[a,b]G. F. R. Sborlini

# Reconstructing parton collisions with machine learning techniques

D. F. Rentería-Estrada  R. J. Hernández-Pinto  P. Zurita

###### Abstract

Having access to the parton-level kinematics is important for understanding
the internal dynamics of particle collisions. Here, we present new results
aiming to an efficient reconstruction of parton collisions using machine-
learning techniques. By simulating the collider events, we related
experimentally-accessible quantities with the momentum fractions of the
involved partons. We used photon-hadron production to exploit the cleanliness
of the photon signal, including up to NLO QCD-QED corrections. Neural networks
led to an outstanding reconstruction efficiency, suggesting a powerful
strategy for unveiling the behaviour of the fundamental bricks of matter in
high-energy collisions.

##  1 Introduction and motivation

In current high-energy experiments, bunches of particles collide, allowing
their fundamental constituents to interact and produce new particles. The
detectors identify the product of the collisions, but it is not
straightforward to understand what is really going on during the process. In
our recent work [1], we explore the application of Machine-Learning (ML)
techniques to model the partonic momentum fractions in terms of
experimentally-accessible variables. In first place, we calculated the
differential hadronic cross-section for photon-hadron production including up
to Next-to-Leading Order (NLO) QCD and Leading-Order (LO) QED corrections.
Using a code based on Monte-Carlo (MC) integration [2, 3], we simulated the
collisions and analyzed the events to determine the correlations among
measurable and partonic quantities. Then, we applied ML algorithms that allow
us to find the momentum fractions of the partons involved in the process in
terms of suitable combinations of the final state momenta.

We tested our proof-of-concept (PoC) with photon-hadron production at
colliders because the photon provides a clean probe to access the parton
kinematics. The aim was to reconstruct the momentum fraction $x$ and $z$ of
the partons coming from the protons and undergoing the fragmentation into a
hadron, respectively. Due to the fact that parton kinematics are not
physically-defined (i.e. it is a model), we provided a quantitative estimation
of their most probable values. For this purpose, we trained Neural Networks
(NN) to predict the MC partonic momentum fractions in terms of external
momenta.

##  2 Computational setup

In order to carry out the calculations, we relied on the well-known
factorization theorem, which implies that the cross-section is described by
the convolution between PDFs, FFs and the partonic cross-section. On top of
that, we took advantage of the smooth-cone isolation [4] with the purpose of
avoiding the introduction of the photon fragmentation (whose accuracy is
considerably lower than the other FFs). So, our starting point for the cross-
section calculation was

$d\sigma_{H_{1}H_{2}\to
h\gamma}=\sum_{a_{1}a_{2}a_{3}}\int\,dx_{1}\,dx_{2}\,dz\,f_{H_{1}/a_{1}}(x_{1},\mu_{I})\,f_{H_{2}/a_{2}}(x_{2},\mu_{I})\,D_{a_{3}/h}(z,\mu_{F})\,d\hat{\sigma}_{a_{1}\,a_{2}\to
a_{3}\gamma}\,,$ (1)

where we included up to NLO QCD and LO QED corrections. For the NLO part, we
applied the FKS method [5] to cancel the infrared singularities. Once the IR-
finite NLO differential cross-section was defined, we established a _binning
strategy_ to discretize the phase-space. In this way, we managed to combine
the real (2-to-3), virtual (2-to-2) and counter-terms (2-to-2) contributions
by defining bins in the measurable variables, i.e. $\bar{{\cal V}}_{\rm
Exp.}=\\{{\bar{p}}_{T}^{\gamma},{\bar{p}}_{T}^{\pi},{\bar{\eta}}^{\gamma},{\bar{\eta}}^{\pi},{\overline{\cos}}(\phi^{\pi}-\phi^{\gamma})\\}$.
Then, we integrated the fully differential cross-section from Eq. (1) in each
bin, obtaining $\sigma_{j}$ for $p_{j}\in\bar{{\cal V}}_{\rm Exp.}$. This step
was crucial, since the higher-order corrections to the cross-sections involve
contributions living in different phase-spaces.

##  3 Parton kinematics reconstruction

For the process that we considered, i.e. $p+p\to\pi+\gamma$, the partonic
momentum fractions $x_{1}$, $x_{2}$ and $z$ are unambiguously fixed by
external particle’s kinematics only at LO. When computing higher-order
corrections, real radiation including extra particles must be included and
momentum conservation equations differs from LO kinematics. In order to obtain
an approximation, we relied on the LO expressions and we found a decent
agreement with the MC momentum fractions, as reported in Ref. [2]. However, in
our recent work [1], we decided to tackle the problem by using more flexible
ML techniques.

Figure 1: Comparison of the real momentum fractions ($y$-axis) and the
reconstructed ones using Neural Networks ($x$-axis), for $x$ (left) and $z$
(right), respectively.

The first step was to introduce the concept of _effective momentum fractions_
at higher-orders. Given a point in the grid,
$p_{j}=\\{{\bar{p}}_{T}^{\gamma},{\bar{p}}_{T}^{\pi},{\bar{\eta}}^{\gamma},{\bar{\eta}}^{\pi},{\overline{\cos}}(\phi^{\pi}-\phi^{\gamma})\\}\in{\bar{\cal
V}}_{\rm Exp}$, we defined

$(x_{1,2})_{j}=\sum_{i}\,(x_{1,2})_{i}\frac{d\sigma_{j}}{dx_{1,2}}(p_{j};(x_{1,2})_{i})\,,\quad(z)_{j}=\sum_{i}\,z_{i}\frac{d\sigma_{j}}{dz}(p_{j};z_{i})\,,$
(2)

with the purpose of identifying a mean value for the MC momentum fractions
associated to the sum of all the real-emission topologies contributing to the
same measurable final-state configuration (i.e. the same point $p_{j}$). Then,
we used ML to find mappings connecting the points in the grid with the average
value of the momentum fraction per bin, i.e.

$X_{\rm REC}:={\bar{\cal V}}_{\rm
Exp}\longrightarrow\bar{X}_{REAL}=\\{(x)_{j}\\}\,,\quad Z_{\rm
REC}:={\bar{\cal V}}_{\rm Exp}\longrightarrow\bar{Z}_{REAL}=\\{(z)_{j}\\}\,.$
(3)

We explored different approaches, being NN among them. The flexibility of the
NN provided the best reconstruction with less constraints, as shown in the
correlation plot in Fig. 1.

Figure 2: Correlation plots for $x_{1}$ (left) and $z$ (right), with the
associated error bands due to the propagation of the scale uncertainties in
the reconstruction. The Linear Method (LM), Gaussian Regression (GP) and
Neural Networks (NN) were considered.

Finally, we studied the propagation of the scale uncertainties by defining
different datasets, varying the renormalization and factorization scales by a
factor 2 up and down. With these datasets, we trained estimators (i.e. $X_{\rm
REC}^{(\xi)}$, with $\xi$ the energy scale) and evaluated them in each point
of the grid. As reported in Ref.[1], the average reconstruction error turned
out to be $7\%$ for $x$ and $5\%$ for $z$. In Fig. 2, we show the error bands
in the correlation plots, for different ML approaches. The bands are narrower
for smaller values of $x$ and $z$, since the cross-section is higher and the
training is more accurate.

##  4 Conclusions

In this work, we presented a proof-of-concept (PoC) to reconstruct parton-
level kinematics by using ML techniques. The results are in agreement with
previous findings [2], but they required much less human intervention. In
particular, NN did not even need to select a function basis (which is an
important choice in other approaches, such as Gaussian Regression or the
Linear Method). Our results indicate that this PoC could be applied to other
processes, and could be used to impose stricter constraints for PDF/FF
determination.

## Acknowledgments

GS is supported by Programas Propios II (Universidad de Salamanca), EU Horizon
2020 research and innovation program STRONG-2020 project under grant agreement
No. 824093 and H2020-MSCA-COFUND-2020 USAL4EXCELLENCE-PROOPI-391 project under
grant agreement No 101034371. D. F. R.-E. and R. J. H.-P. are supported by
CONACyT (Mexico) through the Project No. 320856 (Paradigmas y Controversias de
la Ciencia 2022), Ciencia de Frontera 2021-2024 and PROFAPI 2022 Grant No.
PRO-A1-024 (Universidad Autónoma de Sinaloa). P.Z. acknowledges support from
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)-
Research Unit FOR 2926, grant number 430915485.

## References

  * [1] D.F. Rentería-Estrada, R.J. Hernández-Pinto, G.F.R. Sborlini and P. Zurita, _Reconstructing partonic kinematics at colliders with Machine Learning_ , 2112.05043.
  * [2] D. de Florian and G.F.R. Sborlini, _Hadron plus photon production in polarized hadronic collisions at next-to-leading order accuracy_ , _Phys. Rev. D_ 83 (2011) 074022 [1011.0486].
  * [3] D.F. Rentería-Estrada, R.J. Hernández-Pinto and G.F.R. Sborlini, _Analysis of the Internal Structure of Hadrons Using Direct Photon Production_ , _Symmetry_ 13 (2021) 942 [2104.14663].
  * [4] S. Frixione, _Isolated photons in perturbative QCD_ , _Phys. Lett. B_ 429 (1998) 369 [hep-ph/9801442].
  * [5] S. Frixione, Z. Kunszt and A. Signer, _Three jet cross-sections to next-to-leading order_ , _Nucl. Phys. B_ 467 (1996) 399 [hep-ph/9512328].


