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## Empirical methods in semiconductor nano-structures design and modelling

#### University of Manchester, UK and ACAM, Dublin, Ireland

#### Organisers

For further information and how to register for Dublin event please gowww.tyndall.ie/ptg/Seminars/ACAMTyndall.html

For further information and how to register for Manchester event please go www.cse.scitech.ac.uk/cecam_at_daresbury/nano_structures.shtml

Theory, modelling, and computational methods for semiconductor materials and nanostructures is a topic of rapid growth and great international interest. A lot of the world-wide effort over the past 50 years in establishing the theoretical foundation of methodologies for calculations of structural, electronic, optical and transport (electrical and thermal) properties of semiconductor nanostructured materials [1-61] are now coming to fruition [62-67].

Modern crystal-growth techniques, such as molecular beam epitaxy or metalorganic chemical-vapour deposition, are capable of producing prescribed crystal structures, sometimes even in defiance of equilibrium bulk thermodynamics.

To correlate desired electronic and optical properties with the structure cannot be efficiently done experimentally solely by trial-and-error methods. Hence computational methods and novel algoritms that combine fast empirical solvers, detailed knowledge of the nanostructure shape and size, chemical composition, and mechanical properties, together with ever increasing computational power available are required to address this fundamental problem.

Among the many empirical methods available, the multi-band k.p (k.p), empirical tight binding (ETB), and empirical pseudopotential (EPM) methods are proving invaluable in the design and modelling of modern optoelectronic devices and semiconductor nanostructures.

After the pioneering work on electronic structure of Luttinger and Kohn [1] and Kane [2,5] on the perturbative approach, the k.p method, for electronic structure of bulk and strained material in the vicinity of the characteristic points in the Brillouin zone (usually the Gamma point), with only a limited number of states in the expansion, now days the implementation of the k.p Hamiltonian constantly evolves and now encompasses 16 [12], 24 or even 30 [28] [28a]. The latest development of the k.p theory, using accurate parameterizations obtained from many body perturbation theory in the GW approximation, are capable of describing the electronic structure of the whole Brillouin zone of III-V materials [p4]. Recent efforts on parameterization of the k.p Hamiltonian, for wurtzite structures also include the use of DFT calculation with optimised effective potentials (OEP) [p5]. The most commonly used implementation of the k.p Hamiltonian now days is the 8-band k.p Hamiltonian, which takes into account mixing of the valence band states between themselves, mixing of the lowest conduction band state with few top valence band states, and also takes into account the effect of the spin-orbit interaction [31]. The Spin orbit effect, although regularly ignored in first-principles calculations, is crucial in low energy gap materials like InAs or InSb [5], and in alloys with heavy atoms, like lead containing binaries PbSe etc [29]. The wide use of the 8-band k.p method also owes to the transferability of the Hamiltonian over the whole family of the III-V materials [p6]. With new computer architecture the 8-band k.p Hamiltonian is now regularly used for calculation of the electronic structure of semiconductor nanostructures: quantum dots, quantum dashes, quantum wires, etc [18,19,21,22,24,25,33,37,n1,n4,n5]. The flexibility of the 8-band k.p model also allows for relatively easy rotation of the kinetic, strain and piezoelectric part of the Hamiltonian to allow research of nanostructures grown on high Miller index planes [15,33]. In the plane waves implementation, the multiband k.p code for QD structure, was recently successfully used to relate absorption characteristics and optimal QD arrangements in Intermediate Band Solar Cells (IBSC) based on QD arrays [37]

The development of atomistic empirical potential methods for Molecular Dynamics (MD) and Molecular Statics (MS) has allowed structural simulations of low dimensional III-V semiconductor materials to become a field that has attracted a substantial amount of interest in recent years [38-47] There are several reasons for this, the most important probably being that the electronic properties of lattice mismatched epitaxial semiconductor layers are strongly affected by their structural properties [48]. Hence the reliable determination of quantities such as the elastic properties and the resulting strain is a fundamental prerequisite for implementing any accurate description of the bandstructure and the associated energy levels [49].

The state-of-the-art at present is represented by the well known Abel-Tersoff potentials (ATPs) [50-55], Stillinger Weber potential, [56] and the Keating-Valence Force Field [4] method. Despite their wide use some fundamental questions need to be resolved, particularly in the inefficiency and inaccuracy of these potentials in describing the phonon spectrum [58]. Current attempts at solving these problems concentrate on superimposing the essence of the π-bonding to the σ-bonding in order to capture more of the physical and chemical behaviour [59]. This however might create an extremely computational intensive problem if the potential is to be used within large MD problems. Hence specialized approaches to parallelization of these routines are extremely timely.

Of the empirical approaches to calculate the bandstructure of semiconductor materials Tight Binding (TB) has for a long time been regarded as a very accurate but not ideal alternative to k.p and empirical pseudopotential method (EPM), mainly because of the computational demand. The TB approach is in origin an approximated ab initio technique that relies on constructing the band structure from a truncated linear combination of atomic orbitals (LCAO). However it was proposed that the method could also be used as a semi empirical approach when the Hamiltonian matrix elements in the basis of the atomic orbitals are used as fitting parameters. This opened the way for the Empirical Tight Binding (ETB) approach which is generally found in two “flavours” for applications to semiconductor nanostructures: either using the sp3s* valence shells and interactions up to second nearest neighbours, or the sp3d5s* ones and only nearest neighbour interactions. The s* is a virtual orbital, that can be viewed as an excited s orbital, introduced by Vogl et al, [10] to mimic, in nearest neighbour models, the effect of the higher momentum orbitals.

