1
Supplementary Note 1
.
The proposed physical mechanism of modulation of optical response
We perform
numerical electrostatic calculations (Device
Lumerical,
Lumerical Solutions, Inc.) to
determine the carrier distribution in
the
top TiN electrode of
the
TiN
/SiO
2
/Ag heterostructure
(Fig. 1)
as a
function of applied bias
. This is achieved
by numerically solving the Poisson and drift
-
diffusion equations
(
Supplementary
Fig. 1). We assume that TiN has a bandgap of
E
bg
= 3.4
e
V, electron affinity of
χ
= 8.3
e
V
,
effective electron mass of
m
* = 1.2
m
e
. The
DC
permittivity and electron mobility of TiN are chosen as 9.3
and 1 cm
2
V
-
1
s
-
1
, respectively.
The work functions of TiN films
grown under different deposition conditions
have been previously
reported
1
. However, the carrier concentration of
TiN films has not been specified in
the cited
work
1
. Since the
carrier concentration of TiN strongly depends on fabrication conditions, it is not possible to accurately
calculate the electron affinity of TiN by relying on the reported work function values. Since the cited paper
discusses
possibility of using TiN as a gate electrode, we assume that TiN used in
that
work is heavily doped.
Hence,
in the present study we assume
that the work function of TiN with carrier concentration of 1.8×10
22
cm
-
3
is 6
e
V. In
the
device calculations, we
used the mesh size of 0.05 nm.
2
Supplementary
Figure 1
|
Simulated electron density in TiN as a function of position and applied bias.
When we apply
a
positive bias, an electron accumulation layer is formed in TiN at
the
TiN/SiO
2
interface. At
negative biases of
–
1.8 V and below
,
an electron depletion layer is formed in TiN at
the
TiN/SiO
2
interface.
The optical modulation mechanism is based on
a
metal
-
oxide
-
semiconductor (MOS) field
-
effect dynamics.
For the assumed value of elect
ron affinity, at zero bias, we have
an
electron accumulation in
the
TiN.
3
Supplementary Note 2
. Characterization of fabricated TiN films
For each TiN/SiO
2
/Ag device, we fabricate additional large area control samples for performing Hall
measurements and ellipsometry. The control samples consist of
a
Si substrate, followed by
a
35
nm
-
thick SiO
2
layer
, and an ultrathin TiN film. The carrier concentration
(
N
)
and electron mobility of each TiN film have
been identified via Hall measurements (
Supplementary
Table 1).
N
(cm
-
3
)
5.9 × 10
20
2.6 × 10
21
1.8 × 10
22
4.1 × 10
22
Mobility (cm
2
V
-
1
s
-
1
)
0.558
0.35
0.059
5.8
Ellipsometry fitting model
2 Lorentz
oscillators
1 Drude and
2 Lorentz
oscillators
1 Drude and
2 Lorentz
oscillators
1 Drude and
3 Lorentz
oscillators
Optical property
Optically
dielectric
Optically
dielectric
ENZ
Optically
plasmonic
Film thickness (nm)
8
34
7
46
Substrate
35 nm SiO
2
/Si
35 nm
SiO
2
/Si
9 nm SiO
2
/Ag
35 nm SiO
2
/Si
Applied
DC
Power (W)/
voltage (V)
150/465
150/484
250/514
252/682
Ar/N
2
flow rate
5/0
5/0
5/0
5/0
Base Pressure (torr)
4.0 × 10
-
7
4.0 × 10
-
7
2.6 × 10
-
6
3.1 × 10
-
6
Gas pressure (mtorr)
5
5
3
3
Supplementary
Table 1
|
Material properties and growth parameters of TiN films.
We deposit
thin TiN
films via DC sputtering. When sputtering TiN films we fix the Ar/N
2
flow rate, and the properties of the
resulting films are controlled by changing DC power and voltage.
4
We measure
the
complex dielectric permittivity of
the
fabricated TiN films by using spectroscopic
ellipsometry. We fit
the
ell
ipsometrically measured data
by
using
the
Drude
-
Lorentz model
2
, which is a sum
of
a
Drude term and two
or three
Lorentz oscillators
:
̃
∑
(1
)
We identify the values of
the
free parameters incorporated in the model by fitting them to
the
ellipsometry data.
The
re are three free paramete
rs in the Drude
terms of
Supplementary
Eq
uation
1 including
the damping factor
Γ
,
plasma frequency
, and
. The plasma frequency
relates to the electron effective
mass
m*
and
the
carrier
density
of the film
N
as follows:
√
.
