Warping a single Mn acceptor wavefunction by straining the GaAs host

Transcription

1 ARTICLES Warping a single Mn acceptor wavefunction by straining the GaAs host A. M. YAKUNIN 1 *,A.YU.SILOV 1,P.M.KOENRAAD 1,J.-M.TANG 2,M.E.FLATTÉ 2,J.-L.PRIMUS 3, W. VAN ROY 3,J.DEBOECK 3,A.M.MONAKHOV 4,K.S.ROMANOV 4,I.E.PANAIOTTI 4 ANDN.S.AVERKIEV 4 1 COBRA Inter-University Research Institute, Eindhoven University of Technology, PO Box 513, NL-5600MB Eindhoven, The Netherlands 2 Optical Science and Technology Center and Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA 3 IMEC, Kapeldreef 75, B-3001 Leuven, Belgium 4 Ioffe Physico-Technical Institute, St Petersburg , Russia * a.m.yakunin@tue.nl Published online: 10 June 2007; doi: /nmat1936 Transition-metal dopants such as Mn determine the ferromagnetism in dilute magnetic semiconductors such as Ga 1 x Mn x As. Recently, the acceptor states of Mn dopants in GaAs were found to be highly anisotropic owing to the symmetry of the host crystal. Here, we show how the shape of such a state can be modified by local strain. The Mn acceptors near InAs quantum dots are mapped at room temperature by scanning tunnelling microscopy. Dramatic distortions and a reduction in the symmetry of the wavefunction of the hole bound to the Mn acceptor are observed originating from strain induced by quantum dots. Calculations of the acceptorstate wavefunction in the presence of strain, within a tight-binding model and within an effective-mass model, agree with the experimentally observed shape. The magnetic easy axes of strained lightly doped Ga 1 x Mn x As can be explained on the basis of the observed local density of states for the single Mn spin. The material Ga 1 x Mn x As, the most thoroughly studied ferromagnetic dilute magnetic semiconductor, is a model for exploring the relationship between magnetic, mechanical and semiconducting properties 1 3. The magnetic easy axis of epitaxially grown films can lie in-plane or perpendicular to the film, depending on lattice deformation (strain), hole doping and temperature 1,4 6. These observations, as well as those of the magnetoelastic coupling and magnetic anisotropy 7, seem consistent with models of hole-mediated ferromagnetism 8,9.However,strong magnetic interactions between two Mn spins up to 1 nm apart in the extreme dilute limit have recently been demonstrated 10, and ferromagnetism has long been known to occur even in Ga 1 x Mn x As samples that show insulating behaviour 11 (although with low Curie temperatures). Thus, it is an open and relevant question whether the magnetic anisotropy of Ga 1 x Mn x As deep in the insulating phase can be explained on the basis of the properties of the individual magnetic Mn acceptor. Recent scanning tunnelling microscopy (STM) measurements have shown a pronounced anisotropy of the shape of the wavefunction of a hole bound to an individual Mn acceptor even in unstrained GaAs (ref. 12), an anisotropy that is preserved for very short Mn Mn separations (typical of high-curie-temperature Ga 1 x Mn x As) 13 and leads to anisotropic spin spin couplings 10. Here, the wavefunction of a hole bound to an individual Mn acceptor in a region of highly strained GaAs is spatially mapped by STM topography measurements at room temperature. Two models previously used to describe the anisotropic shape of the unstrained Mn acceptor hole state, a tight-binding model (TBM) 12,14 and an envelope-function effective-mass model (EFM) 12, provide agreement with the shape of the Mn acceptor state in strained GaAs. These experimental and theoretical results confirm that the properties of the valence band of the GaAs host are responsible for the anisotropy of the Mn acceptor states and define their symmetry. The dramatic distortion of the acceptor state with strain suggests additional effects of strain on the magnetic coupling within Ga 1 x Mn x As, as well as on macroscopic properties such as the Curie temperature. We find that strain induces an orientation-dependent binding energy for the Mn acceptor state, which is consistent with a magnetic easy axis perpendicular to the direction(s) of compressive strain, consistent with experimental measurements in low-doped Ga 1 x Mn x As. Changes in acceptor-state wavefunctions have also been suggested to explain strain-induced modifications of the