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Energy consumption in optical modulators for
interconnects
David A. B. Miller*
Ginzton Laboratory, Stanford University, 348 Via Pueblo Mall, Stanford California, 943054088, USA
*dabm@ee.stanford.edu
Abstract: We analyze energy consumption in optical modulators operated
in depletion and intended for lowpower interconnect applications. We
include dynamic dissipation from charging modulator capacitance and net
energy consumption from absorption and photocurrent, both in reverse and
small forward bias. We show that dynamic dissipation can be independent
of static bias, though only with specific kinds of bias circuits. We derive
simple expressions for the effects of photocurrent on energy consumption,
valid in both reverse and small forward bias. Though electroabsorption
modulators with large reverse bias have substantial energy penalties from
photocurrent dissipation, we argue that modulator diodes with thin
depletion regions and operating in small reverse and/or forward bias could
have little or no such photocurrent energy penalty, even conceivably being
more energyefficient than an ideal lossless modulator.
©2012 Optical Society of America
OCIS codes: (250.4110) Modulators; (200.4650) Optical interconnects.
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1. Introduction
Because of the quantum nature of optical detection [1], optics could reduce the energy for
transmitting information even at short distances inside computers and information switching
and processing machines [1–3]. Electrical interconnects in practice have to charge the
capacitances in electrical lines to the signal voltage. Optics avoids that electrostatic energy,
requiring only the energy to drive the optical transmitter devices (e.g., lasers or modulators)
and run any receiver amplifiers [3–5].
Especially for short distances, such as connections to or even within electronic chips, the
energy targets for optical transmitter devices are aggressive – possibly ~100 fJ/bit for longer
offchip distances, 10’s of fJ/bit for dense offchip connections and a few fJ/bit for global on
chip connections [3]. Modulators are particularly attractive for low energy transmitters
because, unlike lasers, they do not have a threshold that could limit the minimum operating
energy, and they may be easier to integrate monolithically with silicon.
There are various modulator approaches for integration with silicon [6]. Examples include
the predominantly electrorefractive carrier injection, carrier accumulation, and carrier
depletion devices using silicon as the active medium – especially lowenergy, highspeed
silicon ring [7–12] or disk resonators [13], electroabsorptive devices using GeSi [14], Ge [15–
17], or Ge quantum wells [18–25] on silicon, and rings with electrooptic polymers [26].
Carrier depletion and electroabsorptive devices typically utilize diode structures in reverse or
small forward bias. Such biasing conditions avoid dissipation from forward currents and
speed limits from carrier recombination times and can enable highspeed lowenergy
operation.
The physical mechanisms of electroabsorption (FranzKeldysh Effect (FKE) [27, 28] and,
especially, the related [29] QuantumConfined Stark Effect (QCSE) in quantum wells [30–
33]) are particularly strong, allowing relatively low energies even without resonators. By
contrast, lowenergy Si modulators typically require preciselytuned highQ resonators (e.g.,
Q ~10,000). As we clarify below, for example, one recent such QCSE device [25] without
such resonators has a dynamic dissipation of ~0.75fJ, lower than any reported Si device.
Gebased modulator devices are promising for integration with siliconbased electronics,
where Ge is already used extensively [34]. Modulators, unlike lasers, also appear to be
relatively tolerant of the crystal defects that arise under latticemismatched epitaxial growth
[35], with even IIIV’s on Si successfully demonstrated [35].
In this paper, we analyze energy in depletionbased modulator devices so that we might
better predict overall energy performance, design devices and drive circuits for minimum
energy, and provide a fair comparison between approaches. Our analysis suggests promising
directions especially for electroabsorptive devices, such as low or even forward biasing.
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Various mechanisms contribute to energy consumption in reverse or slightly forward
biased diode devices, including, (i) charging and discharging of the device capacitance
(dynamic dissipation), and, in electroabsorptive devices, the “static” dissipations from (ii)
absorption of optical power and (iii) dissipation from photocurrent flow. In quantumwell
electroabsorption modulators in particular static biasing can increase the sensitivity to small
additional voltage changes. Additionally, because the absorption edge can remain quite abrupt
even with large bias [23, 36], such bias allows the modulator’s operating wavelength to be
voltagetuned while still allowing low drive swing [19, 23, 36]. Such static bias can, however,
lead to additional dissipation from photocurrent and to some consequences for the dynamic
dissipation of the modulator that we discuss below. Silicon resonator modulators [6, 13] can
also require significant powers and energies per bit for tuning, though we will not discuss this
here.
We presume the modulators can be modeled as a fixed capacitor with a parallel current
source from any photocurrent. For the pin diodes typically used in electroabsorption
devices, the thickness of the iregion largely determines the capacitance, with only small
voltage dependence of capacitance from depletion into the p and n regions. Fixed capacitance
is a substantial approximation for some silicon carrier depletion modulators that may not use
pin diodes, but our model should allow reasonable comparisons. This model also covers
electrorefractive (electrooptic) devices in the form of either biased insulating materials, such
as electrooptic polymers [26], or diodes; presumably these would have little if any
photocurrent.