Here,
q
is the
electron
charge
,
is the
dielectric permittivity of vacuum
.
Ea
ch Lorentz oscillator in
Supplementary
Eq
uation
1
contains three fitting
parameters: the oscillator strength
, the damping factor
, and the oscillator position
. Here, the index
j
numerates the Lorentz oscillators (
j
=1,2,3).
Supplementary
Table 2 summarizes the values of the obtained
fitting parameters that have been used to produce Fig. 2.
N
(cm
-
3
)
(
e
V)
Γ
(
e
V)
(
e
V)
(
e
V)
(
e
V)
(
e
V)
(
e
V)
(
e
V)
5.9 × 10
20
–
–
5.9
3.8
2.1
3.8
1.2
1.6
–
–
–
2.0
2.6 × 10
21
4.6
17.3
13.3
1.0
1.8
6.6
4.0
2.0
–
–
–
1.7
1.8 × 10
22
5.15
102
22
0.6
1.6
10
4.0
0.8
–
–
–
4
4.1 × 10
22
6.9
1.04
0.4
2.4
0.3
8.9
3.5
2.4
1.1
2.6
0.7
1.3
Supplementary Table 2
|
The Drude
-
Lorentz fitting parameters for the complex dielectric permittivity
of TiN.
These values have been used to produce Fig. 2.
5
Supplementary
Figure 2 | Reflectance modulation measured via focused beam spectroscopic
ellipsometry.
and
for the ENZ
-
TiN/SiO
2
/Ag heterostructure as a function of wavelength at different
voltages. In this measurement, the incidence angle is 60°. We observe modulation of
and
under applied
electrical bias. When electrical bias increases
from −1 V to +1 V, we observe a monotonic increase of
and
monotonic decrease of
.
The i
nset shows the SEM image of the device and the normal incident beam size of
focused beam spectroscopic ellipsometry.
The s
cale bar
is
500 μm.
We also perform
ellipsometry of our
ENZ
-
TiN/SiO
2
/Ag
heterostructure
under applied bias. However,
the
obtained data is not amenable to unambiguous interpretation due to
the
reasons described below. The area
of our gate
-
tunable
ENZ
-
TiN/SiO
2
/Ag
heterostructure is only 1 mm
2
. The reason why we choose such small
device area is that the thickness of
our
SiO
2
gate dielectric is only 9 nm. Larger area
gate
-
tunable
ENZ
-
TiN/SiO
2
/Ag
heterostructures would typically exhibit high leakage current upon biasing due to defects of gate
die
lectric. When performing ellipsometry it is important that
the
incoming light is reflected form the working
area of the device only. Hence, during ellipsometry measurements we
have to use
the
focused light beam.
However, the beam spot
is
still larger than
the working area of the device, especially for larger incidence
angles. By using this approach we obtain
and
for different incidence angles.
and
are defined by the
following formula:
r
p
/
r
s
=tan(
)e
i
,
where
r
p
and
r
s
are the reflection coefficients
of the p
-
and s
-
polarized
light, respectively.
However, we
are not
able to obtain a reliable ellipsometry fit to extract dielectric
permittivity of TiN. Nevertheless, we have acquired
and
for a given incidence angle (60°) and different
applied biases
(
Supplementary
Fig. 2)
. We observe optical modulation around 430 nm where
and
exhibit
maximum and minimum, respectively. Finally, we would like to note that since
the
ellipsometry
is performed
6
several months after fabricating the sample, the observed
optical modulation is modest as compared to the
result shown in Fig. 3 of the manuscript.
As we have previously mentioned, because of
the
small device area, we are not able to obtain reliable
ellipsometry fits for
the
ENZ
-
TiN
,
which is incorporated in
the
ENZ
-
TiN/SiO
2
/Ag heterostructure. We
identify
the
dielectric permittivity of
the
TiN in
the
TiN/SiO
2
/Ag heterostructure by fitting
the
parameters of
the
Drude
-
Lorenz model to
the
reflectance spectrum. The reflectance spectrum
taken under normal incidence
wi
th 5X objective (Olympus,
with numerical aperture of
0.14)
, which
focus
es
a supercontinuum
Fianium
laser
down
to a small spot of 3
m in diameter
.
This spot size
is much smaller than the working area of our device
,
which is
1 mm
2
. When fitting
the
parameters of
the
Drude
-
Lorenz model to
the
reflectance data, we
use
the
electron mobility and carrier concentration values derived via Hall measurements (
Supplementary
Table 1).
