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

Recently, a more general, spin-polarized version of a DFT-based generalized gradient approximation (ϭGGA) and its modification by inclusion of the Hubbard potential (ϭGGA+U) were used (82) to calculate geometry, and electronic and magnetic properties of Mn atoms incorporated in bulk GaAs. The electron-electron interactions were described using the Perdew– Burke–Ernzerhof exchange–correlation potential (83), and electron-ion interactions were described by ultrasoft pseudopotentials (84). It was proved that inclusion of the Hubbard potential into calculations (σGGA + U) leads to contraction of geometrical parameters by about 2% as compared to those calculated without the Hubbard potential by σGGA method. The σGGA + U calculations of Ref. 78 have used the value 4.0 eV for the Hubbard potential, which is usually used in LDA+U calculations, and which is chosen to be close to the value U=3.5 eV obtained on the basis of experimental data for photoemission for Ga1-xMnxAs with small values of the impurity concentrations x (85). In both cases of σGGA and σGGA+U, a Mn impurity atom introduces a d-hole, and the majority spin state of the impurity lies at 0.25 eV above the Fermi level. These theoretical data correlate with experimental observations for bulk GaAs systems with low concentration of Mn, and constitute improvement of the previous LDA+U results.

It has become obvious, that being a non-variational theory by its nature (86), DFT cannot be applied successfully to predict properties of transition metal semiconductor (87) and magnetic systems without ad hoc schemes (such as KKR), uncontrolled approximations (such as LDA, GGA or σGGA), pseudopotentials, and adjustable parameters, such as the Hubbard potential U.

All these improvements and adjustments are incorporated to match experimental observations, and are not derivable from the first principles. They are necessary because in the framework of DFT one cannot introduce a self-consistent scheme of controlled approximations to the “real” exchange-correlation functional. Even more such ad hoc manipulations are required to apply DFT-based methods to strongly correlated nanoscale systems, such as layered DMS systems, or their QW and/or QD assemblies. Moreover, one cannot predict results of any of DFT-based theoretical schemes, until ad hoc theoretical approximations for the exchange-correlation functionals and adjustable parameters steaming from experimental data are incorporated into such theoretical schemes.

During two recent decades, another class of theoretical approaches to the electronic structure-property correlations in spatially inhomogeneous and nanoscale magnetic systems has emerged. These approaches have been developed originally to describe quantum phenomena (88 – 94), such as Coulomb blockade, in large nanoscale systems, and are focused on electron dynamics. They use a rigorous quantum statistical mechanical basis (see, for example, Refs. 95 and 96), but include numerous uncontrolled approximations, ad hoc assumptions and intuitive models, such as those explored in Refs. (97, 98). These approaches has received significant attention in literature in conjunction with emerging quantum computing technologies (99), because spin states of electrons residing on magnetic QDs are considered as the most stable realization of quantums of information qubits (100 – 102). Being derived semiphenomenologically from a theoretical basis developed either for bulk systems, or for simple quantum mechanical models, these approaches (103 – 111) tend to provide physically incorrect predictions even for mesoscopic cases, such as tunneling junctions, because they do not include adequate consideration of the electron spin interactions, quantum confinement effects, QD-toQD and QD-to-environment coupling. However, such interactions, effects and coupling are responsible for the major mechanisms of ferromagnetism in DMSs. By their nature, such models do not allow first-principle predictions of the electronic structure and magneto-electronic properties of layered or nanoscale magnetic systems. Fortunately, in the case of small nanoscale systems one can exploit rigorous methods of quantum statistical mechanics (QSM) directly using QSM theory-based “quantum chemistry” software packages, such as GAMESS, GAUSSIAN or Molpro, as described in the following section.

