# A place for two-dimensional plasmonics in electromagnetic wave detection

Dmitry Mylnikov  Laboratory of 2d Materials for Optoelectonics, Moscow
Institute of Physics and Technology, Dolgoprudny 141700, Russia  Dmitry
Svintsov<EMAIL_ADDRESS>Laboratory of 2d Materials for Optoelectonics,
Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia

###### Abstract

Plasmons in two-dimensional electron systems (2DES) feature ultra-strong
confinement and are expected to efficiently mediate the interactions between
light and charge carriers. Despite these expectations, the electromagnetic
detectors exploiting 2d plasmon resonance have been so far inferior to their
non-resonant counterparts. Here, we theoretically analyse the origin of these
failures, and suggest a proper niche for 2d plasmonics in electromagnetic wave
detection. We find that a confined 2DES supporting plasmon resonance has an
upper limit of absorption cross-section, which is identical to that of simple
metallic dipole antenna. Small size of plasmonic resonators implies their weak
dipole moments and impeded coupling to free-space radiation. Achieving the
’dipole limit’ of absorption cross-section in isolated 2DES is possible either
at unrealistically long carrier momentum relaxation times, or at resonant
frequencies below units of terahertz. We further show that amendment of even
small metal contacts to 2DES promotes the coupling and reduces the fundamental
mode frequency. The contacted resonators can still have deep-subwavelength
size. They can be merged into compact arrays of detectors, where signals from
elements tuned to different frequencies are summed up. Such arrays may find
applications in multi-channel wireless communications, hyper-spectral imaging,
and energy harvesting.

###### pacs:

Valid PACS appear here

##  I Introduction

Two-dimensional plasmons represent a remarkable example of electromagnetic
waves confined to scales two orders of magnitude below the free-space photon
wavelength [1]. It is commonly articulated that confinement allows 2d plasmons
to mediate strong light-matter interactions [2, 3, 4]. What is meant by this
mediation remains largely unclear. The fundamental studies reveal intriguing
properties of 2d plasmons [5, 6, 7, 8], yet do not show a distinct promise for
optoelectronics.

This long-reigning existential vacuum was filled by a proposal to use 2d
plasmons for high-responsivity electromagnetic detection [9]. The idea partly
followed the path of bulk plasmonics, with its established applications in
photodetectors and solar cells [10]. More specifically, Ref. [9] suggested a
confined 2d electron system (2DES) to act simultaneously as a resonant cavity
and rectifier (detector) for 2d plasmons. The estimated detector response to
free-space illumination scaled quadratically with plasmon quality factor, and
promised sensitive detection for practically important terahertz range [11].
The idea ignited a booming research in the field of 2d plasmon-enhanced
detectors, with further proposals based on different materials [12, 13, 14,
15, 16, 17], rectification mechanisms [18, 19, 20, 21], and coupling schemes
[22, 23].

So far, the experimentally realised 2d plasmonic detectors [24, 25, 26, 27,
28, 29, 30, 31, 32] did not demonstrate substantial responsivity enhancement
compared to their non-resonant counterparts [33, 34, 35, 36]. Even if
plasmonics was in play, and responsivity demonstrated clear maxima at plasmon
frequencies [24, 25, 30, 32], their height was not impressive for practical
applications. Electronic quality of 2d systems and extra mechanisms of plasmon
loss were typically blamed as limiting factors to the strength of plasmon
resonance [26]. But even in cleanest 2d systems with electron mobility
exceeding millions centimeters squared per volt per second, excitation of
plasmons lead not to resonant enhancement of responsivity, but to responsivity
oscillations [25, 30].

These numerous experiments stimulate a critical re-consideration of advantages
which 2d plasmonics can bring to electromagnetic wave detection. In this
paper, we’ll show that plasmonic detectors based on confined 2DES possess a
fundamental limit of electromagnetic absorption cross-section order of
$\sigma_{\lambda}=\lambda_{0}^{2}/4\pi$, where $\lambda_{0}$ is the free-space
wavelength [37]. This limit is identical for resonant plasmonic detectors and
non-resonant structures matched to conventional antennas [38]; hence, a single
plasmonic detector can’t help increasing responsivity. Further, reaching this
maximum cross-section in plasmonic systems is far more challenging, compared
to antenna-coupled detectors. The reason is small size of 2D plasmonic
resonators, their small dipole moment, and weak coupling to free-space
radiation.

Figure 1: Two-dimensional plasmonic resonators (a) Isolated layer of two-
dimensional electronic system (graphene is shown as example) (b) 2DES
connected to metallic leads (c) Multi-colour array of plasmonic detectors

We explore the ways to achieve perfect coupling between confined 2d plasmons
and free-space radiation. We find that metal contacts, being an inevitable
part of any photodetector, strongly modify the absorption cross-section,
matching conditions, and resonant frequencies. At fixed size of 2DES, the
electron mobility providing best matching scales inversely proportionally to
the square of contacts’ length. Loosely speaking, a good plasmonic detector
should have carefully engineered metal contacts, but not the highest-mobility
2d channel.

Despite the emerging identity between responsvity of plasmonic and non-
plasmonic detectors, we find a suitable niche for 2d plasmonic detectors.
Exploiting the deep-subwavelength confinement, one can ’pack’ a large number
of detectors even in the minimally-focused light spot size of area
$\sim\lambda_{0}^{2}$. Each detector can have absorption cross-section
reaching $\sigma_{\lambda}$. Thus, the net signal collected from an array can
largely exceed the signal from one non-plasmonic antenna-matched device
(having a comparable area). Importantly, additive character of cross-section
works if only plasmonic detectors within a spot are tuned to slightly
different resonant frequencies. Applications of such detectors can lie in
hyper-spectral imaging [39], multi-channel wireless communication [40], and
electromagnetic energy harvesting [41].

