Embedded lead telluride quantum dots
At one prominent example (CdTe-Quantum-Dot) the capabilities of ab initio software in describing and predicting structural and electronic properties (luminescence) of nanostructured materials are demonstrated.
Introduction
The modern fabrication techniques of electronic, optoelectronic, and sensoric devices has renewed the interest in self-organization and selfassembly of nano-structures like quantum dots (QDs) or quantum wires embedded in solid matrices. A variety of structures and materials is found to be promising for future applications in optoelectronics or nanoelectronics. In these cases the confinement of the electron or the hole wave functions are of particular interest, since they exhibit properties similar to those of single atoms or molecules.
In many cases the self-organization process can only be understood in detail with the aid of modern computational calculations at an atomic scale. In particular it is very difficult to predict consequences of certain changes of the experimental setup without the knowledge of the underlying physical and chemical processes.
Material System
The availability of light sources in the mid-infrared spectral region is crucial for many applications, e.g., molecular spectroscopy and gas-sensor systems for environmental monitoring or medical diagnostics. Furthermore, it has been demonstrated that PbTe QDs embedded in a CdTe host material exhibit an intense room-temperature mid-infrared luminescence signal [1]. This makes this system an promissing candidate for future applications [2]. For these reasons we will use the PbTe/CdTe QD system as prototypical example for the modeling of self-organization processes in nano-structures.
Prediction of the quantum dot shape
- Figure 1: Theoretical prediction of the ECS (right-hand-side) and the atomistic structure (left-hand-side) of an embedded PbTe quantum dot [4].
To predict the shape (thermodynamic equilibrium crystal shape- ECS) of PbTe QDs embedded in a CdTe matrix we have calculated in a first step the interface energies between PbTe and CdTe for several interface orientations. Based on Wulff’s theorem [3] the ECS is determined by the minimal interface energy of the QD under the constraint of a fixed volume and a fixed number of atoms. The anisotropy of the interface energies with respect to the interface orientation (typically denoted by the Miller indices hkl) leads to a ECS different from a sphere (see right-hand-side of Fig.1).
- Figure 2: Experimental HRTEM image (from the group of Prof. F. Schäffler University Linz) of a PbTe quantum dot embedded in a CdTe matix [2].
In a second step an atomistic model of the dot-matrix system can be constructed according to the calculated ECS. Based on the quantum mechanical Hellmann-Feynman forces the atomistic geometry of the PbTe quantum dot has been optimized further. The resulting geometry is schematically shown on the left-hand-side of Fig. 1. Beside small interface rumpling effects the formation of Pb-Te bilayers along the [111] direction is observed.1 The high reliability of the theoretical predictions can be demonstrated by a comparison with experimentally observed high-resolution transmission electron microscopy (HRTEM) images. As can be seen in Fig. 2 even at an atomic resolution the theoretical prediction is in excellent agreement with the experimental observations.
Moreover the performed ab initio calculations could clarify the source of the rumpling effect at the QD interfaces and the bilayer formation within the PbTe QDs. The reason for the latter effect is an electric dipole field induced by the polar CdTe/PbTe(100) interfaces. The termination of the interfaces either with cations (Cd2+) or anions (Te2-) leads to interface charges. As shown in Fig.1 all Cd-terimated (100) interfaces are located at one side of the PbTe QD, while all Te-terminated (100) interfaces are located at the opposite QD side resulting in the observed electric dipole field [5].
1This effect is hardly visible in Fig.1, however, a detailed analysis of the bond length distribution with the PbTe QD reveals the bilayer formation [5]
Consequences for the electronic properties
In general the electronic properties of nano-structured systems are affected by two main effects:
- The quantum confinement effect
The small size of the embedded QD leads to spatially confined (within the PbTe QD) electron and hole wave functions. As a result of this restriction the electronic band gap is typically increased by Econf. - The quantum confined Stark effect
The existence of an electric field leads to the localization of the electron and hole wave function at opposite sides of the nanostructure. As a result the electronic band gap is typically reduced by Δstark.
- Figure 3: Schematic picture of the influence of quantum confinement and quantum confined Stark effect on the electronic band gap [4].
The combination of these effects is schematically shown in Fig.3. The electric field leads to a linear slope of the electrostatic potential within the QD region. Hence, the electron and hole wavefunctions are spatially separated and the band gap is reduced. This has major consequences for the recombination rate of generated electron-hole pairs, i.e. the photoluminescence signal is strongly influenced by the details of the electronic properties of QD system. Moreover the measurement of electric fields inside embedded QDs is a hard task and in some cases even not possible, because matrix materials with large dielectric constants (like e.g. CdTe) do effectively screen the electric fields of the QDs. In such cases ab initio calculations are one of the most reliable methods to access the electronic and optical properties. In the considered example it was possible to explain an experimentally observed, completely unexpected photoluminescence intensity drop at low temperatures in the PbTe/CdTe system [4].
Summary
The example PbTe QDs embedded in a CdTe matrix has shown that ab initio calculations are convenient methods to determine structural, electronic and optical properties of nano-structured materials. In particular one can calculate material properties, which are experimentally hardly or even not accessible. Finally, the following advantages of ab initio calculations for determining properties of nano-structured materials can be summarized:
- massively reduction of costs due to the knowledge of electrical and optical properties before the material system is assembled for the first time
- simultaneously determination of properties of multiple material systems with a few ab initio calculations -> prediction of new material systems
- access to material properties, which are hardly measurable -> saving expensive experiments
Contact ...
Dipl.-Physicist
Dr. R. Leitsmann
Phone: +49 (0)371 / 5347 - 553
Email: 
leitsmann(at)matcalc.de
Further reading ...
[1] W. Heiss et al., Appl. Phys. Lett. 88, 192109 (2006)
[2] R.Leitsmann et al., New J. Phys. 8, 317 (2006)
[3] G.Wulff, Z. Kristallgr. Mineral. 34, 449 (1901)
[4] R.Leitsmann and F. Bechstedt, Phys. Rev. B 80, 165402 (2009)
[5] R.Leitsmann et al., Phys. Rev. B 78, 205324 (2008)