Holgado, D. Moran, Á. Navarro Tobar, A. Pérez-Calero Yzquierdo, J. Puerta
Pelayo, I. Redondo, L. Romero, S. Sánchez Navas, L. Urda Gómez, C. Willmott
Universidad Autónoma de Madrid, Madrid, Spain
J.F. de Trocóniz, R. Reyes-Almanza Universidad de Oviedo, Instituto
Universitario de Ciencias y Tecnologías Espaciales de Asturias (ICTEA),
Oviedo, Spain
B. Alvarez Gonzalez, J. Cuevas, C. Erice, J. Fernandez Menendez, S. Folgueras,
I. Gonzalez Caballero, J.R. González Fernández, E. Palencia Cortezon, C. Ramón
Álvarez, J. Ripoll Sau, V. Rodríguez Bouza, A. Trapote, N. Trevisani Instituto
de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, Santander, Spain
J.A. Brochero Cifuentes, I.J. Cabrillo, A. Calderon, J. Duarte Campderros, M.
Fernandez, C. Fernandez Madrazo, P.J. Fernández Manteca, A. García Alonso, G.
Gomez, C. Martinez Rivero, P. Martinez Ruiz del Arbol, F. Matorras, P.
Matorras Cuevas, J. Piedra Gomez, C. Prieels, T. Rodrigo, A. Ruiz-Jimeno, L.
Scodellaro, I. Vila, J.M. Vizan Garcia University of Colombo, Colombo, Sri
Lanka
MK Jayananda, B. Kailasapathy60, D.U.J. Sonnadara, DDC Wickramarathna
University of Ruhuna, Department of Physics, Matara, Sri Lanka
W.G.D. Dharmaratna, K. Liyanage, N. Perera, N. Wickramage CERN, European
Organization for Nuclear Research, Geneva, Switzerland
T.K. Aarrestad, D. Abbaneo, J. Alimena, E. Auffray, G. Auzinger, J. Baechler,
P. Baillon${}^{\textrm{\textdagger}}$, D. Barney, J. Bendavid, M. Bianco, A.
Bocci, T. Camporesi, M. Capeans Garrido, G. Cerminara, S.S. Chhibra, M.
Cipriani, L. Cristella, D. d’Enterria, A. Dabrowski, N. Daci, A. David, A. De
Roeck, M.M. Defranchis, M. Deile, M. Dobson, M. Dünser, N. Dupont, A. Elliott-
Peisert, N. Emriskova, F. Fallavollita61, D. Fasanella, S. Fiorendi, A.
Florent, G. Franzoni, W. Funk, S. Giani, D. Gigi, K. Gill, F. Glege, L.
Gouskos, M. Haranko, J. Hegeman, Y. Iiyama, V. Innocente, T. James, P. Janot,
J. Kaspar, J. Kieseler, M. Komm, N. Kratochwil, C. Lange, S. Laurila, P.
Lecoq, K. Long, C. Lourenço, L. Malgeri, S. Mallios, M. Mannelli, A.C. Marini,
F. Meijers, S. Mersi, E. Meschi, F. Moortgat, M. Mulders, S. Orfanelli, L.
Orsini, F. Pantaleo, L. Pape, E. Perez, M. Peruzzi, A. Petrilli, G.
Petrucciani, A. Pfeiffer, M. Pierini, D. Piparo, M. Pitt, H. Qu, T. Quast, D.
Rabady, A. Racz, G. Reales Gutiérrez, M. Rieger, M. Rovere, H. Sakulin, J.
Salfeld-Nebgen, S. Scarfi, C. Schäfer, C. Schwick, M. Selvaggi, A. Sharma, P.
Silva, W. Snoeys, P. Sphicas62, S. Summers, K. Tatar, V.R. Tavolaro, D.
Treille, A. Tsirou, G.P. Van Onsem, M. Verzetti, J. Wanczyk63, K.A. Wozniak,
W.D. Zeuner Paul Scherrer Institut, Villigen, Switzerland
L. Caminada64, A. Ebrahimi, W. Erdmann, R. Horisberger, Q. Ingram, H.C.
Kaestli, D. Kotlinski, U. Langenegger, M. Missiroli, T. Rohe ETH Zurich -
Institute for Particle Physics and Astrophysics (IPA), Zurich, Switzerland
K. Androsov63, M. Backhaus, P. Berger, A. Calandri, N. Chernyavskaya, A. De
Cosa, G. Dissertori, M. Dittmar, M. Donegà, C. Dorfer, F. Eble, K. Gedia, F.
Glessgen, T.A. Gómez Espinosa, C. Grab, D. Hits, W. Lustermann, A.-M. Lyon,
R.A. Manzoni, C. Martin Perez, M.T. Meinhard, F. Nessi-Tedaldi, J. Niedziela,
F. Pauss, V. Perovic, S. Pigazzini, M.G. Ratti, M. Reichmann, C. Reissel, T.
Reitenspiess, B. Ristic, D. Ruini, D.A. Sanz Becerra, M. Schönenberger, V.
Stampf, J. Steggemann63, R. Wallny, D.H. Zhu Universität Zürich, Zurich,
Switzerland
C. Amsler65, P. Bärtschi, C. Botta, D. Brzhechko, M.F. Canelli, K. Cormier, A.
De Wit, R. Del Burgo, J.K. Heikkilä, M. Huwiler, W. Jin, A. Jofrehei, B.
Kilminster, S. Leontsinis, S.P. Liechti, A. Macchiolo, P. Meiring, V.M.
Mikuni, U. Molinatti, I. Neutelings, A. Reimers, P. Robmann, S. Sanchez Cruz,
K. Schweiger, Y. Takahashi National Central University, Chung-Li, Taiwan
C. Adloff66, C.M. Kuo, W. Lin, A. Roy, T. Sarkar37, S.S. Yu National Taiwan
University (NTU), Taipei, Taiwan
L. Ceard, Y. Chao, K.F. Chen, P.H. Chen, W.-S. Hou, Y.y. Li, R.-S. Lu, E.
Paganis, A. Psallidas, A. Steen, H.y. Wu, E. Yazgan, P.r. Yu Chulalongkorn
University, Faculty of Science, Department of Physics, Bangkok, Thailand
B. Asavapibhop, C. Asawatangtrakuldee, N. Srimanobhas Çukurova University,
Physics Department, Science and Art Faculty, Adana, Turkey
F. Boran, S. Damarseckin67, Z.S. Demiroglu, F. Dolek, I. Dumanoglu68, E.
Eskut, Y. Guler, E. Gurpinar Guler69, I. Hos70, C. Isik, O. Kara, A. Kayis
Topaksu, U. Kiminsu, G. Onengut, K. Ozdemir71, A. Polatoz, A.E. Simsek, B.
Tali72, U.G. Tok, S. Turkcapar, I.S. Zorbakir, C. Zorbilmez Middle East
Technical University, Physics Department, Ankara, Turkey
B. Isildak73, G. Karapinar74, K. Ocalan75, M. Yalvac76 Bogazici University,
Istanbul, Turkey
B. Akgun, I.O. Atakisi, E. Gülmez, M. Kaya77, O. Kaya78, Ö. Özçelik, S.
Tekten79, E.A. Yetkin80 Istanbul Technical University, Istanbul, Turkey
A. Cakir, K. Cankocak68, Y. Komurcu, S. Sen81 Istanbul University, Istanbul,
Turkey
S. Cerci72, B. Kaynak, S. Ozkorucuklu, D. Sunar Cerci72 Institute for
Scintillation Materials of National Academy of Science of Ukraine, Kharkov,
Ukraine
B. Grynyov National Scientific Center, Kharkov Institute of Physics and
Technology, Kharkov, Ukraine
L. Levchuk University of Bristol, Bristol, United Kingdom
D. Anthony, E. Bhal, S. Bologna, J.J. Brooke, A. Bundock, E. Clement, D.
Cussans, H. Flacher, J. Goldstein, G.P. Heath, H.F. Heath, M.l. Holmberg82, L.
Kreczko, B. Krikler, S. Paramesvaran, S. Seif El Nasr-Storey, V.J. Smith, N.
Stylianou83, K. Walkingshaw Pass, R. White Rutherford Appleton Laboratory,
Didcot, United Kingdom
K.W. Bell, A. Belyaev84, C. Brew, R.M. Brown, D.J.A. Cockerill, C. Cooke, K.V.
Ellis, K. Harder, S. Harper, J. Linacre, K. Manolopoulos, D.M. Newbold, E.
Olaiya, D. Petyt, T. Reis, T. Schuh, C.H. Shepherd-Themistocleous, I.R.
Tomalin, T. Williams Imperial College, London, United Kingdom
R. Bainbridge, P. Bloch, S. Bonomally, J. Borg, S. Breeze, O. Buchmuller, V.
Cepaitis, G.S. Chahal85, D. Colling, P. Dauncey, G. Davies, M. Della Negra, S.
Fayer, G. Fedi, G. Hall, M.H. Hassanshahi, G. Iles, J. Langford, L. Lyons,
A.-M. Magnan, S. Malik, A. Martelli, D.G. Monk, J. Nash86, M. Pesaresi, D.M.
Raymond, A. Richards, A. Rose, E. Scott, C. Seez, A. Shtipliyski, A. Tapper,
K. Uchida, T. Virdee20, M. Vojinovic, N. Wardle, S.N. Webb, D. Winterbottom,
A.G. Zecchinelli Brunel University, Uxbridge, United Kingdom
K. Coldham, J.E. Cole, A. Khan, P. Kyberd, I.D. Reid, L. Teodorescu, S. Zahid
Baylor University, Waco, Texas, USA
S. Abdullin, A. Brinkerhoff, B. Caraway, J. Dittmann, K. Hatakeyama, A.R.
Kanuganti, B. McMaster, N. Pastika, M. Saunders, S. Sawant, C. Sutantawibul,
J. Wilson Catholic University of America, Washington, DC, USA
R. Bartek, A. Dominguez, R. Uniyal, A.M. Vargas Hernandez The University of
Alabama, Tuscaloosa, Alabama, USA
A. Buccilli, S.I. Cooper, D. Di Croce, S.V. Gleyzer, C. Henderson, C.U. Perez,
P. Rumerio87, C. West Boston University, Boston, Massachusetts, USA
A. Akpinar, A. Albert, D. Arcaro, C. Cosby, Z. Demiragli, E. Fontanesi, D.
Gastler, J. Rohlf, K. Salyer, D. Sperka, D. Spitzbart, I. Suarez, A. Tsatsos,
S. Yuan, D. Zou Brown University, Providence, Rhode Island, USA
G. Benelli, B. Burkle, X. Coubez21, D. Cutts, M. Hadley, U. Heintz, J.M.
Hogan88, G. Landsberg, K.T. Lau, M. Lukasik, J. Luo, M. Narain, S. Sagir89, E.
Usai, W.Y. Wong, X. Yan, D. Yu, W. Zhang University of California, Davis,
Davis, California, USA
J. Bonilla, C. Brainerd, R. Breedon, M. Calderon De La Barca Sanchez, M.
Chertok, J. Conway, P.T. Cox, R. Erbacher, G. Haza, F. Jensen, O. Kukral, R.
Lander, M. Mulhearn, D. Pellett, B. Regnery, D. Taylor, Y. Yao, F. Zhang
University of California, Los Angeles, California, USA
M. Bachtis, R. Cousins, A. Datta, D. Hamilton, J. Hauser, M. Ignatenko, M.A.
Iqbal, T. Lam, W.A. Nash, S. Regnard, D. Saltzberg, B. Stone, V. Valuev
University of California, Riverside, Riverside, California, USA
K. Burt, Y. Chen, R. Clare, J.W. Gary, M. Gordon, G. Hanson, G. Karapostoli,
O.R. Long, N. Manganelli, M. Olmedo Negrete, W. Si, S. Wimpenny, Y. Zhang
University of California, San Diego, La Jolla, California, USA
J.G. Branson, P. Chang, S. Cittolin, S. Cooperstein, N. Deelen, D. Diaz, J.
Duarte, R. Gerosa, L. Giannini, D. Gilbert, J. Guiang, R. Kansal, V.
Krutelyov, R. Lee, J. Letts, M. Masciovecchio, S. May, M. Pieri, B.V. Sathia
Narayanan, V. Sharma, M. Tadel, A. Vartak, F. Würthwein, Y. Xiang, A. Yagil
University of California, Santa Barbara - Department of Physics, Santa
Barbara, California, USA
N. Amin, C. Campagnari, M. Citron, A. Dorsett, V. Dutta, J. Incandela, M.
Kilpatrick, J. Kim, B. Marsh, H. Mei, M. Oshiro, M. Quinnan, J. Richman, U.
Sarica, J. Sheplock, D. Stuart, S. Wang California Institute of Technology,
Pasadena, California, USA
A. Bornheim, O. Cerri, I. Dutta, J.M. Lawhorn, N. Lu, J. Mao, H.B. Newman,
T.Q. Nguyen, M. Spiropulu, J.R. Vlimant, C. Wang, S. Xie, Z. Zhang, R.Y. Zhu
Carnegie Mellon University, Pittsburgh, Pennsylvania, USA
J. Alison, S. An, M.B. Andrews, P. Bryant, T. Ferguson, A. Harilal, C. Liu, T.
Mudholkar, M. Paulini, A. Sanchez University of Colorado Boulder, Boulder,
Colorado, USA
J.P. Cumalat, W.T. Ford, A. Hassani, E. MacDonald, R. Patel, A. Perloff, C.
Savard, K. Stenson, K.A. Ulmer, S.R. Wagner Cornell University, Ithaca, New
York, USA
J. Alexander, S. Bright-thonney, Y. Cheng, D.J. Cranshaw, S. Hogan, J. Monroy,
J.R. Patterson, D. Quach, J. Reichert, M. Reid, A. Ryd, W. Sun, J. Thom, P.
Wittich, R. Zou Fermi National Accelerator Laboratory, Batavia, Illinois, USA
M. Albrow, M. Alyari, G. Apollinari, A. Apresyan, A. Apyan, S. Banerjee,
L.A.T. Bauerdick, D. Berry, J. Berryhill, P.C. Bhat, K. Burkett, J.N. Butler,
A. Canepa, G.B. Cerati, H.W.K. Cheung, F. Chlebana, M. Cremonesi, K.F. Di
Petrillo, V.D. Elvira, Y. Feng, J. Freeman, Z. Gecse, L. Gray, D. Green, S.
Grünendahl, O. Gutsche, R.M. Harris, R. Heller, T.C. Herwig, J. Hirschauer, B.
Jayatilaka, S. Jindariani, M. Johnson, U. Joshi, T. Klijnsma, B. Klima, K.H.M.
Kwok, S. Lammel, D. Lincoln, R. Lipton, T. Liu, C. Madrid, K. Maeshima, C.
Mantilla, D. Mason, P. McBride, P. Merkel, S. Mrenna, S. Nahn, J. Ngadiuba, V.
O’Dell, V. Papadimitriou, K. Pedro, C. Pena57, O. Prokofyev, F. Ravera, A.
Reinsvold Hall, L. Ristori, B. Schneider, E. Sexton-Kennedy, N. Smith, A.
Soha, W.J. Spalding, L. Spiegel, S. Stoynev, J. Strait, L. Taylor, S. Tkaczyk,
N.V. Tran, L. Uplegger, E.W. Vaandering, H.A. Weber University of Florida,
Gainesville, Florida, USA
D. Acosta, P. Avery, D. Bourilkov, L. Cadamuro, V. Cherepanov, F. Errico, R.D.
Field, D. Guerrero, B.M. Joshi, M. Kim, E. Koenig, J. Konigsberg, A. Korytov,
K.H. Lo, K. Matchev, N. Menendez, G. Mitselmakher, A. Muthirakalayil Madhu, N.
Rawal, D. Rosenzweig, S. Rosenzweig, K. Shi, J. Sturdy, J. Wang, E. Yigitbasi,
X. Zuo Florida State University, Tallahassee, Florida, USA
T. Adams, A. Askew, R. Habibullah, V. Hagopian, K.F. Johnson, R. Khurana, T.
Kolberg, G. Martinez, H. Prosper, C. Schiber, O. Viazlo, R. Yohay, J. Zhang
Florida Institute of Technology, Melbourne, Florida, USA
M.M. Baarmand, S. Butalla, T. Elkafrawy90, M. Hohlmann, R. Kumar Verma, D.
Noonan, M. Rahmani, F. Yumiceva University of Illinois at Chicago (UIC),
Chicago, Illinois, USA
M.R. Adams, H. Becerril Gonzalez, R. Cavanaugh, X. Chen, S. Dittmer, O.
Evdokimov, C.E. Gerber, D.A. Hangal, D.J. Hofman, A.H. Merrit, C. Mills, G.
Oh, T. Roy, S. Rudrabhatla, M.B. Tonjes, N. Varelas, J. Viinikainen, X. Wang,
Z. Wu, Z. Ye The University of Iowa, Iowa City, Iowa, USA
M. Alhusseini, K. Dilsiz91, R.P. Gandrajula, O.K. Köseyan, J.-P. Merlo, A.
Mestvirishvili92, J. Nachtman, H. Ogul93, Y. Onel, A. Penzo, C. Snyder, E.
Tiras94 Johns Hopkins University, Baltimore, Maryland, USA
O. Amram, B. Blumenfeld, L. Corcodilos, J. Davis, M. Eminizer, A.V. Gritsan,
