# Wurtzite quantum wires with strong spatial confinement: polarization
anisotropies in single wire spectroscopy

Viola Zeller<EMAIL_ADDRESS>Institut für Experimentelle und Angewandte
Physik, Universität Regensburg, D-93040 Regensburg, Germany  Nadine Mundigl
Institut für Experimentelle und Angewandte Physik, Universität Regensburg,
D-93040 Regensburg, Germany  Paulo E. Faria Junior  Institut für Theoretische
Physik, Universität Regensburg, D-93040 Regensburg, Germany  Jaroslav Fabian
Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg,
Germany  Christian Schüller  Institut für Experimentelle und Angewandte
Physik, Universität Regensburg, D-93040 Regensburg, Germany  Dominique
Bougeard<EMAIL_ADDRESS>Institut für Experimentelle und Angewandte
Physik, Universität Regensburg, D-93040 Regensburg, Germany

###### Abstract

We report GaAs/AlGaAs nanowires in the one-dimensional (1D) quantum limit. The
ultrathin wurtzite GaAs cores between 20-40 nm induce large confinement
energies of several tens of meV, allowing us to experimentally resolve up to
four well separated subband excitations in microphotoluminescence
spectroscopy. Our detailed experimental and theoretical polarization-resolved
study reveals a strong diameter-dependent anisotropy of these transitions: We
demonstrate that the polarization of the detected photoluminescence is
governed by the symmetry of the wurtzite 1D quantum wire subbands on the one
hand, but also by the dielectric mismatch of the wires with the surrounding
material on the other hand. The latter effect leads to a strong attenuation of
perpendicularly polarized light in thin dielectric wires, making the thickness
of the AlGaAs shell an important factor in the observed polarization behavior.
Including the dielectric mismatch to our k.p-based simulated polarization-
resolved spectra of purely wurtzite GaAs quantum wires, we find an excellent
agreement between experiment and theory.

###### keywords:

GaAs quantum wires, wurtzite, polarization

IR,NMR,UV

##  1 Introduction

One dimensional (1D) quantum wires stand out amongst spatially quantum-
confined solid-state structures because of unique quantized properties such as
fully ballistic transport 1, efficient light-matter interaction with
particular relevance for the realization of lasers 2, 3, strong manifestations
of correlation, 4 or new quasiparticle states 5 and dispersion effects 6 in
spin-orbit coupled systems. At the same time, the experimental exploration of
these rich physics is challenged by the availability of suitable solid-state
structures. One challenge is the realization of wires with confinement
energies large enough to operate experiments in the 1D quantum limit, in
particular for optical spectroscopy. In this regard, catalytically self-
assembled ultrathin GaAs nanowires 7, 8, 9, 10 with very pure crystal phases
11 have recently opened new perspectives. In this letter, we report 1D
semiconductor quantum wires revealing up to four well separated subband
transitions in single wire photoluminescence (PL) with several tens of meV
confinement energies. Our core/shell nanowires feature ultrathin GaAs cores
below 40 nm and an AlGaAs shell for which we have varied the thickness. The
experimental polarization-resolved spectra and observed confinement energies
match very well with our k.p simulations for purely wurtzite nanowires,
including band-mixing in the valence bands. Our experimental and theoretical
study also clearly demonstrates the importance of the AlGaAs shell thickness
in single wire polarization-resolved studies: The intrinsic electronic
polarization selection rules are overlayed by extrinsic, classical
polarization anisotropies, resulting from the dielectric mismatch between the
GaAs/AlGaAs nanowire and its environment. We find that AlGaAs shell
thicknesses of several hundred nanometers are required to experimentally
observe the pure wurtzite electronic 1D quantum wire polarization behavior for
several quantum confined subbands.

