# Strain engineering of the electronic states of silicon-based quantum
emitters

A. Ristori  Department of Physics and Astronomy, University of Florence, Via
G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy  European Laboratory for
Non-Linear Spectroscopy (LENS), Via N. Carrara 1, I-50019 Sesto Fiorentino
(FI), Italy  M. Khoury  Aix Marseille Univ, CNRS, Université de Toulon, IM2NP,
UMR 7334, F-13397 Marseille, France  M. Salvalaglio  Institute of Scientific
Computing, TU Dresden, 01062 Dresden, Germany  Dresden Center for Intelligent
Materials (DCIM), TU Dresden, 01062 Dresden, Germany  A. Filippatos  Dresden
Center for Intelligent Materials (DCIM), TU Dresden, 01062 Dresden, Germany
M. Amato  Laboratoire de Physique des Solides, Université Paris-Saclay, CNRS,
Orsay 91405, France  T. Herzig  Division of Applied Quantum Systems, Felix-
Bloch Institute for Solid-State Physics, University Leipzig, Linnéstrasse 5,
04103 Leipzig, Germany  J. Meijer  Division of Applied Quantum Systems, Felix-
Bloch Institute for Solid-State Physics, University Leipzig, Linnéstrasse 5,
04103 Leipzig, Germany  S. Pezzagna  Division of Applied Quantum Systems,
Felix-Bloch Institute for Solid-State Physics, University Leipzig,
Linnéstrasse 5, 04103 Leipzig, Germany  D. Hannani  Aix Marseille Univ, CNRS,
Université de Toulon, IM2NP, UMR 7334, F-13397 Marseille, France  M. Bollani
Istituto di Fotonica e Nanotecnologie-Consiglio Nazionale delle Ricerche,
Laboratory for Nanostructure Epitaxy and Spintronics on Silicon, Via Anzani
42, 22100 Como, Italy  C. Barri  L-NESS, Dipartimento di Fisica, Politecnico
di Milano, 20133 Como, Italy  C. M. Ruiz  Aix Marseille Univ, CNRS, Université
de Toulon, IM2NP, UMR 7334, F-13397 Marseille, France  N. Granchi  Department
of Physics and Astronomy, University of Florence, Via G. Sansone 1, I-50019
Sesto Fiorentino (FI), Italy  European Laboratory for Non-Linear Spectroscopy
(LENS), Via N. Carrara 1, I-50019 Sesto Fiorentino (FI), Italy  F. Intonti
Department of Physics and Astronomy, University of Florence, Via G. Sansone 1,
I-50019 Sesto Fiorentino (FI), Italy  European Laboratory for Non-Linear
Spectroscopy (LENS), Via N. Carrara 1, I-50019 Sesto Fiorentino (FI), Italy
M. Abbarchi  Aix Marseille Univ, CNRS, Université de Toulon, IM2NP, UMR 7334,
F-13397 Marseille, France  Solnil, 95 Rue de la République, 13002 Marseille,
France<EMAIL_ADDRESS>F. Biccari  European Laboratory for Non-Linear
Spectroscopy (LENS), Via N. Carrara 1, I-50019 Sesto Fiorentino (FI), Italy
Department of Physics and Astronomy, University of Florence, Via G. Sansone 1,
I-50019 Sesto Fiorentino (FI), Italy

###### Abstract

Light-emitting complex defects in silicon have been considered a potential
platform for quantum technologies based on spin and photon degrees of freedom
working at telecom wavelengths. Their integration in complex devices is still
in its infancy, and it was mostly focused on light extraction and guiding.
Here we address the control of the electronic states of carbon-related
impurities (G-centers) via strain engineering. By embedding them in patches of
silicon on insulator and topping them with SiN, symmetry breaking along [001]
and [110] directions is demonstrated, resulting in a controlled splitting of
the zero phonon line (ZPL), as accounted for by the piezospectroscopic
theoretical framework. The splitting can be as large as 18 meV and it is
finely tuned by selecting patch size or by moving in different positions on
the patch. Some of the split, strained ZPLs are almost fully polarized and
their overall intensity is enhanced up to 7 times with respect to the flat
areas, whereas their recombination dynamics is slightly affected. Our
technique can be extended to other impurities and Si-based devices such as
suspended bridges, photonic crystal microcavities, Mie resonators, and
integrated photonic circuits.

Carbon impurities in silicon, G-centers, strain engineering

††preprint: AIP/123-QED

##  I Introduction

Light emitters in silicon (Si) based on complex impurities [1, 2] are
currently scrutinized for their applicability as photon and spin quantum-bits
[3]. Their appeal is manyfold: 1) they are nominally identical and in ensemble
emission they have a sharp zero phonon line (ZPL) featuring a broadening of
about 10 $\mu$eV in conventional Si and less than 1 $\mu$eV (about 10-3 nm) in
spin-less, isotopically purified 28Si [2]; 2) their ZPL lies below the Si
band-gap, in the near-infrared range (1.15 to 1.45 $\mu$m), covering the
telecommunication O, E, and S bands [4, 5]; 3) their recombination lifetime
can be as short as a few ns [6, 7, 8]; 4) they are stable in temperature and
time (no bleaching nor blinking) [9, 7] and can be detected up 120 K [10]; 5)
they have well-defined polarization axes [9, 5, 7].

Beyond these intrinsic properties that are unmatched by their counterparts in
other materials, the possibility to exploit them in quantum technologies is
highly entrancing, provided the advantages that this solid-state platform
offers. Silicon technology steps on all the nano-fabrication methods developed
in the last 50 years for electronic devices and it is, by far, more advanced
than the technology applied to any other material: fabrication of electronic
components can be provided in an industrial production chain with high
material purity (99.9999999$\%$, nine nines), large wafers up to 17 inches,
silicon on insulator (SOI) wafers up to 12 inches, p and n doping, top-down
lithography with nanometric resolution (e.g. based on deep-UV and plasma
etching), availability of isotopically purified 28Si wafers [11, 12, 13, 14,
15, 16].

The renewed interest in this class of Si-based emitters in the context of
quantum technologies resulted in several breakthroughs over the last few years
with the demonstration of: 1) single-photon emission from a large zoology of
well-known (e.g. G-, W-, T-centers) and unknown defects [9, 17, 5, 7, 18, 8];
2) photon coalescence [19]; 3) spin control [20, 21, 22]; 4) integration in
photonic devices, such as Mie resonators [22, 10, 23], integrated photonic
circuits [24, 18, 19], ring resonators [25], and photonic crystals [26, 27]
providing Purcell effect; 5) position control of the emitters with localized
ion implant [28]; 6) coherent population trapping and Autler-Townes splitting
[29].

Most of these works focused on defect creation, enhancement of light emission,
extraction, and guiding, in order to better detect and manipulate these single
photon sources. A further step to control their properties requires a precise
tuning of the photoluminescence frequency and its polarization (e.g. for
setting the coupling of the ZPL with a photonic resonance).

In this paper, we show that a large and tunable splitting of the ZPL of
G-centers can be simply obtained by etching square patches in a [001]-oriented
SOI wafer followed by deposition of SiN via plasma-enhanced chemical vapor
deposition (PE-CVD). Compressive and tensile strain is obtained by adjusting
the plasma frequency during SiN deposition, as confirmed by Raman
spectroscopy. Symmetry breaking of the silicon unit cell along the vertical
[001] direction is obtained at the center of the square patches, whose size
sets the magnitude of the strain and thus the corresponding ZPL splitting in
two main lines. Moving from the center of a patch towards the suspended
(leaning) part, the symmetry breaking occurs also along the $\langle
110\rangle$ directions leading to a splitting of the ZPL in four lines. Finite
element methods calculations of the strain coupled to the piezospectroscopic
theory [30] confirms the overall picture observed in photoluminescence
measurements. Partial or total polarization of the split ZPLs accounts for the
alignment of the emitting dipoles along specific crystallographic directions.
Embedding the emitters in these structures entails an intensity enhancement of
their photoluminescence up to a factor of 7 with respect to the flat
counterpart, which is ascribed to a larger extraction of light, as confirmed
by time-resolved experiments and finite element simulations.

##  II Results

###  II.1 Sample fabrication

Figure 1:  Sample fabrication. a) A 125 nm thick SOI sample is implanted with
carbon ions and recrystallized by annealing at high temperature (1000 ∘ C for
10 seconds in N2 atmosphere). By optical lithography, square patches with
variable side lengths are defined. The buried oxide underneath the SOI patches
was partially removed via chemical etching using a buried-oxide etcher (BOE).
SiN is deposited via plasma-enhanced chemical vapor deposition (PE-CVD).
G-center activation is performed via proton implant. Details of the sample
fabrication are provided in the dedicated section and in Table 1. b) Scanning
electron micrograph (SEM) of SOI patches before SiN deposition. c) SEM of a
strained SOI patch with SiN atop.

A detailed description of sample fabrication is provided in the dedicated
section at the end of the paper (Section IV.1). Here we refer to Figure 1
where the main fabrication steps and some real examples are shown: a 125 nm
thick SOI is implanted with C ions and recrystallized by annealing. The
underlying SiO2 is partially removed with a diluted HF acid. The samples are
then topped with SiN and etched by optical lithography in square patches
having different side length $L$. Finally, proton implant is used to activate
the emitting G-centers.

In this work, we study three samples with etched patches. The details of their
fabrication conditions are provided in Table 1.