We start by analyzing the dynamic dissipation associated with charging and discharging
the capacitance, and continue with the static dissipation from absorption and photocurrent.
Finally, after summarizing dissipation results, we discuss implications for device design.
2. Dissipation from capacitive charging and discharging
In charging and discharging a capacitor, the unavoidable dissipation is associated with the
current flowing through the series resistance in the circuit. Here, we consider all the series
resistances (including those of any driver circuit and of the modulator contacts) to be gathered
into one effective series resistance R. This gathering together makes no difference to the final
energy consumption in our approach. As will be clarified in Section 2.2, the resistor may also
be nonlinear (i.e., depending on voltage) without changing our energy results, so this lumped
resistance also models the transistor output characteristics as far as those matter here.
2.1 Model drive circuit
To analyze energy consumption properly, we need to consider the circuit that charges and
discharges the modulator capacitance C, at least in a simplified form. Figure 1(a) shows an
example circuit based on a CMOS inverter, which swings its output between 0 V and
additional (reverse) bias of magnitude
B
V is also applied to the modulator. Figure 1(b) shows
an equivalent circuit where we have replaced the output stage with a switch, which can be
toggled between 0V and
DD
V
, together with our lumped series resistor R. The total charge
flowing through the modulator is represented by the current source
both photocurrent and leakage current. For simplicity we neglect leakage here, though it could
easily be added if needed. In Fig. 1, we also show an optional bypass capacitor
important function we explain later. For static dissipations, we neglect any voltage drop
across the resistor R from the current flow
DD
V
. An
RI , which could include
BP
C
, whose
RI R
RI , presuming sufficiently small resistances.
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n
ip
“top”
“bottom”
light out
light in
0 V
VB
VDD
logic
level in
CBP
VB
VDD
0 V
CBP
R
C
.
“high”
“low”
VOUT
IR
+

+

(a)
(b)
Fig. 1. (a) Example drive circuit for a pin modulator diode. (b) Equivalent circuit.
2.2 Dynamic energy calculation
Suppose then we change the switch from “low” to “high”. The voltage across C will change
from
B
V to
B DD
VV
+
. Because the electrostatic stored energy in a capacitor C at a voltage V is
2
(1/ 2)CV , the change in energy in the capacitor is
()
2
22
(1/ 2)(1/ 2)
CESB DDBDD DDB
ECVVV CVCV V
∆=+−=+
(1)
Note here that the total energy involved in changing the voltage across the capacitor by an
amount
DD
V
is not simply
(1/ 2)
DD
CV
.
The charge Q required to change the voltage on the capacitor C by
charge has been supplied by the
DD
V
voltage source at a voltage
has provided an energy
DD
VDDDD
Q
EVCV
∆==
through the bias voltage supply onto the bottom plate of the capacitor C . This charge flows at
a voltage
B
V , and so there is an energy from the bias supply
recognize as the last term on the right in Eq. (1).
The energy dissipated in flowing a charge
δδ
=
. When flowing a charge Q
δ
onto a capacitor C, the resulting change in voltage
δ
on the capacitor is such that QC V
δδ
=
. In the circuit, the voltage V across the resistor R
is
DD OUT
VVV
=−
. Hence we find the standard result that the total energy dissipated in the
resistor in charging the capacitor C from 0V to
2
DD
V
, so that voltage supply
=
also has to flow
is
DD
Q
CV
=
.This
DD
V
2
. The total charge
DD
Q
CV
BBDDB
Q
EVCV V
∆==
, which we
Q
δ
through a resistor at a voltage V is
EV Q
V
DD
V
is
()
2
0
(1/ 2)
DD
V
R DDOUT
V
OUT DD
ECVdVCV
∆=−=
∫
(2)
Note that this energy is independent of the value of the resistance R, and does not require
any particular relation between current and voltage for that resistor; the resistor may be
nonlinear.
In summary, in charging the capacitor from a “top” voltage of 0V to
−
, the VDD supply provides energy
which is provided to the capacitor, and half of which is dissipated in the series resistance, and
the VB (bias) supply provides an energy
B
E
∆
energy change in the capacitor as in by Eq. (1). Connecting the switch subsequently to 0V
discharges the capacitor, leading to a further dissipation
This argument leads to a question: what happens to the energy
pass the charge Q back through the bias voltage supply? Is that energy also dissipated? In
DD
V
while holding
=
, half of the bottom at a bias voltage of
B
V
2
DD
VDD
E CV
∆
DDB
CV V
=
to the capacitor to give the total
2
(1/ 2)
DD
R
ECV
∆
∆=
in the resistor.
