4 Mn in 128 cell: - disorder inside unit cell - images are taken care of (unwanted order!) - Mn concentration – 0.0625 (6.25 %) - Different from 1 Mn in 32 atoms unit cell or 2 Mn in 64 atoms unit cell Ga Mn J1J1 JiJi

28. 4 Mn in 128 atoms unit cell Ab initio results Ferro = -553.737 eV Flip 1 = -553.547 eV Flip 2 = -553.586 eV Flip 3 = -553.302 eV Flip 4 = -553.508 eV Ferro - lowest energy configuration  1-Ferro = 0.190 eV  2-Ferro = 0.151 eV  3-Ferro = 0.435 eV  4-Ferro = 0.229 eV

29. 4 Mn in 128 atoms unit cell Heisenberg Hamiltonian results Ferro = Flip 1 = Flip 2 = Flip 3 = Flip 4 =  1-Ferro =  2-Ferro =  3-Ferro =  4-Ferro = For the particular realization, the Hamiltonian is

30. 2 Mn in 128 atoms unit cell Classical x Quantum Heisenberg Hamiltonian results ClassicalQuantum J1J1 -23.2-19.3 J2J2 -10.4-8.7 J3J3 -13.6-11.3 J4J4 -5.6-4.7 J5J5 -2.6-2.2 J6J6 -4.4-3.7 J (meV) Same trend, Classical or Quantum

31. 2 Mn x 4 Mn in 128 atoms unit cell Classical Heisenberg Hamiltonian 2Mn4Mn J1J1 -23.2-12.6 J2J2 -10.4- J3J3 -13.6-2.8 J4J4 -5.6-4.8 J5J5 -2.60.1 J6J6 -4.4- J (meV)

32. 2 Mn x 4 Mn in 128 atoms unit cell Classical Heisenberg Hamiltonian 2Mn(4Mn) 1 (4Mn) 2 J1J1 -23.2-12.6-13.0 J2J2 -10.4--4.7 J3J3 -13.6-2.8-6.0 J4J4 -5.6-4.8- J5J5 -2.60.1-1.3 J6J6 -4.4-- J (meV)

33. 2 Mn x 3Mn x 4 Mn in 128 atoms unit cell Classical Heisenberg Hamiltonian 2Mn3Mn(4Mn) 1 (4Mn) 2 J1J1 -23.2-19.2-12.6-13.0 J2J2 -10.4---4.7 J3J3 -13.6-8.4-2.8-6.0 J4J4 -5.6--4.8- J5J5 -2.60.1-1.3 J6J6 -4.4--- J (meV) 2Mn: x = 0.03125 3Mn: x = 0.046875 4Mn: x = 0.0625

34. 2 Mn x 3Mn x 4 Mn in 128 atoms unit cell Large reduction in the values of some of the J’s – Possible reasons: -Effective Heisenberg Hamiltonian may not be appropriate to describe “magnetic” excitations -Effective Hamiltonian ok to describe low-energy magnetic excitations, but our spin flip excitations may have too high an energy (non-collinear spin ab initio calculations?) -Disorder and/or concentration may have an important effect in the effective J couplings

35. Next steps (1): Perform more calculations with random structures – obtain a distribution for effective J’s Perform similar calculations for different Mn concentrations Non-collinear spin calculations If we conclude that we have a physically correct description through effective J’s + classical Heisenberg Hamiltonian, perform calculations for T > 0 (Monte Carlo) Next steps (2): Study (ab initio) how defects (e.g., interstitial Mn) change this picture by placing them in the, for example, 4 Mn in 128 atoms supercell – local disorder + defects

36. Conclusions: Effective mass descriptions (and improvements thereof) not reliable Effective Mn-Mn interactions not RKKY Disorder strongly influences effective Mn-Mn interactions; simple model? Heterostructures:  -doping, Be co- doping

38. Mn-hole exchange coupling J hd = 0.083 eV; 250 atoms, x = 0.008 J hd = 0.11 eV; 128 atoms, x = 0.0156

39. We have performed total energy calculations based on the density-functional theory (DFT) within the generalized-gradient approximation (GCA) for the exchange-correlation potential. The electron-ion interactions are described using ultra-soft pseudopotentials and plane wave expansion up to 200 eV as implemented in the VASP code. We used a 128-atom and 250-atom fcc supercell and the L-point for the Brillouin sampling. The positions of all atoms in the supercell were relaxed until all the force components were smaller than 0.05 eV/Å.

