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Abstract—An overview is provided of waveguide-coupled

graphene optoelectronics. A review of the optical properties of

graphene is first provided and a motivation for waveguide-

coupled graphene optoelectronics is given. This motivation is

largely based upon the increased interaction length that can be

achieved using such geometries. A derivation of the optical

absorption for graphene interacting with a guided waveguide

mode wave is provided. Device concepts for waveguide-coupled

graphene optoelectronic devices, including optical modulators,

photodetectors, and polarizers operating in the near- and mid-

infrared regime are then described. This discussion provides a

specific emphasis on the effect of disorder on the expected

performance and energy consumption of graphene-based optical

modulators. Finally, an outlook for future areas of exploration is

given.

Index Terms—Waveguides, Optical Communication,

Nanotechnology

I. INTRODUCTION

A. Graphene background and optical properties

HE unique and extraordinary properties of graphene

have led to consideration of this two-dimensional (2D)

allotrope of carbon for a wide range of device applications

[1-5]. Among its exceptional properties are high mobility [6],

thermal conductivity [7], and mechanical strength [8]; a

gapless linear dispersion relation [9]; and symmetric

ambipolar conductivity [2]. However, perhaps some of the

most interesting properties of graphene are related to its

optical characteristics. For instance, graphene is an ultra-

wideband absorber [10], which is a direct result of its gaplass

band structure; has tunable inter-band absorption [11] as a

well as saturable absorption [12] at high power levels, both

consequences of the low density of states in graphene; and

extremely-high and tunable absorption of terahertz radiation

[13], a consequence of the high dc conductivity and variable

density of states. While the original studies on graphene were

performed on small (e.g. ~ 20 x 20 m2) flakes created by

mechanical exfoliation, the pioneering work in [14]

demonstrated that large sheets of single and multi-layer

graphene can be synthesized readily on metal substrates.

Metal-synthesized graphene can then be transferred onto

arbitrary substrates, allowing wafer-scale processing, thereby

Submited on May 13, 2013. This work was supported in part by the NSF

under Grant No. ECCS-1124831, the AFOSR under Award No. FA9550-12-1-0338, and the University of Minnesota College of Science and Engineering.

S. J. Koester and M. Li are with the University of Minnesota-Twin Cities,

200 Union St. SE, Minneapolis, MN 55455, USA. (e-mails: [email protected] , [email protected]).

making practical device applications within reach. Among the

potential optoelectronic device applications of graphene

include optical modulators [15-17], photodetectors [18,19],

polarization controllers [20] and ultrafast pulsed lasers

[12,21].

The primary mechanisms that govern optical absorption in

graphene are inter-band and intra-band optical transitions [22].

These basic mechanisms, described in detail later, are depicted

pictorially in Fig. 1. Inter-band transitions occur when an

electron in the “valance” band of graphene absorbs a photon

and is excited to an empty state in the “conduction” band with

the same momentum. Such a transition can only occur if there

is a filled state at an energy, = ħ / 2 and an empty state at

= +ħ / 2, where is the energy relative to the Dirac energy.

This means that if (at zero temperature) the absolute value of

the chemical potential, |c| is less than ħ / 2, then inter-band

absorption can occur, whereas if |c| is greater than ħ / 2,

inter-band transitions are blocked. This is the so-called Pauli

blocking mechanism, and is depicted in Fig. 1(a). The intra-

band absorption mechanism is shown in Fig. 1(b). These

optical transitions, which require a phonon-assisted process,

occur within the band and are most important at longer

wavelengths (e.g. far-infrared to terahertz, etc.) and at high

carrier concentrations. As the chemical potential approaches

the Dirac energy (c 0), the intra-band transitions

probability also tends toward zero. Waveguide-coupled

graphene optoelectronic devices tend to operate in the near- to

mid-infrared wavelength range, and therefore, the inter-band

optical absorption mechanism tends to dominate in most

circumstances.

Waveguide-Coupled Graphene Optoelectronics

Steven J. Koester, Senior Member, IEEE, and Mo Li, Member, IEEE

T

Fig. 1. Cartoon depicting (a) inter-band and (b) intra-band absorption mechanisms in graphene.




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In the remainder of this paper we describe how the optical

properties of graphene can be utilized to realize practical

optoelectronic devices. We first describe the detailed optical

properties of graphene and motivate the reason for integration

of graphene with planar photonic wave-guides. Next, we

describe the results of optical absorption measurements of

graphene on Si waveguides that confirm the theoretical

prediction that complete optical absorption can be obtained in

these structures. We next describe the device applications of

graphene-on-waveguide structures, and specifically, review

the status in the field on waveguide-coupled optical

modulators, photodetectors and polarizers. Finally, we

describe future trends and opportunities for graphene-based

waveguide-coupled optoelectronic devices.

II. MOTIVATION FOR WAVEGUIDE COUPLING

Planar photonic waveguides, as well as other photonic

elements including photonic band gap structures and various

types of optical resonators and cavities, are building blocks of

photonic integrated circuits (PIC) and optoelectronic systems.