=
BDDB
ECV V
when we
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other words, is the total energy dissipated per complete charge/discharge cycle just the energy
dissipated in the resistor R, that is
2
2
DISSRDD
EECV
∆= ∆=
(3)
or does it also include the energy supplied by the bias supply, i.e., a total of
2
DISSBR DDB
EECV V
∆= ∆+
? The answer depends on the physical nature of the bias voltage
source. If we regard the bias voltage as a nonrechargeable battery, the energy cannot usefully
be put back into the battery for future use, and will be dissipated in a variety of chemical
processes in the battery, generating heat; the energy dissipated per cycle is then
There is, however, a simple way to make the bias supply energy
essentially completely recoverable, which is to use a large bypass capacitor
bias supply, possibly formally decoupled from the actual bias supply by a series resistor
and/or inductor (not shown). Such bypass is commonplace for reducing the effective source
impedance of power supplies, though the use for the bias energy recovery discussed here may
be less obvious. Assuming the capacitance
BP
C
would not change appreciably during a charge/discharge cycle, this capacitor can be charged
and discharged without additional loss, functioning as a perfectly rechargeable battery; all the
resistive loss is already accounted for in the loss in the lumped series resistance R. Hence,
with the use of a bypass capacitor, the energy dissipated per complete charge/discharge cycle
is as given in Eq. (3). Such capacitance may already be present in the circuit in supply line
capacitance. Alternatively, even without a bypass capacitor this charge flow might simply act
to reduce the overall current flowing out of the bias supply for other dissipative reasons (e.g.,
for photocurrent), reducing overall dissipation by
effectively recovering this energy
B
E
∆
.
In driving a modulator with a real digital signal, the number of charge/discharge cycles
depends on the pattern of bits. When sending actual digital data, it is reasonable to expect
equal numbers of “ones” and “zeros”. In nonreturntozero (NRZ) signaling, in which we
have changes in the modulator state (i.e., charge or discharge) only when the digital signal
changes, there are four possible sequences of bits, all of which would be presumably equally
likely  00, 01, 11, 10 [13]. In two of these sequences (00 and 11) there is no change of
state, so no energy is dissipated in charging the capacitor. In the other two (01 and 10), half
of a charge/discharge cycle is involved in each such transition. On the average, therefore, in
an effectively random sequence of bits, there is one complete charge/discharge cycle every 4
bits. So, the total dissipated energy per bit would be (1/ 4)
capacitor over the bias supply) [13, 15], i.e., the dynamic energy per bit becomes
DISSB
=
across the
E
∆
.
BDDB
E CV V
∆
BP
C
is sufficiently large that the voltage across it
B
E
∆
and achieving the same end result of
DISS
E
∆
(when using a bypass
2
(1/ 4)
bitDD
E CV
∆=
(4)
This relation has been used by recent authors [13, 15]. We have shown here that this
number remains valid even in the presence of d.c. bias on the modulator, provided a bypass
capacitor is used on the bias supply. We have also given our result in terms of VDD, the supply
voltage, rather than just a peaktopeak drive voltage,
additional energy dissipated in the resistances in the driver circuit.
pp
V . If
pp
DD
VV
≠
, then we can expect
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Table 1.
Modulator type C
Drive
VDD
Mod.
Depth
Bias
VB
2
1
4
DD
CV
Launch Eff.
β
Energy Mag.
1/β
Si disk [13] 12fF 1V 3.2dB – 3fJ 20–26% 3.8–4.9
GeSi FKE [14] 11fF 3V 8dB 4V 25fJ 4% 25
Ge FKE [15] 25fF 4V 7.5dB 0V 100fJ 6.3% 16
Ge QCSE [25] 3fF 1V 3dB 4V 0.75fJ 5.6% 18
Proposed Ge QCSE
design [23]
24fF 1V 5dB 0V 6fJ 17% 6
Table 1 summarizes dynamic dissipations
modulators or modulator designs on silicon substrates using silicon [13] and/or germanium
[14, 15, 23, 25], materials. Voltages and capacitances come directly from the respective
papers, except the capacitance of the silicon disk modulator [13] is inferred from the
switching energy. (The other quoted factors β and 1/ β arise from optical absorption and
photocurrent, as discussed in Section 3.) For comparison also, a recent electrooptic polymer
ring modulator [26] in silicon slot guides has an estimated capacitance of ~27 fF, which, at a
drive voltage of 3V swing would have a comparable
2
(1/ 4)
DD
CV
for various published low energy
2
(1/ 4)
DD
CV
energy of ~60 fJ/bit.
2.3 Adiabatic operation
The concept of charging and discharging a capacitor without substantial loss in a non
repetitive signal has been discussed extensively in considering adiabatic electronics (e.g., Ref
[37].). If the VDD supply itself is cycled, for example by repetitively ramping up to a
maximum and then ramping down again, then a capacitor can be charged by connecting it to
the supply on the upswing of the voltage and can be discharged by connecting it on the
downswing. Provided the ramp cycle is long compared to the RC time of the capacitor and its
series resistance, essentially arbitrarily small energy need be dissipated to charge or discharge
the capacitor (e.g.,
(1/ 2)( / )
DD
ADs
E CVRC t
∆=
[37] for a voltage supply ramped linearly from
0V to VDD over a time ts; this energy can be arbitrarily small if
approaches will be used commercially is still an open question, in part because they may
require low clock rates, but they could largely eliminate dynamic dissipation.