So far, the impedance matching requirements for confined 2d plasmons haven’t
yet been realised, though they received much attention in metal-insulator and
nanoparticle plasmonics [42, 43]. Recently, it was shown that short wavelength
of propagating 2d plasmons implies their low excitation efficiency by small
scatterers [44]. Interplay of Ohmic and radiative damping was recently studied
for 2d plasmons in discs [45], but practically important absorption cross
section were not analysed. General constraints on extinction and absorption
were recently derived in [46], but applications were demonstrated only for
bulk plasmonic structures. The absorption cross sections limited by material
loss in 2DES were analysed in [47], but developed quasistatic theory did not
reproduce the dipole cross-section limit.

Further on, we shall theoretically elucidate the strategies for making a
plasmonic photodetector with maximum radiation absorption cross-section. We do
not focus on particular photocurrent generation mechanism (be it
thermoelectric, bolometric, photovoltaic, or resistive self-mixing), as all
they are bounded by the absorbed power. The simulations are performed with CST
Microwave Studio package, and supported by simple analytical estimates if
possible.

##  II Absorption limits by confined 2d plasmons

We start by examining the electromagnetic absorption by isolated two-
dimensional systems without metal contacts [Fig. 1(a)]. This case, being not
much practical for detectors, provides useful insights into matching
conditions. The properties of 2DES are described in terms of kinetic
inductance per square $L_{\square}$ and active resistance $R_{\square}$, such
that sheet impedance $Z_{\square}=i\omega L_{\square}+R_{\square}$. The
frequency dependence of $Z$ can be arbitrary, and may originate from intra-
and interband material processes; all we need in simulations is just numerical
value of $Z$ at excitation frequency $\omega$. If the conductivity is
described by Drude model, $L_{\square}$ is linked to sheet carrier density
$n_{s}$ as $L_{\square}=m/n_{s}e^{2}$, and resistance
$R_{\square}=m/n_{s}e^{2}\tau$, where $\tau$ is the momentum relaxation time.
Ths 2DES has a square shape with side $g$.

Figure 2: Electromagnetic absorption by confined 2d plasmons (a) Absorption
cross-section section spectrum of a square 2DES sheet ($g=w=0.5$ $\mu$m) at
different momentum relaxation times $\tau$. Sheet inductance $L_{\square}=20$
pH (b) Absorption cross-section at fundamental plasmonic resonance vs momentum
relaxation time at variable size $g$: emergence of optimum momentum relaxation
$\tau_{\rm opt}$ (c,d) Dependence of resonant frequency $f_{\rm res}$ and
matching time $\tau_{\rm opt}$ on 2DES size $g$ for different values of sheet
inductance $L_{\square}=10$, 20 and 40 pH (controlled by carrier density).
Dots in (d) are the results of numerical simulations, lines are calculated
using analytical expression (1)

The calculated absorption spectrum of a square piece of 2DES ($g=0.5$ $\mu$m)
with kinetic inductance $L_{\square}=20$ pH and moderate momentum relaxation
time $\tau=0.1$ ps is shown in Fig. 2. The parameters correspond to graphene
with Fermi energy $E_{F}\approx 0.4$ eV and mobility $\mu=2.4\times 10^{3}$
cm2/V s 111For GaAs quantum well with $m^{*}=0.067m_{0}$ this inductance
corresponds to sheet density $n_{s}=1.2\times 10^{13}$ cm-2, while $\tau=0.1$
ps corresponds to mobility $\mu\approx 2600$ cm2/V s.. The plasmon resonance
at $f_{\rm res}\approx 15$ THz manifests as a clear absorption maximum, as the
high-quality conditions are well fulfilled, $\omega_{\rm res}\tau=10\gg 1$.
The cross section largely exceeds the geometrical area of 2DES $\sigma\approx
10g^{2}$.

Would one obtain larger resonant absorption if the electronic quality of 2DES
would be increased? We examine it by evaluating the cross-sections for a
sequence of relaxation times, $\tau=1$, $10$ and $100$ ps. The resonant
absorption $\sigma(f_{\rm res})$ initially grows, reaches a maximum at
$\tau_{\rm opt}\approx 50$ ps, and then rolls down. The fundamental limits of
cross-section are best visualised if plotted in units of
$\sigma_{\lambda}=\lambda_{0}^{2}/4\pi$ [Fig. 2 (b)]. The cross-section at
optimum relaxation time is order of $(1...1.5)\sigma_{\lambda}$, which is a
characteristic of dipole resonance limited by radiative decay [37].

The optimum relaxation time $\tau_{\rm opt}\sim 10...100$ ps for reaching the
best absorption at terahertz frequencies is quite large for realistic 2DES, as
it corresponds to mobilities $\mu_{\rm opt}\approx(0.2...2)\times 10^{6}$
cm2/(V s). The very existence of optimum time hints that the resonant
absorption maximum corresponds to some kind of matching. Further on, the
optimum relaxation time becomes longer once the size of 2DES is decreased and
its resonant frequency goes up. It suggests that matching problem is
associated with small size of plasmonic resonator, short wavelength of 2d
plasmons, and ultimately weak radiation coupling of plasmonic systems.

To test this idea and get a simple estimate of optimum relaxation time, we
equate the radiation losses $P_{\rm rad}$ and Joule losses $P_{\rm J}$ in a
piece of 2DES with high-frequency current distribution ${\bf j}_{\omega}({\bf
r})$ (Appendix A). This results in

$\frac{4\pi}{3}\left(\frac{g}{\lambda_{0}}\right)^{2}Z_{0}\langle{\bf
j}_{\omega}({\bf r})\rangle^{2}=2R_{\square}\langle{\bf j}^{2}_{\omega}({\bf
r})\rangle,$ (1)

where $Z_{0}=377$ $\Omega$ is the free-space impedance, and angular brackets
$\langle...\rangle$ denote averaging over the sample area. For a fundamental
mode with nearly-sinusoidal current distribution one can estimate $\langle{\bf
j}^{2}_{\omega}({\bf r})\rangle/\langle{\bf j}_{\omega}({\bf
r})\rangle^{2}=\pi^{2}/8\sim 1$. The inverse quadratic scaling of radiated
power with wavelength is in agreement with recent experimental measurements
[49].