As
seen
in
Supplementary
Fig. 3, the calculated and measured reflectance are
in good agreement. As
demonstrated in
Fig. 2,
the
real part of
the
dielectric permittivity of our TiN shows double
-
ENZ crossing that
is in good agreement with recent
literature reports
3
.
Supplementary
Figure 3 | Measured and calculated reflectance of
the
epsilon
-
near
-
zero (
ENZ
)
-
Ti N
/SiO
2
/Ag heterostructure.
The dielectric permittivity of
the
ENZ
-
TiN is extracted from reflectance
measurements.
7
When analyzing
the
ellipsometry data, we use two or three Lorentz oscillators to fit the data obtained
from the fabricated films. The key issue
here is that optical properties of TiN films strongly depend on the
film stoichiometry
, that is
the
ratio
of titanium and nitrogen
, impurities
, such as
residual oxygen
or oxygen
introduced due to
post growth oxidation, grain size
,
and density/porosity
2
.
The gr
a
in size affects the mean
fr
ee path of the conduction electrons, while the density/porosity influences the conduction electron density.
All these factors likely differ from one film to another resulting in different values of fitting parameters of a
Drude
-
Lorentz model, such as elect
ron mobility and frequencies of Lorentz oscillators. In particular, it has
been previously shown that the spectral position of the zero crossing (ENZ point) is an indicator of film
stoichiometry. It has been shown that for stoichiometric TiN films this cro
ssing occurs at 2.65
e
V (
=
468 nm)
4
. As
seen
in
Fig. 2a, none of our films has an ENZ crossing at
=
468 nm. Hence, all
the
films reported in this
work are non
-
stoichiometric. Moreover, it has been recently shown that amount of residual oxygen in the
sputtering chamber can dramatically affect optical properties of TiN films
3,5,6
. Based on these reports, we
expect that each fabricated film has different chemical and str
uctural composition. The review article
2
discusses the relation between the band structure of TiN obtained via density functional theory (DFT)
calculations and the spectral positions of Lorentz oscillators in an ellipsometry fit. As
seen
in
Supplementary
Table 2 of the mentioned review artic
le
2
, the spectral positions of Loren
tz oscillators show significant
variation. Moreover, the reported spectral position values do not always agree with
the
DFT calculations. The
reason for this is that the DFT calculations are performed for stoichiometric TiN
,
while the films shown in our
wo
rk as well as a number of films shown in the review article
2
are non
-
stoichiometric. Moreover, we perform
compositional analyses on
135 nm
-
thick TiN films deposited on Si substrates via DC sputtering. The
sputtering conditions
are
chosen to be identical to those used to deposit TiN in the tu
nable ENZ
-
TiN/SiO
2
/Ag
heterostructure. We
find
that the stoichiometry of our film is given as TiN
0.8
O
0.2
, where oxygen has been
introduced into our film unintentionally. This result is consistent with previous reports
5,6
. Hence, most likely,
7 nm
-
thick TiN films incorporated in our TiN/SiO
2
/Ag films also include a significant amount of oxygen
impurities. Based on this, it is n
ot straightforward to relate positions of Lorentz oscillators to the band
structure of our TiN, due to limited knowledge of the band structure of the actual film. To summarize, the
8
choice of the model that we use to fit the ellipsometry data
(
two
versus
th
ree Lorentz oscillators, etc.) is
largely
dictated by the composition and structure of the films, which varies
. Due to a limited knowledge of the
dependence of the TiN band structure on the material stoichiometry and the amount of incorporated oxygen
impur
ities, we are not able to perform a direct comparison between the positions of Lorentz oscillators and the
material band structure.
One can obtain indirect evidence of compositional and structural differences in sputtered TiN films
when analyzing the dep
endence of the imaginary part of the measured dielectric permittivity Im(
ɛ
) on the
carrier concentration of the films. Indeed, Im(
ɛ
) monotonically increases with the carrier concentration for
carrier concentrations less than or equal to 1.8 × 10
22
cm
-
3
and abruptly decreases for the sample with
the
carrier concentration
of
1.8 × 10
22
cm
-
3
. This kind of behavior has been previously reported in literature
5
. In
the mentioned paper, Im(
ɛ
) of
the
“intermediately doped” TiN is grea
ter than Im(
ɛ
) of
the
“dielectric” TiN
5
.