Table 1: List of samples and fabrication parameters: carbon implant energy and corresponding depth, and carbon dose; PE-CVD deposition conditions for SiN including high-frequency plasma time fraction (HF$\%$), high-frequency and low-frequency plasma power (HF, LF) corresponding type of strain (compressive or tensile, C or T), SiN thickness as measured by ellipsometry; proton implant energy and corresponding depth, and proton dose Sample | C ions implant |  | PE-CVD | Proton implant |
---|---|---|---|---|---
| Energy (Depth) | Dose | HF Plasma | Plasma Power | Strain | SiN | Energy (Depth) | Dose
| keV (nm) |  $10^{14}$ cm-2 | time % | HF, LF (W) | C, T | nm | keV (nm) |  $10^{14}$ cm-2
A | 6 (25) | 5.5 | 0 | 0, 100 | C | 108 | 90 (830) | 1
B | 6 (25) | 5.5 | 100 | 300, 0 | T | 102 | 90 (830) | 1
C | 30 (100) | 6.5 | 70 | 20, 80 | T | 100 | 90 (830) | 1

###  II.2 Finite element method simulations of strained patches

Figure 2: Simulation of the elastic field. a) Geometry in a perspective view
(left) and 2D cross section highlighting the different materials (right).
Three representative points are marked: “center", “boundary" between the
leaning and suspended SiN/Si bilayer, and “edge" in the leaning part of the
patch. b) Strain field in a ($\bar{1}10$) cross-section of the Si layer.
Thickness is magnified ($\times 2$) for visualization purposes. c) Strain
field components in the SOI layer as a function of the depth ($d$ corresponds
to the $z$ direction [$00\bar{1}$]) from the SiN/SOI interface ($d$ = 0) at
three $(x,y)$ points (center, boundary, and edge, as illustrated in panel a).
d) Strain field component in the (001) plane at $d=62.5$ nm. e) Normalized
distributions of implanted carbon ions ($\Phi$) as a function of the depth in
the SOI layer and in the SiO2 BOX for two beam energies, namely 6 keV and 30
keV, obtained by SRIM software[31].

We provide a theoretical description of the elastic field in the strained
patches via finite element method (FEM) calculations. Details of the method
are provided in Section IV.1. This analysis assesses the SOI deformation due
to the action of the SiN stressor, thus allowing for the detailed explanation
of the photoluminescence results, and independently complements the Raman
spectroscopy results reported in the Supplemental Information provided at the
end of this document. Without loss of generality, we focus first on a system
where the length of the side of the membrane ($L$) is 10 $\mu$m (Figure 2).
Moreover, we consider a SiN layer featuring a compressive in-plane deformation
resulting in strain-field components $\varepsilon_{\tilde{x}\tilde{x}}$,
$\varepsilon_{\tilde{y}\tilde{y}}$ and
$\varepsilon_{\tilde{z}\tilde{z}}\equiv\varepsilon_{zz}$ with
$\tilde{x}=[110]$, $\tilde{y}=[1\bar{1}0]$, and $z=\tilde{z}=[001]$
conveniently chosen to discuss effects on the Si crystal unit cell (while the
canonical frame of reference is considered in the following when referring to
$x$,$y$,$z$ directions).

For a flat film configuration, a tetragonal distortion of the stressor would
occur with a linear out-of-plane displacement, no in-plane deformation, and no
strain partitioning [32] (not shown), as actually measured by Raman
spectroscopy for $L=50~{}\mu$m (thus approaching the flat film case, see the
Supplemental Information provided at the end of this document).

For smaller membranes, additional free surfaces introduce extra elastic
relaxation with strain partitioning among the stressor and the substrate, i.e.
the Si layer. In particular, the SiN stressor induces an in-plane strain of
the opposite sign in the Si layer and an out-of-plane strain with the same
sign of the stressor in-plane strain, as observed in general for
heterostructures exhibiting lateral free surfaces [33, 34] (Figure 2b)).

At the center of the structure ("center" in Figure 2a), owing to the small
relative thickness of the SiN/Si bilayer compared to their lateral size, a
film-like configuration is obtained, and all the strain components vary
negligibly along $z$, top panel) and
$\varepsilon_{\tilde{x}\tilde{x}}=\varepsilon_{\tilde{y}\tilde{y}}$ (Figure
2c). As such, an isotropic in-plane deformation is achieved: in the plane of
the patch the symmetry of the Si unit cell is preserved, whereas it is broken
along $z$ under the action of $\varepsilon_{zz}$ that differs from the in-
plane value.

When moving towards the edge of the SiO2 pedestal ("boundary", Figure 2b)-d)),
the crystal unit cells are deformed asymmetrically along $\tilde{x}$ and
$\tilde{y}$, owing to different actions of free surfaces with normal along
these directions (Figure 2b)). Similar asymmetric behavior is obtained in the
free-standing part of the SiN/SOI bilayer. Here, significant variations of the
strain values occur along the vertical direction (Figure 2c), central and
bottom panels).

The deformation close to the edges of the membrane can be interpreted in terms
of the relaxation induced by a stressor on a suspended substrate with lateral
free surfaces, known to lead to a bending of the bilayer [35]. A bending
downwards is expected for the relaxation of an initially flat bilayer with a
compressive strain of the upper layer.

Thus, both in-plane and out-of-plane symmetry-breaking is achieved when moving
far from the center. This generally holds true over the whole square area of
the membrane featuring additional relaxed region at the corners owing to the
proximity of free surfaces with normal along both $\tilde{x}$ and $\tilde{y}$;
see $\varepsilon_{\tilde{x}\tilde{x}}$ and $\varepsilon_{\tilde{z}\tilde{z}}$
in the (001) plane at a depth of 62.5 nm in the Si layer in Figure 2d)
($\varepsilon_{\tilde{y}\tilde{y}}$ would correspond to
$\varepsilon_{\tilde{x}\tilde{x}}$ rotated by 90∘). Therefore, we can conclude
that properties depending on the symmetry breaking along the [001]
crystallographic direction (i.e. the $z$ direction) are then expected to
emerge everywhere. Instead, properties depending on the symmetry breaking
along the $\langle 110\rangle$ directions, are expected to emerge close to and
at the suspended part of the patches.

Note that the G-centers are placed at different depths $d$ along the vertical
z-axis according to the implant energy, with a relatively broad distribution
owing to the implant straggling (Figure 2e)). At the center of the patch, all
the emitters experience the same strain field, irrespective of their implant
depth, offering an ideal scenario for strain engineering of electronic states.
Moreover, for the sizes considered in this investigation, the strain at the
center of the membranes is found to scale as $1/L^{2}$, so the strain can be
well controlled with the lateral side. This information is also supported by
finite element simulation of the strain field as a function of the patch side
(not shown). However, emitters placed far from the center (close to the edge
of the pedestal and in the suspended part) and at different depths would
experience different strains. Proper weighting should then be considered to
account for the effect on the electronic states.

The information gathered with these simulations is used as input for the
piezospectroscopic model used to evaluate the splitting of the ZPLs of the
G-centers in membranes having different lateral sizes $L$. The details of the
piezospectroscopic model are provided in Section IV.1, while the results are
discussed in the following section.

###  II.3  Photoluminescence spectroscopy of strained G-centers

We now address the effect of strain on the G-center ZPL by studying their
emission from SOI patches having different $L$ and topped with SiN in
different conditions. First, we study the emission from the center of patches
(uniaxial symmetry breaking) and later the case of emitters far from the
center (biaxial symmetry breaking). The simple case of flat SOI topped with
SiN (not yet etched in small patches) is analyzed in detail in the
Supplementary Information provided at the end of this document for samples A,
B, and C, and compared to the unstrained case.

Figure 3: Symmetry breaking along [001] direction. a) Photoluminescence
spectra collected at the center of the membrane for different square sides for
sample C. The vertical dotted line highlights the energy of the ZPL emission
in unstrained samples. The experimental data are represented as square dots
whereas the red lines are Gaussian fits. The inset shows the scheme of the
membrane highlighting the position of the excitation/collection spot. b) Time-
resolved photoluminescence of the ZPL for unstrained SOI, HE and LE components
from sample A. The experimental data are represented as square dots whereas
the red line is a mono-exponential decay fit. c) Energy of HE and LE ZPL peaks
for samples A, B, and C as a function of 1/L2. The horizontal dotted line
highlights the energy of the ZPL emission in unstrained samples. Figure 4:
Theoretical energy shifts for [100] symmetry breaking. Calculated $E_{\rm HE}$
and $E_{\rm LE}$ energy components at the central point (x = 0 and y = 0) for
different values of the eigenstrain, $\varepsilon_{0}$ and at a depth of 25
nm, as a function of $1/L^{2}$, with $L$ the side length of the membranes. The
energy of the unstrained ZPL is highlighted. The data are derived by inserting
in Eq. 4 the strain field simulated with FEM and the Bkl coefficients from
Ref. [36]. The dependence with $1/L^{2}$ was obtained by calculating $E_{\rm
HE}$ and $E_{\rm LE}$ at the central point for $L=6$ to 16 $\mu$m and then
fitted for larger values. The trend at a depth of 100 nm (not shown here) is
similar. See more detail on the piezospectroscopic model that is described at
the end of this paper in Section IV.1.

Symmetry breaking along [001] direction. Micro-photoluminescence experiments
at the center of the patch showcase the typical behavior of G-centers under
applied uniaxial stress [37] (Figure 3a)). At this central point, as assessed
by calculations illustrated in Section II.2, the in-plane strain is isotropic
(i.e. $\varepsilon_{xx}=\varepsilon_{yy}$ or, equivalently
$\varepsilon_{\tilde{x}\tilde{x}}=\varepsilon_{\tilde{y}\tilde{y}}$), and the
symmetry breaking of the Si cubic cell occurs only in the vertical direction
[001] under the action of $\varepsilon_{zz}$ that differs from the in-plane
strain. According to Ref. [30], and Eq. 4, a double peak appears in the
spectrum with two lines having similar brightness and an energy separation
$\Delta E_{\rm HE}+\Delta E_{\rm LE}$ that increases for smaller patch sizes.
The splittings in energy between the two new lines as a function of $1/L^{2}$
(with $L$ patch side) are approximately linear for all the samples (Figure
3c)) but with larger slopes for HE ZPLs and smaller slopes for LE ZPLs. Thus,
the largest splitting is observed for the smallest patches and can be as large
as 18 meV.

Time-resolved spectroscopy on both HE and LE ZPLs shows only small deviations
of the recombination lifetime with respect to the unstrained case (Figure 3b))
that is in between 6 and 4 ns. These values are similar to those recently
reported for individual G-centers in isotopically purified 28SOI [8] and for
ensemble in 220 nm thick SOI [6]. Thus, the strain magnitude attained in these
conditions does not modify the recombination dynamics, as also confirmed by
the broadening of the HE and LE split lines (FWHM about 1.2 meV) which is not
far from the unstrained case.

