Methods

Most of the work performed in our group is based on one of the following numerical methods:

Density Functional Theory
Ab initio Molecular Dynamics
GW and Bethe Salpeter
Valence Force Field Method
Empirical Pseudopotentials: Strained Linear Combination of Bulk Bands
Empirical Pseudopotentials: Plane Wave Expansion
Configuration Interaction

Since 2008 we have been developing the following methods and codes:

Atomic Effective Potentials (AEPs)
Plane Wave Pseudopotential code for 100,000 atoms: code PLANC
Efficient calculation of Exchange and Coulomb Matrix Elements: cmat_atomistic
Efficient calculation of Electron-Phonon Coupling for Clusters
Adiabatic Bond Charge Model
Hartree-Fock Fock code

Density Functional Theory (DFT)

An introduction to DFT can be found here. We use DFT for systems up to 1000 atoms on parallel platforms such as in Garching (RZG) or at the High Performance Computing Center Stuttgart (HLRS). For our largest calculations we use CPMD as it scales very well up to 512 tasks. For calculations of linear response, either phonons or piezoelectric properties we use ABINIT that has the most advanced functionality for energy derivatives and GW capabilities. We use Quantum Espresso when we want to address spectroscopic properties (see GW and Bethe Salpeter section). When we don't trust pseudopotentials or want to test approximations, we use the all-electron ELK code.

Ab initio Molecular Dynamics (AIMD)

We use ab initio molecular dynamics to study the transport properties of ions in condensed matter. We mostly use the CPMD code and also performed some benchmark tests with CP2K.

GW and Bethe Salpeter

We have recently started to use GW and the Bethe Salpeter approach for the calculation of the optical properties of bulk or bulk-like systems. This fully ab-initio approach is only feasible for very small systems, and already the computation of Graphene is hard to converge. However, this is the best available approach for periodic systems. We are presently using the Yambo code.

Valence Force Field Method (VFF)

In most cases, the system of interest is too large to be handled by standard density functional theory and we make use of a classical atomistic force field model, the Valence Force Field (VFF) method, including bond bending, bond stretching and bond bending-bond stretching interactions:

\text{[math]}

where Δd_ij² =

². Here R_i is the coordinate of atom i, d_ij⁰ is the ideal (unrelaxed) bond distance between the atoms i and j, and θ_jik⁰ is the ideal (unrelaxed) angle of the bond j - i - k. The ∑ ^nn_i denotes summation over the nearest neighbors of atom i. The bond stretching, bond angle bending, and bond-length/bond-angle interaction coefficients α_ij⁽¹⁾(≡ α), β_jik, σ_jik are directly related to the elastic constants in a pure zincblende structure. The second-order bond stretching coefficient α⁽²⁾ is related to the pressure derivative of the Young’s modulus by

, where B = (C₁₁ + 2C₁₂)∕3.

An atomic force field is similar to continuum elasticity approaches in that both methods are based on the elastic constants, {C_ij}, of the underlying bulk materials. However, atomistic approaches are superior to continuum methods in two ways, (a) they can contain anharmonic effects, and (b) they capture the correct point group symmetry. The calculation of the energy and forces from expressions such as Eq. (1) can be performed within seconds for millions of atoms, allowing for a manageable strain minimization of large nanostructure.

Empirical Pseudopotentials: Strained Linear Combination of Bulk Bands (SLCBB)

Empirical pseudopotentials in combination with the SLCBB code has been mostly used for the calculation of the electronic properties of epitaxial nanostructures (self-assembled quantum dots, interface fluctuation quantum dots, quantum wells). With this approach we have treated up to 8 million atoms. A review of the methods presented here can be found in Electronic excitations in nanostructures: An empirical pseudopotential based approach J. Phys. Cond. Mat. .21, 023202 (2009) (link).

The SLCBB method is designed to calculate the single particle eigenfunctions and eigenvalues of large nanostructures like self-assembled quantum dots. The SLCBB method uses a basis set of strained Bloch orbitals of the underlying bulk:

$\begin{displaymath} \psi ({\bf x}) = \sum_n^{N_B} \sum_k^{N_k} \sum_{\sigma}^{\uparrow,\downarrow} C_{k,n,\sigma} ~ \phi^0_{k,n}({\bf x}) \end{displaymath}$

(1)

$\begin{displaymath} \phi^0_{k,n}({\bf x})= \frac{1}{\sqrt{N}} u_{k,n}({\bf x}) e^{i {\bf k}\cdot {\bf x}} \end{displaymath}$

(2)

