Dispersion Relations in Heavily-Doped Nanostructures

664 Pages·2016·7.504 MB·English

by  Kamakhya Prasad Ghatak (auth.)

Springer Tracts in Modern Physics 265
Kamakhya Prasad Ghatak
Dispersion
Relations in
Heavily-Doped
Nanostructures
Springer Tracts in Modern Physics
Volume 265
Honorary Editor
G. Höhler, Karlsruhe, Germany
Series editors
Yan Chen, Shanghai, China
Atsushi Fujimori, Tokyo, Japan
Johann H. Kühn, Karlsruhe, Germany
Thomas Müller, Karlsruhe, Germany
Frank Steiner, Ulm, Germany
William C. Stwalley, Storrs, CT, USA
Joachim E. Trümper, Garching, Germany
Peter Wölfle, Karlsruhe, Germany
Ulrike Woggon, Berlin, Germany
Springer Tracts in Modern Physics
Springer Tracts in Modern Physics provides comprehensive and critical reviews of topics of current
interestinphysics.Thefollowingfieldsareemphasized:ElementaryParticlePhysics,CondensedMatter
Physics, Light Matter Interaction, Atomic and Molecular Physics, Complex Systems, Fundamental
Astrophysics.
Suitablereviewsofotherfieldscanalsobeaccepted.Theeditorsencourageprospectiveauthorsto
correspond with them in advance of submittinga manuscript. For reviews of topics belongingto the
abovementionedfields,theyshouldaddresstheresponsibleeditoraslistedbelow.
Specialoffer:Forallclientswithaprintstandingorderweofferfreeaccesstotheelectronicvolumes
oftheSeriespublishedinthecurrentyear.
ElementaryParticlePhysics CondensedMatterPhysics
JohannH.Kühn YanChen
InstitutfürTheoretischeTeilchenphysik FudanUniversity
KarlsruheInstitutfürTechnologieKIT DepartmentofPhysics
Postfach6980 2250SonghuRoad,
76049Karlsruhe,Germany Shanghai,China400438
Email:[email protected] Email:[email protected]
www-ttp.physik.uni-karlsruhe.de/*jk www.physics.fudan.edu.cn/tps/branch/cqc/en/people/
faculty/
ThomasMüller
InstitutfürExperimentelleKernphysik AtsushiFujimori
KarlsruheInstitutfürTechnologieKIT EditorforThePacificRim
Postfach6980 DepartmentofPhysics
76049Karlsruhe,Germany UniversityofTokyo
Email:[email protected] 7-3-1Hongo,Bunkyo-ku
www-ekp.physik.uni-karlsruhe.de Tokyo113-0033,Japan
Email:[email protected]
ComplexSystems http://wyvern.phys.s.u-tokyo.ac.jp/welcome_en.html
FrankSteiner
InstitutfürTheoretischePhysik PeterWölfle
UniversitätUlm InstitutfürTheoriederKondensiertenMaterie
Albert-Einstein-Allee11 KarlsruheInstitutfürTechnologieKIT
89069Ulm,Germany Postfach6980
Email:[email protected] 76049Karlsruhe,Germany
www.physik.uni-ulm.de/theo/qc/group.html Email:peter.woelfl[email protected]
www-tkm.physik.uni-karlsruhe.de
FundamentalAstrophysics
JoachimE.Trümper Atomic,MolecularandOpticalPhysics
Max-Planck-InstitutfürExtraterrestrischePhysik WilliamC.Stwalley
Postfach1312 UniversityofConnecticut
85741Garching,Germany DepartmentofPhysics
Email:[email protected] 2152HillsideRoad,U-3046
www.mpe-garching.mpg.de/index.html Storrs,CT06269-3046,USA
Email:[email protected]
SolidStateandOpticalPhysics www-phys.uconn.edu/faculty/stwalley.html
UlrikeWoggon
InstitutfürOptikundAtomarePhysik
TechnischeUniversitätBerlin
Straßedes17.Juni135
10623Berlin,Germany
Email:[email protected]
www.ioap.tu-berlin.de
More information about this series at http://www.springer.com/series/426
Kamakhya Prasad Ghatak
Dispersion Relations
in Heavily-Doped
Nanostructures
123
Kamakhya PrasadGhatak
Department ofBasic ScienceandHumanities
Institute of Engineering andManagement
Salt Lake,Kolkata
India
ISSN 0081-3869 ISSN 1615-0430 (electronic)
SpringerTracts inModern Physics
ISBN978-3-319-20999-9 ISBN978-3-319-21000-1 (eBook)
DOI 10.1007/978-3-319-21000-1
LibraryofCongressControlNumber:2015943034
SpringerChamHeidelbergNewYorkDordrechtLondon
©SpringerInternationalPublishingSwitzerland2016
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
foranyerrorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia
(www.springer.com)
Courage in adversity, patience in prosperity,
modesty in fame, indifference in own glory,
deep love for foes, staying in solitude,
all these are naturally found in wise persons
Knowledge is proud, he knows too much, but
the wise is humble, he knows no more
What lies before us and what lies behind us
are tiny matters as compared to what lies
WITHIN US
Smile and passion are real debit cards of life.