## 2. VIRTUAL SYNTHESIS OF SMALL QUANTUM DOTS OF III – V

Latest resonant tunneling spectroscopy studies of a variety of GaMnAs surface layers (112) have proven that GaAs band structure is not significantly affected by the presence of Mn substitutional defects for a range of Mn concentrations from 6% to 15%. It appeared, that the GaAs valence band (VB) does not merge with the impurity band, and that the exchange splitting of VB is in the range of several millielectronvolts even for the layers with TC as high as 154º K. At the same time, the ferromagnetic state was more pronounced than that of bulk (Ga,Mn)As and (Ga,Mn)As (26). Moreover, new model studies (113) show that similar to atoms and nuclei QDs of DMS exhibit formation of electron “shells”, and that their magnetism depends on shell occupancy.

There exist nanoscale potential fluctuations in (Ga,Mn)As/GaAs heterostructures that lead to formation of electrostatic QDs (114). Transversal Kerr effect spectroscopy studies (115) identified that MnAs inclusions of 10 nm to 40 nm in linear dimensions were responsible for a strong resonant band in the energy range from 0.5 eV to 2.7 eV in GaMnAs and InMnAs layers formed by laser ablation on GaAs and InAs surfaces. Observed transitions from quasi-two dimensional (2D) to three dimensional (3D) In0.85Mn0.15As structures develop gradually and are rather slow even at 270º C (116). At the same time, electron spin relaxation time in MnAs nanoparticles formed in GaAs lattice was found to be as long as 10 µs at 2º K (117), in contrast to known relaxation times of the order 100 ns in other QD structures and several picoseconds in bulk semiconductor systems (102).

These latest experimental data are in contradiction with the existing conventional models of mechanisms responsible for magnetism in and the band structure of nanoscale DMS systems, as described in Sec. 1. In particular, the major assumption that holes occupy GaAs-like valence band used to obtain successful predictions (118) of magnetization of GaMnAs as a function of the magnetic field at TC contradicts to experimental results of Ref. 112 and modeling results of Ref. 113. Thus, with advance of spintronics and quantum computing, more accurate QSM-based methods are required to evaluate spin entanglement and decoherence, and spin transport properties in small QDs and their heterostructures.

In this chapter the first principle QSM methods free from any assumptions concerning mechanisms and the band structure of the studied systems have been used to synthesize virtually a range of 14-atomic QDs of GaAs and InAs containing one or two Mn or V atoms and study their electronic and magnetic properties. The virtual synthesis method of Refs. 119 – 121 as realized by the GAMESS software package and discussed in Chapter 3 have been applied to minimize the total energy of several atomic clusters composed of tetrahedral symmetry elements (14 - atomic pyramids) of the GaAs and InAs zincblende lattices as described in Chapter 4 to obtain Ga10As4 and In10As4 molecules. Original pyramidal frames of these clusters are built of 10 Ga or In atoms, and four As atoms are placed at ¼ of the cube body diagonals in their bulk zincblende lattices (that is, inside of the pyramidal frames composed of the Ga or In atoms). The initial covalent radii of Ga, In and As atoms in these clusters have been adopted from experiment: 1.26 Ǻ, 1.44 Ǻ and 1.18 Ǻ, respectively. [Properties of these molecules are discussed in Chapter 4 in details.] Such clusters were synthesized at conditions mimicking quantum confinement (when spatial constraints were applied to the centers of mass of clusters’ atoms) and in “vacuum”, that is, in the absence of any constraints (external fields or “foreign” atoms interacting with the clusters’ atoms). At the next step, one or two As atoms have been replaced by one or two vanadium or manganese atoms, without any changes to the positions of the remaining Ga or In and As atoms. The total energy of the so built “diluted magnetic semiconductor” clusters has been minimized in the presence and in the absence of spatial constraints applied to the clusters’ atoms to obtain new pre-designed and vacuum molecules, respectively. Note again, that spatial constraints applied to the centers of mass of the clusters’ atoms have been incorporated to reflect effects of quantum confinement on the molecules synthesized in such a confinement. The total energy of the clusters so built have been minimized using Hartree-Fock (HF) and restricted open shell Hartree Fock (ROHF), leading to the emergence of pre-designed molecules Ga10As3V, Ga10As2V2, In10As3V, In10As2V2, and In10As3Mn. Thus, the Schrödinger equation has been solved numerically using GAMESS software to obtain the ground state of the corresponding molecules in the presence of the boundary conditions realized as the spatial constraints applied to positions of the centers of mass of the clusters’ atoms. Once the pre-designed molecules were obtained, the corresponding “vacuum” molecules have been developed by relaxing the spatial constraints applied to the atomic positions in the pre-designed clusters, and subsequent optimization (solving the corresponding Schrödinger equations in the absence of the spatial constraints). Electronic and magnetic properties of the virtually synthesized molecules are discussed below in this chapter.

