# «SMALL QUANTUM DOTS OF DILUTED MAGNETIC III-V SEMICONDACTOR COMPOUNDS Liudmila A. Pozhar PermaNature, Birmingham, AL 35242 Home Address: 149 Essex ...»

CHAPTER 5

## SMALL QUANTUM DOTS OF DILUTED MAGNETIC

## III-V SEMICONDACTOR COMPOUNDS

Liudmila A. Pozhar

PermaNature, Birmingham, AL 35242

Home Address: 149 Essex Drive, Sterrett, AL 35147

Tel: (205) 678-0934

E-mail: lpozhar@yahoo.com; pozharla@yahoo.com

Running Title: Small QDs of III-V DMS

## SUMMARY

In this chapter quantum many body theoretical methods have been used to study properties of GaAs - and InAs - based, small semiconductor compound quantum dots (QDs) containing manganese or vanadium atoms. Interest to such systems has grown since experimental synthesis of nanoscale magnetic semiconductors, that is, nanoscale semiconductor compounds with enhanced magnetic properties. This enhancement is achieved by several methods, and in particular by doping common semiconductor compounds with "magnetic" atoms, such as Mn or V. Experimental studies indicate that the electron spin density in the case of thin nanoscale “magnetic” semiconductor films and QDs may be delocalized. As described in this chapter, quantum many body theory-based, computational synthesis (i.e., virtual synthesis) of tetrahedral symmetry GaAs and InAs small pyramidal QDs doped with sabstitutional Mn or V atoms proves that such QDs are small "magnetic" molecules that indeed, possess delocalized and polarized electron spin density. Such delocalization provides a physical mechanism responsible for stabilization of these nanoscale molecular magnets, and leads to the development of what can be described as spin-polarized “holes” of the electron charge deficit. In some QDs, numerical values of the electron spin density distribution are relatively large, indicating that such semiconductor systems may be used as nanomaterials for spintronic and magneto-optical sensor applications.

Key words: nanostructure, quantum dot, molecular magnet, virtual synthesis, magnetic properties, electronic properties, spin-polarized hole, bonding, electron spin density, magnetooptics

1. INTRODUCTION Since a discovery of ferromagnetism in bulk Mn-doped GaAs and InAs systems (1- 4), III-V and II-VI - based diluted magnetic semiconductor (DMS) have been vigorously studied both by experimental and theoretical means. Such systems exhibit a variety of electronic and magnetooptical properties defined by carrier-mediated ferromagnetism, including spin polarization of holes, turning on and off the magnetic phase by changing hole concentration at constant temperature, hysteresis of Hall resistance as a function of an external magnetic field and gate voltage below the Curie temperature (Tc), electric-field assisted magnetization reversal, magnetization enhancement by circularly polarized light, current-induced magnetization switching, a strong increase of resistance with lowering temperature, colossal negative magnetoresistance, current-induced domain wall motion, etc. (see Refs. 5 - 7 and references therein). These properties are exceptionally important in materials for quantum electronics, spintronics and quantum information processing, as they allow control of carriers' spin generation, relaxation and deflection, and magneto-optical properties of the materials, using not only external magnetic fields, but also external electric fields and/or current, light, carrier concentration, and quantum confinement effects.

Unfortunately, bulk DMS systems possess very low Curie temperature (TC) that heavily limits their technological use. Thus, synthesis of DMS-based structures with room temperature TC and higher has become one of the central issues of application of DMS-based materials in novel spintronics and quantum information processing technologies. In particular, following the existing practical experience and the p-d Zener model-based predictions (8, 9), state-of-the-art low temperature molecular beam epitaxy (MBE) and other thin film synthesis methods, layered device development techniques, and numerous methods of synthesis of quantum wire (QW) and quantum dot (QD) - based nanostructures have been established over the years (see, for example, Ref. 10 - 16) to realize low dimensional DMS-based nanostructures where DMS components possess enhanced TC up to 173 K (17 - 26). Recently, room temperature ferromagnetism was reported in Mn-doped InGaAs and GaAsSb nanowires (27), and MBEbased formation of (Ga,Mn)As crystalline nanowires with Tc=190 K was achieved (28).