The matching equation (1) has serious implications. The value of $\tau_{\rm
opt}$ should be so large that Ohmic resistance $R_{\square}$ is
$(\lambda_{0}/g)^{2}$ times smaller than free-space impedance. We note that
sheet resistances below $Z_{0}$ were reached only recently in high-quality
GaAs quantum wells [50], while typical graphene sheets have $R_{\square}\sim
1$ kOhm. The factor $(g/\lambda_{0})^{2}$ requiring the smallness of Ohmic
resistance is nothing but the measure of sub-wavelength compression achieved
in a plasmonic resonator.

Scaling of optimum relaxation time with size of plasmon resonator $g$ can be
derived once the scaling of resonant frequency is known. As an estimate, one
can ’’quantize’’ the dispersion of 2d plasmons in extended system $\omega_{\rm
pl}(q)=\sqrt{q/2\varepsilon_{0}L_{\square}}$ at the characteristic wave vector
$q=\pi/g$. As a result, $f_{\rm res}=(2\pi)^{-1}\omega_{\rm pl}(\pi/g)$
becomes inversely proportional to the square root of size, $f_{\rm res}\propto
g^{-1/2}$, while optimum momentum relaxation time scales approximately as
$1/g$. These trends are supported well by our numerical simulations
demonstrated in Fig. 2 c and d.

The inverse scaling of matching time $\tau_{\rm opt}$ with size of plasmonic
structure $g$ suggests that strong light absorption and high-responsivity
detection may be achieved only at THz and sub-THz frequencies with very high
quality of 2DES. Already for mid-infrared applications ($\lambda_{0}\approx 3$
$\mu$m), the matching time $\tau_{\rm opt}\sim 2$ ns exceeds the achievable
values by three orders of magnitude. We shall further show that these severe
matching conditions can be modified via engineering of metal contacts to 2DES.

##  III Matching 2d plasmons to free space with metal contacts

Figure 3: Engineering the metal leads for coupling 2d plasmons to free-space
radiation (a) Absorption cross section of 2DES sheet of size $g=w=0.5$ $\mu$m
with metal arms. Total device length $L_{a}=4$ $\mu$m, inductance
$L_{\square}=20$ pH, $\tau=\tau_{opt}\approx 12$ ps. (b) Frequency scaling of
2DES (blue) and antenna (orange) impedance in the vicinity of fundamental
resonance (c) Distribution of induced currents $j_{x}(x,y=0)$ along the
contacted 2DES at resonance with increasing the length of metal arms. $\tau=1$
ps (d-f) Scaling of resonant frequency and optimum momentum relaxation time in
contacted 2DES as functions of device length $L_{a}$ (d), 2DES size $g$ (e)
and net device width $w$ (f)

Metal contacts are inevitably present in any semiconductor-based
photodetector, and may affect the device electrodynamics. Previously, a strong
effect of contacts on eigenmodes of 2d plasmons was demonstrated
experimentally [51, 52]. Here, we show that metal contacts of moderate length
have a strong impact on radiative coupling of plasmonic devices, and
considerably reduce $\tau_{\rm opt}$.

Figure 3 substantiates our suggestions by showing the spectrum of cross
section (a), the resonant frequency, and optimum momentum relaxation time (c)
for 2DES with finite-length metal arms. The first example shown in Fig. 3(a)
shows that amendment of metal arms to square-shaped 2DES with moderate
relaxation time $\tau=12$ ps considerably enhances the cross-section.
Eventually, it can become close to the fundamental limit of
$\sigma_{\lambda}$.

It is instructive to track the properties of fundamental plasmonic mode of
2DES with increase in metal arm length $L_{a}$. Our simulations show that
resonant frequency scales approximately as $f_{\rm res}\propto L_{a}^{-1/2}$,
while ’matched’ relaxation time scale as $\tau_{\rm opt}\propto L_{a}^{-1}$ if
the arm length exceeds the size of 2DES, $L_{a}\gg g$. Already for $L_{a}=10$
$\mu$m and $g\approx 0.5$ $\mu$m, the optimum momentum relaxation time can
take quite a realistic value $\sim 3$ ps. For a similar contact-less 2DES, the
matched $\tau_{\rm opt}$ is roughly an order of magnitude longer.

The above comparison of matching conditions in contacted and contact-less 2DES
was performed at given 2DES size. Of course, larger-area 2DES also have
shorter relaxation time, just as small 2DES with large metal contacts.
Comparing the values of $\tau_{\rm opt}$ in contacted and contact-less 2DES at
fixed resonant frequency, we find them to be nearly identical. From viewpoint
of practical photo-detection, it is desirable to use small 2DES with large
metal arms, as this leads to enhancement of power density released in 2d
channel.

It is further possible to provide a simple model governing the matching of
plasmonic resonators with the aid of contacts. A hint toward such a model lies
in a very simple distribution of currents in contacted 2DES. While uncontacted
2DES at resonance has a nearly sinusoidal current distribution, this profile
is quickly flattened toward constant value as $L_{a}$ goes up [Fig. 3 (c)]. It
implies that effects of distributed inductance and capacitance are unimportant
for contacted resonators, and simple lumped models can work well.

The desired lumped model is simply a series connection of 2DES with impedance
$Z_{\rm 2des}=(g/w)Z_{\square}$ and short metal dipole antenna with with
impedance $Z_{A}=R_{\rm rad}+(i\omega C_{A})^{-1}$. Imaginary part of 2DES
impedance is mainly due to kinetic inductance of 2D electrons. The antenna
impedance is mainly governed by its capacitive gap. The resonant frequency in
this case is simply given by [52]

$\omega_{0}=\frac{1}{\sqrt{L_{\square}C_{a}}}\sqrt{\frac{w}{g}}.$ (2)

To complete the model, we adopt the expressions for capacitance and radiative
resistance of a short dipole antenna [38],
$C_{a}\approx\frac{\pi}{2}\varepsilon_{0}L_{a}\ln^{-1}[L_{a}/w_{a}-1]$ and
$R_{\rm rad}=\frac{\pi}{6}Z_{0}(L_{a}/\lambda_{0})^{2}$. The matching
condition, in such model, reads simply as $R_{\square}g/w=R_{\rm rad}$, or, in
extended form:

