now takes the following form

$\displaystyle Y_{\eta}=\frac{1}{2(\eta^{2}+1)}(Z_{1}+\eta Z_{2}-Z_{3}+\eta
Z_{4})(\bar{Z}_{1}-\eta\bar{Z}_{2}+\bar{Z}_{3}+\eta\bar{Z}_{4})\,.$ (D.2)

The holographic computation is similar to what we did in section 5. Therefore
we only give the final results for
$C_{\mathcal{D}_{M}\mathcal{D}_{M}\mathcal{O}_{L}}$ and $D^{(p)}_{M|L}$. As a
consistency check, we also compute combination
$\sum_{p=-\frac{L}{2}}^{\frac{L}{2}}(-1)^{p}D^{(p)}_{M|L}$ and compare with
the results in subsection 5.2.

  * •

$L=1$.

$\displaystyle\mathcal{C}_{\mathcal{D}_{M}\mathcal{D}_{M}\mathcal{O}_{1}}=\left(\frac{\lambda}{2\pi^{2}}\right)^{\frac{1}{4}}\sqrt{3}\pi\left(\frac{1}{\eta}+\eta\right)\left(\sqrt{1-4\alpha^{4}}-4\alpha^{4}\text{arcsech}\left(2\alpha^{2}\right)\right)\,,$
(D.3)

from which we can read off

$\displaystyle
D^{(-1/2)}_{M|1}=D^{(1/2)}_{M|1}=\left(\frac{\lambda}{2\pi^{2}}\right)^{\frac{1}{4}}\sqrt{3}\pi\left(\sqrt{1-4\alpha^{4}}-4\alpha^{4}\text{arcsech}\left(2\alpha^{2}\right)\right)\,,$
(D.4)

and find that

$\displaystyle\sum_{p=-\frac{1}{2}}^{\frac{1}{2}}(-1)^{p}D^{(p)}_{M|1}=0\,.$
(D.5)

  * •

$L=2$.

$\displaystyle\mathcal{C}_{\mathcal{D}_{M}\mathcal{D}_{M}\mathcal{O}_{2}}=$
$\displaystyle-\left(\frac{\lambda}{2\pi^{2}}\right)^{\frac{1}{4}}\frac{8\sqrt{5}}{3\eta^{2}}\left(\sqrt{1-4\alpha^{4}}\left(4\alpha^{4}\left(5\eta^{4}+12\eta^{2}+5\right)+\eta^{4}+1\right)\right.$
(D.6) $\displaystyle-$
$\displaystyle\left.24\alpha^{4}\left(\eta^{2}+1\right)^{2}\text{arcsech}\left(2\alpha^{2}\right)\right)\,,$

from which we can read off

$\displaystyle D^{(-1)}_{M|2}=$ $\displaystyle D^{(1)}_{M|2}$ $\displaystyle=$
$\displaystyle-\left(\frac{\lambda}{2\pi^{2}}\right)^{\frac{1}{4}}\left(\frac{160}{3}\sqrt{5(1-4\alpha^{4})}\alpha^{4}+\frac{8}{3}\sqrt{5(1-4\alpha^{4})}-64\sqrt{5}\alpha^{4}\text{arcsech}\left(2\alpha^{2}\right)\right)\,,$
(D.7) $\displaystyle D^{(0)}_{M|2}=$
$\displaystyle-\left(\frac{\lambda}{2\pi^{2}}\right)^{\frac{1}{4}}128\sqrt{5}\alpha^{4}\left(\sqrt{(1-4\alpha^{4})}-\text{arcsech}\left(2\alpha^{2}\right)\right)\,,$

and check that

$\displaystyle\sum_{p=-1}^{1}(-1)^{p}D^{(p)}_{M|2}=\left(\frac{\lambda}{2\pi^{2}}\right)^{\frac{1}{4}}\frac{16\sqrt{5}}{3}(1-4\alpha^{4})^{\frac{3}{2}}\,.$
(D.8)

  * •

$L=3$.

$\displaystyle\mathcal{C}_{\mathcal{D}_{M}\mathcal{D}_{M}\mathcal{O}_{3}}=$
$\displaystyle-\left(\frac{\lambda}{2\pi^{2}}\right)^{\frac{1}{4}}\frac{\sqrt{7}\pi}{\eta^{3}}\left(\eta^{2}+1\right)\left(-\sqrt{1-4\alpha^{4}}\left(2\alpha^{4}\left(52\eta^{4}+137\eta^{2}+52\right)\right.\right.$
(D.9) $\displaystyle+$
$\displaystyle\left.\eta^{4}-\eta^{2}+1\right)+72\alpha^{4}\left(\alpha^{4}\left(2\eta^{4}+7\eta^{2}+2\right)\right.$
$\displaystyle+$
$\displaystyle\left.\left.\left(\eta^{2}+1\right)^{2}\right)\text{arcsech}\left(2\alpha^{2}\right)\right)\,,$

from which we can read off

$\displaystyle D^{(-\frac{3}{2})}_{M|3}=$
$\displaystyle\,D^{(\frac{3}{2})}_{M|3}$ (D.10) $\displaystyle=$
$\displaystyle-\left(\frac{\lambda}{2\pi^{2}}\right)^{\frac{1}{4}}\sqrt{7}\pi\left(72\alpha^{4}\left(2\alpha^{4}+1\right)\text{arcsech}\left(2\alpha^{2}\right)-\sqrt{1-4\alpha^{4}}\left(104\alpha^{4}+1\right)\right)\,,$
$\displaystyle D^{(-\frac{1}{2})}_{M|3}=$
$\displaystyle\,D^{(\frac{1}{2})}_{M|3}=\left(\frac{\lambda}{2\pi^{2}}\right)^{\frac{1}{4}}54\sqrt{7}\pi\alpha^{4}\left(4\left(3\alpha^{4}+1\right)\text{arcsech}\left(2\alpha^{2}\right)-7\sqrt{1-4\alpha^{4}}\right)\,,$

and check that

$\displaystyle\sum_{p=-\frac{3}{2}}^{\frac{3}{2}}(-1)^{p}D^{(p)}_{M|3}=0\,.$

We can see that the combination
$\sum_{p=-\frac{L}{2}}^{\frac{L}{2}}(-1)^{p}D^{(p)}_{M|L}$ indeed reproduce
exactly the results in the twisted translated frame given in subsection 5.2
for $L=1,2,3$. Generalization to higher $L$ is straightforward.

## References

  * [1] P. Yang, Y. Jiang, S. Komatsu and J.-B. Wu, Three-Point Functions in ABJM and Bethe Ansatz, 2103.15840.
  * [2] Y. Jiang, S. Komatsu, J.-B. Wu and P. Yang, Structure Constants in ABJM and Integrable Bootstrap, 21xx.xxxxx.
  * [3] D. Bak, B. Chen and J.-B. Wu, Holographic Correlation Functions for Open Strings and Branes, JHEP 06 (2011) 014 [1103.2024].
  * [4] A. Bissi, C. Kristjansen, D. Young and K. Zoubos, Holographic three-point functions of giant gravitons, JHEP 06 (2011) 085 [1103.4079].
  * [5] P. Caputa, R. de Mello Koch and K. Zoubos, Extremal versus Non-Extremal Correlators with Giant Gravitons, JHEP 08 (2012) 143 [1204.4172].
  * [6] H. Lin, Giant gravitons and correlators, JHEP 12 (2012) 011 [1209.6624].
  * [7] S. Hirano, C. Kristjansen and D. Young, Giant Gravitons on $AdS_{4}\times\mathbb{C}P^{3}$ and their Holographic Three-point Functions, JHEP 07 (2012) 006 [1205.1959].
  * [8] C. Kristjansen, S. Mori and D. Young, On the Regularization of Extremal Three-point Functions Involving Giant Gravitons, Phys. Lett. B 750 (2015) 379–383 [1507.03965].
  * [9] N. Drukker and J. Plefka, Superprotected n-point correlation functions of local operators in N=4 super Yang-Mills, JHEP 04 (2009) 052 [0901.3653].
  * [10] M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiral primary 3-point functions, JHEP 07 (2012) 137 [1203.1036].
  * [11] R. de Mello Koch and R. Gwyn, Giant graviton correlators from dual SU(N) super Yang-Mills theory, JHEP 11 (2004) 081 [hep-th/0410236].
  * [12] T. W. Brown, Half-BPS SU(N) correlators in N=4 SYM, JHEP 07 (2008) 044 [hep-th/0703202].
  * [13] F. Aprile, J. Drummond, P. Heslop, H. Paul, F. Sanfilippo, M. Santagata and A. Stewart, Single particle operators and their correlators in free $\mathcal{N}$ = 4 SYM, JHEP 11 (2020) 072 [2007.09395].
  * [14] S. Corley, A. Jevicki and S. Ramgoolam, Exact correlators of giant gravitons from dual N=4 SYM theory, Adv. Theor. Math. Phys. 5 (2002) 809–839 [hep-th/0111222].
  * [15] Z. Bajnok, R. A. Janik and A. Wereszczyński, HHL correlators, orbit averaging and form factors, JHEP 09 (2014) 050 [1404.4556].
  * [16] S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, On the CFT Operator Spectrum at Large Global Charge, JHEP 12 (2015) 071 [1505.01537].
  * [17] A. Monin, D. Pirtskhalava, R. Rattazzi and F. K. Seibold, Semiclassics, Goldstone Bosons and CFT data, JHEP 06 (2017) 011 [1611.02912].
  * [18] Z. Bajnok and R. A. Janik, Classical limit of diagonal form factors and HHL correlators, JHEP 01 (2017) 063 [1607.02830].
  * [19] Z. Komargodski, M. Mezei, S. Pal and A. Raviv-Moshe, Spontaneously Broken Boosts in CFTs, 2102.12583.
  * [20] K. Zarembo, Holographic three-point functions of semiclassical states, JHEP 09 (2010) 030 [1008.1059].
  * [21] M. S. Costa, R. Monteiro, J. E. Santos and D. Zoakos, On three-point correlation functions in the gauge/gravity duality, JHEP 11 (2010) 141 [1008.1070].
  * [22] Y. Jiang, S. Komatsu and E. Vescovi, Structure constants in $\mathcal{N}$ = 4 SYM at finite coupling as worldsheet g-function, JHEP 07 (2020), no. 07 037 [1906.07733].
  * [23] Y. Jiang, S. Komatsu and E. Vescovi, Exact Three-Point Functions of Determinant Operators in Planar $N=4$ Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 123 (2019), no. 19 191601 [1907.11242].
  * [24] G. Chen, R. de Mello Koch, M. Kim and H. J. Van Zyl, Absorption of closed strings by giant gravitons, JHEP 10 (2019) 133 [1908.03553].
  * [25] G. Chen, R. De Mello Koch, M. Kim and H. J. Van Zyl, Structure constants of heavy operators in ABJM and ABJ theory, Phys. Rev. D 100 (2019), no. 8 086019 [1909.03215].
  * [26] P. D. B. Collins, An Introduction to Regge Theory and High-Energy Physics.  Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK, 5, 2009.
  * [27] M. S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [1209.4355].
  * [28] S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [1703.00278].
  * [29] E. D’Hoker, J. Erdmenger, D. Z. Freedman and M. Perez-Victoria, Near extremal correlators and vanishing supergravity couplings in AdS / CFT, Nucl. Phys. B 589 (2000) 3–37 [hep-th/0003218].
  * [30] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, Extremal correlators in the AdS / CFT correspondence, hep-th/9908160.
  * [31] D. Z. Freedman, K. Pilch, S. S. Pufu and N. P. Warner, Boundary Terms and Three-Point Functions: An AdS/CFT Puzzle Resolved, JHEP 06 (2017) 053 [1611.01888].
  * [32] G. Arutyunov and S. Frolov, On the correspondence between gravity fields and CFT operators, JHEP 04 (2000) 017 [hep-th/0003038].
  * [33] D. Giovannoni, J. Murugan and A. Prinsloo, The Giant graviton on $AdS_{4}xCP^{3}$ \- another step towards the emergence of geometry, JHEP 12 (2011) 003 [1108.3084].
  * [34] J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring Three-Point Functions and Integrability II. Weak/strong coupling match, JHEP 09 (2011) 029 [1104.5501].
  * [35] E. K. Sklyanin, Separation of variables - new trends, Prog. Theor. Phys. Suppl. 118 (1995) 35–60 [solv-int/9504001].
  * [36] Y. Kazama, S. Komatsu and T. Nishimura, Classical Integrability for Three-point Functions: Cognate Structure at Weak and Strong Couplings, JHEP 10 (2016) 042 [1603.03164]. [Erratum: JHEP 02, 047 (2018)].
  * [37] Y. Kazama and S. Komatsu, Wave functions and correlation functions for GKP strings from integrability, JHEP 09 (2012) 022 [1205.6060].
  * [38] Y. Kazama and S. Komatsu, Three-point functions in the SU(2) sector at strong coupling, JHEP 03 (2014) 052 [1312.3727].
  * [39] B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N=4 SYM Theory, 1505.06745.
  * [40] B. Basso and D.-L. Zhong, Three-point functions at strong coupling in the BMN limit, JHEP 04 (2020) 076 [1907.01534].
  * [41] M. Srednicki, Thermal fluctuations in quantized chaotic systems, J. Phys. A 29 (1996) L75–L79 [chao-dyn/9511001].
  * [42] M. Beşken, S. Datta and P. Kraus, Quantum thermalization and Virasoro symmetry, J. Stat. Mech. 2006 (2020) 063104 [1907.06661].
  * [43] A. Belin and J. de Boer, Random Statistics of OPE Coefficients and Euclidean Wormholes, 2006.05499.
  * [44] H. Lin, O. Lunin and J. M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174].
  * [45] K. Skenderis and M. Taylor, Anatomy of bubbling solutions, JHEP 09 (2007) 019 [0706.0216].
  * [46] R. de Mello Koch, Geometries from Young Diagrams, JHEP 11 (2008) 061 [0806.0685].
  * [47] L. Grant, L. Maoz, J. Marsano, K. Papadodimas and V. S. Rychkov, Minisuperspace quantization of ’Bubbling AdS’ and free fermion droplets, JHEP 08 (2005) 025 [hep-th/0505079].
  * [48] L. Maoz and V. S. Rychkov, Geometry quantization from supergravity: The Case of ’Bubbling AdS’, JHEP 08 (2005) 096 [hep-th/0508059].
  * [49] A. Dymarsky and S. Sugishita, KdV-charged black holes, JHEP 05 (2020) 041 [2002.08368].
  * [50] F. A. Smirnov, Quasiclassical study of form-factors in finite volume, hep-th/9802132.
  * [51] N. Gromov, S. Schafer-Nameki and P. Vieira, Efficient precision quantization in AdS/CFT, JHEP 12 (2008) 013 [0807.4752].
  * [52] S. W. Hawking, M. J. Perry and A. Strominger, Soft Hair on Black Holes, Phys. Rev. Lett. 116 (2016), no. 23 231301 [1601.00921].
  * [53] S. W. Hawking, M. J. Perry and A. Strominger, Superrotation Charge and Supertranslation Hair on Black Holes, JHEP 05 (2017) 161 [1611.09175].
  * [54] H. Afshar, D. Grumiller and M. M. Sheikh-Jabbari, Near horizon soft hair as microstates of three dimensional black holes, Phys. Rev. D 96 (2017), no. 8 084032 [1607.00009].
  * [55] M. M. Sheikh-Jabbari and H. Yavartanoo, Horizon Fluffs: Near Horizon Soft Hairs as Microstates of Generic AdS3 Black Holes, Phys. Rev. D 95 (2017), no. 4 044007 [1608.01293].
  * [56] H. Afshar, D. Grumiller, M. M. Sheikh-Jabbari and H. Yavartanoo, Horizon fluff, semi-classical black hole microstates — Log-corrections to BTZ entropy and black hole/particle correspondence, JHEP 08 (2017) 087 [1705.06257].
  * [57] J. Cotler and K. Jensen, A theory of reparameterizations for AdS3 gravity, JHEP 02 (2019) 079 [1808.03263].
  * [58] O. Lunin and S. D. Mathur, AdS / CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342–394 [hep-th/0109154].
  * [59] M. Shigemori, Superstrata, Gen. Rel. Grav. 52 (2020), no. 5 51 [2002.01592].
  * [60] A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99–104 [hep-th/9601029].
  * [61] M. Shigemori, Counting Superstrata, JHEP 10 (2019) 017 [1907.03878].
  * [62] S. Raju and P. Shrivastava, Critique of the fuzzball program, Phys. Rev. D 99 (2019), no. 6 066009 [1804.10616].
  * [63] S. Giusto, E. Moscato and R. Russo, AdS3 holography for 1/4 and 1/8 BPS geometries, JHEP 11 (2015) 004 [1507.00945].
  * [64] I. Bena, S. Giusto, E. J. Martinec, R. Russo, M. Shigemori, D. Turton and N. P. Warner, Asymptotically-flat supergravity solutions deep inside the black-hole regime, JHEP 02 (2018) 014 [1711.10474].
  * [65] S. Giusto, S. Rawash and D. Turton, Ads3 holography at dimension two, JHEP 07 (2019) 171 [1904.12880].
  * [66] J. Garcia i Tormo and M. Taylor, One point functions for black hole microstates, Gen. Rel. Grav. 51 (2019), no. 7 89 [1904.10200].
  * [67] K. Skenderis and M. Taylor, Fuzzball solutions and D1-D5 microstates, Phys. Rev. Lett. 98 (2007) 071601 [hep-th/0609154].
  * [68] I. Kanitscheider, K. Skenderis and M. Taylor, Holographic anatomy of fuzzballs, JHEP 04 (2007) 023 [hep-th/0611171].
  * [69] I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [0704.0690].
  * [70] M. Taylor, Matching of correlators in AdS(3) / CFT(2), JHEP 06 (2008) 010 [0709.1838].