All this phenomenology is representative of all the investigated samples,
irrespective of the nature and magnitude of the applied strain.

The energy shifts calculated at the patches center through the
piezospectroscopic theory (described at the end of this paper in Section
IV.1), using as inputs the strain field obtained by FEM calculations (see
Section II.2), confirm the almost linear dependence of $\Delta E$ with
$1/L^{2}$ (Figure 4): HE and LE ZPLs shift with different slopes (larger for
HE and smaller for LE) as a function of 1/$L^{2}$ as also measured on the
patches and unlike previous reports on bulk samples [37, 38].

From a quantitative point of view, the values of the strain field in the
structure depend on the eigenstrain $\varepsilon_{0}$ (see also Section IV.1),
namely on the magnitude of the strain induced by the SiN layer. A reasonable
agreement between theory and experiments can be found for $\varepsilon_{0}$ =
1$\%$ for the LE split ZPLs, which is consistent with the nominal strain
expected by the fabrication procedure, whereas the measured HE slope is larger
than the prediction. Several factors can lead to discrepancies between the
piezospectroscopic theory and the measured splittings: the model holds for
bulk crystals and does not account for the complexity of the strain field in
ultra-thin patches; measuring the photoluminescence at cryogenic temperature
can further change the overall strain distribution with respect to the FDTD
model reported here owing to the different thermal expansion coefficients of
the SiN stressor, SOI patch, and underlying BOX pedestal. As such, a
quantitative agreement between theory and experiments goes beyond the aim of
this work.

Figure 5:  Symmetry breaking along [001] and [110] direction. a)
Photoluminescence scan from patch side-to-side passing through the center (as
highlighted in the inset). The horizontal, dashed lines highlight the edge of
the SiO2 pedestal. The vertical, red line highlights the ZPL energy of
unstrained G-centers. b) Photoluminescence spectrum at the side of the patch.
The experimental data are shown as squares, whereas the red lines are Gaussian
fits. The insets show a scheme of the membrane and the position of the
excitation/collection spot.

Symmetry breaking along [001] and [110] direction. Moving from the center of
the patch towards its side the Si unit cell is deformed differently along in-
plane directions $\langle 110\rangle$ (i.e.
$\varepsilon_{\tilde{x}\tilde{x}}\neq\varepsilon_{\tilde{y}\tilde{y}}$) with
still $\varepsilon_{\tilde{z}\tilde{z}}\neq 0$ (Figure 2). Moreover, depending
on the position on the membrane and on the depth of the emitters (Table 1),
the overall strain field applied to the G-centers at different depths can be
very different (e.g. $\varepsilon_{\tilde{x}\tilde{x}}$ in the suspended part,
at 25 and 100 nm, can change sign, Figure 2 b)).

This complex strain field across the patches is reflected in the micro-
photoluminescence spectra, as shown in detail for sample B, with G-centers
implanted at a depth of about 25 nm from the surface in Figure 5. In the inner
part (not-leaning, between the vertical dashed lines in Figure 5 a)), close to
the center, the overall picture still resembles the properties of the central
part (splitting in two lines of the ZPL). This points to a dominant role of
symmetry breaking along the [001] direction with respect to other ones. When
moving from the patch center, the two ZPLs undergo a spectral shift to higher
energy under the action of the increasing strain. Moreover, owing to i) the
onset of symmetry breaking in other directions, ii) the changes of the strain
field across the SOI thickness, and iii) the straggling of the ion implant
(Figure 2e)), HE and LE ZPLs undergo a spectral broadening when far from the
center (up to about 1.8 meV). A clear splitting in four components, that is
compatible with symmetry breaking along the [110] direction in addition to
that along the [001], is obtained in the suspended part close to the edge of
the membrane (see also Figure 5b)). A similar picture is found at the angle of
the patches, in their suspended parts, where the ZPL is composite and splits
into four main components (not shown).

Figure 6:  Theoretical energy shift for symmetry breaking along [001] and
[110] directions. The plot showcases the shift of the four, split ZPLs as a
function of position from the patch center to the edge as obtained from the
piezospectroscopic model. We consider a patch having 14 $\mu$m side at a depth
of 25 nm with $\varepsilon_{0}=0.3\%$. The vertical, black, dashed line
highlights the end of the SiO2 pedestal extending up to 5000 nm. The inset
shows the scheme of the patch and the dashed, blue line highlights the
position of the scan along the [110] direction considered in the model.

As in the previous case, this symmetry-breaking picture is supported by the
piezospectroscopic model. Moving from the center towards the patch side, the
strain field is not only composed of diagonal components, and the full
piezospectroscopic Eq. 2 has to be considered. By inserting in this equation
the simulated strain field and considering the four coefficients estimated in
Ref. [36], one obtains different energy shifts (e.g. for a patch with
$L=14~{}\mu$m with C implant at a depth of 25 nm, Figure 6): moving towards
the side of the patch the number of components becomes four as observed in
experiments.

We observe that, using the full piezospectroscopic model, at the patch center
there are three split lines whereas in the experiments only two are visible.
We interpret this as a combination of line broadening (much larger than the
splitting), limited spectral resolution in experiments, and, potentially,
different intensities associated with the split lines [37, 38, 30].

####  II.3.1  Polarization degree of the zero phonon lines

Figure 7: Polarization-resolved photoluminescence. a) Polar plot of the HE and
LE ZPLs from the central point of a membrane. b) Polar plot of the $\alpha$,
$\beta$, $\gamma$ and $\delta$ ZPLs from the side of a patch. The insets show
a scheme of the patch with the excitation/collection spot. All the data are
relative to a patch with $L=12.1~{}\mu$m on sample B.

Polarization-dependent micro-photoluminescence spectra of G-centers are
registered for unstrained and strained samples (Figure 7). For unstrained
G-centers (not covered by SiN), the ZPL is un-polarized, reflecting an even
distribution of the emitting dipoles along the [110] and [1-10], in-plane
directions (not shown) [39]. At the center of the membrane, the LE line shows
a negligible degree of polarization, whereas, for the HE one, it depends on
the applied strain: in sample B, for $L=13.2~{}\mu$m is about 3%, for
$L=12.1~{}\mu$m is 8% and for $L=8.5~{}\mu$m is 50% (here we only report the
case of $L=12.1~{}\mu$m). The polarization direction of the HE line is close
to the [1-1-3] direction, that in our reference frame corresponds to +30
degrees (Figure 7a)).

At the edge of the patch, in the leaning part, un-polarized to almost fully-
polarized ZPLs (up to 70$\%$) are found, with the latter conditions typically
observed for the high-energy component of the quadruplet $\delta$ that is
oriented along the in-plane [12-1] direction (about 115∘ in our reference
frame). Owing to a partial overlap of the split lines and their broadening, in
these latter cases (e.g. as in Figure 6), it is not easy to extract the exact
value of the polarization degree and our estimate is a lower bound. Similar
results are observed at the corners of the membrane where the $\delta$ line
has a polarization degree exceeding 50$\%$ (not shown). These observations
qualitatively agree with previous reports of polarized photoluminescence on
strained G-centers [37, 38, 30].

###  II.4 Photoluminescence enhancement

Figure 8: Emission intensity. Black squares: experimental emission enhancement
from sample B. The data show the spectrally-integrated intensity of an
individual spectrum collected at the center of the SiN/SOI membrane normalized
by its counterpart from the flat areas (the collection spot is determined by
the system lateral resolution and is about 1.1 $\mu$m). Red circles: FDTD
simulations showing the spectrally integrated emission intensity normalized by
its counterpart from the flat SOI.

For sample B, we compare the emission intensity from the center of the patches
with that from the flat SiN/SOI counterpart (black squares in Figure 8). We
observe a clear trend showing an increased emission with a maximum enhancement
of about a factor of 7 whereas, for larger values of $L$, the effect is
reduced.

Provided the negligible impact of strain on the recombination lifetime that
cannot justify the observed enhancement (Figure 3b)) we look for a possible
explanation in a photonic effect springing from the modified environment of
the emitters in the membranes with respect to the flat SiN/SOI system. Through
FDTD simulations (see a description of the simulation in Section IV.1) we
compare the emission of a dipole placed at the center of the patch with
respect to the same dipole in an infinite SiN/SOI (red circles in Figure 7).
The trend of the simulated data corresponds well to the one observed in the
experiments, supporting the idea that the origin of the enhancement resides in
an augmented light extraction springing from the finite size of the patches.

Experiment and simulations display different enhancement factors (larger for
experiments) that can be ascribed to the different conditions considered.
Simulations consider only one dipole laying in the xy plane at a defined
height whereas in rality we observe the emission from an ensemble of dipoles
(excited by a laser spot and detected in a confocal configuration from about
1.1 $\mu$m), featuring several orientations and placed at slightly different
depth in the SOI (Figure 2c)).

##  III  Discussion

The phenomenology presented here is similar to what was shown in early studies
on G-centers [36, 37, 38, 40, 41, 42] and, more generally, to other color
centers that belong to the symmetry group of the first monoclinic type C1h in
cubic crystals [43, 30]. The photoluminescence splitting in a doublet that we
find at the center of the patches mimics the effect of uniaxial strain along
the [001] direction (no in-plane symmetry breaking). At the side of the
patches, in the suspended part, where the Si cell symmetry is lifted also
along the [110] direction, four split lines appear. These experimental
findings qualitatively agree with predictions obtained through the
piezospectroscopic model developed for strained cubic crystals [30]. We showed
this control with a very simple micro-structure, avoiding the use of bulky
samples and experimental setups making use of hydraulic pistons to apply
stress to the samples [37, 38]. It is also worth pointing out that, beyond the
present proof of principle that was limited to relatively large patches ($L$
up to about 8 $\mu$m), the reduction of $L$ to about 1 $\mu$m m with
conventional etching methods, or even below this value (e.g. exploiting solid
state dewetting [10, 44]), is possible. This could lead, potentially, to a
stress that is more than one order of magnitude larger than what we show here,
achieving strain regimes not attained so far.

