-
Theoretical investigation of ultracompact and athermal Si
electro-optic modulator based on Cu-TiO2-Si hybrid plasmonic donut
resonator
Shiyang Zhu,* G. Q. Lo, and D. L. Kwong Institute of
Microelectronics, A*STAR (Agency for Science, Technology and
Research), 11 Science Park Road,
Science Park-II, 117685 Singapore *[email protected]
Abstract: An ultracompact silicon electro-optic modulator
operating at 1550-nm telecom wavelengths is proposed and analyzed
theoretically, which consists of a Cu-TiO2-Si hybrid plasmonic
donut resonator evanescently coupled with a conventional Si channel
waveguide. Owing to a negative thermo-optic coefficient of TiO2
(~-1.8 104 K1), the real part of effective modal index of the
curved Cu-TiO2-Si hybrid waveguide can be temperature-independent
(i.e., athermal) if the TiO2 interlayer and the beneath Si core
have a certain thickness ratio. A voltage applied between the
ring-shaped Cu cap and a cylinder metal electrode positioned at the
center of the donut, which makes Ohmic contact to Si, induces a
~1-nm-thick free-electron accumulation layer at the TiO2/Si
interface. The optical field intensity in this thin accumulation
layer is significantly enhanced if the accumulation concentration
is sufficiently large (i.e., > ~6 1020 cm3), which in turn
modulates both the resonance wavelengths and the extinction ratio
of the donut resonator simultaneously. For a modulator with the
total footprint inclusive electrodes of ~8.6 m2, 50-nm-thick TiO2,
and 160-nm-thick Si core, FDTD simulation predicts that it has an
insertion loss of ~2 dB, a modulation depth of ~8 dB at a voltage
swing of ~6 V, a speed-of-response of ~35 GHz, and a switching
energy of ~0.45 pJ/bit, and it is athermal around room temperature.
The modulators performances can be further improved by optimization
of the coupling strength between the bus waveguide and the donut
resonator. 2013 Optical Society of America OCIS codes: (250.7360)
Waveguide modulators; (240.6680) Surface plasmons; (250.5403)
Plasmonics; (130.3120) Integrated optics devices.
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1. Introduction
A CMOS compatible integrated Si electro-optical (EO) modulator
is a key component in Si electronic photonic integrated circuits
(EPICs) [1]. Many kinds of Si EO modulators have been reported
recently [2,3], mostly relying on the free-carrier dispersion
effect of Si to modulate the Si refractive index based on a MOS
capacitor [4, 5], a PIN diode [6], or a PN junction [7]. Either a
Mach-Zehnder interferometer (MZI) [4,7] or a waveguide-ring
resonator (WRR) [5,6] is utilized to convert the phase variation
into the intensity modulation. The WRR modulators offer smaller
footprints than the MZI modulators but at the price of narrower
optical bandwidth, higher temperature sensitivity due to the
relatively large thermo-optic (TO) coefficient of Si (~1.8 104 K1),
and limited modulation speed due to the long photon lifetime in the
resonator if the resonator has a very high quality factor (Q
value).
One approach to suppress the temperature sensitivity of Si
resonators is by overlaying a polymer coating with a negative TO
coefficient [8], but polymers are currently not compatible with
CMOS process. Another approach is by overcoupling the ring
resonator to a balanced MZI [9], but it requires complex design and
sacrifices the footprint. Moreover, due to the fundamental
diffraction limit of light propagation along Si waveguides, the WRR
modulators
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
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| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
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are still quite large as compared with the nanoscale electronic
devices. The minimum bending radius is ~1.5 m for Si single-mode
channel waveguides [10] and is usually larger than ~5 m for Si rib
waveguides in which the EO modulators are implemented. The total
footprint of Si WRR modulators inclusive of the electrodes is
usually larger than ~200 m2.
A technology emerging recently which can scale down the
dimension of optical devices far beyond the diffraction limit is
plasmonics, which deals with surface plasmon polariton (SPP) signal
propagating along the metal-dielectric interfaces [11]. Several
ultracompact plasmonic EO modulators have been proposed and/or
demonstrated [12,13]. However, they mostly rely on active materials
other than Si and/or require non-standard CMOS techniques for
fabrication. For ease of implementation into the exiting Si EPICs,
it is preferred to use Si as the active material and the modulator
is waveguide-based. A horizontal Cu-insulator-Si-insulator-Cu
nanoplasmonic waveguide is a plasmonic waveguide enabling to
realize plasmonic modulator [14], which has a MOS capacitor
structure and the Si core can be used as the active material.
