# Hyperbolic phonon-polariton electroluminescence in graphene-hBN van der
Waals heterostructures

Qiushi Guo1,2,3∗, Iliya Esin4∗,†, Cheng Li∗, Chen Chen1, Song Liu5, James H.
Edgar5, Selina Zhou6, Eugene Demler7, Gil Refael4, Fengnian Xia1,†
1Department of Electrical Engineering, Yale University, New Haven 06511, USA.
2Photonics Initiative, Advanced Science Research Center, City University of
New York, NY, USA
3Physics Program, Graduate Center, City University of New York, New York, NY,
USA
4Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125, USA
5Tim Taylor Department of Chemical Engineering, Kansas State University,
Manhattan, Kansas 66506, USA
6Department of Electrical Engineering, California Institute of Technology,
Pasadena, California 91125, USA
7Institute for Theoretical Physics, ETH Zurich, Zürich, Switzerland ∗These
authors contributed equally to this work.
†Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Phonon-polaritons are electromagnetic waves resulting from the coherent
coupling of photons with optical phonons in polar dielectrics. Due to their
exceptional ability to confine electric fields to deep subwavelength scales
with low loss, they are uniquely poised to enable a suite of applications
beyond the reach of conventional photonics, such as sub-diffraction imaging
and near-field energy transfer. The conventional approach to exciting phonon-
polaritons through optical methods, however, necessitates costly mid-infrared
and terahertz coherent light sources along with near-field scanning probes,
and generally leads to low excitation efficiency due to the substantial
momentum mismatch between phonon-polaritons and free-space photons. Here, we
demonstrate that under proper conditions, phonon-polaritons can be excited
all-electrically by flowing charge carriers. Specifically, in hexagonal boron
nitride (hBN)/graphene heterostructures, by electrically driving charge
carriers in ultra-high-mobility graphene out of equilibrium, we observe bright
electroluminescence of hBN’s hyperbolic phonon-polaritons (HPhPs) at mid-IR
frequencies. The HPhP electroluminescence shows a temperature and carrier
density dependence distinct from black-body or super-Planckian thermal
emission. Moreover, the carrier density dependence of HPhP electroluminescence
spectra reveals that HPhP electroluminescence can arise from both inter-band
transition and intra-band Cherenkov radiation of charge carriers in graphene.
The HPhP electroluminescence offers fundamentally new avenues for realizing
electrically-pumped, tunable mid-IR and THz phonon-polariton lasers, and
efficient cooling of electronic devices.

Phonon-polaritons (PhPs) are electromagnetic waves that arise from the
coherent interaction between photons and optical phonons (i.e., quanta of
lattice vibrations) in polar dielectrics at mid-infrared (mid-IR) and
terahertz (THz) frequencies. They allow for the confinement of electromagnetic
fields and energy to length scales significantly smaller than the free photon
wavelength, but without incurring significant Ohmic losses often associated
with plasmon-polaritons1. These unique properties make PhPs attractive for
various applications, including sub-wavelength waveguiding2, near-field sub-
diffraction imaging and focusing3, enhanced near-field heat transport4; 5; 6,
and controlling the coherence of thermal emission7; 8; 9. Usually, PhP modes
are excited using optical techniques, which not only necessitates costly mid-
IR and THz coherent light sources and near-field scanning probes, but also
suffers from low excitation efficiency due to the substantial momentum
mismatch between PhPs and free-space photons10. Although all-electrical
methods to excite plasmons-polaritons11 and exciton-polaritons12; 13; 14 have
been demonstrated, an efficient, all-electric approach to excite and control
PhPs still remains elusive.

Figure 1: Concept of HPhP electroluminescence from non-equilibrium charge
carriers in graphene (a) Schematic of HPhP electroluminescence in an
hBN/graphene/hBN vdW heterostructure. (b) Illustration of the Zener-Klein
tunneling mechanism and Fermi surface tilting. Under a lateral electrical
field pointing to the left, electrons in the valence band of graphene can
tunnel into the conduction band, generating additional electron-hole pairs in
the conduction and valence bands. Meanwhile, the lateral electric field leads
to the tilting of graphene’s Fermi surface in momentum space. (c) Graphene
charge carrier distribution in equilibrium state. (d) HPhP electroluminescence
under a low lateral electric field. (e) HPhP electroluminescence under a high
lateral electric field. In both d and e, red arrows denote HPhP emission due
to inter-band electron transitions, and blue arrows denote HPhP emission due
to intra-band Cherenkov radiations.

In fact, electron-phonon interactions are ubiquitous in conducting materials,
in which drifting electrons with sufficient kinetic energy can emit optical
phonon modes (OPs) in surrounding polar dielectrics via long-range Fröhlich
coupling15. Given that PhPs originate from the hybridization of OPs and
photons, this raises an intriguing question: Can PhPs be emitted by drifting
electrons? In optically isotropic polar dielectrics, such as $\rm SiO_{2}$ and
SiC, photons are unable to effectively hybridize with phonons in the bulk of
the material. This limitation arises from the fact that they have comparable
energies only in a narrow range of momenta, which is given by the phonon
frequency divided by the speed of light. Consequently, PhP modes are confined
to the surface, where the photons are allowed to have high in-plane momenta
and low frequencies 16. This constraint has thus far precluded the observation
of PhP emission by drifting electrons. Interestingly, such a constrain can be
overcome when considering a conducting material surrounded by a hyperbolic
polar dielectric, which has in-plane ($\epsilon_{xx}$) and out-of-plane
permittivities ($\epsilon_{zz}$) of different signs within the Reststrahlen
bands (i.e. $\epsilon_{xx}\epsilon_{zz}<0$). As a distinguishing feature of
hyperbolic polar dielectric in the Reststrahlen bands, deep-subwavelength,
ray-like electromagnetic modes of arbitrarily large momentum (known as
hyperbolic phonon polaritons (HPhPs)) can propagate inside the volume of the
dielectric17; 3; 18; 19. With proper out-coupling schemes, these HPhPs can be
scattered into the free space20; 21. Furthermore, these HPhP modes possess a
high density of states in momentum space2; 22, which greatly enhances their
interaction with drifting electrons.

The graphene/hexagonal boron nitride (hBN) van der Waals (vdW) heterostructure
represents an ideal platform for observing the HPhP emission by drifting
electrons. First, hBN is a natural hyperbolic polar dielectric, which supports
low-loss HPhPs at mid-IR frequencies23. Second, the graphene/hBN interface is
ultra-clean, yielding reduced impurity scattering and significantly higher
carrier mobility and drift velocity in graphene24; 25. Third, the electron-
acoustic phonon scattering is intrinsically weak in graphene/hBN vdW
heterostructure, thus rendering the electron-HPhP scattering dominant 22; 26.
Prior studies have shown that in hBN-encapsulated graphene, the cooling rate
of hot carriers in graphene becomes exceptionally high 27; 28; 22; 29,
suggesting that the presence of hBN’s HPhPs can be responsible for the
efficient cooling of hot carriers in graphene. Yet, the direct observation of
HPhP electroluminescence has not been reported. In this work, we employ mid-IR
emission spectroscopy to reveal the hyperbolic phonon-polariton
electroluminescence within graphene/hBN van der Waals heterostructures.

