Semirelativity in Semiconductors: a Review

1.6 MB·

by  Wlodek Zawadzki

Semirelativity in Semiconductors: a Review
Wlodek Zawadzki∗
Instituteof Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland
An analogy between behavior of electrons in narrow-gap semiconductors (NGS) and relativistic
electrons in vacuum is reviewed. Energy band structures ε(k) are considered for various NGS
materials and their correspondence to the energy-momentum relationinspecial relativityis em-
phasized. It is indicated that special relativity for vacuum is analogous to a two-band k·p
7 description for NGS. The maximum electron velocity in NGS is u ≃ 1×108cm/s, which corre-
1 sponds to the light velocity in vacuum. An effective mass of charge carriers in semiconductors
0 is introduced, relating their velocity to quasimomentum and it is shown that this mass depends
2 on electron energy (or velocity) in a way similar to the mass of free relativistic electrons. In
Hg1−xCdxTealloysonecanreachvanishingenergygapatwhichelectronsandlightholesbecome
n three-dimensional massless Dirac fermions. A wavelength λz is defined for NGS, in analogy to
a the Compton wavelength in relativistic quantum mechanics. It is estimated that λz is on the
J orderoftensofAngstromsintypicalsemiconductingmaterialswhichisexperimentallyconfirmed

4 intunnelingexperiments onenergy dispersioninthe forbiddengap. Statistical properties of the
2 electron gas in NGS are calculated and their similarity is demonstrated to those of the Juttner
gasofrelativisticparticles. InterbandelectrontunnelinginNGSisdescribedandshowntobein
] close analogy to the predicted but unobserved tunneling between negative and positive energies
i resultingfromtheDiracequationforfreeelectrons.
c
It is demonstrated that the relativistic analogy holds fororbital and spinproperties of electrons
s
in the presence of an external magnetic field. In particular, it is shown that the spin magnetic
-
l momentofbothNGSelectrons andrelativisticelectrons approaches zerowithincreasingenergy.
r
t This conclusionis confirmed experimentallyfor NGS. Electrons incrossedelectric andmagnetic
m fields are described theoretically and experimentally. It is only the two-band description for
NGS, equivalent to the Dirac or Klein-Gordon equations for free particles, that gives a correct
.
t account of experimental results in this situation. A transverse Doppler shift in the cyclotron
a
resonanceobserved incrossedfieldsinInSbindicates that thereexistsatimedilatationbetween
m
an oscillating electron and an observer. The phenomenon of Zitterbewegung (ZB, trembling
- motion) for electrons in NGS is considered theoretically, following the original proposition of
d Schro¨dinger for free relativistic electrons in vacuum. The two descriptions are in close analogy,
n butthefrequencyofZBforelectronsinNGSisordersofmagnitudeloweranditsamplitudeorders
o of magnitude higher making possible experimental observations in semiconductors considerably
c more favorable. Finally, graphene and carbon nanotubes, as well as topological insulators are
[ considered in the framework of relativistic analogy. These systems, with their linear energy-
quasimomentum dispersions,illustratethe extreme semirelativisticregime. Experimental results
1 for the energy dispersions and the Landau quantizations in the presence of a magnetic field are
v quoted and their analogy to behavior of free relativistic electrons is discussed. Approximations
7 andrestrictionsoftherelativisticanalogyareemphasized. Ontheotherhand,itisindicatedthat
6 invarioussituations itisconsiderablyeasiertoobservesemirelativisticeffects insemiconductors
0 thantherelativisticeffects invacuum.
7
0
.
1 I. INTRODUCTION field,spinmagneticmomentsofchargecarrierscomeinto
0 play, characterized by the corresponding spin g-factors.
7 Effective values of g∗ in semiconductors can be dramati-
1 Since the beginning of solid state physics, theoret- callydifferentfromthe free-electronvalueg =+2dueto
v: ical description of movable electrons and holes used the effect of spin-orbit interactionand small energy gap,
i physical quantities borrowed from free particles. In as first demonstrated by Roth et al. (1959).