Electro-absorption (EA) and phase modulations have been
experimentally demonstrated based on this plasmonic waveguide
[15,16], but a relatively large driving voltage is required to
reach 3-dB modulation in the EA modulators or -phase shift in the
MZI modulators. Another feasible plasmonic waveguide is a vertical
Cu-insulator-Si hybrid plasmonic waveguide (HPW) [17,18]. WRRs with
radius of ~1-2 m and Q-value of ~200-300 have been experimentally
demonstrated based on the Cu-SiO2-Si HPW [19]. Theoretically, the
radius of such plasmonic WRRs can be reduced to submicron (e.g.,
~0.8 m) [20]. Moreover, if TiO2, which has a negative TO
coefficient of ~-1.8 104 K1 and is transparent at near-infrared
wavelengths [21], is used as the insulator between the Cu-cap and
the Si core, the plasmonic WRRs can be athermal. TiO2 is also used
as a gate dielectric in MOS electronics, whose dielectric constant
ranges from 4 to 86 depending on the detailed fabrication processes
[22]. The Cu-TiO2-Si HPW is also a MOS capacitor, thus enabling a
voltage to be applied between the Cu cap and the Si core to
modulate its propagation property. It is expected that the WRR
modulators based on this hybrid plasmonic WRR may overcome the
abovementioned two critical issues of the conventional WRR
modulators, i.e., footprint miniaturization and
temperature-sensitivity suppression. This paper presents a
systematical investigation on ultracompact WRR modulators based on
the Cu-TiO2-Si HPW.
2. Device structure
The structure of the proposed modular is shown in Fig. 1
schematically. It consists of a Cu-TiO2-Si hybrid plasmonic donut
resonator and a bus waveguide. The modulator can be seamlessly
inserted in a dense Si channel waveguide-based photonic circuit. To
reduce the insertion loss, the bus waveguide is a conventional Si
channel waveguide. To reduce the overall footprint, a donut rather
than a ring is used, thus the electrode for Si Ohmic contact can be
positioned at the center of the donut and the donut forms a
standard MOS capacitor (the other electrode is the Cu cap).
The cylindrical electrode for Si Ohmic contact has radius of r0.
The outer radius of the Si donut is R and the inner radius is (R
WP), where WP is the width of Si core of the curved hybrid
plasmonic waveguide. The separation between the resonator and the
bus waveguide is gap. The Si height is H, the slab thickness in the
Si donut is tslab, and the TiO2 thickness is tox. The Cu-cap
thickness is set to be much larger than the penetration depth of
the SPP mode in the metal (~26 nm). Because the plasmonic mode can
only be excited by the electric field of optical mode perpendicular
to the metal/dielectric interface, the proposed modulator is valid
only for the transverse magnetic (TM)-polarized light.
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
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Rgap
(a) (b)
Si
substrate buried SiO2
n-Si
CuTiO2
AlUSG-SiO2
Cu
r0R
WPtslab
H
tox
V
Light input Through
Cu
Cu cap
gap
Woxr0
Fig. 1. (a) Top view, and (b) cross-sectional view of the
proposed Si plasmonic resonator modulator, the bus waveguide is a
conventional single-mode Si channel waveguide and the resonator is
a Cu-TiO2-Si hybrid plasmonic donut with two electrodes located at
the Cu cap and center-donut, respectively. The structural
parameters are also indicated.
The modulators are fabricated on silicon-on-insulator wafers.
The Si pattern of the bus and donut waveguides are defined by
partially dry etching of Si down to tslab using a thin SiO2 layer
as the etching mask, following by dry etching of the remaining Si
down to the buried SiO2 using both SiO2 and photo-resistor as the
etching mask. Using this etching method, there is no misalignment
issue between the inner and outer rings of the donut. After Si
patterning, a thick SiO2 is deposited and a ring-shaped window is
opened to expose the surface of the Si core. There exists possible
misalignment between the SiO2 window and the beneath Si core due to
fabrication imperfection. Here, the SiO2 window (hence the width of
the TiO2/Cu cap, Wox) is intentionally designed to be larger than
the beneath Si core by WP in each side, thus Wox = WP + 2WP. TiO2
is then deposited on the Si core through the windows, followed by
Cu deposition and Cu chemical mechanical polishing (CMP) to remove
TiO2 and Cu outside the windows. The structural parameters are
initially chosen based on our experience [23], as listed in Table
1. Then, one of these parameters is varied while the others keep
the same to investigate its effect on the whole performance.
Table 1. The initial parameter setting of the plasmonic EO
modulator
Radius, R 1.5 m Si height, H 220 nm Si slab height, tslab 50 nm
Si core width, WP 200 nm TiO2 thickness, tox 50 nm SiO2 window
wider than the Si core in each side, WP 50 nm Si bus waveguide
width, WSi 400 nm Separation between the donut and the bus
waveguide, gap 10 nm Radius of the middle Cu cylinder, r0 0.5 m
Doping in the Si donut, ND n-type, 5 1018 cm3 Doping in the center
of the donut for Ohmic contact 2 1020 cm3
The refractive indices of Si, SiO2, TiO2, and Cu depend both on
wavelength and temperature. For simplification, the indices as well
as the TO coefficients at 1550-nm wavelength and room temperature
(RT) are used here, as listed in Table 2. The validity of these
optical parameters has been verified as the calculated propagation
losses of plasmonic
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
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waveguides agree well with the experimental results measured at
1550-nm wavelength and RT [23]. Be noted that the quantitative
results in this paper are accurate only near 1550-nm wavelengths
and room temperature.