S. Kyriacou, P. Maksimovic, J. Roskes, M. Swartz, T.Á. Vámi The University of
Kansas, Lawrence, Kansas, USA
A. Abreu, J. Anguiano, C. Baldenegro Barrera, P. Baringer, A. Bean, A.
Bylinkin, Z. Flowers, T. Isidori, S. Khalil, J. King, G. Krintiras, A.
Kropivnitskaya, M. Lazarovits, C. Lindsey, J. Marquez, N. Minafra, M. Murray,
M. Nickel, C. Rogan, C. Royon, R. Salvatico, S. Sanders, E. Schmitz, C. Smith,
J.D. Tapia Takaki, Q. Wang, Z. Warner, J. Williams, G. Wilson Kansas State
University, Manhattan, Kansas, USA
S. Duric, A. Ivanov, K. Kaadze, D. Kim, Y. Maravin, T. Mitchell, A. Modak, K.
Nam Lawrence Livermore National Laboratory, Livermore, California, USA
F. Rebassoo, D. Wright University of Maryland, College Park, Maryland, USA
E. Adams, A. Baden, O. Baron, A. Belloni, S.C. Eno, N.J. Hadley, S. Jabeen,
R.G. Kellogg, T. Koeth, A.C. Mignerey, S. Nabili, C. Palmer, M. Seidel, A.
Skuja, L. Wang, K. Wong Massachusetts Institute of Technology, Cambridge,
Massachusetts, USA
D. Abercrombie, G. Andreassi, R. Bi, S. Brandt, W. Busza, I.A. Cali, Y. Chen,
M. D’Alfonso, J. Eysermans, C. Freer, G. Gomez Ceballos, M. Goncharov, P.
Harris, M. Hu, M. Klute, D. Kovalskyi, J. Krupa, Y.-J. Lee, B. Maier, C.
Mironov, C. Paus, D. Rankin, C. Roland, G. Roland, Z. Shi, G.S.F. Stephans, J.
Wang, Z. Wang, B. Wyslouch University of Minnesota, Minneapolis, Minnesota,
USA
R.M. Chatterjee, A. Evans, P. Hansen, J. Hiltbrand, Sh. Jain, M. Krohn, Y.
Kubota, J. Mans, M. Revering, R. Rusack, R. Saradhy, N. Schroeder, N. Strobbe,
M.A. Wadud University of Nebraska-Lincoln, Lincoln, Nebraska, USA
K. Bloom, M. Bryson, S. Chauhan, D.R. Claes, C. Fangmeier, L. Finco, F. Golf,
C. Joo, I. Kravchenko, M. Musich, I. Reed, J.E. Siado, G.R.
Snow${}^{\textrm{\textdagger}}$, W. Tabb, F. Yan State University of New York
at Buffalo, Buffalo, New York, USA
G. Agarwal, H. Bandyopadhyay, L. Hay, I. Iashvili, A. Kharchilava, C. McLean,
D. Nguyen, J. Pekkanen, S. Rappoccio, A. Williams Northeastern University,
Boston, Massachusetts, USA
G. Alverson, E. Barberis, Y. Haddad, A. Hortiangtham, J. Li, G. Madigan, B.
Marzocchi, D.M. Morse, V. Nguyen, T. Orimoto, A. Parker, L. Skinnari, A.
Tishelman-Charny, T. Wamorkar, B. Wang, A. Wisecarver, D. Wood Northwestern
University, Evanston, Illinois, USA
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Liu, N. Odell, M.H. Schmitt, M. Velasco University of Notre Dame, Notre Dame,
Indiana, USA
R. Band, R. Bucci, A. Das, N. Dev, R. Goldouzian, M. Hildreth, K. Hurtado
Anampa, C. Jessop, K. Lannon, J. Lawrence, N. Loukas, D. Lutton, N. Marinelli,
I. Mcalister, T. McCauley, F. Meng, K. Mohrman, Y. Musienko50, R. Ruchti, P.
Siddireddy, A. Townsend, M. Wayne, A. Wightman, M. Wolf, M. Zarucki, L. Zygala
The Ohio State University, Columbus, Ohio, USA
B. Bylsma, B. Cardwell, L.S. Durkin, B. Francis, C. Hill, M. Nunez Ornelas, K.
Wei, B.L. Winer, B.R. Yates Princeton University, Princeton, New Jersey, USA
F.M. Addesa, B. Bonham, P. Das, G. Dezoort, P. Elmer, A. Frankenthal, B.
Greenberg, N. Haubrich, S. Higginbotham, A. Kalogeropoulos, G. Kopp, S. Kwan,
D. Lange, M.T. Lucchini, D. Marlow, K. Mei, I. Ojalvo, J. Olsen, D. Stickland,
C. Tully University of Puerto Rico, Mayaguez, Puerto Rico, USA
S. Malik, S. Norberg Purdue University, West Lafayette, Indiana, USA
A.S. Bakshi, V.E. Barnes, R. Chawla, S. Das, L. Gutay, M. Jones, A.W. Jung, S.
Karmarkar, M. Liu, G. Negro, N. Neumeister, G. Paspalaki, C.C. Peng, S.
Piperov, A. Purohit, J.F. Schulte, M. Stojanovic16, J. Thieman, F. Wang, R.
Xiao, W. Xie Purdue University Northwest, Hammond, Indiana, USA
J. Dolen, N. Parashar Rice University, Houston, Texas, USA
A. Baty, M. Decaro, S. Dildick, K.M. Ecklund, S. Freed, P. Gardner, F.J.M.
Geurts, A. Kumar, W. Li, B.P. Padley, R. Redjimi, W. Shi, A.G. Stahl Leiton,
S. Yang, L. Zhang, Y. Zhang University of Rochester, Rochester, New York, USA
A. Bodek, P. de Barbaro, R. Demina, J.L. Dulemba, C. Fallon, T. Ferbel, M.
Galanti, A. Garcia-Bellido, O. Hindrichs, A. Khukhunaishvili, E. Ranken, R.
Taus Rutgers, The State University of New Jersey, Piscataway, New Jersey, USA
B. Chiarito, J.P. Chou, A. Gandrakota, Y. Gershtein, E. Halkiadakis, A. Hart,
M. Heindl, O. Karacheban24, I. Laflotte, A. Lath, R. Montalvo, K. Nash, M.
Osherson, S. Salur, S. Schnetzer, S. Somalwar, R. Stone, S.A. Thayil, S.
Thomas, H. Wang University of Tennessee, Knoxville, Tennessee, USA
H. Acharya, A.G. Delannoy, S. Spanier Texas A&M University, College Station,
Texas, USA
O. Bouhali95, M. Dalchenko, A. Delgado, R. Eusebi, J. Gilmore, T. Huang, T.
Kamon96, H. Kim, S. Luo, S. Malhotra, R. Mueller, D. Overton, D. Rathjens, A.
Safonov Texas Tech University, Lubbock, Texas, USA
N. Akchurin, J. Damgov, V. Hegde, S. Kunori, K. Lamichhane, S.W. Lee, T.
Mengke, S. Muthumuni, T. Peltola, I. Volobouev, Z. Wang, A. Whitbeck
Vanderbilt University, Nashville, Tennessee, USA
E. Appelt, S. Greene, A. Gurrola, W. Johns, A. Melo, H. Ni, K. Padeken, F.
Romeo, P. Sheldon, S. Tuo, J. Velkovska University of Virginia,
Charlottesville, Virginia, USA
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Ledovskoy, A. Li, C. Neu, B. Tannenwald, S. White, E. Wolfe Wayne State
University, Detroit, Michigan, USA
N. Poudyal University of Wisconsin - Madison, Madison, WI, Wisconsin, USA
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Everaerts, F. Fienga, C. Galloni, H. He, M. Herndon, A. Hervé, U. Hussain, A.
Lanaro, A. Loeliger, R. Loveless, J. Madhusudanan Sreekala, A. Mallampalli, A.
Mohammadi, D. Pinna, A. Savin, V. Shang, V. Sharma, W.H. Smith, D. Teague, S.
Trembath-reichert, W. Vetens †: Deceased
1: Also at TU Wien, Wien, Austria
2: Also at Institute of Basic and Applied Sciences, Faculty of Engineering,
Arab Academy for Science, Technology and Maritime Transport, Alexandria, Egypt
3: Also at Université Libre de Bruxelles, Bruxelles, Belgium
4: Also at Universidade Estadual de Campinas, Campinas, Brazil
5: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil
6: Also at University of Chinese Academy of Sciences, Beijing, China
7: Also at Department of Physics, Tsinghua University, Beijing, China
8: Also at UFMS, Nova Andradina, Brazil
9: Also at Nanjing Normal University Department of Physics, Nanjing, China
10: Now at The University of Iowa, Iowa City, Iowa, USA
11: Also at Institute for Theoretical and Experimental Physics named by A.I.
Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia
12: Also at Joint Institute for Nuclear Research, Dubna, Russia
13: Also at Cairo University, Cairo, Egypt
14: Also at Suez University, Suez, Egypt
15: Now at British University in Egypt, Cairo, Egypt
16: Also at Purdue University, West Lafayette, Indiana, USA
17: Also at Université de Haute Alsace, Mulhouse, France
18: Also at Ilia State University, Tbilisi, Georgia
19: Also at Erzincan Binali Yildirim University, Erzincan, Turkey
20: Also at CERN, European Organization for Nuclear Research, Geneva,
Switzerland
21: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
22: Also at University of Hamburg, Hamburg, Germany
23: Also at Isfahan University of Technology, Isfahan, Iran
24: Also at Brandenburg University of Technology, Cottbus, Germany
25: Also at Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State
University, Moscow, Russia
26: Also at Physics Department, Faculty of Science, Assiut University, Assiut,
Egypt
27: Also at Karoly Robert Campus, MATE Institute of Technology, Gyongyos,
Hungary
28: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary
29: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary
30: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary
31: Also at Wigner Research Centre for Physics, Budapest, Hungary
32: Also at IIT Bhubaneswar, Bhubaneswar, India
33: Also at Institute of Physics, Bhubaneswar, India
34: Also at G.H.G. Khalsa College, Punjab, India
35: Also at Shoolini University, Solan, India
36: Also at University of Hyderabad, Hyderabad, India
37: Also at University of Visva-Bharati, Santiniketan, India
38: Also at Indian Institute of Technology (IIT), Mumbai, India
39: Also at Deutsches Elektronen-Synchrotron, Hamburg, Germany
40: Also at Sharif University of Technology, Tehran, Iran
41: Also at Department of Physics, University of Science and Technology of
Mazandaran, Behshahr, Iran
42: Now at INFN Sezione di Bari, Bari, Italy, Università di Bari, Bari, Italy,
Politecnico di Bari, Bari, Italy
43: Also at Italian National Agency for New Technologies, Energy and
Sustainable Economic Development, Bologna, Italy
44: Also at Centro Siciliano di Fisica Nucleare e di Struttura Della Materia,
Catania, Italy
45: Also at Università di Napoli ’Federico II’, Napoli, Italy
46: Also at Consiglio Nazionale delle Ricerche - Istituto Officina dei
Materiali, PERUGIA, Italy
47: Also at Riga Technical University, Riga, Latvia
48: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
49: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
50: Also at Institute for Nuclear Research, Moscow, Russia
51: Now at National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
52: Also at Institute of Nuclear Physics of the Uzbekistan Academy of
Sciences, Tashkent, Uzbekistan
53: Also at St. Petersburg State Polytechnical University, St. Petersburg,
Russia
54: Also at University of Florida, Gainesville, Florida, USA
55: Also at Imperial College, London, United Kingdom
56: Also at P.N. Lebedev Physical Institute, Moscow, Russia
57: Also at California Institute of Technology, Pasadena, California, USA
58: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia
59: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia
60: Also at Trincomalee Campus, Eastern University, Sri Lanka, Nilaveli, Sri
Lanka
61: Also at INFN Sezione di Pavia, Pavia, Italy, Università di Pavia, Pavia,
Italy
62: Also at National and Kapodistrian University of Athens, Athens, Greece
63: Also at Ecole Polytechnique Fédérale Lausanne, Lausanne, Switzerland
64: Also at Universität Zürich, Zurich, Switzerland
65: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria
66: Also at Laboratoire d’Annecy-le-Vieux de Physique des Particules,
IN2P3-CNRS, Annecy-le-Vieux, France
67: Also at Şırnak University, Sirnak, Turkey
68: Also at Near East University, Research Center of Experimental Health
Science, Nicosia, Turkey
69: Also at Konya Technical University, Konya, Turkey
70: Also at Istanbul University - Cerrahpasa, Faculty of Engineering,
Istanbul, Turkey
71: Also at Piri Reis University, Istanbul, Turkey
72: Also at Adiyaman University, Adiyaman, Turkey
73: Also at Ozyegin University, Istanbul, Turkey
74: Also at Izmir Institute of Technology, Izmir, Turkey
75: Also at Necmettin Erbakan University, Konya, Turkey
76: Also at Bozok Universitetesi Rektörlügü, Yozgat, Turkey
77: Also at Marmara University, Istanbul, Turkey
78: Also at Milli Savunma University, Istanbul, Turkey
79: Also at Kafkas University, Kars, Turkey
80: Also at Istanbul Bilgi University, Istanbul, Turkey
81: Also at Hacettepe University, Ankara, Turkey
82: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom
83: Also at Vrije Universiteit Brussel, Brussel, Belgium
84: Also at School of Physics and Astronomy, University of Southampton,
Southampton, United Kingdom
85: Also at IPPP Durham University, Durham, United Kingdom
86: Also at Monash University, Faculty of Science, Clayton, Australia
87: Also at Università di Torino, TORINO, Italy
88: Also at Bethel University, St. Paul, Minneapolis, USA
89: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey
90: Also at Ain Shams University, Cairo, Egypt
91: Also at Bingol University, Bingol, Turkey
92: Also at Georgian Technical University, Tbilisi, Georgia
93: Also at Sinop University, Sinop, Turkey
94: Also at Erciyes University, KAYSERI, Turkey
95: Also at Texas A&M University at Qatar, Doha, Qatar
96: Also at Kyungpook National University, Daegu, Korea