The integrated circuits for light can guide and route optical

signals on a chip to leverage the unparalleled bandwidth

available in the optical spectrum for communications. In

particular, utilizing silicon’s high index of refraction and

transparency in the important near-infrared band for

telecommunication, silicon photonics promises to integrate

PIC with CMOS circuitry, leading to large scale integrated

(LSI) optoelectronic systems for optical interconnections,

chip-to-chip and intra-chip, and for telecommunication and

optical signal processing applications [23]. Silicon, as an

indirect band-gap semiconductor and centrosymmetric crystal,

however, lacks many important properties that are critical for

active optoelectronics functionalities. Other silicon compatible

materials including silicon nitride (SiN) and silicon oxynitride

(SiON) also can only be used as passive optical materials. To

achieve infrared light emission and photodetection with high

efficiency, the only viable approaches are hybrid integration of

active optoelectronic materials, such as III-V compound

semiconductors and germanium, onto the passive silicon

photonic platform [24,25]. Adding new materials to silicon

photonics allows the best properties of different materials to

be fully exploited in highly integrated, multi-functional

optoelectronic systems.

The 2D structure of graphene and the planar configuration

of silicon photonics are inherently compatible with each other.

Coplanar integration can be readily achieved by transferring

and laminating graphene on top of silicon photonic substrates,

which can be planarized with cladding materials. Such

integration, on one hand, allows graphene’s novel

optoelectronic properties to be effectively utilized in photonic

devices. On the other hand, silicon photonic circuits can

provide a versatile platform to investigate and characterize the

fundamental optical properties of graphene. Although

conventional free-space optical measurement systems have

been instrumental in investigating these properties, practical

application requires graphene to be integrated with existing

photonic technology. Furthermore, coplanar integration

eliminates the restriction of the interaction length between

optical field and graphene, which is limited to the thickness of

graphene (3.3 Å per monolayer) for normal-incident light.

With sufficient length of interaction, graphene induced

optoelectronic effects can become strong and graphene’s

optical properties can be more accurately measured.

Specifically, using graphene integrated on a silicon photonic

waveguide, electrically tunable optical absorption has be

achieved and exploited to create an electro-absorptive optical

modulator [26]. Also, it is well known that graphene has a

universal fractional optical absorption of 2.3% per monolayer

over a wideband of optical frequency for normal incidence,

due to the linear band structure. Converted to absorption

coefficient using graphene’s thickness of 3.3 Å, this

absorption corresponds to an extremely large value of 70.5

m-1

(or 306 dB/m). But with 97.7% transmission through

such a thin film of one atomic layer, graphene is virtually

transparent. When used for photodetection, this low absolute

absorption diminishes the external quantum efficiency of a

graphene photodetector, even though internal efficiency can

be very high [18]. Coplanar integration with a waveguide is an

obvious solution to this problem, and both theoretical and

experiments results on these structures will be described.

III. WAVEGUIDE-COUPLED GRAPHENE OPTICAL DEVICES

A. Optical properties of graphene on waveguides

When graphene is laminated on top of a photonic

waveguide, its coupling with the wave-guide mode induces

optical absorption and dispersion, which can be characterized

by the real and imaginary parts of graphene’s dynamic

conductivity (), respectively. Both inter-band and intra-

band transitions in graphene are responsible for the

conductivity and their relative contributions depend on the

optical frequency, , and the chemical potential, . The total

2D conductivity is complex and can be expressed as

= intra + inter = + i. To relate this 2D conductivity to

the volume permittivity, it is suggested to divide the 2D

conductivity by the thickness of graphene, , as

= 0 + i/ = (0 /) + i/ in the calculation and

then take the limit of 0 [27]. Because of its purely 2D

nature, however, care must be given not to treat graphene

simply as a thin sheet of isotropic material in calculations and

simulation.

Using the Kubo formula [22], the intra-band and inter-band

conductivity of graphene are given by

2

intra 22ln 1

2c Bk TcB

B

e k Ti e

i k T

, (1)

and

2

inter 2 22 0

2

2 4

d die i f fd

i

, (2)




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where fd() = 1/{exp[ c)/kBT]+1} is the Fermi-Dirac

distribution function [28, 29]. Fig. 2 plots the real and the

imaginary parts of the conductivities as a function of optical

wavelength at room temperature and several values of

chemical potentials that correspond to reasonable levels of

carrier concentration in graphene. Here, the charged particle

scattering parameter in graphene is assumed to be

ħ = 5 meV. It can be seen that at telecom wavelengths ( ~

1.55 m), the real part of the conductivity is dominated by the

inter-band contribution with a universal value of

0 = e2/4ħ = 6.08 × 10

-5

-1 over a broad spectral range when

ħ > 2|c|. This real conductivity leads to optical absorption

through inter-band transition, which generates photocarriers

and thus is relevant to photodetection. Most notably, when

ħ < 2|c|, the inter-band conductivity drops significantly due

to Pauli blocking. Thus, using electrostatic gating to control

the chemical potential and tune the optical absorption in

graphene is the principle to implementing graphene based

electro-absorptive modulators [15-17]. At the same time, if ħ

> 2|c|, the optical absorption of graphene is strong

considering its one-atomic-layer thickness. With its high

intrinsic carrier mobility, the strong inter-band absorption

indicates graphene can be used for high-speed photodetection

over a wide spectral range.

The imaginary parts of inter-band and intra-band

conductivities are on the similar order of magnitude and can

have different signs in the regime considered here. This fact

plays an important role in determining the mode (TE or TM)

of surface waves that can propagate on graphene [30]. Such an

effect has been utilized to realize a fiber integrated graphene

polarizer [20], which will be discussed in detail later.