2
stRC
≫
. Whether such
3. Energy dissipation from photocurrent
Electroabsorptive devices necessarily absorb some fraction of the input power since that is
how they modulate the signal. This absorption leads to additional mechanisms for energy
consumption. We seem to have two separate kinds of mechanisms – (i) directly absorbed
power from the incident light beam, and (ii) additional power dissipation from photocurrent
I
flowing over a biased region. We expect that, with some large reverse bias voltage
there will be a dissipated power
PCTOT
I V
∼
; this is generally true, as we verify below, and it
has been included in previous analyses [4, 38]. Here, however, we need a more detailed
discussion when considering devices run with low or forward bias voltages, for two reasons:
(a) some of the dissipation of the directly absorbed optical power also involves the movement
of charge and (b) some of the absorbed power may be recoverable in diode structures because
PC TOT
V
,
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they can behave as photovoltaic cells. To understand all the energy dissipations, we need to
look separately at reverse and forward bias.
3.1 Reverse bias dissipation
Most electroabsorption modulators use diode structures, with the active material being in a
nominally undoped or intrinsic iregion between pdoped and ndoped contact regions. Such a
structure allows the necessary high electric fields (typically up to ~105 V/cm) to be applied
under reverse bias without substantial static current flow (in the absence of photocurrent).
Under reverse bias, it is also common for all or nearly all of the absorbed energy to
generate photocurrent, at one electronhole pair per photon absorbed in the intrinsic region.
Photons absorbed in the contact regions may also lead to photocurrent, with carriers diffusing
into the iregion, but we neglect such absorption here. In QCSE modulators, the contact
regions are typically of larger bandgap so that they do not absorb the signal photons anyway.
The reason for the high photocurrent collection efficiency in reverse bias is that the field
typically sweeps the carriers out of the iregion in a short time, e.g., picoseconds to tens of
picoseconds, much shorter than typical recombination times of nanoseconds or longer. Even
in quantum well structures, where the barriers can hold the carriers within the wells, the
carrier emission (by thermionic and/or tunneling) is typically also on picosecond to 100’s of
picosecond timescales [39–41]. Consequently, photocurrent collection often saturates near
100% at low biases (e.g., 1 – 2V or less) as the iregion is depleted. It is possible to suppress
the photocurrent, for example by ion implantation [42], and hence also the dissipation
associated with it, though such approaches can compromise the electroabsorption [42].
We consider first the case of an unbiased homojunction photodiode (Fig. 2(a)), showing
the band structure in real space for a homojunction with bandgap energy EG. We make the
simplifying approximation that the Fermi levels in the p and n regions are at the valence and
conduction band edges. We assume for the moment that the photon energy
the case for most electroabsorptive mechanisms since the changes in absorption all occur near
to the bandgap energy. In Fig. 2(a), we show a photon being absorbed, generating an electron
in the conduction band and a hole in the valence band.
If we hold the overall voltage across the device at zero volts (so that the Fermi levels
remain the same at both sides of the device), then the electron will move downhill to the right
into the nregion, dissipating energy as it does so, for example through electronphonon
scattering, and ending up with an energy approximately equal to the Fermi energy
nregion. Similarly, the hole will move uphill into the pregion on the left, dissipating energy
through scattering until it also ends up with energy
ω
ℏ
is dissipated, all by scattering of the charge carriers within the diode, generating heat in
the diode. One electron of current flows through the external circuit also, but this current
flows at zero voltage in this example case, so no energy is dissipated in or sourced from the
external electrical circuit. The energy dissipated here has all come from the energy in the
photon to start with, so it also corresponds to the absorbed optical energy within the diode.
G
E
ω
ℏ≃
, which is
F
E in the
F
E . The net result is that a total energy of
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(a)(b)
(c)
Fig. 2. Diode band diagrams at (a) zero bias; (b) reverse bias
TOT
V
; (c) forward bias
F
V .
If we now reverse bias the diode by a voltage of magnitude VTOT, as in Fig. 2(b), the
electron (hole) scatters downhill (uphill) to the nregion (pregion) until it ends up at the
electron (hole) quasiFermilevel
Fe
E (
Fh
E ), dissipating a total energy
hole pair of ω
ℏ
(the absorbed optical energy) plus
diss
E
δ
for the electron
TOT
eV
, i.e.,
dissTOT
E eV
δω
ℏ
=+
(5)
where e is the magnitude of the electron charge. The flow of an electron of charge through the
external circuit requires the power supplies to contribute the energy
inside the diode by electron and hole scattering.