Figure 2: Room-temperature charge carrier transport in ultra-clean graphene
encapsulated by h10BN (a) Gate-dependent graphene resistivity measured at room
temperature. The inset shows the optical image of the experimental device with
a hall-bar geometry, which has a channel width of 5 $\mu$m. (b) Extracted
room-temperature carrier mobility of 10B hBN encapsulated graphene as a
function of charge carrier density. (c) Measured room-temperature current
density ($J$) versus lateral electric field ($F$) at various gate voltages
($V_{\mathrm{G}}$). (c) Extracted saturation velocity ($v_{\mathrm{sat}}$)
versus carrier density. Blue symbols are experimental data. The solid blue
curve is an estimation of $v_{\mathrm{sat}}$ by a canonic drift velocity model
considering the lowest energy optical phonon mode of h10BN, which has an
energy of $\hbar\omega_{\mathrm{OP}}=105$ meV.

Figure 1a depicts the concept of HPhP electroluminescence in an
hBN/graphene/hBN vdW heterostructure. In this setup, a single-layer of
graphene sheet is encapsulated by two hBN slabs located below and above the
graphene sheet. When a lateral electric field is applied along the graphene
channel, charge carriers in graphene (red spheres) are accelerated. Under
certain conditions, they can efficiently dissipate their kinetic energy by
emitting HPhP modes (bright yellow rays) in nearby hBN slabs. These HPhP modes
in hBN slabs, which possess large momentum, can be subsequently out-coupled to
the free space via the nano-scatters on top of the hBN. As illustrated in Fig.
1b, the effect of the lateral electric field is two-fold. First, it generates
additional electron-hole pairs in graphene through the Zener-Klein tunneling
mechanism30; 27; 31 with the rate proportional to $\sim F^{3/2}$ 32. Second,
it leads to an energy difference between the forward populated and unpopulated
charge carrier states (manifested by the tilting of graphene’s Fermi surface
in momentum space33), providing a finite charge carrier drift velocity
$v_{\text{d}}$.

Figure 3: Mid-IR HPhP electroluminescence in h10BN/graphene/h10BN
heterostructure. (a) Illustration of HPhP electroluminescence. The
experimental device consists of a h10BN/graphene/h10BN heterostructure
fabricated on a diamond substrate. Edge contacts are formed between the
graphene channel and Au/Cr metal leads. Au nanodiscs (diameter = 700 nm,
height = 100 nm, pitch= 1 $\mu$m) on top of hBN layer enable out-coupling of
HPhPs to free space. (b) Optical microscope image of the fabricated device.
Scale bar: 10 $\mu$m. (c) Simulated near-field distribution of a HPhP mode at
1400 cm-1 excited by a dipole emitter placed within the graphene layer. The
sharp edges of Au nanodiscs scatter the deep-subwavelength, ray-like HPhP mode
into the free space. Scale bar: 300 $\mu$m. (d) Simulated far-field emission
spectra with (red) and without (blue) Au nanodiscs on the top of the
heterostructure. Measured (e) and calculated (g) $V_{\mathrm{DS}}$ dependence
of HPhP electroluminescent spectrum at $V_{\mathrm{G}}$=0. Measured (f) and
calculated (h) temperature dependence of HPhP electroluminescent spectrum
under $V_{\mathrm{DS}}=16$ V and $V_{\mathrm{G}}$=0. White dashed lines in e,
f, g and h mark the boundaries of type I ($785-845$ cm-1) and type II
($1394.5-1650$ cm-1) Reststrahlen bands of h10BN.

The HPhP electroluminescence under the lateral electric field can be described
as a spontaneous emission process of HPhP modes by charge carriers in
graphene. This process adheres to the principles of both energy and momentum
conservation, with the emission rate determined by Fermi’s golden rule. Figure
1d illustrates the HPhP electroluminescence under a low lateral electric
field. Compared to the equilibrium state (Fig. 1c), the Zener-Klein tunneling
mechanism generates additional electrons and holes in the conduction band and
valence band. Due to the additional holes in the valence band, electrons in
the conduction band can undergo a transition to the valence band, leading to
the emission of HPhP modes via inter-band transition (red arrow). However,
since the energy difference between forward and backward-populated electrons
is less than the energy of HPhPs, drifting electrons in graphene cannot emit
HPhP modes through an intra-band process (blue arrow) because there are no
available final states for such transitions to occur. As shown in Fig. 1d,
under a sufficiently high lateral electric field, there will be more electron-
hole pairs generated. Meanwhile, the energy difference between forward and
backward-populated electrons can exceed the energy of HPhPs. In this case, the
HPhPs electroluminescence can be more efficient since charge carriers can emit
HPhPs through both inter-band transitions and intra-band transitions, as shown
in Fig. 1e. The intra-band emission is analogous to phonon Cherenkov emission,
originally describing a mechanism of acoustic phonon emission when the drift
velocity exceeds the speed of sound 34; 35; 36; 37; 38; 39. However, the HPhP
emission rate is much higher than the conventional phonon-assisted electron-
hole scattering, due to the larger density of states available for an HPhP
emission.

To validate the strong interactions between charge carriers in graphene and
HPhPs, we first measure the electrical transport characteristics of a hBN-
encapsulated graphene device fabricated by a vdW assembly technique24. Here,
we used 10B isotopically enriched hBN (h10BN)40 for graphene encapsulation.
Compared to naturally abundant hBN, h10BN offers around threefold improvement
in HPhP lifetime41. In our device, the thickness of the SiO2 substrate, the
bottom hBN, and the top hBN are 300 nm, 35 nm, and 42 nm, respectively. To
eliminate the effect of metal/graphene contact resistance, we employed a four-
terminal (Hall-bar) device structure (inset of Fig. 2a) to measure charge
carrier transport in the graphene. Figure 2a shows the measured room-
temperature resistivity of the device as a function of gate voltage
($V_{\mathrm{G}}$) at a low bias voltage of 10 mV. The transport
characteristics indicate the graphene is remarkably pristine, yielding a room-
temperature carrier mobility ($\mu_{\mathrm{e}}$) exceeding 100,000 cm2 V-1
s-1 at a carrier density of $\sim 1\times 10^{12}$ cm-2 (Fig. 2b), which is
close to the acoustic phonon-limited mobility for intrinsic graphene42.

Using the same device, we further measured the current density ($J$) as a
function of lateral electric field ($F$) at room temperature. As shown in Fig.
2c, under a low lateral electric field ($F<0.05$ V/$\mu$m), the current
density scales almost linearly with the electric field. In this regime, the
carrier drift velocity is mainly limited by the electron-acoustic phonon
scattering 42. However, as the lateral electric field strength increases, the
device exhibits current density saturation 43 since the carrier drift velocity
$v_{\text{d}}$ reaches its maximum achievable value, determined by the group
velocity of graphene’s Dirac cone ($v_{\text{F}}=10^{6}$ m/s) 44. As shown in
Fig. 2c, at even higher lateral electric fields, the current further
saturates. This phenomenon occurs because the forward and backward-populated
charge carriers can accumulate a sufficient energy difference. Such an energy
difference enables the forward-populated carriers to dissipate their kinetic
energy by emitting optical phonons, leading to a further reduction of drift
velocity (see Supplementary Information Section I for detailed analysis). In
Fig. 2d, we plot the saturation velocities ($v_{\mathrm{sat}}$) (blue symbols
) of graphene at various carrier concentrations. By comparing the measured
$v_{\mathrm{sat}}$ with a canonical drift velocity model25 (blue solid curve),
we found that the $v_{\mathrm{sat}}$ of graphene is primarily limited by the
remote OP modes of h10BN ($\omega_{\mathrm{OP}}$= 105 meV). This finding
strongly suggests that, under high lateral electric fields, charge carriers in
graphene interact strongly with OP modes in hBN, raising the intriguing
possibility of observing HPhP electroluminescence.