photoluminescence intensity associated with Mn and other acceptors in GaAs ; our measurements provide concrete results about the distorted acceptor state to guide such investigations. The use of strain to manipulate the magnetic properties of a single Mn acceptor may also provide a pathway, through the use of nanoelectromechanical systems 7, towards ionic-spin-based quantum computing in semiconductors 19. The sizable symmetry-breaking strains required to observably lower the symmetry of the crystal lattice and distort the wavefunction of the acceptor-state hole are challenging to apply. Hydrostatic strains would only change the overall size of the acceptor wavefunction without lowering its symmetry. Embedding a quantum dot (QD) of one lattice constant (for example, InAs) within a host of another lattice constant (for example, GaAs) generates, in addition to hydrostatic strain fields, long-range uniaxial strain fields 20,21 that can exceed 2%. Thus, Mn acceptors 512 nature materials VOL 6 JULY

2 ARTICLES embedded near these InAs QDs experience uniaxial strains of about 100 times larger than those that can now be generated externally, either locally with the STM s piezo-drivers 22 24, or through the entire sample with a vise 25. Moreover, many different strain configurations are possible near InAs QDs, and it is even possible to image multiple Mn acceptors in the same image in differing local strain environments. An understanding of the response of these acceptors to strain requires a brief description of the symmetries at the (110) GaAs surface and of the Mn acceptor state. On the (110) surface of a GaAs crystal, the only symmetry element remaining of the bulk crystal s T d point-group symmetry is reflection in the (1 10) plane. The Mn acceptor wavefunction in unstrained GaAs is symmetric under this reflection 12, independent of the Mn location under the surface (for a detailed analysis see ref. 26). Uniaxial stress along either the [001] or [110] directions does not change this symmetry. We find that for such strains the symmetry of the Mn STM image is indistinguishable from that of the undeformed case. Stress applied in directions other than these breaks the (1 10) reflection symmetry and the highly distorted Mn wavefunctions observed have no remaining symmetry. A large number of quantum dots were observed with Mn acceptors distributed randomly in the GaAs region. The STM image in Fig. 1a shows one case where the QD is cleaved approximately through its centre and two Mn atoms (Mn1 and Mn2) are located at the same depth from the surface, but in regions of differing local strain. For the Mn atoms that were located off the central axis of the InAs QD, such as Mn2 in Fig. 1a, the strain breaks the symmetry of the Mn wavefunction. The wavefunction of this manganese acceptor has no symmetry axis. Those Mn atoms located far from the dot, or near the central axis of the dot, retained their symmetry. In Fig. 1a, the other manganese acceptor (Mn1) is located in nearly strain-free GaAs and exhibits mirror symmetry in the (1 10) plane (and approximate mirror symmetry in the (001) plane). In Fig. 1b, a second QD is shown with a Mn atom on the other side (Mn3). The stress axis for Mn3, relative to that of Mn1, is reflected in the QD plane, so the distortion of the wavefunction is reflected as well. Further examples of Mn in locally strained environments are shown in the Supplementary Information. In Fig. 1c,d, a high-pass Fourier filter was applied to highlight the effect of the symmetry breaking on the Mn acceptor wavefunctions (more details of the filter are given in the Supplementary Information). The two unfiltered Mn topographies from Fig. 1a are enlarged in Fig. 2a,b and line cuts are presented along the [1 10] direction at a distance of 3.5 nm from the centre of the Mn wavefunction. The line cuts clearly show that the topography of Mn1 possesses symmetry with respect to the (1 10) reflection plane. The topography of Mn2, however, exhibits a significant redistribution of the local density of states (LDOS) from one side of the (1 10) plane passing through the impurity site to the other. The LDOS is enhanced along the lower side and suppressed along the upper side. We now describe our calculations of the strain-modified LDOS for Mn in strained GaAs within a TBM. Our tight-binding