arXiv:cond-mat/0503382v1 [cond-mat.mtrl-sci] 16 Mar 2005 Disorder, spin-orbit, and interaction eﬀects in dilute Ga 1-x Mn x As Gregory A. Fiete 1,2 , Gergely Zar´ and 1,3 , Kedar Damle 1,4 , and C. Pascu Moca 3 1 Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA 3 Research Institute of Physics, Technical University Budapest, Budapest, H-1521, Hungary 4 Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA and Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005 India (Dated: February 2, 2008) We derive an eﬀective Hamiltonian for Ga1-xMnxAs in the dilute limit, where Ga1-xMnxAs can be described in terms of spin F =3/2 polarons hopping between the Mn sites and coupled to the local Mn spins. We determine the parameters of our model from microscopic calculations using both a variational method and an exact diagonalization within the so-called spherical approximation. Our approach treats the extremely large Coulomb interaction in a non-perturbative way, and captures the eﬀects of strong spin-orbit coupling and Mn positional disorder. We study the eﬀective Hamiltonian in a mean ﬁeld and variational calculation, including the eﬀects of interactions between the holes at both zero and ﬁnite temperature. We study the resulting magnetic properties, such as the magnetization and spin disorder manifest in the generically non-collinear magnetic state. We ﬁnd a well formed impurity band fairly well separated from the valence band up to xactive < ∼ 0.015 for which ﬁnite size scaling studies of the participation ratios indicate a localization transition, even in the presence of strong on-site interactions, where xactive <xnom is the fraction of magnetically active Mn. We study the localization transition as a function of hole concentration, Mn positional disorder, and interaction strength between the holes. PACS numbers: 75.30.-m,75.47.-m,75.50.Pp I. INTRODUCTION Recently there has been a surge of interest in the more than 30 year old ﬁeld of diluted magnetic semiconductors 1 that has been largely motivated by the potential application of these materials in spin-based computation 2,3,4,5 devices. In particular, the discovery of ferromagnetism in low-temperature molecular beam epi- taxy (MBE) grown Ga 1−x Mn x As has generated renewed interest. 6 In this material Curie temperatures as high as T c ≈ 160K have been observed. 7 In this paper we focus on one of the most studied mag- netic semiconductors, Ga 1−x Mn x As, though most of our calculations carry over to other p-doped III-V magnetic semiconductors. In Ga 1−x Mn x As substitutional Mn 2+ play a fundamental role: They provide local spin S =5/2 moments and they dope holes into the lattice. 8 Since the Mn 2+ ions are negatively charged compared to Ga 3+ , in the very dilute limit they bind these holes forming an acceptor level with a binding energy E b ≈ 112meV. 8 As the Mn concentration increases, these acceptor states start to overlap and form an impurity band, which for even larger Mn concentrations merges with the valence band. Though the actual concentration at which the im- purity band disappears is not known, according to op- tical conductivity measurements, 9,10 and ellipsometry 11 this impurity band seems to persist up to nominal Mn concentrations as high as x nom ≈ 0.05. Angle resolved photoemission (ARPES) data, 12,13,14 scanning tunneling microscope (STM) results 15,16 and the fact that even “metallic” samples feature a resistivity upturn at low temperature 17 suggest that for smaller concentrations (and maybe even for relatively large nominal concentra- tions) one may be able to describe Ga 1−x Mn x As in terms of an impurity band. 18,19,20,21 While optical conductiv- ity results 9,10 and ellipsometry results 11 are suggestive of the presence of an impurity band in moderately doped samples, 9,10 an interpretation based on band to band transitions is also possible. 22 We remark, however, that these materials are extremely dirty 23 –the mean free path is estimated to be of the order of the Fermi wavelength– and therefore it is not clear if the latter approach is ap- propriate. Also, ARPES data indicate that the chemical potential of “insulating” samples lies inside the gap, 12 contradicting a band theory-based interpretation of the optical conductivity data. A detailed understanding of an impurity band model begins with the knowledge of a single Mn acceptor state. 24 The physics of the isolated Mn 2+ + hole sys- tem is well understood: 8 In the absence of the Mn 2+ core spin, the ground state of the bound hole at the acceptor level is four-fold degenerate and well described in terms of a F =3/2 state. For most purposes, only the fourfold degenerate F =3/2 acceptor levels need be considered in the dilute limit even in the presence of the Mn 2+ core spin. As evidenced by infrared spectroscopy, 8 the eﬀect of the S =5/2 Mn core spin on the holes is well-described by a simple exchange Hamiltonian: 1 H exch = G  S ·  F, (1) with G ≈ 5 meV. 8 The bound hole (acceptor) states within the F =3/2 multiplet are not Hydrodgenic due to a signiﬁcant d-wave component of the bound state wavefunction. 19,25,26,27