Next, we study the HPhP electroluminescence in h10BN/Graphene/h10BN
heterostructures using mid-IR emission spectroscopy (see Supplementary
Information Section II for details) and an experimental device illustrated in
Fig. 3a. Figure 3b shows an optical image of our fabricated device. We
fabricate the device on a diamond substrate since the diamond does not support
any optical phonon modes within the frequency range of interest. In addition,
due to its exceptionally high thermal conductivity, diamond can efficiently
dissipate the lattice heat generated in graphene and hBN, effectively
minimizing the influence of thermal emission in our experimental observations.
Compared to the device we used for the transport measurements (inset of Fig.
2a), the device has a larger graphene area of $\sim$140 $\mu$m2, with a
graphene channel length of 7 $\mu$m. This augmented active region ensures
sufficient emission power and a substantial signal-to-noise ratio. In
addition, as shown in Fig. 3a and b, we fabricated an Au nanodiscs array
(diameter = 700 nm, height = 100 nm, pitch = 1 $\mu$m) on top of the hBN layer
in order to significantly enhance the out-coupling of HPhPs into free space.
As shown in the near-field electromagnetic simulation (Fig. 3c), the sharp
bottom edges of Au nanodiscs can compensate for the large momentum mismatch
between the HPhP modes and free-space electromagnetic waves, thus scattering
the propagating HPhP mode in hBN into free space45; 20. Moreover, the
simulated emission spectra shown in Fig. 3d further verify that our designed
nanodiscs array is capable of scattering the HPhPs into the far field.

Figure 3e shows the measured electroluminescence spectrum as a function of
$V_{\mathrm{DS}}$ at zero $V_{\mathrm{G}}$ and an environmental temperature of
83 K. Notably, broadband emission features in type I Reststrahlen band can be
detected at a modest $V_{\mathrm{DS}}$ of 6 V, which corresponds to a lateral
electric field ($F$) of $\sim$0.9 V/$\mu$m and an electrical power of $\sim$20
mW. Furthermore, as $V_{\mathrm{DS}}$ exceeds 9 V, broadband emission features
within the type-II Reststrahlen bands are detectable. In addition, the
intensity of electroluminescence in the type I Reststrahlen band is always
stronger than that in the type II Reststrahlen band. Remarkably, as shown in
Fig. 3f, under an applied $V_{\mathrm{DS}}$ of 16 V, the intensity of
electroluminescence remains nearly unchanged even as we elevate the
environmental temperature from 83 K to 273 K. This phenomenon strongly
suggests that the observed electroluminescence cannot be attributed to
traditional black-body or Super-Planckian thermal radiation process, since
they typically exhibit a strong dependence on the temperature of the
emitter22; 46. To substantiate this point, we also investigated the
traditional thermal radiation process by conducting a control experiment using
another hBN-encapsulated graphene device without Au nanodiscs to out-couple
HPhPs. With this device, the emission features emerge at a significantly
higher applied electrical power ($>500$ mW), thus showing very low emission
efficiency. More importantly, the emission intensity dramatically decreases at
lower environmental temperatures, which is in stark contrast to our
observation here (see Supplementary Information Section III for details).

Figure 4: Gate-voltage dependence of HPhP electroluminescence (a) Illustration
of the experimental device in which h10BN/graphene/h10BN heterostructure was
fabricated on SiO2 (90 nm)/silicon substrate. The doping concentration of
graphene is controlled by the back-gate voltage applied to the silicon
substrate. (b) Optical image of the experimental device for gate-voltage
dependent HPhP emission measurements. Scale bar: 10 $\mu$m. (c) Measured gate-
voltage dependence of HPhP electroluminescence spectrum at 100 K. In the
measurement, the $V_{\mathrm{DS}}$ was fixed at 16 V. (d) Calculated gate-
voltage dependence of e-HPhP scattering rate at a $V_{\mathrm{DS}}$ of 16 V.
(e) Illustration of HPhP electroluminescence at a low $V_{\mathrm{G}}$ (left),
an intermediate $V_{\mathrm{G}}$ (middle), and a high $V_{\mathrm{G}}$
(right).

To gain deeper insights into these behaviors, we model the emission by kinetic
equation 39 for the HPhPs occupations
$b_{{\boldsymbol{q}},a}({\boldsymbol{r}},t)$, where $a=({\alpha},n)$ with
${\alpha}=I,II$ denoting the Restrahlen band, $n=0,1,2...$ denoting the branch
of HPhP modes, and ${\boldsymbol{q}}$ is the wavevector. The kinetic equation
can be written as

$\frac{\partial b_{{\boldsymbol{q}},a}}{\partial t}={\Gamma}^{\rm
amp}_{{\boldsymbol{q}},a}b_{{\boldsymbol{q}},a}+{\gamma}^{\rm
em}_{{\boldsymbol{q}},a}-\frac{b_{{\boldsymbol{q}},a}-b^{0}_{{\boldsymbol{q}},a}}{{\tau}_{p}}-{\boldsymbol{v}}_{s,{\boldsymbol{q}},a}\cdot\nabla
b_{{\boldsymbol{q}},a}$ (1)

where $\Gamma_{{\boldsymbol{q}},a}^{\rm amp}$ and
${\gamma}_{{\boldsymbol{q}},a}^{\rm em}$ denote the amplification and
spontaneous emission rates of HPhP modes due to the emission and absorption of
HPhPs by the non-equilibrium electron and hole populations in graphene.
${\boldsymbol{v}}_{s,{\boldsymbol{q}},a}=\partial_{\boldsymbol{k}}{\omega}_{{\boldsymbol{q}},a}$
is the group velocity of the HPhP modes, and ${\omega}_{{\boldsymbol{q}},a}$
is the angular frequency of HPhP modes,
$b_{{\boldsymbol{q}},a}^{0}=(e^{\hbar{\omega}_{{\boldsymbol{q}},a}/k_{B}T}-1)^{-1}$
is the equilibrium distribution of HPhPs. $1/\tau_{p}$ is the loss rate of
HPhP considering phonon-phonon interactions, phonon-impurity interactions, and
out-coupling to free space. The detailed calculations of
$\Gamma_{{\boldsymbol{q}},a}^{\rm amp}$ and $\gamma_{{\boldsymbol{q}},a}^{\rm
em}$ are described in Supplementary Information Sections IV to VI. In our
theoretical calculation, we accounted for all energy and momentum allowed
e-HPhP scattering processes, given the non-equilibrium electronic
distribution. The average emission rate per frequency ${\omega}$ in the steady
state is then given by

$I_{\rm
rad}({\omega})=\int\frac{d^{2}{\boldsymbol{q}}}{(2{\pi})^{2}}\sum_{a}\frac{\hbar{\omega}_{{\boldsymbol{q}},a}}{{\tau}_{\rm
rad}}\bar{b}_{{\boldsymbol{q}},a}{\delta}({\omega}-{\omega}_{{\boldsymbol{q}},a}),$
(2)

where $\tau_{\rm rad}$ denotes the coupling of the HPhPs modes to free space
assisted by the Au nanodiscs, and
$\bar{b}_{{\boldsymbol{q}},a}=\mathcal{A}^{{}^{-1}}\int
d^{2}{\boldsymbol{r}}b_{{\boldsymbol{q}},a}({\boldsymbol{r}})$ is the HPhP
occupation averaged over the sample area $\mathcal{A}$.