# A Dual-Band 28/38-GHz Power Amplifier With Inter-Band Suppression in 22-nm
FD-SOI CMOS for Multi-Standard mm-Wave 5G Communications

Abbas Nasri, Alireza Yousefi, and Reza Nikandish∗ ∗ Corresponding authorA.
Nasri was with the School of Electrical and Electronic Engineering, University
College Dublin, Dublin D04 V1W8, Ireland (e-mail: abbas.nasri@ieee.org).A.
Yousefi is with the Syntiant Corporation, Irvine, CA 92618 USA.R. Nikandish is
with the School of Electrical and Electronic Engineering, University College
Dublin, Dublin D04 V1W8, Ireland (e-mail: reza.nikandish@ieee.org).

###### Abstract

In this article, we present a dual-band 28/38-GHz power amplifier (PA) with
inter-band suppression for millimeter-wave 5G communications. The dual-band
operation is achieved using a center-tapped transformer network with an extra
resonator which can provide optimum load impedance of the transistor in the
two bands and synthesize a short-circuit between the two bands. This feature
suppresses the PA signal emissions in the inter band, commonly allocated for
other applications. A design procedure is developed for the proposed matching
network including physical limits on the quality factor and the coupling
coefficient of the transformer. The PA is designed using a 22-nm fully-
depleted silicon-on-insulator (FD-SOI) CMOS process. The transistor stacking
and a four-path transformer parallel-series power combining techniques are
used to achieve high output power using the low-voltage process. The PA
achieves simulated performance of 22.6/22.0 dBm saturated output power,
19.8/20.0 dBm output power at 1-dB gain compression, and 33/32 % maximum
power-added efficiency (PAE) at 28/38 GHz. The inter-band suppression is 6 dB
at 33 GHz.

###### Index Terms:

CMOS, dual-band, fifth generation (5G), fully-depleted silicon-on-insulator
(FD-SOI), millimeter-wave, multi-band, power amplifier (PA), transformer.

##  I Introduction

Millimeter-Wave (mm-wave) frequency bands are of paramount importance in the
fifth generation (5G) and the future sixth generation (6G) networks where the
broad spectrum availability can open up opportunities for new high-capacity
communication applications [1, 2]. The development of innovative system and
circuit architectures is essential to leverage exciting potentials of the mm-
wave spectrum. A number of mm-wave bands are allocated for mobile
communications, e.g., 28 GHz, 38 GHz, 60 GHz, 140 GHz, and several radio
transceivers [3, 4, 5, 6, 7] and circuit components [8, 10, 14, 11, 12, 15,
13, 9] operating in these bands are presented. There is an increasing quest
for the concurrent multi-band circuits to develop universal and multi-standard
products in smaller chip area and with lower cost [16, 18, 17, 20, 22, 19, 21,
6, 23, 25, 24, 26, 27].

There are three main approaches to realize the multi-band mm-wave circuits. In
the most straightforward design method, multiple independent circuits each
operating in one of the frequency band are used to realize the multi-band
circuit. The operating band is selected by switches integrated in the
circuits. This approach offers higher reliability and robustness and, as a
result, is commonly used in commercial products. However, this leads to larger
chip area, higher fabrication cost, and lower performance due to high
insertion loss and low isolation of the switches in mm-wave bands.

Another approach to multi-band operation is through broadband circuits
covering multiple bands [25, 24, 26, 27]. Examples include a 29–57-GHz class-
AB power amplifier (PA) using a fourth-order matching network in 28-nm CMOS
[24], a 0.4–31.6 GHz distributed PA in 22-nm fully-depleted silicon-on-
insulator (FD-SOI) CMOS [25], and a frequency reconfigurable dual-path
40–65-GHz PA in 130-nm SiGe BiCMOS [26]. This approach offers a lower
sensitivity to modeling inaccuracies and process variations, but suffers from
two major issues. First, a broadband circuit usually requires complicated
impedance matching networks which their high insertion loss due to the low
quality factor of on-chip passive components can degrade the circuit
performance. Second, the broadband PAs can transmit spurious signals in
undesired communication bands, i.e., out-of-band emissions, while the
broadband low-noise amplifiers (LNAs) can receive blocker signals and noise
present in unintended bands. It is, therefore, essential to use additional
filtering with such broadband circuits which leads to extra loss and higher
system cost.

In the third approach, the multi-band circuits are realized using dedicated
multi-resonance circuit architectures [19, 23, 18, 17, 20, 16, 21]. The design
of these multi-band circuits requires special attention to the circuit
functionality to develop impedance matching networks which can provide the
required conditions at multiple frequencies. This approach has received
increased interests recently, where a number of developments include a dual-
band 28/38-GHz PA in 250-nm SiGe BiCMOS [17], a tri-band 28/37/39-GHz Doherty
PA with reconfigurable matching networks in 130-nm SiGe BiCMOS [18], a dual-
band 28/38-GHz PA in 22-nm FD-SOI CMOS [20], and a dual-band 27/33-GHz
amplifier with transformer-feedback neutralization of the gate-drain
capacitance in 100-nm GaAs pHEMT [23]. The main challenge of this approach is
the development of multi-band circuits which can provide a good performance in
the presence of practical limitations of the process, e.g., low quality factor
of passive elements, parasitic capacitances, modeling inaccuracies, and
process variations.

In this article, we present a dual-band 28/38-GHz PA using a transformer-based
impedance matching network. The transformer network, which provides the
impedance matching and differential-to-single ended transformation, is center-
tapped with an extra resonator. This resonator enables the transformer network
to operate at two frequency bands and provide the inter-band suppression. This
attenuates the PA out-of-band emissions in the frequency bands allocated for
other communication applications.

The article is organized as follows. In Section II, we present the dual-band
matching network architecture, principles of operation, and a detailed
analysis of the impacts of circuit imperfections on its performance. In
Section III, we discuss the circuit design of a dual-band PA using the
developed transformer network realized in 22-nm FD-SOI CMOS process. The post-
layout simulation results of the PA are presented in Section IV and the
conclusions are discussed in Section V.

##  II Dual-Band Transformer Network

###  II-A Proposed Network

The proposed transformer network for dual-band impedance matching and inter-
band suppression is shown in Fig. 1(a). The secondary winding of the
transformer is center-tapped with a reactive resonator network. We will
discuss the selection criteria for this resonator circuit later. The source
impedance of the transformer network is modeled by the optimum load resistance
of the transistor $R_{opt}$ in parallel with a capacitance $C_{p}$ which
should be absorbed into the transformer network. The capacitance $C_{p}$
comprises the output-referred parasitic capacitance of the transistor and the
primary parasitic capacitance of the transformer. The load impedance of the
network includes the load resistance $R_{L}$ in parallel with a capacitance
$C_{s}$, which comprises secondary parasitic capacitance of the transformer
and parasitic capacitance of the output signal pad.

The transformer turn ratio is selected such that it transforms the load
impedance $R_{L}$ to the optimum resistance $R_{opt}$, while the transformer
inductances absorb the parasitic capacitances $C_{p}$ and $C_{s}$. These
conditions should be concurrently satisfied at the lower and upper
frequencies, $\omega_{L}$ and $\omega_{H}$, to realize a dual-band impedance
matching network

Figure 1: (a) Dual-band transformer network with center-tap resonator. (b)
Four possible three-element circuits as the resonator.

$Z_{in}(j\omega_{L})=Z_{in}(j\omega_{H})=R_{opt}.$ (1)

It should be noted that the resistive part of the input impedance can be set
to the optimum resistance by the proper selection of the transformer turns
ratio as

$n\approx\sqrt{\frac{R_{opt}}{R_{L}}}.$ (2)

However, it is challenging to satisfy the reactive part conditions. A
conventional transformer cannot absorb the parasitic capacitances in two
frequency bands. A double-tuned transformer network which can provide two
peaks in its frequency response is useful as interstage network of broadband
amplifiers and resonator of oscillators [24, 28, 29]. This network has
disadvantages of loss, imbalanced magnitudes in the two peaks, and tuning
complexity as the ratio of the two peak frequencies is mainly controlled by
the transformer coupling coefficient. Therefore, we propose the transformer
network of Fig. 1(a) with the center-tap impedance of $Z_{T}(j\omega)$ which
should provide three conditions:

  1. 1.