The importance of our results in the renewed context of quantum technologies
with light-emitters in silicon is two-fold: 1) the possibility to control the
splitting in a large range of energies (equivalent to a stress up to about 0.6
GPa [37, 38]), allowing to tune the emission and, 2) the possibility to
obtain, even from the ensemble, photoluminescence lines with a very large
degree of linear polarization (as also reported in previous studies [37, 38,
30]), demonstrating that an ensemble of emitters shares the same energy and
the same dipole orientation.

The straggling of the ion implant might play a detrimental role for exploiting
these emitters as quantum devices. On one hand, symmetry breaking along the
[001] direction at the center of the membrane entails a uniform strain
landscape in each of the three spatial directions. As such, whatever its
position along z, an emitter will undergo the same splitting, shift, and
polarization orientation as all the others. On the other end, for strain along
the [110] direction (far from the center), the stress strongly depends on the
position along the vertical z-axis: emitters created at different depth would
undergo a different energy shift and splitting. Thus, in view of obtaining
identical emitters (e.g. for quantum communication protocols), a reduction of
struggling might be a hard requirement. This observation is also important for
coupling the emitters to a localized photonic resonance [19]. Similar
considerations hold for the high-temperature annealing step (required for re-
crystallization after implant), which might spread the emitters over long
distances along the vertical z-axis. However, in our case, where 1000 ∘C
annealing for only 20 seconds was used, we do not expect a relevant re-
distribution of carbon. In fact, even for strong changes of the strain profile
along the SOI thickness (e.g. at the patch sides), for low-energy implants at
about 25 nm depth, where the straggling is rather limited and all the emitters
are within the first 50 nm from the surface, the photoluminescence showcases
rather sharp ZPLs: the measured line-widths in sample B at the edge of the
patch is 1-2 meV, not far from 0.9 meV found in the flat, non -strained
counterpart. These observations indirectly confirm that for low-energy implant
annealing at 1000 ∘C for 20 seconds does not affect too much the position of
the implanted C ions in the SOI, unlike what was recently suggested based on
theoretical simulations [19].

A potential element of concern for strained emitters could be the change in
the underlying oscillator strength, impacting their brightness. Although a
complete understanding of these features is beyond the aim of this work, we
note that, from time-resolved photoluminescence experiments, the lifetime is
weakly affected by the strain. Moreover, when monitoring the overall emission
intensity, we observe larger intensity values from the patches with respect to
the flat areas. For smaller patches, this enhancement is almost one order of
magnitude and it might be even larger for smaller ones. This observation is
important in the context of photoluminescence-intensity correlation with
individual emitters (e.g. with Hanbury Brown and Twiss or Hong-Ou-Mandel
interferometers), where light extraction is crucial for shortening the
integration time of a measurement. In spite of its simplicity and lack of
Purcell effect, our method which uses only low-resolution optical lithography,
allows for a relevant increase in the collection efficiency.

The merit of our approach that relies on SOI lithography and deposition of a
stressor film, is its compatibility with other devices, such as waveguides
[18, 19, 24, 25] and Mie resonators [10, 22, 23], that are relevant to all the
other quantum emitters in Si [9, 17, 5, 7, 18, 8]. Our method can be
generalized using suspended bridges providing uni-axial and bi-axial strain in
arbitrary, in-plane directions [45, 46, 47, 48, 49]. Similar results can be
obtained by deposition of stressors [50] or by deposition of Si atop a
strained substrate (e.g. made of SiGe) [51] opening the way to finely control
the position of the ZPL of the emitters (e.g. with respect to a photonic
resonance of a resonant cavity). For instance, this possibility is central for
tuning several independent sources to the same energy in order to produce
coalescent photons, as shown for III-V-based quantum-emitters [52]. In the
same context, the evidence of an almost fully-polarized line measured on the
ensemble is a promising signature for aligning all the emitting dipoles along
the same axis. For unstrained G-centers in fact, the dipole orientation can be
aligned along the crystallographic directions [110], [101], and [011],
depending on the position of the self-interstitial Si atom [39] providing
unpolarized light from ensemble emission. Our method suggests the possibility
to engineer the polarization axes, providing at least one split ZPL with the
same orientation for all the emitters.

Finally, the possibility to control the degeneracy of the electronic level in
G-centers, and more generally in this class of impurities in silicon, might be
relevant for the manipulation of their spin features. Symmetry breaking by
application of strain could be the way to split the metastable triplet state
of the G-center [53, 54] and thus encode spin-based quantum bits [55, 56, 57,
58]. The fast recombination time of G-centers with respect to similar emitters
in Si (e.g. the T-center [20, 21, 22]) would provide a more efficient read-out
of the spin degree of freedom providing a spin-photon interface.

##  IV Conclusions

In conclusion, we showed that the splitting of the zero-phonon-line of the
G-centers ensemble can be controlled producing doublets and quadruplets having
large splittings, up to 18 meV, that is more than one order of magnitude
larger than the broadening of the photoluminescence peaks. The splitting
amplitude can be controlled by selecting a specific patch size. Symmetry
breaking along [001] and [110] can be obtained respectively a the center of
square patches and in their suspended parts. The merit of our approach is its
simplicity, as it relies on conventional SiN deposition (chemical vapor
deposition) and lithography (optical lithography and plasma etching). It can
be further engineered by changing patch geometry and orientation with respect
to the crystallographic axes, SiN thickness and, most importantly, it can be
easily extended to most common photonic devices. It will be possible to apply
the same principle to other color centers in Si and access the fine features
of the underlying recombination dynamics allowing to exploit of the spin
degree of freedom for quantum technologies.

###  IV.1 Methods

Fabrication of light-emitting, strained membranes. The sample fabrication is
described in Figure 1 and the corresponding features are detailed in Table 1.

The samples are obtained by dicing a 125 nm thick SOI laying atop a 2 $\mu$m
thick buried oxide (BOX) on bulk Si (from Soitec) in 1 cm $\times$ 1 cm parts.
They were first implanted with carbon ions at a depth of 25 nm (samples A and
B) and 100 nm (sample C) following a well-established procedure [59, 6]. After
the implant, all the samples were flash-annealed in N2 atmosphere for 20 s at
1000∘ C to cure the radiation damages.

Each sample was patterned by optical lithography following this process:
\- Chemical cleaning in a sonic bath, first with acetone and then with
ethanol;
\- Spin-coating of positive photo-resist (MICROPOSIT S1813) at 3000 rpm
forming a layer of about 1.5 $\mu$m;
-Photo-lithography via UV laser system (Dilase 250 by Cloe). The pattern used for photoluminescence spectroscopy of strained G-centers consists of square patches whose size is tuned from about 15 to 5 $\mu$m;
\- Resist development (using a solution from MICROPOSIT MF-319) to remove the
parts that were not exposed to the UV laser;
\- Pattern etching with a CF4 plasma in a plasma-enhanced chemical vapor
deposition chamber (PE-CVD, from Oxford);
\- Sample cleaning with acetone;
\- Partial etching of the BOX by immersion in an aqueous solution containing
10% of Buffered Oxide Etch (BOE 10:1) to form partially-suspended membranes;
\- Deposition of a $\sim 100~{}$nm thick SiN layer via PE-CVD. Three
depositions are performed using different parameters changing the nature of
the strain (tensile or compressive) applied to the SOI patches (Table 1). This
is done by tuning the time percentage of high- and low-frequency plasma
(HF$\%$, which represents the fraction of time in which the high-frequency
pulse is turned on with respect to the total pulse time expressed as a
percentage). Changing HF$\%$ results in a compressive (${\rm HF}<60\%$),
tensile ($\rm HF>65\%$) or almost unstrained ($\rm 60\%<HF<65\%$) SiN [60].
Ellipsometry (ESM-300 by Wollam) was systematically used to measure the
thickness of the SiN layers deposited atop the SOI. The strained SiN affects
the SOI patches resulting in a bent membrane, upwards for tensile strain, or
downwards for compressive strain (Figure 1c));

\- Implant with protons, to form the G-centers by inducing the interstitial Si
[61]. The beam energy was set to 90 keV in order to stop the protons within
the BOX, avoiding the activation of unwanted emitters in the SOI or in the
underlying bulk Si.

Raman spectroscopy setup. Micro-Raman spectroscopy is performed at room
temperature and is used to assess the stress in the flat samples and in the
SOI patches. The setup consists of an Horiba-Jobin Yvon HR800-UV Raman
spectrometer mounting a 1800 gpmm grating. It is characterized by a spectral
resolution of about 0.5 cm-1. The dispersed signal is then collected with a
Jobin Yvon Synapse Si-based CCD camera. The samples are placed on a platform
equipped with a $x-y$ translation stage to scan the sample surface and
characterized by a spatial precision of about 250 nm. The samples are excited
by a He:Ne laser ($\lambda=632.8$ nm) through a 50$\times$ objective lens
(Olympus SLMPLN50X) with a numerical aperture NA $=0.35$ mounted on a
metallographic microscope (from Olympus). The Raman signal from the sample is
collected via the same objective in a back-scattering configuration.

Micro-photoluminescence spectroscopy setup. The micro-photoluminescence
spectra are detected with a custom-made confocal microscope setup in a
backscattering configuration. The samples are placed in a low-vibration, He-
flow cryostat (Janis ST-500), mounted on a $x-y$ translation stage (Physik
Instruments) to scan the sample surface with fine control on the position of
about 250 nm. The samples’ temperature is fixed at about $10~{}$K.

The excitation sources are a CW diode-pumped solid-state laser emitting at 532
nm (CNI MLL-III-532) and a mode-locked Ti:Sapphire tunable laser (Spectra
Physics Tsunami, 700-900 nm spectral range, 200 fs pulse duration, 12.2 ns
pulse period).

The photoluminescence from the samples is collected by an infinity-corrected
Mitutoyo 100$\times$ objective lens (NIR, $\mathrm{NA}=0.7$), separated from
the excitation by a dichroic mirror, spectrally dispersed by a spectrograph
(Acton SP2300i) mounting a 600 gr/mm grating. The photoluminescence is
detected by an InGaAs array (Princeton Instruments OMA V-512) kept at
$-100^{\circ}~{}$C by a cold finger immersed in liquid nitrogen.