Paynowanduselater.Butegoandvindictive
attitude are like credit cards of life.
Use now and pay later.
Iron does not become steel without through
fire. I must be prepared to undergo the test of
fire, if I wish to enjoy the success in life.
To be the best, I must extract the best from
myself
This book is dedicated to Dr. Amarnath
Chakravarti, Ex. Professor and the Head
of the Department of the Institute of Radio
Physics and Electronics of the University
of Calcutta, the Ph.D. Supervisor and
Philosopher of the present author.
Sir, your teaching such creativity bring,
I must not change my state with a King
Preface
ThecreationofNanoElectronics,thesubsetofthegeneralizedsetPhysics,isbased
on the following two important concepts:
(cid:129) Thesymmetryofthewavevectorspaceofchargecarriersinelectronicmaterials
having various band structures is being reduced from a 3D closed surface to a
quantized2Dclosedsurface,quantizednon-parabolasandfullyquantizedwave
vector space leading to the formation of 0D systems such as ultrathin films
(UFs), doping superlattices, inversion and accumulation layers, quantum wells
(QWs), quantum well superlattices, carbon nanotubes, nanowires (NWs),
quantum wire superlattices, magnetic quantization, magneto size quantization,
quantum dots (QDs), magneto inversion and accumulation layers, magneto
quantumwellsuperlattices,magnetoNIPIs,quantumdotsuperlatticesandother
field aided nanostructures.
(cid:129) The advent of modern experimental methods, namely Fine Line Lithography
(FLL), Metallo-Organic Chemical Vapor Deposition (MOCVD), Molecular
Beam Epitaxy (MBE), etc., for fabricating low-dimensional nanostructured
systems.
Quantum confined materials have gained much interest in modern physics
because of their importance to unlock both new scientific revelations and multi-
dimensional altogether unheard of technological applications. In UFs, quantization
of the motion of carriers in the direction perpendicular to the surface exhibits the
two-dimensional carrier motion of charge carriers, and the third direction is being
quantized.Anotherone-dimensionalstructureknownasNWshasbeenproposedto
investigate the physical properties in these materials, where the carrier gas is
quantized in two transverse directions and they can move only in the longitudinal
direction. As the concept of quantization increases from 1D to 3D, the degree of
freedom of the free carriers decreases drastically and the total density-of-states
(DOS) function changesfromHeavisidestepfunctiontotheDirac’sdeltafunction
forming QDs which, in turn, depend on the carrier energy spectra in different
materials. An enormous range of important applications of such low-dimensional
ix
x Preface
structuresformodernphysicsinthequantumregime,togetherwitharapidincrease
incomputingpower,havegeneratedconsiderableinterestinthestudyoftheoptical
properties of quantum effect devices based on various new materials of reduced
dimensionality. Examples of such new applications include quantum switches,
quantum registers, quantum sensors, heterojunction field-effect, quantum logic
gates, quantum well and quantum wire transistors, quantum cascade lasers,
high-frequency microwave circuits, high-speed digital networks, high-resolution
terahertz spectroscopy, advanced integrated circuits, superlattice photo-oscillator,
superlattice photo-cathodes, resonant tunneling diodes and transistors, superlattice
coolers, thermoelectric devices, thin film transistors, micro-optical systems,
intermediate-band solar cells, high performance infrared imaging systems, optical
modulators, optical switching systems, single electron/molecule electronics,
nanotube-based diodes, and other nanoelectronic devices [1–14].