At the lower frequency $\omega_{L}$, the resonator should operate as an open-
circuit. The transformer’s primary and secondary inductances are
$L_{p}=L_{p1}+L_{p2}$ and $L_{s}=L_{s1}+L_{s2}$, and resonate with the
parasitic capacitances.

  2. 2.

At the higher frequency $\omega_{H}$, the resonator should be equivalent with
a short-circuit to effectively decrease the inductances of the transformer to
$L_{p1}$ and $L_{s1}$. This enables the transformer to absorb the parasitic
capacitances at the higher frequency.

  3. 3.

At the inter-band frequency $\omega_{SC}$, it should provide a proper
reactance which, along with the transformer, creates a short-circuit at output
port of the transformer. This realizes the expected inter-band suppression.

The discussed conditions can be summarized as follows.

$\begin{cases}Z_{T}(j\omega_{L})=\infty\ \\\ Z_{T}(j\omega_{H})=0\ \\\
Z_{out}(j\omega_{SC})=0\end{cases}$ (3)

The first two conditions are only dependent on the resonator network’s
elements, while the third condition is also related to the transformer’s
elements. We develop a design approach for the dual-band matching network.

In the dual-band transformer circuit of Fig. 1(a), several circuit structures
can be envisioned as the resonator network. We consider the four resonator
circuits shown in Fig. 1(b) as the possible realizable circuits using three
reactive elements. These circuits have a zero and a pole in their impedance
frequency response. The requirements of the resonator network given in (3)
indicate that the circuit should ideally have a pole at $\omega_{L}$ and a
zero at $\omega_{H}$. This condition suggests to use the network I or II with
$\omega_{z}>\omega_{p}$. However, these networks include two inductors which
can lead to higher loss and larger chip area compared to the networks III and
IV with only one inductor. We can use the network III or IV which their pole
is set at $\omega_{L}$ and the impedance at $\omega_{H}$ is practically small
enough such that it can be approximated by zero. This condition can be
considered as $|Z_{T}(j\omega_{H})|\ll R_{L}$ or, for the sake of future
derivations, as follows

$|Z_{T}(j\omega_{H})|={\delta}R_{L},$ (4)

where $\delta$ is a constant which can be set in the typical range of 0.01–0.1
depending on the required accuracy and circuit element values. For the
resonator network, we select the network III and derive the design criteria.

###  II-B Design Approach

The impedance of the resonator network III in Fig. 1(b) can be derived as

$Z_{T}(j\omega)=\frac{1-\omega^{2}L_{ts}(C_{ts}+C_{t1s})}{1-\omega^{2}L_{ts}C_{ts}}\frac{1}{j\omega
C_{ts1}}.$ (5)

The resonator network should have a pole at $\omega_{L}$, leading to

$\omega_{L}^{2}L_{ts}C_{ts}=1.$ (6)

The resonator network should meet the condition (4) at $\omega_{H}$ which
using (5) and (6) can be derived as

$\frac{\left(1+\frac{C_{ts1}}{C_{ts}}\right)\omega_{H}^{2}-\omega_{L}^{2}}{\omega_{H}^{2}-\omega_{L}^{2}}\frac{1}{\omega_{H}C_{ts1}R_{L}}=\delta.$
(7)

This can be solved for $C_{ts1}$ as follows

$C_{ts1}=\frac{C_{ts}}{{\delta}\omega_{H}C_{ts}R_{L}-\alpha},$ (8)

where $\alpha=\omega_{H}^{2}/(\omega_{H}^{2}-\omega_{L}^{2})$. This indicates
that the parameter $\delta$ should satisfy the condition
$\delta>\alpha/(\omega_{H}C_{ts}R_{L})$ and cannot be selected arbitrarily
small. Therefore, the minimum theoretical impedance of the resonator in (4) is
derived as

$|Z_{T}(j\omega_{H})|_{\rm
min}=\frac{\omega_{H}^{2}}{\omega_{H}^{2}-\omega_{L}^{2}}\frac{1}{\omega_{H}C_{ts}}.$
(9)

The short-circuit frequency of the transformer network should meet the
condition $\omega_{L}<\omega_{SC}<\omega_{H}$. We can consider it as the mid
logarithmic distance between the lower and higher frequencies, leading to

$\frac{\omega_{SC}}{\omega_{L}}=\frac{\omega_{H}}{\omega_{SC}}\Rightarrow\omega_{SC}=\sqrt{\omega_{L}\omega_{H}}.$
(10)

The impedance of the resonator network at the short-circuit frequency can be
derived using (5), (6), and (10) as

$Z_{T}(j\omega_{SC})=\frac{\left(1+\frac{C_{ts1}}{C_{ts}}\right)\omega_{H}-\omega_{L}}{\omega_{H}-\omega_{L}}\frac{1}{j\omega_{SC}C_{ts1}},$
(11)

which is equivalent with a capacitance given by

$C_{sc}=\frac{(\omega_{H}-\omega_{L})C_{ts1}}{\left(1+\frac{C_{ts1}}{C_{ts}}\right)\omega_{H}-\omega_{L}}.$
(12)

This capacitance can generate a resonance with the transformer to produce a
short-circuit under certain conditions. The output impedance of the
transformer network should be derived to apply the inter-band suppression
condition, $Z_{out}(j\omega_{SC})=0$. Using the circuit of Fig. 1 with the
resonator network III, assuming $L_{p1}=L_{p2}=\frac{1}{2}L_{p}$ and
$L_{s1}=L_{s2}=\frac{1}{2}L_{s}$, and setting $Z_{out}(j\omega_{SC})=0$, it
can be shown that the following condition should be satisfied

$Z_{T}(j\omega_{SC})+\frac{1}{4}j\omega_{SC}L_{s}=0.$ (13)

Using (11) and (12), this condition can be derived as

$\frac{1}{4}\omega_{SC}^{2}L_{s}C_{sc}=1.$ (14)

The resonator network can be designed using the conditions given by (6), (9),
and (14). There are extra degrees of freedom in this system of equations which
provide more design flexibility. The transformer network can be designed using
the approximate input-referred equivalent circuit of the transformer (assuming
$K_{m}\approx 1$) comprising a parallel RLC circuit with the elements

$L_{in}\approx L_{p}$ (15) $C_{in}\approx C_{p}+\frac{1}{n^{2}}C_{s}$ (16)
$R_{in}\approx n^{2}R_{L}.$ (17)

The network is designed such that $L_{in}$ and $C_{in}$ resonate at the lower
frequency $\omega_{L}$, while $R_{in}=R_{opt}$. The resonator network
effectively reduces the inductances to $L_{p1}$ and $L_{s1}$ at the higher
frequency $\omega_{H}$, enabling the network to resonate at both $\omega_{L}$
and $\omega_{H}$. An important advantage of this network is that the
transformer coupling coefficient is not a design parameter and the transformer
layout can be optimized for the maximum coupling. This is in contrary with the
conventional double-tuned transform network which usually should be designed
with a low coupling coefficient to achieve broadband response at the cost of a
lower gain [24, 28, 29].

We use the transducer power gain as the performance metric for the transformer
network such that the impact of losses can also be evaluated [30]. This is
defined as the ratio of the output power delivered to the load to the
available source power and can be derived as

$G_{T}(\omega)=\frac{4R_{opt}R_{L}|Z_{21}|^{2}}{|(Z_{11}+R_{opt})(Z_{22}+R_{L})-Z_{12}Z_{21}|^{2}},$
(18)

where $Z_{ij}$ denotes the impedance parameter of the transformer two-port
network.

In Fig. 2, $G_{T}(\omega)$ of the dual-band transformer network is shown. The
network is realized using ideal lossless elements and perfect transformer
coupling ($K_{m}=1$). The network is designed for the lower frequency of 28
GHz, the upper frequency of 38 GHz, and the short-circuit frequency of
$f_{SC}=\sqrt{f_{L}f_{H}}\approx$ 33 GHz.

Figure 2: Transfer function of the dual-band 28/38-GHz transformer network
realized using ideal elements.

In Fig. 3, the impact of changing the capacitance $C_{ts}$ on the frequency
response is shown. This indicates that the short-circuit frequency can be
controlled by using a digitally controlled or switched $C_{ts}$. The network
has no notch for $C_{ts}=0$, while the notch is shifted toward the lower
frequencies by increasing $C_{ts}$.

Figure 3:  Transfer function of the dual-band transformer with swept
${C_{ts}}$. The notch frequency can be controlled by ${C_{ts}}$.

###  II-C Impact of Transformer Loss

The transformer loss arises from the limited quality factor of inductors, due
to metal and substrate losses, and the imperfect transformer coupling. It is
assumed that the primary and secondary inductors of the transformer have the
quality factor of $Q_{\rm XFMR}$. This can be modeled by resistances
$r_{p}=\omega L_{p}/Q_{\rm XFMR}$ and $r_{s}=\omega L_{s}/Q_{\rm XFMR}$ in
series with $L_{p}$ and $L_{s}$, respectively. For the inductor of the
resonator network with a quality factor of $Q_{T}$, the loss can be similarly
modeled as a resistance $r_{ts}=\omega L_{ts}/Q_{T}$ in series with the
inductor $L_{ts}$.

In Fig. 4, the impact of transformer quality factor on the transfer function
is illustrated. The insertion loss in the two pass bands increases by lowering
$Q_{\rm XFMR}$. For high $Q_{\rm XFMR}$, the insertion loss is limited by the
resonator network quality factor $Q_{T}$ which is assumed to be 30 in this
simulation. A transformer coupling coefficient of 1 is used to focus on the
effects of the transformer quality factor. As will be discussed in Section
III, the transformer physical structure is an hexagonal spiral to comply with
design rules of the process. This limits its quality factor to about 20–25.

In Fig. 5, the transfer function versus frequency is shown for different
transformer coupling coefficients $K_{m}$. The insertion loss increase when
$K_{m}$ is reduced from 0.8 to 0.5. The impact is more significant in the
upper band.

Figure 4: Transfer function of the dual-band transformer for different values
of the transformer quality factor (assuming $K_{m}=1$ and $Q_{T}=30$). Figure
5: Transfer function of the dual-band transformer with swept coupling
coefficient $K_{m}$ (assuming $Q_{T}=30$).

###  II-D Impact of Resonator Loss

The resonator loss is mainly caused by the limited quality factor of the
inductor $L_{ts}$. This is modeled by a resistance $r_{ts}=\omega
L_{ts}/Q_{T}$ in series with the inductor $L_{ts}$ in the network III of Fig.
1(b). The effects of $Q_{T}$ on the transfer function of the transformer
network are evaluated in Fig. 6. The resonator quality factor has significant
impact on the suppression at the notch frequency. For example, to achieve 10
dB suppression, the quality factor of the resonator inductor should be at
least 40 which is difficult to reach in this low-resistivity substrate
process. Fortunately, the resonator inductor can be implemented as a straight
transmission line, unlike the hexagonal spiral shape transformer, which
features a higher quality factor compared to the transformer.

Figure 6:  Transfer function of the dual-band transformer network for
different values of the inductor quality factor $Q_{T}$ in the resonator
network.

##  III Power Amplifier Circuit Design

The dual-band PA architecture and circuit details are shown in Fig. 7. The PA
comprises four power cells (PA1–4) combined in the parallel-series
configuration. The power cells are matched to 50 $\Omega$ using the dual-band
transformer network center-tapped with the resonator ($\rm XFMR_{o}$). The
parallel combining of output signals from two transformers $\rm XFMR_{o}$ also
transforms the impedance level to 25 $\Omega$ which is then returned to 50
$\Omega$ through the series combining by the transformer $\rm XFMR_{out}$. The
input power is divided between the power cells using the transformers $\rm
XFMR_{i}$ and the transmission lines $\rm TL_{in}$. The power cells are
realized using differential double-stacked structure shown in Fig. 7. We
discuss details of the PA circuit design.

Figure 7: Schematic of the dual-band power amplifier.

###  III-A FD-SOI CMOS Process

The PA is implemented in the Global Foundries 22-nm fully-depleted silicon-on-
insulator (FD-SOI) CMOS process (22FDX). The process structure is shown in
Fig. 8 which includes one Aluminum top thick metal layer, 2 thick and 7 thin
metal Copper layers. The substrate has a low resistivity (7 $\rm\Omega.cm$)
unlike the conventional SOI processes. The transformers in the PA circuit are
implemented on the two top copper layers (IA and OI) which have a lower
resistivity compared to the top Aluminum layer (LB). The circuit capacitors
are realized as the metal-oxide-metal (MOM) capacitors using stacked thin
metal layers (M1-2 and C1-5).