TOT
eV
that is dissipated
3.2 Forward bias dissipation
We could run a diode modulator into moderate forward bias
is “small” – that is, sufficiently low that there is negligible forward current flow. We could
choose
B
V to be a forward bias of magnitude
biased by an amount
DDF
VV
−
in the “high” state (when the “top” of the modulator is
connected to
DD
V
) and forward biased by an amount
the modulator is connected to 0V). When forward biased, the electron and hole would
dissipate an energy
F
eV
ω −
ℏ
within the diode as they move to their respective contacts, and
an energy
F
eV would be put back into the bias supply – a photovoltaic power generation.
Presuming the bias supply is reversible (e.g., by bypass capacitance), this energy can be
recovered. Hence, per photon, the dissipated energy from photocurrent when the device is
under a (small) forward bias of magnitude
F
V is
expression as Eq. (5) if we understand
TOT
V
= −
negative as required to represent a forward bias.
F
V (Fig. 2(c)) provided that
F
V
()
FB
VV
= −
. The diode would then be reverse
F
V in the “low” state (when the “top” of
F
eV
ω −
ℏ
, which gives us exactly the same
, i.e., the “reverse” bias voltage is now
F
V
3.2 Indirect gap semiconductors
In the case of a homojunction made from an indirect bandgap material like Ge, the electrical
band gap energy is the indirect band gap and the optical absorption of interest takes place near
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the higher direct band gap energy, as sketched in Fig. 3(a) for the zero bias case. The answers
above (i.e., Eq. (5)) remain unchanged. The photogenerated electron merely scatters down
into the lower indirect conduction band edge, a process that likely takes a short time (e.g., <
200 fs [41]). Still the entire photon energy is dissipated in the diode (or partly in the circuit in
the forward biased case).
(a)(b)
Fig. 3. Diode band structures for (a) indirect gap materials and (b) heterostructures.
3.3 Heterojunction diodes
Figure 3(b) shows a heterostructure for the zero bias case with band offsets in the conduction
and valence bands between the larger band gap p and n regions and a lower band gap material
within the iregion. Here we show the case where the iregion starts sufficiently within the
larger gap material that there is no charge accumulation in the narrower gap material. If the i
region starts immediately with the narrower gap material, the bands may kink near the
heterointerfaces, with the possible formation of charge accumulation layers just inside the i
region. Still, in either case the electron will move downhill into the nregion ending up near
the electron Fermi level there and similarly for the hole falling up into the pregion. The
precise process by which the electron and hole cross through any “kink” regions will vary
with the detailed designs and doping densities but the ultimate result will be the same; still the
entire photon energy is dissipated in the diode (or partly in the circuit in the forward biased
case). A quantum well structure has essentially the same behaviors as the simple
heterostructure of Fig. 3(b). Whether charge accumulation occurs in the quantum wells
nearest to the doped contacts or in the “barrier” materials just beside the doped contacts
depends on the detailed design of the structure, but still the entire photon energy is dissipated.
3.4 Expression for energy dissipation
In all these cases of different structures, the magnitude of the total photocharge generated (of
electrons or holes) is
()/
PC absTOT
QeEV
ω
=
ℏ
energy during some particular bit period for modulator under a total bias voltage
the additional energy associated with flowing
Q
Hence regardless of whether we are in reverse or small forward bias, or whether we have a
homojunction, a heterojunction, or a quantum well structure, we can use the same expression
for the energy dissipation. Specifically, the total energy
flow and from net absorption of optical energy in the device when the device is under a total
bias voltage VTOT is given by
()()
PCATOTabsTOTTOTPC
EVEVVQ
=+
, where
()
absTOT
EV
is the absorbed optical
TOT
V
, and
PC
over a voltage
TOT
V
is
PC TOT
Q V
.
PCA
E
dissipated from photocurrent
()()
1//
absTOTTOT
EVVe
ω
ℏ
=+
(6)
where we are assuming perfect photocurrent collection efficiency, our convention is that a
positive
TOT
V
corresponds to a reverse bias and
/ e
ω
ℏ
is a voltage numerically equal to the
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photon energy in electronvolts. Unlike the dynamic capacitive energy case, adiabatic
techniques cannot eliminate this energy because it is dissipated by scattering as the charge
carriers move inside the device itself.
3.5 Energy per bit from absorption and photocurrent dissipation
To understand the energy per bit, first we have to define some terms. In the conceptually
simplest optical communications transmitter we would turn on a light source of power P for a
bit time tbit to send a “one” and turn it off to send no power for time tbit to send a “zero”. In a
bit stream with equal numbers of “ones” and “zeros”, the average launched energy per bit
would then be
/ 2
bittransbit
EPt
=
, and the average launched power would be
hypothetical perfectly efficient light source, on the average the total energy
/ 2
P
. For a
bittot
E
required to
launch such an energy per bit would also be
light. Defining an energy launching efficiency β as
/ 2
bit
Pt
since all the energy goes into the emitted
/
bittrans
E
bittot
E
β =
(7)
for this ideal transmitter,
1
β = . Knowing this efficiency, then the total energy
bittot
E
/
β
we need
to supply per bit to launch an average energy per bit of
In a modulatorbased transmitter, the output light may not turn off completely, so we
consider the useful launched power or energy to be the difference between the power
energy
a.c. coupled receiver, this difference in powers or energies between “one” and a “zero” signals
is what matters. Some other receiver designs intended for interconnects, such as the
integrating doublesampling design [43], similarly work from this difference. Specifically,
then, for the useful launched energy per bit we would have the average
bittrans
E
is
bittot
E
bittrans
E
=
.
