1. Time-dependent Schrödinger equation to describe the GWP dynamics:

1. Development of Numerical Methods to Simulate Electron Dynamics in Real Time Stephen Blama and Jia-An Yan Department of Physics, Astronomy and Geosciences, Towson University, Towson, MD 21252 Calculation Box Setup Transmission through Multiple Potential Barriers Introduction As a second example, we apply our method to study a GWP propagating through a double-barrier and a triple-barrier structure. These potential barrier structures can be realized by semiconductor heterostructures or superlattices grown by molecular beam epitaxy (MBE) or chemical vapor deposition (CVD) in experiment. The multiple barrier system is an important model to understand the electron transport in electronic devices, providing a useful tool to design nano-electronic and opto-electronic devices. The interactions of electrons with matter are of fundamental interest and are central to many technologies from transistors, diodes, and dosimeters to sophisticated imaging, lasers, and quantum computing. In particular, the realization of electron’s wave-particle dual character has given birth to one of the most powerful imaging instruments: the electron microscope. The ability of an electron microscope to resolve three-dimensional (3D) structures on the atomic scale is continuing to affect different fields, including condensed matter physics, materials science, chemistry, and biology. Electrons are the key to the development of novel technologies, such as electron holography, the electron-impact chemistry [1-3], and the 4D ultrafast electron microscope [4]. A conceptual understanding of the interactions of electrons and electron dynamics in real time will allow us to acquire deep insights into electron tunneling, scattering and diffraction which may be widely applied to developing novel technologies. Schematic plot of the numerical setup for a GWP scattered by potential barriers. The total length L = 140 Å. The width of a single rectangular barrier is W = 2.5 Å. For multiple barriers, the separation between them is d = W = 2.5 Å. The potential barrier height is V0 =10.0 eV. xt= 45 Å and xr = -50 Å are the locations to record the transmitted and reflected wave packet, respectively. The complex adsorbing potential (CAP) is applied at both the left and right boundaries to minimize the reflection. The calculated transmission coefficients T(E) as a function of energy have been shown to the left. The barrier width W = 2.5 Å , and the separation d = 2.5Å. Insets schematically show respectively the potential barrier structure. In both cases, the transmission coefficient oscillates as a function of the incident electron energy E, with smaller period than in the single barrier structure when E > 10 eV = V0(barrier height). This is a quantum mechanical effect. More importantly, even when E < 10 eV, we see transmission peaks of 1 for specific energy. This is a resonant tunneling resulting from the resonance between the incident plane wave components and the quasi-level of the potential well formed between two barriers. 4D electron imaging. (Ref. [4]). GWP Propagating in Free Space First, we consider the GWP propagating in free space. The analytical solution is available to compare with the numerical results. The figure to the left shows the evolution of |φ(x,E)|2for various GWP widths α0 (from0.5to5.0 Å) as a function of energy E. Here x = 5 Å is assumed in the calculations. One can see that the probability density changes smoothly with respect to energy E when the width is large enough (in this case, α0 >= 2.0 Å). In quantum mechanics, a wave packet (WP) refers to a waveform concentrated in a well-defined region of space, and is a natural combination of the wave and particle duality of an electron [5]. Owing to the relative mathematical simplicity, the WP expressed by a Gaussian function (GWP) has been a popular way to describe the initial state of many quantum systems. Study of WP dynamics is also of interest for visualizing the quantum mechanics from the pedagogical point of view. The developed method can also be applied for studying complicated systems. The left plot shows the transmission of a GWP through a 11-barrier structure. The result is consistent with those obtained by advanced Green’s function method, showing that the developed method is efficient and accurate to study the electron dynamics in 1D. Electron density distribution in an atomic corral. Image originally created by IBM Corporation. Obtained by Fourier transformation of the GWP wave function, the probability density |φ(x,E)|2 as a function of x (in Å) and E is shown to the right. The GWP width is chosen to be α0 = 5 Å. The initial average kinetic energy is 10 eV (i.e., k0 = 1.62/Å). For a specific energy, the |φ(x,E)|2 exhibits plateaus after the starting point, implying the plane wave nature of the transformed wave function. Clearly, this corresponds essentially to the stationary solution to TDSE in free space. Summary We developed a numerical method to study the electron dynamics in real time, by propagating the GWP in real space according to the time-dependent Schrödinger equation using the Crank-Nicolson method. We showed that the real-time evolution of the GWP with specific width and initial momentum intrinsically manifests the electron transport within the corresponding energy range. By performing a Fourier transformation of the GWP wave function from time to the energy domain, the transmission coefficients as a function of energy can be extracted within a single run. We demonstrate this method by numerically calculating the electron transmission coefficients through various 1D potential barriers. The numerical results for various 1D potential barrier systems agree very well with analytical results when available. The extension of this wave packet method to study electron transport and diffraction phenomena in two-dimension and three-dimension is straightforward [6]. The main purpose of this work is to develop an efficient numerical method to study the electron dynamics in real time, using one-dimensional (1D) barriers as model systems. In the proposed method, a GWP is propagated in real space and in real time by solving the time-dependent Schrödinger equation (TDSE) based on the Crank-Nicolson method. Interestingly, the transport properties in the energy domain can be extracted from a single run of the GWP propagation by Fourier transformation from time to energy domain. Method • 1. Time-dependent Schrödinger equation to describe the GWP dynamics: • 2. Finite difference method to numerically solve the TDSE: • 3. Crank-Nicolson method to solve the TDSE: • Gaussian wave packet form at t = 0 (initial momentum ħk0): • 5. Fourier transformation from time to energy domain: Transmission through a Single Potential Barrier Next we study the electron scattered by a single potential barrier. The GWP is propagated through a single potential barrier. The wave packet width is set to α0 = 3.0 Å. Propagation time step Δt = 0.003 fs and k0 = 1.62/Å. The total propagation time t = 40000Δt = 120 fs.The rectangular potential barrier has been indicated as squares in the calculation box setup. The barrier height is 10 eV, and the width is W = 2.5 Å. After we obtain the GWP in energy space, as shown below (left), we can calculate the transmission coefficients as: Acknowledgements This work was supported by the Fisher College of Science and Math, Towson University. We would like to thank them for their research and travel grants. References G. Hanel, S. Denifl, P. Scheier, M. Probst, B. Farizon, M. Farizon, E. Illenberger, and T. D. Märk, Phys. Rev. Lett. 90, 188104 (2003). B. Boudaïffa, P. Cloutier, D. Hunting, M. A. Huels, and L. Sanche, Science 287, 1658 (2000). 3. L. G. Caron and L. Sanche, Phys. Rev. Lett. 91, 113201 (2003). 4. A. H. Zewail, Science 328, 187 (2010). 5. David J. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall; 2nd edition, 2004). 6. Jia-An Yan, J. A. Driscoll, B. K. Wyatt and K. Varga, and S.T. Pantelides, Phys. Rev. B 84, 224117 (2011). The above plot presents a comparison between the calculated T(E) and the exact analytical solution. The wave packet width is α0 = 3.0 Å. The calculated results agree very well with analytical solutions.