$R_{\square}\frac{g}{w}=\frac{\pi}{6}Z_{0}\left(\frac{L_{a}}{\lambda_{0}}\right)^{2}.$
(3)

The above analytical expressions confirm the numerically observed scaling of
resonant frequency and matching time, and correct them by a logarithmic factor
$f_{\rm res}\propto[L_{a}/\ln(L_{a}/w_{a})]^{-1/2}$ and $\tau_{\rm
opt}\propto[L_{a}\ln(L_{a}/w_{a})]^{-1}$. The lumped theory predicts the
values of $f_{\rm res}$ and $\tau_{\rm opt}$ very well, as seen from Fig. 3
(d), (e) by comparison of solid and dashed curves. Another option for
increased matching of 2d plasmonic resonators lies in increase of arms’ width
$w$. As soon as $w\ll\lambda_{0}$, increased width leads to quadratic
enhancement of radiating dipole moment. This makes $\tau_{\rm opt}$ drop
abruptly as $w$ increases, as shown in Fig. 3 (f). Ammending the antenna
reciprocity theorem to the lumped model (Appendix B), one can further
analytically predict the value of $\sigma$, as demonstrated in Fig. 3 with
dashed line.

The above matching options enable, in principle, achieving large absorption
cross-sections by plasmonic photodetectors while keeping their deep sub-
wavelength confinement. Indeed, the size of optimally-matched contacted
structures in considered in Fig. 3 did not exceed $0.5$ $\mu$m $\times 4$
$\mu$m. Small areas of plasmonic detectors allow sparing the space on a chip,
compared to large antenna-coupled detectors. This is a purely economic (yet
important) benefit. Still, the absorption cross-sections (and responsivities)
of a single plasmonic and a single antenna-coupled non-plasmonic detectors are
bounded by the same values.

Figure 4: Single and multi-color arrays of plasmonic photodetectors (a)
Absorbance spectrum $\alpha(f)$ of a single-colour array, i.e. a square-
lattice of of identical plasmonic detectors. Different lines correspond to
different cell periods $D$. Inset: resonant absorbance vs period (b)
Absorption spectrum of the four-colour array with optimum cell period $D=10$
$\mu$m. The absorbances of individual detectors are non-interfering and each
reaches maximum value of $0.5$. Inductance $L_{\square}=20$ pH, 2DES lengths
$g=300,\,310,\,320,\,330$ nm, width $w=500$ nm, total length $L_{a}=1$ $\mu$m,
$\tau=\tau_{opt}=23$ ps

##  IV Exploiting deep-subwavelength confinement for photodetection

While a single 2d plasmonic resonator and an optimally-loaded dipole antenna
have the same absorption cross-sections, it may be tempting to exploit the
small size of plasmonic cavities for design of detector arrays. By connecting
the devices in series (or in parallel), one may progressively increase the
photovoltage (or photocurrent) generated by such an assembly. One can estimate
the maximum number of plasmonic devices per optimally focused light spot as
$N\sim(\lambda_{0}/\lambda_{\rm pl})^{2}$. Thus, one may expect an $N$-fold
enhancement of plasmonic meta-device responsivity, compared to non-plasmonic
device using classical antenna focusing.

Such packing is hardly possible if all detectors are tuned to the same
resonant frequency. The emerging complexities for plasmonic arrays are
illustrated in Fig. 4. Here, we plot the absorbance $\alpha(f)$ by a square-
lattice array of plasmonic detectors tuned to $f_{0}\sim 13$ THz vs the
lattice period $D$. The absorbance is defined as the cross-section per unit
cell area, $\alpha=\sigma/D^{2}$. As the packaging becomes denser, the
absorbance first goes up ($D\lesssim 10$ $\mu$m). By further increasing the
density, we observe a strong degradation of $\alpha$. Instructively, the
optimal cell period appears to be order of $\sigma^{1/2}_{\lambda}\approx 6.5$
$\mu$m. It implies that plasmonic device at a resonance strongly distorts the
field in a large nearby area $\sim\sigma_{\lambda}$, and exploiting the
’compactness’ advantage of plasmonics becomes impossible for single-frequency
detection. Thus, the ’effectively occupied area’ by plasmonic cavity is still
order of $\sigma_{\lambda}$, and hence order of size of a classical antenna.

The situation becomes more encouraging once we recall that large absorption
cross-section occurs only at the resonant frequency. One may thus use the
$\sigma_{\lambda}$-circle near the 2d plasmonic resonator for placing other
detectors tuned to different frequencies. Dissimilar devices do not hinder
each other’s absorption, provided their line width
$\delta\omega\sim\tau^{-1}_{\rm opt}$ exceeds the spacing between resonant
frequencies. Figure 4 (b) substantiates this idea and compares the absorbance
of single-color array (orange line) with that of four-color array of the same
period (blue line). Though formal lattice periods for these arrays are equal,
the area per element in multi-colour case is four times smaller. Each of four
tuned devices fully exploits its resonant cross-section.

We may suggest three application domains for such detector arrays. First of
all, they can be exploited for sensitive detection of signals comprised of
several wavelengths. This situation is common for multi-channel wireless
communications. The plasmonic detector array resonantly responding to all $M$
communication wavelengths occupies the space $\lesssim\sigma_{\lambda}$.
Achieving the same result with classical dipole antennas would require the
area of $M\sigma_{\lambda}$, which becomes highly impractical for $M\gg 1$.
Second, the arrays of plasmonic detectors connected in series can effectively
generate photovoltage upon illumination by broadband signals. The limiting
example of such broadband signal is the background radiation noise, nowadays
considered as a source for energy harvesting. Finally, plasmonic arrays can
find applications for compact hyper-spectral imagers, where signal from each
resonating element is proportional to light intensity at given frequency.

##  V Discussion and conclusions

The presented study was aimed to indicate a possible niche for 2D plasmonics
in photodetection applications. We tried to dispel a common view that high
quality factor for 2d plasmons is a necessary and sufficient condition for
making a good plasmonic photodetector. Instead, we have shown that a good
plasmonic photodetector requires impedance matching between radiative and
Ohmic resistances. This is most simply achieved by engineering metal contacts
to 2DES. In some situations (like at the mid-infrared frequencies), large
electron momentum relaxation time are indeed essential. The quality
requirements to 2DES are, however, becoming less important for THz and sub-THz
frequencies.