##  2 Experimental and theoretical details

###  2.1 Nanowire growth

The investigated core/shell wurtzite GaAs quantum wires were grown by solid
source molecular-beam epitaxy (MBE) on semi-insulating GaAs(111)B substrates
via Au-catalyzed vapour-liquid-solid growth. Prior to growth, the substrates
were covered with a very thin Au-layer of nominally 0.05
$\mathrm{\SIUnitSymbolAngstrom}\text{/}$ which forms Au/Ga-droplets of $\sim$
30 nm diameter. The nanowire growth procedure was separated into two steps: In
the first step, we synthesized the ultrathin GaAs cores at a substrate
temperature of 540 °C for 95 min, a Ga rate of 0.7
$\mathrm{\SIUnitSymbolAngstrom}\text{/}$/s and an As4/Ga ratio of 2. This step
yields vertically grown wires with high wurtzite crystal phase purity7, 12,
11, an uniform length of 2 µm average and an average diameter of 30 nm ($\pm$
10 nm), thin enough to observe 1D spatial quantum confinement effects 7. In
the second step, the growth temperature was lowered to 460 °C while the As4/Ga
ratio was increased to $\sim$ 4, enabling a lateral overgrowth of the
ultrathin GaAs core with an Al0.38Ga0.62As shell. In this way we fabricated
several wafers with different AlGaAs shell thicknesses. Figure 1 shows
scanning electron micrographs (SEM) of as-grown nanowires without shell (a)
and with various shell thicknesses corresponding to a lateral overgrowth time
of 13 min (b), 32 min (c) and 63 min (d). All nanowire shells were finally
capped with nominally 5 nm GaAs to prevent the oxidation of the AlGaAs. We
produced nanowires with total core/shell diameters ranging from 50 nm to 230
nm as determined by SEM, which we refer to as dielectric diameter in the
following, as the GaAs core and the Al0.38Ga0.62As shell have similar
dielectric constants. The larger bandgap of the Al0.38Ga0.62As shell confines
all photocarriers to the GaAs core and suppresses non-radiative recombination
caused by surface states, which is why we obtain a sufficient radiative
efficiency in the photoluminescence (PL) even for the thinnest wires.

Figure 1: Representative scanning electron micrographs of quantum wires from
four different growth runs. Pure GaAs core without AlGaAs shell (a) and with
different AlGaAs shell thicknesses (b-d). The total diameter of the wires
amounts to 30 nm (a), 70 nm (b), 160 nm (c) and 220 nm (d) with lengths
ranging from 2 µm (a) to 3 µm (d). Scale bar is 250 nm.

###  2.2 Optical spectroscopy

For PL experiments, the nanowires were randomly dispersed onto
$300\,\text{nm}$ thermal SiO2 formed on p-doped silicon chips by carefully
brushing the growth substrate over the chip surface. In our confocal
microphotoluminescence (µ-PL) setup, the samples were excited by a pulsed
semiconductor diode laser in a He-flow cryostat at 4.2 K. We used two
different PicoQuant lasers with peak energies of 1.8 eV (690 nm) or 3.1 eV
(405 nm), $\sim$ 50 ps pulse duration and a variable repetition frequency. The
laser beam was focused onto the sample through a 100x microscope objective
with a numerical aperture of NA = 0.8, resulting in a spot diameter of $\sim$
1 µm, hence allowing us to address single nanowires lying on SiO2. The PL
signal was collected through the same objective and focused onto the entrance
slit of a 75 cm spectrometer, dispersed by a 150 mm-1 grating and detected
with a charge coupled device (CCD) camera. The exciting laser light was cut
off by a suitable long-pass filter.

###  2.3 Theoretical modeling

The conduction and valence band states of the pure GaAs wurtzite nanowires are
calculated via the k.p framework, within the formalism of the envelope
function approximation and the plane wave expansion to treat the hexagonal
quantum confinement. The valence band k.p Hamiltonian incorporates band-mixing
effects of heavy, light, and crystal field split-off holes, which are crucial
to properly model the intrinsic polarization anisotropies of optical
spectra13, 14, 15, 16. The conduction band is treated with a parabolic
dispersion. Specific details of the k.p-based calculations can be found in
Refs. 17, 7, 18. We use the valence band parameters and the electron effective
masses given in Ref. 19 (from the GW-LDA data set, without including the spin
splitting terms in the valence band).

The optical signatures of the different subband excitations in the nanowires
are investigated by calculating the absorption spectra20, 21

$\alpha^{a}(\hbar\omega)=C_{0}\underset{c,v,k_{z}}{\sum}\left|p^{a}_{cvk_{z}}\right|^{2}\,F_{cvk_{z}}\,\delta\left(\hbar\omega_{cvk_{z}}\\!-\\!\hbar\omega\right),$
(1)