The process offers multiple types of transistors with different threshold and
breakdown voltages. Super-low threshold voltage (SLVT) transistors are used in
this design to benefit from their high transconductance $g_{m}$, unity current
gain frequency $f_{T}$, and unity unilateral power gain frequency $f_{\rm
max}$. The SLVT NMOS transistor features 0.25 V threshold voltage, 0.8 V
nominal supply voltage, $f_{T}$ of 350 GHz, and $f_{max}$ of 370 GHz. The
process also provides body bias possibility to adjust threshold voltage of
transistors which has not been used in this design.

Figure 8: The 22-nm FD-SOI CMOS process structure.

###  III-B Stacked Power Cell

The circuit schematic of the power cells is shown in Fig. 7. The low breakdown
voltage of the thin-oxide transistors limits their supply voltage and output
power capability. Therefore, two transistors are stacked to increase the
maximum supply voltage from nominal 0.8 V to 1.6 V. In the double-stacked
amplifier, the gate node of the top transistors $M_{3,4}$ is biased through a
large resistor $R_{1}$ operating roughly as open-circuit for the RF signal,
unlike the Cascode amplifier in which the gate of top transistors is RF
grounded. An accurately designed capacitor $C_{1}$ is included in the gate of
top transistor to control the gate-source voltage swing of this device. This
results in higher output power and efficiency in the stacked amplifier. The
input transistors $M_{1,2}$ are biased at the class-AB to improve their gain
and power-added efficiency (PAE).

The power cell is realized as a differential amplifier, shown in Fig. 9(a),
where the cross-connected neutralization capacitors $C_{neut}$ are used to
cancel the gate-drain capacitors and achieve unconditional stability.
Furthermore, an inductor $L_{s}$ is placed in the common node of the input
transistors pair to improve the common-mode stability. This inductor has no
effect in the differential mode, as shown in Fig. 9(b), while it reduces gain
of the transistors in the common mode, as can be inferred from Fig. 9(c).
Layout of the power cell is shown in Fig. 10.

Figure 9: (a) Power cell differential amplifier with stability network, (b)
differential-mode circuit, (c) common-mode circuit. Figure 10: Layout of the
power cell.

###  III-C Size of Transistors

The gate length of all transistors is set at the minimum length of the
process, 20 nm. to achieve the best RF performance. The width of transistors
is selected based on the output power (14 dBm at 1-dB gain compression) and
optimum load resistance (close to 50 $\Omega$) requirements for each power
cell. The width is determined using the load-pull and source-pull simulations
performed on the power cell. The load-pull simulation results at the two
frequencies 28 GHz and 38 GHz are shown in Fig. 11. The optimum output power
is 14.8/14.4 dBm at 28/38 GHz.

(a)

(b)

Figure 11: The load-pull simulation result for output power at 1-dB gain
compression of the power cell: (a) 28 GHz, (b) 38 GHz. The optimum output
power is 14.8/14.4 dBm at 28/38 GHz.

###  III-D Output Power Combiner

The output power combiner which also transforms the load resistance to the
optimum load resistance of transistors is realized using four dual-band
transformer networks and a transformer as output power combiner. The real part
of the optimum load impedance of the power cells is about 50 $\Omega$. Thus,
the dual-band transformer $\rm XFMR_{o}$ has a 1:1 turn ratio. The parallel
combining of two power cells transforms the resistance level to 25 $\Omega$
which is then converted to 50 $\Omega$ using the series combining. This allows
the output transformer $\rm XFMR_{out}$ also be realized as a 1:1 transformer.
This realizes a parallel-series power combining which can ideally convert the
14 dBm output power of the power cell to 20 dBm output power. The output
network layout is shown in Fig. 13.

The power combiner performance is simulated using the EMX planar 3D
electromagnetic (EM) simulator. The extracted inductance and quality factor of
the dual-band transformers are shown in Fig. 12. The inductance is about 200
pH and the quality factor is 22–25. Simulated scattering parameters of the
dual-band transformer network are shown in Fig. 14. The insertion loss is
about 1 dB, the inter-band suppression is 6 dB, and the input/output return
losses are higher than 15 dB.

(a)

(b)

Figure 12: Extracted (a) inductance and (b) quality factor of the output
transformer $\rm XFMR_{o}$. Figure 13: Layout of the output power combiner and
impedance matching network.

(a)

(b)

Figure 14:  Simulated scattering parameters of the dual-band transformer with
the center-tap resonator network: (a) $S_{11}$ and $S_{22}$, (b) $S_{21}$.

###  III-E Input Power Splitter

The input power splitter also serving as the input impedance matching network
is shown in Fig. 15. This network comprises two single-to-differential
transformers and two transmission lines. The input impedance of the
transformers should be 100 $\Omega$ to provide an input impedance 50 $\Omega$
with the two-way power splitter. The input transformers match the 100 $\Omega$
resistance to the optimum source impedance of the transistors.

The input network is designed with a broad bandwidth that covers the lower and
upper bands. Scattering parameters of the input network are shown in Fig. 16,
which indicates an insertion loss lower than 1.2 dB in the target bands.

Figure 15: Layout of the input power splitter and impedance matching network.
Figure 16: Simulated scattering parameters of the input network. Insertion
loss is lower than 1.2 dB in the target band.

##  IV Power Amplifier Simulation Results

The layout of the PA is shown in Fig. 17, where the chip measures 0.5 mm
$\times$ 0.9 mm. The PA is biased at the supply voltage of $\rm
V_{DD}=1.6\,V$, the gate bias voltage of $\rm V_{G}=0.35\,V$ for the input
transistors, and $\rm V_{stack}=1.2\,V$ for the stack devices. The PA consumes
$\rm 80\,mA$ drain current in the quiescent condition. The PA small-signal and
large-signal simulations are performed using the 22-nm FD-SOI process design
kit (PDK) for active devices and full electromagnetic simulations of the
passive devices.

Figure 17: Layout of the PA in 22-nm FD-SOI process.

###  IV-A Small-Signal Simulations

Simulated scattering parameters of the PA are shown in Fig. 18. The small-
signal gain is 16.0 dB at 28 GHz and 15.5 dB at 38 GHz. The inter-band
suppression is around 6 dB which has been limited by low quality factors of
the transformers and the resonator. It is expected to achieve higher inter-
band suppression using a process with higher passives quality factor. The
input return loss is higher than 10 dB is both bands which indicate a good
input impedance matching. The output return loss is 13 dB at the lower and 8
dB at the higher band. Furthermore, the stability K factor (Rollet criterion),
shown in Fig. 19, indicates that the PA is unconditionally stable.

(a)

(b)

Figure 18: Simulated S-parameter for proposed dual-band output matching
transformer: (a) $S_{21}$, (b) $S_{11}$ and $S_{22}$. Figure 19: Simulated
stability K factor of the PA.

###  IV-B Large-Signal Simulations

Simulations results for the large-signal performance of the PA are shown in
Fig. 20. The saturated output power $\rm P_{sat}$ is 22.6 dBm at 28 GHz and
22.0 dBm at 38 GHz. The output-input characteristics indicate that the output
power is saturated at around 4-dB gain compression. The output power at 1-dB
gain compression $\rm P_{1dB}$ is 19.8/20.0 dBm at 28/38 GHz, about 2–3 dB
lower than the saturated power. Furthermore, the maximum PAE reads 33/32% at
28/38 GHz.

(a)

(b)

Figure 20: Simulated saturated output power, PAE, and gain versus input power
at: (a) 28 GHz (b) 38 GHz.

###  IV-C Performance Comparison

A performance comparison of the designed dual-band PA with state-of-the-art
mm-wave CMOS PAs operating in similar frequency bands is presented in Table I.
We have indicated the references which are based on simulation results [27,
31, 32] with an asterisk symbol to make a fair comparison. The saturated
output power $\rm P_{sat}$ of the PA in this work is over 3 dB higher than the
PAs based on simulations results, while $\rm PAE_{max}$ is also higher than
these works. Furthermore, $\rm P_{sat}$ is higher than that of the fabricated
PAs except for [10] with single-band operation, 37% higher supply voltage, and
6$\times$ chip area. $\rm PAE_{max}$ is competitive with state-of-the-art even
though it is challenging to achieve high efficiency for such a high power
level. Furthermore, the proposed inter-band suppression technique for the
dual-band PA has not been considered in the literature.

TABLE I: Performance comparison with mm-wave CMOS PAs. | f (GHz) |  $\mathrm{P_{1dB}}$ (dBm) |  $\mathrm{P_{sat}}$ (dBm) |  $\mathrm{PAE_{max}}$ (%) | Gain (dB) |  $\mathrm{V_{DD}}$ (V) |  Area (mm2) | Process
---|---|---|---|---|---|---|---|---
[17] | 28/38 | 17.8/15.8 | 18.8/17.0 | 25.0/17.5 | 25.7/21.4 | 3.3 | 0.94 | 250-nm BiCMOS
[16] | 26.5/37 | 18.7/18.6 | 20.2/19.1 | 33.6/32.0 | 13.0/10.7 | NA | 0.11 | 28-nm CMOS
[20] | 28/38 | 15.7/15.2 | 19.4/19.2 | 26.4/25.1 | 31.7/29.2 | 2.7 | 0.26 | 22-nm FD-SOI CMOS
[27]∗ | 30/40 | 13.7/13.6 | 17.3/16.4 | 29.2/22.7 | 14.9/13.3 | NA | NA | 22-nm FD-SOI CMOS
[31]∗ | 24/28 | 14.0/15.4 | 17.8/17.3 | 32.0/29.2 | 26.5/22.1 | NA | NA | 22-nm FD-SOI CMOS
[32]∗ | 28 | 15.4 | 18.7 | 12.4 | 17.5 | 2.0 | 0.66 | 28-nm CMOS
[33] | 30 | 13.2 | 14.0 | 35.5 | 15.7 | 1.0 |  0.16∗∗ | 28-nm CMOS
[12] | 39 | 21.5 | 26.0 | 27.7 | 38 | 2.2 | 2.96 | 28-nm CMOS
[10] | 39 | 16.6 | 18.3 | 29.1 | 12.5 | 1.8 | 0.48 | 16-nm FinFET CMOS
[5] | 28 | 18.2 | 19.4 | 34.3 | 32 | 2.4 | 0.10 | 22-nm FD-SOI CMOS
[34] | 28 | 20.7 | 21.5 | 26.0 | 20.4 | 1.8 | 0.5 | 28-nm CMOS
[35] | 28 | 17.4 | 18.0 | 26.7 | 19.0 | 1.0 | 0.16 | 65-nm CMOS
This Work∗ | 28/38 | 19.8/20.0 | 22.6/22.0 | 33.0/32.0 | 16.0/15.5 | 1.6 | 0.45 | 22-nm FD-SOI CMOS
∗Simulation results ∗∗Active area |  |  |  |  |  |

##  V Conclusion

We presented a dual-band transformer network with inter-band suppression which
is enabled by a resonator in the center-tap of the transformer. A theory was
developed to provide insights into the circuit operation, design guidelines,
and impacts of imperfect on-chip circuit elements. A proof-of-concept
28/38-GHz power amplifier (PA) was designed using the proposed network in
22-nm Fully-Depleted Silicon-on-Insulator (FD-SOI) CMOS process. The PA
achieved state-of-the-art output power through multiple techniques including
the transistor stacking, sizing the power cells for 50-$\Omega$ optimum
resistance, and four-path parallel-series transformer combining. The inter-
band suppression is 6 dB at 33 GHz which is limited by quality factor of
inductors and transformers. It can be improved, in a future research, through
applying the inter-band suppression in the input transformers, using a
modified resonator circuit, and using a process with higher quality factor of
inductors.

## Acknowledgment

The authors would like to thank Global Foundries for the PDK support.

