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

$-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi }{\partial x^{2}}}+V(x)\Psi (x)=i\hbar {\frac {\partial \Psi }{\partial t}}$

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 $m=1$ and $\hbar =1$, and your discrete time and space grid should be defined using:

$\Delta t=5\times 10^{-7};\ \Delta x=5\times 10^{-4}$.

Take the incident wave packet to be of the form

$\Psi (x,t=0)={\frac {1}{(2\pi )^{1/4}{\sqrt {\sigma }}}}e^{-(x-x_{0})^{2}/4\sigma ^{2}}e^{ik_{0}(x-x_{0})}$

with parameters

$\sigma ^{2}=0.00025;\ k_{0}=700;\ x_{0}=0.3$.

Let the height of the potential barrier be $V_{0}=2k_{0}^{2}$, the center of the wave packet is at $x_{0}$ and the barrier itself itself starts at $x_{b}=0.6$. Plot the time evolution of the wave packet for $x<1.0$.

(a) How does the tunneling behavior depend on the width of the barrier? Consider the barrier widths in the range $d=0.001$ to $d=0.1$.

(b) Investigate tunneling through two barriers of the same width $d=0.001$ spaced at a distance $L=0.01$. In this case, you should vary $k_{0}$ and $\sigma$ 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 where you plot probability density $|\Psi (x)|^{2}$ in front of the barrier; colliding with the barrier; and reflecting and/or tunneling through the barrier.

Part II for PHYS660 students only: Propagation of spin-polarized wave packet through 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:

${\hat {H}}={\frac {{\hat {p}}_{x}^{2}+{\hat {p}}_{y}^{2}}{2m^{*}}}+{\frac {\alpha }{\hbar }}({\hat {p}}_{y}{\hat {\sigma }}_{x}-{\hat {p}}_{x}{\hat {\sigma }}_{y})$

where the momentum operator in 2D is $({\hat {p}}_{x},{\hat {p}}_{y})=(-i\hbar \partial /\partial x,-i\hbar \partial /\partial y)$ and ${\hat {\sigma }}_{x}$ and ${\hat {\sigma }}_{y}$ are the Pauli matrices.

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

$\mathbf {H} =\sum _{\langle \mathbf {mm^{\prime }} \rangle }\mathbf {t} _{\mathbf {mm^{\prime }} }|\mathbf {m} \rangle \langle \mathbf {m} ^{\prime }|$

where

$\mathbf {t} _{\mathbf {mm} ^{\prime }}=-t_{O}\mathbf {I} -it_{\mathrm {SO} }{\hat {\sigma }}_{y}\ \mathrm {for} \ \mathbf {m} =\mathbf {m} '+\mathbf {e} _{x}$

$\mathbf {t} _{\mathbf {mm} ^{\prime }}=-t_{O}\mathbf {I} +it_{\mathrm {SO} }{\hat {\sigma }}_{x}\ \mathrm {for} \ \mathbf {m} =\mathbf {m} '+\mathbf {e} _{y}$

makes it possible to study on the computer how spin-polarized electronic wave packet propagates through such wire. Here $\mathbf {I} =\mathrm {ones} (2)$ is the unit $2\times 2$ matrix. The sites $\mathbf {m}$ and $\mathbf {m} ^{\prime }$ coupled by nonzero $\mathbf {t} _{\mathbf {mm} ^{\prime }}$ are nearest neighbors, and $\mathbf {e} _{x}$ and $\mathbf {e} _{y}$ are unit vectors along the x- and y-direction, respectively.

Assume that $t_{\mathrm {SO} }=0.1t_{O}$ and $t_{O}=1$ sets the unit of energy (typically in eV), as well as that wire is described on the square lattice $L_{x}\times L_{y}=160\times 31$ sites. The region with $t_{\mathrm {SO} }\neq 0$ is located between sites whose x-coordinate is 31 to 130.

The initial wave packet injected from the left is chosen as:

$\Psi (x,y;t=0)=C\sin \left({\frac {\pi y}{(L_{y}+1)a}}\right)e^{ik_{x}-\delta k_{x}^{2}x^{2}/4}\otimes {\begin{pmatrix}1\\0\end{pmatrix}}$

where C is the normalization constant obtained from $\Psi ^{\dagger }\Psi =1$, $k_{x}a=0.44$ and $\delta k_{x}a=0.1$.

Since Hamiltonian $\mathbf {H}$ is time-independent and Crank–Nicolson algorithm does not work in 2D, here you can use the exact eigenenergies and eigenstates:

$\mathbf {H} |E_{n}\rangle =E_{n}|E_{n}\rangle$

to evolve the wave packet above according to:

$|\Psi (t)\rangle =\sum _{n}e^{-iE_{n}t/\hbar }|E_{n}\rangle \langle E_{n}|\Psi (t=0)\rangle$

Plot the local spin density $S_{\mathbf {m} }^{z}$ carried by the wave packet and the magnitude of the corresponding polarization vector ${\frac {\hbar }{2}}\mathbf {P} =\sum _{\mathbf {m} }\mathbf {S} _{\mathbf {m} }$ as a function of time (akin to Fig. 3 in Ref. [3]). Showing how $|\mathbf {P} |$ decays below one signifies spin decoherence due to SO coupling. Experimentally, local spin density carried by a wave packet can be created by a pump laser beam and then detected by after some propagation by the probe laser beam as discussed in Ref. [4].

References

• [1] D. D. Awschalom and M. E. Flatté, Challenges for semiconductor spintronics, Nature Phys. 3, 153 (2007). [PDF]
• [2] 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.
• [3] 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]