# Electrically tunable plasmonic metasurface as a matrix of nanoantennas

LUIS ANGEL MAYORAL-ASTORGA  School of Electrical Engineering and Computer
Science, University of Ottawa, Ottawa, ON K1N 6N5, Canada  NEXQT Institute,
University of Ottawa, 25 Templeton Street, Ottawa, ON K1N 6N5, Canada  MASOUD
SHABANINEZHAD  School of Electrical Engineering and Computer Science,
University of Ottawa, Ottawa, ON K1N 6N5, Canada  NEXQT Institute, University
of Ottawa, 25 Templeton Street, Ottawa, ON K1N 6N5, Canada  HOWARD NORTHFIELD
School of Electrical Engineering and Computer Science, University of Ottawa,
Ottawa, ON K1N 6N5, Canada  NEXQT Institute, University of Ottawa, 25
Templeton Street, Ottawa, ON K1N 6N5, Canada  SPYRIDON NTAIS  School of
Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON
K1N 6N5, Canada  NEXQT Institute, University of Ottawa, 25 Templeton Street,
Ottawa, ON K1N 6N5, Canada  SABAA RASHID  School of Electrical Engineering and
Computer Science, University of Ottawa, Ottawa, ON K1N 6N5, Canada  Canadian
Centre for Electron Microscopy, McMaster University, Hamilton, Canada  EWA
LISICKA-SKRZEK  School of Electrical Engineering and Computer Science,
University of Ottawa, Ottawa, ON K1N 6N5, Canada  Canadian Centre for Electron
Microscopy, McMaster University, Hamilton, Canada  HAMID MEHRVAR  Huawei
Technologies Canada, Ottawa, Canada  ERIC BERNIER  Huawei Technologies Canada,
Ottawa, Canada  DOMINIC GOODWILL  Huawei Technologies Canada, Ottawa, Canada
LORA RAMUNNO  NEXQT Institute, University of Ottawa, 25 Templeton Street,
Ottawa, ON K1N 6N5, Canada  Department of Physics, University of Ottawa,
Ottawa, ON K1N 6N5, Canada  PIERRE BERINI  School of Electrical Engineering
and Computer Science, University of Ottawa, Ottawa, ON K1N 6N5, Canada  NEXQT
Institute, University of Ottawa, 25 Templeton Street, Ottawa, ON K1N 6N5,
Canada  Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5,
Canada

###### Abstract

We report the fabrication and characterization of a plasmonic metasurface
comprising electrically-contacted sub-wavelength gold dipole nanoantennas,
conformally coated by a thin hafnia film, an indium tin oxide layer and a
backside mirror, forming metal-oxide-semiconductor (MOS) capacitors, for use
as an electrically-tunable reflectarray or metasurface. By voltage biasing the
nanoantennas through metallic connectors and leveraging the carrier refraction
effect in the MOS capacitors, our measurements demonstrate phase control in
reflection over a range of about 30 degrees, with a constant magnitude of
reflection coefficient of 0.5, and the absence of secondary lobes.
Comprehensive electromagnetic and quantum carrier models of the structure are
developed and are in excellent agreement with the measurements. The
metasurface holds promise for use as an optical phased array.

##  1 Introduction

Electrically-tunable metasurfaces hold significant promise for numerous
applications, including dynamic beam forming and focusing, beam steering, and,
generally, wavefront shaping [1, 2, 3].

Beam steering, for example, is key to LIDAR scanners for self-driving cars and
autonomous machines, but is typically achieved using mechanical means, leading
to bulky and slow systems which precludes their use in cases that require a
high refresh rate. Nanophotonic approaches hold promise for LIDAR applications
[4],

particularly to implement beam steering devices based on optical phased
arrays[5, 6]. Compared to mechanical scanning, optical phased arrays
exploiting electronic tuning can enable significantly faster scanning speeds.

An ideal beam steering metasurface should possess several key characteristics
[6]. It should feature a large aperture, allowing for the steering of light
across a wide range of angles, without generating diffraction orders.
Simultaneously, it should maintain the beam at a constant magnitude with low
loss throughout the steering process. In addition, such a surface should be
reliable, compact, cost-effective, and solid-state, eliminating the need for
mechanical components. It should also offer rapid scanning capabilities to
enable swift operations. Optical phased arrays for beam scanning can be
implemented using tunable metasurfaces [6]. Such structures typically consist
of a planar arrangement of phase-tunable coherent light emitters, where an
individual emitter is referred to as a pixel. A pixel can take various forms
and be arranged to form a metasurface that operates in reflection
(reflectarray) or in transmission (transmitarray).

Optical phased arrays based on plasmonic nanoantennas hold promise for meeting
the requirements of optical beamsteering. Resonant plasmonic nanoantennas
support strongly enhanced fields which confer a high sensitivity to local
perturbations in refractive index. They are also nanoscale structures,
enabling pixels to be dimensioned to sub-wavelength scales which precludes the
emergence of grating (secondary) lobes. Moreover, metal nanoantennas can also
serve as device electrodes, ensuring strong optical overlap with the active
region.

Integrating plasmonic nanostructures with a semiconductor enables exploitation
of the carrier refraction effect through electrical biasing in an appropriate
device configuration. Altering the carrier density by applying a gate voltage
provides a pathway to dynamically and actively tune the phase, amplitude and
polarization of the emerging light [7, 8, 9, 10, 6]. A promising approach to
dynamically tune a metasurface for beam steering involves manipulating the
optical properties of a semiconductor used in its construction. Transparent
conductive oxides (TCOs), particularly Indium Tin Oxide (ITO), are of strong
interest as this material offers high transparency in the visible and near-
infrared spectrum, coupled with excellent electrical conductivity, making it
an ideal candidate for applications in photonics and optoelectronics.

Implementing devices that replicate the function of a turning mirror to
control a beam generated by a nearby source involves the pixelation and
application of a controllable phase gradient across a reflective area. Such a
device, termed a tunable reflectarray, allows for steering a monochromatic
beam to any desired angle of reflection. To achieve this, it is desirable that
the phase of each pixel over the beam cross-section to be continuously
variable from $0$ to $2\pi$ without affecting the incident light’s intensity.

The versatility of the tunable reflectarray concept makes it a powerful tool.
Much of the ongoing research in beam steering devices, particularly those
involving optical metasurfaces or plasmonic nanostructures, is focused on the
development of reflectarrays constructed with tunable pixels, incorporating
ITO as electro-optical active material.

Huang _et al._ [11] achieved a phase change of 184° at a wavelength of 1573 nm
for a single-gated device comprised of 50 nm thick gold strips on 5 nm of
Al2O3, on 20 nm of ITO, on 80 nm of Au. However, their structure displayed a
strong dependence of reflectance on the bias voltage. Park _et al._ [12]
reported phase tuning of 180° in the reflected field using a device stack
comprised of Au, Al2O3, ITO and Au, operating at a wavelength of 6 µm. Double-
gated devices were reported where the metasurface elements were controlled
with two independent voltages driving two MOS field effect regions, thereby
enabling wider phase tunability [13]. They employed ITO as the active
metasurface material and a composite hafnia/alumina gate dielectric. The
device exhibited a continuous phase shift from 0 to 303° at a wavelength of
$\lambda$ = 1550 nm, however, it displayed a significant dependence of
reflectance on the bias voltage.

More recently, Kim _et al._ [14] introduced a beam steering device composed of
a 5 nm thick ITO layer sandwiched between two insulator layers (Al2O3/HfO2).
By employing two separate bias controls, they managed to control both the
amplitude and the phase from 0 to 3600. They reported a side mode suppression
ratio (SMSR) of 2.7 dB. An experimental demonstration of beam steering with a
metasurface array operating at 1.3 $\mu$m was also recently presented [15].
The metasurface unit cell consisted of a metal–dielectric–oxide structure with
indium tin oxide as an active layer, modulated by using top fan-out
electrodes. The metasurface array was pixelated in two-dimensions and
exhibited a phase change of 137°, allowing for two-dimensional beam steering
to an angle of 7.3°. The side mode suppression ratio (SMSR) ranges from -9 to
6 dB.

Here we focus on the development of a reflectarray following a nanoscale pixel
design based on a gold dipole nanoantenna that enables phase control in
reflection, explored theoretically in previous work [7]. We demonstrate phase
tuning with a constant magnitude of reflection and the absence of secondary
lobes due to the sub-wavelength dimensions of the pixels. In the forthcoming
sections, we describe and discuss the device concept, our fabrication
techniques and our test set-up, and we present measurements compared with
theoretical results based on electromagnetic and quantum carrier models of the
structure. We give brief conclusions in the last section of the paper.

##  2 Description and operation of the reflectarray

Figure 1(A) illustrates the plasmonic unit cell (pixel) of dimensions $a_{x}$
by $a_{y}$ employed in our reflectarray. The unit cell exhibits periodicity
along both the $x$ and $y$ directions. The pixel design comprises a gold
dipole nanoantenna, formed by two rods of length $L_{d}$, width $W$, and
thickness $t_{Au}$. These rods are separated by a gap $g$ and intersected by
two perpendicular gold connectors of width $W_{c}$, positioned at an edge-to-
edge distance of $d_{c}$ from each other, as shown in Fig. 1(C). The
dimensions of the dipole and connectors within the plasmonic unit cell are
presented in the table inset in Fig. 1.

Figure 1: Sketch of the sub-wavelength unit cell used in the reflectarray,
including the dimensions used in the theoretical model of the experimentally
realised structure. (A) Panoramic view of the complete material stack of the
unit cell, (B) 1D MOS capacitor enabling electrical control of the optical
performance of the pixel, and (C) panoramic view of gold dipole nanoantennas
along with connectors.

Within this unit cell, a metal dipole nanoantenna is positioned with its long
axis aligned in the $x$-direction. This configuration enables the structure to
resonate in its fundamental mode at the desired wavelength when excited by a
plane wave or beam that is polarized in the $x$-direction and propagating in
the $z$-direction (upward through the SiO2 substrate). In the case of this
particular polarization (or any polarization in the $z-x$ plane), electrical
contacts can be strategically placed along each segment of the antenna in the
$y$-direction as shown in Fig. 1. This arrangement produces a nanoantenna
array that can be addressed column-wise when viewing the $x-y$ plane (_cf._
Fig. 4(H)). Thus, the structure is designed to serve not only as an antenna
but also as device electrodes.