1P or
1 E launched for a “one” and the power
0P or energy
0
E launched for a “zero”. To an
10/ 2
E
bittrans
EE
=
−
(8)
We presume that, after accounting for any coupling losses into the modulator and for any
other losses in the modulator other than the optical absorption that takes place within it (e.g.,
optical scattering losses), there is an optical input energy per bit period of
transmitted, absorbed, reflected, deflected, or some combination of the these.
For some ideal modulator in a simple “on”/“off” signaling (as in nonreturntozero (NRZ)
signaling), we can imagine that all of the incident optical energy per bit
the “one” state and none in the “zero” state (so
at the modulator optical input in every bit period. Hence, even if the only energy involved was
the incident optical energy, the launching efficiency for such an ideal modulator would be
0.5
β =
.
Now, for a real modulator, we can usefully define the optical (power or energy) absorption
of the modulator as a function of bias voltage as (
the input energy
ino
E
that is not transmitted by the modulator. (For example, for a modulator
that can be described by an effective absorption coefficient α over a length L, we would have
() 1 exp[() ]
TOTTOT
VVL
ηα
= −−
). For a given voltage bias
transmitted optical energy is
ino
E
, which can be
ino
E
is transmitted in
would be present
/ 2
bittrans
E
ino
E
=
). Now,
ino
E
)
TOT
V
η
; equivalently, this is the fraction of
TOT
V
on the modulator, the
()()
1
outoTOTTOTino
EVVE
η
=−
(9)
and the absorbed energy
()
absTOT
EV
is
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()()
absTOTTOTino
EVVE
η
=
(10)
Whether a logic “one” corresponds to a “high” voltage on the modulator or a “low” one
depends on the details of the modulator. Typically for a FKE modulator or a QCSE modulator
with simple “rectangular” wells, we choose to operate with photon energies below the band
gap energy, in which case the absorption typically increases with increasing voltage. We can
call such a modulator “normally on” because at low reverse bias voltage it is in its more
transmitting state. Quantum well modulators can also be operated at somewhat shorter
wavelengths (larger photon energies), especially with coupled well designs, in which case
potentially useful modulation can be obtained in which the absorption decreases with
increasing (reverse bias) voltage [44,45]. We can call such a modulator “normally off” (it is in
its low transmission state at low reverse bias). The use of the modulus in Eq. (8) allows for
both possibilities, however. Using Eqs. (9) and (10) we have
()()
(1/ 2)/ 2
bittrans
E
outoDDB outoBinohilo
EVVEVE
ηη
=+−=−
(11)
where
()
hiDDB
VV
ηη
=+
and
()
loB
V
ηη
=
.
The total energy put into the modulator in a given bit period is the sum of the optical input
energy,
ino
E
, and the energy dissipated from photocurrent. The energy
includes both the photocurrent dissipation energy and any absorbed optical energy, so to get
the total energy put into the modulator in a bit period, we only have to add the optical energy
that is not absorbed, which is the transmitted energy
()
PCATOT
EV
already
(
+
)
outo TOT
EV
; the total energy put into the
()
TOT
V
. Adding up the total
modulator in a given bit period is therefore
energy put into the modulator in each of the two bias conditions and dividing by 2 to get the
average, we have the average energy to launch a bit when the optical input energy is
()
PCATOTo out
EEV
o in
E
:
()()()()
{}
(1/ 2)
bittot
E
PCADDBoutoDDBPCABoutoB
EVVEVVEVEV
=+++++
(12)
Substituting from Eqs. (6), (9) and (10) gives, after some algebra,
[]
2/ 2
bittot
E
inohihi
“photocurrent
, dimensionless numbers that correspond to
lolo
E
η µ
+
η µ
=
convenient
/ (
B
V
=
+
(13)
Here
(
=
we
V
+
have
)/ (
ℏ
defined
and
dissipation multipliers”
/ )
e
hiDDB
V
µ
the respective “high” and “low” total voltages across the diode, expressed in voltage units of
/ e
ω
ℏ
(i.e., equal to the photon energy in electronVolts). Finally, using Eqs. (7) and (8) we
obtain the energy launching efficiency
ω
/ )
e
lo
µω
ℏ
2
hilo
η µ
+
hihilolo
η
η µ
+
η
β
−
=
(14)
Note, incidentally, that the top line in Eq. (14) is just the magnitude of the change in
transmission
T
∆
between the “low” and “high” states of the modulator. If we are not
collecting all the photocurrent or none is generated in the “low” and “high” states, we can
proportionately reduce the multiplication factors
for no photocurrent). For a modulator that does not have photocurrent dissipation (such as a
silicon ring or disk [13]) but does nonetheless have only limited change in transmission
between the two states, instead of the ideal of
β =
lo
µ and
hi
µ respectively (reducing to zero
0.5
, we would therefore have
/ 2
T
β = ∆
(15)
which we can use for comparison of different kinds of modulators.