$\begin{displaymath} u_{k,n}({\bf x})= \frac{1}{\sqrt{V_0}} \sum_G^{N_G} A_{k,n}({\bf G}) e^{i {\bf G} \cdot {\bf x}} \end{displaymath}$

(3)

where

, ${\bf x}$ and ${\bf k}$ stand for the band index, the real space coordinate and the k-vector index. This basis is highly optimized because it is adapted to the system under investigations. The SLCBB method has been used extensively in the past several years to investigate In

Ga $_{1-x}$ As quantum dots embedded in GaAs. For this special case the SLCBB basis would consist of the strained InAs Bloch functions (the InAs quantum dot is under compressive strain) and unstrained GaAs Bloch functions (the surrounding GaAs matrix is mostly unstrained). Often, physical intuition helps in choosing an adequate Bloch functions. The Bloch functions are generated for a certain number of ${\bf k}$ points that can be centered around the $\Gamma$ -point for a strongly direct band gap material or around

or other points of the Brillouin zone or around a combination of points, e.g., $\Gamma$ and

points. The basis must always be tested for convergence with respect to the number of ${\bf k}$ -points used, the number of bands and the number of different Bloch function types. Typically the SLCBB method does not scale like plane wave methods with the cube of the number of plane-waves (

) but it scales in a different way. A million atom supercell with pure InAs, for instance, will require for the computation of the band gap a similar effort than for the computation of the band gap in an 8 atom supercell. The scaling is proportional to the size of the basis needed, which does not only depend on system size but on the type of system (unlike the plane-wave basis sets). In the SLCBB method, the Hamiltonian is expensive and is fully stored. It is dense but rather small. We mostly get away with matrix sizes below 100,000x100,000.

The eigenvalue problem is presently solved via the Arnoldi Restart ARPACK method around a certain reference energy. This procedure differs drastically from ab-initio method that require the calculations of all the bands up to the Fermi energy. Here only information about the band edges (few conduction band states near the CBM and few valence band states near the VBM), see our description of the EPM method. The SLCBB method has recently been extended to treat electric fields and to include the effects of piezoelectricity (see our description of Piezo effects here).

Recent applications of the SLCBB method can be found in the research section.

Empirical Pseudopotentials: Plane Wave Expansion

With the increase of the computational power and the code developments we did on our code PLANC (see PLANC development), we are now able to treat increasingly large systems, without having to resort to the basis set approximation implemented in SLCBB. Besides being more accurate, the methods is also more convenient and less consuming in terms of human resources, that we value the most. The code can be used nearly blindly, once the potentials have been properly generated (see section on AEPs).

Configuration Interaction (CI)

Once the quasiparticle eigenfunctions have been calculated using PLANC or SLCBB, we follow the configuration interaction (CI) method to obtain the excitations (such as an exciton) of the system. At this point, the correlations of the ground state are assumed to be decoupled from the correlations of the excitation. This is justified by the Brillouin theorem stating that there is no coupling between the Hartree-Fock (HF) ground state and the single-exciton (singles) excitation |Φ_{h_i,e_j}⟩. Note that we are not starting from the HF ground state, as in the Brillouin theorem, but from the solution of the quasiparticle equation. It can still be shown that |Φ_{h_i,e_j}⟩ and |Φ₀⟩ are decoupled. There is, however, coupling between the higher excitations such as double-exciton excitation (doubles) and the ground state. These are neglected in our approach, which can be justified by the fact that doubles are energetically remote from the ground state.

The correlations in the excitation are treated at the level of singles only, i.e., only single-exciton excitation where one hole in the valence band and one electron in the conduction band are created, are allowed to interact. Formally, the correlated exciton wave function can be constructed from a set of Slater determinants:

† † |Φhi,ej⟩ = bhicej|Φ0⟩

(1)

where b_{h_i}^† is the creation operator for holes and c_{e_j}^† the creation operator for electrons. The Slater determinants |Φ_{h_i,e_j}⟩ can be calculated from anti-symmetrized products of single-particle wave functions ψ_i.

The exciton wave functions |Ψ⟩ are expanded in terms of this determinantal basis set:

∑ |Ψ ⟩ = A (hi,ej)|Φhi,ej⟩ , hi,ej

(2)

where A are the expansion coefficients and we use i to index hole states and j to index electron states. The Slater rules allow us to express the matrix elements between Slater determinants in terms of one– and two–center integrals:

⟨Φ |H |Φ ⟩ = (ɛ - ɛ )δ δ + ⟨e h′|v|h e ′⟩ - ⟨eh ′|v|e ′h ⟩ , hi,ej hi′,ej′ ej hi hihi′ ejej′ j i i j j i j i

