Table Of ContentMathematics in Industry 31
The European Consortium for Mathematics in Industry
Vito Dario Camiola
Giovanni Mascali
Vittorio Romano
Charge
Transport in Low
Dimensional
Semiconductor
Structures
The Maximum Entropy Approach
Mathematics in Industry
The European Consortium for Mathematics in Industry
Volume 31
ManagingEditor
MichaelGünther,UniversityofWuppertal,Wuppertal,Germany
SeriesEditors
LuisL.Bonilla,UniversityCarlosIIIMadrid,Escuela,Leganes,Spain
OtmarScherzer,UniversityofVienna,Vienna,Austria
WilSchilders,EindhovenUniversityofTechnology,Eindhoven,TheNetherlands
The ECMI subseries of the Mathematics in Industry series is a project of The
European Consortium for Mathematics in Industry. Mathematics in Industry
focusesontheresearchandeducationalaspectsofmathematicsusedinindustryand
otherbusinessenterprises.BooksforMathematicsinIndustryareinthefollowing
categories: research monographs, problem-oriented multi-author collections,
textbookswithaproblem-orientedapproach,conferenceproceedings.Relevanceto
the actualpracticaluse of mathematicsin industry is the distinguishingfeatureof
thebooksintheMathematicsinIndustryseries.
Moreinformationaboutthissubseriesathttp://www.springer.com/series/4651
Vito Dario Camiola (cid:129) Giovanni Mascali (cid:129)
Vittorio Romano
Charge Transport in Low
Dimensional Semiconductor
Structures
The Maximum Entropy Approach
VitoDarioCamiola GiovanniMascali
DepartmentofMathematics DepartmentofMathematics
andComputerScience andComputerScience
UniversityofCatania UniversityofCalabria
Catania,Italy ArcavacatadiRende,Italy
VittorioRomano
DepartmentofMathematics
andComputerScience
UniversityofCatania
Catania,Italy
ISSN1612-3956 ISSN2198-3283 (electronic)
MathematicsinIndustry
TheEuropeanConsortiumforMathematicsinIndustry
ISBN978-3-030-35992-8 ISBN978-3-030-35993-5 (eBook)
https://doi.org/10.1007/978-3-030-35993-5
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Tothememoryofprof.AngeloMarcelloAnile
Preface
Nissunaumanainvestigazionesipòdimandareverascienzia
s’essanonpassaperlematematichedimostrazioni,
esetudiraichelescienzie,cheprincipianoefiniscononella
mente,abbianoverità,questononsiconcede,masiniega,
permolteragioni,eprima,
cheintalidiscorsimentalinonaccadeesperienzia,sanzala
qualenulladàdisécertezza.
LeonardodaVinci
Quelliches’innamorandipraticasanzascienzia
soncome’lnocchierch’entrainnaviliosenzatimoneobussola,
chemaihacertezzadovesivada.
LeonardodaVinci
In 1948 W. Shockley, W. Brattain and J. Bardeen invented the first transistor at
the Bell laboratories and in 1958 Jack Kilby, an engineer at Texas Instruments,
announced the creation of the first integrated circuit. Since then, progression in
electronicengineeringhasmeantminiaturization.
By continuing to shrink the dimension of electronic devices, the effects of
quantumconfinementbecome more and more relevant. In particular,these effects
need to be well understood in metal-oxide-semiconductor field-effect transistors
(MOSFETs)andinthemorerecentdoublegateMOSFETs(DG-MOSFETs),which
are the backbone of the modern integrated circuits. The DG-MOSFET, with its
extra gate, is considered one of the most appropriate structures for minimizing,
in nanometerdevices, the short channeleffects which cause a deteriorationin the
transistor performance—thisthanks to the enhanced gate control(two gates) over
thechannelandtothereducedsiliconlayerthickness.
In a DG-MOSFET the potential between the two gates and the oxide layers
confinestheelectronsinthetransversaldirection,producingaquantumwellwhose
lengthiscomparabletothedeBrogliewavelength.A similareffectisalsopresent
inhetero-structureslikeAlGa-Ga[22,62,63,126].
Recent years have also witnessed great interest in 2D materials for their
promising applications. The most investigated material is graphene, but of late
the single-layer transition metal dichalcogenides (TMDCs), such as molybdenum
vii
viii Preface
disulfide and tungsten diselenide, and black phosphorus have received a certain
amountofattention(seeforexample[204]).
Graphene is considered a potential new material for future applications in
nanoelectronic [20, 48, 55, 121, 151, 172] and optoelectronic devices [204] due
to the fact that it has very good mechanical properties, is an excellent heat and
electricity conductor, and also has noteworthy optical properties. It is a two-
dimensionalallotropeofcarbonwhichconsistsofcarbonatomstightlypackedintoa
honeycombhexagonallattice.Thankstothisstructureithas,asfirstapproximation,
aconicalnotcurvedbandstructure,sotheeffectivemassoftheelectronsiszeroand
chargecarriersexhibitaphoton-likebehavior.
Inapurelyquantumapproach,electricpropertiesofnanoscaledevices,suchas
current-voltage curves, can be computed via the non-equilibrium Green function
[50,200].Anotherwaytotackletheproblemisintheframeworkofquantumkinetic
theory via the Wigner function, which gives the macroscopic physical quantities
of interest as expectation values [99, 136, 177]. Other approaches are based on
themasterequation[75].However,instructureslikeDG-MOSFETs,theconfining
effect is in the direction transversal to the oxide, while in the other directionsthe
electronsflowfreelyfromthesourcetothedrainandtheelectrontransportcanbe
treatedsemiclassicallywhenthetypicallongitudinallengthisoftheorderofafew
tens of nanometers. In fact, in these conditions the electrons, as waves, achieve
equilibrium along the confining direction in a time which is much shorter than
the typical transport time, so that one can adopt a quasi-static description along
the confining direction by a coupled Schrödinger–Poisson system which leads to
a subband decomposition. The transport along the longitudinal direction can be
describedbyasemiclassicalBoltzmannequationforeachsubband.