Figures 3g and h show the $V_{\mathrm{DS}}$ dependence and temperature
dependence of averaged HPhP emission rate calculated by equation 2. The
theoretical results agree with our experiments, indicating that the e-HPhP
scattering process is responsible for the observed electroluminescence. The
higher observed emission intensity in the type I Reststrahlen band can be
attributed to the lower energy threshold needed to excite this type of HPhP
mode. Moreover, the emission intensity does not change significantly with
temperature, because the temperature mostly affects the mobility of the
carriers 42, which has a weak effect on the e-HPhP scattering rate.
Furthermore, according to our theoretical analysis, we did not observe a net
gain of HPhP along the graphene channel. This is presumably due to the level
of the disorder in graphene and the strong out-coupling of HPhP modes to the
free space.

As HPhP electroluminescence arises from electron-HPhP scattering, its
intensity can be significantly influenced by the Fermi level of graphene. This
is because the Fermi surface of graphene governs the available momentum space
for e-HPhP scattering events. To investigate this, we experimentally study
$V_{\mathrm{G}}$ of the HPhP electroluminescence using another device
illustrated in Fig. 4a. For this device, the h10BN/Graphene/h10BN
heterostructure was fabricated on SiO2 (90 nm)/silicon substrate and the Fermi
level of graphene can be controlled by the back-gate voltage
($V_{\mathrm{G}}$) applied on the silicon substrate. Figure 4b shows the
optical image of the fabricated device, which has a graphene channel length of
7 $\mu$m.

Fig. 4c shows the measurement results under a fixed lateral bias voltage of 16
V. Again, we observed broadband HPhP emission features in both type-I and
type-II Reststrahlen bands, despite that the observed emission intensity in
type-I Reststrahlen is weaker than that of type-II in this case. This can be
ascribed to the high optical loss of SiO2 substrate around 800 cm-1 due to the
presence of an optical phonon mode47. Interestingly, we found the HPhP
emission in type-I and type-II Reststrahlen bands behave quite differently in
response to the gate voltage ($V_{\mathrm{G}})$. As $V_{\mathrm{G}}$ was tuned
from -2 V to 22 V, the intensity of type-I HPhP emission first decreases, and
then substantially increases. However, the type-II HPhP emission monotonously
increases with the gate voltage. This behavior, which is well captured by the
theoretically calculated HPhP emission rate shown in Fig. 4d, is attributed to
the different responses of inter-band and intra-band emission processes to the
gate voltage. Specifically, when $V_{\mathrm{G}}$ $\sim$ 0, the system
predominantly emits type-I HPhPs through the inter-band processes, whereas the
intra-band emission process is weak due to the small Fermi surface and
momentum space, as illustrated in 4e. As $V_{\mathrm{G}}$ increases, the
density of holes decreases, leading to a suppression of the inter-band
emission process and a strong reduction of type-I HPhPs emission. Further
increasing the $V_{\mathrm{G}}$ resulting in stronger intra-band emission of
type-I HPhPs due to larger available momentum space. In contrast, type-II
HPhPs are predominantly emitted through inter-band processes within the
parameter range of the experiment. Due to the higher energy of type-II HPhPs,
their emission is activated only for higher values of $V_{\rm G}$, giving rise
to a growth of emission with the $V_{\rm G}$. For even higher values of
$V_{\rm G}$, the type-II HPhP emission rate starts to decrease because of the
decreasing density of holes, similar to type-I HPhPs.

To summarize, our study presents the first observation of HPhP
electroluminescence in graphene/hBN van der Waals heterostructures, which also
marks the first experimental demonstration of exciting phonon-polariton waves
exclusively through electrical methods. This observed HPhP electroluminescence
emerges under a strongly biased non-equilibrium electronic state in graphene,
which supports both intra-band Cherenkov and inter-band Klein-Zener emission
processes. Notably, HPhP electroluminescence exhibits a broadband emission
spectrum within the two Reststrahlen bands of hBN, characterized by
temperature independence that distinguishes it from traditional black-body and
Super-Planckian thermal radiation. Moreover, our study further reveals that
the charge carrier density in graphene can modulate the respective
contributions of intra-band Cherenkov and inter-band Klein-Zener emission
processes to the HPhP electroluminescence.

Although we have not observed a net gain of HPhP along the device channel (see
Supplementary Information Section VI for the theoretical efficiency analysis)
in this study, achieving a net gain holds significant promise by minimizing
HPhP outcoupling to free space and further increasing the driving current.
Furthermore, by introducing the feedback of HPhP modes using polaritonic nano-
resonators17; 48; 49, the HPhP electroluminescence can potentially become
coherent, thus leading to the electrically driven HPhP lasing. Looking
forward, our reported HPhP electroluminescence mechanism can be harnessed to
develop conceptionally new and power-efficient electrically-pumped amplifiers,
and polaritonic lasers at mid-IR and THz frequencies for sensing and
communication applications. On a different front, the HPhP electroluminescence
or lasing can serve as an efficient mechanism for cooling hot carriers in
silicon or III-V semiconductor-based CMOS, RF, and power transistors, which
can greatly alleviate the thermal bottleneck of current CMOS, RF, and power
electronics.

## Methods

Device Fabrication. h10BN crystals are synthesizes using a method described in
Ref. 40. Graphene and h10BN flakes were obtained through mechanical
exfoliation. The process of device fabrication began by assembling a
h10BN/graphene/h10BN heterostructure on a 90 nm SiO2/Si (or diamond)
substrate, following the method described in Ref. 24. Subsequently, the sample
underwent annealing at 600 ∘C for 6 hours in a flowing hydrogen/argon
environment. This annealing step served to eliminate impurities and enhance
the carrier mobility of graphene. To define the device channel and metal
electrodes, a 100 kV electron beam lithography system and PMMA as a resist
were employed. The heterostructure was partially etched using SF6 and O2
plasma in an Oxford Plasmalab 100 reactive ion etching (RIE) system. Following
this, Cr/Au electrodes (5 nm/45 nm) were thermally evaporated to form edge
contacts with the graphene. To fabricate the Au nanodiscs on top of the
heterostructure, another e-beam lithography step was used to define the
pattern. Subsequently, we etched the top h10BN by $\sim$4 nm by CH3 and O2
plasma. This step resulted in the formation of h10BN nano-slots, which
stabilized the Au nanodiscs. Finally, the Au discs were fabricated by
evaporating 5 nm Cr and 100 nm Au using an e-beam evaporator, followed by a
lift-off process in acetone.