(3)

with the two center integrals

∫ ∫ ⟨e h |v|h ′e ′⟩ = ψ ⋆(r )ψ⋆(r )v(r,r )ψ ′(r )ψ ′(r )dr dr , j i i j j e i h e h i e j h e h

(4)

using v(_e,_h) for the screened Coulomb interaction described in the next section. The last term in Eq. (3) describes the direct Coulomb integrals and the one before last the exchange integrals. The formalism described from Eq. (1) to Eq. (4) can be generalized to the case of an arbitrary number of electrons and holes and is not limited to the case of excitons. For the case of multiexcitons or charged excitons, the subspace of Slater determinants included in Eq. (1) has been restricted to excitations that conserve the number of electrons and the number of holes. For instance, a biexciton state has been constructed from Slater determinants with two electrons and two holes (double excitation), neglecting the coupling to Slater determinants with one electron and one hole (singles). This coupling is non–zero, but small since the energy difference between the single and the double excitations is approximately given by the band gap. This generalization from the exciton case to an arbitrary number of electrons and holes, represents one of the advantages of this approach. In our numerical treatment, equations such as (3) are not directly implemented but rather the action of creation and annihilation operators in a general second quantization form.

More details about the methods can be found in Electronic excitations in nanostructures: An empirical pseudopotential based approach J. Phys. Cond. Mat. .21, 023202 (2009) (link).

Atomic Effective Pseudopotentials (AEPs)

We are developing a new type of pseudopotentials aimed at the calculation of electronic and optical properties of large structures, i.e., a large number of atoms. The potentials are based on the effective potential obtained from DFT pseudopotential calculations of bulk and superlattices, hence their name. They are NOT aimed at the calculation of the total energy. This allows us to solve an inner-eigenvalue problem for only a few bands close to the band gap. The energy cut-off we use are the DFT energy cut-offs. The potentials are non-local in a real-space Kleineman-Bylander sense. They are presently used in reciprocal space (periodic boundary condition) and in a full-real space implementation (PLANC). The potentials contain spin-orbit coupling. An empirical band-gap corrections can be used. We are presently working in a DFG sponsored project on the development of magnetic pseudopotentials.

Pseudopotential Large Atomic Nanostructure Calculations: PLANC

Since 2008 we are developing the code PLANC. In parallel with the development of the atomic effective pseudopotentials (AEPs), one of the main occupations of the group since 2008 is the development of a modern order(N) code to be used with the AEPs. This code has presently the following features:

Fortran95 modular high performance code.
Implementation of three types of solvers: 1) A Davidson solver used with the folded spectrum method 2) An Arnoldi restart algorithm able to address the inner-problem directly and 3) A direct diagonalization method that we can use for small systems and as test.
The implementation of non-local potentials in four different flavors 1) Semi-local implementation in G-space. 2) Kleinman-Bylander implementation in G-space with full storage of the projectors. 3) Kleinman-Bylander implementation in G-space with on-the-fly calculation of the structure factors. 4) Kleinman-Bylander implementation in R-space, which works in Order(N).
Dual space representation of the local potentials via Fast Fourier Transformations. We have the option to use the FFTW library or our own implementation.
Parallelization through openMP.

Efficient calculation of Exchange and Coulomb Matrix Elements

We are developing three different approaches to calcluate the Coulomb and Exchange integrals. 1) From SLCBB wave fucntions expressed in the basis of strained Bloch wave functions. In this appraoch we use a projection formalism and expresse the million-atom wave functions in the basis of Bloch functions at the Gamma point. 2) From wave functions expressed in a plane wave basis 3) From real space wave functions obtained from the real-space version of PLANC.

Efficient calculation of Electron-Phonon Coupling for Clusters

We are calculating the electron-phonon coupling matrix elements from DFT wave functions for specific phonon modes. We are presently using this cabpality for up to 1000 atoms, the limitation being the electronic wave functions obtained from DFT. We are working on this scheme for PLANC wave functions wiht more then 100,000 atoms.

Development of interatomic potentials for phonons in III-V materials.

We implemented the adiabatic bond charge model into a modern code. The current version (v0.1) can calculate the phonon states of bulk silicon and III-V semiconductor such as GaAs. We intended to use it for the calculation of clusters, but found it to be inapropriate. We derived our own interatomic potentials that we can use for failry large structures in combination with GULP.

Hartree Fock code

We devleoped a self consistent Hartree-Fock code that uses the quasiparticle wave functions obtained from SLCBB as basis set.This allows us to perform a self consistent optimization of the basis functions used in the calculation of the Coulomb and Exchange integrals that subsequently enter the CI code. The improvement of the results is only marginal for self-assembled quantum dots but become larger for interface fluctution quantum dots with a large lateral extent.