Table 2. Optical parameters of Si, SiO2, TiO2, and Cu at 1550-nm
and RT
Real part of index, n dn/dT (K1) Imaginary part of index, k
dk/dT (K1) Si 3.455 1.86 104 0 0
SiO2 1.445 0.1 104 0 0 TiO2 [21] 2.2 1.8 104 0 0 Cu [24] 0.282
4.1 104 11.05 4.3 104
3. Thermo-optic simulations
The effective modal indices of the curved Cu-TiO2-Si HPWs are
calculated using the eigenmode expansive (EME) method [25]. The Si
core has an asymmetric rib structure to mimic the donut resonator
shown in Fig. 1(b). The electrical field intensity distribution of
1550-nm fundamental TM mode in the curved HPW with parameters as
listed in Table 1 is depicted in Fig. 2(a), showing that the
electric field is enhanced in the TiO2 layer as well as the Si core
just beneath the TiO2 layer. The lateral confinement is well
provided by the Si core as the extended TiO2 region (i.e., over the
Si core region laterally) contains weak electric field. The
effective modal index is calculated to be 2.275 + i0.00577,
corresponding to a propagation loss of 0.203 dB/m, which is close
to the experimental result [23]. The ratios of optical intensity in
the TiO2 layer, the Si rib, the Si slab, and the surrounding SiO2
cladding layer are 41.3%, 42.0%, 0.04%, and 16.7%, respectively. As
expected, the thin Si slab has negligible effect on the optical
mode.
Cu
TiO2
Si
SiO2
(a) (b)-20 -15 -10 -5 0 5 10 15 20
2.270
2.272
2.274
2.276
2.278
2.516
2.518
2.520
2.522
2.524
5.4
5.6
5.8
6.0
6.2
9.2
9.4
9.6
9.8
10.0
tox=50nm
Rea
l par
t of e
ffect
ive
mod
al in
dex,
nef
f
Temperature deviated from RT (C)
tox=10nm
RT
x10-3
Imag
inar
y pa
rt of
effe
ctiv
e m
odal
inde
x, k
eff
Fig. 2. (a) Electric intensity distribution of the fundamental
TM mode at 1550 nm in the curved Cu-TiO2-Si HPW with structural
parameters as listed in Table 1; (b) The real part (neff) and
imaginary part (keff) of the effective modal index of two curved
HPWs as a function of temperature, the TiO2 thicknesses of these
two HPWs are 10 and 50 nm, respectively.
The real (neff) and imaginary part (keff) of effective modal
indices are plotted in Fig. 2(b) as a function of temperature in
the range of 20C deviated from RT for two curved HPWs which have
TiO2 thicknesses tox of 10 nm and 50 nm, respectively. Both neff
and keff depend on temperature almost linearly. Thus, TO
coefficients of the effective modal index (i.e., dneff/dT and
dkeff/dT) can be deduced from only two temperature points. neff
determines the resonant wavelengths (r) and keff determines the
extinction ratio (ER) of WRRs [26]. Since dkeff/dT is relatively
small for our HPWs, e.g., ~1.5 105 K1 for the 10-nm-TiO2 HPW and
~8.0 106 K1 for the 50-nm-TiO2 HPW, as read from Fig. 2(b), a WRR
can be claimed to be athermal if its dneff/dT is zero even its
dkeff/dT is not zero.
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
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Figure 3 plots dneff/dT versus tox for curved Cu-TiO2-Si HPWs.
As expected, dneff/dT decreases monotonically with tox increasing
because the ratio of optical intensity in the TiO2 layer increases.
However, the slope of the dneff/dT~tox curves decreases with tox
increasing. This observation can be explained by the fact that the
hybrid mode shown in Fig. 2(a) is a superposition of a pure SPP
mode located at the Cu/TiO2 interface (i.e., waveguide without the
Si core) and a pure optical mode located at the Si core (i.e.,
waveguide without the metal). The hybrid mode becomes more
optical-like when tox increasing [27]. In the extreme case when tox
is sufficiently larger (e.g., > ~200 nm), it behaves as a pure
optical mode as the conventional Si waveguide with TiO2 behaving as
a cladding layer, thus dneff/dT will be independent on tox. One
sees from Fig. 3 that the dneff/dT~tox curve depends on the Si core
width WP weakly, while depends on the Si height H strongly. The
critical tox at which dneff/dT = 0 (i.e., athermal point) depends
on WP weakly, while increases with H increasing. Electrically, a
thin gate dielectric is preferred to reduce the driving voltage.
Optically, the height of the Si waveguide should be thick enough
for vertical optical confinement. To balance the electric and
optical requirements, our modulator is set to be H = 160 nm and tox
= 50 nm. It is athermal as read from Fig. 3(b).
0 10 20 30 40 50 60 70 80 90 100-10
-5
0
5
10
15
20
25
0 10 20 30 40 50 60 70 80 90 100-10
-5
0
5
10
15
20
25
dnef
f/dT
(10
-5/C
)
TiO2 thickness, tox (nm)
WP=100nm WP=200nm WP=300nm
(a) (b) TiO2 thickness, tox (nm)
dnef
f/dT
(10
-5/C
)
H=340nm H=280nm H=220nm H=160nm
Fig. 3. The dneff/dT value of curved Cu-TiO2-Si HPW as a
function of the TiO2 thickness for (a) HPWs with H = 220 nm and WP
of 100, 200, and 300 nm respectively, and (b) HPWs with WP = 200 nm
and H of 340, 280, 220, and 160 nm respectively. The other
structural parameters are as listed in Table 1. The athermal point
is defined when dneff/dT = 0.