in which the superindex $a$ refers to the light polarization along $x$
(perpendicular) or $z$ (parallel to the nanowire axis) direction, the index
$c(v)$ labels the conduction (valence) subbands, $k_{z}$ is the wave vector
along the nanowire axis, $p^{a}_{cvk_{z}}=\left\langle
c,k_{z}\left|\vec{p}\cdot{\hat{a}}\right|v,k_{z}\right\rangle$ is the dipole
matrix element for the polarization of the light $\hat{a}$ which incorporates
the valence band mixing and anisotropic Kane parameters16, 22,
$F_{cvk_{z}}=\left(f_{c,k_{z}}-f_{v,k_{z}}\right)$, $f_{c(v),k_{z}}$ is the
Fermi-Dirac distribution for the electron occupancy in the conduction
(valence) subbands, $\hbar$ is the Planck’s constant, $\omega_{cvk_{z}}$ is
the interband transition frequency and $\delta$ is the Dirac delta function.
The constant $C_{0}$ is
$C_{0}=4\pi^{2}e^{2}/(c_{l}n_{r}\varepsilon_{0}m_{0}^{2}\omega\Omega)$, in
which $e$ is the electron charge, $c_{l}$ is the speed of light, $n_{r}$ is
the refractive index of the material, $\varepsilon_{0}$ is the vacuum
permittivity, $m_{0}$ is the free electron mass, and $\Omega$ is the nanowire
volume. In our calculations, we assume undoped systems at nearly zero
temperature, so that $\left(f_{c\vec{k}}-f_{v\vec{k}}\right)=1.$23 To include
broadening effects due to finite carriers’ lifetimes20, 24, we replace the
Dirac delta function by a Lorentzian function with a full width at half-
maximum of 5 meV.

##  3 Results and Discussion

###  3.1 Observation of wurtzite subbands

We have investigated several tens of single quantum wires with different core
and shell diameters. These wires were optically pre-selected to display a
defect-free emission, which is correlated with a high crystal phase purity.
12, 11, 25 A characteristic excitation power series of the PL emission of a
single wire is shown in Figure 2a. While at low excitation power the emission
(light brown curve) only displays the single narrow PL peak representative of
the ground state emission (labeled "1" in 2a), up to four peaks arise in the
spectrum as we gradually increase the excitation power, labeled "1" to "4".
Note that the emission peaks 1-4 do not significantly change their energetic
position with increasing excitation level.

In Fig. 2b, we summarize the peak energies observed in 37 single wires for the
up to four peaks which arise with increasing excitation power. These wires
span core diameters between approximately 20-40 nm. The figure displays the
peak energies of the peaks 2-4 as a function of the energy of their
corresponding ground state emission peak 1. As a consequence of increasing
spatial quantum confinement, the ground state PL energy of peak 1 continuously
increases from 1.515 eV to 1.548 eV for decreasing core diameters.
Furthermore, for all wires, the peak energies for peaks 2-4 clearly increase
steadily, along with the corresponding ground state energy. All three peak
energies 2-4 follow a roughly linear evolution as a function of the ground
state energy 1, indicated as a guide to the eye for each peak in Fig. 2b. The
slightly larger scattering of the peak energies of peaks 3 and 4 is a result
of some of these peaks being broader than peaks 1 and 2, increasing the error
on our peak value determination. The observed relationship of peaks 2-4 to
peak 1 suggests that, like peak 1, the emission energies of peaks 2-4 are
governed by the same core diameter dependence for all 37 studied quantum
wires. At the same time, we have observed that for a given core diameter, the
energies for peaks 1-4 are independent of the shell thickness, i.e. of the
dielectric diameter of the wire. We interpret these observations as a strong
indication for peaks 2-4 to represent emission peaks from excited quantum wire
subbands, successively filled with increasing excitation power. We emphasize
that the 37 quantum wires shown in Fig. 2b feature different dielectric
diameters. As a consequence, all the different shapes shown in Fig. 1b-d are
present in 2b. Hence, while each wire in principle geometrically represents a
cavity, we exclude peaks 1-4 to represent Fabry-Pérot resonances: It would be
highly unlikely to observe the behavior documented in 2b for Fabry-Pérot
resonances in 37 wires with different dielectric diameters, tip shapes and
wire lengths.

Figure 2: Observation of subband transitions in wurtzite quantum wires. (a)
Spectra of a single quantum wire at increasing excitation intensities. With
higher excitation, the subbands gradually fill up and appear as discernible
peaks in the PL spectrum. The individual spectra are normalized to their
maximum. (b) Subband energies of the investigated nanowires at high excitation
power as a function of the ground state emission energy (at low excitation
power). The linear dependence between the energies are indicative of subband
transitions. The lines are guides for the eye.

###  3.2 Polarization resolved analysis

Being able to resolve the emission of excited quantum wire subbands, opens up
the possibility to experimentally uncover the polarization behavior of quantum
wires in a wurtzite crystal: In bulk wurtzite GaAs, the degeneracy between the
heavy hole (hh) and light hole (lh) valence band is lifted. As a result, the
PL emission of bulk nanowires has been shown to be dominated by transitions
from the hh-band, inducing a preferred polarization perpendicular to the
nanowire axis. 25, 11, 26, 27, 28, 29 Adding spatial quantum confinement, we
have recently discussed a realistic model of the energetic subband structure
of GaAs wurtzite quantum wires in the context of intersubband transitions
resolved by resonant inelastic light scattering 18.

