Nonlocal effects on magnetism in the diluted magnetic semiconductor Ga$_{1}$$_{−}$ x Mn x As Unjong Yu, 1, 2 Abdol-Madjid Nili, 1 Karlis Mikelsons, 1,3 Brian Moritz, 4 Juana Moreno, 1 and Mark Jarrell 1 $^{1}$Department of Physics and Astronomy & Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA $^{2}$Department of Applied Physics, Gwangju Institute of Science and Technology, Gwangju 500-712, Korea $^{3}$Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA 4 Stanford Institute for Materials and Energy Science, SLAC National Accelerator Laboratory and Stanford University, Stanford, CA 94305, USA (Dated: October 30, 2018) The magnetic properties of the diluted magnetic semiconductor Ga 1 − x Mn x As are studied within the dynamical cluster approximation. We use the k$_{·}$p Hamiltonian to describe the electronic struc- ture of GaAs with spin-orbit coupling and strain effects. We show that nonlocal effects are essential for explaining the experimentally observed transition temperature and saturation magnetization. We also demonstrate that the cluster anisotropy is very strong and induces rotational frustration and a cube-edge direction magnetic anisotropy at low temperature. With this, we explain the temperature-driven spin reorientation in this system. PACS numbers: 75.50.Pp, 75.30.Gw, 78.55.Cr The discovery of high temperature ferromagnetism in diluted magnetic semiconductors (DMS) has stimulated a great deal of attention [1]. The interest in these materials is due to possible applications in spintronics [2] as the source of a spin polarized current or as the base material for a chip that can simultaneously store and process data. In spite of extensive studies, our understanding of fer- romagnetism in these systems is far from complete [3]. There are a few serious difficulties in the theoretical study of DMS: (i) The magnetic interaction between lo- cal magnetic moments and itinerant carrier spin, which is responsible for the high transition temperature ( T$_{c}$ ), is strong and outside the Ruderman-Kittel-Kasuya-Yosida (RKKY) regime. (ii) There exists strong disorder from the random distribution of magnetic ions. (iii) Nonlo- cal effects are expected to be crucial judging from the spatially oscillating and anisotropic magnetic interaction predicted in theory [4–7] and observed in experiments [8]. The mean-field study by Dietl et al. [9] captures the main features of DMS systems qualitatively and some even quantitatively. However, it ignores strong corre- lations, disorder effects, and spatial fluctuations, and fails to describe some DMS materials, such that subse- quent studies have brought their approach into question [10, 11]. Studies [12] based on the dynamical mean-field theory (DMFT) [13] have made considerable improve- ments by including strong correlation and disorder ef- fects. However, the local nature of the DMFT presents severe limitations when studying this system. The effects of short-range fluctuations and spatial correlations were shown to be important in the classical Heisenberg model [14]. In this letter, we show that nonlocal effects may be equally important for the itinerant carriers, which me- diate the effective interaction between local moments in DMS. The dynamical cluster approximation (DCA) [15] sys- tematically incorporates nonlocal effects as the cluster size ( N$_{c}$ ) increases while retaining strong correlations. where the first term describes the electronic structure of the host material (GaAs) in the k · p approximation and the second term introduces a magnetic interaction between the carrier spin ( J ) and the local magnetic mo- ment ( S ) of Mn at position R$_{I}$ . The large magnitude of the Mn magnetic moment ( S = 5 / 2) allows us to treat it classically. This model is generally accepted to describe DMS [4, 5, 9, 12, 17, 18], since a mean-field treatment of the Hamiltonian [9, 17, 18] is able to explain many physical properties of the system. For the k · p Hamilto- nian ( H$_{k}$$_{·}$$_{p}$ ), we adopt a 4 × 4 Luttinger-Kohn model de- scribing heavy and light hole bands with spin-orbit cou- pling, but ignoring the conduction and split-off bands. We use the Luttinger parameters γ$_{1}$ = 6 . 98, γ$_{2}$ = 2 . 06, and γ$_{3}$ = 2 . 93 [19]. Biaxial strain is included in H$_{k}$$_{·}$$_{p}$ through the strain tensor ε$_{xx}$ = ε$_{yy}$ = ε$_{0}$ = Δ a/a and ε$_{zz}$ = ( − 2 c$_{12}$/c$_{11}$ ) ε$_{0}$ with the ratio of elastic stiffness The model Hamiltonian we adopt is When N$_{c}$ = 1, the DCA is equivalent to the DMFT, and exact results are approached as N$_{c}$ → ∞ . Since all the possible disorder configurations are considered in a cluster, the DCA is also a better approximation for disor- der average than the coherent potential approximation or DMFT by including multi-impurity scattering terms [16]. Thus, the DCA is an ideal method for studying DMS sys- tems. In this Letter, we study the magnetic properties of the prototypical DMS system Ga$_{1}$$_{−}$$_{x}$Mn$_{x}$As using the DCA and the k · p method, which describes the non- interacting band structure of pure GaAs. We show that nonlocal effects are very important for properly capturing the magnitude of T$_{c}$ , the saturation magnetization, and the magnetic anisotropy of this material. In particular, we show that the strong cluster anisotropy is responsible for the magnetic anisotropy along the cube-edge direction and the spin reorientation at low temperature. H = H$_{k}$$_{·}$$_{p}$ + J$_{c}$ ∑ I S ( R$_{I}$ ) · J ( R$_{I}$ ) , (1) arXiv:1001.1716v1 [cond-mat.str-el] 11 Jan 2010