Ground state details of the constitutive atoms are listed in Table 1.

TABLE I. Ground States of Semiconductor Compound Atoms

3. PRE-DESIGNED AND VACUUM MOLECULES In10As3Mn The case of the In10As3Mn molecules is very special from several points of view. In particular, findings obtained in the process of virtual synthesis of such molecules may help shed light of the origin of some controversies between the existing semi-phenomenological theoretical predictions and experimental observations specific to InAs nanoscale systems.

In contrast to the majority of the studied zincblende-derived molecules (see Chapter 4 and subsequent sections of the current chapter for details), the shape of the vacuum In10As3Mn molecule visibly deviates from pyramidal shape of its pre-designed counterpart (Figs. 1 and 2).

Fig. 1. (Color online) The pre-designed pyramidal molecule In10As3Mn: (a) front view; (b) top view;

(c) atomic positions. Indium atoms are yellow, As red, and Mn blue. In (a) and (b) all dimensions are approximately to scale; atomic dimensions roughly correspond to their respective covalent radii. In (c) atomic dimensions are reduced.

Fig. 2. (Color online) The vacuum molecule In10As3Mn. (a) and (b): side views; (c) front view.

Indium atoms are yellow, As red, and Mn blue. In (a) and (b) all dimensions are approximately to scale; atomic dimensions roughly correspond to their respective covalent radii.

As compared to the pre-designed pyramid of Fig. 1, in the vacuum molecule In atoms moved somewhat closer toward each other and Mn atom (Fig 2a). This is enough for the vacuum molecule to lose the pyramidal shape. The position adjustment allows stabilization of the vacuum In10As3Mn molecule in the absence of any spatial constraints and external fields. The electron charge density distribution (CDD) and molecular electronic potential (MEP) of these molecules are shown in Figs. 3 and 4.

Fig. 3. The molecular electrostatic potential (MEP) of the pre-designed molecule In10As3Mn for several isosurfaces of the electron charge density distribution (CDD) calculated for fractions (isovalues) of the CDD maximum value 4.25145 (arbitrary units): (a) 0.001; (b) 0.02; (c) and (d) 0.05; (e) 0.08; (f) 0.1. The color coding scheme is shown in each figure. In atoms are yellow, As red and Mn blue. All dimensions in (a) to (d) are to scale; atomic dimensions roughly correspond to the atoms’ covalent radii. In (e) and (f) atomic dimensions are reduced to show the MEP surface structure.

Fig. 4. The electron charge density distribution [(a) and (b); CDD], and molecular electrostatic potential (MEP) of the vacuum molecule In10As3Mn for several isosurfaces of the CDD calculated for fractions (isovalues) of the CDD maximum value 4.61805 (arbitrary units). (a) and (b): 0.01 and 0.05, respectively; (c) to (f): MEP for CDD isovalues 0.001, 0.05, 0.1 and 0.1, respectively. The color coding scheme for MEP surfaces is shown in each figure. Indium atoms are yellow, As red and Mn blue. All dimensions in (a) to (d) are to scale; atomic dimensions roughly correspond to the atoms’ covalent radii. In (e) and (f) atomic dimensions are reduced to show the MEP surface structure.

Both In10As3Mn molecules have similarly shaped CDD and MEP surfaces, and almost the same maximum and minimum values of CDD. However, the CDD and MEP surfaces of the vacuum molecule reflect significant loss of tetrahedral symmetry (compare the surfaces in Fig.

3a and Fig. 4c, for example). In both cases electron charge is delocalized, spread through the space occupied by the molecule and reaches beyond it. The electron charge density still is about