Figure 3a sketches the electronic subband structure of a 1D quantum wire for
the conduction (grey), the dominantly heavy hole (red) and the dominantly
light hole (blue) bands. Note that dominantly hh transitions (red) only
contribute to emission and absorption with perpendicular dipoles, while lh
transitions (blue) contribute to both, perpendicular and parallel
polarization. Hence, the character of the involved electronic transitions can
be expected to strongly affect the polarization behavior of experimental
emission and absorption spectra.

In Fig. 3b, we show the realistic simulation of a polarization-resolved
absorption spectrum for a GaAs quantum wire with 26 nm core diameter. The
spectrum is structured by the Van Hove singularities of the joint density of
states (jDOS) for a 1D quantum object weighted by the allowed optical
transitions. These singularities are resonances of the transitions highlighted
by the arrows in 3a and are offset with respect to the bulk wurtzite GaAs
bandgap set here to $E_{\mathrm{gap}}=0$ meV. As can be seen in Fig. 3b, our
calculation for a wurtzite quantum wire with a 26 nm core diameter predicts
the emission and absorption for energies up to 200 meV above the bulk wurtzite
bandgap to be dominated by perpendicular transition dipoles.

Figure 3: Polarization selection rules in 1D-confined wurtzite GaAs. (a)
Sketch of the subband dispersion resulting from size quantization. The
electronic conduction band (gray) and the non-degenerate heavy hole (red) and
light hole (blue) valence bands are split into subbands. The arrows indicate
the allowed dipole transitions with their respective allowed directions of
polarization. (b) Calculated, polarization resolved absorption spectra of a
wurtzite GaAs nanowire with 26 nm core diameter, showing typical Van Hove
singularities of a 1D quantum object.

To test this prediction experimentally, we inserted a rotatable linear
polarizer in the detection path and analyzed the polarization of the emitted
light in x- and z-direction. A schematic of the setup is shown in Figure 4a.
We now discuss a series of quantum wires for which the energy of peaks 1-3
display excellent match with the first Van Hove singularities in Fig. 3b. From
this excellent match between the simulation and the energy positions of the PL
peaks, we conclude that all the wires discussed in the following have a core
diameter of 26 nm.

In Fig. 4b we start with the polarization-resolved spectra (x- and
z-direction) of a quantum wire which features a comparatively thin dielectric
diameter of 45 nm, determined by SEM. In accordance with the predicted
dominant hh character from our simulation, the polarization of the ground
state transition at 1.537 eV clearly shows a preferential direction
perpendicular to the wire axis in the PL spectrum. Intriguingly, however, the
dominant polarization of the emitted light changes from perpendicular to
parallel at an energy of $\sim$ 1.55 eV. The parallel polarized PL then
clearly dominates the spectrum above $\sim$ 1.55 eV. Hence, at first sight,
the polarization in the experiment is in discrepancy with our simulation in
Figure 3b, which is solely based on interband transitions from the electronic
1D subband structure. Note, however and as mentioned before, that the
energetic positions of peaks 1-3 in 4b are in excellent match with 3b.

This discrepancy in the polarization behavior between experiment and
simulation, motivates us to consider possible extrinsic impacts on the
polarization behavior of the emission and absorption in nanowires and to
explore whether they might add to or even outweigh the prediction of our
intrinsic electronic simulation in certain energy regions of the spectrum. In
this context, it has been discussed that the high aspect ratio, combined with
the large refractive index contrast between the wire and its environment
(vacuum), causes a strong polarization anisotropy. This dielectric mismatch
effect favors the absorption and emission of parallel polarized light. For
instance, when a dielectric cylinder with the dielectric constant $\epsilon$
is placed into an external electric field $\mathbf{E_{0}}$, the component
normal to the wire axis is essentially suppressed inside the wire: 30, 19, 31,
13, 28

$E_{\perp}=\frac{2\epsilon_{0}}{\epsilon+\epsilon_{0}}E_{0\perp},$ (2)

where $\epsilon_{0}$ is the vacuum permittivity, $E_{\perp}$ and $E_{0\perp}$
the normal electric field components inside and outside the wire,
respectively. At the same time, the field component parallel to the wire is
not reduced: $E_{\parallel}=E_{0\parallel}$. Though eq. (2) was initially
derived for static electric fields, it was shown to also be valid for
electromagnetic waves, as long as their wavelength can be considered to be
much larger than the diameter of the wire, $\lambda\gg d$.30, 19, 31, 13 Given
that $E_{0\parallel}=E_{0\perp}$, the damping constant for the perpendicular
field inside the wire amounts to

$\delta=\frac{I_{x}}{I_{z}}=\frac{|E_{\perp}|^{2}}{|E_{\parallel}|^{2}}=\Big{|}\frac{2\epsilon_{0}}{\epsilon+\epsilon_{0}}\Big{|}^{2}.$
(3)

