Radiative decay rates in Si crystallites with a donor ion investigated
A team of Lobachevsky University researchers headed by Prof. Vladimir Burdov investigates radiative properties of silicon nanocrystals having sizes of the order of one nanometer, with phosphorus or lithium impurity ion introduced into them.
The scientists also examine the possibility of improving the optical characteristics of nanocrystals by introducing shallow donors, as well as the temperature dependence of these characteristics.
For this purpose, scientists calculate the rates (inverse lifetimes) of interband transitions accompanied by the emission of a photon whose energy is close to that of the nanocrystal optical gap.
The calculation of the electronic structure of nanocrystals, wave functions, and matrix elements of the optical transition is performed within the framework of the time-dependent density functional theory (the Casida approach). At zero temperature, only the ground electron-hole transition is possible - from the lower energy level of the conduction band to the upper state of the valence band, as shown schematically in Fig. 2.
At finite temperatures, electrons can occupy higher levels in the conduction band, and the transition to the valence band accompanied by photon emission can also occur from these levels.
In this case, the electron-hole transition can be characterized by some statistical average value of the recombination rate, which depends on the temperature.
The presence of a donor in a nanocrystal is manifested in two ways. First, the electric field of the donor ion modifies the electronic wave functions and, as a consequence, the magnitude of the matrix element of the optical transition and the radiative recombination rate. Second, the donor atom emits an "extra" electron into the nanocrystal, which populates the lower level in the conduction band.
As a result, after laser excitation of the system accompanied by the emergence of an electron-hole pair, two electrons can be present simultaneously in the conduction band, as shown in Fig. 2. This automatically opens the channel for nonradiative de-excitation through the Auger process, which, as a rule, is much faster than the radiative interband transition.
However, since the process of laser pumping and the process of nanocrystal formation with the subsequent introduction of donors are strongly separated in time, one can hope that the electron given by the donor to the nanocrystal will have time to escape into the surrounding matrix and find some deep level there.
Accordingly, by the time the laser pumping starts, the conduction band of the nanocrystal will be free, and the excited electron-hole pair will be able to perform a radiative transition, since the channels of nonradiative relaxation will be "shut off".
The calculation of the inverse lifetime of the radiative transition is performed within the framework of the nonstationary perturbation theory, the "Fermi golden rule". The results of the calculation are presented in Fig. 3, which shows the temperature dependence of the radiative recombination rates in silicon nanocrystals.
As can be seen from the figure, a sharp increase of the recombination rate occurs with increasing temperature for both nanocrystals containing lithium. This rise is due to the fact that excited states of the conduction band (at finite temperatures) participate in the interband transitions, which occur faster than the ground electron-hole transition (in Fig. 3, its rate corresponds to zero temperature). In nanocrystals containing phosphorus, all the interband transitions in which excited states of the conduction band are involved have substantially greater energies than the ground transition: even room temperatures are not sufficient to involve these states into interband dynamics.
For this reason, radiative transitions in nanocrystals with phosphorus have the rates that are close to the rate of the ground electron-hole transition, and depend weakly on the temperature.
Fig. 4 shows the temperature dependences of the ratio of the radiative recombination rates in a nanocrystal with and without the donor ion for all four nanocrystals under study.
According to Vladimir Burdov, the ground electron-hole transition (which corresponds to zero temperature on the graph) is substantially accelerated in a Si46H60P nanocrystal containing a phosphorus ion. The introduction of a lithium ion, on the contrary, considerably slows down the radiative transition between the ground states in the bands, which can be seen in Fig.4.
The situation changes dramatically as the temperature is increased to room temperature. In both nanocrystals containing lithium, the radiative recombination rate becomes larger than in a "pure" nanocrystal. In the Si46H60P nanocrystal, the radiative transition rate becomes several times smaller; nevertheless, it still remains much higher than in the corresponding nanocrystal without phosphorus, Si47H60. When phosphorus is introduced into the Si34H36P nanocrystal, its emissivity does not improve: the radiative recombination rate remains approximately one order of magnitude lower compared with the corresponding "pure" Si35H36 nanocrystal at any temperature.
"On the whole, it can be stated that the introduction of small phosphorus or lithium donors into small-sized silicon nanocrystals can in many cases accelerate the radiative interband transitions, which is certainly a positive factor from the point of view of improving the nanocrystals' optical properties," Vladimir Burdov comments.
It should be noted that the enhancing luminescence intensity or the acceleration of interband radiative transitions due to the introduction of phosphorus or lithium into silicon nanocrystals was observed experimentally by different research groups.
Si30H40Li nanocrystals (top left), Si42H64Li (top right), Si34H36P (bottom left) and Si46H60P (bottom right). Large gray-blue spheres are silicon atoms; small gray spheres - hydrogen atoms that saturate dangling surface bonds; lilac central spheres - lithium ions; orange central spheres - phosphorus ions.
The ground electron-hole transition from the lower energy level of the conduction band to the upper state of the valence band at zero temperature.
As a result, after laser excitation of the system accompanied by the emergence of an electron-hole pair, there can be two electrons simultaneously in the conduction band.
Temperature dependence of the growth of recombination rate for both nanocrystals with lithium. This growth is due to the participation of excited states of the conduction band (at finite temperatures) in the interband transitions, which occur faster than the ground electron-hole transition.
Dependence of the rate of the ground electron-hole transition (corresponding to zero temperature on the graph) in a Si46H60P nanocrystal due to the introduction of a phosphorus ion.