X
practice it meant that the motion of electrons and
r holes was described in analogy to free charged parti- However, the effects of periodic potential are not lim-
a
cles, with the free mass replaced by an effective mass ited to the effective values of m∗ and g∗, particularly in
which reflected the influence of periodic lattice poten- narrowgapsemiconductors(NGS). Kane(1957)was the
tial. A theoretical basis for the effective-mass approach first to show that, in the energy bands of narrow-gap
was provided by Luttinger and Kohn (1955) followed material InSb, one deals with a nonquadratic ε(k) de-
by Zak and Zawadzki (1966). It turned out, somewhat pendence, where ε is the electron energy and ℏk is the
counter-intuitively, that the effective masses m∗ of elec- absolute value of quasimomentum. This feature, called
tronsinsemiconductorsareconsiderablysmallerthanthe band’s nonparabolicity, also results from small value of
free electron mass. In the presence of external magnetic the gap and can be quite well described by a so called
two-bandmodel of interacting conduction and light-hole
bands. ItwasremarkedbyKeldysh(1963),Wolff(1964),
Zawadzkiand Lax(1966),Aronov and Pikus(1966)that
∗Electronicaddress: [email protected] the two-band (or two-level) model of the k p theory
·
2
is analogous to the Dirac equation for relativistic elec- band’s extremum (we take k=0 for simplicity). It then
trons in vacuum. For over 50 years this analogy be- follows from Eq. (1) that the LK amplitudes satisfy the
tween the behavior of electrons in narrow gap semicon- eigenvalue equations
ductors and that of relativistic electrons in vacuum has
been traced for various physical situations, culminating pˆ2
+V (r) u =ε u (4)
0 l0 l0 l0
inrecentyearswiththediscoveriesofgrapheneandtopo- (cid:20)2m0 (cid:21)
logicalinsulators. The present article reviews this effort,
where ε is the energy of the l’th band at k = 0. The
see also Zawadzki(1970, 1997). l0
amplitudes u are orthonormal
We describe various “semirelativistic” effects for elec- l0
trons in semiconductors and indicate corresponding fea- 1
tures for relativistic electrons with appropriate refer- ul′0 ul0 =δl′l (5)
Ωh | i
ences. We mention cases in which, in contrast to obser-
vations in semiconductors, relativistic effects in vacuum whereΩisthevolumeoftheunitcellandtheintegration
have not been observed. Some of the basic derivations is over Ω. One proves that the LK functions represent a
for semiconductors are given without going into details, complete set, with l running over all energy bands. This
others are simplified in order to make basic ideas more means that one can expand the Bloch state for the nth
easily accessible. Limitations of the relativistic analogy bandintermsoftheLKfunctions(indexnisomittedon
are emphasized. An effort has been made to quote the both sides)
important semiconductor work on the subject.
The review is organized as follows. First, we intro- Ψk = cl(k)uik·rul0(r) (6)
duce the relativistic analogy and follow its consequences l
X
for semiconductor electrons in absence of external fields.
where the sum runs over all the bands l. One performs
Next,semiconductorelectronsareconsideredinthepres-
the elementary operations following from Eqs. (1) and
ence of a magnetic field and in crossedelectric and mag-
(6), and uses Eq. (4). Multiplying both sides by u∗ /Ω
netic fields. In the third part we describe separately l′0
forl′ =1,2,3,...andintegratingoverΩ,oneobtainsthe
graphene, carbon nanotubes, and topological insulators,
k p theory in the form
astheyarecharacterizedbyreduceddimensionalitiesand ·
possessquite specialproperties. Thereviewis concluded ℏ2k2 ℏk
by a discussion. εl0 ε+ δl′l+ pl′l cl =0 (7)
− 2m m
l (cid:20)(cid:18) 0(cid:19) 0 (cid:21)
X
wherel′ =1,2,3,...andthemomentummatrixelements
II. BAND STRUCTURES IN SEMICONDUCTORS.
are
RELATIVISTIC ANALOGY
1
We begin by considering charge carriers in semi- pl′l = ul′0 pˆ ul0 . (8)
Ωh | | i
conductors in the absence of external fields. In this
case one can use for their description the k p theory Equation (7) represents the k p theory for the ener-
(Luttinger and Kohn,1955). Wepresentither·einasim- giesεn(k)andtheexpansioncoeffi·cientscnl fortheBloch
ple form neglecting spin effects, in order to come across stateΨnk. Thephenomenologicalparametersareεl0 and
with the basic idea. The initial eigenvalue equation for pl′l. Ifallthe bandsareincluded,set(7)containsnoap-
an electron in a periodic potential reads proximations.