The spatial resolution of the micro-photoluminescence setup is about 1.1
$\mu$m, and the spectral resolution is about 350 $\mu$ eV.

Time-resolved PL (TR-PL) measurements were performed using the time-correlated
single photon counting (TCSPC) technique using an InGaAs/InP APD (ID Quantique
ID230) and a time correlator (ID Quantique ID900) interfaced with the PC. The
time resolution of the system is about 200 ps.

Polarization measurements are performed by placing a rotatable half-wave plate
and a fixed polarizer along the collection path.

Finite element method simulations of strain distribution. The strain
distribution in the system is computed by 3D Finite Element Method (FEM)
calculations based on the linear elasticity theory [47, 62] exploiting the
commercial FEM package COMSOL Multiphysics. To model the experimental
structure under investigation, we consider a square bilayer formed by a SiN
layer of 100 nm atop a 125 nm thick silicon layer. This layer is supported at
the center by a SiO2 pillar with a square base. The whole structure is placed
on a silicon substrate. The suspended part of the SiN/Si bilayer, from its
edge to the edge of the SiO2 pedestal, is 2 $\mu$m (independently of the
lateral extension of the membrane). Simulations are performed for structures
with a center-to-edge extension of the SiN on SOI ranging between 6 and 16
$\mu$m whereas the bottom SiO2 (see Figure 2a), mimicking the geometry shown
in Figure 1b) and c)).

The elastic field is then computed by the FEM simulations to satisfy the
mechanical equilibrium condition (without external forces),
$\nabla\cdot\bm{\sigma}=0$, the free surface boundary condition,
$\bm{\sigma}\cdot\widehat{\mathbf{n}}=0$ with $\widehat{\mathbf{n}}$ the
normal to the free surface(s), and the Dirichlet boundary condition
$\mathbf{u}=0$ with $\mathbf{u}$ the displacement field for the silicon
substrate similarly to previous investigations of mechanical stress in
heteroepitaxial systems [34, 63].

The stress field is defined as
$\bm{\sigma}=\mathbf{C}:(\bm{\varepsilon}-\varepsilon_{0}\mathbf{I})$ with
$\bm{\varepsilon}=(1/2)(\nabla\mathbf{u}+(\nabla\mathbf{u})^{\rm T})$ the
strain field and $\varepsilon_{0}$ the eigenstrain [64], namely minus the
mismatch of the SiN layer with respect to the Si layer induced by the growth
process. $\varepsilon_{0}$ is found to vary according to the fabrication
condition in the range $\pm 1\%$. The elastic constant tensor $\mathbf{C}$ is
assumed to be isotropic for the purposes of this work. Expressed in terms of
Young modulus (E) and Poisson ratio ($\nu$) they are: $E_{\rm Si}$=$130\,{\rm
GPa}$, $\nu_{\rm Si}$=$0.27$, $E_{\rm SiN}$=$250\,{\rm GPa}$, $\nu_{\rm
SiN}$=$0.23$, $E_{\rm SiO_{2}}$=$70\,{\rm GPa}$, $\nu_{\rm SiO_{2}}$=$0.17$.

We explicitly consider only the case of a SiN layer under compressive strain.
However, the resulting strain distributions scale linearly with
$\varepsilon_{0}$. Therefore, as we are interested in the ratios between
different strain components, their distributions are provided in relative
units w.r.t such an eigenstrain.

Theoretical piezospectroscopic model. The behavior of non-cubic color centers
in cubic crystals can be described by adopting a piezospectroscopic method
[30, 65, 38] coupling the observed energy splitting of the center spectral
lines with the strain field. The transition energy shift in the emission of a
center, $\Delta E$, can be written as:

$\Delta E=\sum_{k,l}B_{kl}\varepsilon_{kl}$ (1)

where $\varepsilon_{kl}$ are the strain field components (with $kl$ pointing
at the canonical frame of reference), and $B_{kl}$ some coefficients which
form a second rank symmetric tensor [30]. Both experimental [37, 36, 54] and
first principles studies [54, 66, 39] have clearly demonstrated that
G-centers, which consist of two substitutional carbons and one interstitial
silicon atom, behave as first-type monoclinic centers (C1h) with an axis along
the [110] direction. In this case, the number of coefficients $B_{kl}$ is
reduced to four, and Eq. 1 can be rewritten as:

$\Delta
E=B_{1}\varepsilon_{zz}+B_{2}(\varepsilon_{xx}+\varepsilon_{yy})+2B_{3}\varepsilon_{xy}+2B_{4}(\varepsilon_{yz}-\varepsilon_{xz})$
(2)

Depending on the particular strain field, Eq. 2 can be further simplified.
More specifically, in the case of a tensile or compressive deformation along
the [100] direction, the number of split spectral lines is two, whereas it
corresponds to four when the deformation is along the [110] direction. In the
first case, Eq. 2 becomes:

$\Delta E=B_{1}\varepsilon_{zz}+B_{2}(\varepsilon_{xx}+\varepsilon_{yy})$ (3)

Assuming that the energy separation $\Delta E$ is composed by an high energy
shift, $\Delta E_{\rm HE}$, and a low energy shift, $\Delta E_{\rm LE}$
(corresponding to the two split spectral lines) Eq. 4 can be written as:

$\Delta E_{\rm HE}+\Delta E_{\rm
LE}=B_{1}\varepsilon_{zz}+B_{2}(\varepsilon_{xx}+\varepsilon_{yy})$ (4)

if this equality holds, then:

$\begin{split}\Delta E_{\rm HE}&=E_{\rm HE}-E_{0}=B_{1}\varepsilon_{zz}\\\
\Delta E_{\rm LE}&=E_{\rm
LE}-E_{0}=B_{2}(\varepsilon_{xx}+\varepsilon_{yy})\\\ \end{split}$ (5)

where $E_{0}$ is the ZPL (0.969 eV). Eqs. 5 can be finally written as:

$\begin{split}E_{\rm HE}&=B_{1}\varepsilon_{zz}+E_{0}\\\ E_{\rm
LE}&=B_{2}(\varepsilon_{xx}+\varepsilon_{yy})+E_{0}\\\ \end{split}$ (6)

On the other hand, when the deformation is along the [110] direction, the non-
diagonal components of the strain tensor cannot be neglected. In this case,
the number of split lines is four, and Eq. 2 has to be fully taken into
account.

We employed both versions of Eq. 2 (full and simplified one) to theoretically
evaluate the shift of the G-center ZPL moving from the center to the side of
the patches. The strain field, $\varepsilon_{kl}$, is computed by 3D FEM
calculations as described above while $B_{kl}$ coefficients are taken from
previous works on [100] and [110] strained cubic Si [36]. However, in our
study, Eq. 2 assumes a slightly more complex form. Indeed, as is shown in
Figure 2, the strain field $\varepsilon_{kl}$ is a function of the distance
from the center of the patch, $d$. As a consequence, Eq. 2 becomes:

$\begin{split}\Delta
E(d)=&B_{1}\varepsilon_{zz}(d)+B_{2}[\varepsilon_{xx}(d)+\varepsilon_{yy}(d)]~{}+\\\
&+~{}2B_{3}\varepsilon_{xy}(d)+2B_{4}[\varepsilon_{yz}(d)-\varepsilon_{xz}(d)]\end{split}$
(7)

and, in the same way, Eqs. LABEL:energy_shift_100_ELE can be written as:

$\begin{split}E_{\rm HE}(d)&=B_{1}\varepsilon_{zz}(d)+E_{0}\\\ E_{\rm
LE}(d)&=B_{2}[\varepsilon_{xx}(d)+\varepsilon_{yy}(d)]+E_{0}\end{split}$ (8)

In this way, we build a predictive model to prove that the measured ZPL energy
shifts are due to the difference in the strain field between the center and
the side of the patches. Moreover, such a model can also demonstrate that the
dependence of $\Delta E$ on $1/L^{2}$ (where $L$ is the patch side) is linear.

Finite Difference Time Domain simulations of light emission. Calculations of
light emission from the etched patches were performed by Finite Difference
Time Domain (FDTD) method employing the commercial software Lumerical. To
simulate the emission from a single G-center, an x-polarized, spectrally
narrow, light-emitting electric dipole is used, centered around 0.970 eV with
a pulse length of 3000 fs. We simulate two kinds of systems: 1) an infinite
multilayer structure composed by (from top to bottom) a 100 nm thick SiN
layer, a 125 nm thick SOI layer, and a 2 $\mu$m thick SiO2 layer atop a Si
bulk. 2) Square patches of different sides are modeled analogously to the one
used in FEM simulations described before, except made for the strain effect
which is not included in these FDTD simulation domain. In both cases (infinite
multilayer and finite size membranes) the dipole is positioned at 25 nm from
the upper surface of the SOI layer (at the center of the membrane for the
finite patch cases). A 2D monitor geometrically configured to reproduce an
acquisition of NA = 0.7, is positioned atop of the multilayered structures and
collects the signal coming from the system. The intensity is then calculated
by integrating the emission over the monitor area (12 $\mu$m $\times$ 12
$\mu$m). For the sake of thoroughness, we also studied different
configurations by changing the monitor distance from the multilayer and
adjusting the monitor size in order to reproduce the experimental NA of 0.7.
No significant differences were highlighted, and this analysis is not
reported.

## Funding information

This research was funded by the EU H2020 FET-OPEN project NARCISO (No.
828890), the French National Research Agency (ANR) through the projects
ULYSSES (No. ANR-15-CE24-0027-01) and OCTOPUS (No. ANR-18-CE47-0013-01), and
funding from the European Union - Next Generation EU (PNRR) through the
research project National Quantum Science and Technology Institute (NQSTI).
M.B., F.I., and N.G. acknowledge the project PNRR MUR project PE0000023 -
NQSTI. M.A. acknowledges the ANR AMPHORE project (ANR-21-CE09-0007) of the
French Agence Nationale de la Recherche. F.B. acknowledges Fondazione Cassa di
Risparmio di Firenze for funding this work within the projects Photonic Future
2021.1508 and PUPO (co-funded by the University of Florence and the Italian
Ministry of University and Research).