Maximizing and controlling optical absorption in graphene

integrated on waveguides is critical to the development of

highly efficient graphene photodetectors and electro-

absorptive modulators. By calculating the real part of

graphene’s dynamic conductivity, the optical absorption can

be calculated using the Poynting’s theorem of energy

conservation. From Ohm’s law, the surface current density

on graphene is:

s 0 t, ,t tJ r E r , (3)

where Et is the electric field component that is in the plane of

graphene. Thus, the time-averaged Ohmic power dissipation

per unit area Qs (W/m2) in the graphene layer is given by

2*

s t 0 t t 0 ts

1 1, , , ,

2 2Q t t t t r J r E r E r E r E . (4)

The factor of one-half in (4) is due to the fact that the peak

value is used in the electric field vector Et (hereafter the

dependence on position and time will not be explicitly

displayed). Considering incident plane electromagnetic wave

of an arbitrary polarization state, Et, consists of two transverse

components (see Fig. 3 for the definition of the vectorial

directions):

t t1 1 t2 2ˆ ˆE E E t t , (5)

2 2*

0 t t 0 t1 t2s

1 1

2 2Q E E E E , (6)

where 1t̂ and

2t̂ are the two orthogonal unit vectors in the plane

of graphene, while Et1 and Et2 are complex field amplitudes.

The method outlined above can be used to calculate

absorption of a normal-incidence plane optical wave due to

inter-band transitions. In the specific case shown in Fig. 3(a),

the ratio be-tween absorbed power and total incident power is:

*

0 t ts 0

*inc 00 inc inc

1

2 2.3%1

2

Q

I cc

E E

E E

, (7)

Fig. 2. Inter-band (solid lines) and intra-band (dashed lines) contribution to the dynamic conductivity in graphene. The vertical black line marks the

telecommunication wavelength of = 1.55 m.




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where e2/40ħc = 1/137 is the fine-structure constant defined

in MKS system. In the above expression, the relationship

Et = Einc is required by the boundary conditions. Thus, starting

from graphene’s conductivity, we reach the same value of

absorption as obtained using Fresnel’s formulas in the thin

film limit [31,32].

We next consider the absorption by a graphene monolayer

parallel to and at a distance above a propagating waveguide

mode, as is shown in Fig. 3(b). The overlap between the

evanescent field of the waveguide mode and the graphene

layer leads to linear absorption in graphene. Compared to the

graphene absorption, the intrinsic propagation loss of a

strongly guided optical mode in the waveguide is negligible.

Thus, the power P(z) of the waveguide mode propagating

along the z direction decays exponentially as:

0 expP z P z , (8)

where P0 is the waveguide mode power at z = 0 and is the

linear absorption coefficient (m-1

). is defined as the power

absorbed by graphene per unit length normalized to total

waveguide mode power:

s

1 1( )

W

dP zQ x dl

P z dz P z . (9)

Here we assume the graphene layer is parallel to the x-z plane

and has a width of W in the x direction. The expressions in

equation (6) can be used in the integral for TE mode as:

20

0 in-plane 0( ) ( , )2 W

y E x y dxP z

, (10)

where y0 is the position of the plane of graphene.

The mode of a rectangular waveguide has to be solved

numerically using finite element method (FEM) tools. The

field distribution Ein-plane(x, y0) which includes components Ex

and Ez for the quasi-TE waveguide mode, can be used in the

above formulas to calculate the graphene absorption. This

field integration method is intuitive and revealing, but requires

simulation to use a very fine mesh at the graphene layer to

obtain accurate numerical values of the in-plane electrical

field. A more computationally economical and robust way to

calculate the absorption is to use the effective index method.

The complex mode index ñ = n + i can be readily solved

using FEM mode solvers in which the graphene layer is set as

a boundary layer with surface impedance of 0-1

. The linear

absorption coefficient is then calculated from the extinction

coefficient, , as = 4/. In the FEM model, graphene is

incorporated as a boundary described by

)]nEnJHHn ˆˆ(ˆ[)(ˆ21 s

between two regions of

dielectrics. Here n̂ is the surface normal unit vector and is

the complex conductivity of graphene. In Fig. 4, the calculated

linear absorption coefficient is plotted for graphene integrated

on three types of silicon waveguides: 110 nm thick, 220 nm

thick strip waveguides and a horizontal slot waveguide. The

distance h between the graphene layer and the surface of the

waveguides is varied. The evanescent field outside the

waveguide decays exponentially with h, and so does the

absorption coefficient, , of graphene. A larger waveguide

provides higher confinement of the optical mode, and thus less

mode coupling with graphene and less absorption. In order to

enhance graphene absorption, it is therefore beneficial to

sacrifice the mode confinement by reducing the waveguide

size. Furthermore, a slot waveguide can concentrate the

optical field in the void region filled with low index material

where graphene can be inserted. This leads to considerable

improvement in absorption compared with the simple strip

waveguide configuration.

To precisely measure the absorption properties of a

graphene-on-waveguide structure, we have employed an

interferometric method to avoid the need to determine the

absolute optical power levels, which may introduce large

uncertainties due to unreliability of optical alignment in the

experiments [33]. Instead, the extinction ratio measured in the

fringes of the interferometer output is used to determine the

difference of power loss between two interferometer arms of

graphene integrated waveguides. The experimentally

measured values of absorption coefficient were found to agree

with theoretical values well. The highest achievable linear

absorption for a monolayer of graphene in such waveguide

systems is 0.2 dB/m, which means that in a 100-m-long

linear waveguide, graphene can absorb 99% (20 dB insertion

loss) of the optical power. This leads to the possibility that

photodetectors with near-unity external quantum efficiency

and relatively small device footprints could be realized. For

electro-absorptive modulation, an extinction ratio of 3 dB can

be achieved in a device as short at 15 m. Using multiple

layers of graphene on the waveguide can further reduce the

device size.