On the other hand, over a broad wavelength range, Im(
ɛ
) of
the
“metallic” TiN is smaller than Im(
ɛ
) of
the
“intermediately doped” or “dielectric” TiN
5
. It is a
rgued that when the oxygen impurity concentration in TiN
is lessened, the crystallinity of TiN films is improved
. As a result,
Re(
ɛ
) becomes more negative, and Im(
ɛ
)
decreases.
Interestingly, in the visible wavelength range we observe a non
-
monotonic beha
vior of Im(
ɛ
) as a
function of wavelength (Fig. 2b) that is consistent with
the
previously reported functional dependences of
Im(
ɛ
) on wavelength
2,3,5
. On the
other hand, as we observe in our work, Im(
ɛ
) typically grows monotonically
as a function of wavelength in the near
-
infrared wavelength range
2,3,5
.
9
Supplementary
Note 3
.
Eff
ect of carrier concentration of
TiN on reflectance of
TiN/SiO
2
/Ag
heterostructure
s
In each of the fabricated TiN/SiO
2
/Ag heterostructures there are QDs embedded in the SiO
2
layer.
Hence, one might assume that the observed optical modulation is due to the variation of
the
effective
refractive index of InP
QDs embedded in the structure. However, this is not a possible mechanism since only
heterostructures with specific carrier concentration of TiN yield reflectance modulation under applied bias
(
Supplementary
Fig.
4
). This indicates that the presence of TiN
film is crucial for observation of optical
modulation.
Supplementary
Figure
4
|
Reflectance of TiN/SiO
2
/Ag heterostructure
s
as a function of applied bias at
a
quantum dot
emission wavelength
of
λ
=630 nm.
The fabricated TiN
film has different carrier concentration
for each measured heterostructure.
(
a
)
R
eflectance modulation under applied bias when
the
TiN is in
the
epsilon
-
near
-
zero (ENZ) phase (
N
=1.8
×
10
22
cm
−
3
). We observe no detectable reflectance modulation when
TiN
is in
(
b
)
optically plasmonic (
N
=4.1
×
10
22
cm
−
3
) or
(
c
)
optically dielectric phase (
N
=2.3
×
10
20
cm
−
3
).
10
Supplementary Note 4
. Modulation speed of optical response of TiN/SiO
2
/Ag heterostructure
s
We perform
the
modulation frequency measurements four months after the sample was fabricated. As a
result,
by
the time of
the
modulation frequency measurements, our sample
has
already
been
degraded.
Supplementary
Fig
.
5 shows
the
reflectance spectrum of the degraded samp
le taken right before
the
modulation frequency measurements. Degraded samples typically exhibit an increased contact resistance
between the TiN and contact electrode.
Therefore
, we believe, we need to apply a larger electrical bias of ±5
V to observe refle
ctance modulation. We would like to point out that the application of higher voltages does
not necessary imply that the built
-
in electric field inside the heterostructure is going to be larger. Moreover,
the reflectance characteristics of the degraded samp
le have also been modified; likely, due to sample oxidation
(compare Fig. 3a and
Supplementary
Fig.
5). Despite sample degradation, the sample still exhibits optical
modulation under applied bias. We perform reflectance modulation measurements using the degraded sample.
Importantly, we observe a modulation frequency of 20 MHz, and the detected reflectanc
e values show a
perfect match with the reflectance values measured in
Supplementary
Fig
.
5. Finally, we would like to note
that the samples can be protected by depositing a thin encapsulation layer
, such as
Al
2
O
3
.
Supplementary
Figure 5
|
Modulation spe
d of TiN/SiO
2
/Ag heterostructure.
We measure
the
reflectance
of
the
TiN/SiO
2
/Ag heterostructure under modulating bias that changes between
5 V and 5 V with
a
modulation frequency of 20 MHz
.
T
he
modulation amplitude varies between 50% and 63%. Our detectio
n
frequency was limited by the Si photodetector response time. This implies that
the
modulation frequency of
our device could potentially be higher than 20 MHz.
11
Supplementary Note 5
. Calculation of reflectance
under
applied bias
As it
has been
shown in
Supplementary Note 3
,
meeting ENZ condition in the TiN film is crucial for
observation of optical modulation.