It can be seen that, apart from the free-electron term
pˆ2 ℏ2k2/2m , the structure of Eq. (7) looks very much like
+V (r) Ψ=εΨ (1) 0
0
2m theDiracequation(DE):ithasband-edgeenergiesonthe
(cid:20) 0 (cid:21)
diagonaland quasiomomenta ℏk off the diagonal(in DE
where pˆ is the momentum operator, m is the free elec-
0 one deals with the momenta). This similarity provides
tronmass,V (r)istheperiodicpotentialofacrystallat-
0 the basis for the relativistic analogy.
tice and ε is the energy. It is well known that solutions
Initsfull form(7),the k p matrixhasinfinite dimen-
of Eq. (1) are the Bloch functions ·
sions, so finding its complete solutions is not a practi-
Ψnk =eik·runk(r) (2) cal task. Following Luttinger and Kohn (1955) one can
separate the energy bands by using second-orderpertur-
in which ℏk is the quasimomentum and unk(r) is the bation theory and show that, in the parabolic ε(k) ap-
periodic Bloch amplitude for the n’th energy band. The proximation, each band is characterized by an effective
BlochstatesareexpandedintermsoftheLuttinger-Kohn mass. However,one can take a different route, first indi-
(LK) functions cated by Kane (1957). Namely, one can neglect distant
χlk =eik·rul0(r) (3) bandsandfindapproximatesolutionsforafinitenumber
ofbandsseparatedbysmallenergygaps. Thisprocedure
where u are the periodic LK amplitudes given by the correspondstothe perturbationtheoryfornearlydegen-
l0
Bloch amplitudes taken at the k value corresponding erateenergylevels. The simplestapproximationwithout
3
spinisthetwo-bandmodel(ortwo-levelmodel)forInSb- respectively, the LK amplitudes
type materials in which one takes into account at k = 0
the conduction level of s-like symmetry and the valence u =iS, (9)
10
triple-degeneratelevelofp-likesymmetry,separatedfrom
each other by the energy gap ε . This approximation is u =(X iY)/√2, u =Z, u =(X +iY)/√2.
g 20 30 40
−
justified if the gap ε is distinctly smaller than energy (10)
g
separations from other bands. We choose the zero of en-
ergy in the middle of the gap. Thus, at k = 0, one has Taking into accountsymmetry of the momentum matrix
for the conduction and valence levels ε /2 and ε /2, elements,oneobtainsthereducedk pmatrixintheform
g g
− ·
ε′+ε /2 Cℏk Cℏk Cℏk c
g − z + 1
−
 Cℏk ε′ ε /2 0 0 c 
+ g 2
− − =0 , (11)
 Cℏk 0 ε′ ε /2 0 c 
 z − − g  3 
 Cℏk− 0 0 ε′ εg/2c4 
 − −  
  
where C = ( i/m ) S pˆ X is a real number, ε′ = ε
0 x
− h | | i −
ℏ2k2/2m and k = (k ik )/√2. One solves set (11)
0 ± x y
±
and obtains four energy roots
ε 2 ℏ2k2 21
ε′ = g +ε (12)
1,2 ± 2 g2m∗
(cid:20)(cid:16) (cid:17) 0(cid:21)
ε′ = ε /2, (13)
3,4 − g
1 2C2
= . (14)
m∗ ε
0 g
The roots ε′ correspond to the conduction and light-
1,2
hole bands, while ε′ correspond to two heavy-hole
3,4
bands. It turns out that 1/m∗ of Eq. (14) is usually
0
muchlargerthan1/m , so thatone canneglectthe free-
0
electron term in Eq. (12) and put ε′(k) ε(k) and
ε′ ε /2. In this approximation we hav≃e finally for
3,4 ≃ − g
the conduction and light-hole bands
ε 2 ℏ2k2 12
g
ε = +ε (15)
1,2 ± 2 g2m∗
(cid:20)(cid:16) (cid:17) 0(cid:21)
where m∗, given by Eq. (14), is the effective mass of
0
electrons and light holes. In the above model the result-
ing bands are spherical, i.e. the energy depends on the
absolute value of k, not on its direction.
