Graphene geometric diodes for terahertz rectennas Zixu Zhu, Saumil Joshi, Sachit Grover

J. Phys. D: Appl. Phys. 46 (2013) 185101 (6pp)
Graphene geometric diodes for terahertz
Zixu Zhu, Saumil Joshi, Sachit Grover1 and Garret Moddel
Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder,
CO 80309-0425, USA
Received 30 November 2012, in final form 15 March 2013
Published 15 April 2013
Online at
We demonstrate a new thin-film graphene diode called a geometric diode that relies on
geometric asymmetry to provide rectification at 28 THz. The geometric diode is coupled to an
optical antenna to form a rectenna that rectifies incoming radiation. This is the first reported
graphene-based antenna-coupled diode working at 28 THz, and potentially at optical
frequencies. The planar structure of the geometric diode provides a low RC time constant, on
the order of 10−15 s, required for operation at optical frequencies, and a low impedance for
efficient power transfer from the antenna. Fabricated geometric diodes show asymmetric
current–voltage characteristics consistent with Monte Carlo simulations for the devices.
Rectennas employing the geometric diode coupled to metal and graphene antennas rectify
10.6 µm radiation, corresponding to an operating frequency of 28 THz. The graphene bowtie
antenna is the first demonstrated functional antenna made using graphene. Its response
indicates that graphene is a suitable terahertz resonator material. Applications for this
terahertz diode include terahertz-wave and optical detection, ultra-high-speed electronics and
optical power conversion.
(Some figures may appear in colour only in the online journal)
1. Introduction
2. Principle of operation
An optical rectenna incorporates an antenna and a diode to
convert high-frequency electromagnetic fields to direct current
(dc) power [1]. Figure 1 shows the configuration of an optical
rectenna system, in which light is absorbed by an optical
antenna and converted to alternating current (ac). An ultrafast
diode rectifies the ac current to provide dc power to a load.
The two primary requirements for the diode are ultra-highspeed response and an impedance that is well matched with
that of the antenna. The most widely investigated diode for
terahertz rectennas is the metal–insulator–metal (MIM) diode,
which is limited in frequency response because of fundamental
RC constraints in parallel-plate devices [2]. We propose and
demonstrate a new type of diode, called a geometric diode, that
does not suffer from the RC constraints of parallel-plate devices
because of its planar structure. Because it is formed from a
conductive material, the resistance of the geometric diode is
also sufficiently low to match the antenna impedance [3].
The fundamental physical requirement for the geometric diode
is that its critical dimensions are on the order of the mean-free
path length (MFPL) of charge carriers in the material. On
this scale, the charge carrier transport within the diode can be
considered ballistic, such that the boundaries and the geometry
of the device have substantial impact on the charge movement
[4]. High-frequency ballistic devices operating at 50 GHz
based on geometric asymmetry in semiconductors have been
demonstrated [5]. In contrast, the geometric diode is formed
from a conductive material which supplies the ballistic charge
carriers. We chose graphene as the thin-film material because
its MFPL is significantly greater than that of metals. This
allows for devices that are sufficiently large to be fabricated
within the capabilities of current lithography techniques.
Furthermore, as we shall see, the limiting frequency for charge
transport in graphene is higher than what has been obtained in
semiconductors [6].
The operation of a geometric diode is shown in
figure 2(a). The critical region of the device is the inverse
Present address: National Center for Photovoltaics, National Renewable
Energy Lab., Golden, CO 80401, USA.
© 2013 IOP Publishing Ltd
Printed in the UK & the USA
J. Phys. D: Appl. Phys. 46 (2013) 185101
Z Zhu et al
shown in figure 3. The MFPL is the key material parameter
used in the simulation, and is assumed to be 200 nm, a typical
value for high-quality graphene.