Finally, we note that although the absorption effect is

strong, the presence of graphene layers on the waveguides

induces negligible dispersive effect to the mode profile and

real index of the waveguide modes. Simulations have

confirmed this to be valid for up to ~ 10 layers of graphene

layers in the middle of a slot waveguide. This assumption is

still valid up to ~ 50 graphene layers. Therefore, it is

reasonable to assume that in any practical device geometry,

the dispersion effect induced by graphene is negligible. Stated

another way, although the effective volume refractive index of

Fig. 3. Schematics showing the electric fields of (a) the normal incident optical field onto a graphene sheet and (b) the propagating field along a

graphene sheet in a guided mode.




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graphene can be high: reff

~ 0/0 ~ 17, due to the atomic

thickness of graphene, the aggregate effect is too small to

disturb a well-guided optical mode in the waveguide

significantly. Surface modes, however, can exist on graphene

and can have significant effects on the polarization states of

the optical mode, which will be discussed in the latter section.

B. Optical modulators

Electro-absorption optical modulators are perhaps the most

compelling and obvious optoelectronic application of

graphene due to the tunable optical absorption that can be

achieved in graphene. Furthermore, as described above,

waveguide geometries are particularly well suited for optical

modulators, due to the fact that nearly complete optical

absorption can be achieved in a graphene-on-waveguide

geometry. However, design of waveguide-coupled graphene

modulators brings about significant challenges, particularly

regarding how to gate graphene without incurring significant

optical losses. These challenges are different than those

incurred using Si optical modulators. In Si, electro-optic

effects are relatively weak, and therefore modulation is

achieved using plasma dispersion effects, where injected

electrons and holes induce slight changes in the Si refractive

index [34]. By utilizing a Mach-Zehnder or ring resonator

design, a tunable phase shift can be induced thus leading to

tunable destructive or constructive interference that creates the

optical modulation signal [35]. Si/SiGe optical modulators

have also been proposed using either the Franz-Keldish effect

or the quantum-confined stark effect, which increases the

electro-optic effects and thus allows more compact modulator

designs. In all of these structures, the electrical contact region

is distinct from the optically active region of the device.

However, in graphene-based optical modulators, graphene

serves the dual role of both electrode and optically active

material. Given this relatively unique mechanism for action,

novel designs are needed for realizing optimized optical

modulators.

The first experimental results on graphene-on-waveguide

optical modulators were reported by Liu, et al. [17]. In this

device structure, a graphene optical modulator is created by

transferring graphene above a silicon waveguide and then

gating the graphene through the waveguide itself. Gating

action is achieved by making a separate contact to the silicon

waveguide through a thin, heavily-doped Si region adjacent to

the waveguide. The basic principle of this device operation

was established and modulators operating at 3.5 V with

extinction ratio (ER) and bandwidth (BW) of ~ 3 dB and

1.2 GHz, respectively, were demonstrated. While this was the

first demonstration of a waveguide-coupled graphene-based

modulator, the design utilized in that work has several

shortcomings. First of all, the device requires electrical

contact to be made through the silicon waveguide itself, and

therefore, the Si needs to be doped and contacted using a thin,

heavily-doped extension region. The high doping can

introduce free carrier absorption in the Si waveguide, leading

to increased insertion loss. It also introduces an extra level of

process complexity. The silicon-gated design, by its very

definition, also requires electrical contact to the silicon, and

therefore, if high speeds are desired, silicon with high

crystalline quality is needed and other dielectric materials

cannot be used as the waveguide material. This negates one of

the key advantages of graphene for optoelectronic

applications, that of its transferability onto arbitrary of

substrates and its performance in a broad range of the optical

spectrum.

Due to the limitations of the waveguide-gated modulator

design described by Liu et al., more recent work has focused

on designs where graphene itself is utilized as a transparent

gate electrode, which then modulates the absorption in a

second graphene layer. This graphene-on-graphene geometry

can significantly improve the modulator speed, reduce the

insertion loss, simplify the device design, and allow

modulation of both near- and mid-infrared optical signals. A

basic graphene-on-graphene optical modulator design is

shown in Fig. 5 [36]. This device structure consists of a

waveguide (which is generally silicon, but could also be made

out of numerous other materials depending upon the

application), a dielectric spacer layer above the waveguide, a

first layer of graphene, a second dielectric layer and then a

second layer of graphene. The lower dielectric layer should be

as thin as possible to ensure optimal absorption in the

graphene, while the thickness of the insulator separating the

two graphene layers can be optimized, depending upon the

requirements of the application. Such dielectric layers could

consist of SiO2, high- dielectrics such as HfO2, or other

materials such as hexagonal boron nitride (BN) [37]. Ohmic

contacts to the two graphene layers are made to the side of the

waveguide at sufficient distances to avoid excessive optical

losses by absorption in the metal contacts. A standard silicon-

on-insulator (SOI) substrate can be used to fabricate the

waveguide layer for use in a spectral range from = 1.1 μm to

mid-infrared with wavelength up to = 3.6 μm. However, due

to the fact that no electrical contact is made to the waveguide

material, much greater flexibility in the choice of the

waveguide and cladding materials exists compared to standard

Fig. 4. Graphene integrated in three types of silicon waveguides: 110-nm

and 220-nm-thick strip waveguides and horizontal slot waveguide. The

contour plot of horizontal electrical field shows the fundamental TE modes

of the waveguides. Graphene absorption coefficients up to 0.2 dB/m can

be obtained with the slot waveguide.




