Supplementary Material
Elliptical Quantum Dots as On-Demand Single Photons Sources with Deterministic
Polarization States
Chu-Hsiang Teng,1 Lei Zhang,2 Tyler A. Hill,2 Brandon Demory,1 Hui Deng,2 and Pei-Cheng Ku1*
1
Department of Electrical Engineering and Computer Science, University of Michigan, 1301 Beal Ave., Ann
Arbor, MI 48105, USA
2
Department of Mechanical Engineering, University of Michigan, 2350 Hayward St., Ann Arbor, MI 48105, USA
Sample Fabrication
The site-controlled InGaN QDs used in this work were fabricated by a top-down process.1 A sample consisting
of a single unintentionally-doped (uid) In0.11Ga0.89N quantum well and a uid 10-nm GaN barrier layer was
epitaxially grown on a 3-µm thick GaN template on c-plane sapphire using metal-organic chemical vapor
deposition. The sample was then patterned by electron-beam lithography (JEOL JBX-6300FS) using the 950PMMA A2 photoresist (PR). After PR development, chromium was deposited by e-beam evaporator (Enerjet
Evaporator) and lifted off. Inductively-coupled plasma reactive ion etching (ICP-RIE) using LAM9400 with Cl2
and Ar was employed to form GaN nanopillars with one InGaN QD embedded in each nanopillar.
Photoluminescence Measurement Setup
The A 390-nm wavelength femtosecond mode-locked laser with 80-MHz repetition rate and 150-fs pulse
duration was used as the excitation source. It had an incident angle ~ 50° from the normal direction of sample
surface, and the spot size was ~ 40 μm in diameter. A pair of confocal lenses and a pinhole were inserted when
single QD measurement was carried out. The pinhole allowed spatial selection of a circular area with a 0.8-μm
diameter. The spectral resolution was 0.1 nm at 400-nm wavelength. The combination of a rotating half wave plate
and a fixed linear polarizer was used to resolve the spectra at different polarization angles. The second-order
photon correlation (g(2)) was measured using a Hanbury Brown-Twiss interferometer.
Numerical Simulation
*
peicheng@umich.edu
1
Electronic band structures and wave functions of InGaN QDs were calculated by a semiconductor simulation
package, nextnano3.2 First, strain-relaxation due to the nanopillar geometry was considered, and the strin profile
was calculated by minimizing the strain energy. Secondly, the strain-induced bandstructure modification was taken
into account in order to get correct electron and hole wavefunctions. The electron state was obtained by solving
the conduction band Schrödinger equation based on effective mass model. The valence band states were simulated
based on the strain-involved 6×6 k·p model and the envelope function approximation. The strain-involved
Hamiltonian was solved in px, py, and pz basis, denoted as |X>, |Y>, and |Z>. After the envelope functions associated
with |X>, |Y>, |Z>, and the electron state |S> were solved, the matrix elements associated with x, y, and z
polarizations were derived using the following equations3:
x-polarized: |𝑀𝑥 |2 = |⟨Ψ𝑒 |𝑝𝑥 |𝛹ℎ ⟩|2 ≅ |⟨S|𝑝𝑥 |𝑋⟩|2 |⟨𝜑𝑒 |𝜑ℎ,𝑥 ⟩|
2
2
2
2
y-polarized: |𝑀𝑦 | = |⟨Ψ𝑒 |𝑝𝑦 |𝛹ℎ ⟩| ≅ |⟨S|𝑝𝑦 |𝑌⟩| |⟨𝜑𝑒 |𝜑ℎ,𝑦 ⟩|
2
2
z-polarized: |𝑀𝑧 |2 = |⟨Ψ𝑒 |𝑝𝑧 |𝛹ℎ ⟩|2 ≅ |⟨S|𝑝𝑧 |𝑍⟩|2 |⟨𝜑𝑒 |𝜑ℎ,𝑧 ⟩|
where 𝑝𝑥 , 𝑝𝑦 , and 𝑝𝑧 are momentum operators, S, X, Y, and Z are the Bloch basis, 𝜑𝑒 and 𝜑ℎ are the envelope
functions of the electron and hole states respectively, and Ψ𝑒 = 𝜑𝑒 · 𝑆 and Ψ𝑒 = 𝜑ℎ,𝑥 · X + 𝜑ℎ,𝑦 · Y + 𝜑ℎ,𝑧 · Z.
Spin degeneracy and Coulomb interaction were not included in our simulations.
Several QD materials were simulated. The z axes were assumed to be the [0001] and [001] directions for
wurtzite and zinc-blende QD materials, respectively, and the x axes corresponded to [101̅0] and [100]. Material
parameters were taken from Reference 4, 5, 6, and 7. The degree of linear polarization (DLP) was determined
by:
2
DLP =
|𝑀𝑦 | − |𝑀𝑥 |2
2
|𝑀𝑦 | + |𝑀𝑥 |2
All QDs were assumed to be elliptical with their short-axis lengths fixed at 22 nm while the long-axis lengths
were varied from 22 nm to 44 nm.
References
1
L.-K. Lee and P.-C. Ku, physica status solidi (c) 9, 609 (2012).
2
2
S. Birner, T. Zibold, T. Andlauer, T. Kubis, M. Sabathil, A. Trellakis, and P. Vogl, IEEE Trans. Electron Devices
54, 2137 (2007).
3
F. Rossi, Theory of Semiconductor Quantum Devices: Microscopic Modeling and Simulation Strategies
(Springer, 2010).
4
I. Vurgaftman and J. R. Meyer, Journal of Applied Physics 94, 3675 (2003).
5
I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, Journal of Applied Physics 89, 5815 (2001).
6
W. Shan, J. J. Song, H. Luo, and J. K. Furdyna, Physical Review B 50, 8012 (1994).
7
J. Piprek, Semiconductor optoelectronic devices: Introduction to physics and simulation (Academic Press, 2003)
3