Exciton and Biexciton Energies in GaN / AlN Quantum Dots. G. Hönig , A. Schliwa , D. Bimberg , A. Hoffmann Teilprojekt A5. Institut für Festkörperphysik Technische Universität Berlin. [email protected] c-plane GaN / AlN QDs. Vertical electron -hole separation - PowerPoint PPT Presentation
Exciton and Biexciton Energies in GaN/AlN Quantum DotsG. Hnig, A. Schliwa, D. Bimberg, A. Hoffmann
Institut fr FestkrperphysikTechnische Universitt [email protected]
1c-plane GaN/AlN QDs
Vertical electron-hole separation Redshift of X- X energy inverse proportional to QD-heightNr.
We have heard a lot about the polarization fields in wurtzite c-plane nanostructures before. So let me only quickly summarize: We have in case of GaN/AlN Quantum Dots strong pyro and piezo-electric fields, which tilts the confining bandstructure of the Quantum Dot. Most of you know this as the quantum confined stark effect, leading to an decrease of recombination energies, a spatial separation of electron and hole densities in side the Quantum Dot, and also a stronger confinement in growth direction, besides the classical energy shift due to polarization of the charge densities by the field. The spatial separation of the charge carrier densities clearly dominate properties of Nitride Quantum Dots.2Outline Exciton States
Biexciton Complex in Relation to Exciton
A New Approach: Wigner-crystalsNr.
I have decided to focus my talk on 3 points: The properties of Excitons and Biexcitons in GaN/AlN QDs and at the end I will give you an introduction to new idea we have developed, which might be able to declare some special experimental findings, we talk about in the first two sections of this talk. We will start now with the exciton.3Electron Hole Interaction
Direct Coulomb Interaction related to charges attractive between electron and hole
directexchangeExchange Interaction related to spins repulsive between electron and hole( if spins are antiparallel, if parallel = 0 ) splits up energies related to spin orientation
If we confine one electron and one hole in a single QD, it forms the easiest particle complex, able to recombine under emission of a photon: An Exciton. As already mentioned, in the QDs we are interested today, these charge carriers are separated by the strong built-in electric fields, with the electron at the top and the hole at the bottom. The hole has a much bigger effective mass than the electron, leading to a smaller orbital size. Electron and hole ofcourse interact with eachother. This can be described the Coulomb operator which is very similar to the classic coulomb interaction of particles, except for using operators. To calculate the energy of the interaction, we use the Coulomb-Operator on the two-particle wavefunction, which you can think of in the easies case as a slater-determinant. By doing so, you find two non-vanishing terms. One of it describing the classical interaction of the charge densities of electron and hole state. Here we call it the direct Coulomb interaction. The other looks similar but has exchanged indices. This one has no classical analouge, but it is related to the spins of the particles. While the direct interaction between negatively charged electron and positively charged hole is attractive, the exchange interaction between electron and hole is repulsive, if their spins are antiparallel! Therefore the exchange interaction lifts up the energy of excitons, where electron and holes have antiparallel spin orientations compared to excitons with parallel spin orientations.4Root of 4 Exciton StatesDiagonalization of the Hamilton-matrix leads to the eigenstates:
&Imagine the particle states stem from the conduction and heavy hole valence band, respectively. (neglecting the intermixed band structure)
The spins would be:
Total spin of exciton states: BRIGHT: 1 DARK: 2Because of the repulsive nature of exchange interaction between electron and hole, BRIGHT states have always higher energy than DARK states!Nr.
If we calculate the matrix of the coulomb operator on basis of the four possible spin orientations of the exciton and diagonalising this matrix afterwards, we find these two pairs of eigenstates of the coulomb operator. Neglecting interamixing bands, the spins of electron and hole correspond to these values, which gives us for the two pairs of eigenstates total spins of of +-1 or +-2. Together with the information, that a single emitted photon must have total spin of +-1, only these two states are able to decay into a single photon. Therefor we call these states the bright states and these the dark states of the exciton. Due to the exchange interaction, existing between electron and hole of these states, they are lifte up in energy and always energetically above the dark states. If the wavefunctions of electron and hole have a symmetry which is below c3v, which means 3 fold rotational symmetry, the bright states split up and will decay into photons, which show linear polarization parallel or orthogonal to the longer axis of ist wavefunctions. An elongation of the wavefunctions can by achieved by an non symmetric confinement potential, for example by an elongated quantum dot structure.5Exciton and Biexciton States
3,4 optically active(BRIGHT)1,2 optically inactive(DARK)Nr.