The gold antennas and contacts are supported by a glass (fused silica)
substrate. They were deposited on a 3 nm layer of chromium to ensure
adherence, as shown in magenta in Figs. 1(A)-(B). Additionally, a 3 nm layer
of chromium was deposited onto the gold antennas to facilitate the subsecuent
oxide layer. Following this, the structure is further conformally coated by a
7 nm layer of insulating oxide (hafnia, HfO2), followed by a 75 nm thick layer
of indium tin oxide (ITO) as the semiconductor, and finally by a ground Ohmic
contact designated as the backside metal (BSM) also functioning as an optical
mirror, comprised of the metal stack 3 nm Cr + 50 nm Au + 500 nm Cu + 50 nm
Pt. This layer arrangement forms electrically a MOS (metal oxide
semiconductor) capacitor structure, operating optically as a resonant
nanoantenna on an insulator/ITO stack, on a mirror.

The device is constructed such that ITO fills the gap between the antenna
arms, which is where the optical fields are enhanced and the structure is most
sensitive. When the MOS structure is driven into strong accumulation, a high
electron concentration layer is formed within a confined area approximately 4
nm away from the interface of the ITO and the insulator. We refer to the
accumulation layer as the perturbed layer in ITO, indicated in red in Fig.
1(B). Furthermore, ITO undergoes a transition through epsilon-near-zero (ENZ)
in this layer as accumulation takes place, resulting in a significant
perturbation of the antenna resonance, including its phase.

The antenna unit cells (pixels), of dimensions $a_{x}$ by $a_{y}$, are
distributed in a periodic fashion over a designated area measuring $10\times
10$ $\mu$m2, centered within an electrical fan-out structure to contact an
array column-wise. The selection of pixel periodicity is crucial, and is
chosen small enough to satisfy the condition $a_{x}$, $a_{y}$ $\leq\lambda/2$,
where $\lambda$ represents the wavelength of operation in the medium through
which light is incident on the nanoantennas (glass substrate). By respecting
this condition, the arrangement ensures the absence of grating diffraction
orders in reflection.

##  3 Materials and methods

###  3.1 Fabrication process

The fabrication process employed three distinct lithography techniques. A
trilayer contact photolithography process was employed to create relatively
large features via lift-off, such as a fan-out structure, contact fingers and
pads, and the backside metallization. For specific details of this fabrication
process, refer to [16]. This technique provides better resolution than
conventional bilayer photolithography lift-off processes. A traditional
single-layer photolithography process was used to create a wet etch mask to
define ITO regions. A Helium ion beam lithography technique was utilized to
fabricate the smallest features via lift-off, such as the nano-antennas and
its crossing contacts to fan-out structures. This technique provides better
resolution than e-beam lithography while minimising proximity effects. A
detailed description of our Helium ion beam lithography process can be found
in [17].

The complete fabrication flow is depicted in Fig. 2. The fabrication process
begins with the patterning of the first metal layer (FML) on a fused silica
substrate using a trilayer photolithography process [16], as illustrated in
Fig. 2(A). The FML was deposited via evaporation as the metal stack 3 nm Cr +
28 nm Au + 2 nm Cr. The top layer of Cr was added to improve adhesion for the
subsequent oxide application, as illustrated in Fig. 2(A)-(B). It was then
lifted-off such that the desired features remained on the substrate, as shown
in Fig. 2(C). The metal features defined in this step include alignment marks
for subsequent process steps and to facilitate experimental characterization,
a fan-out structure, contact pads to connect to a nanoantenna array, and
various process control structures. Then, the nanoantenna arrays and crossing
contacts were fabricated using a Helium ion beam lithography and liftoff
process [17], as the metal stack 3 nm Cr + 55 nm Au + 3 nm Cr, as shown in
Fig. 2(D). Electron beam lithography was employed to expose alignment marks
into the resist enabling Helium ion beam visibility and the overlay of the
nanoantennas onto the FML. This was necessary since the FML cannot be imaged
through PMMA with a helium ion beam unlike an electron beam. The crossing
contacts were aligned and overlapped with the fan out features on the FML to
ensure electrical continuity of the nanoantennas to the contact pads. Then we
used an atomic layer deposition (ALD) process to conformally deposit a 7 nm
thick layer of hafnia, as shown in Fig. 2(E). This was followed by the
deposition of a 75 nm thick ITO layer via sputtering, which was then annealed
at 350 ${}^{\text{o}}$C in $\text{O}_{2}$, as shown in Fig. 2(F). A wet etch
mask was then defined photolithographically, as shown in Fig. 2(G), the ITO
layer was etched in a bath of TE-100 etchant (Transene Inc.), and the
photoresist mask was stripped, as shown in Fig. 2(H). The ITO patterns thus
formed consisted of disks centered above the nanoantenna arrays and their
crossing contacts. Then we defined the BSM using a trilayer photolithograpy
process[16], and we removed the exposed hafnia layer using a reactive ion
etching (RIE) process prior to BSM deposition, as shown in Fig. 2(I). The BSM
was then deposited via evaporation as the metal stack 3 nm Cr + 50 nm Au + 500
nm Cu + 50 nm Pt, as shown in Fig. 2(J), followed by lift-off to form complete
structures, as shown in Fig. 2(K). The BSM pattern consisted of circular
patches on the ITO disks defining the Ohmic contacts and mirrors thereon, and
overlapped metal on the fan-out and pad structures to reduce the resistance
and facilitate probing.

Figure 2: Fabrication process flow. (A) Trilayer photolithography stack to
define FML on fused silica substrate; (B) FML deposition by evaporation (3 nm
Cr + 28 nm Au); (C) Lift-off; (D) Fabrication of nanoantenna arrays by Helium
ion beam lithography, metal evaporation, and lift-off (3 nm Cr + 55 nm Au + 3
nm Cr); (E) Atomic layer deposition of 7 nm thick hafnia layer; (F) Sputter
deposition of 75 nm thick ITO layer followed by annealing at 350
${}^{\text{o}}$C in O2; (G) Wet etch mask application; (H) ITO wet etch in
bath of TE-100 (Transene); (I) Trilayer photolithography stack to define BSM,
followed by removal of exposed hafnia via reactive ion etching (RIE); (J) BSM
deposition by evaporation (3 nm Cr + 50 nm Au + 500 nm Cu + 50 nm Pt); (K)
Lift-off, revealing completed device.

Witness samples were included during the deposition of hafnia and ITO for the
purpose of evaluating the films independently. The hafnia layer had a density
of 9.61268 g/cm3 and a thickness of 7.76 $\pm$ 0.258 nm, confirmed using X-ray
reflectometry (XRR). The electronic properties of the ITO layer were
determined using an Ecopia HMS-3000 Hall effect measurement system in Van der
Pauw configuration. The optical properties of the deposited ITO and hafnia
layers were extracted from ellipsometry measurements using a spectroscopic
ellipsometer (UVISEL Plus, Horiba Scientific). See Section LABEL:sup-
section:ITOandHafniaCha under supplementary material for details.

###  3.2 Electro-optic test set-up and characterization techniques

To investigate the reflectarray’s electro-optical performance, we used two
custom-built confocal microscopes. Fig. 3(A) shows the setup used to measure
the magnitude of the reflection coefficient $|\Gamma(V)|$, and Fig. 3(B) shows
the setup used to measure the relative phase shift $\phi(V)$, both at normal
incidence and at bias voltage $V$. In both setups, The input light beam was
delivered through a polarization-maintaining single-mode optical fiber (PMOF)
and collimated using a fiber collimation package (F220APC-1550, Thorlabs). A
polarizer (LPNIR100, Thorlabs) was inserted to fix the polarization
orientation parallel to the nano-antennas. The incident beam was focused onto
the reflectarray using a plan apochromatic NIR objective (IR18091161
50$\times$, NA = 0.42, BoliOptics). The reflected beam was redirected by
beamsplitter BS1 (BSW29, Thorlabs) to the measurement instruments. For the
setup shown in Fig. 3(A), beamsplitter BS2 allowed monitoring the beam spot
using an IR camera (Micronviewer 7290A, Electrophysics) while taking power
measurements (81525A, Agilent/HP). For the setup shown in Fig. 3(B),
beamsplitter BS1 and mirror M produced a reference beam used to interfere with
the beam reflected by the reflectarray on the IR camera (this setup
effectively integrates a Michelson interferometer at the input of the confocal
microscope).

Figure 3: Confocal microscope for measuring (A) the magnitude of the
reflection coefficient $|\Gamma(V)|$, and (B) its relative phase $\phi(V)$.

We used two different light sources for our measurements. An incoherent light
source was used to measure the magnitude of the reflection coefficient (Fig.
3(A)), whereas a coherent light source was used to measure the relative phase
shift (Fig. 3(B)). The incoherent light was delivered by a super-continuum
white light source (SuperK Extreme, NKT Photonics) with an acousto-optic
tunable filter (SuperK SELECT, NKT Photonics), whereas the coherent light was
delivered by a tunable laser (TUNICS-Plus CL, 1510 nm to 1640 nm,
Photonetics).

The sub-wavelength reflectarray was designed to reflect the incident beam
without secondary lobes (which would be apparent in our set-up on images
captured by the IR camera). It is desirable for the incident beam spot to be
smaller than the area of the reflectarray (10$\times$10 $\mu$m2) to minimize
diffraction from its edges[18].

To ensure this, the beam spot diameter was measured for both of our sources
(super-continuum and the tunable laser) using the knife-edge method[19],
resulting in a 8.5 $\mu$m diameter, fulfilling this requirement.

To apply bias voltages to the device under test (DUT), we used a custom probe
card (Accuprobe) to simultaneously contact the 42 pads of a DUT: 2 pads
connected in common to the device’s ground, and 40 pads connected columnwise
to the nanoantenna array to apply the gate voltages. Alignment of the probe
card to a DUT was facilitated by a miniaturised microscope camera positioned
on the backside of the DUT, as depicted in Fig. 3. We used a programmable
logic controller (PLC) (Productivity2000, Automation Direct) with analog
output modules (P2-08DA-2, 8 outputs, Automation Direct) and two 16 bit
digital to analog converters (P2-16DA-2, 32 outputs, Automation Direct) to
apply a bias voltage individually to every pad of the DUT. Each analog output
had a voltage resolution of 0.305 V/count, with a maximum possible count of
65536 (16 bits), and a maximum/minimum DC output voltage of $\pm$10 V. The
maximum available current per output was 10 mA.

We measured the magnitude of the relative reflection coefficient $|\Gamma(V)|$
at different bias voltages using the setup shown in Fig. 3(A). $|\Gamma(V)|$
was taken as $\sqrt{P_{m}/P_{sub}}$, where $P_{m}$ is the power reflected by
the reflectarray, and $P_{sub}$ is the power reflected by a nearby region of
the substrate without nanoantennas but coated with hafnia, ITO and the
backside Au mirror. $P_{sub}$ can be used as a reference power because the
hafnia and ITO layers are very thin and the backside mirror is a good
approximation to a perfect mirror at the wavelengths of interest. Optical
density filter 2 (OD 2) was adjusted to avoid saturating the IR camera. We
kept the applied bias voltage equal and constant on all pads while sweeping
the wavelength.

The phase shift measurement relied on the intensity change resulting from the
interference between the beam reflected by the Device Under Test (DUT) and the
reference beam reflected by mirror M (Fig. 3(B)). First we model the
overlapping beams as two interfering waves. We write the wave representing the
reference beam as $w_{1}(z,t)=A\cos{(kz-\omega t)}$, and the wave representing
the beam reflected by the reflectarray as
$w_{2}(z,t)=|\Gamma(V)|B\cos{(kz-\omega t+\phi(V))}$. Setting $z=0$ as the
camera location and $t=0$ yields $w_{1}=A$ and
$w_{2}=|\Gamma(V)|B\cos{(\phi(V))}$. The intensity of the interference of the
beams on the camera can be expressed

$\begin{split}I(V)=&[A+|\Gamma(V)|B\cos{(\phi(V))}]^{2}\\\
=&A^{2}+2AB|\Gamma(V)|\cos{(\phi(V))}+\\\
&|\Gamma(V)|^{2}B^{2}\cos^{2}{(\phi(V))},\end{split}$ (1)

Initially, the bias voltage was set to zero. To achieve equal intensity for
the reflected and reference beams on the camera, OD 1 and OD 2 were carefully
adjusted. This involved individually blocking one beam’s optical path while
adjusting the Optical Density (OD) filter of the other.