In our study, we have omitted the consideration of photodetectors based on
grating-coupled 2DES [27, 22]. For an infinite grating, the maximum absorbance
at plasmon resonance does not exceed 50 %, which is the same as for 2d film
perfectly matched to free space. The gain from plasmonic effects here is not
as essential as for confined 2DES. Absorption cross-sections by finite
grating-coupled 2DES [23] requires a separate consideration, but we expect the
matching requirements similar to Eq. (1) to appear in that case. For the same
reason, we haven’t considered the 2d photodetectors with arrays of plasmonic
particles for enhanced absorption [53, 54]. The latter also have no relation
to 2d plasmonics.

It would be supercilious to say that our proposed multi-color arrays based on
2D plasmons are an only niche for 2D plasmonics in photodetection. Of course,
future inventions can’t be predicted. Ultra-high confinement is not the only
important property of 2d plasmons, another to be mentioned is the electric and
magnetic tunability of resonant frequency. The latter enables the design of
all-electrical radiation spectrometers. High responsivity and tunability are
essential for spectroscopy, while deep-subwavelength size is not essential. In
this niche, it may be beneficial to use 2DES with moderate mobility optimally
matched with large ($\sim\lambda_{0}$) antenna arms. It is possible to show
(Appendix C) that tunability of resonant frequency persists in this case,
while relaxation time requirements are softened.

It is tempting to reveal why previous proposals of the resonant plasmonic
detectors [9, 16, 18, 17] strongly overestimated their responsivities. The
reason lies in assumption that incident electromagnetic field can be mapped to
a perfect voltage source with zero internal resistance. Assuming very long
momentum relaxation time, one could obtain arbitrarily large $Q$-factor of
plasmon resonance. In reality, radiative resistance $R_{\rm rad}$ of all
plasmonic systems is finite. Moreover, radiative resistance is linked to the
ability of detector to receive power from free space. By releasing the
assumption of voltage source ideality, one immediately realizes the existence
of limiting responsivity for plasmonic radiation detectors.

This work was supported by the grant MK-1035.2021.4 of the President of
Russian Federation. The authors thank V. Muravev for fruitful discussions.

##  Appendix A Matching the absorbed and radiated power

We derive the matching condition by equating the absorbed ($P_{\rm abs}$) and
radiated ($P_{\rm rad}$) powers in confined 2DES with current distribution
${\bf j}_{\omega}({\bf r})$. We start with power radiated by a small dipole,

$P_{\rm rad}=Z_{0}\frac{\omega^{4}}{3\pi c^{2}}|{\bf d}_{\omega}|^{2}.$ (4)

We re-express the dipole moment via current distribution using its definition,
${\bf d}_{\omega}=\int{{\bf r}\rho_{\omega}({\bf r})d^{2}{\bf r}}$, and
continuity equation $-i\omega\rho_{\omega}+\partial_{\bf r}{\bf j}=0$. This
results in:

${\bf d}_{\omega}=\frac{1}{i\omega}\int{{\bf j}_{\omega}({\bf r})d{\bf r}}.$
(5)

Introducing (5) into (4), and expressing the frequency via wavelength,
$\omega=2\pi c/\lambda_{0}$, we find:

$P_{\rm rad}=Z_{0}\frac{4\pi}{3\lambda_{0}^{2}}\left|\int{{\bf
j}_{\omega}({\bf r})d{\bf r}}\right|^{2},$ (6)

which is equivalent to the left-hand side of Eq. (1) of the main text. As for
absorbed power, it is simply given by Joule’s law:

$P_{\rm abs}=2{\rm Re}[Z_{\rm 2des}(\omega)]\int{|{\bf j}_{\omega}({\bf
r})|^{2}d{\bf r}}.$ (7)

For Drude model of conductivity, ${\rm Re}[Z_{\rm 2des}(\omega)]$ is nothing
but dc sheet resistance $R_{\square}$. This brings us exactly to Eq. (1) of
the main text.

##  Appendix B Analytical model for absorption of contacted 2d plasmonic
device

The developed model for resonant frequencies and matching conditions can be
further extended to a fully analytical estimates of absorption cross-section.
It is based on reciprocity theorem for antennas [38], and enables to present a
cross-section of contacted plasmonic resonator as

$\sigma=\frac{\lambda_{0}^{2}}{4\pi}e_{\rm match}D(\theta,\phi),$ (8)

where $D(\theta,\phi)$ is the antenna directivity and

$e_{\rm match}=\frac{4R_{\rm rad}R_{\rm 2des}}{|Z_{A}+Z^{*}_{\rm 2des}|^{2}}$
(9)

is the matching efficiency which turns to unity for a perfectly-matched
device. For dipole radiation, which is always the case for fundamental mode of
sub-wavelength plasmonic resonator, $D(\theta,\phi)=3/2\sin^{2}\theta$, and
$\theta$ is the angle between dipole direction and wave vector of incident
(radiated) light. We observe that the antenna model for cross-sections works
very well for prediction of absorption cross-sections, as seen from comparison
of dashed and solid lines in Fig. 3a.

##  Appendix C A tunable resonant detector with low-mobility 2DES

Another key advantage of 2D plasmonic resonators, apart from deep-sub-
wavelength size, is the tuning of eigen-frequency upon variations of sheet
conductivity $Z_{\rm 2des}$. The latter is enabled in situ by electric field
effect or transverse magnetic field. Frequency tuning enables the realisation
of all-electrical plasmonic radiation spectrometers. The deep confinement for
such devices is unimportant, while large absorption cross-section is still
highly desirable.