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SOI-based PICs. For instance, polysilicon or amorphous

silicon could be utilized for the waveguide, which may be

advantageous for integration into the far back-end of silicon

CMOS circuits. In addition, at longer waveguides ( >

3.6 μm) the absorption loss in silicon dioxide becomes

significant, and therefore, other waveguide / cladding material

combinations could also be possible, Ge/SiNx, Ge-on-Si, and

chalcogenide glasses [38].

For graphene-on-graphene optical modulators to operate

properly, the doping in graphene must be controlled. Doping

control can allow the chemical potential in the lower graphene

layer to be positioned at the optimal energy of + ħ / 2, as well

as allow the top graphene layer to be tuned so that it does not

absorb light or can modulate light in phase with the lower

graphene layer. However, since such doping control is not

necessarily possible or easily achieved, it is likely that a DC

bias will be needed in order to tune the graphene into its

optimal mode of operation.

In our previous work, we calculated the bandwidth,

exaction ratio and insertion loss of graphene-on-graphene

modulators at various doping levels for different wavelengths.

Here, we show refinements of these calculations based upon

our theoretical and experimental calculations for the

absorption coefficients in waveguide-coupled graphene. For

waveguide-coupled optical modulators, the RC-limited

bandwidth can be calculated from the total series resistance

and capacitance of the modulator:

ms

BCR

f2

13 , (11)

where Rs is the series resistance and Cm is the modulator

capacitance. The capacitance, in turn, is determined by the

series combination of the quantum capacitance, cQ, in each

graphene layer and the oxide capacitance per unit area, cox =

3.90 / EOT associated with the dielectric separating the two

graphene layers. Here 0 is the permittivity of free space and

EOT is the SiO2-equivalent oxide thickness. The total

capacitance, Cm, can then be express as

mm

Qox

Qox

m LWcc

ccC

5.0

5.0, (12)

where Wm and Lm are the modulator width and length,

respectively. Assuming the series resistance is composed of

components associated with the contact resistance, Rc and

graphene sheet resistance, Rsh, then the RC-limited bandwidth

can be expressed as:

Qox

Qox

mextmshc

Bcc

cc

WWWRRf

2

4

13

, (13)

where Wext is the distance between the metal contacts and the

waveguide edge. It is instructive to point out that, to first

order, the bandwidth does not depend upon the modulator

length, but does decrease with decreasing EOT. In general,

the insertion loss, L, and the extinction ratio, ER, can be

expressed (in dB) as:

maxlog10 TL , (14)

and

minmax /log10 TTER , (15)

where Tmax (Tmin) is the maximum (minimum) transmission

coefficient within a modulator voltage cycle. Previously, we

showed that modulator bandwidths in excess of 100 GHz

(30 GHz) can be achieved at = 1.55 m (3.5 m) for a

graphene separation thicknesses of 20 nm and driving voltages

of 8 V (5 V). In Fig. 6, we show modified results where the

graphene absorption coefficients have been updated based

upon the results in [39]. The optimal extinction ratio is

obtained when there is no doping in graphene, so that when a

bias is applied, the graphene layers modulate each other in

cooperative fashion. However, the highest bandwidth occurs

when the top graphene layer is either heavily n- or p-type

doped, and thus had the lowest resistance. High insertion

losses occur when the top graphene layer is absorbing, and the

lowest extinction ratio occurs when the absorption modulation

in the top and bottom graphene layers is out of phase.

A key performance concern for graphene-on-graphene

optical modulators based upon chemical vapor deposited

(CVD) graphene is random disorder in graphene [40]. In our

prior work, we utilized a random potential model to calculate

the effect of disorder on the modulator performance. More

recently, we have implemented an effective temperature

model to account for the disorder and verified this model

using capacitance-voltage measurements of metal-oxide-

graphene capacitors [41]. In this model, the quantum

capacitance per unit area, cQ, is expressed as:

,2

cosh12ln8

22

2

effF

eff

QkT

hc

vh

kTec (16)

where

22

0 TTTeff . (17)

Fig. 5. Structure of the basic graphene-on-graphene modulator integrated on a Si/SiO2 waveguide. The overlay shows the typical mode profile of the

waveguide’s fundamental TE mode [36].




7

In (17), T is the actual temperature and T0 is an empirical

parameter that expresses the degree of disorder induced by

random potential fluctuations. The effect of increasing values

of T0 are also shown in Fig. 6, where it can be observed that

disorder has a strong effect on the insertion loss and extinction

ratios that can be achieved in these devices. In particular,

increasing disorder reduces the extinction ratio and increases

the insertion loss, with both trends arising from the increasing

“difficulty” for the gate voltage to completely shut off

absorption in the graphene layers. In order to recover a fixed

value of extinction ratio, higher drive voltages are needed, and

the implications of this on the energy performance of the

modulators will be discussed in the following section. It is

interesting to note that increasing T0 actually slightly improves

the bandwidth in our calculations due to the increasing carrier

concentration at fixed bias. However, this trend could be an

artifact of the calculations due to the fact that the carrier

mobility, , was kept constant with increasing T0, while in

reality, and T0 could be interdependent.

Recently, Liu, et al. [42] reported experimental results of

graphene-on-graphene modulators where the graphene layers

were separated by Al2O3. Since the layers in that work

utilized CVD graphene, the results provide a good test for our

simple modeling results. In [42], 6.5 dB extinction ratio was

achieved for a voltage swing of roughly 6 V, along with an

insertion loss of -4 dB and bandwidth of 1 GHz. Utilizing our

model, we are able to match this using a using a value of T0 ~

600 K. This value is consistent with T0 values extracted from

CVD graphene on metal-oxide-graphene capacitors [41]. We

also calculate a 2 dB insertion loss, but predict a 20 GHz

bandwidth assuming Rc = 2 -mm and = 4000 cm2/Vs. The

bandwidth discrepancy is likely due to higher series resistance

arising from either higher contact resistance than the modeling

assumption or due to partial breakage of graphene over the

waveguide edge, a phenomenon often observed in our group

for graphene transferred onto patterned samples.