The measured real and imaginary parts of
the
dielectric permittivity of TiN
are shown in Fig. 2. Although
when we increase the TiN carrier concent
ration,
TiN undergoes transition from
optically dielectric to optically plasmonic phase, the imaginary part of
its
dielectric permittivity does not
increase in a monotonic fashion (Fig. 2b). This indicates that to accurately model the bias
-
dependent optica
l
response of
the
TiN film in
the
TiN/SiO
2
/Ag heterostructure, one cannot simply change the carrier
concentration in the
Drude
-
Lorentz model
while keeping
i
ntact the parameters of the Lorentz oscillators
defined at zero bias. Therefore, in our simulations we describe the accumulation layer of TiN by
the
experimentally measured dielectric permittivity of
the
heavily doped TiN film.
Next, we calculate
the
reflectance of
the
TiN/SiO
2
/Ag heterostructure
.
In our simulations,
we assume
layer thicknesses
identical to those identified from
the
transmission electron microscopy image (Fig. 1). The
calculated TiN/SiO
2
/Ag he
terostructure consists of 80 nm
-
thick Ag
back reflector, 9 nm
-
thick SiO
2
, and 7 nm
-
thick TiN. We assume that upon application of electrical bia
s, a 1 nm
-
thick modulated TiN layer is formed in
TiN at TiN/SiO
2
interface. For example, when applying
a
positive bias a 1 nm
-
thick charge accumulation
la
yer is formed in TiN at TiN/SiO
2
interface.
As a result
, the modulated la
yer becomes optically plasmonic
and
the
reflectance
of the heterostructure is increased
(
Supplementary
Fig.
6
).
On the other hand, under
negative bias, the permittivity of the modulat
ed TiN layer moves toward the ENZ region
that results
in the
suppression of the reflectance.
Finally
,
we note that
in our simulations the mesh size is 2 nm
i
n the directions
along the layers of TiN/SiO
2
/Ag heterostructure
,
while for the direction normal to the layers the mesh size is
0.5 nm.
12
Supplementary
Figure
6
|
Calculated reflectance
modulation of TiN/SiO
2
/Ag heterostructure
.
C
alculated
reflectance
spectrum of
the
TiN/SiO
2
/Ag heterostructure when
the
TiN is in
the
epsilon
-
near
-
zero (
ENZ
)
region (
N
=1.8
×
10
22
cm
−
3
). The reflectance
increases
when
we apply
a
positive bias
.
The dielectric
permittivity of
the
TiN
films is shown in
Fig. 2.
Our calculations show that
the
observed reflectance modulation cannot be fully explained by simply
changing the carrier concentration in the Drude term of the Drude
-
Lorentz model that defines the dielectric
permittivity of TiN. The carrier concentration of our ENZ
-
TiN film (
N
=
1.8 × 10
2
2
cm
-
3
) is at least 2 orders of
magnitude higher as compared to
the
carrier concentrations that one typically deals with in ITO
-
based gate
-
tunable devices
7
(
N
=
3 × 10
20
cm
-
3
). Therefore, a number of different effects may contribute to the observed
optical modulation. One possible effect is the electron effective mass’s dependence on applied voltage due to
nonparabolicity of conduction band. Another possible contributor is the
dependence of the electron mobility
on applied bias. In
Supplementary
Fig. 6 we have assumed that the dielectric permittivity of 1 nm thick
accumulation layer in TiN is given by the measured dielectric permittivity of TiN film with carrier
concentration o
f
N
= 4.1 × 10
22
cm
-
3
(Fig. 2). This assumption is, strictly speaking, not correct; however, it
allows us to estimate the sensitivity of the calculated reflectance with respect to
the
variation of
the
refractive
index of
the
1 nm
-
thick TiN layer
,
which is l
ocated at the interface of TiN and SiO
2
(
Supplementary
Fig. 6).
To summarize, due to high carrier concentration of TiN films, modifying carrier concentration in the Drude
term of the Drude
-
Lorentz model cannot reproduce experimentally observed reflectance
modulation.
13
Supplementary Note 6
.
Modulation of PL intensity
of InP QDs embedded in
active/passive
heterostructure
s
Supplementary
Figure 7
|
Modulation of
photoluminescence (
PL
)
intensity of InP
quantum dots
embedded in the gated Ti N/SiO
2
/Ag active plasmonic heterostructure.
(a)
PL intensity spectra for
different gate voltages in a linear scale (the same PL spectra
are
show
n
in Fig. 4a).
(b)
PL spectra in a log
scale.
(c)
Differential PL spectra. Red line: (PL(1V)
–
PL(0V))/PL(0V) and blue l
ine: (PL(
–
1V)
–
PL(0V))/PL(0V).