We show that a functional all-electric spectrometer (or tunable resonant
detector) can be realised with ’dirty’ 2d electron systems and resonant metal
antennas with size $\sim\lambda_{0}$. Figure 5 substantiates our findings by
showing the absorption spectrum of 2DES-loaded antenna ($L_{a}=16$ $\mu$m,
$g=w=8$ $\mu$m) at various 2D inductances. Instructively, the cross-section at
the second resonance reaches $(1...3)\sigma_{\lambda}$, while the momentum
relaxation time is moderate, $\tau=20$ fs. The resonant frequency goes down
with increasing $L_{\square}$ (inset), as it typically occurs in plasmonic
systems.

Figure 5: Resonant and tunable absorption by low-mobility 2DES. Calculated
spectrum of absorption cross section for small ($g=0.1$ $\mu$m) moderate-
quality 2DES ($\tau=20$ fs) coupled to large antenna $L_{a}=16$ $\mu$m, $w=8$
$\mu$m. The highest peak occurs at frequency where $\lambda_{0}\sim L_{a}$

The origin of large cross section and wide frequency tuning in this example
come from the special design of arms having the length order of $\lambda_{0}$.
Electric current has a node in the middle of the structure, and reaches large
values in the metal arms. This enables matching of high-impedance structures
to free space. On the other hand, bare antenna impedance strongly varies with
frequency once its length is close to $\lambda_{0}$, and small variations of
load impedance can lead to pronounced variations of the net resonant
frequency.

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# PAGE: A POSITION-AWARE GRAPH-BASED MODEL FOR EMOTION CAUSE ENTAILMENT IN
CONVERSATION

###### Abstract

Conversational Causal Emotion Entailment (${\rm C}_{2}{\rm E}_{2}$) is a task
that aims at recognizing the causes corresponding to a target emotion in a
conversation. The order of utterances in the conversation affects the causal
inference. However, most current position encoding strategies ignore the order
relation among utterances and speakers. To address the issue, we devise a
novel position-aware graph to encode the entire conversation, fully modeling
causal relations among utterances. The comprehensive experiments show that our
method consistently achieves state-of-the-art performance on two challenging
test sets, proving the effectiveness of our model. Our source code is
available on Github111https://github.com/XiaojieGu/PAGE.

Index Terms—  Emotion cause entailment, graph neural networks, position
encoding

##  1 Introduction

For a target utterance transmitting a specific emotion, the ${\rm C}_{2}{\rm
E}_{2}$ aims to identify the causal utterances from the conversation history
responsible for the target emotion. This novel task is essential to design
current dialogue agents, such as empathetic response [1] and emotion
counseling [2]. Besides, it provides a potential way to improve the
interpretability of affect-based models [3].

Lee et al. [4] first proposed emotion cause extraction (ECE), who pointed out
the importance of this task and considered it in a sentence-level
classification paradigm. Early research employed rule-based strategies [5] and
traditional machine learning approaches [6, 7] to deal with this problem. As a
further step toward the ECE, ${\rm C}_{2}{\rm E}_{2}$ considers a more
challenging conversation scenario. So far, only a few studies [3, 8] have put
the finger on ${\rm C}_{2}{\rm E}_{2}$. Porial et al. [3] solely paired one
target utterance with other utterances, which loses contextual information and
breaks the causal relationship between utterances. Li et al. [8] utilized
commonsense knowledge to facilitate causes recognization. From our insight
view, the contextual information has a significant effect on utterance
understanding, and it is difficult to determine whether they have a
corresponding causality without considering the position between them [9, 10].
For that, a naive way is to directly concatenate the absolute position
embedding with the utterance representation [11, 12]. However, this scheme
constantly leads to the aggregation of uninformative context, bringing
inference noise (i.e., the causal-irrelevant context). To this end, Xia et al.
[13] utilized a multi-head attention mechanism to weigh the position
information. Ding et al. [9] set the window size to consider only the
adjacency of target utterances. While these position encoding strategies
target the documents rather than conversations. Therefore, they neglected the
vital inter-speaker dependency, which is essential in understanding
conversation [14].

To address the above issues, we propose PAGE (Position-Aware Graph-based model
for Emotion cause entailment), in which we devise a novel relative position
encoding schema to distinguish utterances of different speakers for better
reasoning. Intuitively, relative position plays a vital role in conversation-
based causal inference. For example, there are two causal-relevant utterances
from distinct speakers, namely “ _Hey, you wanna see a movie tomorrow?_ ” and
“ _Sounds like a good plan._ ”. If the order is reversed, we probably do not
realize that “a good plan” refers to “see a movie”. Therefore, we construct
the position relationship based on the relative distance between the different
utterances of the inter-speaker. Furthermore, there is an explicit topological
relationship between the emotion and the cause [15], so we leverage graph
neural networks to encode the entire conversation context. We evaluate our
approach on the latest benchmark dataset. The experimental results show that
our method gains competitive performance compared with other strong baselines.
In summary, the main contributions of this paper are as follows:

$\bullet$ We propose a position encoding strategy that can enhance emotion
cause entailment and the understanding of conversation context in ${\rm
C}_{2}{\rm E}_{2}$. We design a novel position-aware graph to better aggregate
the entire conversation.

$\bullet$ We conduct comprehensive experiments to demonstrate the
effectiveness of our approach and provide a thorough analysis. Our model
achieves state-of-the-art performance on the benchmark dataset, especially
improving 8.1% absolute in F1 score on the IE test set.

Fig. 1: Overall architecture of the PAGE.

##  2 METHODOLOGY

As shown in Figure 1, our framework consists of three components, namely
Utterances Encoding with Emotion (§ 2.1), Position-aware Graph (PaG) (§ 2.2)
and Causal Classifier (§ 2.3). Given a conversation
$C=\\{u_{1},u_{2},\ldots,u_{k}\\}$ where there are target utterances $u_{t}\in
C$ of speaker $S_{t}\in\\{A,B\\}$. Each $u_{t}$ is labeled with a specific
non-neutral emotion (e.g., “happiness”, “anger”, “ sadness”). Our goal is to
identify the causal utterances in the conversation history of $u_{t}$. That
is, whether a $u_{i}$ ( $i\leq t$) is a cause for $u_{t}$ or not.