4. Electrical simulations
A semiconductor device simulation software MEDICI is used to
obtain the two-dimensional (2D) dynamic free carrier distribution
in the MOS capacitor at different biases, as in the case of
conventional Si MOS modulators [28]. The dielectric constant of
TiO2 is set to 80, which is reachable for a high-quality TiO2 film
[22]. The Si core of the resonator is n-type doped with
concentration (ND) of 5 1018 cm3 in the rib and slab region and 2
1020 cm3 in the contact region for good Ohmic contact. Auger
recombination, Shockley-Hall-Read recombination, surface
recombination, Fermi-Dirac statics, and the Modified Local Density
Approximation (MLDA) method in the MEDICI are included to account
for the heavy doping and the quantum confinement effect on the
carrier concentration near the TiO2/Si interface [29]. The 2D free
carrier distribution in the MOS capacitor under 5-V bias is shown
in Fig. 4(a). The accumulated electrons are located near the
TiO2/Si interface. To see the 2D distribution more clearly, the
figure near the interface is enlarged, as shown in Fig. 4(b). We
can see that the electron concentration contours are almost in
parallel with the TiO2/Si interface. Therefore, the 2D distribution
of free electron distribution N(x,y) can be simplified by a
one-dimensional
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
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(1D) distribution N(y). Figure 4(c) plots 1D electron
distributions along y-axis for the Cu-TiO2-Si MOS capacitor under
different biases ranging from 1 V to 8 V. At the 1 V bias, the
electrons are depleted from the interface. The depletion width Wdep
( 0.5DN
) is ~16.3 nm when ND = 5 1018 cm3. With the gate voltage
increasing, the electrons are accumulated at the interface,
maximizing at a short distance (~0.3-0.5 nm) away from the
interface due to the quantum mechanical effect and then decreasing
to ND quickly with the distance from the interface increasing. The
free-electron distribution approaches to the interface more closely
when V increases. These results agree well with the experimental
observation [30]. As a first approximation, the electron
distribution is approximated by a step function to define an
accumulation layer (AcL) which has width of tAcL and average
concentration of NAcL as:
0 0( ) .d dAcL FB dox AcL AcL
N V V Ee t t e t
= =
(1)
where 0 is the vacuum permittivity, d is the dielectric constant
of the gate dielectric, e is electronic charge, VFB is the
flat-band voltage, and Ed is the electric field in the gate
dielectric. Here, we simply assume tAcL = 1nm. At large V, tAcL
will be smaller than 1 nm and NAcL will be larger than that
predicted from Eq. (1). The achievable NAcL depends on the
breakdown field of the gate dielectric, and it can be larger than
1020 cm3 for modern CMOS devices.
V
TiO2SiO2 Metal
21020 cm-3 n-Si51018 cm-3 n-Si
5.01018
TiO2
7.91018
1.310192.010193.210195.010191.310203.21020
at V = 5V
0.0 0.5 1.0 1.5 2.0 2.5 10 20 30101610171018
1019
1020
1021tAcL
8V6V
4V
3V
2V
1V
0V
Ele
ctro
n co
ncen
tratio
n (c
m-3)
Distance from the TiO2/Si interface, y (nm)
-1V
Wdep
51018cm-3 n-Si
50nm TiO2, =80
Cu
V
y
(a)
(b) (c) Fig. 4. (a) Two-dimensional electron distribution of
Cu-TiO2-Si MOS capacitor under 5-V bias. Free electrons are
accumulated near the TiO2/Si interface. (b) Electron concentration
contour near the TiO2/Si interface, different color represents
different concentration. (c) One-dimensional electron distribution
along y-axis of Cu-TiO2-Si capacitor as shown schematically in the
inset under different biases. The depletion width Wdep and the
accumulation layer thickness tAcL are also indicated.
For transient state simulations, the gate voltage V is increased
from 0 to 8 V with the ramp time of the gate voltage of 10 fs. The
free electron concentration in the 1-nm AcL is plotted in Fig. 5 as
a function of time. Rise time (tr) of the electron concentration is
defined as the time for the electron concentration to increase from
10% to 90%, and fall time (tf) is defined as the time for the
electron concentration to drop from 90% to 10%. In the case of Si
n+-contact just
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
12705
-
below the electrode, as shown in Fig. 4(a), the sum of these
times (s = tr + tf) is ~29 ps. The modulation speed estimated from
the inverse of s is ~34 GHz. The speed can be improved by
shortening the distance between the accumulation layer and the n+
contact. In the case that the n+ contact in the Si slab is extended
to the Si rib, both tr and tf decrease, as shown by the dash curve
in Fig. 5. s is read to ~9.3 ps for this MOS capacitor, which
corresponds to a ~107-GHz modulation speed. However, the
propagation loss of the curved HPW will increase from 0.27 dB/m to
0.31 dB/m when the doping level in the Si slab is increased from 5
1018 cm3 to 2 1020 cm3 (while keeping the doping level in the Si
rib to be 5 1018 cm3). To balance the propagation loss and the
speed, the n+ doping may be extended to a certain location between
the Si rib and the electrode.