4. Material and fabrication
The material used to fabricate the geometric diode must meet
two requirements. The first is the ability to withstand high
current density through the neck, up to 107 A cm−2 . This
number comes from the value of VDS required to observe
asymmetry in the I (V ) characteristic. For metal devices,
this VDS is on the order of 0.1 V. Assuming the device has
a 100 matched resistance to antenna, the current density
in the neck region with a cross section area of 100 ×
100 nm2 is 107 A cm−2 . The second requirement of the
material is a sufficiently large charge carrier MFPL. Being
a planar conductive thin-film device, metal was our initial
choice. However, the MFPL of charge carriers in metals at
room temperature is 10–30 nm [7], which would necessitate
impractically small neck geometries. Furthermore, we found
that at high current densities metals with small junctions suffer
from electromigration. To circumvent these disadvantages we
incorporated graphene instead. Graphene devices working at
terahertz have been shown theoretically and also demonstrated
[11–13]. The MFPL in graphene can be up to 1 µm,
corresponding to a carrier mobility of 200 000 cm2 V−1 s−1
[14]. Graphene can withstand a current density of up to
108 A cm−2 [14].
We used mechanical exfoliation to apply graphene flakes
onto an oxidized silicon substrate [8], consisting of a thermally
grown 90 nm SiO2 layer on a silicon wafer having a resistivity
of 1–5 cm. The diode electrodes were formed in a fourpoint-probe configuration [15] to avoid the effects of contact
resistance. Dc I (V ) measurements were carried out by
measuring the voltage drop between the inner contacts while
applying a current through the outer contacts. The metal
contacts (15 nm Cr/40 nm Au) were thermally evaporated and
lifted off from an NR9-1000P photo resist. Electron-beam
lithography (JEOL JBX-9300FS E-beam writer) on a ma-N
negative E-beam resist followed by an oxygen plasma etch
was used to pattern the graphene. The neck width was
approximately 75 nm, with the MFPL measured to be ∼45 nm
[9]. To prevent heating of the graphene and hysteresis due
to charge build-up [16], a pulsed voltage was applied to the
outer contacts using a Keithley 2612 SourceMeter set to the
four-wire mode.
The conic band structure of graphene allows the carrier
concentration to be controlled by the gate voltage. The charge
neutral point (CNP) occurs for a gate voltage at which the
electron and hole concentrations are equal, corresponding to a
conductivity minimum. The substrate formed the gate. For our
graphene, the CNP was obtained for a gate voltage of 24 V, as
shown in figure 4. The suppression of the current at 0 V gate
voltage indicates that the graphene has a small mesoscopic
CNP inhomogeneity [17–19]. When the gate voltage is below
the CNP voltage, holes are the majority charge carriers, and for
gate voltages above the CNP voltage, electrons are the majority
charge carriers. Because the forward direction in geometric
Figure 1. Schematic of a rectenna. An incident electromagnetic
wave is received by the antenna, and rectified by the circuit
containing diode and a low-pass filter, providing a dc voltage to
the load.
arrowhead-shaped constriction (the neck region). The left-toright moving carriers have a higher probability of reflecting
off the diagonal walls on the left-hand side of the neck and
channelling through the arrowhead region than the right-to-left
moving carriers, which are more likely to be blocked by the flat
walls on the right-hand side of the arrowhead. This difference
in probability causes dissimilar current levels for forward
(driving left-to-right motion) and reverse (driving right-to-left
motion) bias voltages. An atomic force microscope (AFM)
image of a fabricated device is shown in figure 2(b).
3. Monte Carlo simulation
The current–voltage (I (V )) asymmetry of the diode was
modelled using a Monte Carlo simulation based on the Drude
model to simulate the movement of charge within the device
[7]. In the simulation, the charges have a randomly directed
Fermi velocity within each collision time and reflect specularly
at the edges of the device. The Fermi velocity of charge carriers
in graphene was calculated to be about 106 m s−1 based on a
backscattering MFPL of 45 nm, corresponding to a collision
time of 5×10−14 s. The MFPL was experimentally determined
from the conductivity versus gate voltage measured in a region
adjacent to the diode [8]. The backscattering MFPL was
obtained by multiplying the elastic MFPL by π/2, which is an
averaging factor for scattering in two dimensions. Because the
backscattering MFPL—as opposed to the elastic MFPL—is a
closer approximation to the inelastic MFPL used in the Drude
model, it is the number that we have used in the simulation [9].