calculations treat a uniform uniaxial compressive stress along either a 111 or a 100 axis. We assume that the Mn acceptor potential is identical to that in unstrained GaAs. Thus, the change in shape comes entirely from the change in the properties of the bulk GaAs crystal when it is strained. The overlap parameters as well as the structure factors in the tight-binding hamiltonian are modified according to the atomic displacements as described in the Methods section. The on-site energies remain the same. To calculate the acceptor-state wavefunction, we follow the Koster Slater procedure described in ref. 14. The real-space Green s functions are calculated for uniformly strained GaAs with a 5 mev energy resolution. a c GaAs host 4 nm Mn1 Mn1 (X shape) InAs QD1 - [001] Mn2 Mn2 (S shape) - [110] b d Mn3 InAs QD2 Mn3 (mirrored S shape) Figure 1 Constant-current STM image. a,b, Two manganese acceptors, Mn1 and Mn2 (a), and one manganese acceptor, Mn3 (b), in regions of differing local strain produced by a QD. Uniaxial stress originating from the InAs QD is oriented approximately along [001] for Mn1, [ 111] near Mn2 and [ 11 1] near Mn3. c,d,parts of images a and b after a high-pass spatial Fourier filter was applied. a c Measured height (Å) Mn1 m2 m1 - [110] - [001] m1 (a) Mn Distance along [110] direction (nm) b m2 Mn2 (b) Mn2 m2 m1 2 nm Figure 2 Comparison of the topography (as measured) for manganese dopants in the strained and unstrained GaAs regions of Fig. 1a. a, Mn1. b, Mn2. c,line profiles along the vertical lines in a and b. The horizontal dashed lines indicate the symmetry axis for the unstrained Mn wavefunction. Strains 1% are weak enough that the momentum integrations required to obtain the real-space Green s functions can be carried out in the Brillouin zone of unstrained GaAs. Then the impurity Green s functions are obtained by solving the Dyson equation in real space. Modified hopping matrix elements and bond angles produce greater light-hole character (with the orbital quantization axis nature materials VOL 6 JULY

3 ARTICLES a c b Figure 3 Calculated logarithm of Mn LDOS. a d, For unstrained (a,b) and strained (c,d) GaAs using the EFM (a,c) and TBM (b,d). The Mn dopants are located five layers below the surface ( 1 nm). For the TBM, the LDOS at each atomic site is spatially distributed according to a normalized gaussian with a 2 Å width. For each model, the LDOS cross-sections are normalized by the maximum value of the unstrained LDOS. parallel to the compressive strain) in the acceptor ground state, and hence lead to an extension of the acceptor state along the direction of compressive stress and a contraction along the perpendicular directions. This is analogous, for this acceptor-state theory, to the energy splitting between zone-centre light and heavy holes in bulk models that treat the crystal momentum as well defined 8,9. The effect of strain on the LDOS can also be calculated within an envelope-function model for Mn Ga by applying the zerorange potential method 27 and assuming uniform uniaxial strain. For deep impurities such as Mn in GaAs, this method gives the wavefunction s asymptotic form for localized states for distances greater than one lattice constant. As the exchange potential dominates and its range is at least five times shorter than the hole Bohr radius, most of the wavefunction is located outside the range of the impurity potential, and thus the symmetry of strained GaAs defines the symmetry of the acceptor state. The Schrödinger equation for a zero-range potential model is solved for a hamiltonian, H, which is the sum of the Luttinger hamiltonian, H L, for the cubic crystal ( H L = γ ) 2 γ 2 p 2 2γ 2 p 2 J { } 2 i i 2γ 3 p i p j Ji,J j, where γ 1 = 7.65,γ 2 = 2.41 and γ 3 = 3.28 are the Luttinger parameters of GaAs 28, and the Bir Pikus hamiltonian, H BP, for deformations 29, H BP = aɛ + b J 2 i (ɛ ii 13 ) ɛ + 1 d { } Ji,J j ɛij, (1) i 3 i>j where ɛ = i ɛ ii. The LDOS for the Mn in unstrained GaAs results from the sum of the probability density of four degenerate states. In the presence of stress along the [ 111] direction and the perturbative hamiltonian given by equation (1), the four solutions split into two doublets each of whose shape is independent of the magnitude of the stress and valence-band parameters a,b and d. The