Mid-infrared electroluminescence measurement. The detailed mid-infrared
electroluminescence measurement procedure is described in Supplementary
Information Section II. In the measurement, the sample was wired-bonded to a
chip carrier and was placed on a temperature-controllable stage (Linkham
HFS600-PB4). The gate voltage to the device was applied to the device using a
Keithley 2612 source meter. In this setup, instead of employing a DC source-
drain bias voltage, a 34.5 kHz square wave with phase synchronization to the
lock-in amplifier was utilized as the source-drain voltage bias. The square
wave was generated by an arbitrary function generator (AFG) and amplified by a
custom-made voltage amplifier. Once the square wave was applied to the device,
the MCT detector collected both a DC environmental thermal emission background
signal and an AC emission signal from the device. The AC voltage output from
the MCT detector underwent filtering through a narrow pass-band band-pass
filter (34.2 - 34.8 kHz). The filtered output voltage signal was then fed into
the lock-in amplifier with a time constant set to 1 ms. The DC output signal
from the lock-in amplifier was directed to the DC input of the FTIR (Fourier-
transform infrared spectroscopy). The FTIR was used to form the interferogram
and acquire the IR emission spectra after performing the Fourier transform on
the data.

Electromagnetic simulations. The optical response of graphene-hBN
heterostructures was simulated using Ansys Lumerical FDTD (2023b). A 3-D FDTD
model was constructed by specifying the thickness and geometry of hBN slabs,
graphene, and Au discs. Periodic boundary conditions were imposed on two sides
and absorption boundary conditions (PML) were imposed both on the top and the
bottom of the simulation region. A graphene layer was modeled using a uniaxial
anisotropic permittivity assuming that the graphene layer has a finite
thickness (C (graphene)-Falkovsky (mid-IR) material model). In the model,
graphene’s conductivity is described by a Drude-like expression50 which is
valid for long-wavelength mid-IR. The the uniaxial anisotropic dielectric
function of h10BN is modeled by

$\epsilon_{\mu}(\omega)=\epsilon_{\infty,\mu}\left({\frac{{\omega_{\mathrm{LO,\mu}}^{2}-\omega^{2}-i\Gamma_{\mu}\omega}}{{\omega_{\mathrm{TO,\mu}}^{2}-\omega^{2}-i\Gamma_{\mu}\omega}}}\right)$
(3)

where $\mu$ stands for in-plane (xy) or out-of-plane (z) direction. $\omega$,
$\omega_{\mathrm{TO}}$, $\omega_{\mathrm{LO}}$, $\Gamma$ and
$\epsilon_{\infty}$ represent the optical frequency, the TO and LO phonon
frequencies, the phonon damping, and the high-frequency permittivity,
respectively. According to Ref. 41, for h10BN, $\epsilon_{\infty,xy}$=5.1,
$\epsilon_{\infty,z}$=2.5, $\omega_{\mathrm{LO,xy}}$=1650 cm-1,
$\omega_{\mathrm{LO,z}}$=845 cm-1, $\omega_{\mathrm{TO,xy}}$=1394.5 cm-1,
$\omega_{\mathrm{TO,z}}$=785 cm-1, $\Gamma_{xy}$=1.8 cm-1, $\Gamma_{z}$=1
cm-1. To model and simulate the excitation and far-field emission of HPhPs
(Fig. 3b and c), an electric dipole emitter located within the graphene layer
was used as the source. The emitter’s bandwidth was chosen to range from 500
to 2000 cm-1, and its orientation was set to be in the plane of graphene in
order to emulate the electron-HPhP scattering. In order to reduce the
simulation time consumption, we set the single graphene layer to be 1 nm in
thickness and the corresponding permittivity of graphene was normalized to
match the thickness of 1 nm. The mesh size of the simulation was set to 1 nm.

## Data Availability

The data that support the plots within this paper and other findings of this
study are available from the corresponding author upon reasonable request.

## Code Availability

The computer codes used to perform the theoretical analysis and simulations in
this paper are available from the corresponding author upon reasonable
request.

## Acknowledgements

The authors would like to thank Renwen Yu, Javier Garcia de Abajo, Jacob
Khurgin, and Francisco Guinea for insightful discussions. Research at Yale
University was supported by the National Science Foundation CAREER Award
(ECCS-1552461) and Yale University. E.D. acknowledges support from the SNSF
project 200021-212899 and the ARO grant number W911NF-21-1-0184. J.H.E.
acknowledges the support of the Materials Engineering and Processing program
of the National Science Foundation, Award Number CMMI 1538127. G.R. expresses
gratitude for the support by the Simons Foundation, and the ARO MURI Grant No.
W911NF-16-1-0361. I.E. is grateful for support from the Simons Foundation and
the Institute of Quantum Information and Matter.

## Authors Contributions

Q.G. conceived the idea. Q.G. and C.L. fabricated the devices and performed
the measurements with assistance from C.C. I.E. performed the theoretical
modeling and physical interpretation of the carrier transport and the HPhP
electroluminescence, with inputs from G.R. and E.D. S.L. and J.H.E.
synthesized the h10BN. Q.G. performed the optical simulation with assistance
from S.Z. Q.G. and I.E. wrote the manuscript with inputs from all authors.
E.D., G.R., and F.X. supervised the project.

## Competing Interests

The authors declare no competing interests.

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political communities they have reached. Other studies also consider the use
of entropy measures over distributions representing the proportion of users
that browse given information sources [5]. Some studies, for example [135],
explicitly consider networks of information items (e.g., blog posts) and the
concepts that they use.

As with the example of a generic recommender systems, we present in a
schematic fashion (in Table 4) different diversity-related concepts of
interest for research questions in the field of social networks and media
studies, along with quantities that can be computed as network diversity
measures.