Given that nanowires rarely fulfill the condition $\lambda\gg d$, we go beyond
Eq. 3 by estimating the attenuation of the perpendicular electric field
$\delta=I_{x}/I_{y}$ as a function of nanowire diameter by finite-element-
method (FEM) based modeling (see Supplementary Information for a detailed
description). For the wire discussed in Fig. 4b, we find $\delta=0.016$. Fig.
4c then displays again our calculated absorption spectra discussed in Fig. 3b,
but, in addition, the x-polarized spectrum has now been multiplied with
$\delta=0.016$, to take into account the dielectric mismatch effect for a wire
with a diameter of 45 nm. We have shifted the absorption spectrum rigidly so
that the first singularity coincides with the ground state PL peak of Fig. 4b.
Clearly, the simulated absorption spectrum Fig. 4c predicts a preferred
z-polarization above 1.55 eV, while x-polarization dominates below this
energy. As a consequence, the simulation is now in excellent agreement with
the characteristic features of the experimental emission spectra: not only do
the energetic positions of the first four PL peaks match with calculated
electronic subband excitations, but also the switching from a dominant
x-polarization to a dominant z-polarization around 1.55 eV is correctly
predicted. Note that we analyzed several nanowires with dielectric diameters
around 45 nm and observed this switching of the dominant polarization for all
of them, even for core diameters smaller or larger than the discussed 26 nm.
Hence, we interpret the strong attenuation of the perpendicular transitions in
these wires as a consequence of strong dielectric mismatch.

Figure 4: Polarization resolved analysis of thin quantum wires. (a) Simplified
schematic of the measurement setup. (b) Experimental PL emission spectra
analyzed perpendicular (xpol, red) and parallel (zpol, black) to the wire
axis. (c) Calculated perpendicularly (red) and parallely (black) polarized
absorption spectra. The x-polarized spectrum (red) was multiplied with the
attenuation constant $\delta=0.016$ according to our simulations.

Our model system of wurtzite GaAs quantum wires with variable shell thickness
allows us to further specify the effect of the dielectric mismatch on the
polarization behavior as a function of the dielectric diameter. In particular,
the attenuation of the perpendicular electric field component should gradually
be weakened when increasing the dielectric diameter. Thus, we now discuss the
polarization-resolved spectra for a much thicker wire featuring a dielectric
diameter of 220 nm. At the same time, compared to Figure 4b, we kept the GaAs
quantum wire core diameter constant at 26 nm. From our FEM simulations we
calculate $\delta=0.47$ for a dielectric diameter of 220 nm. In Figure 5a, we
show the experimental polarization-resolved PL spectra and compare them to the
corresponding, calculated absorption spectra, plotted in 5b. Like in Fig. 4c,
the x-polarized simulated spectrum was multiplied by the attenuation factor,
here $\delta=0.47$. The three major PL peaks of the experiment agree very well
with the first Van Hove singularities predicted for this 26 nm quantum wire
core. At the same time, contrary to the thinner 45 nm dielectric diameter,
this 220 nm dielectric diameter reveals a preferred polarization perpendicular
to the wire axis over the entire experimentally covered energy range, in
excellent consistency with the simulated spectra.

Note that the experimental parallel spectrum reveals two peaks in the energy
range of 1.50 eV - 1.60 eV which are not predicted by the simulation in Fig.
5b. As we discuss in more detail in the Supplementary Information, although
our experimental configuration excludes the collection of perpendicularly
y-polarized emitted light, we attribute these peaks to stem from y-polarized
emitted light which is scattered into z-polarization primarily at the nanowire
end facets. Given that the wave guiding of perpendicularly polarized light is
more efficient for thicker dielectric wire diameters, we expect this
scattering effect to be more pronounced in our wire with 220 nm diameter than
in the 45 nm wire. Hence, the two peaks are clearly visible in the Fig. 5a in
the 220 nm wire, while their occurrence is less obvious in the 45 nm wire in
Fig. 4b.

To demonstrate that the dominant modulation mechanism of the intrinsic
polarization-resolved emission of our quantum wires is the dielectric
diameter-dependent attenuation, we finally compare the 45 nm, the 220 nm and a
wire with an intermediate dielectric diameter thickness of 110 nm in Fig. 5c
and 5d (constant core diameter, 26 nm). We plot the energy-dependent degree of
linear polarization (DLP) defined as

$\mathrm{DLP}=\frac{(I_{x}-I_{z})}{(I_{x}+I_{z})},$ (4)

so that the pure x-polarization of the light yields DLP = +1 and pure
z-polarization DLP = -1.

