Debye Screening Length: Effects of Nanostructured Materials

403 Pages·2014·7.845 MB·English

by  Kamakhya Prasad Ghatak, Sitangshu Bhattacharya (auth.)

Springer Tracts in Modern Physics 255
Kamakhya Prasad Ghatak
Sitangshu Bhattacharya
Debye Screening
Length
Effects of Nanostructured Materials
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Kamakhya Prasad Ghatak
Sitangshu Bhattacharya
Debye Screening Length
Effects of Nanostructured Materials
123
Kamakhya PrasadGhatak Sitangshu Bhattacharya
Department of Electronics Department of ElectricalEngineering
and CommunicationEngineering Shiv NadarUniversity
National InstituteofTechnology Noida,Uttar Pradesh
Agartala, Tripura West India
India
ISSN 0081-3869 ISSN 1615-0430 (electronic)
ISBN 978-3-319-01338-1 ISBN 978-3-319-01339-8 (eBook)
DOI 10.1007/978-3-319-01339-8
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From the desk of the first author:
I stand in front of my wife See,
And clearly she wants nothing from me,
But when she speaks through her eyes,
The very ‘‘I’’ becomes the sea
From the desk of the second author:
I dedicate this monograph to my beautiful
and wonderful wife Rekha Verma, the source
ofinspirationforplayinginthiscreativeplay
Preface
The unification of the asymmetry of the wave vector space of charge carriers in
semiconductors with modern techniques of fabricating nanostructured materials
such as MBE, MOCVD, and FLL in one, two, and three dimensions (such as
quantum wells (QWs), NIPI structures, inversion and accumulation layers, quan-
tum well superlattices, carbon nanotubes, quantum wires, quantum wire super-
lattices,magneticquantization,magnetosizequantization,quantumdots,magneto
inversion and accumulation layers, magneto NIPIs, magneto quantum well su-
perlattices, quantum dot superlattices, and other field aided low-dimensional
systems spawn not only useful quantum effect devices but also unearth new
concepts in the realm of low-dimensional solid-state science and related disci-
plines. It is worth remarking that these semiconductor nanostructures occupy a
central position in the entire arena of nanoscience and technology by their own
right and find extensive applications in quantum registers, quantum switches,
quantumsensors,quantumlogicgates,heterojunctionfieldeffectdevices,quantum
well and quantum wire transistors, quantum cascade lasers, high-speed digital
networks, high-frequency microwave circuits, high-resolution terahertz spectros-
copy, superlattice photo-oscillator, advanced integrated circuits, superlattice
photocathodes, resonant tunneling diodes and transistors, thermoelectric devices,
super-lattice coolers, thin film transistors, intermediate-band solar cells, micro-
optical systems, high performance infrared imaging systems, bandpass filters,
thermal sensors, optical modulators, optical switching systems, single electron/
molecule electronics, nanotube-based diodes, and other nanoelectronic devices.
Knowledge regarding these quantized structures may be gained from original
research contributions in scientific journals, various patents, proceedings of con-
ferences,reviewarticles,anddifferentresearchmonographs[1–5]respectively.In
this context, it may be noted that the available reports on the said areas cannot
afford to cover even an entire chapter excluding a few pages on the Debye
screening length (DSL) for nanostructured materials.
It is well known that the DSL of the carriers in semiconductors is a very
importantquantitycharacterizingthescreeningoftheCoulombfieldoftheionized
impurity centers by the free carriers [6]. It affects many of the special features of
modern nanodevices, the carrier mobilities under different mechanisms of scat-
tering, and the carrier plasmas in semiconductors [7–9]. The DSL is a good
approximation for the accurate self-consistent screening in presence of band tails
vii
viii Preface
andisalsousedtoillustratetheinteractionbetweenthecollidingcarriersinAuger
effectinsolids[6].TheclassicalvalueoftheDSLisequalto[e k T/(e2n )]1/2(e ,
sc B 0 sc
k , T, e, and n are the semiconductor permittivity, the Boltzmann’s constant, the
B 0
temperature, the magnitude of the carrier charge, and the electron concentration,
respectively, which is valid for both the carriers. In this conventional form, the
DSLdecreaseswithincreasingcarrierconcentrationataconstanttemperatureand
thisrelationholdsonlyundertheconditionofcarriernon-degeneracy.Ontheother
hand, under the condition of extreme degeneracy, the expression of DSL for
p
materials having parabolic energy bands can be written as L ¼ðp2=3(cid:2)h ffieffiffiffiffiÞ
D sc
p
ðeg1=331=6n1=6 ffimffiffiffiffiffiÞ(cid:2)1((cid:2)h;m and g are Dirac constant, effective electron mass at
v 0 c c v
the edge of the conduction band and valley degeneracy respectively. Thus we
observedthatinthiscasetheresultisindependentoftemperature,butdependson
n ,g andm .Besides,theindicesofinverseelectronvariationchangesfromhalf
0 v c
in the former case to one-sixth in the latter case. Since the performance of the
electron devices at the device terminals and the speed of operation of modern
switching transistors are significantly influenced by the degree of carrier degen-
eracy present in these devices, the simplest way of analyzing such devices taking
into account the degeneracy of the band is to use the appropriate DSL to express
theperformanceatthedeviceterminalandswitchingspeedintermsofthecarrier
concentration [10].