## References

  * [1] T. S. Rappaport et al., ”Millimeter Wave Mobile Communications for 5G Cellular: It Will Work!,” in IEEE Access, vol. 1, pp. 335-349, 2013, doi: 10.1109/ACCESS.2013.2260813.
  * [2] T. S. Rappaport et al., ”Wireless Communications and Applications Above 100 GHz: Opportunities and Challenges for 6G and Beyond,” in IEEE Access, vol. 7, pp. 78729-78757, 2019, doi: 10.1109/ACCESS.2019.2921522.
  * [3] J. Pang et al., ”A 28-GHz CMOS Phased-Array Transceiver Based on LO Phase-Shifting Architecture With Gain Invariant Phase Tuning for 5G New Radio,” in IEEE Journal of Solid-State Circuits, vol. 54, no. 5, pp. 1228-1242, May 2019, doi: 10.1109/JSSC.2019.2899734.
  * [4] M. Lokhandwala, L. Gao and G. M. Rebeiz, ”A High-Power 24–40-GHz Transmit–Receive Front End for Phased Arrays in 45-nm CMOS SOI,” in IEEE Transactions on Microwave Theory and Techniques, vol. 68, no. 11, pp. 4775-4786, Nov. 2020, doi: 10.1109/TMTT.2020.2998011.
  * [5] X. Tang et al., ”Design and Analysis of a 28 GHz T/R Front-End Module in 22-nm FD-SOI CMOS Technology,” in IEEE Transactions on Microwave Theory and Techniques, vol. 69, no. 6, pp. 2841-2853, June 2021, doi: 10.1109/TMTT.2021.3059891.
  * [6] S. Wang and G. M. Rebeiz, ”Dual-Band 28-and 39-GHz Phased Arrays for Multistandard 5G Applications,” in IEEE Transactions on Microwave Theory and Techniques, 2022, doi: 10.1109/TMTT.2022.3220272.
  * [7] A. Karakuzulu, W. A. Ahmad, D. Kissinger and A. Malignaggi, ”A Four-Channel Bidirectional D-Band Phased-Array Transceiver for 200 Gb/s 6G Wireless Communications in a 130-nm BiCMOS Technology,” in IEEE Journal of Solid-State Circuits, vol. 58, no. 5, pp. 1310-1322, May 2023, doi: 10.1109/JSSC.2022.3232948.
  * [8] V. Camarchia, R. Quaglia, A. Piacibello, D. P. Nguyen, H. Wang and A. Pham, ”A Review of Technologies and Design Techniques of Millimeter-Wave Power Amplifiers,” in IEEE Transactions on Microwave Theory and Techniques, vol. 68, no. 7, pp. 2957-2983, July 2020, doi: 10.1109/TMTT.2020.2989792.
  * [9] S. Callender, S. Pellerano and C. Hull, ”An E-Band Power Amplifier With 26.3% PAE and 24-GHz Bandwidth in 22-nm FinFET CMOS,” in IEEE Journal of Solid-State Circuits, vol. 54, no. 5, pp. 1266-1273, May 2019, doi: 10.1109/JSSC.2019.2899493.
  * [10] S. Daneshgar et al., ”High-Power Generation for mm-Wave 5G Power Amplifiers in Deep Submicrometer Planar and FinFET Bulk CMOS,” in IEEE Transactions on Microwave Theory and Techniques, vol. 68, no. 6, pp. 2041-2056, June 2020, doi: 10.1109/TMTT.2020.2990638.
  * [11] S. Y. Mortazavi and K. Koh, ”Integrated Inverse Class-F Silicon Power Amplifiers for High Power Efficiency at Microwave and mm-Wave,” in IEEE Journal of Solid-State Circuits, vol. 51, no. 10, pp. 2420-2434, Oct. 2016, doi: 10.1109/JSSC.2016.2588470.
  * [12] K. Dasgupta, S. Daneshgar, C. Thakkar, J. Jaussi and B. Casper, ”A 26 dBm 39 GHz Power Amplifier with 26.6% PAE for 5G Applications in 28nm bulk CMOS,” 2019 IEEE Radio Frequency Integrated Circuits Symposium (RFIC), 2019, pp. 235-238, doi: 10.1109/RFIC.2019.8701834.
  * [13] F. Wang and H. Wang, ”A Broadband Linear Ultra-Compact mm-Wave Power Amplifier With Distributed-Balun Output Network: Analysis and Design,” in IEEE Journal of Solid-State Circuits, vol. 56, no. 8, pp. 2308-2323, Aug. 2021, doi: 10.1109/JSSC.2021.3078485.
  * [14] A. Sarkar, F. Aryanfar and B. A. Floyd, ”A 28-GHz SiGe BiCMOS PA With 32% Efficiency and 23-dBm Output Power,” in IEEE Journal of Solid-State Circuits, vol. 52, no. 6, pp. 1680-1686, June 2017, doi: 10.1109/JSSC.2017.2686585.
  * [15] M. Pashaeifar, L. C. N. de Vreede and M. S. Alavi, ”A Millimeter-Wave CMOS Series-Doherty Power Amplifier With Post-Silicon Inter-Stage Passive Validation,” in IEEE Journal of Solid-State Circuits, vol. 57, no. 10, pp. 2999-3013, Oct. 2022, doi: 10.1109/JSSC.2022.3175685.
  * [16] J. Lee, J. -S. Paek and S. Hong, ”Millimeter-Wave Frequency Reconfigurable Dual-Band CMOS Power Amplifier for 5G Communication Radios,” in IEEE Transactions on Microwave Theory and Techniques, vol. 70, no. 1, pp. 801-812, Jan. 2022, doi: 10.1109/TMTT.2021.3122533.
  * [17] K. Ding, D. M. W. Leenaerts and H. Gao, ”A 28/38 GHz Dual-Band Power Amplifier for 5G Communication,” in IEEE Transactions on Microwave Theory and Techniques, doi: 10.1109/TMTT.2021.3134641.
  * [18] S. Hu, F. Wang and H. Wang, ”A 28-/37-/39-GHz Linear Doherty Power Amplifier in Silicon for 5G Applications,” in IEEE Journal of Solid-State Circuits, vol. 54, no. 6, pp. 1586-1599, June 2019, doi: 10.1109/JSSC.2019.2902307.
  * [19] H. Hashemi and A. Hajimiri, ”Concurrent multiband low-noise amplifiers-theory, design, and applications,” in IEEE Transactions on Microwave Theory and Techniques, vol. 50, no. 1, pp. 288-301, Jan. 2002, doi: 10.1109/22.981282.
  * [20] X. Xu et al., ”A 28 GHz and 38 GHz High-Gain Dual-Band Power Amplifier for 5G Wireless Systems in 22 nm FD-SOI CMOS,” 2020 50th European Microwave Conference (EuMC), 2021, pp. 174-177, doi: 10.23919/EuMC48046.2021.9338181.
  * [21] G. Nikandish, E. Babakrpur and A. Medi, ”A Harmonic Termination Technique for Single- and Multi-Band High-Efficiency Class-F MMIC Power Amplifiers,” in IEEE Transactions on Microwave Theory and Techniques, vol. 62, no. 5, pp. 1212-1220, May 2014, doi: 10.1109/TMTT.2014.2315591.
  * [22] D. d. Rio et al., ”Multi-Gbps Tri-Band 28/38/60-GHz CMOS Transmitter for Millimeter-Wave Radio System-on-Chip,” 2019 IEEE MTT-S International Microwave Symposium (IMS), 2019, pp. 488-491, doi: 10.1109/MWSYM.2019.8701020.
  * [23] G. Nikandish and A. Medi, ”Transformer-Feedback Dual-Band Neutralization Technique,” in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 64, no. 5, pp. 495-499, May 2017, doi: 10.1109/TCSII.2016.2581919.
  * [24] M. Vigilante and P. Reynaert, ”A Wideband Class-AB Power Amplifier With 29–57-GHz AM–PM Compensation in 0.9-V 28-nm Bulk CMOS,” in IEEE Journal of Solid-State Circuits, vol. 53, no. 5, pp. 1288-1301, May 2018, doi: 10.1109/JSSC.2017.2778275.
  * [25] U. Celik and P. Reynaert, ”Robust, Efficient Distributed Power Amplifier Achieving 96 Gbit/s With 10 dBm Average Output Power and 3.7% PAE in 22-nm FD-SOI,” in IEEE Journal of Solid-State Circuits, vol. 56, no. 2, pp. 382-391, Feb. 2021, doi: 10.1109/JSSC.2020.3020229.
  * [26] C. R. Chappidi and K. Sengupta, ”Frequency Reconfigurable mm-Wave Power Amplifier With Active Impedance Synthesis in an Asymmetrical Non-Isolated Combiner: Analysis and Design,” in IEEE Journal of Solid-State Circuits, vol. 52, no. 8, pp. 1990-2008, Aug. 2017, doi: 10.1109/JSSC.2017.2686843.
  * [27] J. C. Mayeda, J. Lopez and D. Y. C. Lie, ”Highly-Efficient Broadband Medium Power Amplifier Design in 22nm CMOS FD-SOI for mm-Wave 5G,” 2020 IEEE Texas Symposium on Wireless and Microwave Circuits and Systems (WMCS), Waco, TX, USA, 2020, pp. 1-4, doi: 10.1109/WMCS49442.2020.9172413.
  * [28] A. Mazzanti and A. Bevilacqua, ”Second-Order Equivalent Circuits for the Design of Doubly-Tuned Transformer Matching Networks,” in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 65, no. 12, pp. 4157-4168, Dec. 2018, doi: 10.1109/TCSI.2018.2846029.
  * [29] A. Bevilacqua and A. Mazzanti, ”Doubly-Tuned Transformer Networks: A Tutorial,” in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 68, no. 2, pp. 550-555, Feb. 2021, doi: 10.1109/TCSII.2020.3046021.
  * [30] G. R. Nikandish, A. Nasri, A. Yousefi, A. Zhu and R. B. Staszewski, ”A Broadband Fully Integrated Power Amplifier Using Waveform Shaping Multi-Resonance Harmonic Matching Network,” in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 69, no. 1, pp. 2-15, Jan. 2022, doi: 10.1109/TCSI.2021.3095708.
  * [31] J. C. Mayeda, J. Tsay, D. Y. C. Lie and J. Lopez, ”Effective AM-PM Cancellation with Body Bias for 5G CMOS Power Amplifier Design in 22nm FD-SOI,” 2019 IEEE International Symposium on Circuits and Systems (ISCAS), Sapporo, Japan, 2019, pp. 1-4, doi: 10.1109/ISCAS.2019.8702159.
  * [32] B. Moret, V. Knopik and E. Kerherve, ”A 28GHz self-contained power amplifier for 5G applications in 28nm FD-SOI CMOS,” 2017 IEEE 8th Latin American Symposium on Circuits & Systems (LASCAS), Bariloche, Argentina, 2017, pp. 1-4, doi: 10.1109/LASCAS.2017.7948059.
  * [33] S. Shakib, H. Park, J. Dunworth, V. Aparin and K. Entesari, ”A Highly Efficient and Linear Power Amplifier for 28-GHz 5G Phased Array Radios in 28-nm CMOS,” in IEEE Journal of Solid-State Circuits, vol. 51, no. 12, pp. 3020-3036, Dec. 2016, doi: 10.1109/JSSC.2016.2606584.
  * [34] D. Manente, F. Padovan, D. Seebacher, M. Bassi and A. Bevilacqua, ”A 28-GHz Stacked Power Amplifier with 20.7-dBm Output P1dB in 28-nm Bulk CMOS,” in IEEE Solid-State Circuits Letters, vol. 3, pp. 170-173, 2020, doi: 10.1109/LSSC.2020.3009973.
  * [35] C. Yu, J. Feng and D. Zhao, ”A 28-GHz CMOS Broadband Single-Path Power Amplifier with 17.4-dBm P1dB for 5G Phased-Array,” ESSCIRC 2018 - IEEE 44th European Solid State Circuits Conference (ESSCIRC), 2018, pp. 38-41, doi: 10.1109/ESSCIRC.2018.8494246.


# Lattice Fluid Dynamics: Thirty-five years Down the Road

Sauro Succi  Sauro Succi (1,2,3)
(1) Center for Life Nano-Neuro Science at La Sapienza,
Italian Institute of Technology, Viale Regina Elena 295, 00161, Roma, Italy,
(2) Physics Department, Harvard University,
Oxford Str 17, 02138, Cambridge, USA,
(3) Mechanical Engineering Department,
University College London,Gower Street, London WC1E 6BT (UK)

# Lattice Fluid Dynamics: Thirty-five Years Down the Road

Sauro Succi  Sauro Succi (1,2,3)
(1) Center for Life Nano-Neuro Science at La Sapienza,
Italian Institute of Technology, Viale Regina Elena 295, 00161, Roma, Italy,
(2) Physics Department, Harvard University,
Oxford Str 17, 02138, Cambridge, USA,
(3) Mechanical Engineering Department,
University College London,Gower Street, London WC1E 6BT (UK)

###### Abstract

We comment on the role and impact of lattice fluid dynamics on the general
landscape of computational fluid dynamics. Starting from the historical
development of Lattice Gas Cellular Automata, we move on to a cursory
appraisal of the main applications of the Lattice Boltzmann method and
challenges ahead.