Fig. 5c shows the experimentally determined DLP for all three wires, as
extracted from the polarization-resolved PL spectra. We compare these traces
to the DLP extracted from our calculated absorption spectra shown in Fig. 5d,
for which the x-polarization was modulated with the attenuation $\delta$-value
corresponding to each dielectric diameter. In order to align the energy scale,
the energy of the ground state transition was set to zero for all curves.

The simulation quite precisely matches the energetic positions and magnitudes
of the experimental DLP variations for all three wires, as prominently seen
for the two distinct peak-like features for the 45 nm and 110 nm wires. As can
be seen in Fig. 3b, for our representative example of 26 nm quantum wire core
diameter, the significant hh-lh splitting induced by wurtzite GaAs and by the
quantum confinement leads to negligible parallel polarized contributions in
absorption and emission in the energy window 0-100 meV considered in Fig.
5c,d. Hence, wires with a dielectric diameter thick enough to display
negligible attenuation, i.e. a $\delta$-factor close to 1, delivers a positive
DLP, close to DLP=+1. From the comparison of Fig. 5c,d, we see that this is
still true even for $\delta=0.47$ in wires with a dielectric diameter of 220
nm. As mentioned before, we attribute the discrepancy in absolute DLP values
between experiment and simulation to scattering effects in the experiments,
which are not included in our simulations.

Although the parallel polarized contribution is negligible between 0-100 meV
in the electronic absorption and emission spectra, our simulation predicts
that the lh-character - which produces parallel polarization - is non-zero. It
actually increases from 0 towards 100 meV depending on the strength of the
dielectric mismatch effect. Remarkably, the electromagnetic phenomenon of the
dielectric mismatch effect and its attenuation of $E_{\perp}$ allows to reveal
this non-zero contribution of the parallel polarization experimentally. The
DLP traces for the wires with 110 nm and 4 nm show that the experimental
visibility of the spectral structure of the parallel polarized contributions
increases with decreasing dielectric wire diameter, i.e. with decreasing
$\delta$-value. This excellent match between simulations and experiments
highlights that the three DLP spectra are the result of a subtle superposition
of the polarization behavior imposed by the electronic transitions and the
attenuation of the perpendicular field components due to the diameter-
dependent dielectric mismatch effect.

Figure 5: (a) Polarization resolved PL emission spectra of a quantum wire with
a dielectric diameter of 220 nm and a core diameter of 26 nm. (b)
Corresponding calculated absorption spectra of a 26 nm wire where the
x-polarized spectrum was multiplied with $\delta=0.47$. (c) Energy dependent
DLP curves of three wires with a 26 nm core and different dielectric diameters
of d = 45 nm, d = 110 nm and d = 220 nm respectively. (d) Energy dependent DLP
curves of the calculated absorption spectra with different $\delta$-values
found from our FEM simulations for the corresponding diameters in (c).

##  4 Conclusion

In conclusion, our MBE grown core/shell nanowires reproducibly provide
ultrathin wurtzite GaAs cores. Such cores ease experiments in the 1D quantum
limit 2, 3, 32, 4, 7, given their large confinement energies of several tens
of meV. We found an excellent agreement between polarization-resolved
photoluminescence spectra and k.p-based simulations for the excitation of up
to four well-separated subband transitions. At the same time, while it does
not modify the 1D quantum confinement of the GaAs core, we have revealed a
significant role of the thickness of the AlGaAs shell in polarization-resolved
spectroscopy of such 1D quantum wires. Our experiments and simulations show
that for shells below thicknesses around 200 nm, the dielectric mismatch
effect will significantly overlay the intrinsic electronic 1D wurtzite
selection rules. Comparatively large shell thicknesses above 200 nm were
required to observe the polarization behavior predicted by the electronic
selection rules for up to four 1D subband excitations.

We acknowledge the financial support of the Deutsche Forschungsgemeinschaft
through Project ID 422 31469 5032-SFB1277 (Subprojects A01, B05, B11).