energy i d i>j splitting separating the two doublets does depend on the stress, however, and at room temperature for reasonable strains ( 1%) only the lowest-energy doublet is occupied by a hole. The principal differences between this zero-range potential method and the TBM are the character of the hamiltonian (continuum Luttinger in the limit of strong spin orbit interaction instead of lattice tight binding) and the character of the potential (non-magnetic zero-range versus spin-dependent nearest neighbours (a T d -symmetric potential)). As the potential considered in the zero-range potential model is independent of spin, its conclusions about the effect of strain on the acceptor wavefunction shape also apply to non-magnetic acceptor states. Figure 3 shows the calculated LDOS for Mn acceptors in unstrained and strained GaAs, within both the TBM and the envelope-function model. Both are in qualitative agreement with the experiment, indicating an extension of the wavefunction extent along the [ 111] direction and a contraction of the wavefunction extent along the [1 12] direction. As described above, the source of the distorted shape for the TBM is the increased light-hole character of the acceptor state (with the quantization axis parallel to the compressive stress). This distortion of the acceptor wavefunction implies a breaking of cubic symmetry for the magnetic easy axis. The dominant orbital angular-momentum character of the Bloch functions of the acceptor state is aligned perpendicular to the long axis of the acceptor state and (in the ground state) is oriented antiparallel to the Mn core spin. This spin orbit correlation between acceptor wavefunction shape and spin orientation has previously been suggested as a method for measuring the Mn spin orientation 30. We calculate from the TBM that for (001) growth of in-plane 0.1% compressively strained Ga 1 x Mn x As films (x 0.02), this correlation produces a 6 mev lower energy for Mn spins oriented perpendicular to the film plane compared with Mn spins in the plane, and therefore a perpendicular magnetic anisotropy. For in-plane tensile-strained films, the easy axis lies in the plane. For other strain directions, as described in the Supplementary Information, the acceptor-level energy splittings can be described with equation (1) with a = 1.1eV,b = 6.5 ev and d = 8.2 ev. The easy axis described above for (001) in-plane compressively strained films agrees with low-temperature measurements on low-hole-concentration (Al,Ga,Mn)As (ref. 4) and on low-doped Ga 1 x Mn x As (refs 5,6) films. The strain-induced anisotropy energy of a single Mn described above is produced only if the acceptor state is occupied by a hole. Perpendicular magnetic anisotropy is only seen in refs 4 6 in samples whose hole concentration is substantially lower than the Mn concentration. The sample in ref. 6 with the most persistent perpendicular anisotropy had a hole concentration 1/20 of the Mn concentration. In addition, given that the valenceband-edge energy splittings calculated within this TBM are known to be too large by a typical factor of 2, we estimate the anisotropy energy per Mn for these low-doped Ga 1 x Mn x As to be of the order of mev per Mn, comparable to values seen in high-doped Ga 1 x Mn x As. For high-doped samples, however, the presence of increased amounts of Mn and holes lead to a switch in the magnetic easy axis that is beyond description by our model here, and may originate from the interaction between Mn pairs or larger groups. We have observed the strain-distorted acceptor-level shape of substitutional Mn in GaAs and explained the shape in the context of the TBM and EFM. We have also connected this singleacceptor-level distortion with the magnetic easy axis in low-doped magnetic films. This suggests that some properties of Ga 1 x Mn x As films, previously traced to hole-mediated ferromagnetism treated in mean field theory, may be understood in the context of the physics of single magnetic acceptors. 514 nature materials VOL 6 JULY