To further understand the trade-offs associated with the

modulator structural parameters and random potential

fluctuations, we have analyzed the bandwidth efficiency trade-

offs associated with graphene-based optical modulators. For

Fig. 6. (a) Calculated values at = 1.55 m for the (a) extinction ratio, (b) insertion loss and (c) RC-limited bandwidth for graphene-on-graphene optical

modulators assuming the dimensional parameters in [36] for values of the disorder parameter, T0, ranging from 0 to 1500 K in steps of 300 K. (d)-(f) same

parameters as in (a)-(c) except at = 3.5 m. The linear absorption coefficients have been adjusted compared to [36] based upon the calculations in [39].




8

this work, an extension of our results in [39], we assumed that

graphene is optimally doped (or that an appropriate DC bias

was applied), such that both the top and bottom graphene

layers are ideally tuned to have cooperative absorption

modulation. Further assuming that metallic losses and free

carrier absorption can be ignored, relations for the modulator

insertion loss, L, and extinction ratio, ER, (in dB) can be

derived and expressed as:

m

om

e

LL

1

2explog10

, (18)

1

12explog10

m

m

ome

eLER , (19)

where 0 is the linear absorption coefficient, while the

dimensionless term, m, describes the effectiveness of an

applied voltage at creating a potential change in the graphene,

and can be approximated by the expression:

oxQefft

pp

ccV

Vm

22

1. (20)

In the quantum capacitance limit (cQ << cox), m Vpp/4Vt-eff,

where Vpp is the peak-to-peak modulator voltage, and Vt-eff =

kTeff/e is the effective thermal voltage, including the effective

temperature defined in (17). It can be further shown that for a

fixed insertion loss of L = 3 dB that an extinction ratio of

ER = 3 dB can be achieved when m = 4.ln(2). At room

temperature, and with T0 = 0, this corresponds to a peak-to-

peak voltage swing of only 72 mV.

Finally, assuming a random bit stream, the minimum

energy of modulation can be expressed as:

2

min4

1ppmVCE . (21)

Previously, in [39], we calculated the minimum energy, and

bandwidth-energy trade-offs associated with graphene-based

optical modulators assuming pristine graphene. Here we

include the effects of random potential fluctuations and the

Fig. 7. (a) Calculated minimum switching energy, Emin, for graphene-on-one graphene optical modulators vs. wavelength in the quantum capacitance limit for

different values of the disorder parameter, T0. (b) Calculated Emin vs. EOT for different values of T0. (c) Calculated minimum peak-to-peak drive voltage vs. EOT for different values of T0. (d) Minimum energy-delay product for different values of T0 vs. wavelength. Here EOT = 1 nm. All calculations in (a)-(d)

assume graphene layers on TE-mode Si/SiO2 waveguides with optimized width and length adjusted for ER = 3 dB and L = -3 dB.




9

results are summarized in Fig. 7. In all plots shown in Fig. 7,

the insertion loss and extinction ratio are fixed at L = 3 dB

and ER = 3 dB, respectively.

In Fig. 7(a), the minimum switching energy in the quantum

capacitance limit (EOT 0) is plotted vs. wavelength for

increasing disorder, represented by increasing T0. The

minimum energy at = 1.55 m is less than 1 fJ/bit, which is

largely due to the extremely-small voltage needed for

modulation. The increasing energy at longer wavelengths is a

result of the wider waveguides needed at longer wavelengths

as well as the reduced linear absorption coefficient.

Increasing the disorder is shown increase the energy

consumption required to provide fixed performance, with

Emin ~ 4 fJ/bit for T0 = 600 K, but still less than 10 fJ/bit for

T0 = 900 K. Of course, having EOT 0 is impractical and

some thickness of dielectric is needed for practical devices.

Due to the fact that it is the Fermi-level movement within

graphene that leads to optical modulation, any voltage drop in

the region between the layers is “wasted.” Therefore, the

switching energy increases with increasing EOT as shown in

Fig. 7(b). Nevertheless, for realistic values of EOT = 2 nm and

T0 = 600 K, values of Emin ~ 10 fJ/bit remain achievable. Fig.

7(c) shows the associated voltages need to achieve L = 3 dB

and ER = 3 dB, respectively, and the plot indicates that

increasing disorder and EOT increase the modulation voltages

required. Finally, even though reducing the value of EOT

improves the energy efficiency of the modulator, because of

the fact that the resistance does not scale as the capacitance is

increased, the bandwidth is reduced with lower EOT. This

results in a fundamental bandwidth-efficiency trade-off, and

values of the energy-delay product calculated for different

values of T0 and at different wavelengths are shown in

Fig. 7(d). For these calculations, EOT was fixed at 1 nm. A

key potential advantage of graphene-based optical modulators

is their low voltage of operation compared to other types of

modulators, such as many Si and Si/Ge based devices.

Therefore, minimization of the random potential fluctuations,

and thus T0, will be critical in the future to realize the full

benefits of graphene-based optical modulators.

C. Photodetectors and other devices

The strong and broadband inter-band absorption and high

room-temperature carrier mobility in graphene suggest it can

be utilized for broadband, ultrafast photodetection. It has been

predicted that the operational wavelength range of graphene

photodetectors should range from 300 nm to 6 m or longer.