However, reached values of TC for ingenious layered structures based on Mn-doped (Ga,Mn)As, that is expected to be the highest TC DMS among III-V candidates, are significantly lower than those predicted by the existing theoretical models of hole-mediated ferromagnetism in bulk, spatially homogeneous Mn-doped GaAs and InAs systems. In particular, the p-d Zener model (8,

9) predicts TC=345 K for a spatially homogeneous sample Ga0.8Mn0.2As with high Mn concentration from Ref. 11, while the observed TC for that structure is 118 K. Because the p-d Zener model is designed to describe DMSs in the low impurity concentration limit, it also fails to predict (even qualitatively) an increase in TC of samples with high Mn concentration (11).

Similarly, an improved approach (29) that uses a semi-phenomenological model predicts unrealistic values of TC above 500 K in structures with remote doping, such as heterostructures of Ref. 24. [In such heterostructures impurities are introduced as a so-called δ layer on one side of a heterojunction, so impurities and carriers are not in the same channel. This provides for the carrier mobility up to three orders of magnitude higher than that in heterostructures where impurities and carriers share the channel.] It is important to note that the exiting theoretical approaches have been developed to describe ferromagnetism (FM) in bulk, homogeneous DMDs with low impurity concentration (30, 31). Correspondingly, they incorporate assumptions that are not applicable to quantumconfined systems and nanoscale heterostructures, such as those composed of layered films, QWs and QDs of a few nanometers in linear dimensions. In such systems carrier motion is quantized in the direction(s) of confinement, which is not incorporated into the models’ formalism.

Moreover, a small number of atoms (2 to 4, in the case of a confinement with at least one linear dimension of about 1 nm) require consideration of many-particle quantum problem. In the absence of a rigorous, first principle quantum theory of solids, and in particular semiconductors, necessary to fully account for quantum effects, the semi-phenomenological band theory of semiconductors was modified to produce numerous models designed to quantify observed electronic structure and properties of semiconductor films, quantum wells, thick QWs, large QDs, and their heterostructures (30, 32, 33).

One of such models due to Zener (34), called the p-d Zener model, has been very successful in the case of bulk DMSs (8, 9), and with some modifications has also been successfully applied to thick DMS films and layered systems (see, for example, Refs. 31 and references therein). This model makes use of the fact that in DMSs the Fermi energy EF lies, as a rule, within the majority t-band of the "magnetic" impurity (such as Mn atoms) which is t2g symmetric. Moreover, d-states of magnetic impurity atoms lie below the valence p-states of the host semiconductor structure, like in (Ga,Mn)As. Such p- and d-states hybridize pushing the majority valence states to high energies, and the minority valence states to lower energies. This mechanism, called kinetic p-d exchange, leads to appearance of holes in the valence band, and thus to generation of a significant magnetic moment μ per an impurity atom (μ = 5μB in the case of Mn impurity, where μB is the Bohr magneton) leading to hole-mediated ferromagnetism. At the same time, the majority valence band becomes spin-polarized with a much smaller magnetic moment per an impurity atom (equal to -μB per a Mn atom) due to weak d-d interactions between impurity electrons. Thus, the majority (d) - majority (p) orbit hybridization (the p-d exchange) produces FM coupling that leads to a decrease in the total energy of the system.

Importantly, the p-d Zener model exploits the virtual crystal and molecular field approximations, the limit of low hole and impurity concentrations (with the hole concentration much smaller than that of the impurities), and an assumption of weak p-d coupling (35). In particular, when the p-d hybridization is large, the coupling becomes strong, and both approximations become invalid (36, 37). Also, the p-d Zener model is expected to fail in the case of nanometer-thin DMS films, thin QWs and small QDs where the concentration of impurity atoms may reach tens of percent. Moreover, for DMS systems possessing large characteristic dimensions, but heterogeneous in nature (that is, featuring regions of the higher and smaller impurity concentration, or different magnetic phases) this model is also invalid. Indeed, the p-d Zener model predicts the Curie temperature TC ≈ 345º K in the case of a FM-homogeneous Ga1-xMnxAs film with the Mn concentration x over 10%, while the experimental value is approximately 118º K (12). According to this model, TC increases with the increasing hole concentration. However, when the hole concentration exceeds that of the impurities, experimentally observed TC of 5 nm thin Ga1-xMnxAs from Ref. 11 tends to decrease with an increase in the hole concentration. Moreover, the p-d Zener model does not account for many other experimental data (14, 38 - 41) on magnetic properties of DMS films. There were attempts to modify this model to account for new experimental observations (18, 42), but they could not accommodate for emerging experimental evidence indicating that FM mechanisms other than the p-d Zener exchange, significantly contributed to magnetic properties of DMSs.