The proposed EO modulator is a MOS capacitor working between the
depletion and accumulation states. The switching energy Es per bit
of the MOS modulator can be roughly estimated as:
2 21 1 .2 2s dep dep accu accu
E C V C V= + (2)
where Cdep and Caccu are capacitances under the depletion and
accumulation states respectively. Because Cdep is smaller than
Caccu and Vdep is smaller than Vaccu, Es is mainly determined by
the second term of Eq. (2), namely the accumulation state. Caccu
can be
approximated to the gate oxide capacitance as 0
oxaccu
d
tC A
, where A is the active area.
For the modulator with structural parameters as listed in Table
1, A is 1.76 m2 and Caccu is ~25 fF. If the driving voltage is 6 V,
Es is estimated to be ~0.45 pJ.
10-12 10-11 10-10 10-91018
1019
1020
1021
10-12 10-11 10-10 10-9
0
2
4
6
8
NAc
L(cm
-3)
Time (s)
n+-contact just below the electrode: s=29ps
n+-contact extended to the Si rib: s=9.3ps
Gat
e vo
ltage
(V)
Fig. 5. (a) The transient response of the electron concentration
in the 1-nm-thick AcL of the Cu-TiO2-Si MOS capacitor at the gate
voltage variation between 0 and 8 V. Rise and fall time are defined
as 10% to 90% time period. The solid curve is for a MOS capacitor
with n+-contact just below the electrode as shown in Fig. 4(a) and
the dash curve is for a MOS capacitor with the n+-contact extended
to the Si rib.
5. Electro-optic simulations
The modification of Si reflective index (nSi + ikSi) at 1550 nm
depends on the free carrier concentration (Ne for electrons and Nh
for holes) almost linearly as:
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accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
12706
-
( )0.822 18
18 18
8.8 10 8.5 10 ,
8.5 10 6 10 .
e h
e h
n N N
N N
= + = +
(3)
However, when the free carrier concentration becomes very large,
e.g., > ~1020 cm3, n and will deviate significantly from the
above linear dependence. Instead, the well-known Drude model can be
used to estimate Si complex permittivity () and index at this
region [31]:
( ) ( )2 *
2 02
/ ( )( ) .1 / ( )
DSi Si
N e mn k ii
= + = +
(4)
where ( = 11.7) is the Si static permittivity, m* ( = 0.272 m0)
is the electron effective mass, and is the electron relaxation
time. The calculated nSi and kSi are plotted in Fig. 6(a) as a
function of ND at = 1550 nm ( = 1.2 1015 s1).
0 1 2 3 4 5 6 7 8 9 100.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0
0.2
0.4
0.6
0.8
1 2 3 4 5 6 7 8 9 10
-0.020
-0.016
-0.012
-0.008
-0.004
0.000
0.000
0.004
0.008
0.012
0.016
0.020
0.024
Electron concentration, NAcL (1020cm-3)
Rea
l par
t of S
i ind
ex, n
Si
at =1550nm
Imag
inar
y pa
rt of
Si i
ndex
, kSi
NAcL in 1-nm AcL (1020cm-3)
n eff m
odifi
catio
n, n
eff-n
eff(d
ep)
WP=100nm WP=200nm WP=300nm
Depleti
on
flat-ba
nd
(a) (b)
k eff m
odifi
catio
n, k
eff-k
eff(d
ep)
Fig. 6. (a) The real part (nSi) and imaginary part (kSi) of Si
refractive index as a function of free electron concentration in
the range of 1 102010 1020 cm3, calculated based on the Drude
model. (b) Modification of the calculated effective modal index of
the curved Cu-TiO2-Si HPWs as a function of NAcL in the 1-nm AcL
(compared with that in the depletion state).
The effective modal index is calculated as a function of NAcL in
the 1-nm AcL for curved Cu-TiO2-Si HPWs with WP of 100, 200, and
300 nm, respectively (the Si height H is 160 nm and the other
parameters are as listed in Table 1). The results are shown in Fig.
6(b). The index of the 5 1018 cm3-doped Si is set to 3.4506 +
i0.00123 based on Eq. (3) and the index of the depleted Si is
3.455. We can see that the real part of effective modal index
decreases and the imaginary part increases almost exponentially
with NAcL increasing, which can be explained by the increase of
optical modal confinement in the 1-nm AcL. For example, Fig. 7(a)
shows electric field (|Ey|) 2D distribution of fundamental TM mode
in the curved Cu-TiO2-Si HPW with NAcL = 1 1020 cm3. To see the
field distribution in the 1-nm-thick AcL more clearly, normalized
|Ey| 1D distributions along the y-axis at x = 0, shown as the dash
line in Fig. 7(a), are plotted in Fig. 7(b) for HPWs with NAcL = 1
1020 cm3 and 8 1020 cm3 respectively. In the case of NAcL = 8 1020
cm3, the electric field in the 1-nm AcL is dramatically enhanced.
Because the continuity of electric displacement normal to the
interfaces makes Ey is inversely proportional to the permittivity
roughly, the Si permittivity becomes smaller than the permittivity
of TiO2 when NAcL is larger than ~6 1020 cm3 as read from 6(a),
which results in dramatic enhancement of optical field in this thin
layer. To see this point more clearly, Fig. 7(c) plots the optical
intensity ratio in the 1-nm AcL, which is defined as the optical
power contained in this region over the total optical power, as a
function of NAcL. From Figs. 6(c) and 7(c), one sees that the
modulation efficiency is WP dependent. The HPW with 100-nm WP has
smaller modulation efficiency than the HPWs with larger WPs
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
12707
-
while the HPW with 200-nm WP provides similar modulation
efficiency as HPW with 300-nm WP. This is because the lateral
confinement is weak when WP is too small (e.g., < ~100 nm) while
it becomes saturated when WP is large enough (e.g., > ~200 nm).