Table 4: Schematic representation of diversity-related concepts in social networks and media studies, and the corresponding network diversity measures that can be used to address them in quantitative studies. Examples of concepts expressible in research questions |  Corresponding network diversity measures
---|---
Collective diversity of affiliations of users |  $D_{\alpha}\left(X_{\text{users}}\xrightarrow{E_{\text{identify}}}X_{\text{affiliations}}\right)$
Collective affiliation diversity of users through the contents they share |  $D_{\alpha}\left(X_{\text{users}}\xrightarrow{E_{\text{share}}}X_{\text{posts}}\xrightarrow{E_{\text{link}}}X_{\text{articles}}\xrightarrow{E_{\text{belong}}}X_{\text{medias}}\xrightarrow{E_{\text{affiliate}}}X_{\text{affiliations}}\right)$
Individual topic diversity of user $u\in V_{\text{users}}$ through posting |  $D_{\alpha}\left(X_{\text{users}}\xrightarrow{E_{\text{post}}}X_{\text{posts}}\xrightarrow{E_{\text{include}}}X_{\text{topics}}\>|\>X_{\text{users}}=u\right)$
Individual topic diversity of user $u\in V_{\text{users}}$ through posting of followers |  $D_{\alpha}\left(X_{\text{users}}\xrightarrow{E_{\text{follow}}^{\intercal}}X_{\text{users}}^{\prime}\xrightarrow{E_{\text{post}}}X_{\text{posts}}\xrightarrow{E_{\text{include}}}X_{\text{topics}}\>|\>X_{\text{users}}=u\right)$
Individual affiliation diversity of users mentioned by user $u\in V_{\text{users}}$ |  $D_{\alpha}\left(X_{\text{users}}\xrightarrow{E_{\text{post}}}X_{\text{posts}}\xrightarrow{E_{\text{mention}}}X_{\text{users}}^{\prime}\xrightarrow{E_{\text{identify}}}X_{\text{affiliations}}\>|\>X_{\text{users}}=u\right)$
Diversity of affiliation groups that treat topic $t\in V_{\text{topics}}$ in articles |  $D_{\alpha}\left(X_{\text{affiliations}}\>|\>X_{\text{affiliations}}\xrightarrow{E_{\text{affiliate}}^{\intercal}}X_{\text{medias}}\xrightarrow{E_{\text{belong}}^{\intercal}}X_{\text{articles}}\xrightarrow{E_{\text{treat}}}V_{\text{topics}}=t\right)$
Affiliation diversity of users according to who they mention in their posts relative to their own identified political affiliation |  $D_{\alpha}\left(X_{\text{users}}\xrightarrow{E_{\text{post}}}X_{\text{posts}}\xrightarrow{E_{\text{mention}}}X_{\text{users}}^{\prime}\xrightarrow{E_{\text{identify}}}X_{\text{affiliations}}\>\|\>X_{\text{users}}\xrightarrow{E_{\text{identify}}}X_{\text{affiliations}}\right)$

###  5.5 Ecology

Diversity is useful in ecology, as identified and commented in Section 2.2.
Many advances in diversity measures come from this community (e.g., [13]). One
prominent concept in this domain is the diversity of species in a habitat. For
the computation of quantitative indices of this diversity, individuals from
different species are counted or their number is estimated. From their
apportionment into the species present in a habitat, diversity is then
computed and reported.

Interactions among organisms are also of interest in ecology. These can be
treated using graph representations and models. One of such interactions, also
related to diversity, is represented by so-called _food webs_ [136]: network
models that describe species that feed on other species. In the past, there
have been efforts to use graph formalisms to treat food webs [137, 138].
Similarly, other relations between species have been described using graphs,
such as parasitation [139]. Another subject of interest in ecology is the
description of habitats and their interconnectedness; there have been several
approaches using graph theory to describe these connections [140, 141].

We suggest that all these elements present in ecology can be treated using
heterogeneous information networks. Using network diversity measures,
different concepts related to diversity can be computed. Let us consider for
example a heterogeneous information network with vertex types for habitats
($V_{\text{habitats}}$), for individuals ($V_{\text{individuals}}$), for
species ($V_{\text{species}}$), for genera ($V_{\text{genera}}$), for families
($V_{\text{families}}$), and so on as needed.

Let us also consider for our example several edge types. Edge type
$E_{\text{connect}}$ is that of edges between habitats, indicating whether an
individual can access a given habitat from another one. Edge type
$E_{\text{inhabit}}$ is used to represent which individuals inhabit which
habitats. Edge types $E_{\text{eat}}$ and $E_{\text{parasite}}$ are used to
represent relations between species; which species eat which species, and
similarly for parasitation. Edge type $E_{\text{belong,1}}$ is used to
represent which individual belongs to which species. Finally, edge types
$E_{\text{belong,2}}$ and $E_{\text{belong,3}}$ contain edges indicating which
specie belongs to which genus, and which genus belongs to which family
(species, genera, and families are three of the eight major taxonomic ranks in
biological classification).

This setting can accommodate the common practices of measurement of –bio–
diversity of a habitat $h\in V_{\text{habitats}}$, which in network diversity
measures finds the expression

$D_{\alpha}\left(X_{\text{habitats}}\xrightarrow{E_{\text{inhabit}}^{\intercal}}X_{\text{individuals}}\xrightarrow{E_{\text{belong,1}}}X_{\text{species}}\>|\>X_{\text{habitats}}=h\right),$

with $\alpha=0$ giving the richness biodiversity, and $\alpha=\infty$ the
Berger-Parker biodiversity of the habitat. Figure 15 illustrates the network
schema of the described heterogeneous information network, along with a table
of diversity-related concepts and their expressions as network diversity
measures.

$V_{\text{habitats}}$$V_{\text{individuals}}$$V_{\text{species}}$$V_{\text{genera}}$$V_{\text{families}}$$\cdots$$E_{\text{inhabit}}$$E_{\text{belong,1}}$$E_{\text{belong,2}}$$E_{\text{belong,3}}$$E_{\text{belong,4}}$$E_{\text{eat}}$$E_{\text{parasite}}$$E_{\text{connect}}$

Examples of concepts expressible in research questions |  Corresponding network diversity measures
---|---
Species diversity in habitat $h\in V_{\text{habitats}}$ |  $D_{\alpha}\left(X_{\text{species}}\>|\>X_{\text{species}}\xrightarrow{E_{\text{belong,1}}^{\intercal}}X_{\text{individuals}}\xrightarrow{E_{\text{inhabit}}}X_{\text{habitats}}=h\right)$
Genera diversity in habitat $h\in V_{\text{habitats}}$ |  $D_{\alpha}\left(X_{\text{genera}}\>|\>X_{\text{genera}}\xrightarrow{E_{\text{belong,2}}^{\intercal}}X_{\text{species}}\xrightarrow{E_{\text{belong,1}}^{\intercal}}X_{\text{individuals}}\xrightarrow{E_{\text{inhabit}}}X_{\text{habitats}}=h\right)$
Species diversity of habitats adjacent to those where a species $s\in V_{\text{species}}$ is present |  $D_{\alpha}\left(X_{\text{species}}\xrightarrow{E_{\text{belong,1}}^{\intercal}}X_{\text{individuals}}\xrightarrow{E_{\text{inhabit}}}X_{\text{habitats}}\xrightarrow{E_{\text{connect}}}X_{\text{habitats}}^{\prime}\right.$ $\left.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\xrightarrow{E_{\text{inhabit}}^{\intercal}}X_{\text{individuals}}^{\prime}\xrightarrow{E_{\text{belong,1}}}X_{\text{species}}^{\prime}\>|\>X_{\text{species}}=s\right)$
Species diversity of the predators of species that parasite a species $s\in V_{\text{species}}$ |  $D_{\alpha}\left(X_{\text{species}}\>|\>X_{\text{species}}\xrightarrow{E_{\text{eat}}}X_{\text{species}}^{\prime}\xrightarrow{E_{\text{parasite}}}X_{\text{species}}^{\prime\prime}=s\right)$
Diversity in habitat $h_{1}\in V_{\text{habitats}}$ relative to another habitat $h_{2}\in V_{\text{habitats}}$ |  $D_{\alpha}\left(X_{\text{habitats}}\xrightarrow{E_{\text{inhabit}}^{\intercal}}X_{\text{individuals}}\xrightarrow{E_{\text{belong,1}}}X_{\text{species}}\>|\>X_{\text{habitats}}=h_{1}\>\|\>\right.$ $\left.\quad\quad\quad\quad X_{\text{habitats}}^{\prime}\xrightarrow{E_{\text{inhabit}}^{\intercal}}X_{\text{individuals}}^{\prime}\xrightarrow{E_{\text{belong,1}}}X_{\text{species}}^{\prime}\>|\>X_{\text{habitats}}^{\prime}=h_{2}\right)$

Figure 15: Network schema of an example from ecology, and table with examples
of diversity-related concepts and their expression as network diversity
measures.