Although many new effects in quantized structures have already been reported,
the interest in further research of different other aspects of such quantum-confined
materials is becoming increasingly important. One such significant concept is the
Dispersion Relations (DRs) of carriers in semiconductors and their nanostructures,
which occupies a singular position in the arena of Modern Physics and related
disciplines in general and whose importance [15–36] has already been established
since the inception of the theory of band structure of Solid State Physics. The
concept of DRs is offundamental importance for not only the characterization of
semiconductor nanostructures, but also for the study of carrier transport in semi-
conductors and their quantized counterparts through proper formulation of the
Boltzmann Transport equation which, in turn, needs the corresponding carrier
energy spectra of the heavily doped materials and is still one of the open research
problems. It is important to note that six important transport quantities,
namelytheeffectivecarriermass(ECM),density-of-states(DOS)function,the
sub-band energy and the measurement of band gap in the presence of strong
light waves, intense electric field and heavy doping are in disguise in the very
important concept of DR. Besides, the acoustic mobility limited momentum
relaxation time is inversely proportional to the respective DOS function of a
particular semiconductor and integral over the DOS function leads to carrier
statistics under the condition of extreme carrier degeneracy which, in turn, is
connected to the 25 important transport topics of quantum effect devices,
namely the Landau Dia and Pauli’s Para Magnetic Susceptibilities [37], the
Einstein’s Photoemission [38], the Einstein Relation [39], the Debye Screening
Length [40], the Generalized Raman gain [41], the Normalized Hall coefficient
[42], the Fowler-Nordheim Field Emission [43], the Gate Capacitance [44], the
Thermoelectric Power [45], the Plasma Frequency [46], the Magneto-Thermal
effect in Quantized Structures [47], the Activity coefficient [48], the Reflection
coefficient [49], the Heat Capacity [50], the Faraday rotation [51], the Optical
Effective Mass [52], the Carrier contribution to the elastic constants [53], the
Diffusion coefficient of the minority carriers [54], the Nonlinear optical response
[55], the Third order nonlinear optical susceptibility [56], the Righi-Leduc coeffi-
cient[57],theElectricSusceptibility[58],theElectricSusceptibilityMass[59],the
Preface xi
Electron Diffusion Thermo-power [60] and the Hydrostatic Piezo-resistance
Coefficient [61] respectively.
ThepresentmonographsolelyinvestigatesDRsinheavilydopednanostructures
of nonlinear optical, III–V, II–VI, gallium phosphide, germanium, platinum anti-
monide, stressed, IV–VI, lead germanium telluride, tellurium, II–V, zinc and cad-
mium diphosphides, bismuth telluride, III–V, II–VI, IV–VI and HgTe/CdTe
quantumwellHDsuperlatticeswithgradedinterfacesundermagneticquantization,
III–V, II–VI, IV–VI and HgTe/CdTe HD effective mass superlattices under mag-
netic quantization, quantum confined effective mass superlattices and superlattices
ofHDoptoelectronicmaterialswithgradedinterfacesinadditiontootherquantized
systems.Incidentally,evenafter40yearsofcontinuouseffort,weseethatcomplete
investigation of the DR comprising the whole set of HD materials and allied sci-
encesisreallyaseaandpermanentlyenjoysthedomainofimpossibilitytheorems.
DRs have different forms for different materials and change under one-, two- and
three-dimensional quantum confinement of charge carriers. It is rather curious to
notethatforthe31importantconcepts,only6monographshavebeenwritten[62–
67] and the remaining 25 books will appear in the future, hopefully from the
readersofthisbook.Inthiscontext,itmaybementionedthattheavailablereports
on the said areas cannot afford to cover even an entire chapter containing detailed
investigations on DRs in semiconductors and their quantized structures.
Itisworthremarkingthattheeffectsofcrossedelectricandquantizingmagnetic
fields on thetransport properties ofsemiconductorshavingvarious bandstructures
have been relatively less investigated compared to the corresponding magnetic
quantization, although the study of cross-fields is offundamental importance with
respecttotheadditionofnewphysicsandrelatedexperimentalfindingsinmodern
quantum effect devices. It is well known that in the presence of electric field (E )
0
alongx-axisandthequantizingmagneticfield(B)alongz-axis,theDRsofcarriers
insemiconductorsbecomemodified,forwhichthecarriermovesinboththezandy
directions respectively. The motion along y direction is purely due to the presence
ofE alongx-axisandintheabsenceofanelectricfield,theeffectiveelectronmass
0
along y-axis tends to infinity indicating the fact that the electron motion along
y-axis is forbidden. The effective electron mass of the isotropic, bulk semicon-
ductors having parabolic energy bands exhibit mass anisotropy in the presence of
cross-fields and this anisotropy depends on the electron energy, the magnetic
quantum number, the electric and the magnetic fields respectively, although the
effective electron mass along z-axis is a constant quantity. In 1966, Zawadzki and
Lax[68]derivedtheexpressionofDRforIII–Vsemiconductorsinaccordancewith
the two-band model of Kane under cross-fields configuration, which generates the
interest to study this particular topic of solid-state science in general [69].
Itiswellknownthatheavydopingandcarrierdegeneracyarethekeystounlock
the important properties of semiconductors; they are especially instrumental in
dictating thecharacteristics ofOhmicand Schottkycontactsrespectively[70]. It is
an amazing fact that although heavily doped semiconductors (HDS) have been
investigatedintheliterature,thestudyofthecorrespondingDRsofHDSisstillone
of the open research problems. We have obtained the exact E-k dispersion