##  1 Introduction

Computational Fluid Dynamics (CFD) has been around for as long as computers
exist, starting with von Neumann’s program to simulate the weather on the
ENIAC machine (1950’s) and even earlier, with 1922 Richardson’s description of
”human” computers computing the weather by hand (he estimated that 64,000
human calculators, each calculating at a speed of 0.01 Flops/s, would be
sufficient to predict the weather in real time
(https://www.altair.com/c2r/ws2017/weather-forecasting-gets-real-thanks-high-
performance-computing). Leaving aside human calculators, electronic ones have
made a spectacular ride till current days, from the few hundred Flops of ENIAC
to the current few hundred Petaflops of the Top 1 IBM-Nvidia Summit computer.
Sixteen orders of magnitude in 70 years, close to a sustained Moore’s law rate
(doubling every 1.5 years)! Amazingly, CFD has been consistently on the
forefront of such spectacular ride and continues to do so to this day. Finite
Differences served as the tool of the trade in 60-70’s, later to be replaced
by finite-volumes and, to a lesser extent, by Finite-Elements as well on
account of the need of handling complex geometries, a must for most
engineering problems. On a more fundamental side, spectral methods, have taken
the lions’s share of CFD for the numerical study of homogeneous fluid
turbulence, and keep holding their pole position to this day. All of these
methods respond to the same strategy: Discretization of the continuum
equations of fluid mechanics. That is, they start from the Navier-Stokes
equations and represent them on a suitable grid in real or reciprocal space,
sometimes both. This is the mainstream numerics for basically any classical
continuum field equations, but in the mid 80’s a qualitatively different
approach was put forward by a small group of inspired researchers in France
and the USA.

##  2 Lattice Gas Cellular Automata

In the mid 80’s a very different approach was proposed: instead of discretise
the continuum fluid equations, one rather devises a fictitious particle
dynamics which would recover the continuum picture in the infrared limit:
large scales as compared to the particle mean free path [1, 2]. At first, this
sounds like a self-inflicted pain, for it is known that molecular dynamics can
barely reach micrometric regions in space and milliseconds in time, even on
the largest supercomputers. The key, though, is that particle are not real
molecules but ”effective” ones, i.e. they represent a large collection of real
molecules, where large means basically the number of molecules in a single
computational cell.

It turns out that if the dynamics of these fictitious particles can be
designed in such a way as to preserve the basic mass-momentum-energy
conservation laws, relinquishing all the hydrodynamically irrelevant molecular
details, this fictitious particle dynamics (FPD) would prove competitive
against the mainstream discretization approach.

A spectacular example in point is the famous Lattice Gas Cellular Automata
(LGCA) first proposed by Frisch, Hasslacher and Pomeau in 1986 [1] with
substantial early contributions by Stephen Wolfram as well [2]. In equations:

$n_{i}(x+c_{i},t+1)-n_{i}(x,t)=C_{i}[n]$ (1)

where $n_{i}={0,1}$ is a boolean occupation number indicating the
absence(presence) of a particle at site $x$ and time $t$ with discrete
velocity $c_{i}$ (vector indices relaxed for simplicity). The LHS of the above
equation represents the free-streaming of a boolean particle with velocity
$c_{i}$ from site $x$ at time $t$ to site $x_{i}=x+c_{i}$ at time $t+1$. The
streaming is synchronous, in that all sites $x_{i}$ still belong to the
uniform regular lattice. This step is exact, in that it implies an error-free
transfer of information from site $x$ to $x_{i}$, no loss in between. The RHS
codes for the local particle collisions at site $x$ and time $t$.

The beauty is that it does not have to conform to any detail of molecular
interactions, but only to the main conservation laws: mass-momentum-energy
(the latter not being part of the original formulation) as well as rotational
invariance.

Remarkably, the above prescriptions can be encoded in Boolean form too. To be
precise, the collision operator takes the values $C_{i}={-1,0,+1}$
corresponding to annihilation, no-action, generation of a particle with
velocity $c_{i}$ at site $x$ and time $t$. A typical collision $(i,j)\to(k,l)$
is then coded by a fourth-order boolean polynomial of the form:

$C_{ij\to kl}=n_{i}\cdot n_{j}\cdot\bar{n}_{k}\cdot\bar{n}_{l}$

where $\cdot$ is a logical AND and overbar denote logical NOT (turning 0 into
1 and viceversa). Note that the above term involves a four-body interaction
because boolean particles are fermions, hence the collision can take place
only on condition that the final states are empty.

The major appeal of LGCA rest with its fully Boolean nature, hence literally
round-off freedom, as well as their outstanding amenability to parallel
computing. The expectations were high, to the point that the Washington Post
maintained that the method should be ”kept off the Soviet hands” [3] (for the
record, back in 1986 the Berlin wall was still standing).

Despite the tantalizing promises, the LGCA did not make to the mainstream CFD,
for a series of conceptual and technological reasons, including statistical
noise, low collisional rates and exponential complexity of the collision rule
with the number of discrete states.

In hindsight, the main show-stopper was the latter. Indeed the complexity of
the collision operator, scales like $2^{b}$ for a LGCA with $b$ discrete
states per site ($b$ stands for ”bits”). It turns out that in three spatial
dimensions the minimum suitable hydrodynamic lattice is a four-dimensional
Face Centered HyperCube (FCHC), featuring $b=24$ and resulting in
$O(2^{24}\sim 16$ millions) boolean operations at each lattice site and every
time step. Too much to compete with ”standard” methods, requiring of the order
of hundreds of floating point operations for the same task. In other words,
LGCA proved unviable because of their exponential computational intensity.

Heroic efforts were put in place, mostly by the French school to mitigate the
problem, i.e. minimise the number of collisions for a given value of the
Reynolds number [4]. Even so, the result did not change the ultimate sentence
of computational unviability, so that practical interest in LGCA started to
flag down in the early 90’s.

It would be a gross mistake to identify this computational ”failure” with a
scientific flop. Quite on the contrary, LGCA was an overly productive idea, as
it opened the way to the possibility that fluids can be simulated using a
suitably stylized microscopic approach rather than by discretizing the
continuum equations of fluids.

Among others, this idea gave birth to the highly successful offspin known as
Lattice Boltzmann (LB) method [5, 6, 7].

Even though in hindsight it appears ”obvious” that LB could have been
generated by direct discretization of the continuum Boltzmann kinetic
equation, an observation that led a significant number of scientists to
dismiss the LGCA and the historical LB route, as needlessly complicated. This
is very easy, but only in hindsight: the fact remains that historically this
opportunity was realised and seized only thanks to the LGCA work. Till then,
discrete velocity models, which predated LGCA by two-three decades [8], were
never meant to be used for fluid-dynamic purposes!

##  3 Entry Lattice Boltzmann

Be as it may, historically LB developed in the wake of LGCA.

The basic idea of LB is to replace effective particles with corresponding
probability distribution functions (PDF), along the same line which takes
molecular dynamics to Boltzmann’s kinetic theory. On the lattice though the
task is littered with deadly catches, namely badly broken symmetries that
would prevent the recovery of the correct equations of fluids in the continuum
limit. The Lattice Boltzmann (LB) method made its earliest chronological
appearance in 1988 [9], its first computationally viable realization being
published just months later [10]. Ever since, LB has marked a tremendous
growth in methods and applications [11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25] over an amazingly broad spectrum of problems across
different regimes and scales of motion, literally from astrophysical jets[26]
all the way down to quark-gluon plasmas [27].

##  4 LB in a nutshell

The LB equation (LBE) reads as follows:

$f_{i}(\vec{x}+\vec{c}_{i},t+1)-f_{i}(\vec{x},t)=-\Omega_{ij}(f_{j}-f_{j}^{eq})(\vec{x};t)+S_{i}(\vec{x};t)$
(2)

where the lattice time step is made unit for simplicity. In the above,
$f_{i}(\vec{x},t)\equiv f(\vec{x},\vec{v}=\vec{c}_{i};t),\;i=0,b$ denotes the
probability of finding a particle at lattice site $\vec{x}$ and time $t$ with
a molecular velocity $\vec{v}=\vec{c}_{i}$. Here, $\vec{c}_{i}$ denotes a set
of discrete velocities, which must exhibit enough symmetry to obey the mass-
momentum (energy) conservation rules, along with rotational invariance
(isotropy).

Typical two and three dimensional lattices are shown in Fig. 1.

Figure 1: The workhorse lattices in two and three dimensions, with $9$ and
$19$ discrete velocities, respectively.

The left hand side of eq. (2) denotes free-streaming, while the right hand
side is the collision step, which consists of a short-range component, driving
the system towards a local equilibrium $f_{i}^{eq}$ and a soft-core source of
momentum $S_{i}(\vec{x};t)$. The latter is in fact more general and can
represent the source of any macroscopic quantity relevant to the physics in
point. This is a major asset of LB, one that permits to include complex
mesoscale physics often beyond the realm of continuum fluids, at a minimum
cost in terms of programming and computing overheads.

The local equilibrium corresponds to a finite-order truncation of a local
Maxwellian:

$f_{i}^{eq}=w_{i}\rho(1+\beta u_{i}+\frac{\beta^{2}}{2}q_{i})$ (3)

where $\beta=1/c_{s}^{2}$, $u_{i}=\vec{u}\cdot\vec{c}_{i}$ and
$q_{i}=u_{i}^{2}-u^{2}c_{s}^{2}$, $c_{s}$ and $\vec{u}$ being the lattice
sound speed and the local flow velocity, respectively. In the above $w_{i}$ is
a set of weights normalized to unity, the discrete analogue of the absolute
Maxwellian in continuum velocity space. The truncation is an unavoidable
consequence of lattice discreteness, which permits to recover Galilean
invariance only to a finite order in the Mach number $Ma=|\vec{u}|/c_{s}$. The
relevant hydrodynamic quantities are computed as simple linear and local
combinations of the discrete distributions, namely:

$\rho(\vec{x};t)=\sum_{i}f_{i}(\vec{x};t),\;\;\;\rho\vec{u}(\vec{x};t)=\sum_{i}\vec{c}_{i}f_{i}(\vec{x};t)$
(4)

where $\rho$ is the fluid density. Higher order moments deliver the fluid
pressure and stress tensor in the form of linear and local combinations of the
discrete populations, which proves very convenient for simulation purposes.
Formally, The LB eq. (2) is nothing but a set of finite-difference equations,
and yet one with great power inside. This power stems mainly from four basic
ingredients, namely: i) Exact free-streaming, ii) Local lattice equilibria,
iii) Tunable relaxation matrix, iv) Flexible external source,  Before moving
on to these items, we hasten to add that the eq. (2) can be shown to converge
to the (quasi-incompressible) Navier-Stokes equations in the usual limit of
small Knudsen numbers, $Kn=\lambda/L\ll 1$, i.e. small mean free path versus
the typical scale of variation of hydrodynamic quantities. This is also a
statement of weak departure from local equilibrium. Technically, this entails
a Taylor expansion in the lattice time step, as combined with a double
expansion in low Knudsen and Mach number. The tool of the trade is the
Chapman-Enskog asymptotics [28].

##  5 Mainstream LB applications

As mentioned above, at the time of this writing LB is a massive bibliographic
presence in fluid dynamics and most allied fields, particularly soft flowing
matter [6] and the physics/biology interface [29]. It would be utterly
impossible to cover this vast ground in any single paper, hence here we shall
just outline the main applications in very broad strokes.

###  5.1 Macroscopic flows

Historically, Lattice Boltzmann was developed as a computational alternative
to the discretization of the Navier-Stokes equations of continuum fluid
dynamics, the main target being high-Reynolds turbulent flows in complex
geometries [30]. The initial hopes were that LB would offer a better
resolution of the near-grid scales or even provide a natural subgrid model via
the extra degrees of freedom of the kinetic representation (then dubbed ”ghost
modes”) versus the hydrodynamic one [31].

Figure 2:  Two examples of fully-turbulent lattice Boltzmann simulations for
automotive and aeronautical applications. (Courtesy of Dassault Systemes)

This turned out not to be the case, but prepared the ground for subsequent
developments which have met with significant success, as witnessed by
existence of a number of open source [32] and commercial codes, particularly
POWERFLOW, developed by EXA Corporation and recently acquired by Dassault
Systemes [33]. The academic side has also witnessed remarkable progress,
mostly but not exclusively in connection, with the development of the entropic
method [34] and Large-Eddy simulations [35, 36, 37]. More recently, LB has
also been used to study turbulent flows with suspended bodies, a topic of
great relevance for energy and environment [38]. Another successful
application of LB to macro-hydrodynamics are flows in porous media, with
accompanying heterogeneous chemical reactions [39, 40], which draw major
benefits from the LB ability to deal efficiently with grossly irregular
geometries.

###  5.2 Multiphase and colloidal flows

Two areas where LB has made a real difference are multiphase/multi-component
flows and flows with suspended bodies [41]. In the former case LB offers the
major benefit of simplicity: interfaces need no explicit tracking but emerge
spontaneously, informed by the corresponding mesoscale forces which are
implemented as soft source terms. This benefits comes at the prize of several
limitations, most of which have been however significantly mitigated in the
course of time [42, 43, 44].