On the other hand, the modulator with smaller WP has a smaller
active area, which means a faster modulation speed. To balance the
optical and electric performance, WP is set to be 200 nm for our
modulator.
y (n
m)
Cu TiO2 Si SiO2
1-nm AcL
(a) (b) (c)1 2 3 4 5 6 7 8 9
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Inte
nsity
ratio
in 1
-nm
AcL
(%)
NAcL in the 1-nm AcL (1020cm-3)
100nm 200nm 300nm
0.0
0.5
1.0
-20 -10 0 10 20 30 40 50 100 200 3000.0
0.5
1.0
Nor
mal
ized
ele
ctric
fiel
d, E
y NAcL=11020cm-3
Distance along y at x = 0 (nm)
NAcL=81020cm-3
Fig. 7. (a) The electric field (|Ey|) distribution of the
fundamental TM mode in the curved Cu-TiO2-Si HPW, (2) |Ey|
distribution along y-axis at x = 0, as shown by the dash line in
Fig. 7(a) in the cases of NAcL = 1 1020 cm3 and 8 1020 cm3,
respectively, and (c) Optical intensity ratio in the 1-nm-thick AcL
as a function of NAcL for HPWs with WP of 100, 200, and 300 nm,
respectively.
It has been experimentally demonstrated that the transmission
spectrum of HPW-based WRRs can in general be expressed as
[18,19]:
2 2
2 2
2 cos( ) .1 2 cos
t tTt t
+
=
+ (5)
where ( )2 / 2gn R = is the phase change around the ring, 2 is
the power loss factor per roundtrip around the ring, ( )expt t i=
is the self-coupling coefficient, and ng is the group index. 2 at
different states can be calculated from keff read from Fig. 6(b).
ng in the depletion state is calculated to ~3.782, close to the
experimental result [19]. Because it is difficult to calculate the
ng value accurately, we simply assume that the free-carrier effect
induced ng modification is the same as the neff modification,
namely ng = neff, thus ng at different states and different NAcLs
can also be read from Fig. 6(b). Since ng is larger than neff, this
assumption may underestimate the modification efficiency of our
modulator. The self-coupling coefficient t is related with the
cross-coupling coefficient k as 2 2 1t k+ = . The coupling between
the bus waveguide and the resonator depends on the separation
between them (gap) and the effective modal index difference between
the bus waveguide and the curved plasmonic waveguide [26]. The |t|
value can be varied in a large range from over-coupling (|t| < )
to under-coupling (|t| > ), depending on the detailed structural
parameters of WRRs [19]. The relationship between the coupling
strength and the WRRs structural parameters has been well studied
[26], and will not be studied in detail in this paper. Instead, we
simply assume |t| to a certain value and assume 0 = (i.e., no phase
shift due to coupling). Figure 8(a) plots the calculated spectra of
our modulator in the case of |t| = 0.8. Because is ~0.75 in the
depletion state, the resonator is slightly under-coupling with ER
of ~18 dB. With NAcL increasing, ng decreases and increases
simultaneously, which leads to r blue-shift and ER reduction as the
resonator becomes more under-coupled. In contrast, for conventional
Si WRR modulators, only the shift of r occurs and ER is not changed
because is almost independent on the applied voltage. One can see
from Fig. 8(a) that the EO induced modulation (modulation depth) is
enhanced in the region of wavelength larger than r due to this
additional modification, while in the region of wavelength smaller
than r, the
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
12708
-
modulation depth is weakened. Figure 8(b) plots the calculated
spectra in the case of over-coupling by assuming |t| = 0.5 which
has ER of ~8 dB in the depletion state. With NAcL increasing, the
resonance wavelength is blue-shifted and the ER increases. ER
becomes very large when approaches to |t|. The modulation depth is
enhanced in the region of wavelength smaller than r and is weakened
in the region of wavelength larger than r due to this additional
modification. It indicates that the performance of our plasmonic
modulators depends on |t| more sensitively than the conventional Si
WRR modulators. Moreover, because of the relatively large
propagation loss of the plasmonic waveguide (hence the small value)
as compared with the conventional Si waveguide, our plasmonic
modulators require strong coupling (hence the small |t| value)
between the bus waveguide and the resonator to meet the critical
coupling condition.
1500 1520 1540 1560 1580 1600-18
-16
-14
-12
-10
-8
-6
-4
-2
0
1500 1520 1540 1560 1580 1600-40-30-20-16
-14
-12
-10
-8
-6
-4
-2
0
Tran
smis
sion
(dB
)
Wavelength (nm)
Depletion Flat-band
Accumulation 11020cm-3
21020cm-3
31020cm-3
41020cm-3
51020cm-3
61020cm-3
71020cm-3
81020cm-3
At undercoupling, |t|=0.8
(a) (b)
Depletion Flat-band
Accumulation 11020cm-3
21020cm-3
31020cm-3
41020cm-3
51020cm-3
61020cm-3
71020cm-3
81020cm-3
Tran
smis
sion
(dB
)
Wavelength (nm)
At overcoupling, |t|=0.5
Fig. 8. Transmission spectra of the plasmonic donut modulator at
depletion, flat-band, and accumulation states with NAcL ranging
from 1 1020 cm3 to 8 1020 cm3, calculated based on Eq. (5). (a) At
under-coupling with |t| = 0.8 and (b) At over-coupling with |t| =
0.5.