###  2.1 Utterances Encoding With Emotion

For an utterance $u_{n}=\\{w_{1},w_{2},\ldots,w_{m}\\}$ consisting of $m$
words, we add two special tokens $[CLS]$ and $[SEP]$, at the beginning and end
of it, respectively. Then, we use Roberta [16] to conduct sentence-level
encoding and take the hidden state of the last layer as word-level
representation:

$\displaystyle
h_{w}=\textrm{RoBERTa}\left([CLS],w_{1},w_{2},\ldots,w_{m},[SEP]\right),$ (1)

where $h_{w}\in\mathbb{R}^{m\times{d_{R}}}$ and $d_{R}=768$ denotes the
dimension of word-level representation in RoBERTa. Then, we get utterance
representation $h_{u}\in\mathbb{R}^{d_{u}}$ by a linear projection
$W_{u}\in\mathbb{R}^{d_{R}\times d_{u}}$ on $h_{w}$.

Emotion information in the conversation can pass among speakers [17], which is
beneficial for entailment [18]; thus, we concatenate emotion embedding with
the utterance representation, i.e., $h_{c}=h_{e}\oplus h_{u}$. To capture the
utterance features from multiple aspects, we employ a multi-head self-
attention mechanism [19], which contributes to sentence-level sentiment
analysis [20]. The value of the Q, K, and V vectors are the same as $h_{c}$:

$\displaystyle
head_{N}=\textrm{softmax}\left(\frac{QK^{T}}{\sqrt{d_{u}}}\right)V,$ (2)

where $head_{N}\in\mathbb{R}^{\frac{d_{u}}{N}}$ and N is the number of head.
We concatenate heads together to get the attention output $h_{a}$ and add it
to $h_{c}$ by $x=h_{a}+h_{c}$. Then, followed by multilayer perceptron (MLP)
with a single hidden layer and a residual connection:

$\displaystyle h_{n}$ $\displaystyle=\sigma\left(MLP(x)\right)+x,$ (3)

where the output dimension of the MLP is $\mathbb{R}^{d_{u}}$ and
$\sigma(\cdot)$ is a sigmoid function.

###  2.2 Position-aware Graph

To alleviate long-term dependency between utterances in long conversations, we
utilize graph neural networks to perform position-aware encoding. We design a
directed graph denoted as $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{R})$.
The utterance $u_{n}\in\mathcal{V}$ in the conversation denotes a node, whose
initial representation is $h_{n}$, and $r_{o,t}\in\mathcal{R}$ is the type of
an edge $(u_{o},r_{o,t},u_{t})\in\mathcal{E}$, where $u_{t}$ denotes the
target utterance and $u_{o}$ ( $1\leq o\leq n$ ) denotes other utterances.

The relative position plays a significant role in the transformation of causal
information between utterances; thus, we represent the type of edges between
nodes in relative position relations. Moreover, the sequence relationship
between the utterances of the same or different speakers facilitates the
information understanding in the utterances and enhances entailment. For this,
we construct a positional relation $r_{o,t}$ based on a relative distance
$D_{o,t}$ that denotes the distance between the target utterance $u_{t}$ and a
neighboring utterance $u_{o}$:

$\displaystyle D_{o,t}=\begin{cases}\frac{o-t}{2}&{S_{o}=S_{t}}\\\
-1&{S_{o}\neq S_{t}\ and\ t=o\pm 1}.\\\ \frac{o-t-1}{2}&{others}\end{cases}$
(4)

Fig. 2: An example of our relative position encoding. The target utterance is
$u_{5}$ and window size $w=1$. Purple represents past utterances, orange
represents target utterance and cream represents future utterances.

As the example shown in Figure 2, the target utterance $u_{5}$ has the same
relative distance from both surrounding utterances $u_{3}$ and $u_{4}$ in the
case of the same or different speakers, i.e., $D_{3,5}=D_{4,5}$, they are
considered to be in the same positional relationship, i.e., $r_{3,5}=r_{4,5}$.
For the utterance $u_{j}$ where $j>t$, the utterances after the target
utterance are not dominant compared to the parts before the target utterance
[21], but they still can serve as additional context information to
conversation understanding. We cannot completely ignore them, so we treat the
parts after the target utterance as the same relative position relation. In
this paper, we set it to 1. Furthermore, previous work indicates that most
causes are located near the emotion [10], and if two utterances are far apart,
then the causal relationship between them would diminish. We do not need to
give them the same attention compared to those near the target utterance. To
alleviate this issue, we set a window size in which the relative position
relation between $u_{t}$ and $u_{j}$ is set to the same. In short, we get
relative position relation $r_{o,t}$ from $u_{o}$ to $u_{t}$ by:

$\displaystyle r_{o,t}=\begin{cases}-w&{D_{o,t}<-w}\\\ D_{o,t}&{D_{o,t}\geq-w\
and\ o\leq t}.\\\ 1&{o>t}\end{cases}$ (5)

We use R-GCN [22], which can integrate different relationships between nodes
to get the final utterance representation which is beneficial to the position-
aware transformation:

$\displaystyle
h_{t}^{{}^{\prime}}=\sigma\left(\sum_{r\in\mathcal{R}}\sum_{o\in\mathcal{N}_{t}^{r}}\frac{1}{c_{t,r}}W_{r}h_{o}+W_{0}h_{t}\right),$
(6)

where $\mathcal{N}_{t}^{r}$ denotes the set of neighboring nodes of node $t$
under the relationship $r$, $c_{t,r}$ is a regularization constant,
$W_{r}\in\mathbb{R}^{{d_{u}}\times{d_{u}}}$ is the trainable parameter to
transform the neighborhood node $o$ with relationship $r$.

###  2.3 Causal Classifier

We concatenate the final represent representations of the target utterance
$u_{t}$ and the other utterance $u_{o}$, then employ MLP with a single hidden
layer and a sigmoid function to yield logits $\hat{p}_{o,t}$ and use cross-
entropy loss function to do binary classification:

$\displaystyle\hat{p}_{o,t}$
$\displaystyle=\sigma\left(\textrm{MLP}\left(h_{o}^{{}^{\prime}}\oplus
h_{t}^{{}^{\prime}}\right)\right).$ (7)

##  3 experiment

###  3.1 Experiment Setting

Test set | Conv. | Utt. | Avg. | Pos. Pairs | Neg. Pairs
---|---|---|---|---|---
DD | 225 | 2,405 | 10 | 1,894 | 26,814
IE | 16 | 665 | 41 | 1,080 | 11,305
Table 1: The statistics of RECCON test set, where ”DD” and ”IE” stands for the
RECCON-DD and RECCON-IE test sets, respectively. Avg. represents the number of
utterances per conversation on average.