## Acknowledgements

We acknowledge the Nanotecmat platform of the IM2NP Institute of Marseille.

## Conflict of interest

The authors declare no conflict of interest

## References

  * Davies  _et al._ [2006] G. Davies, S. Hayama, L. Murin, R. Krause-Rehberg, V. Bondarenko, A. Sengupta, C. Davia, and A. Karpenko, Physical Review B 73, 165202 (2006).
  * Chartrand  _et al._ [2018] C. Chartrand, L. Bergeron, K. Morse, H. Riemann, N. Abrosimov, P. Becker, H.-J. Pohl, S. Simmons, and M. Thewalt, Physical Review B 98, 195201 (2018).
  * Khoury and Abbarchi [2022] M. Khoury and M. Abbarchi, Journal of Applied Physics 131, 200901 (2022).
  * Lohrmann  _et al._ [2017] A. Lohrmann, B. Johnson, J. McCallum, and S. Castelletto, Reports on Progress in Physics 80, 034502 (2017).
  * Baron  _et al._ [2022a] Y. Baron, A. Durand, P. Udvarhelyi, T. Herzig, M. Khoury, S. Pezzagna, J. Meijer, I. Robert-Philip, M. Abbarchi, J.-M. Hartmann, _et al._ , ACS Photonics 9, 2337 (2022a).
  * Beaufils  _et al._ [2018] C. Beaufils, W. Redjem, E. Rousseau, V. Jacques, A. Y. Kuznetsov, C. Raynaud, C. Voisin, A. Benali, T. Herzig, S. Pezzagna, J. Meijer, M. Abbarchi, and G. Cassabois, Physical Review B 97, 035303 (2018).
  * Durand  _et al._ [2021] A. Durand, Y. Baron, W. Redjem, T. Herzig, A. Benali, S. Pezzagna, J. Meijer, A. Y. Kuznetsov, J.-M. Gérard, I. Robert-Philip, M. Abbarchi, V. Jacques, G. Cassabois, and A. Dreau, Physical Review Letters 126, 083602 (2021).
  * Baron  _et al._ [2022b] Y. Baron, A. Durand, T. Herzig, M. Khoury, S. Pezzagna, J. Meijer, I. Robert-Philip, M. Abbarchi, J.-M. Hartmann, S. Reboh, J.-M. Gérard, V. Jacques, G. Cassabois, and A. Dréau, Applied Physics Letters 121, 084003 (2022b), https://doi.org/10.1063/5.0097407 .
  * Redjem  _et al._ [2020] W. Redjem, A. Durand, T. Herzig, A. Benali, S. Pezzagna, J. Meijer, A. Y. Kuznetsov, H. Nguyen, S. Cueff, J.-M. Gérard, _et al._ , Nature Electronics 3, 738 (2020).
  * Khoury  _et al._ [2022] M. Khoury, H. Quard, T. Herzig, J. Meijer, S. Pezzagna, S. Cueff, M. Abbarchi, H. S. Nguyen, N. Chauvin, and T. Wood, Advanced Optical Materials 10, 2201295 (2022).
  * Lo  _et al._ [2009] C. C. Lo, A. Persaud, S. Dhuey, D. Olynick, F. Borondics, M. C. Martin, H. A. Bechtel, J. Bokor, and T. Schenkel, Semiconductor science and technology 24, 105022 (2009).
  * Itoh and Watanabe [2014] K. M. Itoh and H. Watanabe, MRS communications 4, 143 (2014).
  * Becker  _et al._ [2006] P. Becker, D. Schiel, H. Pohl, A. Kaliteevski, O. Godisov, M. Churbanov, G. Devyatykh, A. Gusev, A. Bulanov, S. Adamchik, _et al._ , Measurement Science and Technology 17, 1854 (2006).
  * Maurand  _et al._ [2016] R. Maurand, X. Jehl, D. Kotekar-Patil, A. Corna, H. Bohuslavskyi, R. Laviéville, L. Hutin, S. Barraud, M. Vinet, M. Sanquer, _et al._ , Nature communications 7, 1 (2016).
  * Mazzocchi  _et al._ [2019] V. Mazzocchi, P. Sennikov, A. Bulanov, M. Churbanov, B. Bertrand, L. Hutin, J. Barnes, M. Drozdov, J. Hartmann, and M. Sanquer, Journal of Crystal Growth 509, 1 (2019).
  * Sabbagh  _et al._ [2019] D. Sabbagh, N. Thomas, J. Torres, R. Pillarisetty, P. Amin, H. George, K. Singh, A. Budrevich, M. Robinson, D. Merrill, _et al._ , Physical Review Applied 12, 014013 (2019).
  * Hollenbach  _et al._ [2020] M. Hollenbach, Y. Berencén, U. Kentsch, M. Helm, and G. V. Astakhov, Optics Express 28, 26111 (2020).
  * Prabhu  _et al._ [2022] M. Prabhu, C. Errando-Herranz, L. De Santis, I. Christen, C. Chen, and D. R. Englund, arXiv preprint arXiv:2202.02342 (2022).
  * Komza  _et al._ [2022] L. Komza, P. Samutpraphoot, M. Odeh, Y.-L. Tang, M. Mathew, J. Chang, H. Song, M.-K. Kim, Y. Xiong, G. Hautier, _et al._ , arXiv preprint arXiv:2211.09305 (2022).
  * Bergeron  _et al._ [2020a] L. Bergeron, C. Chartrand, A. Kurkjian, K. Morse, H. Riemann, N. Abrosimov, P. Becker, H.-J. Pohl, M. Thewalt, and S. Simmons, Physical Review X Quantum 1, 020301 (2020a).
  * Bergeron  _et al._ [2020b] L. Bergeron, C. Chartrand, A. Kurkjian, K. Morse, H. Riemann, N. Abrosimov, P. Becker, H.-J. Pohl, M. Thewalt, and S. Simmons, arXiv preprint arXiv:2006.08794 (2020b).
  * Higginbottom  _et al._ [2022] D. B. Higginbottom, A. T. Kurkjian, C. Chartrand, M. Kazemi, N. A. Brunelle, E. R. MacQuarrie, J. R. Klein, N. R. Lee-Hone, J. Stacho, M. Ruether, _et al._ , Nature 607, 266 (2022).
  * Hollenbach  _et al._ [2022a] M. Hollenbach, N. S. Jagtap, C. Fowley, J. Baratech, V. Guardia-Arce, U. Kentsch, A. Eichler-Volf, N. V. Abrosimov, A. Erbe, C. Shin, _et al._ , Journal of Applied Physics 132, 033101 (2022a).
  * DeAbreu  _et al._ [2022] A. DeAbreu, C. Bowness, A. Alizadeh, C. Chartrand, N. Brunelle, E. MacQuarrie, N. Lee-Hone, M. Ruether, M. Kazemi, A. Kurkjian, _et al._ , arXiv preprint arXiv:2209.14260 (2022).
  * Lefaucher  _et al._ [2022] B. Lefaucher, J.-B. Jager, V. Calvo, A. Durand, Y. Baron, F. Cache, V. Jacques, I. Robert-Philip, G. Cassabois, T. Herzig, _et al._ , arXiv preprint arXiv:2210.05485 (2022).
  * Errando Herranz  _et al._ [2022] C. Errando Herranz, C. Gerlach, L. De Santis, C. Panuski, M. Prabhu, H. Raniwala, I. Christen, and D. Englund, Bulletin of the American Physical Society (2022).
  * Redjem  _et al._ [2023] W. Redjem, Y. Zhiyenbayev, W. Qarony, V. Ivanov, C. Papapanos, W. Liu, K. Jhuria, Z. A. Balushi, S. Dhuey, A. Schwartzberg, L. Tan, T. Schenkel, and B. Kanté, All-silicon quantum light source by embedding an atomic emissive center in a nanophotonic cavity (2023).
  * Hollenbach  _et al._ [2022b] M. Hollenbach, N. Klingner, N. S. Jagtap, L. Bischoff, C. Fowley, U. Kentsch, G. Hlawacek, A. Erbe, N. V. Abrosimov, M. Helm, _et al._ , Nature Communications 13, 1 (2022b).
  * Higginbottom  _et al._ [2023] D. B. Higginbottom, F. K. Asadi, C. Chartrand, J.-W. Ji, L. Bergeron, M. L. Thewalt, C. Simon, and S. Simmons, PRX Quantum 4, 020308 (2023).
  * Kaplyanskii [1967] A. Kaplyanskii, Le Journal de Physique Colloques 28, C4 (1967).
  * Ziegler  _et al._ [2010] J. F. Ziegler, M. D. Ziegler, and J. P. Biersack, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 268, 1818 (2010).
  * Ayers  _et al._ [2016] J. E. Ayers, T. Kujofsa, P. Rago, and J. Raphael,  _Heteroepitaxy of semiconductors: theory, growth, and characterization_ (CRC press, 2016).
  * Glas [2006] F. Glas, Phys. Rev. B 74, 121302 (2006).
  * Salvalaglio and Montalenti [2014] M. Salvalaglio and F. Montalenti, Journal of Applied Physics 116, 104306 (2014).
  * Freund  _et al._ [1999] L. B. Freund, J. A. Floro, and E. Chason, Applied Physics Letters 74, 1987 (1999).
  * Tkachev and Mudryi [1978] V. Tkachev and A. Mudryi, Journal of Applied Spectroscopy 29, 1485 (1978).
  * Foy  _et al._ [1981] C. P. Foy, M. C. do Carmo, G. Davies, and E. C. Lightowlers, Journal of Physics C: Solid State Physics 14, L7 (1981).
  * Thonke  _et al._ [1981] K. Thonke, H. Klemisch, J. Weber, and R. Sauer, Physical Review B 24, 5874 (1981).
  * Udvarhelyi  _et al._ [2021] P. Udvarhelyi, B. Somogyi, G. Thiering, and A. Gali, Physical review letters 127, 196402 (2021).
  * Yukhnevich [1965a] A. Yukhnevich, Sov. Phys. Solid State 7, 322 (1965a).
  * Yukhnevich [1965b] A. Yukhnevich, Fiz. Tverd. Tela 7 (1965b).
  * Yuknevich and Mudryi [1973] A. V. Yuknevich and A. V. Mudryi, Fiz. Tekh. Poluprovdn 7, 1215 (1973, [Sov. Phys. Semicond. 7, 815 (1973)).
  * Baumann  _et al._ [1967] G. Baumann, F. Lanzl, W. Von Der Osten, and W. Waidelich, Le Journal de Physique Colloques 28, C4 (1967).
  * Toliopoulos  _et al._ [2020] D. Toliopoulos, M. Khoury, M. Bouabdellaoui, N. Granchi, J.-B. Claude, A. Benali, I. Berbezier, D. Hannani, A. Ronda, J. Wenger, M. Bollani, M. Gurioli, S. Sanguinetti, F. Intonti, and M. Abbarchi, Optics Express 28, 37734 (2020).
  * Ureña  _et al._ [2013] F. Ureña, S. H. Olsen, and J.-P. Raskin, Journal of Applied Physics 114, 144507 (2013).
  * Süess  _et al._ [2013] M. J. Süess, R. Geiger, R. Minamisawa, G. Schiefler, J. Frigerio, D. Chrastina, G. Isella, R. Spolenak, J. Faist, and H. Sigg, Nature Photonics 7, 466 (2013).
  * Bollani  _et al._ [2015] M. Bollani, D. Chrastina, L. Gagliano, L. Rossetto, D. Scopece, M. Barget, V. Mondiali, J. Frigerio, M. Lodari, F. Pezzoli, _et al._ , Applied Physics Letters 107, 083101 (2015).
  * Barget  _et al._ [2016] M. R. Barget, M. Lodari, M. Borriello, V. Mondiali, D. Chrastina, M. Bollani, and E. Bonera, Applied Physics Letters 109, 133109 (2016).
  * Gassenq  _et al._ [2017] A. Gassenq, S. Tardif, K. Guilloy, I. Duchemin, N. Pauc, J. M. Hartmann, D. Rouchon, J. Widiez, Y. M. Niquet, L. Milord, _et al._ , Journal of Applied Physics 121, 055702 (2017).
  * Scopece  _et al._ [2014] D. Scopece, F. Montalenti, M. Bollani, D. Chrastina, and E. Bonera, Semiconductor Science and Technology 29, 095012 (2014).
  * David  _et al._ [2020] T. David, I. Berbezier, J.-N. Aqua, M. Abbarchi, A. Ronda, N. Pons, F. Domart, P. Costaganna, G. Uren, and L. Favre, ACS Applied Materials & Interfaces 13, 1807 (2020).
  * Grim  _et al._ [2019] J. Q. Grim, A. S. Bracker, M. Zalalutdinov, S. G. Carter, A. C. Kozen, M. Kim, C. S. Kim, J. T. Mlack, M. Yakes, B. Lee, _et al._ , Nature materials 18, 963 (2019).
  * O’donnell  _et al._ [1983] K. O’donnell, K. Lee, and G. Watkins, Physica B+ C 116, 258 (1983).
  * Ivanov  _et al._ [2022] V. Ivanov, J. Simoni, Y. Lee, W. Liu, K. Jhuria, W. Redjem, Y. Zhiyenbayev, C. Papapanos, W. Qarony, B. Kanté, _et al._ , Physical Review B 106, 134107 (2022).
  * Arroyo-Camejo  _et al._ [2014] S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasubramanian, Nature communications 5, 1 (2014).
  * Zu  _et al._ [2014] C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, Nature 514, 72 (2014).
  * Foletti  _et al._ [2009] S. Foletti, H. Bluhm, D. Mahalu, V. Umansky, and A. Yacoby, Nature Physics 5, 903 (2009).
  * Wu  _et al._ [2014] X. Wu, D. R. Ward, J. Prance, D. Kim, J. K. Gamble, R. Mohr, Z. Shi, D. Savage, M. Lagally, M. Friesen, _et al._ , Proceedings of the National Academy of Sciences 111, 11938 (2014).
  * Berhanuddin  _et al._ [2012a] D. D. Berhanuddin, M. A. Lourenço, R. M. Gwilliam, and K. P. Homewood, Advanced Functional Materials 22, 2709 (2012a).
  * Karouta  _et al._ [2012] F. Karouta, K. Vora, J. Tian, and C. Jagadish, Journal of Physics D: Applied Physics 45, 445301 (2012).
  * Berhanuddin  _et al._ [2012b] D. D. Berhanuddin, M. A. Lourenço, R. M. Gwilliam, and K. P. Homewood, Advanced Functional Materials 22, 2709 (2012b).
  * Landau  _et al._ [1986] L. D. Landau, E. M. Lifšic, E. M. Lifshitz, A. M. Kosevich, and L. P. Pitaevskii,  _Theory of elasticity: volume 7_ , Vol. 7 (Elsevier, 1986).
  * Bergamaschini  _et al._ [2016] R. Bergamaschini, M. Salvalaglio, R. Backofen, A. Voigt, and F. Montalenti, Advances in Physics: X 1, 331 (2016).
  * Kinoshita and Mura [1971] N. Kinoshita and T. Mura, Physica status solidi (a) 5, 759 (1971).
  * Hughes [1966] A. Hughes, Proc. Phys. Soc. 87, 535 (1966).
  * Timerkaeva  _et al._ [2018] D. Timerkaeva, C. Attaccalite, G. Brenet, D. Caliste, and P. Pochet, J. Appl. Phys. 123, 161421 (2018).
  * Ogura  _et al._ [2009] A. Ogura, H. Saitoh, D. Kosemura, Y. Kakemura, T. Yoshida, M. Takei, T. Koganezawa, I. Hirosawa, M. Kohno, T. Nishita, _et al._ , Electrochemical and Solid-State Letters 12, H117 (2009).
  * Suess  _et al._ [2014] M. Suess, R. Minamisawa, R. Geiger, K. Bourdelle, H. Sigg, and R. Spolenak, Nano letters 14, 1249 (2014).