4 ARTICLES METHODS EXPERIMENTAL METHODS Our sample was grown on a (001) substrate at 485 C. The sample structure contained three layers of QDs formed by depositing 1.7, 1.9 and 2.1 monolayers of InAs. The layers were separated by 80 nm GaAs spacers to prevent strain-driven coupling and were covered by a 100 nm cap. A constant Mn flux was maintained during the growth of the whole structure to achieve the Mn doping concentration N Mn cm 3 in GaAs. The sample was cleaved under ultrahigh vacuum conditions to obtain an atomically flat clean (110) surface.theldosofmnacceptorslocatedunderthesurfaceatadepthof about nm was probed witha sharp W tip by tunnellingthrough the Mn acceptor state as described in ref. 31. THEORETICAL METHODS The local electronic structure around the Mn atoms in GaAs is calculated using a 16-band sp 3 TBM for the GaAs host with spin orbit interaction 32.TheMn atoms are modelled with an effective potential consisting of an on-site Coulomb term of 1 ev and a spin-polarized term, non-zero for only the p states at the first nearest-neighbour As sites, of ev to describe the hybridization between the Mn d-orbitals and As p-orbitals. The effective interaction of the d-orbitals and the As s-orbitals is set to zero. These values produce the correct acceptor-state binding energy in bulk GaAs. The spin of the Mn 3d electrons is assumed to be aligned along a given axis. The LDOS were obtained by calculating the imaginary part of the lattice Green s functions in real space with an energy linewidth of 5 mev. Further details are available in ref. 14. We assume that the Mn acceptor potential in strained GaAs is identical to that in unstrained GaAs. Thus, the change in shape comes entirely from the change in the properties of the bulk GaAs crystal when it is strained. We calculate the strain tensor as a function of the applied stress to obtain the displacements of the first-nearest-neighbour atoms. The overlap parameters as well as the structure factors in the tight-binding hamiltonian for GaAs are modified according to these displacements as described below. The on-site energies remain the same. For a uniform stress along the [ 111] direction, the strain is described byatensor, α β β ɛ = β α β, β β α where α/β = 2C 44 /(C 11 +2C 12 ) = for GaAs. C 11, C 12 and C 44 are the continuum elastic constants of GaAs. The relative displacement of the As and Ga sublattices along the diagonal, which can be affected by so-called internal strain, must also be specified to fully determine the displacement of the atoms in the zinc-blende lattice. The displacements of the four first-nearest neighbours, including internal strain, are l i = ɛ l i 2βηl 1, where i = 1...4, l 1 is the bond vector along the stress direction and the internal strain parameter η = 0.65 for GaAs (ref. 33). The five overlap matrix elements, V ssσ, V spσ, V psσ, V ppσ and V ppπ, are modified according to l 2 scaling 34,where l is the bond length. Received 28 November 2006; accepted 16 May 2007; published 10 June References 1. Ohno, H. Making nonmagnetic semiconductors ferromagnetic. Science 281, (1998). 2. Awschalom, D. D., Samarth, N. & Loss, D. (eds) in Semiconductor Spintronics and Quantum Computation (Springer, Berlin, 2002). 3. MacDonald, A. H., Schiffer, P. & Samarth, N. Ferromagnetic semiconductors: Moving beyond (Ga, Mn)As. Nature Mater. 4, (2005). 4. Takamura, K., Matsukura, F., Chiba, D. & Ohno, H. Magnetic properties of (Al,Ga,Mn)As. Appl. Phys. Lett. 81, (2002). 5. Sawicki, M. et al. Temperature peculiarities of magnetic anisotropy in (Ga,Mn)As: The role of the hole concentration. J. Supercond. 16, 7 10 (2003). 6. Sawicki, M. et al. Temperature dependent magnetic anisotropy in (Ga,Mn)As layers. Phys.Rev.B70, (2004). 7. Masmanidis, S. C. et al. Nanomechanical measurement of magnetostriction and magnetic anisotropy in (Ga,Mn)As. Phys.Rev.Lett.95, (2005). 8. Abolfath, M., Jungwirth, T., Brum, J. & MacDonald, A. H. Theory of magnetic anisotropy in III 1 xmnxv ferromagnets. Phys.Rev.B63, (2001). 9. Dietl, T., Ohno, H., Matsukura, F., Cibert, J. & Ferrand, D. Hole-mediated ferromagnetism in tetrahedrally coordinated semiconductors. Phys.Rev.B63, (2001). 10. Kitchen, D., Richardella, A., Tang, J.-M., Flatté, M. E. & Yazdani, A. Atom-by-atom substitution of Mn in GaAs and visualization of their hole-mediated interactions.nature 442, (2006). 11. Matsukura, F., Ohno, H., Shen, A. & Sugawara, Y. Transport properties and origin of ferromagnetism in (Ga,Mn)As. Phys.Rev.B57, R2037 R2040 (1998). 12. Yakunin, A. M. et al. Spatial structure of an individual Mn acceptor in GaAs. Phys.Rev.Lett.92, (2004). 