Dataset. We use a recently proposed benchmark dataset for ${\rm C}_{2}{\rm
E}_{2}$, namely RECCON [3], where each utterance in the conversations is
attached with a human-annotated emotion label. It is sampled from two popular
datasets of emotion recognition task in conversation [23, 24]. To
comprehensively evaluate the model, there are two test sets in RECCON: RECCON-
DD and RECCON-IE. It is worth noting that the data source of the RECCON-DD is
the same as those in the training set, while RECCON-IE is not. In other words,
the RECCON-IE stands for a demanding cross-domain evaluation. Specifically, as
shown in Table 1, the average length of each conversation in IE is longer than
in DD. Besides, the emotional shifts in IE conversations are more frequent
than in DD, which requires a more complex emotional understanding capacity of
the model [3]. Following Poria et al. [3], we omit those rare-appeared future
causes and adopt the “Fold-1” of RECCON as the negative samples.

Implementation Details. For training, we choose the cross entropy loss
function and set the learning rate to 3e-5 with a batch size of 4. The
dimension $d_{u}$ of utterance representations and hidden size of MLP are set
to 300, and the number of attention heads and window size are set to 6 and 3,
respectively. For the hyperparameter $c_{t,r}$, we fix it to 2.

###  3.2 Results

Model | DD | IE
---|---|---
Neg. F1 | Pos. F1 | Macro F1 | Neg. F1 | Pos. F1 | Macro F1
Base[3] | 88.74 | 64.28 | 76.51 | 95.67 | 28.02 | 61.85
ECPE-MLL[25] | 94.68 | 48.48 | 71.59 | 93.55 | 20.23 | 57.65
ECPE-2D[9] | 94.96 | 55.50 | 75.23 | 97.39 | 28.67 | 63.03
RankCP[26] | 97.30 | 33.00 | 65.15 | 92.24 | 15.12 | 54.75
${\rm KEC}^{\clubsuit}$ [8] | 95.74($\pm$0.05) | 66.76($\pm$0.33) | 81.25($\pm$0.17) | 86.08($\pm$0.46) | 19.72($\pm$1.71) | 52.9($\pm$0.8)
PAGE | 95.80($\pm$0.06) |  68.80($\pm$0.11) |  82.30($\pm$0.05) | 96.41($\pm$0.25) |  45.96($\pm$0.82) |  71.19($\pm$0.52)
-w/o PaG | 93.36($\pm$0.46) | 52.94($\pm$0.97) | 73.15($\pm$0.31) | 84.53($\pm$2.0) | 21.62($\pm$0.32) | 53.07($\pm$0.89)

Table 2: The results on RECCON. We use Macro F1 as an overall metric, while
the Pos. F1 and Neg. F1 represent F1 score on positive and negative pairs,
respectively. $\clubsuit$: since KEC [8] only report the results on DD, we run
the KEC algorithm on the IE test set and report the results.

We compare our approach with solid baselines. As shown in Table 2, the scores
of the first four baselines are reported by Poria et al. [3], which are the
best-run results among several repeated experiments. For better comparison, we
follow Li et al. [8] (i.e., KEC), reporting the average F1 score and the
corresponding variance over five random runs.

Overall, we achieve state-of-the-art performance on both test sets under two
main metrics (i.e., Pos. F1 and Macro F1), except the Neg. F1 due to the
imbalance in the number of the Pos. and Neg. samples, as shown in Table 1.
Compared to their methods, our relative position encoding strategy can fully
consider inter-speaker dependency, which can effectively enhance the
understanding of utterances. Since the conversations in the IE set are longer
than those in the DD set, as shown in Table 1, the emotion cause shifts more
in the IE set. Therefore, detecting the causes in the IE set is more
challenging than in the DD set. Owing to our PaG structure, which has
advantages in aggregating long-term contextual information, our model still
maintains excellent results on IE, with a notable performance gap compared
with other baselines. Since the conversations in real-world applications are
mostly verbose [27], this promising performance demonstrates the robustness
and practicality of our approach.

To further prove the effectiveness of our position encoding strategy, we
conduct an additional ablation study. As shown in the bottom row of Table 2,
after removing the PaG, the performance drops significantly, especially in
those lengthy conversations of the IE test set. This result indicates that
position is beneficial to emotion cause inference in the conversations, which
is particularly helpful to those long conversations, demonstrating the
outstanding generalization capacity and potential practicality of our model.

###  3.3 Effect of Window Size

Fig. 3: Performance on the DD validation set with varying window size.

The window size may have an underlying effect on the model performance. Hence,
we experiment to investigate the performance fluctuation from various window
sizes. Note that the validation set of RECCON comes from the same source as
the DD test set, while there is no corresponding validation set for IE. Figure
3 presents the results. Theoretically, a larger window size can lead to more
position categories in the graph, but it increases the complexity of the graph
network. While a tiny window represents that only the adjacency of target
utterances has an informative position signal. As shown in Figure 3, when we
increase the window size from 1 to 3, the performance is improved because the
diverse position categories benefit the context understanding. However, the
large window size ($>3$) has adverse effects due to the inference noise from
the long-distance utterances (i.e., causal-irrelevant context). Thus we choose
window size = 3 in the PAGE.

##  4 Conclusion

In this paper, we propose a position-aware graph for ${\rm C}_{2}{\rm E}_{2}$
task, which is beneficial for emotion cause entailment. Specifically, our
framework takes advantage of an effective position encoding strategy
incorporating inter-speaker dependency, thus enhancing the capacity for
complex emotion cause reasoning. Our PAGE model achieves SOTA performance on
two challenging test sets, with 1.1% and 8.1% improvement at Macro F1 compared
to previous models.

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