## Supplemental Information

###  S0.2  Raman spectroscopy of SiN/SOI patches

Figure S1:  Raman spectroscopy of SOI patches. a) From the bottom to the top
panel are displayed Raman spectra of Si bulk, the center of the patch from
samples A, B, and C. The experimental data are shown as square dots. The red
curves are Lorentzian fits the data. The vertical, dotted line highlights the
spectral position of the Raman signal of the bulk Si. The insets show the
scheme of the membrane with a black dot highlighting the position of the
detection. b) Raman spectra from samples A and B (respectively top panels and
bottom panels) detected in the suspended parts, at the corner of the patch,
and at the side. The experimental data are shown as square dots. Red and green
curves represent Lorentzian fits to the data. The vertical, dotted lines
highlight the spectral position of the Raman signal of bulk Si. The insets
show the scheme of the membrane with a dot highlighting the position of the
detection.

Raman spectroscopy is performed at room temperature on samples grown in the
same conditions as those investigated in micro-photoluminescence, on patches
having a side of 50 $\mu$m. They are summarized as follows (Figure S1): i)
Irrespective of the conditions used for SiN deposition (Table 1), in the flat
areas (far from the etched patches), the spectral shift of the SOI Raman
signal is within the resolution of the experimental setup. As such, in these
parts, the strain in the SiN is not relaxed in the SOI that is considered
unstrained (not shown [67]). ii) At the center of the patches the Raman signal
shows a peak broadening very close to that detected on Si (about 2.7 cm-1). In
this position, for SiN grown with plasma HF$\%$ = 100 and 70 (samples B and
C), the strain is tensile, whereas, for plasma HF$\%$ = 0 (sample A), the
strain is compressive. This information is consistent with high-resolution SEM
that shows the bending of the membranes upwards or downwards, respectively for
tensile or compressive strain (e.g. see Figure 1 (c)). For these patches with
50 $\mu$m side, assuming an uniaxial stress, $\sigma$, can be written as
$\sigma$(MPa) = -434 $\Delta\omega$(cm-1) where $\Delta\omega$ is the shift of
the Raman peak with respect to the Si peak [47, 48]. For samples A, B, and C
we find, respectively, a stress of about -143 MPa, 191 MPa, and 87 MPa (Figure
S1 (a)). iii) Raman spectra detected at the edge and at the corner of the
membrane, in its suspended part, show a splitting of the peak in two
components, having similar intensity and broadening (FWHM $\sim$ 2.8 cm-1) and
split of 2.5 to 4.5 cm-1 (Figure 2 (b)). This can be tentatively interpreted
as the presence of a biaxial strain and the onset of a TO phonons associated
with strain in the [110] direction[45, 68].

###  S0.3  Spectroscopy of flat SiN/SOI

We characterize the emission of G-centers from flat areas for strained and
unstrained samples (Figure S2). We observe the typical spectrum composed of a
sharp ZPL (at about 0.969 eV) and a broad phonon sideband. This latter
contribution is composite and we can identify the presence of the TA(X) and
TA(W) phonon modes [6] (Figure S2 (a), inset of the bottom panel). The ZPLs
display a Gaussian lineshape to a good approximation, small shifts (of about
0.2 meV, within the spectral resolution) with respect to the unstrained case
and a similar broadening (0.7 meV to 1.0 meV). These observations highlight
the negligible impact of the SiN deposition on the G-center dynamics in the
flat areas.

Figure S2:  G-centers emission in flat areas. a) Photoluminescence spectra
collected in flat areas for samples A, B, C, and without SiN (from the bottom
to the top panel, respectively). The inset in the bottom panel shows the
G-center spectrum of sample A in the flat area in a logarithmic scale in order
to highlight the presence of the TA(X) and TA(W) phonon lines. b) Blow up of
the ZPL for samples A, B, C, and without SiN (from bottom to top panel,
respectively). The experimental data are represented as square dots whereas
the red line is a Gaussian fit. The vertical line highlights the position of
the ZPL for the un-strained case.