To align both beams on the camera, the tilt mount of mirror M was adjusted
until the position of the reference beam almost perfectly overlapped with the
reflected beam. Subsequently, the translation stage of mirror M was fine-tuned
to achieve total constructive interference. The intensity $I(0)$ was captured
under the conditions that the intensity of individual beams $w_{1}$ and
$w_{2}$ were the same and constructively interfering ($A=|\Gamma(0)|B$ and
$\cos{(\phi(0))}=1$). From equation (1) at V = 0 V, we deduced
$A=\frac{\sqrt{I(0)}}{2}$

Then a non-zero and identical bias voltage was simultaneously applied to all
pads of the device, and $I(V)$ was measured. By assuming that the intensity
change in the resulting interference patterns was solely due to the phase
shift or that the magnitude of the relative reflection coefficient at a given
wavelength is almost constant ($|\Gamma(V)|\approx|\Gamma(0)|$), we solved
$\cos{(\phi(V))}$ in eq. (1) using the quadratic formula:

$\begin{split}0=&(A^{2}-I(V))+2A^{2}\cos{(\phi(V))}+\\\
&A^{2}\cos^{2}{(\phi(V))}.\end{split}$ (2)

The solutions of Eq. (2) are:

$\displaystyle\cos{(\phi(V))}=$ $\displaystyle\frac{-A+\sqrt{I(V)}}{A}$ (3)
$\displaystyle\cos{(\phi(V))}=$ $\displaystyle\frac{-A-\sqrt{I(V)}}{A}$ (4)

where the second solution is dropped for producing values out of range of the
cosine function. Isolating the phase shift $\phi(V)$ from Eq. (3), yields:

$\phi(V)=\cos^{-1}\Big{(}{\frac{\sqrt{I(V)}-A}{A}}\Big{)},$ (5)

where $I(V)$ has been measured and $A$ determined as described above. See
section SLABEL:sup-section:phase_measurement_example under supplementary
material for a an example calculation of the phase shift from the raw data.

###  3.3 Capacitance vs. voltage (C-V) measurements

We measured individual capacitance-voltage (C-V) characteristics for every
nanoantenna column using an HP 428a LCR meter, at an AC voltage of $\pm$30 mV
and a frequency of 10 kHz. The DC bias voltage was increased from -2 V in
steps of 0.05 V until the breakdown voltage (BDV) was reached in accumulation.
The C-V measurements were performed after the electro-optic measurements
because the capacitor structures become compromised when the BDV is reached.

From the C-V characteristics, we determined the dielectric constant (DC
relative permittivity) of hafnia using:

$\epsilon_{ox}=C_{max}\frac{t_{ox}}{A_{p}\epsilon_{0}},$ (6)

where $C_{max}$ is the maximum capacitance measured in accumulation, $t_{ox}$
is the thickness of the hafnia layer, $A_{p}$ is the capacitor area probed by
pad $p$, and $\epsilon_{0}$ is the vacuum permittivity. The breakdown field
(BDF) in MV/cm was calculated using

$\text{BDF}=\text{BDV}\times 10/t_{ox,nm},$ (7)

where the breakdown voltage BDV is in V and $t_{ox,nm}$ is the thickness of
the hafnia layer in nm.

###  3.4 Electrostatic modelling

Classical models such conventional drift-diffusion (CDD) [20, 21] are widely
used to compute the voltage-dependent carrier density profile in the
semiconductor of MOS capacitors. However, when the MOS structure is
incorporated into a plasmonic structure or metasurface, and the semiconductor
is an epsilon-near-zero medium (such as ITO), then effects such as quantum
confinement, quantum tunneling, quantum compressibility and electron
diffraction/interference need to be considered. These factors are crucial to
accurately compute the profile of the carrier density [22, 23, 24]. The
details of the profile significantly impact the accuracy of the resulting
permittivity distribution, particularly regarding the number and location of
epsilon-near-zero points [25]. Here, we used self-consistent Schrödinger-
Poisson (SP) coupling to compute the carrier density profile in ITO under
accumulation [25]. This model is theoretically more rigorous than CDD because
it takes the aforementioned quantum effects into account. We used the
Semiconductor Physics node available under the Semiconductor Module of COMSOL
6.1, and the Schrödinger-Poisson interface with the equilibrium study [26].

In materials like ITO, which are sensitive to local refractive index
perturbations, the SP model shows significant deviations from classical models
in predicting carrier density under accumulation. The Conventional Drift-
Diffusion (CDD) model [25] predicts the highest carrier density at the
HfO2-ITO interface, decreasing exponentially from the boundary. This
prediction contrasts with the SP model, which suggests a lower density at the
interface and a peak within the ITO. The CDD model, treating carriers as a
classical free electron gas, overlooks quantum effects such as quantization,
confinement, and diffraction/interference, which are accounted for in the SP
model. These quantum effects lead to findings like the reduced carrier density
at the HfO2-ITO interface due to quantum confinement and oscillatory profiles
at higher gate voltages. At sufficiently high voltages, the SP model predicts
the carrier density in ITO crossing two ENZ points, contrasting with the
single ENZ point by the CDD model. This discrepancy significantly affects the
permittivity profile and the optical response of the material [25].

The Poisson and Schrödinger equations are coupled second-order nonlinear
differential equations, and solving this system is nontrivial, requiring an
iterative approach. To start the iterations, we computed the potential
distribution throughout the 1D MOS capacitor (Fig. 1(B)) via the CDD and used
it as an input to the SP model. Since the Cr is just 3 nm, we assumed that
voltage is applied to the gold and ignored the Cr in electrostatic simulation.
To obtain the potential distribution, we applied a gate voltage to the metal
and grounded the ITO. The metal work function was set to 5.1 eV, whereas the
electron affinity, bandgap energy, static relative permittivity, and doping
concentration of the ITO were set to 4.8 eV, 2.8 eV, 9.1, and 2.65 $\times
10^{20}$ cm-3, respectively.

The potential distribution obtained from the CDD was then input into the SP
model by updating the potential term therein. In the next step, the carrier
density profile $N(d)$ is computed using the sum of statistically weighted
densities of the eigenfunctions obtained from the updated Schrödinger equation
via [22, 25]

$N(d)=\sum_{i}W_{i}|\phi_{i}(d)|^{2},$ (8)

where $W_{i}$ represents the weight factors, $\phi_{i}$ is the $i^{th}$
normalized eigenfunction associated with eigenenergy $E_{i}$, and $d$ denotes
the distance from the oxide surface. Then, the new potential distribution is
computed by solving Poisson’s equation using the updated charge distribution
$N(d)$ (Eq. (8)) in the charge density term therein. This procedure is
repeated until the electrostatic potential distribution converges. A uniform
mesh of step size 0.1 nm was used to discretize the structure. Further details
on setting up the problem in COMSOL can be found elsewhere [25].

We then used the electron density distribution computed by the SP model $N(d)$
in the Drude equation [27] to obtain the spatially-dependent relative
permittivity in the perturbed ITO:

$\epsilon(d,\omega)=\epsilon_{\infty}-\frac{\omega_{D}(d)^{2}}{\omega^{2}+i\gamma_{D}\omega},$
(9)

where $\epsilon_{\infty}$ is the high-frequency relative permittivity and
$\gamma_{D}$ is the collision frequency, which were taken as 3.92 and
$4.4\times 10^{13}$ rad/sec, respectively [25]. $\omega_{D}(d)$ is the
spatially-dependent plasma frequency of the ITO obtained via

$\omega_{D}(d)=\sqrt{\frac{N(d)}{n_{b}}}\omega_{D,b}$ (10)

where $N(d)$ is obtained for an applied bias voltage of interest.
$\omega_{D,b}$ is the bulk plasma frequency of ITO and was fixed to
$1.55\times 10^{15}$ rad/sec [25].

###  3.5 Optical modelling

To investigate the gate-tunable optical response of the reflectarray, we used
the relative permittivity distribution in ITO computed for different applied
gate voltages as described in Subsection 3.4 as well as the experimentally-
determined dimensions of the nanoantenna array (Fig. 1). The carrier density
in the perturbed layer is a function of distance from the hafnia-ITO
interface, so we projected the 1D profile computed for the 1D MOS (Fig. 1(B),
red region) over the 3D unit cell (Fig. 1(A), red region).

We used the Electromagnetic Waves, Frequency Domain interface of the Wave
Optics module in COMSOL 6.1 for the optical modelling [26], and we computed
the complex reflection coefficient of a unit cell _vs._ wavelength at various
applied bias voltages. The wave equations derived from Maxwell’s equations
were solved by applying material properties and boundary conditions [28]:

$\displaystyle\mathbf{\nabla}\times\mathbf{\nabla}\times\mathbf{E}(x,y,z)-k^{2}\epsilon_{r}(x,y,z)\mu_{r}\mathbf{E}(x,y,z)$
$\displaystyle=0$ (11)
$\displaystyle\mathbf{\nabla}\times\big{(}\frac{1}{\epsilon_{r}(x,y,z)}\mathbf{\nabla}\times\mathbf{H}(x,y,z)\big{)}-k^{2}\mu_{r}\mathbf{H}(x,y,z)$
$\displaystyle=0,$ (12)

where $\mu_{r}$ is the relative permeability (set to 1), $\epsilon_{r}(x,y,z)$
is the relative permittivity distribution describing the structure, $k$ is the
free-space wavenumber, and $\mathbf{E}$ and $\mathbf{H}$ are the electric and
magnetic field vectors.

The structure varies along the $z$-axis and is periodic along the $x$ and $y$
directions (Fig. 1). The propagation and electric field directions of the
incident light were taken along the $+z$ and $x$ directions, respectively.
Periodic boundary conditions were applied on the $x$ and $y$ boundaries, and a
source port was placed along the bottom to excite the structure and record the
reflected wave (the excitation propagates through the SiO2 substrate, in
accord with the experimental situation). The modeling domain in the $z$
direction was truncated and two perfectly matched layers (PMLs) were added to
the top and bottom of the unit cell to absorb the incident waves.

In our optical model, the refractive index of hafnia was taken from
experimental ellipsometry data (see section SLABEL:sup-
section:ITOandHafniaCha), the relative permittivity of Au and Cr were taken
from Johnson and Christy [29] and Rakić _et al._ [30], respectively, and the
voltage-dependent relative permittivity distribution of the ITO - obtained
from our electrostatic calculations in Subsection 4.2 \- was used. We used
mesh mapping along the top boundaries of the perturbed region in ITO and swept
it throughout the 3D domain of this region in a step of 0.1 nm. A tetrahedral
mesh was used for the remaining domains.

##  4 Results and discussion

###  4.1 Fabrication

Completed devices produced by the fabrication process described in Subsection
3.1 are shown in Fig. 4, highlighting various stages in the fabrication flow.
Fig. 4(A) shows a completed device being probed, and the inset depicts the
pad, pin layout and numbering. Fig. 4(B) shows the device after FML formation
and fabrication of the nanoantennas in the central area of the device. Fig.
4(C) shows the central area of the device after hafnia and ITO
deposition/lift-off (central circular patch). In Fig. 4(D), the same area is
shown after BSM formation, which is the final step in the fabrication flow.
Figs. 4(E) and 4(f) present AFM scans taken along the red and green lines
sketched on Fig. 4(D). Fig. 4(E) shows step changes in the material stack
corresponding to the 6.9 nm thick hafnia layer, the 78 nm thick ITO patch, and
the 596 nm thick BSM, whereas Fig. 4(F) shows the step changes in the material
stack corresponding to the 33 nm thick FLM and 595 nm thick BSM. In Fig. 4(H),
a low-magnification helium ion microscope (HIM) image reveals the nanoscale
fan-out structure connecting the columns of the nanoantenna array to the
large-scale fan-out structure on the FML. The high-magnification inset shows
good step coverage of these fingers. Fig. 4(G) is gives a high-magnification
HIM image of a nanoantenna array, from which feature dimensions are measured
and reported in the table inset to Fig. 1.

Figure 4: Images of fabricated devices. (A) Microscope image of complete
device with pins landed on contact pads. Inset: Pad layout of the device and
notation. The same layout was used for the pins of the probe card. (B)
Microscope image (50$\times$) of the first metal layer (FML) and the
nanoantenna array. (C) Microscope image (500$\times$) of the device center
showing the fan-out fingers reaching beneath the circular ITO patch. (D)
Microscope image (500$\times$) of the device center after backside metal
deposition. (E) and (F) AFM scans along two locations showing step changes in
the material stack due to hafnia, ITO and backside metal (scan taken along red
line sketched in Part (D)), and step changes in the material stack due to the
first metal layer and and the backside metal layer (scan taken along green
line sketched in Part (D)). (H) HIM image at low magnification revealing the
nanoantenna array and the nanoscale fan-out fingers connecting the array to
the large-scale fan-out fingers on the FML (high-magnification inset shows
good step coverage of these fingers). (G) HIM image at high magnification of a
few unit cells of the nanoantenna array.