###  5.6 Antitrust and competition law

Many developments and applications of concentration measures are found in the
economics community, antitrust regulation, and competition law. As shown in
Section 2.2, concentration is a concept for which indices are the reciprocal
of those used for the concept of diversity. Concentration or diversity indices
are used to measure the degree to which some firms concentrate the production
of units (or the provision of services) in an industry. Let us consider, for
example, the classification and apportionment of tons of steel produced –in a
given period of time in a given country– by the firms that produced them. From
this apportionment or distribution, concentration of the steel industry can be
quantitatively measured with diversity measures. This is the subject of the
doctoral thesis of O. C. Herfindahl, for which he developed what is now known
as the Herfindahl-Hirschman Index [67]. The quantitative measurement of
concentration of an industry allows for important comparisons to be made by
industry regulators, such as, for example, the degree of concentration of an
industry should a given merger or acquisition be allowed.

This exercise in measurement of industrial concentration, and the detection
and limitation of monopolistic behavior, is made significantly more difficult
by the existence of cross-ownership, or cross-control relation between firms.
Cross-ownership refers to situations in which firms from an industry are
mutually owned in complex, network-like relations (in a simple example between
two firms A and B, firm A owns a part of firm B, and firm B owns a part of
firm A). Cross-control refers, similarly, to situations where firms can name
board members of other firms in the same industry producing complex relations
of control in a network-like fashion.

Specialized economics literature accounts for many case studies that challenge
the application of the aforementioned procedure to regulation [142, 143], and
that address the complex structure of co-ownership networks and their
importance in regulation [144, 145]. This makes graph-theoretical approaches
good candidates for making advances in the measurement of concentration in
industries [146]. In particular, the proposed network diversity measures
provide tools that allow the measurement of many concepts of interest in
antitrust regulation and competition law when dealing with network structures.

To illustrate this, let us consider a heterogeneous information network with
three vertex types: that of vertices representing produced units of services
${V}_{\text{units}}$ (e.g., tons of steel, barrels of oil, or clients of
portable phone services), that of vertices representing firms
${V}_{\text{firms}}$ that produce those units or provide those services, and
that of persons ${V}_{\text{persons}}$ that own the firms. To model the
relations between these entities represented by vertex types, let us consider
five edge types: $E_{\text{produced}}$, linking each units to the firm that
produced them, $E_{\text{own}}$, linking firms to the persons that own them,
$E_{\text{cross-own}}$, linking firms with each other according to cross-
ownership, and similarly, $E_{\text{control}}$ and $E_{\text{cross-control}}$
linking firms with each other according to control and cross-control (for
example, having the right to choose a member of the board of a given firm).
For edge types $E_{\text{own}}$, $E_{\text{cross-own}}$, $E_{\text{control}}$,
and $E_{\text{cross-control}}$, the multiplicity of edges can account for the
units by which property or control is represented, such as, for example,
shares or members of the boards of the firms. For example, if ownership of a
firm is represented by 10 shares, it will have 10 edges, that can belong to
edge types $E_{\text{own}}$ or $E_{\text{cross-own}}$.

In this setting the common measurement of industry diversity is expressed as

$D_{\alpha}\left({X}_{\text{units}}\xrightarrow{E_{\text{produced}}}{X}_{\text{firms}}\right),$

which becomes the Herfindahl-Hirschman Diversity (reciprocal of the
Herfindahl-Hirschman Index) by choosing $\alpha=2$. Figure 16 illustrates the
heterogeneous information network described in this example, along with a
table of concepts related to diversity and expressible using the proposed
network diversity measures.

${V}_{\text{units}}$${V}_{\text{firms}}$${V}_{\text{persons}}$$E_{\text{produced}}$$E_{\text{control}}$$E_{\text{own}}$$E_{\text{cross-
own}}$$E_{\text{cross-control}}$

Examples of concepts expressible in research questions |  Corresponding network diversity measures
---|---
Industry diversity with cross-ownership relations |  $D_{\alpha}\left({X}_{\text{units}}\xrightarrow{E_{\text{produced}}}{X}_{\text{firms}}\xrightarrow{E_{\text{cross-own}}}{X}_{\text{firms}}^{\prime}\right)$
Industry diversity according to persons with cross-ownership relations |  $D_{\alpha}\left({X}_{\text{units}}\xrightarrow{E_{\text{produced}}}{X}_{\text{firms}}\xrightarrow{E_{\text{cross-own}}}{X}_{\text{firms}}^{\prime}\xrightarrow{E_{\text{own}}}{X}_{\text{persons}}\right)$
Diversity of cross-ownership of a firm $f\in{V}_{\text{firms}}$ |  $D_{\alpha}\left({X}_{\text{firms}}\xrightarrow{E_{\text{cross-own}}}{X}_{\text{firms}}^{\prime}\>|\>{X}_{\text{firms}}=f\right)$
Diversity of ownership of firms relative to their diversity of control |  $D_{\alpha}\left({X}_{\text{firms}}\xrightarrow{E_{\text{own}}}{X}_{\text{persons}}\>\|\>{X}_{\text{firms}}^{\prime}\xrightarrow{E_{\text{control}}}{X}_{\text{persons}}^{\prime}\right)$

Figure 16: Network schema of a cross-ownership and cross-control heterogeneous
information network in a setting from _antitrust_ regulation or competition
law, where products are apportioned in the firms that produced them, which can
be cross-owned or cross-controlled by each others.

###  5.7 Scientometrics

Scientometrics, within the field of bibliometrics, studies the measurement and
analysis of scientific literature. Overlapping with information systems,
scientometrics study, for example, the _importance_ of publications in
networks of citations using metrics such as the _Impact Factor_ or the
_Science Citation Index_. In networks including other entities such as
authors, other measurements include the _h-index_ , an index for the
productivity and citation impact of scholars. Recent studies have used
heterogeneous information networks to represent data including other entities,
such as journals and conferences, in order to extract extended measurements
[147].

The study of networks modeling and representing scientific production is of
interest for other reasons too. Diversity of topics explored by scientific
communities is a concept of interest, for example, in public policy [148], and
in general for the understanding and description of the structure of
scientific communities [149, 150]. Another practical application of the
measurement of diversity in citation networks is the maintenance of
classification systems [151].