All this splitting leads to a energetic ordering scheme similar to this. At the top we see the possible biexciton, consisting of two electron hole pairs in its energetic ground state. The biexciton can only have one energy, since all possible spin orientations together form the ground state. The exciton has round about half of the complex energy of the biexciton. We will discuss its exact relative energetic position later in this talk. The ground state energies of the exciton are split up to four levels. The first two consist of electron-hole pairs with parallel spins, and therefore zero exchange energy. While the bright states are energetically higher than the dark states and might show a splitting for unsymmetric wavefunctions. Only the bright states are expected to be visible, and therefore we focus on them. But let us add, thate if the dark states show a splitting, it is expected to be 3 orders of magnitude smaller then that of the bright states, which is not correctly shown in this scheme.6
full linear polarization 180 periodicity 90 phase shift between states fine-structure splitting on the order of meVExperimental FindingsC. Kindel, , Y. Arakawa, G. Hnig, , A. Hoffmann, D. Bimberg Phys. Rev. B 81, 241309(R) (2010)Nr.
For the excitonic bright states their have been experimental and theoretical investigations published, showing clearly separated bright states, with orthogonal, linear polarization directions. Remarkable is, that there is only one line pair visible for each quantum dot. Since we would expect from theory, at least two, if the biexciton is visible. Let me add, that an excited exciton state, with the hole in the next highest state, also would show such an emission line splitting. In this cas, the splitting of exciton lines has a size of about 4 meV. Which is remarkable, since in for example Arsenides, only hundreds of microeV have been reported. This also means, that the energy of exchange interaction has atleast half of the line splitting, so some meV, here at leas 2 meV7Investigations of FSS
comparatively huge FSS on the order of meV height increase (with constant aspect ratio) causes decrease of FSS diameter increase (with constant height) causes decrease of FSS agreement with experimental findings (black squares)C. Kindel, , Y. Arakawa, G. Hnig, , A. Hoffmann, D. Bimberg Phys. Rev. B 81, 241309(R) (2010)
Series are calculated with 20% laterally elongated QD structures.Nr.
This slide shows the variation of fine structure splitting, reported for a couple of GaN/AlN QDs. The clear trends from small to higher splittings is caused by the spatial separation of electron and hole within the quantum dots, and which is reduced for smaller dot sizes. In smaller Dots, therefore the exchange energy between electron and hole is much enhanced. Since in this energy regime, the resolution of experimental setups is about 1 meV, and besides this, also the line widths of emission from bigger quantum dots is enlarged to meV order, there are no experimental results below 3.9 eV of emission energy, which becomes important in the next section of my talk.8Outline Exciton States
Biexciton Complex in Relation to Exciton
A New Approach: Wigner-crystalsNr.
Now we come to the part about the Biexciton.9
Biexciton Relative to ExcitonEbind = 2 EX - EXXNr.
I already told you something about the biexciton in GaN/AlN QDs. Lets say, we look in the energy regime, where the fine-structure of the exciton is not resolveable, and we see only one excitonic emission. Slight changes of the exciton emission energy relative to the biexciton are possible, leading to shifting of the emission line of the biexciton below the line of the exciton emission or higher than that. First case, we call a binding biexciton and define ist binding energy to be positive. While in the second case we define it to be negative. But what influences this biexciton binding energy?10Meanfield Biexciton
ExcitonBiexcitonSpatial electron hole separation leads to higher energy of the biexciton!Correlation Energy & Conditional ProbabilitiesNr.
Compared to the exciton case, with the attractive direct coulomb energy, and possible repulsive exchange energy. The biexciton has additional particle interactions, since we now have four particles interacting with each other. We have something like two excitons. So additionally to the attractive direct coulomb force, we get repulsive energy terms between holes and between electrons, but also additionally attractive parts between electrons and holes. Without the middle arrows, so without interaction between the two confined excitons, the biexciton would have exactly twice the energy of the exciton (this is ofcourse only tru