Distribution of Delay Times in Laser Excited CdSe-ZnS Core-Shell Quantum Dots

Transcription

1 Distribution of Delay Times in Laser Excited CdSe-ZnS Core-Shell Quantum Dots Andrei Vajiac Indiana University South Bend Mathematics, Computer Science Advisor: Pavel Frantsuzov, Physics

2 Abstract This project studies the distribution of delay times between laser excitation and photon emission for individual CdSe-ZnS core-shell quantum dots. The experimental data was obtained from Argon National Labs, where a Time Correlated Single Proton Counting device was used to detect the photon emissions from the quantum dots. The data was analyzed statistically, using Matlab, to determine what, if any, connections exist among the delay times that occur between laser excitation and photon detection and the on/off states of the quantum dots. This follows research performed by Wu, Sun, and Pelton on the distribution of recombination rates for quantum dots.

3 In this communication we present results from the statistical analysis of the delay times between laser excitation and photon emission in CdSe-ZnS Quantum Dots formed in colloidal solution. The data, obtained from Argonne National Laboratory, is the same as was used in Recombination rates for single colloidal quantum dots near a smooth metal film by X. Wu, Y. Sun, and M. Pelton. I. What is a Quantum Dot? Quantum dots, discovered in the 1980s by Alexei Ekimov and Louis E. Brus are very small semiconductors whose properties and characteristics are closely tied to the size and shape of each individual crystal. To be able to understand the specifics of quantum dots, one must first have an understanding of excitons. An exciton is an electrically neutral quasi-particle existent in insulators, semiconductors, and some liquids. It is a bound state of an electron hole and an electron, which are attracted to each other by an electrostatic Coulomb force. Excitons form when an electron is excited from the valence band into the conduction band by a photon. This excitation leaves behind a hole of opposite electric charge to which the electron is attracted by the Coulomb force. In this communication we wish to study the recombination rate of the electron and the electron hole within semiconductor nanocrystals (quantum dots). Now one can say that quantum dots are semiconductors who have their excitons confined in all three spatial dimensions. This quantum confinement effect occurs when the diameter of a particle is of the same magnitude as the wavelength of the electron wave function, hence causing quantum dots electronic and optical properties to be substantially different from those of bulk semiconductors. If the confining dimension is large enough, the bandgap between the valence band and the conduction band remains at its original energy. However, as the confining dimension decreases even further, the bandgap becomes dependent on size; this quantum confinement effect is generally used to explain the results from excitons being confined to a dimension that approaches a measurement known as the exciton Bohr radius. This type of quantum confinement leads to a few new electronic structures. Quantum Wells occur when the

4 electrons or holes are confined in one spatial direction and free propagation is allowed in the other two dimensions. Quantum Wires occur when the electrons or holes are confined in two spatial directions, allowing free propagation in the remaining direction. Finally, Quantum Dots are created when the excitons are confined in all three spatial directions. Since quantum dots can be relatively simply manufactured through many means at many different sizes, this bandgap can thus be altered to whatever size one may need. This allows quantum dots to be produced for various applications. For example, quantum dots have been used quite effectively in producing fluorescent dyes that can be used in medical applications. II. The Data The data was obtained from Argonne National Laboratory and is the same as the data used in Recombination rates for single colloidal quantum dots near a smooth metal film by X. Wu, Y. Sun, and M. Pelton. Quantum Dots were excited using laser pulses and the photon emissions were detected using a Single Photon Counter. The laser used had a 400nm excitation wavelength and the repetition rate was 5MHz. Output pulses from the photon detector, as well as sync pulses from the laser, were sent to a Time Correlated Single Photon Counting (TCPSC) Machine. Since the exact times of the laser pulses is known, the data was comprised of two main parts. First, the time that has elapsed from the beginning of the experiment until the photon detection was known, henceforth referred to as true time. Also, the delay time between the previous laser excitation and the subsequent photon detection was also able to be recorded. Each dataset received contained the results from a different quantum dot being excited over a time span of 10 minutes. Each of these datasets contained anywhere between 2.5 million and 5.5 million detected photons and their respective true times and delay times. III. Data Analysis Method Since the datasets obtained are so large, the statistical toolbox within Matlab was used to analyze the data. Matlab has many different capabilities when it comes to analyzing data in