###  4.2 Electrostatic performance

In Figure 5(A), the graph depicts the computed capacitance per unit area
plotted against the voltage applied to the sub-wavelength reflectarray, with
the data point represented by colored squares. The computations involved three
different values for the DC relative permittivity of hafnia,
$\epsilon_{ox}=12$, $12.4$ and $16.4$, combined with three thickness values,
$t_{ox}=7$, $7.5$ and $8$ nm. These values were chosen to encompass the
measured experimental C-V characteristics, as discussed subsequently in
Section 4.4. Specifically, the combination $\epsilon_{ox}=12$ and $t_{ox}=7$
nm generates theoretical C-V characteristics that are representative of the
average experimental measurements. The C-V theoretical result was obtained by
taking the derivative of the charge per unit area in ITO relative to the
applied voltage, the former computed from the associated carrier density
profiles. The corresponding measured C-V characteristics plotted on Fig. 5(A)
will be discussed in Subsection 4.4.

Figure 5(B) shows the electron carrier density profiles for $\epsilon_{ox}=12$
and $t_{ox}=7$ nm. The electron carrier density was calculated inside the
perturbed region _vs._ distance from the hafnia-ITO interface for the gate
voltages listed in the legend, computed using the SP model. It is clear that
the perturbation extends to about 4 nm into the ITO from the hafnia-ITO
interface ($z=0$). For each applied voltage, a reduction in the carrier
density is noted at the hafnia-ITO interface due to quantization of the
electron states thereon. At gate voltages from -2 to 0 V the perturbed region
is in depletion. By increasing the gate voltage, the carrier density
increases, and the region transitions from depletion to accumulation at 1V. As
noted in Fig. 5(B), for positive voltages, the lower carrier density at the
interface is followed by a clear maximum inside ITO, which grows and shifts
toward the hafnia-ITO interface with increasing voltage and electron
accumulation. The observed oscillatory behavior in the carrier density profile
is due to electron wavefunction diffraction and interference [31].

Figure 5: Electrostatic performance of sub-wavelength reflectarray. (A)
Capacitance per unit area _vs._ applied voltage (C-V curves), measured for
several device columns, identified by pad and pin number. The computed C-V
curve is also shown for comparison. (B) Computed carrier density profile from
the hafnia-ITO interface into ITO for different bias voltages. (C) and (D)
Corresponding computed real and imaginary parts of the relative permittivity
distribution into ITO.

Figures 5(C) and 5(D) show the real and imaginary parts of the relative
permittivity of ITO, respectively, computed by inserting the electron carrier
density profile shown in Fig. 5(B) into the Drude equation (eqs. (9) and
(10)), at the free-space operating wavelength of $\lambda=1560$ nm. As noted
in Figs. 5(C) and 5(D), the real and imaginary parts of the permittivity
approach 4 and 0, respectively, at the hafnia-ITO interface for all gate
voltages, due to electron depletion that develops thereon (Fig. 5(B)), driven
by the quantization of the electron states. A dip (bump) is then clearly noted
in the real (imaginary) part of the permittivity at bias voltages $\geq 0$ V.
This dip (bump) occurs at the maximum of the carrier density in ITO (Fig.
5(B)) and reduces (increases) with increasing gate voltage. The electric
fields become localised and enhanced in regions where the relative
permittivity of ITO becomes small [25].

The permittivity of ITO experiences perturbation through the carrier
refraction effect induced in the material (recall that the ITO is the
semiconductor in MOS capacitors). Voltage gating facilitated by metallic
connectors induces a carrier density perturbation over a thin ITO region near
the nanoantenna under accumulation and depletion. During accumulation, the
carrier density increases, leading to a blue-shift of the epsilon-near-zero
wavelength ($\lambda_{ENZ}$) in the perturbed ITO region. The local changes in
carrier density within the thin ITO layer, induced by accumulation or
depletion in the MOS capacitor, lead to a local variation in permittivity due
to the carrier refraction effect. As this perturbed ITO layer is situated in
the enhanced field region near the plasmonic nanostructure, its refractive
index variation alters the resonance of the nanoantenna, thereby influencing
the reflection properties of the reflectarray, particularly its phase shift.

###  4.3 Electro-optical performance

Theoretical and experimental results summarising the electro-optical
performance of the sub-wavelength reflectarray are presented in Fig. 6. In
Fig. 6(A), the measured wavelength response of $|\Gamma|$ exhibits a prominent
dip at 1560 nm, indicating the resonance wavelength for strong coupling of the
incident beam to the nanoantennas. Notably, $|\Gamma|$ shows minimal variation
across different applied bias voltages.

The experimental resonance wavelength closely matches the simulation results,
thus validating the accuracy of the model. However, it is worth noting that
the experimental $|\Gamma|$ are slightly larger than the calculations. The
difference between them is 0.15 off-resonance and 0.24 on-resonance. This
difference is due to background light present during the measurements and
imperfect coupling due to slight misalignment of the beam to the reflectarray.

Figure 6: Electro-optical performance of sub-wavelength reflectarray. (A)
Measured and computed $|\Gamma|$ response for bias voltages between -2 and 4.5
V. (B) Measured and computed relative phase shift $\phi$ _vs._ bias voltage at
two operating wavelengths. (C) Measured and computed $|\Gamma|$ _vs._ bias
voltage at two operating wavelengths. (D) Intensity contours of beam reflected
from sub-wavelength reflectarray, captured by IR camera at zero bias voltage.
Horizontal and vertical cross-sectional profiles are shown as the white plots.

Figure 6(B) gives the relative phase shift as a function of bias voltage for
two wavelengths of operation, referenced to the phase shift at 0 V, calculated
using eq. (5). The wavelengths of interest are the resonant wavelength (1560
nm) and the maximum wavelength accessible on our tunable laser (1640 nm). On
resonance ($\lambda=1560$ nm), the relative phase shift was measured as
$7.4^{\text{o}}$ at -2 V and $-22.8^{\text{o}}$ at 4.5 V, resulting in a
voltage-tuned phase range of $30^{\text{o}}$. Conversely, off-resonance
($\lambda=1640$ nm), the voltage-tuned phase range is smaller, spanning
$28.6^{\text{o}}$ starting from $9.1^{\text{o}}$ at -2 V and reaching
$-19.5^{\text{o}}$ at 4.5 V.

Figure 6(C) gives the magnitude of the relative reflection coefficient as a
function of bias voltage at the two selected wavelengths. The magnitude of the
reflection coefficient remains flat over the bias voltage range investigated.
Specifically, at the wavelength of 1560 nm, the experimental $|\Gamma|$
exhibits a flatness of 4.9%, about an average of 0.63. In comparison, the
theoretical flatness is 4.4%, about an average of 0.4. Likewise, at the
wavelength of 1640 nm, the experimental $|\Gamma|$ has a flatness of 3.3%
about an average of 0.68, whereas the theoretical flatness is 5.2% about an
average of 0.5. The flatness is defined as the ratio of the maximum deviation
to the average value, multiplied by 100.

The insertion loss (IL) in dB of the reflectarray is defined as IL =
$-20\log_{10}|\Gamma|$. The experimental IL at 1560 nm is 4 dB, and at 1640 nm
it is 3.35 dB. In contrast, the theoretical IL values are higher, 8 dB at 1560
nm and 6 dB at 1640 nm. The difference between them is attributed to
background light present during the measurements and imperfect coupling due to
slight misalignment of the beam to the reflectarray, as mentioned previously.

Figure 6(D) provides a visualization of the reflected beam spot from the
reflectarray under no bias, including horizontal and vertical cross-sections
along the center of the beam spot. Notably, the spot is very clean and absent
of diffraction lobes. The vertical cross-section displays a well-formed
Gaussian distribution, whereas the horizontal cross-section exhibits a weak
diffraction pattern caused by scattering effects from the vertical edges of
the reflectarray. The reflected beam maintained the same appearance as the
bias voltage was varied over the range of Figure 6(C), including the weak
diffraction pattern, owing to the flatness of the magnitude of the reflection
coefficient.

These scattering effects are caused by the incident polarization and boundary
conditions acting on the vertical edges of the reflectarray. Scattering from
metallic edges occurs differently depending on whether the polarisation is
hard or soft - the polarization is said to be hard when the boundary condition
that applies is a Neumann condition, and soft for a Dirichlet condition [32].
The scattering pattern from a metallic edge is broader for the hard
polarization than for the soft polarization, yielding an apparent larger beam
diameter upon reflection. The incident light beam is linearly polarized and
oriented horizontally, so $E_{x}\neq 0$ and $E_{y}=0$. As a consequence the
electric field across the top and bottom (horizontal) edges of the
reflectarray corresponds to the soft polarization, but across the right and
left (vertical) edges it corresponds to the hard polarization yielding a broad
scattering pattern along the $x$-axis. This undesired effect of diffraction
from edges has been previously reported in other experimental work [33, 34].

###  4.4 Electrical characteristics of the hafnia layer

Figure 5(A) illustrates the voltage-dependent specific capacitance (C-V
characteristics) measured for a reflectarray. The measurements exhibit
repeatability and reveal distinct accumulation patterns in the capacitors of
each column, identified by pin number. The raw capacitance measurements were
offset on average by $0.0781$ pF relative to the calibrated probes, leading to
an overestimation of the specific capacitance by $\Delta C_{s}=0.1$ $\mu
F/cm^{2}$. This offset was subtracted from the data presented in Figure 5(A).
More details on this point are given in Section SLABEL:sup-section:rawdata of
the supplementary document. The resulting C-V characteristics highlight
variations in specific capacitance among the columns, suggesting discrepancies
that should ideally converge into a single curve. These deviations are
attributed to variances in the hafnia layer.

Analysis of the measured C-V data allows the dielectric constant (DC relative
permittivity) of the hafnia layer to be extracted using Eq. (6), assuming a
constant hafnia thickness of $t_{ox}=7.76\pm 0.258$ nm (_cf._ Subsection 3.1,
XRR measurements). The dielectric constant was calculated for each pin and
ranged between $\epsilon_{ox}=15.85\pm 0.511$ for pin 21 and
$\epsilon_{ox}=12.43\pm 0.401$ for pin 38, corresponding to the maximum and
minimum capacitance values shown in Fig. 5(A). These pins are positioned at
the top and bottom of the device, respectively, indicating a position-
dependent variation in the permittivity of the hafnia layer across the device.

The dielectric strength of the Hafnia layer was determined using Eq. (7),
yielding a breakdown field (BDF) of 5.735 $\pm$ 0.2 MV/cm. The BDF reported in
the literature for thin (ALD) hafnia is $\approx$10 MV/cm[35]. The low BDF
value measured here for the hafnia layer prevents the MOS structures from
reaching deep accumulation.

The non-uniformity observed in the hafnia layer, along with its low relative
permittivity and BDF, suggests the presence of nanopores and a low density
relative to bulk hafnia. Referring to the fabrication flow, we note that the
metal stack of the nanoantennas, Fig. 2(D), consists of a layer of Cr,
followed by a layer of Au, capped by a thin layer of Cr, implying that the
hafnia layer must nucleate on this surface. We suspect an insufficient surface
density of hydroxyl (OH) functional groups on this metal stack, compromising
hafnia nucleation during the initial cycles of the ALD process and resulting
in reduced density of the film[36, 37]. Further work on improving the
formation of hafnia on such metal stacks would be required.

###  4.5 Epsilon near zero in ITO

The deposited ITO layer exhibits an epsilon-near-zero (ENZ) state in the near-
infrared (NIR) wavelength range, as confirmed by the analysis on witness
samples (refer to Section SLABEL:sup-section:ITOandHafniaCha in the
supplementary material for details). The device’s performance is significantly
influenced by the ENZ wavelength of the ITO layer. This perturbation in ITO
permittivity near the nanoantenna, induced by carrier refraction, is
facilitated by voltage gating, allowing for carrier density modulation through
accumulation and depletion. The increased carrier density leads to a blue-
shift in $\lambda_{ENZ}$ during accumulation impacts the reflectarray’s
reflection properties.

The wavelength at which the refractive index of ITO becomes zero,
$\lambda_{\text{ENZ}}$, can be determined using the following equation:

$\lambda_{\text{ENZ}}=\frac{2\pi c_{0}\sqrt{\epsilon_{\infty}}}{\omega_{D,b}}$
(13)

where $c_{0}$ is the speed of light in vacuum, $\epsilon_{\infty}$ denotes the
dielectric constant of ITO, and $\omega_{D,b}$ corresponds to the Drude
frequency for the bulk. Eq. (13) is derived from the Drude dispersion equation
assuming that $\gamma_{D}\gg\omega_{D,b}$, as shown in Eq.
(LABEL:Sequ:DrudeModel) in the supplementary information file.