Numericalintegrationofthe Boltzmann–Schrödinger–Poissonsystem hasbeen
performed with Monte Carlo methods or deterministic schemes for solving the
transportpart [4, 28, 79,134, 167, 168, 196], but they are very expensive,from a
computationalpointofview,forcomputer-aideddesign(CAD)purposes.Thishas
promptedsubstitution of the Boltzmann equations with macroscopic models such
asdrift-diffusionorenergy-transportones[27,64].
In this monograph, in order to describe the electron transport, models of an
energytransporttypewillbeadopted.Theyarededuced,underasuitablediffusion
scaling, from a system of equations derived from the Boltzmann equations by
using the moment method. The moment equations are closed by resorting to the
maximumentropyprinciple(MEP).Theresultsfitintotheframeworkofextended
thermodynamics[97,153].
In the contextof the Bayesian interpretationof the probability,Jaynes [94,96]
showedthatitispossibletodeveloptheMEPasaninferencemethodtoobtainthe
resultsachievedinthecontextofthestatisticalmechanicsor,moregenerally,inall
cases wherethe complete information,necessaryfor a detaileddescription,is lost
orisnotavailable.
Thetermentropywasintroducedforthefirsttime inclassicalthermodynamics
by Clausius in 1865. In 1948 Shannon [186] used the same term to indicate the
amountofignoranceaboutasysteminthecontextofinformationtheory.Theformal
Preface ix
expressionforthisquantityisthesameasthatofthestatisticalBoltzmannentropy
inthecanonicalensembleandin1957Jaynes[94,96]showedthatitispossibleto
obtainalltheexpressionsofstatisticalmechanicsbymaximizingit.
An approach based on the MEP is also described for the treatment of charge
transportingraphene.Itleadstoahydrodynamicalmodelwhichhasbeentestedby
acomparisonwithdirectsimulationMonteCarloresults.Themodelappearstobe
soundandsufficientlyaccurateforsystematicusein CAD simulatorsforcomplex
electrondevices.
The book is addressed to applied mathematicians, physicists, and electronic
engineers.Itis writtenforgraduateorPhD readersbuttheinitialchaptercontains
a modicum of semiconductor physics, which makes the book self-consistent and
usefulalsoforundergraduatestudents.
The plan of the book is as follows: In Chap.1 an overview of semiconductor
bandstructureand ofthe Boltzmannequationis given,with emphasisonthe case
of confined transport.In Chap.2 the MEP is presented in a generalway, focusing
attentiononitsmeaningasamethodofinference.Chapter3isdevotedtothegeneral
featuresofapplicationof MEPtobulk (three-dimensional)semiconductorswith a
specialfocusontheproblemoftherealizability.
Detailed results for silicon are reported in Chap.4, where in addition the
numerics are discussed and applied to the most common semiconductor devices.
Someformalpropertiesarediscussedinthesubsequentchapter.
In Chap.6 a quantum model is introduced by using a Wigner approach which
allows us to get, under a suitable scaling, explicit quantum corrections to the
semiclassicalmodelpresentedinChap.4.
Theremainingpartsconstitutethecoreofthebook.Themathematicalmodelfor
chargetransportinDG-MOSFETsisformulatedandasuitablenumericalschemeis
presented,alongwithsimulationsincasesofinterestfortheelectricalengineering
design.
ThebookconcludeswithanapplicationofMEPtographene.
Inordertohelpthosewhowanttousethemodelspresentedinthismonograph,
inthelastAppendixdetailednumericalcodesarereportedandcommenteduponfor
thesimulationofaone-dimensionalsilicondiode.Theinterestedreadercaneasily
modifythesecodestodealwithmorecomplexdevices.
Theauthorswishtoacknowledgethestimulatingandfruitfuldiscussionswitha
numberofcolleagues.WewouldliketomentionNauofelBenAbdallah,Giuseppe
Alì, Luigi Barletti, Alexander Blokhin, Luis Bonilla, Carlo De Falco, Massimo
Fischetti,GiovanniFrosali,IreneGamba,CarloJacoboni,AnsgarJüngel,Michael
Junk, Antonino La Magna, Salvatore Fabio Liotta, Armando Majorana, Americo
Marrocco, Omar Morandi, Orazio Muscato, Claudia Negulescu, Paola Pietra,
GiovanniRusso,FerdinandSchürrer,JeanMichelSellier,andFrancescoVecil,and
weapologizeforanymissingnames.
AnexpressionofthanksalsogoestothePhDstudentsMarcoCoco,LilianaLuca,
and GiovanniNastasi, who have read the first version of the book and helped the
authorsinimprovingthepresentationofthetopics.
x Preface
ThebookisdedicatedtothememoryofProf.AngeloMarcelloAnile,whowas
the scientific mentor of two of the authors (G.M. and V.R.) and also indirectly a
guide for V.D.C. The approach to the matter treated in this monograph has been
deeplyinfluencedbyhisviewofscience.
Catania,Italy VitoDarioCamiola
Cosenza,Italy GiovanniMascali
Catania,Italy VittorioRomano
September2019