Below, we illustrate the way proposed network diversity measures can address
some of the concepts relevant to these areas of research by means of an
example. Let us consider a heterogeneous information network consisting of the
following vertex types: authors ${V}_{\text{authors}}$, laboratories
${V}_{\text{lab.}}$ (or affiliation institutions), journals
${V}_{\text{journals}}$, scientific articles ${V}_{\text{papers}}$, keywords
used by these articles ${V}_{\text{keywords}}$, and domains of research
${V}_{\text{domains}}$ (e.g., ecology, economics). We also consider edge types
for representing relations between these entities (see Figure 17): affiliation
$E_{\text{aff.}}$ of authors to institutions, edition (or peer-review)
$E_{\text{edit}}$ of journals by authors, writing $E_{\text{write}}$ of
articles by authors, use of keywords $E_{\text{use}}$ in articles, association
of keywords $E_{\text{belong}}$ with domains, publication $E_{\text{publish}}$
of articles by journals, and declared treatment $E_{\text{treat}}$ of research
domains by journals. Figure 17 illustrates the corresponding heterogeneous
information network, along with a table of concepts related to diversity and
expressible using the proposed network diversity measures.

${V}_{\text{papers}}$${V}_{\text{authors}}$${V}_{\text{journals}}$${V}_{\text{keywords}}$${V}_{\text{lab.}}$${V}_{\text{domains}}$$E_{\text{aff.}}$$E_{\text{write}}$$E_{\text{edit}}$$E_{\text{cite}}$$E_{\text{use}}$$E_{\text{belong}}$$E_{\text{treat}}$$E_{\text{publish}}$

Examples of concepts expressible in research questions |  Corresponding network diversity measures
---|---
Diversity of keywords used by author $a\in{V}_{\text{authors}}$ |  $D_{\alpha}\left({X}_{\text{authors}}\xrightarrow{E_{\text{write}}}{X}_{\text{papers}}\xrightarrow{E_{\text{use}}}{X}_{\text{keywords}}\>|\>{X}_{\text{authors}}=a\right)$
Diversity of domains addressed in publications by author $a\in{V}_{\text{authors}}$ relative to domains he or she addresses in editing (or peer-reviewing) |  $D_{\alpha}\left({X}_{\text{authors}}\xrightarrow{E_{\text{write}}}{X}_{\text{papers}}\xrightarrow{E_{\text{use}}}{X}_{\text{keywords}}\xrightarrow{E_{\text{belong}}}{X}_{\text{domains}}\>|\>{X}_{\text{authors}}=a\>\|\>\right.$
|
$\quad\quad\quad\quad\left.{X}_{\text{authors}}^{\prime}\xrightarrow{E_{\text{edit}}}{X}_{\text{journals}}\xrightarrow{E_{\text{treat}}}{X}_{\text{domains}}^{\prime}\>|\>{X}_{\text{authors}}^{\prime}=a\right)$
Diversity of domains addressed by citations by authors of laboratory $l\in{V}_{\text{lab.}}$ |  $D_{\alpha}\left({X}_{\text{lab.}}\xrightarrow{E_{\text{aff.}}^{\intercal}}{X}_{\text{authors}}\xrightarrow{E_{\text{write}}}{X}_{\text{papers}}\xrightarrow{E_{\text{cite}}}{X}_{\text{papers}}^{\prime}\xrightarrow{E_{\text{use}}}{X}_{\text{keywords}}\xrightarrow{E_{\text{belong}}}{X}_{\text{domains}}\>|\>{X}_{\text{lab.}}=l\right)$

Figure 17: Network schema of a heterogeneous information network in an example
for scientometrics, and table of examples of concepts related to diversity and
expressible using the proposed network diversity measures.

##  6 Conclusions

This article presents a formal framework for the measurement of diversity in
heterogeneous information networks. This allows for the extension of the
application of diversity measures from classification modeled by apportioning
into distributions, to data represented in network structures.

By presenting a concise theory resulting from the imposition of desirable
properties of axioms, we organize diversity measures across a wide spectrum of
domains into a family of functions defined by a single parameter $\alpha$: the
true diversities. Providing a formalism for heterogeneous information networks
and constrained random walks on it, we consider different probability
distributions on which diversity measures are computed. These diversity
measures are related to the structure of the heterogeneous information
network, and thus to the phenomena or objects it represents. Diversity
measures are also related to the different ways in which distributions are
computed, which allows us to distinguish several types of diversities:
collective, individual, mean individual, backward, relative, and projected
diversities. Some of these network diversities relate to existing measures in
the literature, that we framed into a comprehensive framework. But they also
allow for the treatment of new concepts related to diversity in networks. We
provide examples of applications in several domains.

The main contributions of this article are:

  * 1.

The proposition of an axiomatic theory of diversity measures that allows us to
present most of their uses across several domains with a single-parameter
family of functions.

  * 2.

The formalization of concepts and tools to describe and process heterogeneous
information networks, that have been gaining attention in representation
learning and information retrieval communities (in particular in recommender
systems).

  * 3.

The definition of several _network diversity measures_ , resulting from the
application of true diversities to probability distributions that are
computable with the heterogeneous information network formalism. These network
diversity measures allow for the referentiation, expression, and computation
of concepts relevant to diversity in networks, extending the use of diversity
from systems of classification and apportionment to systems best described by
network-structured data.

  * 4.

The mapping of some of the _network diversity measures_ to pre-existing
quantitative measurements that are widespread in different fields, and the
development of new applications through examples in recommender systems,
social media studies, ecology, competition law, and scientometrics.

In addition to providing means of referencing, expression, and computation on
diversity-related concepts in complex data modeled by heterogeneous
information networks, the _network diversity measures_ could be leveraged in
different downstream tasks performed in data mining. Expanding on those
mentioned in the introduction (Section 1), we can now hint at more precise
examples of applications in such tasks. In Recommender Systems, for example,
previous works have used diversities associated with meta paths to avoid over-
fitting in the training stage on data modeled with heterogeneous information
networks [152]. Network diversity measures could allow, in this case, to
target different collective and individual diversities in the function to be
optimized. Other applications use meta paths to consider node similarities in
strategies for node classification [153], and could leverage network diversity
measures to consider parametrization of the weight given to balance and
variety in diversity when used. Research in representation learning could
embed different quantities computable with network diversity measures in
learning strategies where diversity would be leveraged for optimizing
different objective function through training [154]. Finally, we hint to other
applications that would need to be explored in greater detail than it is
possible here, in domains related to heterogeneous information networks and
diversity, such as community detection [155, 156, 157], clustering in networks
[158, 159], and the analysis of time-series resulting from temporal networks
[160].

Many relevant advances in computer sciences hinge on the improvement of
performance metrics through the development of novel algorithms and
methodologies. This article seeks to contribute in the proposal of the metrics
with which advacements are to be compared with the state of art. It is for
this purpose what we propose a detailed discussion of the properties and
implications of the proposed network diversity measures. We hope that this
framework for the application of diversity measures to network structures will
enrich research on diversity in the domains identified in Section 5 and
beyond. Future developments in this line of research might consider the
identification of algebraic structures for network diversity measures, and
their application to case studies.

## Acknowledgement

This work has been partially funded by the European Commission H2020 FETPROACT
2016-2017 program under grant 732942 (ODYCCEUS) and by the French National
Agency of Research (ANR) under grant ANR-15-CE38-0001 (AlgoDiv).

The authors would like to sincerely thank Hông-Lan Botterman and Matthieu
Latapy for their comments, as well as Claire Schaffer for her thorough proof-
reading.

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