5 arrays and as such using Matlab required making a decision between iterative and boolean styles of programming. There are many distinct differences between these two styles, but in general, iterative programs are easier to write and understand. They are similar in style to conventional programming, in particular C++, however it is possible that iterative programs can cause the analysis to run very slowly, since they require actions to be performed in succession over every single data point in the dataset. Matlab also contains a boolean style of programming, which allows data points in an array to be interpreted as boolean values, which allows for much faster calculations. However, the issue with boolean programming is that the style is not as intuitive as the iterative style; the programs many times become harder to write if one does not have much experience with the Matlab system. It is also important to note that although in general one can say that the boolean style produces faster programs than the iterative style, it is not always the case. In particular, there was a program written to put the data in to different time bins which was able to be executed in a few seconds when the program was written iteratively, but when the program was translated to a boolean style, it took several minutes. Therefore much trial and error was required to be able to write the fastest, and best, programs to analyze the data. IV. Data Analysis Initially, after extracting the data from the files received so that they could be imported into Matlab, the data needed to be binned by splitting the 10 minute experiment into smaller time intervals (bins) and then counting the number of detected photons in each bin (using the true time). This sort of analysis allows us to determine times of high photon emission from the quantum dot, known as on-times and times of low photon emission, known as off-times. An example of data binning is shown below:

6 This initial binning analysis provides us with a few questions. Firstly, we want to know why the quantum dot goes into off periods. The quantum dots exhibit the strange characteristic of being in an on state, or cycling from on to off quite rapidly, to going into an off state for periods up to two minutes, and then returning to the on state. This behavior is quite surprising, since quantum dots' activity is usually measured on a nanosecond scale; thus an off state of up to two minutes is really a very long time for the dot to not be emitting very many photons. Another question that is raised is whether or not the delay times of the photons have any sort of connection to the on and off states. In general, we wish to discover whether there are any clues about the quantum dot behavior that can help us better understand and possibly to predict when off states will occur and how long they might last. One attempt was to see if the range of delay times for photon detection that occurred in the off states for the quantum dot was smaller than the range of delay times occurring in the on states. We binned the delay times for the quantum dots in the same fashion as before, when we binned the true times, and obtained the following result:

7 When comparing the above result from the graph of on and off times from before, we saw that our initial prediction was not correct; in fact we found that the range of delay times for the off states was significantly larger than the range of delay times for the on states. From this point we are attempting to see if the distribution of delay times for the entire experiment (for each quantum dot) could be modeled using some sort of fit curve. We endeavor to fit the distribution using an exponential fit, accounting for background noise in the equation. A least-squares method is being used to calculate the best fit. Our first idea for the data fit was to use the following model: ( )

8 However, upon separating the delay times into photons emitted during on times and photons emitted during off times for each experiment, we obtained many similar graphs to the one below: The above represents a histogram of the distribution of delay times for the on states of a particular quantum dot throughout the 10 minute experiment. As we can see, there is a significant amount of background noise in the data (it doesn t seem to approach zero) and thus we need to change our predicted model. We now wish to model the data using the function: ( ) Where B represents the background noise in our data collection. The least squares method is being used to model the data, so we wish to minimize: [( ( )]

9 That is, [( ] Where, in the above, m represents our number of data points. To minimize S, we need to take the partial derivatives with respect to our unknowns: A, B, and tau. After taking these partial derivatives and simplifying, we end up with: By setting all of these partial derivatives to zero, we find the minimum of S, and thus our least squares estimate. Obviously, the previous system of three nonlinear equations is not solvable manually; we are currently writing a Matlab program to be able to solve that linear system, and as such calculate the values of A, B, and tau.