The DSL depends on the density-of-states function which, in turn, is signifi-
cantly affected by the different carrier energy spectra of different semiconductors
having various band structures. In recent years, various energy wave vector dis-
persionrelationsofthecarriersofdifferentmaterialshavebeenproposed[11–28]
which have created the interest in studying the DSL in such quantized structures
under external conditions. It is well known, from the fundamental study of
Landsberg [6], that the DSL for electronic materials having degenerate electron
concentrationisessentiallydeterminedbytheirrespectiveenergybandstructures.
Ithas,therefore,differentvaluesindifferentmaterialsandvarieswiththeelectron
concentration, with the magnitude of the reciprocal quantizing magnetic field
under magnetic quantization, with the quantizing electric field as in inversion
layers,withthenano-thicknessasinquantumwells,withsuperlatticeperiodasin
thequantumconfinedsuperlatticesofsmallgapcompoundswithgradedinterfaces
having various carrier energy spectra. The nature of these variations has been
investigatedintheliteratureby[29–49]andsomeofthesignificantfeatures,which
have emerged from these studies, are:
(a) The DSL in bulk semiconductors decreases with increasing carrier concen-
tration and such variations are significantly influenced by constants of the
energy band spectra;
(b) The DSL decreases with the increasing magnitude of the quantizing electric
field as in inversion layers;
(c) TheDSLoscillateswiththeinversequantizingmagneticfieldundermagnetic
quantization due to the SdH effect;
Preface ix
(d) TheDSLexhibitscompositeoscillationswiththevariouscontrolledquantities
as in superlattices of non-parabolic compounds with graded interfaces.
This book, divided into four parts, contains fifteen chapters, is partially based
onourongoingresearchesontheDSLfrom1980andanattempthasbeenmadeto
presentacrosssectionoftheDSLforwiderangeofnon-parabolicsemiconductors
andtheirnanostructureswithvaryingcarrierenergyspectraundervariousphysical
conditions. The first part deals with the influence of quantum confinement on the
DSLinnon-parabolicsemiconductors.Chapter1investigatestheDSLinquantum
wells of non-parabolic semiconductors and at first we study the DSL in QWs of
nonlinear optical materials on the basis of a generalized electron dispersion law
introducing the anisotropies of the effective masses and the spin-orbit splitting
constants, respectively, together with the inclusion of the crystal field splitting
withintheframeworkofthek.pformalism.TheresultsofIII–V(e.g.,InAs,InSb,
GaAs etc.), ternary (e.g., Hg Cd Te), quaternary (e.g., In Ga As P lattice
1-x x 1-x x 1-y y
matchedtoInP)compoundsformaspecial caseofourgeneralizedanalysisunder
certain limiting conditions. The DSL in QWs of II–VI, Bi, IV–VI, stressed Kane-
type semiconductors, Te, GaP, PtSb Bi Te , Ge, and GaSb materials have also
2, 2 3
been investigated by using the respective appropriate energy band structure. The
importance of the aforementioned semiconductors has also been described in the
same chapter. With the advent of modern experimental techniques offabricating
nanomaterials, it is possible to grow semiconductor superlattices (SLs) composed
of alternative layers of two different degenerate layers with controlled thickness
[50]. These structures have found wide applications inmany new devices such as
photodiodes [51], photoresistors [52], transistors [53], light emitters [54], tunnel-
ing devices [55], etc [56–69]. The investigations of the physical properties of
narrow gap SLs have increased extensively, since they are important for opto-
electronic devices and also since the quality of heterostructures involving narrow
gapmaterialshasbeengreatlyimproved.ItmaybenotedthattheNIPIstructures,
also called the doping superlattices, are crystals with a periodic sequence of
ultrathinfilmlayers[70,71]ofthesamesemiconductorwiththeintrinsiclayerin-
between together with the opposite sign of doping. All the donors will be posi-
tivelychargedandalltheacceptorsnegatively.Thisperiodicspacechargecausesa
periodic space charge potential which quantizes the motions of the carriers in the
z-direction together with the formation of the subband energies. The electronic
structures of the NIPIs differ radically from the corresponding bulk semiconduc-
tors as stated below:
a. Each band is split into mini-bands;
b. The magnitude and the spacing of these mini-bands may be designed by the
choice of the superlattices parameters; and
c. The electron energy spectrum of the NIPI crystal becomes two-dimensional
leading to the step functional dependence of the density-of-states function.