Thus, other semi-phenomenological models, such as an impurity band (IB) model have been developed (43). Yet, even in the case of relatively thick DMS films of Ref. 14, the major issue concerning the hole-mediated FM mechanism remains unresolved. In particular, it is not quite clear where the mediating carriers reside – in a disordered valance band (VB), as stated in Refs. 6 and 8, or in the impurity band that may be detachable from the host VB and retaining dorbit properties of the impurities, such as Mn (43 - 50).

More sophisticated theoretical approaches steaming from the first-principle theoretical basis have been suggested to explain non-conventional origin of ferromagnetism in and complex magnetic properties of DMSs. Most likely, the earliest of them was an approach that used scattering theoretical methods to solve Schrödinger equation (51 - 53) for electrons in a periodic system. However, another approach due to Korringa, Kohn and Rostoker (KKR) introduced in Refs. 54 - 56 became more popular due to its simplicity and mathematical transparency. This approach uses the density functional theory (DFT) mathematical foundation (54) and a smart choice of a reference system to simplify a general DFT procedure of band structure calculations.

Numerous KKR-based models were developed and applied to periodic systems with localized perturbations (57 – 65). The KKR method itself was further improved when the problem of determining the charge density n(r) was solved (66, 67) in terms of the single particle Green’s function G(r1, r2) introduced as the solution of the inhomogeneous Kohn-Sham equation of the DFT with the δ-function source, and the algebraic Dyson equation used to relate the Green’s function of structural defects to the Green’s function of the ideal crystal (68). Further advance of the KKR method is related to inclusion of non-spherical interaction potentials using a shape function technique to model non-spherical parts of the potential, and using the LippmannSchwinger equation to develop an iterative scheme of determination of the Green’s functions from the Dyson equation (68 – 72). A concept of a coherent potential (73, 74) was incorporated to make KKR-based methods applicable to disordered systems and giving rise to so-called the coherent potential approximation (CPA). The use of the local force theorem (75) enabled further successful applications of KKR-CPA models to disordered magnetic systems. According to this “theorem”, in the case of a frozen ground state potential and small perturbations in the charge and magnetization densities of a system, a variation of the total energy of the system is equal to the sum of the single particle energies over all occupied energy states. This theorem simplified evaluation of the exchange interaction energy between two magnetic atoms (76).

Simplification of the exchange-correlation functional by the use of the local density approximation (LDA) made it practical to apply DFT-based methods to calculate the energy level structure of large molecules and the band structure of solids. However, LDA failed in the case of transition metal compounds and DMSs, where it predicted a partially filled d-band with metallic character of the electronic level structure (77). Thus, other adjustments of DFT-based methods were introduced, including the most popular self-interaction correction LDA (SICLDA) and LDA+U (78, 79) methods, with U being the Hubbard potential, to reach at least partial success in predicting the band structure and TC of strongly correlated systems, and in particular, DMSs. In GaAs systems with Mn impurities the Hubbard potential accounts for Coulomb correlation effects that have to be incorporated to calculate more accurately the band structure of DMSs (79), and also leads to an increase in hole delocalization and a decrease in the p–d Zener interaction (78). It also inhibits the double exchange and pushes the d-states to lower energies (80). As a result, in LDA+U approximation the mean-field Curie temperature is a linear function of the carrier concentration (81). Despite important results, it is not clear whether further improvement in prediction of the band structure and magnetic properties of DMS is possible in the framework of LDA and its modifications.