13. Yakunin, A. M. et al. Spatial structure of Mn Mn acceptor pairs in GaAs. Phys.Rev.Lett.95, (2005). 14. Tang, J.-M. & Flatté, M. E. Multiband tight-binding model of local magnetism in Ga 1 xmnxas. Phys. Rev. Lett. 92, (2004). 15. Prince, P. J. Strain dependence of the acceptor binding energy in diamond-type semiconductors. Phys. Rev. 124, (1961). 16. Bir, G. L., Butekov, E. I. & Pikus, G. E. Spin and combined resonance on acceptor centres in Ge and Si type crystals I: Paramagnetic resonance in strained and unstrained crystals. J. Phys. Chem. Solids 24, (1963). 17. Bhargava, R. N. & Nathan, M. I. Stress dependence of photoluminescence in GaAs. Phys. Rev. 161, (1967). 18. Averkiev, N. S., Gutkin, A. A., Kolchanova, N. M. & Reshchnikov, M. A. Influence of uniaxial deformation on the Mn impurity photoluminescence band of GaAs. Sov. Phys. Semicond. 18, (1984). 19. Tang, J.-M., Levy, J. & Flatté, M. E. All-electrical control of single ion spins in a semiconductor. Phys. Rev. Lett. 97, (2006). 20. Stangl, J., Holý, V. & Bauer, G. Structural properties of self-organized semiconductor nanostructures. Rev. Mod. Phys. 76, (2004). 21. Bruls, D. M. et al. Determination of the shape and indium distribution of low-growth-rate InAs quantum dots by cross-sectional scanning tunneling microscopy. Appl. Phys. Lett. 81, (2002). 22. Chilla, E., Rohrbeck, W., Fröhlich, H.-J., Koch, R. & Rieder, K. H. Probing of surface acoustic wave fields by a novel scanning tunneling microscopy technique: Effects of topography. Appl. Phys. Lett. 61, (1992). 23. Yang, J. S., Voigt, P. U. & Koch, R. Nanoscale investigation of longitudinal surface acoustic waves. Appl. Phys. Lett. 82, (2003). 24. Koch, R. & Yang, J. S. Nanoscale imaging of surface acoustic waves by scanning tunneling microscopy. J. Appl. Phys. 97, (2005). 25. Crooker, S. A. & Smith, D. L. Imaging spin flows in semiconductors subject to electric, magnetic, and strain fields. Phys.Rev.Lett.94, (2005). 26. Yakunin, A. M. Thesis, Eindhoven Univ. of Technology, The Netherlands (2005). 27. Averkiev, N. S. & Il inskii, S. Yu. Spin ordering of carriers localized at two deep centers in cubic semiconductors. Phys. Solid State 36, (1994); Fiz. Tverd. Tela (1994). 28. Lawaetz, P. Valence-band parameters in cubic semiconductors. Phys.Rev.B4, (1971). 29. Bir, G. L. & Pikus, G. E. Symmetry and Strain-Induced Effects in Semiconductors (Halsted, Jerusalem, 1974). 30. Tang, J.-M. & Flatté, M. E. Spin-orientation-dependent spatial structure of a magnetic acceptor state in a zinc-blende semiconductor. Phys.Rev.B72, (R) (2005). 31. Yakunin, A. M. et al. Imaging of the (Mn 2+ 3d 5 +hole) complex in GaAs by cross-sectional scanning tunneling microscopy. Physica E 21, (2004). 32. Chadi, D. J. Spin-orbit splitting in crystalline and compositionally disordered semiconductors. Phys. Rev. B 16, (1977). 33. Molinás-Mata, P., Shields, A. J. & Cardona, M. Phonons and internal stresses in IV IV and III V semconductors: The planar bond-charge model. Phys.Rev.B47, (1993). 34. Froyen, S. & Harrison, W. A. Elementary prediction of linear combination of atomic orbitals matrix elements. Phys.Rev.B20, (1979). Acknowledgements This work was supported by the Dutch Foundation for Fundamental Research on Matter (FOM), NanoNed (a technology programme of the Dutch ministry of Economic Affairs via the foundation STW), the ARO MURI DAAD , NSF Grant No. PHY , the Belgian Fund for Scientific Research Flanders (FWO) and the EC GROWTH project FENIKS (G5RD-CT ) as well as RFBR ( ), INTAS and the RF Program of Scientific Schools # and the Scientific Programs of RAS. The authors would like to thank J. H. Wolter for his important contribution to this research in the early stages. We also thank H. Ohno for comments on the manuscript before submission. Correspondence and requests for materials should be addressed to A.M.Y. Supplementary Information accompanies this paper on Competing financial interests The authors declare no competing financial interests. Reprints and permission information is available online at nature materials VOL 6 JULY