The simulation uses a carrier concentration of 1.1×1012 cm−2 ,
which is the value determined from the conductivity versus
gate voltage measurement for a gate voltage of 30 V. When
a simulated dc voltage is applied, a constant field velocity
is added to the thermal velocity based on a charge effective
mass of 0.02me in graphene [10], where me is the electron rest
mass. Although the field velocity is small compared with the
Fermi velocity, a low-noise I (V ) curve shown in figure 3 was
obtained after running the program for a sufficiently long time
corresponding to approximately 106 collisions. The width of
the neck and device geometric asymmetry determine the I (V )
asymmetry. Shrinking the neck width while keeping all other
device dimensions the same increases the I (V ) asymmetry, as
J. Phys. D: Appl. Phys. 46 (2013) 185101
Z Zhu et al
Figure 2. (a) Inverse arrowhead geometric diode structure. The neck width (dneck ) must be on the order of the MFPL for charge carriers in
the material. The charge carriers reflect at the boundaries of the device. On the left-hand side of the neck, the carriers moving to the right
can either directly channel through the neck or reflect off the tapering edges and keep moving forward. On the right-hand side of the neck,
the vertical edge blocks most of the carriers moving to the left. Hence the diode forward direction for carriers is left-to-right. (b) AFM
image of a fabricated graphene geometric diode device. The graphene inverse arrowhead diode is positioned between two metal contacts.
Based on AFM scans the thickness of graphene is between 0.5 and 1 nm. The measured thickness is larger than the ideal 0.35 nm and is due
to the AFM calibration and surface roughness of the substrate. The neck width is 75 nm.
Figure 4. Dirac curve of the geometric diode made by exfoliated
graphene. The CNP is around 24 V. Using the gate field effect
method, we estimate the MFPL of this graphene piece to be ∼45 nm.
Figure 3. Simulated I (V ) curves for a geometric diode with
different neck widths for a fixed shoulder width of 1 µm. Narrower
necks restrict the reverse current more effectively. For VDS value of
0.1 V the charge carrier drift velocity is 2.2 × 104 m s−1 , which is
about 2% of the graphene electron Fermi velocity.
diodes depends only on the geometry and is independent of
carrier type, applying a gate voltage to change the dominant
carrier type should reverse the polarity of the diode. This is, in
fact, the case and it serves as confirmation that the rectification
is due to the geometry [9].
5. 28 THz optical response
Figure 5. Geometric diode rectennas. AFM images of geometric
diodes coupled to (a) a graphene and (b) a metal bowtie antenna.
Because the metal antenna is so much thicker than the graphene
diode in (b), the graphene cannot be seen in the AFM image.
To test the high-frequency operation of the geometric diode,
we formed a rectenna with a 5.1 µm long bowtie antenna in an
edge-fed configuration [20]. The structure is a combination of
two opposing 2.3 µm long triangular sections with a 500 nm
long geometric diode placed at the center of the gap, as
shown in figures 5(a) and (b). Devices with both metal
(15 nm/45 nm thermally evaporated Cr/gold) and graphene
antennas were fabricated. Figure 6 shows the setup of the
optical measurement system at different polarization angles
of the beam relative to the metal antenna/graphene geometric
diode rectenna system. An infrared CO2 laser, operating at a
wavelength of 10.6 µm and chopped at 280 Hz, was used as
the source to measure rectification at optical frequencies. Twopoint measurements were performed on the antenna-coupled
diodes. The voltage and the current were measured separately
using a lock-in amplifier with zero applied bias.
J. Phys. D: Appl. Phys. 46 (2013) 185101
Z Zhu et al
antenna in figure 5(a) is probably too large to resonate at 28
THz, and might be more efficient if it were smaller.
