P1: GVG Journal of Superconductivity: Incorporating Novel Magnetism (JOSC) pp378-josc-367300 January 22, 2002 8:17 Style file version Nov. 07, 2000 Journal of Superconductivity: Incorporating Novel Magnetism, Vol. 15, No. 1, February 2002 (C 2002) Density Functional Calculations for III–V Diluted Ferromagnetic Semiconductors: A Review Stefano Sanvito,1 Gerhard Theurich,2 and Nicola A. Hill1 Received 1 November 2001 In this paper we review the latest achievements of density functional theory in understanding the physics of diluted magnetic semiconductors. We focus on transition-metal-doped III–V semiconductors, which show spontaneous ferromagnetic order at relatively high temperature and good structural compatibility with existing III–V devices. We show that density functional theory is a very powerful tool for (i) studying the effects of local doping defects and disorder on the magnetic properties of these materials, (ii) predicting properties of new materials, and (iii) providing parameters, often not accessible from experiments, for use in model Hamiltonian calculations. Such studies are facilitated by recent advances in numerical implementations of density functional theory, which make the study of systems with a very large number of atoms possible. KEY WORDS: diluted magnetic semiconductor; density functional theory. 1. INTRODUCTION device will open completely new possibilities, with huge potential applications. Hybrid structures, where The possibility of using the spin degree of free- magnetic metals are used to inject spin electrons dom as well as the electronic charge for electronic ap- into semiconductors have been shown to be prob- plications (“spintronics”) has received great attention lematic [3–5]. This is due to the large mismatch be- in the last few years [1]. A prototypical application of tween the resistivities of semiconductors and metals, this concept is the giant magnetoresistance (GMR) ef- which seriously precludes effective spin injection [6]. fect [2], where the electrical resistance of a magnetic Although this fundamental obstacle can be overcome, multilayer is changed by the application of a magnetic for instance by large contact resistances [7], it is nat- field. GMR sensors are already commercially avail- ural to turn attention toward other directions and try able for read heads in high density data storage de- to create all-semiconductor devices. These, of course, vices, with performances increased by 1 order of mag- need magnetic semiconductors. nitude with respect to conventional sensors. However At present several types of diluted magnetic current GMR sensors do not exploit completely the semiconductors (DMSs) are available [8]. Among potential of spintronics. In existing electronic devices these, the DMSs based on III–V semiconductors are (for example, personal computers) there are two main particularly important because of their compatibility elements: the logic components and the data stor- with existing III–V-based technology. III–V DMS are age device. The former are transistors based on semi- obtained by low-temperature molecular beam epitaxy conductor technology, while the latter is essentially a (MBE) deposition of III–V semiconductors with a metallic magnetic film. Of course the ability to com- transition metal such as Mn. The nonequilibrium bine both logic elements and data storage in the same growth is necessary to prevent the formation of addi- tional phases, and in general, only low concentrations of transition-metal ions can be incorporated in the 1Materials Department, University of California, Santa Barbara, California 93106. nonmagnetic matrix. Nevertheless, despite the low 2Department of Chemistry, University of Pennsylvania, concentrations, the systems develop long-range Philadelphia, Pennsylvania 19104. ferromagnetic order with remarkably high Curie 85 0896-1107/02/0200-0085/0 C 2002 Plenum Publishing Corporation P1: GVG Journal of Superconductivity: Incorporating Novel Magnetism (JOSC) pp378-josc-367300 January 22, 2002 8:17 Style file version Nov. 07, 2000 86 Sanvito, Theurich, and Hill temperatures Tc. For the known III–V-based DMS The result of Eq. (2) can be greatly refined by the highest Curie temperatures obtained are Tc including a detailed description of the band structure 30 K for (In,Mn)As [8,9], Tc = 110 K for of the underlying nonmagnetic semiconductors or by (Ga,Mn)As [8], and, very recently, a report of incorporating correlation effects going beyond the Tc = 940 K for (Ga,Mn)N [10]. Moreover, al- mean fieldt approximation. However, it is important though this paper is focused only on the III–V to stress that the use of these models always requires DMS, it is worth mentioning that ferromagnetism parameters that are often difficult to extract from the with Tc often above room temperature has been experiments. For example the experimental value of found in several other DMSs, including Ge1xMnx the exchange constant N0 vary in the range 1–3 eV [11], Cd1xMnxGeP2 [12], Ti1xCoxO2 [13], and [19–24]. Zn1xCoxO [14]. In addition, models based on any kind of mean (Ga,Mn)As, although it has a Tc far below room field approximation naturally fail in describing lo- temperature, is the most widely studied DMS. This is cal effects, which occur on the atomic scale. The fer- due to its structural compatibility with most epitaxi- romagnetism of (Ga,Mn)As is very sensitive to the ally grown III–V’s, which makes it ideal for building sample “history,” such as the growth conditions [8] heterostructures and exploring new device concepts. and eventual after-growth processing [25,26]. Since Examples that have been already achieved include the growth dynamics certainly affects the microscopic spin injection into heterostructures [15] and field- configuration of the samples, this suggests that knowl- effect control of the ferromagnetism [16]. edge of the local chemical environment is crucial for Three important features underlie the ferro- understanding and modeling the properties correctly. magnetic order of (Ga,Mn)As, and these are shared These considerations show that it is essential to by the other Mn-doped III–V DMSs: (i) Mn2+ ions have a microscopic theory providing information to substitute for Ga3+ cations in the zinc blende lattice, the simpler mean field-like models. Density functional providing localized spins (S = 5/2inGa1xMnxAs theory (DFT) [27,28] is to date the most efficient and In1xMnxAs), (ii) there are free holes in the sys- and accurate microscopic theory for describing the tem although the actual concentration is much smaller electronic, magnetic, and structural properties of the than the Mn concentration (despite the nominal va- ground state of electronic systems with a large number lence suggests that the two concentrations should be of degrees of freedom. Recent advances in the numer- identical), and (iii) the hole spins couple antiferro- ical implementations, mainly concentrated in the use magnetically with the Mn spins, owing to a dynamic of improved pseudopotentials [29], of efficient basis p–d coupling. Then the ferromagnetic behavior can be sets [30] and of N order approximations [31], make reasonably well described by the Zener model [17], in possible the study of systems containing several hun- which antiferromagnetic exchange coupling partially dreds of atoms. Such computational capabilities are spin polarizes the holes, which in turn cause an align- required to study the DMSs in the low concentration ment of the local Mn spins. Within the Zener model, limit. The main aim of this paper is to provide a review the interaction Hamiltonian between the hole spin sE of the achievements of DFT in describing the prop- and the Mn spin SE is erties of the DMSs. In particular we will show that DFT is an invaluable tool for (i) studying the effects H =NsESE, (1) 0 of local doping defects and disorder on the magnetic where N0 is the concentration of the cation sites and properties, (ii) predicting properties of new materi- is the p–d exchange integral. N0 is usually called the als, and (iii) providing parameters, often not accessi- exchange constant. If one simply uses the mean field ble from experiments, for use in model Hamiltonian approximation [17,18], in which the magnetizations calculations. of both carriers and Mn ions are considered to be The remainder of the paper is organized as fol- uniform in space, we obtain an expression for Tc: lows. First, we will briefly overview the most recent developments in density functional theory. In the fol- xN S(S+1)2 T = 0 s , (2) lowing section we will discuss the structural prop- c 2 3kB(g B) erties of (Ga,Mn)As and explain why zinc blende where s is the magnetic susceptibility of the free MnAs cannot be grown. Moreover we will also look carriers (holes in this case), g is their g-factor, at spin–orbit effects and explain why this introduces kB is the Boltzmann constant, and B is the Bohr only minor quantitative changes in the exchange con- magneton. stant of (Ga,Mn)As. Then we will move to the low P1: GVG Journal of Superconductivity: Incorporating Novel Magnetism (JOSC) pp378-josc-367300 January 22, 2002 8:17 Style file version Nov. 07, 2000 Density Functional Calculations for III–V Semiconductors 87 dilution limit, calculating the exchange constants and The so-called Kohn–Sham wave functions i are discussing the limitations of mean field models. In the single-particle eigenfunctions that are strictly mean- remaining sections we will consider the effects of the ingful only for determining (r), and the Kohn–Sham local microscopic configuration of the Mn ions and eigenvalues εi are the derivatives of the total energy possible intrinsic defects on the ferromagnetism of with respect to the occupation of state i. Note that they (Ga,Mn)As. In particular we will look at the role of are not strictly related to single-particle excitation en- intrinsic defects and at the transport properties of dig- ergies, although the Kohn–Sham band structure can ital ferromagnetic heterostructures (DFH) [32].