Figure 3: (a) Spontaneous ordering of droplets in a microfluidic channel
([48]). (b) Large-scale simulation of a colloidal flow ([50])

Many LB variants exist today which have found massive use especially in the
area of microfluidics (see below). Among these, a very fruitful option is
offered by the Color Gradient technique [45], in which the idea is to add an
explicit anti-diffusive flux sending particles of each species uphill along
their density gradient instead of against it, thereby promoting the formation
of an interface against the coalescing effect of surface tension. The stress-
jump condition across fluid interfaces can be further augmented with an
immersed-like force modeling the repulsive effect generated by a surfactant
solution absorbed onto the drop interfaces. This contribution can be added to
the collision operator via a suitable forcing term as proposed in [46, 47].
The above approach has been shown to correctly capture highly non trivial
many-droplet configurations, like dense emulsions and foams. As an example,
the extended multi-component model has been shown to reproduce the formation
of ordered droplets clusters in microfluidic channels [48]. As shown in figure
3(a), the droplets, continuously injected within the main channel, undergo a
spontaneous ordering into hexagonal clusters, which is due to a subtle
competition between local, short-range, repulsive interactions (i.e., the
near-contact forces) and the surface tension. Another major field of
application are complex flows with immersed bodies, both rigid and deformable,
in which LB is typically coupled to other methodologies for fluid structure
interactions [41, 49]. To this regard, recent Multi-GPU state-of-the art
implementations of LB codes for multi-component colloidal flows [50] have been
shown to deliver remarkable performances, up to $200$ GLUPS (Giga Lattice
Updates Per Second on a cluster made of $512$ A100 NVIDIA cards), on
computational grids with several billions of lattice points ( see figure 3(b).
These results open attractive prospects for the computational design of new
materials based on colloidal particles.

###  5.3 Micro and nano-hydrodynamics

In the early 1990’s LB was applied exclusively to macroscopic fluid problems,
and even with a vengeance, since some prominent researchers vetoed its use for
anything but this [51]. Yet, precisely in those years, a few groups offered
numerical evidence that the LB scheme can handle moderately non-equilibrium
flows beyond the strict realm of hydrodynamics [52, 53, 55]. This, together
with a boost of activity in the area of non-ideal, multiphase and
multicomponent fluids, has led to a major mainstream of LB applications,
mostly in soft-matter and microfluidics [46, 56, 57, 58, 59, 60, 61, 62].

The key ingredients for this transition from the macro to the micro-
hydrodynamic levels turned out to be the introduction of higher order
lattices, securing the compliance with generalized hydrodynamics beyond the
Navier-Stokes description, as well as the development of suitable boundary
conditions, capable of dealing with non-equilibrium effects associated with
large gradients at solid walls [53, 54]. Very recently, the LB has also been
successfully coupled to off-lattice, particle models to develop novel classes
of fully mesoscale hybrid approaches capable of capturing the physics of
fluids at the micro‐ and nanoscales whenever a continuum representation of the
fluid falls short of providing the complete physical information. In addition,
LBM has also been coupled to Direct Simulation Monte Carlo (DSMC) in view of
efficiently simulating isothermal flows characterized by variable rarefaction
effects [63].

Another major mainstream of current LB research is the study of flows with
suspended bodies, both rigid and deformable ones. This was originated by
pioneering work on colloidal nanoflows, i.e. rigid spheres flowing in a LB
solvent, which was sparked by the development of the so-called fluctuating
Lattice Boltzmann, taking in due account the statistical fluctuations which
characterize hydrodynamic behavior at the nanoscale [64].

The original fluctuating LB (FLB) scheme for colloidal flows has spawn many
subsequent developments, including biopolymers and deformable objects of
biological interest, such as membranes and cells. Methodologically, this was
made possible by the fruitful merge between LB and the Immersed Boundary
Method [49]. Details of this highly technical and utterly important subject
can be found in [41].

##  6 Quantum-relativistic fluids

Even though the mainstream of LB research remains hard-rooted it classical
physics, it should be mentioned that the LB scheme is amenable to remarkable
generalizations in the direction of both quantum and relativistic mechanics.
The former possibility was recognized as early as 1993, building on a formal
analogy between the Boltzmann equations and the Dirac equation of relativistic
quantum mechanics [65, 66, 67]. Incidentally, the corresponding quantum LB
(QLB) scheme was later shown to represent a quantum random walk, as per the
seminal Ahronov et al paper, which appeared just a couple of months ahead of
QLB. The QLB was also shown to be amenable to quantum computing [68].

The extension of the LB scheme to relativistic fluids came much later, as it
dates of 2010 [69, 70] and it has undergone major progress in the last decade
[71]. The relativistic LB requires some non-trivial technical manipulation due
to the fact that, unlike the Newtonian case, the relativistic energy of
massive particles is an irrational function of the particle momentum, i.e.

$E(p)=\sqrt{m^{2}c^{4}+p^{2}c^{2}}$ (5)

This makes the task of developing discrete-velocities schemes less
straightforward than in the non-relativistic case, $E\ll mc^{2}$, because one
can no longer rely on orthogonal polynomials in the continuum, such as
Hermite’s. However, suitable discrete orthogonalization techniques have been
developed over the years, which have permitted to realize fairly efficient
relativistic LB schemes despite the large number of discrete speeds involved
[72].

This has permitted the simulation of relativistic flows, such as electron
transport in graphene and transport phenomena in quark gluon plasmas.
Amazingly, similar schemes have also been used for the simulation of cosmic
neutrinos in astrophysical flows [26]. Relativistic hydrodynamics is still a
small niche as compared to the massive activity in non-relativistic fluids.
However, due to the mounting interest in relativistic fluids at the crossroads
between condensed matter, high-energy physics and gravity [73], it is
reasonable to expect that relativistic LB schemes may gain significant
momentum for the years to come.

##  7 Towards exascale LB

A common thread to all the LB applications is their outstanding amenability to
parallel computing, across virtually any kind of architecture (provided hard
and ingenuous implementation work is spent on the task). Remarkable examples
abound in the literature [32, 74, 75, 76, 77, 78, 79, 80, 81], but here I
shall just refer to two instances I am directly familiar with.

Figure 4: LB across scales: from above galaxies all the way down inside the
proton! Top row: astrorelativistic jets resulting from binary neutron star
mergers, flow around a racing car, blodd flow in human arteries at blood cell
resolution. Middle: a wet foam. Bottom row: double-stranded biopolymer
trasnslocating through a membrane, The electronic cloud around the methane
molecule, the quark-gluon plasma produced by heavy ion collisions. All of
these flows, covering from megaparsecs to femtometers (37 decades), made use
of different variants of the LB method.

First, the use LB as a fast water solvent solver for the study of protein
crowding within the cell, which reached in excess of 20 Pflops on the Titan
supercomputer, but many outstanding performances have been obtained in other
massively parallel applications at all scales of motion [76]. As a recent and
rather spectacular example, I wish to quote the full-scale simulation of the
deep-sea sponge Euplectella Aspergillum, with a fully realistic description of
the highly complex geometrical porous structure at a 10-micron resolution, all
the way up to the centimeters of the full sponge [82]. It is fair to say that
such kind of study would have been impossible without LB. The above
applications witness the ability of suitably optimized LB codes to achieve
Petascale performance on highly non-trivial fluid problems.

The LB assets, exact free streaming, machine-accurate conservative collisions
and flexibility towards inclusion of mesoscale physics are expected to carry
over from the Peta to the Exascale.

However,taking LB towards the exascale will face the usual challenges common
to most numerical methods, namely the need of minimizing the cost of accessing
data [83], offsetting communication overheads by overlapping communication and
computation and, a possibility unexplored so far to the best of this author’s
knowledge, develop fault-tolerant LB schemes. All of the above, by taking into
account the increasing constraints on power consumption, for which machine
learning might prove a valuable ally, typically by optimizing the
reconstruction of fine-grained information from coarse-grain simulations.

##  8 Towards LB Quantum Computing

Quantum computing holds great promises to become the leading simulation
technology for the future. To make the promise come true, however, several
obstacles need to be circumvented, on both hardware and software fronts. As to
latter, one has first to identify the class of problems which can be
formulated in terms of quantum computing algorithms. To date, the most likely
candidates for such a paradigm shift are mostly in the area of quantum
chemistry and material science; understandably so, since quantum computing is
obviously most suited to quantum systems [84]. In this respect, fluid dynamics
faces with two major issues: it is strongly nonlinear and dissipative. Indeed,
quantum computing for fluids is still in its infancy [85, 86], even though
interesting efforts to circumvent the two major barriers have started to
emerge in the recent years. In the following we just sketch the basi ideas.

###  8.1 Handling nonlinearity

As to nonlinearity, an interesting possibility is to advocate Carleman
linearization, which consists in trading non-linearity for linearity in extra-
dimensions.

A simple example says it best.

Consider the logistic equation:

$\frac{dx}{dt}=x(1-x)$ (6)

with the initial condition $x(0)=1/2$.

The quadratic non-linearity can be formally disposed of by defining two
independent variables $x_{1}\equiv x$ and $x_{2}\equiv x^{2}$, so that the
quadratic logistic equation becomes formally linear, i.e.:

$\frac{dx_{1}}{dt}=x_{1}-x_{2}$ (7)

This is obviously open and requires an additional equation for $x_{2}$. By
differentiating $dx^{2}/dt=2xdx/dt=2x^{2}(1-x)$ and defining $x_{3}\equiv
x^{3}$, we obtain

$\frac{dx_{2}}{dt}=2x_{2}-2x_{3}$ (8)

This is still open , as it requires an additional equation for $x_{3}$. It is
therefore apparent that teh Carleman linearization is confronted with an
endless hierarchy which needs to be truncated at some stage.

The first order truncation is $x_{2}=0$, which delivers $x(t)=e^{t}$, just the
initial stage of the exact solution of the logistic equation,
$x(t)=e^{t}/(1+e^{t})$. The second order truncation is $x_{3}=0$, delivers a
$2\times 2$ linear system, whose solution is readily checked to be
$x(t)=e^{t}(1-e^{t})$, still not quite exact but closer than the previous one.
The name of the game is quite clear: by truncating the Carleman series at
level $N$ we obtain a $N\times N$ system of first order linear ODEs, whose
solution captures the $N$-th order expansion of the denominator in the exact
solution. Clearly, the exact solution, meaning by this a solution which holds
exactly and at arbitrarily long times, can only be recovered in the limit
$N\to\infty$. Hence the Carleman linearization buys linearity, but at the
price of an inevitable error which grows in time. Preliminary results indicate
that the truncation error decays fast with the order of truncation [87], but
much more work is required to put this observation on a solid basis for the
actual LB equation. This is certainly a most intersting direction for future
research.

###  8.2 Handling Dissipation

Several techniques are available to handle dissipative effects within a
conservative (Hamiltonian) formulation. One possibility is to augment the
original dissipative system with one (or more) extra-variables, call it
reservoir, absorbing the dissipated energy so that the global system, original
plus reservoir, is by construction conservative.

Another possibility is to represent dissipative operators (the collision
operator in the case of LB) as the weighted sum of two unitaries. Let $C_{ij}$
the Carleman-linearized collision matrix associated with the LB scheme, one
writes

$C_{ij}=U_{ij}+\alpha V_{ij}$ (9)

where $U$ and $V$ are two unitaries and $\alpha$ a numerical factor between
$0$ and $1$. The price for splitting the Carleman matrix into the sum of two
unitaries is that the quantum update may eventually fail, with a probability
which depends on $\alpha$. For the case of linear advection-diffusion
problems, it has been shown that the loss of efficiency can be contained to
some 10 percent on a wide range of $\alpha^{\prime}s$ [88]. Whether the same
is true for the full fluid equations is a totally open question at the time of
this writing, and again an interesting topic for future research.

##  9 Summary: Whither LB?

As we have been illustrating in this short review, over the last three
decades, the LB has made proof of an amazingly flexibility, with a wide
spectrum of applications covering fluid motion across an impressively broad
range of scales, literally from inside protons to the outer Universe (see
figure 4)! The impact on CFD and allied fields is massive, as reflected by its
bibliographic indicators. But even leaving aside from bibliometrics, it is
fair to say that LB has greatly facilitated the inspection of highly complex
flowing states of matter which would be very hard to analyze with other
methods, if possible at all. The main asset behind this success is always the
same: information travels along straight characteristics and the streaming
step is literally exact (zero-roundoff). With the due amount of work, far from
trivial in the case of complex geometries, this permits to achieve high
parallel efficiencies also in the presence of fairly complex geometries and
strong nonlinearities. In the end, this is what makes LB such a powerful tool
for the study of complex states of flowing matter across regimes and scales.
This said, many challenges still remain. Among others, how to control lattice
artifacts in high-Reynolds fluid flows with immersed bodies; how to enhance
the stability of LB schemes in the presence of substantial heat transfer and
compressibility effects; how to handle large density contrasts in multiphase
flows. How to represent large nanoscale fluctuations without hampering
numerical stability. From a more technical, but no less important,
perspective, how to make LB compliant with the forthcoming exascale
architectures, which requires a serious effort in the direction of minimizing
the cost of accessing data [79, 80, 83]. And as mentioned above, how to make
LB compliant with the (hopefully) forthcoming quantum computing architectures.
Summarizing, the original Lattice Gas idea of solving fluid dynamics by means
of fictitious molecules instead of discretizing continuum equation did not
work in its original and most radical (boolean) form. However, it has proved
exceedingly fruitful from the conceptual standpoint, by providing the stepping
stone for a new class of mesoscale methods and particularly the Lattice
Boltzmann method. In hindsight, LB could have been derived as a discrete
velocity version of the continuum Boltzmann equation, with no reference to any
microscale dynamics. However, this does not change the fact that, without the
LGCA inspiration, LB would have been discovered much later, if at all.