This is the first reported graphene bowtie antenna working
at terahertz frequency and it opens the possibility of using
patterned graphene as an optical resonator. Bulk graphene
absorbs 2.3% of incident white light [22]. Patterned graphene
structures could enhance the light–plasmon coupling in
graphene. Plasmon absorption of over 13% at 3 THz has been
reported for graphene micro-ribbon arrays [10]. Since the
graphene optical conductivity cutoff frequency is controlled
by the gate voltage and intrinsic doping level [23], graphene
antenna and resonators operating at terahertz frequencies can
be competitive with conventional 2D resonators made of metal.
The angular dependence of the response indicates that
rectification was not a result of optically generated charge
diffusion or thermoelectric effects due to a difference in the
illumination of the two sides of the junction. Also, no response
was detected from an illuminated diode that was not coupled to
an antenna. Since no gate voltage was applied to the antennacoupled diodes before or during the measurement, no p–n
junctions could have been formed as a result of an applied
field [24]. The geometric diode genuinely responds to and
rectifies 28 THz signals.
We compare the measured maximum current of 190 pA
with the expected current as follows.
Based on the
measured laser beam power and beam width we estimate
an input intensity of 5.6 mW mm−2 over an antenna area of
approximately 37.5 µm2 [25]. The diode/antenna coupling
efficiency is calculated to be 12% [26] for a 3000 diode
and an antenna with a characteristic impedance of 100 ,
and the antenna radiation efficiency is estimated to be 37%
[25]. Although the bowtie antenna efficiency was poor, we
chose it because of its relative ease of fabrication. The
diode responsivity [3], which is defined as half the ratio
of the second derivate to the first derivative of the I (V )
characteristic and is a measure of the dc power out divided
by the ac power in, is a key figure of merit for rectenna
operation. The zero-bias responsivity of the diode used in the
rectenna optical measurement is 0.0285 A W−1 . Combining
the estimated antenna parameters with the input power and
the diode responsivity gives an estimated current of 270 pA,
which is in good agreement with the measured value of 190 pA.
A current of 190 pA for a wavelength of 10.6 µm corresponds
to a quantum efficiency of 0.01%. Design improvements are
expected to improve the antenna and diode efficiencies. In
theory, a similar but smaller geometric diode (50 nm shoulder
width with 10 nm neck width) has been shown to have more
nonlinear and asymmetric I (V ) characteristics [27].
To calculate the capacitance we consider the fringing fields
between the two sides of the geometric diode above the device
in air and below in the 90 nm SiO2 bottom substrate, and which
induce charge storage. To estimate the worst case capacitance
between the two sides of the neck, we assume that there is an
approximately 100 nm by 100 nm air gap in the neck region.
Using the planar thin-film capacitance analysis method [28]
the capacitance of a graphene geometric diode is calculated
to be about a few attofarads. The measured resistance of the
graphene device is approximately 1 k, which gives an overall
Figure 6. Rectenna optical measurement setup. The CO2 laser
beam was chopped at a frequency of 280 Hz and guided to the
device by means of a set of mirrors. A lock-in amplifier, with the
chopping frequency as the reference, and a two-point-probe setup
was used to measure the photocurrent.
Figure 7. Short-circuit current response of a rectenna with a
graphene antenna (green circles) and metal antenna (blue diamonds)
as a function of polarization angle (θ). The graphene rectenna has a
lower current response due to the larger series resistance of its
In figure 7 we show the diode short-circuit current as a
function of beam polarization angle (θ ) relative to the antenna
axis for diodes coupled to metal and graphene antennas.
The cosine squared response exhibited a maximum when
the optical field and the antenna were aligned to each other,
showing that the current was indeed produced by the radiation
coupled through the antenna. When the laser polarization was
aligned with the antenna axis, the output open circuit voltage
was found to be proportional to the input optical laser power.