To estimate $\lambda_{\text{ENZ}}$ of the ITO layer in the reflectarray, we
used the parameters obtained in Section LABEL:sup-section:ITOandHafniaCha of
the supplementary information file. Given that the carrier concentrations
measured via the Hall effect align well with $\omega_{D,b}$ derived from the
ellipsometry fittings, we calculated $\lambda_{\text{ENZ}}$ using an average
$\omega_{D}^{\prime}=(\omega_{D\text{,Hall}}+\omega_{D\text{,Ell}})/2$, where
$\omega_{D\text{,Hall}}$ is the Drude frequency for the bulk calculated using
Eq. (LABEL:Sequ:ncal) based on the carrier concentration measured via the Hall
effect, and $\omega_{D\text{,Ell}}$ represents the Drude frequency for the
bulk obtained from the ellipsometry fitting. Using Eq. (13), the anticipated
range of $\lambda_{\text{ENZ}}$ is from $\lambda_{\text{ENZ}}=1.91$ $\mu$m to
$\lambda_{\text{ENZ}}=2.1$ $\mu$m. As the MOS structures enter the
accumulation region, the carrier density in the perturbed ITO layer increases
(Fig. 5(B)), resulting in a blue-shift of $\lambda_{\text{ENZ}}$ therein. If
the breakdown field of hafnia is sufficiently high, $\lambda_{\text{ENZ}}$ may
transition through the resonance wavelength of the reflectarray, leading to a
significant phase shift in the reflected beam [7]. Due to the low BDF of the
hafnia layer here, it was not possible to apply a sufficiently large bias
voltage for this regime to be attained.

###  4.6 Speed of operation

The response speed of the reflectarray is inherently limited by the time
required for the MOS capacitors to reach full charge under accumulation, then
to discharge when the bias voltage is reduced. The measured specific
capacitances given in Fig. 5(A), the corresponding areas (_cf._ Section
SLABEL:sup-section:pinAreas in the supplementary information file), and the
minimum and maximum pin resistances determined to be $R_{\text{pin}}=1.8$
k$\Omega$ and $R_{\text{pin}}=1.85$ k$\Omega$ (_cf._ Section LABEL:sup-
section:pinResistance in the supplementary information file), were considered
in estimating the speed of operation. Based on these parameters, the time
constant of the capacitors was calculated as
$\tau=R_{\text{pin}}C_{\text{pin}}$, resulting in values ranging from $\tau=$
1.98 to 2.5 ns, from which the maximum frequency of operation is determined to
be 63 MHz (for $\tau=$ 2.5 ns).

##  5 Conclusion

We have successfully fabricated, experimentally characterized, and
theoretically modelled a novel sub-wavelength reflectarray capable of varying
the phase of the reflected beam without modifying its intensity or introducing
scattering effects. The reflectarray pixels consisted of Au dipole
nanoantennas integrated within a MOS capacitor structure. The pixels were
fabricated on a fused Silica substrate, with a conformal coating of thin
hafnia followed by layers of ITO and a Au backside mirror. By utilizing this
MOS capacitor, we were able to modulate the carrier density of ITO at the
interface with hafnia, thereby leading to the observed phase shift in the
reflected beam. The reflectarray demonstrates excellent performance near 1550
nm, producing exclusively the desired specular reflected lobe. The
reflectarray generated a maximum measured phase shift of about 30 degrees at
$\lambda=1560$ nm, limited by the quality of our hafnia insulating film. The
magnitude of reflection coefficient was about 0.5 and remained flat over the
tuning range of the phase.

Research funding: Financial support provided by the Natural Sciences and
Engineering Research Council of Canada (211636) and by Huawei Canada (570551)
are gratefully acknowledged.

Author contributions: Luis Angel Mayoral Astorga obtained and processed the
experimental data. Masoud Shabaninezhad performed all simulations. Howard
Northfield, Ewa Lisicka-Skrzek, Sabaa Rashid and Spyridon Ntais designed
layouts and fabricated the structures. Hamid Mehrvar, Eric Bernier, Dominic
Goodwill, Lora Ramunno and Pierre Berini conceived and directed the research.
All authors contributed to the discussion of the results and preparation of
the manuscript.

Conflict of interest: The authors declare no conflicts of interests.

Data availability: All data generated or analyzed during this study are
included in this published article and its supplementary information file.

## References

  * [1] Mikhail Y. Shalaginov, Sawyer D. Campbell, Sensong An et al., ”Design for quality: reconfigurable flat optics based on active metasurfaces”, _Nanophotonics_ , vol. 9, no. 11, 2020.
  * [2] Arash Nemati, Qian Wang, Minghui Hong and Jinghua Teng, ”Tunable and reconfigurable metasurfaces and metadevices”, _Opto‐Electron Adv._ , vol. 1, no. 5, 2018.
  * [3] Amr M. Shaltout, Vladimir M. Shalaev and Mark L. Brongersma, ”Spatiotemporal light control with active metasurfaces”, _Science_ , vol. 364, no. 6441, 2019.
  * [4] Inki Kim, Renato Juliano Martins, Jaehyuck Jang et al.,”Nanophotonics for light detection and ranging technology”, _Nature Nanotechnoly_ , vol. 16, no. 5, 2021.
  * [5] Martijn J.R. Heck, ”Highly integrated optical phased arrays: photonic integrated circuits for optical beam shaping and beam steering”, _Nanophotonics_ , vol. 6, no. 1, 2017.
  * [6] Pierre Berini, ”Optical beam steering using tunable metasurfaces”, _ACS Photonics_ , vol. 9, no. 7, 2022.
  * [7] Antonino Calá Lesina, Dominic Goodwill, Eric Bernier, Lora Ramunno and Pierre Berini, ”Tunable plasmonic metasurfaces for optical phased arrays”, _IEEE Journal of Selected Topics in Quantum Electronics_ , vol. 27, no. 1, 2021.
  * [8] Manuel Decker, Christian Kremers, Alexander Minovich et al., ”Electro-optical switching by liquid-crystal controlled metasurfaces”, _Optics Express_ , vol. 21, no. 7, 2013.
  * [9] Anthony Olivieri, Chengkun Chen, Saád Hassan, Ewa Lisicka-Skrzek, R. Niall Tait and Pierre Berini, ”Plasmonic nanostructured metal–oxide–semiconductor reflection modulators”, _Nano Letters_ , vol. 15, no. 4m 2015.
  * [10] Krishnan Thyagarajan, Ruzan Sokhoyan, Leonardo Zornberg and Harry A. Atwater, ”Millivolt modulation of plasmonic metasurface optical response via ionic conductance”, _Advanced Materials_ , vol. 29, no. 31, 2017.
  * [11] Yao-Wei Huang, Ho Wai Howard Lee, Ruzan Sokhoyan et al., ”Gate-Tunable Conducting Oxide Metasurfaces”, _Nano Letters_ , vol. 16, no. 9, 2016.
  * [12] Junghyun Park, Ju-Hyung Kang, Soo Jin Kim, Xiaoge Liu, and Mark L. Brongersma, ”Dynamic Reflection Phase and Polarization Control in Metasurfaces”, _Nano Letters_ , vol. 17, no. 1, 2017.
  * [13] Ghazaleh Kafaie Shirmanesh, Ruzan Sokhoyan, Ragip A. Pala, and Harry A. Atwater, ”Dual-Gated Active Metasurface at 1550 nm with Wide (¿300°) Phase Tunability”, _Nano Letters_ , vol. 18, no. 5, 2018.
  * [14] Sun Il Kim, Junghyun Park, Byung Gil Jeong et al., ”Electrically Reconfigurable Active Metasurface for 3D Distance Ranging”, 2020 IEEE International Electron Devices Meeting (IEDM), 2020.
  * [15] Sun Il Kim, Junghyun Park, Byung Gil Jeong et al.,”Two-dimensional beam steering with tunable metasurface in infrared regime”, _Nanophotonics_ , vol. 11, no. 11, 2022.
  * [16] Howard Northfield, Oleksiy Krupin, R. Niall Tait and Pierre Berini, ”Tri-layer contact photolithography process for high-resolution lift-off”, _Microelectronic Engineering_ , vol. 241, no. 11545, 2021
  * [17] Sabaa Rashid, Jaspreet Walia, Howard Northfield et al., ”Helium ion beam lithography and liftoff”,_Nano Futures_ , vol. 5, no. 025003m, 2021.
  * [18] Antonio Cala’ Lesina, Dominic Goodwill, Eric Bernier, Lora Ramunno and Pierre Berini, ”On the performance of optical phased array technology for beam steering: effect of pixel limitations”, _Opt. Express_ , vol. 28, no. 21, 2020.
  * [19] Marcos A. C. de Araújo, Rubens Silva, Emerson de Lima, Daniel P. Pereira and Paulo C. de Oliveira, ”Measurement of Gaussian laser beam radius using the knife-edge technique: improvement on data analysis”, _Applied Optics_ , vol. 48, no. 2, 2009.
  * [20] Innem V. A. K. Reddy, Josep M. Jornet, Alexander Baev and Paras N. Prasad, ”Extreme local field enhancement by hybrid epsilon-near-zero–plasmon mode in thin films of transparent conductive oxides”, _Optics Letters_ , vol. 45, no. 20, 2020.
  * [21] Swati Rajput, Vishal Kaushik, Lalit Singh et al., ”Efficient optical modulation in ring structure based on Silicon-ITO heterojunction with low voltage and high extinction ratio”, _Optics Communication_ , vol. 545, no. 129562, 2023.
  * [22] James H. Luscombe, Ann M. Bouchard and Marshall Luban, ”Electron confinement in quantum nanostructures: Self-consistent poisson-Schrödinger theory”, _Phys. Rev. B_ , vol. 46, no. 10262, 1992.
  * [23] C. Fiegna and A. Abramo, ”Solution of 1-D Schrodinger and Poisson equations double gate SOI MOS” ,SISPAD ’97. 1997 international conference on simulation of semiconductor processes and devices. Technical digest, 1997.
  * [24] Mohammad Sojib, Dhruv Fomra, Vitaliy Avrutin, Ü. Özgür and Nathaniel Kinsey, ”Optimizing epsilon-near-zero based plasmon assisted modulators through surface-to-volume ratio”, _Optics Express_ , vol. 30, no. 11, 2022.
  * [25] Masoud Shabaninezhad, Lora Ramunno and Pierre Berini, ”Tunable plasmonics on epsilon-near-zero materials: the case for a quantum carrier model”, _Optics Express_ , vol. 30, no. 26, 2022.
  * [26] COMSOL Multiphysics©V 6.1, ”COMSOL AB, Stockholm, Sweden”, www.comsol.com.
  * [27] Charles Kittel, _Introduction to solid state physics. Free electron fermi gas._ , Wiley, 2004.
  * [28] Pierre Berini, ”Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures”, _Physical Review B_ , vol. 61, no. 10484, 2000.
  * [29] P. B. Johnson and R. W. Christy, ”Optical constants of the noble metals”, _Phys. Rev. B_ , vol. 6, no. 12, 1972.
  * [30] Aleksandar D. Rakić, Aleksandra B. Djurišić, Jovan M. Elazar and Marian L. Majewski, ”Optical properties of metallic films for vertical-cavity optoelectronic devices”, _Appl. Opt._ , vol. 37, no. 22, 1998.
  * [31] Chien-Wei Lee and Jenn-Gwo Hwu, ”Quantum-mechanical calculation of carrier distribution in MOS accumulation and strong inversion layers”, _AIP Advances_ , vol. 3, no. 10, 2013.
  * [32] Constantine A. Balanis, ”Geometrical theory of diffraction: edge diffraction” in _Advanced engineering electromagnetics_ , Wiley press, 1989.
  * [33] Kenta Suzuki, Kenichi Oguchi, Yasuaki Monnai, Makoto Okano and Shinichi Watanabe, ”Spatio-temporal imaging of terahertz electric-field vectors: observation of polarization-dependent knife-edge diffraction”, _Applied Physics Express_ , vol. 12, no. 052010, 2019.
  * [34] Maryam Al-Shehab, Saba Siadat Mousavi, Maude Amyot-Bourgeois et al., ”Design and construction of a Raman microscope and characterization of plasmon-enhanced Raman scattering in graphene”, _Journal of the Optical Society of America B_ , vol. 36, no. 8, 2019.
  * [35] Saba Siadat Mousavi, Anthony Olivieri and Pierre Berini, ”Fabrication of a high-speed plasmonic reflection/transmission modulator”, _AIP Advances_ , vol. 11, no. 025023, 2021.
  * [36] Paul C. Lemaire, Mariah King and Gregory N. Parsons, ”Understanding inherent substrate selectivity during atomic layer deposition: Effect of surface preparation, hydroxyl density, and metal oxide composition on nucleation mechanisms during tungsten ALD”, _J. Chem. Phys._ , vol. 146, no. 052811, 2017.
  * [37] Véronique Cremers, Riikka L. Puurunen, Jolien Dendooven, ”Conformality in atomic layer deposition: Current status overview of analysis and modelling”, _Appl. Phys. Rev._ , vol. 6, no. 021302, 2019.