5 Supplementary material Warping a single Mn acceptor wave function by straining the GaAs host A. M. Yakunin, A. Yu. Silov, P. M. Koenraad, J.-M. Tang, M. E. Flatté, J.-L. Primus, W. Van Roy, J. De Boeck, A. M. Monakhov, K. S. Romanov, I. E. Panaiotti, N. S. Averkiev I. OVERVIEW The local density of states of individual Mn acceptors near InAs quantum dots in a GaAs host are mapped at room temperature by scanning tunneling microscopy. Dramatic distortions and a reduction in the symmetry of the wave function of the hole bound to the Mn acceptor are observed originating from local strain induced by quantum dots. Calculations of the acceptor state wave function in the presence of strain, within a tight-binding model and within an envelope-function effective mass model, agree with the experimentally-observed shape. The magnetic easy axes of strained lightly-doped Ga 1 x Mn x As can be explained based on the observed local density of states for the single Mn spin. The following supplementary section expands on the information provided in the Article providing additional examples of observed strained distorted Mn, details of the high-pass filter used in Fig. 1 of the Article, and additional details of the tight-binding model used to calculate the wave function shape and the magnetic anisotropy energy. 1

6 II. SYMMETRY OF THE GAAS (110) SURFACE AND UNSTRAINED MN AC- CEPTOR STATE Shown below is the symmetry of the (110) GaAs surface and the Mn wave function. In (a) red are gallium sites and grey are arsenic sites; the larger circles are in even layers below the surface and the smaller circles are in the odd layers below the surface. The shaded triangles show schematically the unstrained acceptor wave function when the center gallium site is occupied by a Mn atom. The appearance of the Mn wavefunction with respect to the Ga and As sublattices is discussed in detail in the Ref. [1]. The unstrained Mn acceptor wave function shape shown in (b) is from Ref. 12 of the Article, and corresponds to a Mn atom located in the fifth layer below the surface. The acceptor wave function has reflection symmetry along the [1 10] direction and approximate reflection symmetry along the [001] direction. These symmetries are preserved unless the strain is not collinear to both the [001] and [1 10] directions. (a) [1 1 0] (b) [00 1] As Mn Ga Ga FIG. 1: (a) Symmetry of the (110) GaAs surface and the (b) Mn wave function. The wave function can be deformed only if the strain axis is not collinear to both [001] or [1 10] directions. 2

7 III. SYMMETRY OF MN ACCEPTOR STATE IN STRAINED GAAS In Fig. 2 four cases of Mn near InAs QDs are shown. They show Mn atoms located in each of the four quadrants around an InAs QD (white arrows) as well as in regions where the strain is weak or does not break the symmetry of the Mn atoms (purple arrows). The Mn atoms in (c) and (d) are shown in Fig. 1 of the Article. (a) (b) (c) (d) FIG. 2: Four different cases of relative arrangement of QD and Mn when Mn wave-function deformation occurs. Deformed Mn wave functions are indicated by white arrows and some isolated undeformed Mn wave functions are indicated by purple arrows. The color scale of each panel is adjusted for the best view. 3

8 IV. HIGH-PASS FILTER In figures 3 and 4 we demonstrate the details of the sequential filtering procedure. The Fourier spectrum of the topographical STM image of a single undisturbed Mn wavefunction is shown in the Fig. 3(a). The central part of the spectrum shows the spatial harmonics related to Mn wavefunction. The remote satellite features marked by a 0 and b 0 correspond to the lattice parameters of the (110) surface nm and 0.4 nm respectively. In the figures 3(b) and 4(b) the main features of the spectrum are separated from the high frequency noise by applying the band pass filter. No modification of the Mn wavefunction envelope is introduced at this stage. In the figures 4(c) and 3(c) the isotropic part of the Mn wavefunction is removed by applying high pass filter with the typical radius of 8-10 lattice constants a 0. An identical filtering sequence was applied to the images with quantum dots to obtain figures 1(aa) and 1(bb) of the Article. Original Band Pass High Pass (a) (b) (c) b 0 a 0 FIG. 3: Different stages of sequential Fourier filtering performed for Mn wavefunction in the strain free GaAs (a) (b) (c). Below, Fig. 4 shows the intermediate results of the filtering procedure for one QD-Mn pair. This example shows that the Mn wavefunction deformation is present at any stage of the filtering. 4