The rectenna with the graphene antenna exhibited a
lower current response due to its much larger antenna series
resistance, which is about 1 k for the graphene antenna
compared with a few ohms for the metal antenna. A recent
simulation of graphene antenna at 1.4 THz indicates that
to achieve high antenna response at terahertz the size of a
graphene antenna needs to be much smaller than the metal
antenna optimized for use at the same frequency [21]. Our
J. Phys. D: Appl. Phys. 46 (2013) 185101
Z Zhu et al
Figure 8. (a) Dc I (V ) characteristics (solid blue) for an exfoliated graphene geometric diode at a gate voltage of 20 V. The Monte Carlo
simulation (dashed green) uses the dimensions of the fabricated device: neck width =75 nm, shoulder width =400 nm, and the measured
MFPL =45 nm. (b) Calculated responsivity [1/2|I (V )/I (V )|] as a function of the applied bias. At 0 V bias, the responsivity is
0.12 A W−1 .
RC time constant of femtoseconds corresponding to a cutoff
frequency of 100 THz. By reducing the area of the non-critical
region around the neck, the RC time constant can be reduced
6. Dc I (V ) measurements and verification of
geometric effect
To investigate the relationship between the 28 THz and dc
responses, we show the measured I (V ) curve for an exfoliated
graphene diode in figure 8. The characteristics exhibit
asymmetry in a direction that is consistent with simulation
results obtained using the actual physical parameters of the
device. A plot of the responsivity versus the applied voltage
is shown in figure 8(b). This device has a responsivity of
0.12 A W−1 at zero bias voltage, and a responsivity over
0.2 A W−1 under bias. These diode parameters correspond to
the measured stand-alone diode I (V ) characteristics, and are
different from those of the antenna-coupled diode in figure 5.
The responsivity is expected to improve substantially as the
ratio of MFPL to neck width is increased, as shown in figure 3.
In graphene, I (V ) nonlinearity can be due to the graphene
breaking down at a high voltage [29]. That is not the case here
for two reasons. First, we have seen asymmetric nonlinearity at
a low voltage not just at a high voltage. Second, the curvature
of the reported nonlinearity of graphene at breakdown voltages
had an opposite curvature direction compared with that of our
diode I (V ) characteristics. At higher voltages, for VDS >
2.5 V, we found the diode I (V ) curvature starts to switch sign,
possibly due to the breakdown phenomenon there.
To further confirm the effect of geometry on the I (V )
asymmetry, we compared the characteristics of geometrically
symmetric and asymmetric junctions. We also fabricated
asymmetric devices using graphene deposited by chemical
vapour deposition (CVD) (provided by P L McEuen’s laboratory at Cornell University) having a shorter MFPL than the
exfoliated graphene devices. To quantify the I (V ) asymmetry
we define the electrical asymmetry as A = |I(V)/I(-V)|. A comparison of the three devices is shown in figure 9. The value of
Figure 9. Electrical asymmetry plots for a CVD-produced graphene
geometric diode (red circles), an exfoliated graphene geometric
diode (green asterisks) and a geometrically symmetric CVD
graphene junction (blue squares). The electrical asymmetry is
defined as A = |I(V)/I(-V)|. CVD graphene, having a shorter MFPL
than the exfoliated graphene, gives a lower asymmetry than the
exfoliated graphene device. The neck width for all the devices is
75 nm, with a shoulder width of 400 nm.
A for a graphene geometrically symmetric junction remains at
unity as compared with the geometrically asymmetric devices,
which exhibit electrical asymmetry. The CVD graphene diode
has a lower electrical asymmetry due to its reduced MFPL. The
plot exhibits the significance of geometric asymmetry and the
MFPL in achieving diode behaviour.
7. Conclusions
We have developed and demonstrated a new kind of diode
for use in high-frequency rectennas. The graphene geometric
diode exhibits dc I (V ) asymmetry and its measured electrical
characteristics are consistent with Monte Carlo simulations.