## References

  * van Wees et al. 1988 van Wees, B. J.; van Houten, H.; Beenakker, C. W. J.; Williamson, J. G.; Kouwenhoven, L. P.; van der Marel, D.; Foxon, C. T. Quantized conductance of point contacts in a two-dimensional electron gas. _Physical Review Letters_ 1988, _60_ , 848–850
  * Kapon et al. 1989 Kapon, E.; Simhony, S.; Bhat, R.; Hwang, D. M. Single quantum wire semiconductor lasers. _Applied Physics Letters_ 1989, _55_ , 2715–2717
  * Asada et al. 1985 Asada, M.; Miyamoto, Y.; Suematsu, Y. Theoretical Gain of Quantum-Well Wire Lasers. _Japanese Journal of Applied Physics_ 1985, _24_ , L95–L97
  * Deshpande et al. 2010 Deshpande, V. V.; Bockrath, M.; Glazman, L. I.; Yacoby, A. Electron liquids and solids in one dimension. _Nature_ 2010, _464_ , 209–216
  * Sau et al. 2010 Sau, J. D.; Tewari, S.; Lutchyn, R. M.; Stanescu, T. D.; Das Sarma, S. Non-Abelian quantum order in spin-orbit-coupled semiconductors: Search for topological Majorana particles in solid-state systems. _Physical Review B_ 2010, _82_ , 214509
  * Rashba 2012 Rashba, E. I. Quantum nanostructures in strongly spin-orbit coupled two-dimensional systems. _Physical Review B_ 2012, _86_ , 125319
  * Dirnberger et al. 2019 Dirnberger, F.; Kammermeier, M.; König, J.; Forsch, M.; Faria Junior, P. E.; Campos, T.; Fabian, J.; Schliemann, J.; Schüller, C.; Korn, T.; Wenk, P.; Bougeard, D. Ultralong spin lifetimes in one-dimensional semiconductor nanowires. _Applied Physics Letters_ 2019, _114_ , 202101
  * Vainorius et al. 2016 Vainorius, N.; Lehmann, S.; Gustafsson, A.; Samuelson, L.; Dick, K. A.; Pistol, M.-E. Wurtzite GaAs Quantum Wires: One-Dimensional Subband Formation. _Nano Letters_ 2016, _16_ , 2774–2780
  * Loitsch et al. 2015 Loitsch, B.; Rudolph, D.; Morkötter, S.; Döblinger, M.; Grimaldi, G.; Hanschke, L.; Matich, S.; Parzinger, E.; Wurstbauer, U.; Abstreiter, G.; Finley, J. J.; Koblmüller, G. Tunable Quantum Confinement in Ultrathin, Optically Active Semiconductor Nanowires Via Reverse-Reaction Growth. _Advanced Materials_ 2015, _27_ , 2195–2202
  * Corfdir et al. 2018 Corfdir, P.; Li, H.; Marquardt, O.; Gao, G.; Molas, M. R.; Zettler, J. K.; van Treeck, D.; Flissikowski, T.; Potemski, M.; Draxl, C.; Trampert, A.; Fernández-Garrido, S.; Grahn, H. T.; Brandt, O. Crystal-Phase Quantum Wires: One-Dimensional Heterostructures with Atomically Flat Interfaces. _Nano Letters_ 2018, _18_ , 247–254
  * Furthmeier et al. 2014 Furthmeier, S.; Dirnberger, F.; Hubmann, J.; Bauer, B.; Korn, T.; Schüller, C.; Zweck, J.; Reiger, E.; Bougeard, D. Long exciton lifetimes in stacking-fault-free wurtzite GaAs nanowires. _Applied Physics Letters_ 2014, _105_ , 222109
  * Dirnberger et al. 2019 Dirnberger, F.; Abujetas, D.; König, J.; Forsch, M.; Koller, T.; Gronwald, I.; Lange, C.; Huber, R.; Schüller, C.; Korn, T.; Sánchez-Gil, J.; Bougeard, D. Tuning Spontaneous Emission through Waveguide Cavity Effects in Semiconductor Nanowires. _Nano Letters_ 2019, _19_ , 7287–7292
  * Wang et al. 2001 Wang, J.; Gudiksen, M. S.; Duan, X.; Cui, Y.; Lieber, C. M. Highly Polarized Photoluminescence and Photodetection from Single Indium Phosphide Nanowires. _Science_ 2001, _293_ , 1455–1457
  * Persson and Xu 2004 Persson, M. P.; Xu, H. Q. Giant polarization anisotropy in optical transitions of free-standing $\mathrm{InP}$ nanowires. _Physical Review B_ 2004, _70_ , 161310
  * Maslov and Ning 2005 Maslov, A. V.; Ning, C. Z. Radius-dependent polarization anisotropy in semiconductor nanowires. _Physical Review B_ 2005, _72_ , 161310
  * Faria Junior et al. 2014 Faria Junior, P. E.; Campos, T.; Sipahi, G. M. Interband polarized absorption in InP polytypic superlattices. _Journal of Applied Physics_ 2014, _116_ , 193501