In the near-IR, graphene detectors have demonstrated

bandwidths as high as 40 GHz in a normal-incidence

configuration. The intrinsic bandwidth of graphene

photodetectors is predicted to be up to 500 GHz [18].

However, normal-incidence graphene photodetectors suffer

from very low responsivity of only 1.5 mA/W when unbiased

and ~ 6 mA/W when a bias voltage is applied [19]. To

improve the responsivity, optical cavities [43] and quantum

dots [44] have been integrated with graphene, but these

approaches sacrifice the detector’s bandwidth or speed. Dark

current is a significant concern for graphene photodetectors.

Due to the gapless nature of graphene, operating a

photodetector with an applied bias voltage results in a large

dark current and is thus undesirable for realistic applications.

Therefore, practical graphene photodetectors must operate at

zero bias. This requirement means that a built-in field needs

to be created between the contacts, so that the photogenerated

carriers can be swept to the electrodes before recombining,

and can be accomplished by using electrodes of different work

functions [19]. Furthermore, the graphene chemical potential

needs to be “tuned” to be near the Dirac energy, to ensure

maximum field penetration into the graphene [19]. In addition

to photovoltaic generation of carriers, photo-thermoelectric

and hot carrier effects also play important roles in graphene

photodetectors and can augment the detection responsivity if

the doping level in the graphene and the contact regions can be

optimally controlled [45,46].

Use of a waveguide integrated photodetector is promising

to augment the responsivity without the compromises of other

approaches. As discussed earlier, the reduced optical field

overlap with graphene can be compensated by a very long

interaction length. An integrated structure similar to that used

in the optical modulator demonstrated by Liu et. al. [17] can

be used. The optical loss due to metal contacts, however,

needs to be minimized.

The Dirac spectrum of electron-hole systems in graphene

also leads to very interesting new surface modes of

electromagnetic waves propagating in graphene that are highly

sensitive to the relative contributions of inter- and intra- band

transitions to the total complex conductivity. The intra-band

contribution of the dynamic conductivity in Equation (1) is

similar to the Drude conductivity of other 2D electron systems

including metal surfaces and 2D electron gases in

semiconductor quantum wells [47]. In these electron systems,

it is well-known that only TM modes (surface plasmons) are

supported and propagate. This can be determined from the

boundary conditions when the imaginary part of conductivity

is positive (intra > 0) [29,30]. In graphene, however, the

inter-band transitions contribute to the dynamic conductivity

with a negative value of imaginary conductivity, as shown in

Fig. 2. When ħ / 2 > |c| inter-band contributions dominate

so the total imaginary conductivity is negative ( < 0) as

shown in Fig. 8. It is predicted that, in this regime, only TE

surface modes can exist while the TM mode is no longer

supported and thus will have a high loss [30]. Bao et al.

utilized such an effect to implement a fiber integrated

graphene polarizer [20]. The device includes monolayer and

few-layer graphene transferred on an optical fiber with an

exposed core. Because CVD-grown graphene with high p-type

doping is used, the condition < 0 is satisfied over a wide

spectral window. Therefore, only the TE modes (guided and

leaky modes) can couple and be weakly guided at the fiber

graphene interface while the TM modes will suffer a high loss.

This difference in propagation loss between two types of

modes leads to polarizing of the input light. An extinction

ratio of 27 dB at telecommunication band is reported although

the 5 dB insertion loss due to graphene is relative high. Such a

loss is inevitable because of inter-band absorption but a sweet




10

spot balancing the polarization extinction ratio and the loss

may exist by controlling the chemical potential in graphene.

To achieve high extinction ratio, the fiber-based polarizer is

more than several millimeters long. Such a large footprint is

due to the large core size of the fiber and consequently the

weak coupling with the graphene layer and the low differential

loss factor between two polarizations = TM – TE. To

reduce the size of the graphene polarizer, graphene can again

be integrated on a nanophotonic waveguide to attain a much

stronger light-graphene interaction. Such a waveguide-

integrated polarizer can achieve high polarization extinction

ratio with a significantly shorter length and the total footprint

can be further reduced by using a spiral waveguide layout

design [48]. Moreover, electrostatic tuning of the chemical

potential can change the sign of . For example, as shown in

Fig. 8, at 1.55 m the sign of changes from negative when

c = 0.4 eV to positive whenc > 0.5 eV, leading to a change

of the supported surface mode (TE or TM) on graphene and

alternation of the polarizer function. Thus, such a waveguide-

integrated polarizer can be electrically controlled with a gate

electrode. Although such an electrically-controlled polarizer

has yet to be demonstrated, a change of the polarizer function

with chemical potential has been observed in a polymer

waveguide-integrated device [49]. Such an integrated tunable

polarization controller is highly desirable to avoid polarization

dependent loss (PDL) in optical communication systems.

IV. OUTLOOK

In principle, the gapless band structure of graphene allows

it to be in resonance with electromagnetic radiation of any

frequency. As reviewed above, graphene has been exploited

for optoelectronic devices working in the visible, near-

infrarad, terahertz and radio-frequency spectral ranges. Its

application in the interesting mid-infrared (mid-IR) range,

however, has not been explored. The mid-IR spectral range,

broadly defined as the wavelength range of 210 μm, has

attracted tremendous scientific and technological interest

recently. The increased interest is spurred by the fast

development in mid-IR quantum cascade lasers (QCLs) and

fiber lasers, which now are available as off-the-shelf products

on the market [50,51]. Mid-IR is very promising for free-space

optical communication and remote sensing because there are

two transparency windows (35 and 813m) of atmosphere

in the mid-IR band; and for absorption spectroscopy to detect

and analyze chemical species for remote sensing and health

diagnosis.