# URYSOHN LEMMAS IN TOPOLOGICAL VECTOR SPACES

Ramkumar. S and Ganesa Moorthy. C
Department of Mathematics, Alagappa University, Karaikudi - 630 003, India.
Email: $1$<EMAIL_ADDRESS>and $2$<EMAIL_ADDRESS>
###### Abstract

Two variations of classical Urysohn lemma for subsets of topological vector
spaces are obtained in this article. The continuous functions constructed in
these lemmas are of quasi-convex type.

## Introduction

The classical technique for the proof of Urysohn’s lemma (See: [1]) is applied
to derive metrization theorem (See: [2]) for topological vector spaces. It is
possible to find the continuous functions of quasi-convex type, when the same
technique is applied on certain subsets of a topological vector spaces. The
Urysohn lemma is derived on normal spaces or on locally compact spaces. So,
two concepts of convex normal and convex regular subsets of a topological
vector spaces are introduced; their properties are studied; and two Urysohn
lemmas are derived in this article.

## Main Results

###### Definition 1

A subset $A$ of a topological vector space is said to be locally convex if
every point has a local base in $A$ consisting of convex subsets of $A$.

###### Definition 2

Let $A$ be a locally convex subset of a topological vector space $X$. A set
$A$ is said to be convex regular, if for a given point $x$ and a given open
convex set $B$(open in $A$) such that $x\in B$, there is an open convex subset
$C$ of $A$ such that $x\in C\subseteq cl\ C\subseteq B$.

###### Definition 3

Let $A$ be a locally convex subset of a topological vector space $X$. A set
$A$ is said to be convex normal, if for a given non empty closed set $A_{0}$
and a given open convex set $B$ such that $A_{0}\subseteq B$, there is an open
convex subset $C$ of $A$ such that $A_{0}\subseteq C\subseteq cl\ C\subseteq
B$.

###### Proposition 4

Every compact convex set in a locally convex topological vector space is
convex normal.

Proof: Let $A_{0}$ be a closed convex subset of a compact convex set $A$ in a
locally convex topological vector space $X$. Let $B$ be an open convex set in
$A$ containing $A_{0}$. To each $x\in A_{0}$, find an open convex
neighbourhood $U_{x}$ of 0 in $X$ such that
$(x+U_{x})\mathop{\cap}A\subseteq(x+cl\ (U_{x}))\mathop{\cap}A\subseteq B$.
Since $A_{0}$ is compact, find a finite subset $\\{x_{1},x_{2}\cdots x_{n}\\}$
of $A_{0}$ such that

$A_{0}\subseteq\mathop{\cup}\limits_{i=1}^{n}(x_{i}+U_{x_{i}})\mathop{\cap}A$
(1)

The convex hull $C$ of right hand side of (1) is open in $A$ and it is
contained in $B$. Note that, any element in $C$ is of the form
$\lambda_{1}y_{1}+\lambda_{2}y_{2}\cdots\lambda_{n}y_{n}$ with $y_{i}\in
x_{i}+U_{x_{i}},\ 0\leq\lambda\leq 1$ and
$\sum\limits_{i=1}^{n}\lambda_{i}=1$. For each $y\in cl\ C$, there is net
$\left(\lambda_{1\delta}y_{1\delta}+\lambda_{2\delta}y_{2\delta}\cdots\lambda_{n\delta}y_{n\delta}\right)_{\delta\in\mathcal{D}}$
in $C$ which converges to $y$. We can find a sub net
$\left(\lambda_{1\alpha}y_{1\alpha}+\lambda_{2\alpha}y_{2\alpha}\cdots\lambda_{n\alpha}y_{n\alpha}\right)_{\alpha\in\mathcal{G}}$
of this net such that $\lambda_{1\alpha}\to\lambda_{1},\ y_{1\alpha}\to
y_{1},\cdots\lambda_{n\alpha}\to\lambda_{n},y_{n\alpha}\to y_{n}$(say). Since
$0\leq\lambda\leq 1$, $\sum\limits_{i=1}^{n}\lambda_{i}=1$, $y_{i}\in
x_{i}+cl\ (U_{x_{i}})$ and the convex hull of
$\mathop{\cup}\limits_{i=1}^{n}(x_{i}+cl\ (U_{x_{i}}))\mathop{\cap}A$ is
contained in $B$, we conclude that $y=\sum\limits_{i=1}^{n}\lambda_{i}y_{i}\in
B$. Thus $cl\ C\subseteq B$. This proves the result.

###### Definition 5

A function $f$ from a convex subset $A$ of real vector space into the real
line is said to be

  1. (i)

convex if $f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y)$, for
every $x,y\in A$, for every $\lambda\in[0,1]$.

  2. (ii)

quasi convex if $\\{x\in A:f(x)<r\\}$(or equivalently $\\{x\in A:f(x)\leq
r\\}$) is a convex set,for every real number $r$.

###### Definition 6

Let $A$ be a convex subset of a topological vector space $X$. The set $A$ is
said to be ‘(quasi) convex completely regular’ if for a given point $x\in A$
and a given open convex set $B\subseteq A$ such that $x\in B$, there is
continuous (quasi)convex function $f:A\to[0,1]$ by $f(x)=0$ and $f(A\backslash
B)=\\{1\\}=f(B^{c})$.

###### Proposition 7

Every convex subset of a locally convex space is convex completely regular,
when it contains more than one point.

Proof: Let $A$ be a locally convex subset of a locally convex space. Let $x\in
A$ and $B$ be an open convex subset of $A$ such that $x\in B$. Without loss of
generality, let us assume that there is a continuous semi norm $p$ on $X$ such
that $\inf\\{p(x-y):y\in A\backslash B\\}\geq 1$. Define a map $f:A\to[0,1]$
by $f(y)=\min\\{p(x-y),\ 1\\}$. Then $f$ is continuous convex mapping such
that $f(x)=0$ and $f(y)=1$, for every $y\in A\backslash B$.

###### First Urysohn lemma 8

Let $A$ be a convex normal subset of a topological vector space $X$. Let
$A_{0}$ be a non empty closed convex subset of $A$ and $B$ be an open convex
subset of $A$ containing $A_{0}$. Then there is a continuous quasi convex
function $f:A\to[0,1]$ such that $f(A_{0})=\\{0\\}$ and $f(A\backslash
B)=\\{1\\}$.

Proof: Let $P$ be the set of all rational numbers in the interval [0,1].
Define $U_{1}=B$. Find an open convex subset $U_{0}$ in $A$ such that
$A\subseteq U_{0}\subseteq cl(U_{0})\subseteq U_{1}\subseteq B$. Find
$(U_{r})_{r\in P}$ of open convex sets such that $`p<q\Rightarrow
cl(U_{p})\subseteq U_{q}$’. Define $f:A\to[0,1]$ by $f(x)=\begin{cases}\inf\
\\{p\in P:x\in U_{p}\\},&\mbox{ if }x\in U_{1}\\\ 1&\mbox{ if }x\notin
U_{1}\end{cases}$

By the proof of the classical Urysohn lemma (See:[1, Theorem 33.1]), $f$ is a
continuous function. Since $\\{x\in A:f(x)\leq\epsilon\\}$ is convex, for
every $\epsilon\geq 0$, $f$ is a quasi convex function.

###### Lemma 9

Let $A$ be a locally convex locally compact subset subset of a locally convex
space $(X,(p_{i})_{i\in I})$. Suppose further that $A$ is complete as a
uniform space with the uniformity induced by $(p_{i})_{i\in I}$. Let $A_{0}$
be a nonempty compact convex subset of $A$ and $B$ be an open(in $A$) convex
subset of $A$ such that $A_{0}\subseteq B$. Then there is an open convex
subset $C$ of $A$ such that $A_{0}\subseteq C\subseteq clC\subseteq B$ and
$clC$ is compact.

Proof: To each $x\in A_{0}$, find an open convex neighbourhood $U_{x}$ of 0 in
$X$ such that
$(x+U_{x})\mathop{\cap}A\subseteq(x+cl(U_{x}))\mathop{\cap}A\subseteq B$, and
$(x+cl(U_{x}))\mathop{\cap}A$ is compact. Find a finite subset
$\\{x_{1},x_{2}\cdots x_{n}\\}$ of $A_{0}$ such that

$A_{0}\subseteq\mathop{\cup}\limits_{i=1}^{n}(x_{i}+U_{x_{i}})\mathop{\cap}A$
(2)

The convex hull $C$ of right hand side of (2) is open in $A$ and it is
contained in $B$. By theorem in [2] and by completeness of $A$, the convex
hull of $\mathop{\cup}\limits_{i=1}^{n}(x_{i}+cl(U_{x_{i}}))\mathop{\cap}A$
has a compact closure. Thus $clC$ is contained in a compact subset of $A$. So,
as in the proof of proposition 4, we see that $clC\subseteq B$. This completes
the proof.

###### Second Urysohn lemma 10

Let $A$ be a locally convex locally compact subset subset of a locally convex
space. Suppose further that $A$ is complete. Let $A_{0}$ be a compact convex
subset of $A$, and $B$ be an open convex subset of $A$ such that
$A_{0}\subseteq B$. Then, there is a continuous function $f:A\to[0,1]$ with
compact support such that $f(A_{0})=\\{1\\},\ f(A\backslash B)=\\{0\\}$, and
$1-f$ is a quasi convex function.

Proof: Find convex open sets $U_{0}$ and $U_{1}$ such that $cl(U_{0})$ and
$cl(U_{1})$ are compact and $A\subseteq U_{0}\subseteq cl(U_{0})\subseteq
U_{1}\subseteq cl(U_{1})\subseteq B$. As in the proof of first Urysohn lemma,
we can find a function $g:A\to[0,1]$ such that $g(A_{0})=\\{0\\},\
g(A-U_{1})=\\{1\\},\ g$ is a quasi convex and $g$ is continuous. The required
function $f$ is $1-g$.

## References

  * [1] James R. Munkres, Topology, 2nd edition, Prentice Hall of India, New Delhi, 2000
  * [2] W. Rudin, _Functional Analysis_ , second ed., International series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991.