##  10 Acknowledgments

This research has received funding from the European Research Council under
the Horizon 2020 Programme Grant Agreement n. 739964 (”COPMAT”). The author is
indebted to very many colleagues and friends in Italy and around the world,
too many to mention without taking the chance of embarrassing omissions. Here
I only wish to thank Andrea Montessori and Mihir Durve for their generous help
in putting this manuscript together.

## References

  * [1] U. Frisch, B. Hasslacher and Y. Pomeau, Phys. Rev. Lett. 56, 1505, (1986)
  * [2] S. Wolfram, J. Stat. Phys. 45, 471, (1986)
  * [3] P.J. Hills, The Washington Post, Nov 19, 1985
  * [4] M. Henon, Complex Systems 1, 763 (1987), https://content.wolfram.com/uploads/sites/13/2018/02/01-4-12.pdf
  * [5] S. Succi, The lattice Boltzmann equation for fluid dynamics and beyond, Oxford Univ. Press (2001)
  * [6] S. Succi, The lattice Boltzmann equation for complex states of flowing matter, Oxford Univ. Press (2018)
  * [7] C. Aidun and J. Clausen, Annu. Rev. Fluid Mech, 42, 439, (2010)
  * [8] J.E. Broadwell, Phys. Fluids , 7 1243–1247 (1964) R. Gatignol, ”Théorie cinétique d’un gaz répartition discrète de vitesses” , Springer (1975); R. Monaco, L. Preziosi, ”Fluid dynamic applications of the discrete Boltzmann equation” , World Sci. (1991) T. Platkowski, R. Illner, survey on the mathematical aspects of the theory” SIAM Review , 30 (2) 213–255 (1988)
  * [9] G. Mc Namara and G. Zanetti, Phys. Rev. Lett., 61, 2332, (1988)
  * [10] F. Higuera and S. Succi, Europhys. Lett. ,8, 517, (1989). For the record, the seeds of the EPL paper can be found in F. Higuera, in Discrete Kinetic Theory, Lattice Gas Dynamics and Foundations of Hydrodynamics, R. Monaco ed., World Scientific, Singapore, p. 162 1989 and S. Succi, R. Benzi, F. Higuera, ibidem, p. 329.
  * [11] F. Higuera and J. Jimenez, Europhys. Lett., 9, 663, (1989)
  * [12] F. Higuera , S. Succi and R. Benzi, Europhys. Lett., 9, 345, (1989)
  * [13] R. Benzi, S. Succi and M. Vergassola, Phys. Rep., 222, 145, (1992)
  * [14] H. Chen, S. Chen and W. Matthaeus, Phys. Rev. A, 45, R5342 (1992)
  * [15] S. Chen, H. Chen et al, Phys. Rev. Lett., 67(27), 3776, (1991)
  * [16] J. M. Koelman, Europhys. Lett., 15(6), 603, (1991)
  * [17] Y. Qian, D. d’Humières and P. Lallemand, Europhys. Lett., 17(6), 479, (1992)
  * [18] D. d’Humieres, I. Ginzburg, M. Krafczyk et al. Phil. Trans. Roy. Soc., 360,437, (2002)
  * [19] I. Karlin, A. Ferrante and H. C. Oettinger, Europhys. Lett.,62(3), (1999)
  * [20] X. He and L.S. Luo, Phys. Rev. E. 56, 6811, (1997)
  * [21] Z. Guo, C. Zheng and B. Shi, Phys. Rev. E, 65, 046308 (2002)
  * [22] S. Succi, Eur. Phys. J. B: 64, 471 (2008)
  * [23] F. Alexander, S. Chen, JD Sterling, Phys. Rev. E, 47, R2249, (1993)
  * [24] X. He, S. Chen, G. Doolen, J. Comp. Phys., 146, 282 (1998)
  * [25] M. Sbragaglia et al, J. Fluid Mech., 628, 299, (2009)
  * [26] L.R. Weih, A. Gabbana, D. Simeoni, L. Rezzolla, S. Succi, R. Tripiccione, MNRAS, 498, Issue 3 (2020).
  * [27] S. Succi, EPL Perspective, 109, 5, 50001 (2015)
  * [28] S. Chapman and T. Cowling, The mathematical theory of non-uniform gases, Cambridge U.P., (1990)
  * [29] S. Succi, Sailing the Ocean of Complexity: Lessons from the Physics-Biology Frontier, Oxford U.P., (2022)
  * [30] H. Chen et al, Science, 301, 633, (2003)
  * [31] Benzi, R., Succi, S., Two-dimensional turbulence with the lattice Boltzmann equation, Journal of Physics A: Mathematical and General, 23(1), L1 (1990).
  * [32] Latt, J., Malaspinas, O., Kontaxakis, D., Parmigiani, A., Lagrava, D., Brogi, F., …, Chopard, B. Computers and Mathematics with Applications, 81, 334-350 (2022)
  * [33] https://www.3ds.com/products-services/simulia/products/powerflow/
  * [34] S.S. Chikatamarla, C.E. Frouzakis, I.V. Karlin et al. J. Fluid Mech. 656, 298, (2010)
  * [35] O. Malaspinas, P. Sagaut, Journal of Fluid Mechanics 700, 514-542 (2010) Toward advanced subgrid models for Lattice-Boltzmann-based Large-eddy simulation: Theoretical formulations
  * [36] P Sagaut, Computers and Mathematics with Applications 59 (7), 2194-2199, (2010)
  * [37] NH Maruthi, C Thantanapally, M Namburi, V Kumaran, S Ansumali, arXiv preprint arXiv:2204.02191(2022)
  * [38] Perlekar, P., Biferale, L., Sbragaglia, M., Srivastava, S., Toschi, F. Physics of Fluids, 24(6), 065101.
  * [39] Q. Kang, P. Lichtner, DR Janecky, Advances in Applied Mathematics and Mechanics 2(5):545-563 (2010)
  * [40] G Falcucci, S Succi, A Montessori, S Melchionna, P Prestininzi, C Barroo, … Microfluidics and Nanofluidics 20 (7), 1-13
  * [41] T. Kruger, The Lattice Boltzmann Method, Springer Verlag (2017)
  * [42] X. Shan and H. Chen, Phys. Rev. E 47, 1815, (1993)
  * [43] M. Swift et al, Phys. Rev. Lett., 75, 830, (1995)
  * [44] G. Falcucci et al., Comm. in Comp. Phys., 9, 269, (2011)
  * [45] Gunstensen, A. K., Rothman, D. H., Zaleski, S. and Zanetti, G. Physical Review A, 43(8), 4320 (1991)
  * [46] A Montessori, M Lauricella, N Tirelli, S Succi, J. of Fluid Mech. 872, 327-347, (2019)
  * [47] Montessori, A., Tiribocchi, A., Lauricella, M., Bonaccorso, F. and Succi, S. Physical Review Fluids, 6(2), 023606 (2020).
  * [48] Montessori, A., Tiribocchi, A., Lauricella, M., Bonaccorso, F. and Succi, S. Soft Matter, 17(9), 2374-2383 (2021).
  * [49] Zhi-Gang Feng, E. Michaelides, J. of Comp. Phys. 195, 602, (2004)
  * [50] Bonaccorso, F., Lauricella, M., Montessori, A., Amati, G., Bernaschi, M., Spiga, F., Tiribocchi A. and Succi, S. , arXiv preprint arXiv:2112.08264 (2021).
  * [51] L.S. Luo, Phys. Rev. Lett. 92, 139401 (2004)
  * [52] X. Shan, X. Yuan and H. Chen, J. Fluid Mech., 550, 413, (2006)
  * [53] S. Ansumali and I.V. Karlin, Phys. Rev. E, 66, 026311 (2002)
  * [54] S.P. Thampi, S. Ansumali, R. Adhikari, Sauro Succi, J. Comp. Phys., 234, 1, (2013)
  * [55] A. Montessori, P. Prestininzi, M. La Rocca, S. Succi, Phys. Rev. E 92 (4), 043308, (2015)
  * [56] B. Duenweg and A. Ladd, Adv. Polym. Sci., 221, 89, (2009)
  * [57] M. Sbragaglia et al, Phys. Rev. Lett. 97, 204503(2006)
  * [58] M. Sbragaglia et al, Phys. Rev. E 75, 026702 (2007)
  * [59] J. Latt and B. Chopard, Math. Comput. Simul., 72, 165 (2006)
  * [60] XD Niu et al, Phys. Rev. E 76, 036711, (2007)
  * [61] R. Benzi, L. Biferale, M. Sbragaglia, S. Succi, F.Toschi, EPL (Europhys. Lett.) 74(4) 651, (2006)
  * [62] A Tiribocchi, A Montessori, M Lauricella, F Bonaccorso, S Succi, S Aime, et al, Nat. Commun. 12 (1), 1-10, (2021)
  * [63] Di Staso, G., Clercx, H. J. H., Succi, S., Toschi, F. . Journal of Computational Science, 17, 357-369 (2016)
  * [64] A. Ladd, J. Fluid Mech., 271, 285 (1994),
  * [65] S. Succi and R. Benzi, Physica D 69, 327, (1993)
  * [66] P. Dellar, D. Lapitski, Phil. Trans. Roy. Soc., 369, 2155 (2011)
  * [67] F. Fillion et al, Phys. Rev. Lett. 111, 160602, (2013)
  * [68] J. Yepez et al., Phys. Rev. Lett. 103, 084501 (2009)
  * [69] M. Mendoza et al, Phys. Rev. Lett., 105, 014502, (2010).
  * [70] M. Mendoza, H.J. Herrmann and S. Succi, Phys. Rev. Lett., 106, 156601, (2011).
  * [71] A. Gabbana, D. Simeoni, S. Succi, R. Tripiccione, Phys. Rep., 7 (2020).
  * [72] A. Gabbana, V. Ambrus D. Simeoni, S. Succi, R. Tripiccione, Nat. Comp. Sci, in press (2022)
  * [73] G. Policastro, D.T. Son and A. Starinets, Phys. Rev. Lett., 87, 081601, (2001)
  * [74] M Bernaschi, S Melchionna, S Succi, M Fyta, E Kaxiras, JK Sircar Computer Physics Communications, 180 (9), 1495-1502 (2009)
  * [75] M Bernaschi, M Fatica, S Melchionna, S Succi, E Kaxiras Concurrency and computation: practice and experience 22 (1), 1-14 (2010)
  * [76] M. Bernaschi, M. Bisson, M. Fatica and S. Melchionna, Proceedings of the International Conference on High-Performance Computing, Networking, Storage and Analysis (2013).
  * [77] MD Mazzeo, PV Coveney Computer Physics Communications 178 (12), 894-914 (2008)
  * [78] Feichtinger, C., Donath, S., Köstler, H., Götz, J., Rüde, U. , Journal of Computational Science, 2(2), 105-112 (2013).
  * [79] S Alowayyed, D Groen, PV Coveney, AG Hoekstra Journal of Computational Science 22, 15-25 (2017)
  * [80] S Succi, G Amati, M Bernaschi, G Falcucci, M Lauricella, A Montessori Computers and Fluids 181, 107-11 (2019)
  * [81] https://walberla.net
  * [82] G Falcucci, G Amati, P Fanelli, VK Krastev, G Polverino, M Porfiri, S Succi, Nature 595 (7868), 537-541, (2021)
  * [83] Shet, A. G., Sorathiya, S. H., Krithivasan, S., Deshpande, A. M., Kaul, B., Sherlekar, S. D., Ansumali, Physical Review E, 88(1), 013314 (2013).
  * [84] R. Feynman, International Journal of Theoretical Physics volume 21, pages 467–488 (1982)
  * [85] F. Gaitan, npj Quantum Information 6:61(2020) ; https://doi.org/10.1038/s41534-020-00291-0
  * [86] Jin-Peng Liu , Herman Kolden Hari K. Krovi, Nuno F. Loureiro, Konstantina Trivis and Andrew M. Childs, PNAS 2021 Vol. 118 No. 35 e2026805118(2021), https://doi.org/10.1073/pnas.2026805118
  * [87] W. Itani and S. Succi, Fluids, 7, 1(24), (2022)
  * [88] A. Mezzacapo, M Sanz, L Lamata, IL Egusquiza,S. Succi and E. Solano, Nat. Sci. Comp. , 5, 1,1(2016)