Project 5

Propagation of Quantum Wave Packets in One and Two Dimensions

Introduction

This project explores tunneling of quantum wave packets in one dimension through a single potential barrier or a double barrier structure where resonant tunneling can be observed at special energies. In quasi-one-dimensional wire with the spin-orbit coupling one can observe spin precession and spin decoherence studied in current research on spintronics [1].

Part I for both PHYS460 and PHYS660 students: Quantum tunneling of spinless wave packets

By numerically solving the time-dependent Schrödinger equation

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V(x) \Psi(x) = i\hbar \frac{\partial \Psi}{\partial t} }

via the Crank-Nicholson algorithm for partial-differential equations in one spatial dimension, study reflection and transmission of a quantum wave packet from a barrier for which the potential energy is greater than the kinetic energy of the incident wave packet. For computer simulation the units should be chosen as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m=1 } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hbar=1 } , and your discrete time and space grid should be defined using:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t= 5 \times 10^{-7}; \ \Delta x = 5 \times 10^{-4} } .

Take the incident wave packet to be of the form

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Psi(x,t=0) = \frac{1}{(2 \pi)^{1/4} \sqrt{\sigma}} e^{-(x-x_0)^2/4\sigma^2}e^{ik_0(x-x_0)} }

with parameters

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma^2=0.00025; \ k_0 = 700; \ x_0=0.3 } .

Let the height of the potential barrier be Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_0 =2 k_0^2 } , the center of the wave packet is at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_0} and the barrier itself itself starts at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_b =0.6 } . Plot the time evolution of the wave packet for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x<1.0 } .

(a) How does the tunneling behavior depend on the width of the barrier? Consider the barrier widths in the range Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=0.001 } to Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=0.1 } .

(b) Investigate tunneling through two barriers of the same width Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=0.001 } spaced at a distance Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L=0.01 } . In this case, you should vary Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_0 } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma } to observe high transmission when the average energy of the wave packet is close to the energy of quasibound states in between the two barriers. Such process, termed resonant tunneling, is the basis of operation of the so-called resonant tunneling diodes.

In both cases (a) and (b) your result should show at least three snapshots of wave packet propagation: in front of the barrier; colliding with the barrier; and reflecting and/or tunneling through the barrier.

Part II for PHYS660 students only: Spin dynamics of spin-polarized wave packet in Rashba quantum wires

In current research on semiconductor spintronics [1], spin-orbit (SO) coupling as the manifestation of relativistic corrections to the Schrodinger equation in solid state systems, play an essential role. This is due to the fact that some types of SO couplings, such as the Rashba one in two-dimensional electron gases, can be controlled by the gate voltage which then allows us to manipulate spins as magnetic degree of freedom via electric fields that are far easier to generate in small volumes and on short time scales.

Consider a two-dimensional (2D) quantum wire in the xy-plane described by the Rashba Hamiltonian:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{H} = \frac{\hat{p}_x^2 + \hat{p}_y^2}{2m^*} + \frac{\alpha}{\hbar}(\hat{\sigma}_x \hat{p}_y - \hat{\sigma}_x \hat{p}_y) }

where the momentum operator in 2D is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\hat{p}_x,\hat{p}_y) = (-i\hbar \partial/\partial x, -i\hbar \partial/\partial y) } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\sigma}_x } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\sigma}_y } are the Pauli matrices.

The discretized version of this Hamiltonian (for discretization procedure see Sec. 6.1 in Ref. [2]):

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{H} = \sum_{\langle \mathbf{mn} \rangle} t_{\mathbf{mn}}^{\sigma \sigma^\prime} |\mathbf{m} \rangle \langle \mathbf{n}| }

where

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{\mathbf{mn}}^{\sigma \sigma^\prime} = }

makes it possible to study on the computer how spin-polarized electronic wave packet propagates through such wire.

Assume that wire is described on the square lattice Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 100 \times 31 } sites and the initial wave packet injected from the left is:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Psi(x,y;t=0)=C\sin \left( \frac{\pi y}{(L_y+1)a} e^{ik_x - \delta k_x^2 x^2/4} \otimes }

References

• [1] D. D. Awschalom and M. E. Flatté, Challenges for semiconductor spintronics, Nature Physics 3, 153 (2007).
• [3] B. K. Nikolić, L. P. Zarbo, and S. Souma, Spin currents in semiconductor nanostructures: A nonequilibrium Green function approach, Chapter 24, page 814-866 in Volume I of The Oxford Handbook of Nanoscience and Technology: Frontiers and Advances, edited by A. V. Narlikar and Y. Y. Fu (Oxford University Press, Oxford, 2010); also available as arXiv:0907.4122.
• [2] B. K. Nikolić, L. P. Zarbo, and S. Welack, Transverse spin-orbit force in the spin Hall effect in ballistic quantum wires, Phys. Rev. B 72, 075335 (2005).[PDF]
• [4] S. A. Crooker and D. L. Smith, Imaging spin flows in semiconductors subject to electric, magnetic, and strain fields, Phys. Rev. Lett. 94, 236601 (2005). [PDF]