The simple calculation based on Eq. (5) represents an ideal
condition of WRRs in which many effects are ignored. To verify the
results observed in Fig. 8, three-dimensional (3D) full-difference
time-domain (FDTD) simulation is performed. To enhance the coupling
between the bus waveguide and the resonator, a race-track shaped
resonator is used. The gap between the bus waveguide and the
resonator is set to 10 nm and the directional coupling length is
set to be 500 nm. The total footprint of the modulator inclusive
electrodes is ~8.6 m2, and the active area A is ~1.96 m2. To
minimize the simulation error during simulation in different
states, only the Si complex index in the 1-nm-thick AcL is changed
as read from Fig. 6(b), while all other settings including the grid
size keep the same. Figure 9(a) plots the transmission spectra for
the modulator under accumulation states with NAcL = 1 1020 cm3 or 6
1020 cm3, which corresponds to a bias of 1.1 V or 6.8 V,
respectively, according to Eq. (1). In the case of NAcL = 1 1020
cm3, ER is ~9 dB near 1550 nm, which corresponds to = ~0.74 and |t|
= ~0.85. When NAcL increases to 6 1020 cm3, the resonant waveguides
are blue-shifted by ~13.5 nm and ER is reduced to be ~2.5 dB
because of the reduction of to ~0.67. At 1546 nm wavelength, the
output power is modified from ~-10 dB to ~-2 dB by increasing NAcL
from 1 1020 to 6 1020 cm3. The optical power distributions in the
modulator with NAcL of 1 1020 and 6 1020 cm3 are shown in Figs.
9(b) and 9(c) respectively, and dynamic light propagation through
the modulator is shown by the attached movie (Media 1). One sees
that in the case of NAcL = 1 1020 cm3, the modulator is almost in
on-resonance as being close to |t|, which results in optical trap
in the resonator and small
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
12709
-
output power of ~-10 dB. In the case of NAcL = 6 1020 cm3, the
modulator is in off-resonator as being larger than |t|, which
results in large output power of ~-2 dB. Thus this modulator
operating at 1546 nm has modulation depth of ~8.0 dB and insertion
loss of ~2 dB. The relatively low insertion loss (as compared with
other plasmonic modulators [15,16]) is benefited from the
conventional Si channel bus waveguide. As the abovementioned, both
the modulation depth and the insertion loss can be further improved
by adjusting the coupling between the bus waveguide and the
resonator.
1400 1450 1500 1550 1600 1650 1700-12
-10
-8
-6
-4
-2
0
Tran
smis
sion
(dB)
Wavelength (nm)
NAcL=11020cm-3
NAcL=61020cm-3
(a)
(b) (c)
NAcL=11020cm-3 NAcL=61020cm-3
Fig. 9. (a) Transmission spectra of the plasmonic modulator
under accumulation with NAcL of 1 1020 cm3 and 6 1020 cm3, obtained
from FDTD simulation; (b) Media 1 the optical power density in the
modulator with 1 1020 cm3 NAcL at = 1546 nm, and (c) The optical
power distribution in the modulator with NAcL = 6 1020 cm3 at =
1546 nm. The output power is modulated from ~-10 dB to ~-2 dB at
this wavelength.
6. Conclusions
In summary, an ultracompact Si WRR modulators based on the
recently developed Cu-insulator-Si HPW is proposed. The modulator
is atheraml when TiO2 with a certain thickness is used as the
insulator. The EO modulation is achieved by free-electron
accumulation near the TiO2/Si interface. Significant modification
of both the real and imaginary parts of effective modal index of
the curved Cu-TiO2-Si HPW can be obtained if NAcL is large enough
(e.g., > ~6 1020 cm3), which leads to the resonance wavelength
blue-shift and the extinction ratio variation simultaneously. The
modulator provides a large modulation depth and a small insertion
loss after optimization of the coupling between the bus waveguide
and the resonator. The modulator offers high speed, up to ~100 GHz
dependent on the doping scheme, and low power consumption of ~0.45
pJ/bit owing to its ultracompact footprint. Moreover, the resonator
can be further scaled down to submicron radius and offers a
relatively large
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
12710
-
fabrication tolerance. These promising performances combined
with the fully CMOS compatibility make the proposed modulator very
attractive for dense Si EPICs.
Appendix
In this appendix, we discuss the further miniaturization of the
ring radius and the fabrication tolerance.
In the above analysis, the radius of the plasmonic resonator is
set to R = 1.5 m. The radius (hence the footprint of modulator) can
be further reduced. Figure 10 plots the effective modal index and
the TO coefficient as a function of the bending radius for curved
Cu-TiO2-Si HPWs with different WPs. The real part of effective
modal index decreases and the imaginary part increases with R
decreasing due to outward shift of the optical mode [20],
especially for the HPW with wider WP. A noticeable variation of the
effective modal index occurs when R is less than ~1.0 m for HPW
with WP of ~200 nm. It indicates that the bending radius of our
modulator can be reduced to ~1.0 m without noticeable performance
degradation. Moreover, the thermo-optic property keeps almost
unchanged for bending radius as small as 0.5 m, as shown in Fig.