9 (a) Original (b) Band Pass High Pass (isotropic) (c) High Pass 2 (isotropic) (d) FIG. 4: Different stages of sequential filtering (a) (b) (c) (d). Figures (c) and (d) differ by the size of the high pass filter. 5

10 V. CORRESPONDENCE OF MN WAVE FUNCTION SHAPE WITH QD STRAIN [1 1 0] [00 1] FIG. 5: Symmetry of the ellipsoidal QD. Stress is outwardly directed from the dot, leading to stress along the [ 111] direction in the blue regions and along the [1 11] direction in the red regions. The wave functions in the red and blue regions are mirrored images of each other. VI. PARAMETERS FOR THE EFFECTIVE SPIN HAMILTONIAN FROM THE TIGHT-BINDING MODEL The first-order modification of the energies of the valence band edge and of the acceptor level due to the strain tensor can be described, by symmetry, with the effective Hamiltonian H = a (ɛ xx + ɛ yy + ɛ zz ) + b [( J 2 x J 2 /3 ) ɛ xx + c.p. ] + d 3 [(J x J y + J y J x ) ɛ xy + c.p.] (1) where c.p. stands for cyclic permutation. For the valence band edge states J = 3/2 and for the acceptor level J = 1. The sign convention for a, b, and d is set so the Hamiltonian describes hole energies. Shown below are the values for the coefficients in Eq. (1) for the valence band edge (hole convention), calculated for several values of the strain along the stress direction. The results for the hydrostatic deformation potential a show greater variability, but those for the shear 6

11 deformation potentials show at most 3% variability or nonlinearity. Thus Eq. (1) appears accurate for strains in this regime. [001] -0.1% -0.05% 0.05% 0.1% a b TABLE I: Values of deformation potentials (in ev) for the valence band edge states for use in Eq. (1) extracted from the application of stress along the [001] direction, producing a strain of 0.1%, 0.05%, 0.05%, or 0.1% along the [001] direction. [111] -0.1% -0.05% 0.05% 0.1% a d TABLE II: Same as Table I but for stress along the [111] direction. The value of a is for the shift of the valence band edge, not the gap of the semiconductor, and hence high quality experimental values for this are not available. The experimental values for b and d are within a factor of 2 of these calculated ones. Deformation potentials can be produced which are closer to the experimental values from tight-binding calculations which include more states, such as s or d states, or which alter the distance scaling of the tight-binding matrix elements from the l 2 scaling we have assumed. As these models introduce many additional band-structure and strain parameters we have retained a smaller basis set and simpler scaling rules. As the deformation potentials we calculate are within a factor of 2 of the experimental ones we do not expect the use of more complex tightbinding models (larger basis sets and less transparent scaling rules) to significantly modify the appearance of the Mn wave function in the strained crystal or the magnetic anisotropy we calculate from this model. 7

12 VII. ACCEPTOR LEVEL ENERGY SHIFTS AND SPLITTINGS FOR ARBI- TRARY STRESS DIRECTIONS The splittings of the acceptor energy levels with strain can be described for arbitrary stress direction with Eq. (1) with deformation potentials determined for the acceptor state. Shown below are the values for the a, b, and d coefficients for the acceptor level in Eq. (1) calculated for several values of the strain along the stress direction. Slightly larger variability or nonlinearity is found for the acceptor level than for the valence band energy shifts. Still, Eq. (1) appears accurate for strains in this regime. Strains approaching 1%, however, produce clearly nonlinear effects. [001] -0.1% -0.05% 0.05% 0.1% a b TABLE III: Values of deformation potentials (in ev) for the acceptor level for use in Eq. (1) extracted from the application of stress along the [001] direction, producing a strain of 0.1%, 0.05%, 0.05%, or 0.1% along the [001] direction. [111] -0.1% -0.05% 0.05% 0.1% a d TABLE IV: Same as Table III but for stress along the [111] direction. These deformation potentials and Eq. (1) can be used to determine the magnetic anisotropy energy for a Mn with a hole in the acceptor state under stress along an arbitrary direction. [1] Andrei M. Yakunin, PhD Thesis: Mn in GaAs studied by X-STM: from a single impurity to ferromagnetic layers, Eindhoven University of Technology (2005). 8