  * Campos et al. 2018 Campos, T.; Faria Junior, P. E.; Gmitra, M.; Sipahi, G. M.; Fabian, J. Spin-orbit coupling effects in zinc-blende InSb and wurtzite InAs nanowires: Realistic calculations with multiband $\mathbf{k}\cdot{}\mathbf{p}$ method. _Physical Review B_ 2018, _97_ , 245402
  * Meier et al. 2021 Meier, S.; Faria Junior, P. E.; Haas, F.; Heller, E.-S.; Dirnberger, F.; Zeller, V.; Korn, T.; Fabian, J.; Bougeard, D.; Schüller, C. Intersubband excitations in ultrathin core-shell nanowires in the one-dimensional quantum limit probed by resonant inelastic light scattering. _Physical Review B_ 2021, _104_ , 235307
  * Cheiwchanchamnangij and Lambrecht 2011 Cheiwchanchamnangij, T.; Lambrecht, W. R. L. Band structure parameters of wurtzite and zinc-blende GaAs under strain in the GW approximation. _Physical Review B_ 2011, _84_ , 035203
  * Chuang 1995 Chuang, S. _Physics of Optoelectronic Devices_ ; Wiley Series in Pure and Applied Optics; Wiley, 1995
  * Cardona and Peter 2005 Cardona, M.; Peter, Y. Y. _Fundamentals of semiconductors_ ; Springer, 2005; Vol. 619
  * Faria Junior et al. 2017 Faria Junior, P. E.; Xu, G.; Chen, Y.-F.; Sipahi, G. M.; Žutić, I. Wurtzite spin lasers. _Physical Review B_ 2017, _95_ , 115301
  * Ravi Kishore et al. 2010 Ravi Kishore, V. V.; Partoens, B.; Peeters, F. M. Electronic structure and optical absorption of $\text{GaAs}/{\text{Al}}_{x}{\text{Ga}}_{1-x}\text{As}$ and ${\text{Al}}_{x}{\text{Ga}}_{1-x}\text{As}/\text{GaAs}$ core-shell nanowires. _Physical Review B_ 2010, _82_ , 235425
  * Chow and Koch 1999 Chow, W. W.; Koch, S. W. _Semiconductor-laser fundamentals: physics of the gain materials_ ; Springer Science & Business Media, 1999
  * Ahtapodov et al. 2012 Ahtapodov, L.; Todorovic, J.; Olk, P.; Mjåland, T.; Slåttnes, P.; Dheeraj, D. L.; Van Helvoort, A. T.; Fimland, B. O.; Weman, H. A story told by a single nanowire: Optical properties of wurtzite GaAs. _Nano Letters_ 2012, _12_ , 6090–6095
  * Ketterer et al. 2011 Ketterer, B.; Heiss, M.; Uccelli, E.; Arbiol, J.; Fontcuberta i Morral, A. Untangling the Electronic Band Structure of Wurtzite GaAs Nanowires by Resonant Raman Spectroscopy. _ACS Nano_ 2011, _5_ , 7585–7592
  * De Luca et al. 2015 De Luca, M.; Zilli, A.; Fonseka, H. A.; Mokkapati, S.; Miriametro, A.; Tan, H. H.; Smith, L. M.; Jagadish, C.; Capizzi, M.; Polimeni, A. Polarized Light Absorption in Wurtzite InP Nanowire Ensembles. _Nano Letters_ 2015, _15_ , 998–1005
  * Chen et al. 2008 Chen, H.-Y.; Yang, Y.-C.; Lin, H.-W.; Chang, S.-C.; Gwo, S. Polarized photoluminescence from single GaN nanorods: Effects of optical confinement. _Optics Express_ 2008, _16_ , 13465
  * Ba Hoang et al. 2010 Ba Hoang, T.; Moses, A. F.; Ahtapodov, L.; Zhou, H.; Dheeraj, D. L.; van Helvoort, A. T. J.; Fimland, B.-O.; Weman, H. Engineering Parallel and Perpendicular Polarized Photoluminescence from a Single Semiconductor Nanowire by Crystal Phase Control. _Nano Letters_ 2010, _10_ , 2927–2933
  * Landau and Lifshitz 1984 Landau, L.; Lifshitz, E. _Electrodynamics of Continuous Media_ , 2nd ed.; Course of Theoretical Physics; Pergamon, 1984; Vol. 8
  * Ruda and Shik 2005 Ruda, H. E.; Shik, A. Polarization-sensitive optical phenomena in semiconducting and metallic nanowires. _Physical Review B_ 2005, _72_ , 115308
  * Furthmeier et al. 2016 Furthmeier, S.; Dirnberger, F.; Gmitra, M.; Bayer, A.; Forsch, M.; Hubmann, J.; Schüller, C.; Reiger, E.; Fabian, J.; Korn, T.; Bougeard, D. Enhanced spin–orbit coupling in core/shell nanowires. _Nature Communications_ 2016, _7_ , 12413