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We show results for the energy launch efficiency β of various modulators in Table 1. We
also give the “energy magnification factor” 1/ β . The detailed calculations are given in the
Appendix.
3.6 Example limiting cases
First, we note that a hypothetical modulator with
dissipation from flow of photocurrent in the low state (for example, by having
0
lo
µ =
) would have an energy launching efficiency
modulator as discussed above.
To see how this energy efficiency of Eq. (14) behaves more generally, let us presume for
simplicity that we are dealing with a high contrast modulator. As mentioned above, with
electroabsorption modulators, we need to distinguish between two cases of “normallyon” and
“normallyoff”.
For a highcontrast modulator in the more common normallyon case, we would have
1
hi
η ≃ , corresponding to essentially no transmission when the “high” voltage is applied.
Then
0
hi
η =
,
1
lo
η = , and no additional
V =
, as expected for an ideal
0
B
so that
0.5
β =
1
2
lo
η µ
+
hilolo
η
β
µ
−
+
≃
(16)
If the modulator was also highly transmitting in its “on” (“low”) bias state, i.e.,
then we obtain the simple formula
0
lo
η ≃
,
()
1/ 2
hi
βµ
+
≃
(17)
This case is simple to understand physically. In the “high” state, we absorb all of
we also have an energy dissipation from photocurrent flow of
have
ino
E
incident energy (which is also transmitted straight through the device). So we have
average total input energy (1/ 2)[(
inohi
E
µ
+
output energy
/ 2
ino
E
, leading to the ratio Eq. (17)
For the normallyoff high contrast case, we would have
ino
E
, and
hiino
E
µ
. In the “low” state, we
)](1/ 2) (2)
ino inoinohi
EEE
µ
+=+
and average
1
lo
η ≃ , leading to
1
2
hi
hihi lo
η
β
η µ
+
µ
−
+
≃
(18)
If the modulator was also highly transmitting in the “high” state (i.e.,
would have
0
hi
η ≃
), then we
()
1/ 2
lo
βµ
+
≃
(19)
3.7 Low energy operating modes for electroabsorptive modulators
One interesting possibility for low energy modulator operation is to run the diode into a small
forward bias that partially cancels the builtin field of the diode. In quantum well diodes with
thin depletion regions (e.g., 150 nm [46]) or very sensitive electroabsorptions (as, for
example, in the asymmetric coupled wells of Ref [45].), clear and strong electroabsorption is
seen in going from zero bias to small forward bias (0.75 V for Ref [46]. and 1 V for Ref
[45].), large enough to make potentially useful modulators.
For example, we could connect the “bottom” terminal of the diode directly to
eliminating the additional reverse bias supply, equivalently making
DD
V
,
BDD
VV
= −
, as in Fig. 4.
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Provided
diode (~band gap energy of the semiconductor in electronVolts), the device will still be in
small forward bias with small forward current. Obviously, then there is zero total voltage
applied across the diode in the “high” state, which makes
negative (
/( / )
loDD
Ve
µω
= −
ℏ
), so any photocurrent in this state is fed back into the bias
supply (or the bypass capacitor
BP
C
across it). Such a scheme is conceivably viable for Ge
quantum well modulators driven from low voltage CMOS. For example, with a
we could forward bias such a Ge quantum well diode with relatively little forward current. If
the depletion region in such a structure was hypothetically ~80 nm thick, which would allow
for several quantum wells, then the built in field in the diode could be
zero bias (presuming a Gelike diode structure with a band gap energy of 0.67 eV), decreasing
to
~ 3.4 10 V/cm
×
with a 0.4 V forward bias. Such field changes are more than enough for
strong electroabsorption with, e.g., 10 nm thick Ge wells [23] and for FKE devices.
DD
V
is significantly smaller in magnitude than the forward turnon voltage of the
0
hi
µ =
.
lo
µ actually becomes
0.4V
DD
V
∼
,
4
~ 8.4 10 V/cm
×
with
4
VDD
0 V
logic
level in
CBP
p
in
“top”
“bottom”
light
out
light
in
Fig. 4. Example drive circuit for a forwardbiased modulator diode.
For a normallyon modulator in the biasing scheme of Fig. 4, there could be some benefit
from this photovoltaic forward bias operation in the “low” state, though in a highcontrast
modulator in the limit where the modulator is also highly transmitting in the “low” state there
would be no actual photocurrent generated in that “low” state because there would be no
absorption in that state. The efficiency would then limit to
0
hi
µ =
and
1
lo
η = ), which is, however, as good as an ideal modulator; here, there would be
no excess dissipation from photocurrent flow.