Rectennas incorporating geometric diodes provide optical
frequency rectification of 10.6 µm wavelength radiation with
both metal and graphene bowtie antennas. The measured
short-circuit currents correspond to the values estimated using
the diode and antenna parameters. Further improvement in
the diode and antenna design is expected to increase device
J. Phys. D: Appl. Phys. 46 (2013) 185101
Z Zhu et al
[14] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S and
Geim A K 2009 Rev. Mod. Phys. 81 109–62
[15] Moddel G, Zhu Z, Grover S and Joshi S 2012 Solid State
Commun. 152 1842–5
[16] Joshi P, Romero H E, Neal A T, Toutam V K and
Tadigadapa S A 2010 J. Phys.: Condens. Matter
22 334214
[17] Connolly M R, Chiou K L, Smith C G, Anderson D,
Jones G A C, Lombardo A, Fasoli A and Ferrari A C 2010
Appl. Phys. Lett. 96 113501
[18] Blake P, Wang R, Morozov S V, Schedin F,
Ponomarenko L A, Zhukov A A, Nair R R,
Grigorieva I V, Novoselov K S and Geim A K 2009
Solid State Commun. 149 1068–71
[19] Lohmann T, von Klitzing K and Smet J H 2009 Nano Lett.
9 1973–79
[20] Weiss M D, Eliasson B J and Moddel G 2003 Patent
No 6664562
[21] Tamagnone M, Gomez-Diaz J S, Mosig J R and
Perruisseau-Carrier J 2012 J. Appl. Phys. 112 114915
[22] Nair R R, Blake P, Grigorenko A N, Novoselov K S,
Booth T J, Stauber T, Peres N M R and Geim A K 2008
Science 320 1308
[23] Li Z Q, Henriksen E A, Jiang Z, Hao Z, Martin M C, Kim P,
Stormer H L and Basov D N 2008 Nature Phys. 4 532–5
[24] Williams J R, DiCarlo L and Marcus C M 2007 Science
317 638–41
[25] González F J and Boreman G D 2005 Infrared Phys. Technol.
46 418–28
[26] Sanchez A, Davis C F Jr, Liu K C and Javan A 1978 J. Appl.
Phys. 49 5270
[27] Dragoman D and Dragoman M 2013 J. Phys. D: Appl. Phys.
46 055306
[28] Zubko O G, Nikol’Skii S P and Vendik M A 1999 Tech. Phys.
44 349–55
[29] Murali Y, Yang Y, Brenner K, Beck T and Meindl J D 2009
Appl. Phys. Lett. 94 243114
The authors gratefully acknowledge assistance in device
preparation from Kendra Krueger and David Doroski. This
work was carried out under a contract from Abengoa Solar,
with initial support from Hub Lab. Device processing was
carried out in part at the Colorado Nanofabrication Laboratory,
and in part at the Cornell NanoScale Facility, both members of
the National Nanotechnology Infrastructure Network, which
is supported by the National Science Foundation (Grant
ECS-0335765). We also thank Jonathan Alden in Professor
P L McEuen’s group in Cornell University for providing the
CVD graphene sample.
[1] Fumeaux C, Herrmann W, Kneubühl F K and Rothuizen H
1998 Infrared Phys. Technol. 39 123-83
[2] Grover S, Dmitriyeva O, Estes M J and Moddel G 2010 IEEE
Trans. Nanotechnol. 9 716-22
[3] Grover S and Moddel G 2011 Solid State Electron. 67 94–9
[4] Datta S 1992 Phys. Rev. B 45 13761–4
[5] Song A M 2002 Appl. Phys. A 75 229–35
[6] Avouris P 2010 Nano Lett. 10 4285–94
[7] Ashcroft N W and Mermin N D 1976 Solid State Physics
(New York: Holt, Rinehart and Winston) chapter 1
[8] Novoselov K S et al 2004 Science 306 666–9
[9] Nayfeh O M 2011 IEEE Trans. Electron Devices 58 2847–53
[10] Ju L et al 2011 Nature Nanotechnol. 6 630–4
[11] Ryzhii V 2006 Japan. J. Appl. Phys. 45 L923–5
[12] Rana F 2008 IEEE Trans. Nanotechnol. 7 91–9
[13] Vicarelli L, Vitiello M S, Coquillat D, Lombardo A,
Ferrari A C, Knap W, Polini M, Pellegrini V and
Tredicucci A 2012 Nature Mater. 11 865–71