40. Isosurfaces for the difference between calculated for the Mn Ga ground state and the GaAs host

41. m(r) =   (r)-   (r) m(r) = +0.5 e - /Å 3

42. Sub-Si n=p

43. n.5p.oo5

44. As Ga a1 t2 As a1a1 t2t2 Mn a1a1 t2t2 t2t2 e

45. F. Matsukura, H. Ohno, A. Shen, and Y. Sugawara, Phys. Rev. B 57, R2037 (1998) MBE at low growth T (200 - 300 O C) on GaAs (001) substrates x = 0.015 – 0.071 200 nm thick Ga 1-x Mn x As samples A. van Esch et al, PRB 56, 13103 (1997) Ga 1-x Mn x As layers grown on GaAs (100) substrates GaAs grown by MBE at low temperatures (200 – 300 O C) samples of 3  m thick with Mn concentrations up to 9% K. W. Edmonds et al, APL 81, 3010 (2002) metallic behavior for 0.015  x  0.08 Ga 1-x Mn x As layers grown on semi-insulating GaAs (001) substrates by low-temperature (180 – 300 O C) MBE using As 2 samples: 45 nm thick S. J. Potashnik et al, APL 79, 1495 (2001) temperature during growth:  250 O C Ga 1-x Mn x As layers: thicknesses in range 110 – 140 nm

46. M. J. Seong et al, PRB 66, 033202 (2002) samples grown as in Potashnik et al:  250 O C and  120 nm used a Raman-scattering intensity analysis of the coupled plasmon-LO phonon mode and the unscreened LO phonon. H. Asklund et al, PRB 66, 115319 (2002) angle-resolved photoemission; 1% - 6% growth temperature of LT-GaAs and GaMnAs was typically 220 0 C Mn concentrations accurate within 0.5 % NOTE THAT T. Hayashi et al, APL 78, 1691 (2001) “a 10 o C difference in the substrate temperature during growth can lead to a considerable difference in the transport properties as well as in magnetism even though there is no difference in the growth mode as observed by electron diffraction

47. 2 Mn atoms as nearest-neighbors (Ga sub-lattice) Antiferromagnetic coupling m(r) = +0.004 e - /Å 3 m(r) = -0.004 e - /Å 3

48. VERY DILUTED DOPING LIMIT: Mn FORMS ACCEPTOR LEVEL 110 meV ABOVE VALENCE BAND ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY OBSERVES IMPURITY BAND NEAR E F. INFRARED MEASUREMENTS OF THE ABSORPTION COEFFICIENT ALSO REVEAL A STRONG RESONANCE NEAR THE ENERGY OF THE Mn ACCEPTOR IN GaAs. E. J. Singley, R. Kawakami, D. D. Awschalom, and D. N. Basov, PRL 89, 097203 (02) conductivity data: estimate the effective mass to be 0.7 mo < m* < 15 m o for the x = 0.052 sample, and larger at all other dopings, which suggest that the carriers do not simply reside in the unaltered GaAs valence band

49.  favor a picture of the electronic structure involving impurity states at E F rather than of holes doped into an unaltered GaAs valence band work obtained by using “complete” Kohn-Luttinger formalism (magnetic anisotropy, strain, etc): M. Abolfath, T. Jungwirth, J. Brum, and A. H. MacDonald, PRB 63, 054418 (2001). T. Dietl, H. Ohno, and F. Matsukura, PRB 63, 195205 (2001).

50. Isosurfaces for the net local magnetization: two Mn Ga defects In (a) and (b) the two Mn are nearest neighbors with their S=5/2 spins alligned parallel and antiparallel, respectively Mn As Mn-Mn 1 st nn Mn As Green=0.004e/A Blue= -0.004e/A