(d)
PL peak position as a function of applied voltage. We observe no detectable wavelength
shift.
14
Supplementary
Figure 8
|
Modulation of the
photoluminescence (
PL
)
intensity of InP
quantum dots
embedded in the gated Ti/SiO
2
/Ag passive heterostructure.
(a)
PL intensity spectra for different gate
voltages in a linear scale (the same PL spectra
are
show
n
in Fig. 4b).
(b)
PL spectra in a log scale.
(c)
Differential PL spectra. Red line: (PL(1V)
–
PL(0V))/PL(0V) and blue line: (PL(
–
1V)
–
PL(0V))/PL(0V).
(d)
PL peak position as a function of applied voltage. We observe no detectable wavelength shift.
15
Supplementary Note 7
.
Calculation of
local density of optical states
(
LDOS
)
We calculate the
local density of optical states (
LDOS
)
in
the
SiO
2
gap of
the
TiN/SiO
2
/Ag
heterostructure via finite difference time domain (FDTD) method. In our calculations we model an InP/ZnS
QD as a point dipole source located in the lossless sphere with a diameter of 4.5 nm and refractive index of
3
.5. In our calculations we assume that the QD is located in the center of
the
SiO
2
layer, and the direction of
the dipole moment is normal to the layers of the heterostructure. The refractive index of SiO
2
is assumed to be
n
SiO2
= 1.47. The optical permitti
vity of Ag and TiN are determined by ellipsometry measurements. It is known
that the LDOS is proportional to the power radiated by a point dipole in the given environment
8
. Thus, we use
FDTD to calculate LDOS enhancement with respect to LDOS in a bulk lossless dielectric with refractive
index of 3.5 (Fig. 4
d
). It is worth mentioning that while calculating the LDOS, one should average out over
different orientations of the dipole moment.
To enhance the amount of observed LDOS modulation, we suggest using a plasmonic cavity
(
Supplementary
Fig. 9a) that can be
formed by placing an appropriately designed patch antenna on top of an
Ag/SiO
2
/TiN heterostructure (akin to previously demonstrated reflectarray antennas
9,10
). It has been shown
that the lifetime of a qua
ntum emitter embedded in the dielectric gap of the patch antenna resonator is very
sensitive to gap thickness
10
. If the resonant wavelength of the cavity is aligned with the emission
wavelength
of the emitter, this configuration will yield a large modulat ion of the PL intensity under an applied bias.
Supplementary
Fig
.
9b shows the distribution of the electric field excited by the point dipole embedded in the
5 nm thick gap of the prop
osed patch antenna structure (
Supplementary
Fig. 9a). The assumed dielectric
permittivity of 7 nm thick TiN film is ident ical to that of ENZ
-
TiN film used in the present work (the film
with carrier concentration
N
= 1.8 ×10
22
cm
-
3
from Fig. 2). In the case
of a positive bias, we assume that a 1 nm
thick accumulation layer is formed in TiN at the TiN/SiO
2
interface. The dielectric permittivity of the
accumulation layer is assumed to be identical to that of the metallic TiN described in the present work (the
f
ilm with carrier concentration
N
= 4.1 × 10
22
cm
-
3
from Fig. 2). The LDOS enhancement for two different
biases is shown in
Supplementary
Fig.
9c
. As demonstrated in
Supplementary
Fig
.
9,
the suggested structure
enables significant LDOS modulation under an applied bias. The simulation shown in
Supplementary
Fig
.
9
16
aims to demonstrate the sensitivity of the lifetime of the emitter with respect to variation of the refractive
index of the mat
erial in the gap, despite the difference in dielectric permittivity between the value assumed for
the TiN accumulation layer and that of metallic TiN derived in the present work (
N
= 4.1 × 10
22
cm
-
3
). Finally,
we would like to note that using
a
patch antenn
a resonator can increase the emission intensity from the de
vice
by two orders of magnitude
9
.
Supplementary
Figure 9 | (a)
Schematic of the proposed patch antenna with a quantum emitter embedded
in a
5 nm
-
thick SiO
2
layer
.
(b)
Calculated electric field distribution in the patch antenna structure. The
depicted electric field is excited by the point dipole oriented normally to the planar layers of Ag, SiO
2
, and
TiN.
(c)
Calculated
z
-
projection of the
local density of optical states
(
LDOS
)
enhancement with respect to
the
value of the
LDOS in the
bulk InP, as a function of the emission wavelength, for zero bias and
a
positive bias.