While both ETB and EPM reduce the need for fitting in the final results by obtaining accurate parameterisations from the bulk and strained crystal properties, the main advantages of using ETB for calculating the electronic levels of nanostructures is that the treatment of the strain induced deviations from the bulk band structure is nearly self consistent: unlike the screened EPMs which depend explicitly on the strain (even if it is only the hydrostatic strain), the ETB parameters scale with increasing bond length (this feature is known as Harrison’s scaling rule) to take into account the hydrostatic strain, while the formulation itself has a strong dependence on the bond angle (through the Bloch phases [exp (-i k.r)] between different nearest neighbour atoms, which clearly takes care of the effects of the shear strain. Furthermore as an atomistic method it is also possible to treat problems like piezoelectricity self consistently, even though a solution to the problem has not yet been clearly identified.

There is no doubt that despite parameterisations being available for all commonly used semiconductors,[60] ETB applied to semiconductor nanostructures still requires significant computational power. In general the use of the sp3d5s* (20 orbitals) approach for N atoms requires the solution of a 20xN order matrix, utilising 22 parameterised two centre integrals,[61,n3] hence the need for parallel clusters machines.

The state-of-the-art tight binding calculations were recently used to investigate influence of electronic structure and multiexciton spectral density on multiple-exciton generation in semiconductor nanocrystals (Si, InAs, PbS) and its possible application in photovoltaic device [36].

The essence of the Empirical Pseudopotential Method (EPM) is to provide non-self consistent band structure calculations for crystals and nanocrystals with a full and accurate description of both occupied and unoccupied states over a large energy range. This is usually achieved by ‘fitting’ the pseudopotential (i.e. potential experienced by chemically active electrons in their valence shell) in momentum (or reciprocal) space. Due to crystal symmetry, this requires only fitting the Fourier components v(G) of the pseudopotential corresponding to only a few shortest reciprocal lattice vectors {G}. For solids with super structures, or for nanocrystals, v(G) corresponding to shorter G vectors can be interpolated from the values for the relevant bulk materials. The earliest successful application of the EPM was made for explaining optical measurement on semiconductors by Cohen and Bergstresser [3] who employed the so-called local EPM and by Chelikowsky and Cohen [9] who employed the so-called non-local EPM, including spin-orbit interaction. More recent successful applications of the EPM for superlattices and nanostructures have been made by several groups [13,20,23,26,p1,n2,n7]. As the EPM method is non-self consistent it is computationally much cheaper and faster to apply than ab-inito methods. Moreover, as it deals with both occupied and unoccupied states on equal footing, it is very useful for studying opto-electronic properties without requiring any further adjustment to the theory of band structure.

Further development of the EPM, particularly desirable for electronic structure calculations of quantum dots includes the “linear combination of bulk bands” (LCBB) variant of the full EPM [20]. In the LCBB method the wave functions of a nanostructure (superlattice, wire, and dot) are expanded as linear combinations of a limited number (n) of the bulk Bloch states of the constituent materials and wave vectors k. This flexibility allows one to use physical intuition in selecting the n and k that are most relevant for a given problem.

The EPM method was successfully used to predict excitonic properties of the novel solar cells based on multi exciton generation (MEG) mechanisms in CdSe QD’s and role of Auger processes in MGE [26]

State-of-the art application of the EPM method as a fast empirical [20,n7] solver in combination with stochastic search engines allows determination of the crystal structure with pre-assigned electronic and optical properties [63]. To correlate desired electronic and optical properties with the structure cannot be efficiently done solely experimentally by trial-and-error methods. Hence the computational solution of the “inverse problem” address the fundamental problem of finding the atomic configuration of a complex, multi-component system that produces a target electronic-structure property.

Electronic structure methods, such as multiband k.p, Empirical Tight Binding and Empirical Pseudopotential, coupled with empirical methods for the determination of the crystal structure (Valence Force Field, Bond Order Potentials) have recently seen much improvement in both the methodology, parameterizations and computational speed. The next step will certainly be that of condensing and interfacing all the individual efforts to produce tools that are of more general use to the wider community. It is therefore now very timely to devote two closely coupled CECAM Workshops to the use of empirical methods for semiconductor nanostructure design and modelling, with the first (ACAM) Workshop focused primarily on computational/technical issues, numerical implementations and parametrisation strategies, followed immediately by the second (STFC Daresbury) Workshop highlighting the scientific issues and demands related to empirical nanostructure design and modelling.

## References

**Ireland**

Eoin O'Reilly (Tyndall National Institute at University College Cork) - Organiser

**United Kingdom**

Max Migliorato (University of Manchester) - Organiser

Gyaneshwar Srivastava (University of Exeter) - Organiser

Stanko Tomic (STFC Daresbury Laboratory) - Organiser