10(b).
0.50 0.75 1.00 1.25 1.50 1.75 2.00-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0
1
2
3
4
5
6
0.50 0.75 1.00 1.25 1.50 1.75 2.00-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
n eff v
arai
tion,
nef
f(R)-
n eff(R
=)
Radius, R (m)
WP= 100 200 300
10-4
(a) (b)
k eff v
aria
tion,
kef
f(R)-
k eff(R
=)
nm nm nm
Radius, R (m)
dnef
f/dT
(10
-5/C
)
WP= 100nm 200nm 300nm
Fig. 10. (a) The effective modal index versus radius of curved
Cu-TiO2-Si plasmonic waveguides with different WPs (compared with
that of the corresponding straight plasmonic waveguide, namely R =
), (b) the thermo-optic coefficient of the real part of effective
modal index dneff/dT versus radius.
The TiO2/Cu cap of the Cu-TiO2-Si waveguide is designed to be
wider than the beneath Si core by WP, as shown in the inset of Fig.
11(a). In literature, the metal/insulator width in the
metal-insulator-Si hybrid plasmonic waveguides is either infinite
[18,19,27] or is the same as the Si core [32]. Here, the effect of
the width of the TiO2/Cu cap is studied. Figure 11 plots the real
and imaginary parts of effective modal index (compared to that with
the infinite wide TiO2/Cu cap, namely WP = ) as a function of WP
for curved Cu-TiO2-Si HPWs with WP of 100, 200, and 300 nm,
respectively. With WP increasing, neff increases first and then
decreases slowing while the keff increases continuously.
Nevertheless, the dependence of the effective modal index on WP is
weak, especially for HPWs with wider WP, in consistent with the
fact that a good lateral confinement can be provided by the beneath
Si core in the Cu-TiO2-Si HPWs when WP is wide enough (> ~200
nm).
In fabrication, the central line of the TiO2/Cu cap may misalign
from the central line of the beneath Si core, as shown
schematically in the inset of Fig. 12(a). In the case that the
TiO2/Cu cap is intentionally designed to be wider than the beneath
Si core by 50 nm in each side, the over-width in one side will be
(50nm - WP) and that in the other side will be (50nm + WP) if the
misalignment is WP. Figure 12 plots the real and imaginary parts of
effective modal index (compared to that without misalignment,
namely WP = 0) as a function of WP for
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
12711
-
curved Cu-TiO2-Si HPWs with WP of 100, 200, and 300 nm,
respectively. Both the real and imaginary parts of the modal index
decrease with |WP| increasing. While in the WP range from 10 nm to
20 nm, the variation of the effective modal index with WP is very
small, especially for HPWs with wider WP. It indicates that our EA
modulator has a relatively large misalignment tolerance of ~15 nm
for the TiO2/Cu cap fabrication. It should be noted that
fabrication of the coupling region will have small tolerance, as
the conventional Si WRR modulators.
0 10 20 30 40 50 60 70 80 90 100-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
0 10 20 30 40 50 60 70 80 90 100-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
WP WPWP
Si
Cu
TiO2
n eff v
aria
tion,
nef
f-nef
f(W
P=
)
WP (nm)
100nm 200nm 300nm
(a) (b) WP (nm)k e
ff var
iatio
n, k
eff-k
eff(
WP=
) (10
-4)
100nm 200nm 300nm
Fig. 11. (a) The real part of effective modal index for curved
Cu-TiO2-Si HPWs versus width difference between the TiO2/Cu cap and
the beneath Si core, WP, as shown schematically in the inset,
compared to that with the infinite wide TiO2/Cu cap, namely WP = ,
(b) The imaginary part of effective modal index versus WP.
-50 -40 -30 -20 -10 0 10 20 30 40 50-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
-50 -40 -30 -20 -10 0 10 20 30 40 50-15
-10
-5
0
5
n eff v
aria
tion,
nef
f-nef
f(W
P'=
0)
WP' (nm)
100nm 200nm 300nm
(a) (b)
50nm-WP 50nm+WPWP
Si
Cu
TiO2
k eff v
aria
tion,
kef
f-kef
f(W
P'=0)
(10
-5)
WP' (nm)
100nm 200nm 300nm
Fig. 12. (a) The real part of effective modal index for curved
Cu-TiO2-Si plasmonic waveguides versus the deviation of central
line of the TiO2/Cu cap from the beneath Si core, WP, as shown
schematically in the inset, compared to that without the deviation,
namely WP = 0, (b) The imaginary part of effective modal index
versus WP.
Acknowledgment
This work was supported by the Science and Engineering Research
Council of A*STAR (Agency for Science, Technology and Research),
Singapore Grant 092-154-0098.
#184420 - $15.00 USD Received 29 Jan 2013; revised 5 Apr 2013;
accepted 5 Apr 2013; published 16 May 2013(C) 2013 OSA 20 May 2013
| Vol. 21, No. 10 | DOI:10.1364/OE.21.012699 | OPTICS EXPRESS
12712