Graphene has a great potential in this important spectral

range. In particular, optical modulators and photodetectors in

the mid-IR are very challenging to develop because the low

photon energy of mid-IR light requires semiconductor

materials with very small gap band. Current mid-IR

photodetectors are based on narrow gap compound

semiconductors such as indium antimonide (InSb), HgCdTe

and their heterostructures, which are expensive to grow,

difficult to integrate and requiring cryogenic cooling to obtain

high sensitivity. The same principle of utilizing graphene’s

strong and tunable inter-band absorption can be applied for

detection and modulation mid-IR optical signals. The photon

energy in the mid-IR is in the range of 120-620 meV so for

inter-band absorption, the chemical potential in graphene

needs to be less than 60-310 meV.

Additional challenges remain for the control of the

graphene properties to optimizing graphene devices’

performance. Random potential fluctuations, in particular,

remain a challenge, particularly for CVD-grown graphene,

where the multi-domain structure creates inherent potential

fluctuations. However, even integration of pristine graphene

on high-K dielectrics offers challenges, because of the

disorder associated with the substrate materials. In addition,

recent work on phonon effects in high-K dielectrics, such as

HfO2 [52], indicate that phonon-assisted optical transitions

may limit modulation depth and insertion loss. Utilization of

2D materials such as BN offers a possibility for improvements

in the optical losses, however BN does not have a high

dielectric constant. Therefore, for low photon energy doping

control remains an issue in graphene-based devices, both for

photodetectors and optical modulators. For modulators,

control of doping levels to precisely target the ideal operating

point would be useful, since application of a DC bias to tune

the modulators is non-ideal due to leakage currents and

reliability issues associated with application of large voltages.

For photodetectors, the ability to achieve low doping levels is

critical since field penetration into graphene is essential in

order to achieve high internal quantum efficiencies.

V. CONCLUSIONS

In conclusion, an overview waveguide-coupled graphene

optoelectronics has been provided. The optical properties of

graphene have been reviewed, and these motivate waveguide-

coupled geometries due to the increased interaction lengths

that can be achieved. A review of the device concepts for

waveguide-coupled graphene optoelectronic devices has been

provided, and these include optical modulators,

photodetectors, and polarizers. Optical modulators operating

in the near- and mid-infrared regime are shown to be capable

Fig. 8. Imaginary part of dynamic conductivity of graphene at different

optical wavelengths and values of the chemical potential, c. When the

imaginary conductivity is negative, graphene is predicted to support TE polarized surface waves; otherwise, TM polarized surface waves are

supported.




11

of bandwidths above 100 GHz and 30 GHz, respectively.

Doping control is shown to be important to maximize the

extinction ratio and minimize insertion loss. Graphene optical

modulators are shown to be capable of energy consumption of

less than 1 fJ/bit in the ideal quantum capacitance limit using

pristine graphene. However, random potential fluctuations

and practical device geometries could increase this value by

roughly a factor of 10. Motivation for waveguide coupling in

other graphene-based devices is also provided and these

devices include photodetectors and optical polarizers.

Graphene devices have tremendous potential as a platform for

broadband photonic integrated circuits and if continued

improvement in the material quality and integration processes

for graphene-on-waveguide structures can be made, then the

outlook is good that such potential can be realized.

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Steven J. Koester (M’96–SM’02)

received the B.S.E.E and M.S.E.E.

degrees from the University of Notre

Dame, Notre Dame, IN, in 1989 and

1991, respectively, and the Ph.D. degree,

in 1995, from the University of California,

Santa Barbara, where his research

involved the study of quantum transport in

InAs quasi-1-D structures.

He has been a Professor of Electrical and Computer

Engineering in the College of Science and Engineering at the

University of Minnesota, in Minneapolis, MN since 2010.

Prior to joining the University of Minnesota, he was a

Research Staff Member with the T. J. Watson Research

Center, IBM Research Division, Yorktown Heights, NY

where his work involved Si/SiGe devices and materials, high-

speed Ge photodetectors, and III-V MOSFETs. His most

recent position at IBM was Manager of Exploratory

Technology where his team investigated novel device and

integration solutions for post-22-nm node CMOS technology.

Dr. Koester’s current research involves investigations into

the device applications of graphene, including novel sensors,

spintronics, and optoelectronic devices. He has authored or

coauthored more than 160 technical publications and

conference presentations, and is the holder of 46 U.S. patents.

He was the general chair of the 2009 Device Research

Conference and is currently an associate editor of IEEE

Electron Device Letters.

Mo Li received the B.S. degree in Physics

from the University of Science and

Technology, China in 2001, the M.S.

degree in Physics from the University of

California, San Diego in 2003, and the

Ph.D. in Applied Physics, from the

California Institute of Technology in

2007.

He has been an Assistant Professor of

Electrical and Computer Engineering in the College of

Science and Engineering at the University of Minnesota, in

Minneapolis, MN since 2010. Prior to joining the University

of Minnesota he was a postdoctoral associate in the

Department of Electrical Engineering at Yale University from

2007 to 2010.

Dr. Li’s current research involves the development of nano-

opto-mechanical systems and waveguide-coupled graphene

optoelectronic devices. He received the McKnight Land-Grant

Professorship of the University of Minnesota in 2013 and the

AFOSR Young Investigator Award in 2012.

.

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