InChap.2theDSLinNIPIstructuresofnonlinearoptical,III–V,II–VI,IV–VI,
andstressedKane-typesemiconductorshasbeenstudied.Inrecentyears,therehas
x Preface
beenconsiderableinterestinthestudyoftheinversionlayerswhichareformedat
the surfaces of semiconductors in metal-oxide-semiconductor field-effect transis-
tors (MOSFET) under the influence of a sufficiently strong electric field applied
perpendicular to the surface by means of a large gate bias. In such layers, the
carriers form a two-dimensional gas and are free to move parallel to the surface
while their motion is quantized in the direction perpendicular to it leading to the
formationofelectricsubbands[72–84].InChap.3theDSLinn-channelinversion
layers of nonlinear optical, III–V, II–VI, IV–VI, stressed Kane-type semicon-
ductors, Ge and GaSb has been investigated. It is worth remarking that under
conditions of extreme degeneracy, electric quantum limit, and parabolic electron
dispersionlawofthecorrespondingbulkmaterials,the2DDSLfor2Dsystemsof
inversionlayers,ultrathinfilmsandNIPIstructures,assumesthecommonformas
L ¼½ð2e p(cid:2)h2Þ=ðe2m g Þ(cid:3). Thus, the result is independent of carrier concentra-
2D sc c v
tion, temperature, and the signature of the nature of the specific 2D system is
totally absent, although for non-parabolic energy bands both the result and con-
clusiondifferwidely.Underconditionofnon-degeneracy,screeninglengthforall
the three aforesaid 2D systems can be written as L ¼½ð2e k TÞ=ðe2n Þ(cid:3) (n is
2D sc B s s
the surface electron concentration per unit area). Thus, we infer that in the pre-
viouscase,theresultisindependentofelectronstatisticsandtemperaturewhereas
in the latter case L varies inversely as n at constant temperature exhibiting
2D s
rectangular hyperbolic variation.
With the advent of nanophotonics, there has been considerable interest in
studying the optical processes in semiconductors and their nanostructures in the
presence of intense light waves [85]. It appears from the literature that the
investigations have been carried out on the assumption that the carrier energy
spectraareinvariantquantitiesinthepresenceofstrongexternalphotoexcitation,
whichisnotfundamentallytrue.Thephysicalpropertiesofsemiconductorsinthe
presence of strong light waves which alter the basic dispersion relations have
relatively been much less investigated in [86, 87] as compared with the cases of
other external fields needed for the characterization of the low-dimensional
semiconductors. The second part of this book studies the influence of light waves
on the DSL in optoelectronic semiconductors and Chap. 4 investigates the influ-
ence of light waves on the DSL in III–V, ternary and quaternary semiconductors
by formulating new electron dispersion relation within the framework of k.p
formalism. Chapter 5 explores the effect of light waves on the DSL for ultra-thin
films of III–V, ternary, and quaternary semiconductors. Chapters 6 and 7 inves-
tigate the opto-DSL under magnetic quantization and also under cross-field con-
figurations of the said materials respectively.
Withtheadventofnanodevices,thebuilt-inelectricfieldbecomessolargethat
the electron energy spectrum changes fundamentally and the solo Chap. 8 in the
third part investigates the DSL under intense electric field in bulk specimens of
III–V, ternary and quaternary semiconductors. This chapter also explores the
influence of electric field on the DSL on the basis of new dispersion law under
magnetic quantization, size quantization, NIPI structures, inversion layers,