For a normallyoff modulator operating at high contrast (
0.5
β =
(see Eq. (16) when
1
lo
η ≃ ) biased as in Fig. 4,
( ) (
/ 2
)
1
hilo
βηµ
−+
≃
(20)
For
/ 2
hi lo
ηµ
< −
, where we remember that here
0
lo
µ <
in forward bias, the energy
launching efficiency β would actually be larger than 50%, better than in an “ideal”
modulator. This improvement compared to the “ideal” comes because of the photovoltaic
energy generation when the modulator is in its “low” (and strongly absorbing) state.
Whether we run in the normallyon or normallyoff modes, driving the modulator into
forward bias, especially in the simple biasing scheme of Fig. 4, can lead to very low energy
dissipations. In either case, under such biasing we can have situations where there is no
energy penalty from the photocurrent generation, with the possibility of even a slight energy
benefit from photocurrent in the “normallyoff” mode.
4. Total energy dissipation
The core result of this analysis can be summarized in one expression. For a modulator run in
reverse bias and/or in a small enough forward bias that any forward current is negligible, or
for an insulating modulator, the minimum total (optical plus net electrical) energy required to
send a bit of information is on the average
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2
(1/ 4)/
bitDDbittrans
EE CV
β
=+
(21)
where we presume a driver that is otherwise as efficient as it can be (other than using
adiabatic schemes). Here, C is the modulator capacitance, and
voltage (presumed to be the same as the driver supply voltage in calculating dissipated
energy).
bittrans
E
is the optical energy or energy difference we want to launch, on average, into
the optical channel to send a bit. β is the optical launch efficiency: if the total of all the
optical input energy and all the energy dissipated by optical absorption and by photocurrent in
the modulator is on the average
bittot
E
for one bit, then
be regarded as an energy magnification factor because it is the number by which the average
launched optical energy per bit has to be multiplied when calculating the total energy used by
the modulator.
We have established one expression for β , Eq. (14), which works for absorptive
modulators, including the effects of absorbed optical power, power dissipation from the flow
of photocurrent and even power recovery from photovoltaic effects in forward biased devices.
Underlying this expression for β is one formula, Eq. (6), for the static energy (i.e., from
absorption and photocurrent) dissipated in the modulator, an expression that holds even for
heterojunctions, including quantum wells, and indirect gap semiconductors. If we regard the
numerator in Eq. (14) for β more generally as the difference in transmission between the two
states of the modulator, and turn off the terms that correspond to power dissipation from
photocurrent, the same expression also works for modulators that generate no photocurrent
(regardless of whether they are absorptive, refractive, or some combination), in which case
the simpler expression Eq. (15) can be used for β . We show example results from published
modulators and designs in Table 1.
DD
V
is the peaktopeak driver
/
bittrans
E
bittot
E
β =
(Eq. (7)). 1/ β can
5. Conclusions
Optical modulators are attractive compared to lasers for lowenergy dense optical
interconnects because they offer thresholdless operation and easier integration with Si. We
analyzed energy in modulators that operate in depletion. The analysis applies to diode devices
run anywhere from small forward bias (small enough that the forward current is negligible) to
large reverse bias, and to insulating modulators. Even with optimally efficient driver circuits,
energy consumption occurs from dynamic dissipation in capacitive charging and discharging,
from absorbing or otherwise disposing of optical energy that is not transmitted, and any
dissipation from flow of photocurrent.
In the dynamic dissipation, we clarified that, provided a bypass capacitor is used on the
bias supply, the dynamic energy per bit can indeed be written as
even though the energy moved in and out of the capacitor can be much larger than this. We
also conclude that this dynamic energy is not fundamental. In principle it can be avoided by
adiabatic operation, though whether this is practical is still an open question in electronic
systems generally.
Our analysis suggests several conclusions for the design of future lowenergy modulators.
First, reducing the optical energy required at the receiver by, for example, reducing the
capacitance of the photodetector and its integration with the transistor circuits helps
interconnect links generally to reduce their operating energy per bit. This reduction is,
however, particularly important for electroabsorptive modulators because they tend to
magnify the energy used through photocurrent dissipation, especially at high bias voltages.
Based on recent results showing low dynamic energies per bit (e.g., 3 fJ/bit [13] and 0.75
fJ/bit [25]), we can conclude that we are going to be able to make modulators with very low
dynamic dissipation. Electroabsorption modulators such as those of Ref [25]. could also
2
(1/ 4)
DD
bit
E CV
∆=
(Eq. (4))
#161542  $15.00 USD
(C) 2012 OSA
Received 17 Jan 2012; revised 24 Feb 2012; accepted 27 Feb 2012; published 1 Mar 2012
12 March 2012 / Vol. 20, No. S2 / OPTICS EXPRESS A307
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