Equation (15) presents an important result. First, for
not too large k values one can expand the square root
and take the first two terms. This gives ε1,2 = (εg/2+ FIG. 1 Energy versus quasimomentum for the two-band
ℏ2k2/2m∗), which is the standard parabolic ap±proxima- model of a semiconductor (schematically). For the Dirac
tion to th0e conduction and light-hole bands. Second, for equation the gap is 2m0c2 and the “bottom of the band”
sufficientlylargekvaluesonecanneglectthefirsttermin mass is m0.
the square root and obtain ε = (ε /2m∗)1/2ℏk. This
1,2 ± g 0
linear k approximationis valid for not too large k values
since it is known that, near the Brillouin zone bound- andthis is our essentialpoint, formula (15) has the “rel-
aries, the ε(k) dependence must be horizontal. Third, ativistic” character. The energy-momentum relation re-
4
sulting from the Dirac equation for relativistic electrons
in vacuum is (Dirac, 1958)
1
2m c2 2 p2 2
ε= 0 +2m c2 . (16)
0
±"(cid:18) 2 (cid:19) 2m0#
It is seen that the dispersion relations (15) and (16)
have identical forms with the following correspondence
2m c2 ε (17)
0 g
→
m m∗ (18)
0 → 0
p ℏk (19)
→
The correspondenceexpressedin Eqs.(15) and (16) is
illustrated in Fig. 1. Energy gaps in semiconductors are
manyordersofmagnitudessmallerthan2m c2 1MeV
0
andtheeffectivemassesm∗areconsiderablysma≃llerthan
0
m . Expression (19) suggests that the quasimomentum
0
of semiconductor electrons corresponds to the momen- FIG. 2 Band structures of mixed crystals Hg1−xCdxTe for
tum of electrons in vacuum. We emphasize that, while increasing composition x which change from the “reversed”
band ordering in HgTe to the “straight” ordering in CdTe
the quasimomentum has indeed many properties of the
momentum, this is not always the case, as we will show (schematically). For zero interaction gap between Γ8 and
Γ6 bands one deals with linear ε(k) dispersions and three-
below in the description of Zitterbewegung. The quasi-
dimensional “massless” electrons and light holes.
momentum appearsfor electronsin solidsinstead ofmo-
mentum because in the presence of a periodic potential
the Bloch states are characterized by this quantity, see
that the relativistic analogy has a rather formal charac-
Eq. (3) and Zawadzki (2013).
ter. Still, it leads to many physical similarities.
One can use the same procedure for HgTe-type
Ternary alloys Hg Cd Te offer an interesting possi-
1−x x
of materials: α-Sn, HgSe, HgS, in which the p-like
bility which also can be interpreted in terms of the rela-
triple-degenerate level is above the s-like level, see
tivistic analogy. CdTe has the “straight” band ordering
Groves and Paul (1963). One obtains solutions of this
of InSb-type while HgTe has the “inverted” band order-
problem by simply replacing ε by ε , where ε is pos-
g 0 0 ing. By producing ternary alloys of these two materials
−
itive. In this case the relativistic analogy still holds for
indifferentchemicalproportionsonecangocontinuously
the conduction and light-hole bands. It is seen that, as
from one band structure to the other. In particular, one
far as the nonparabolicityof the bands is concerned, the
canreachthesituationinwhichtheenergyseparationbe-
role of ε is now played by ε , often called “the interac-
g 0 tween s-like and p-like levels, i.e. the interaction gap, is
tion gap”. The main difference from the previous case
exactly zero, see e.g. Dornhaus and Nimtz (1976). This
is that now at k = 0 the conduction band is degenerate
occurs for x = 0.165 at 300K. The two-band model
with two heavy-hole bands. For this reason the HgTe-
describes this case very well. For ε = 0 the first term
g
type materials are often called zero-gapsemiconductors.
underthesquarerootinEq.(15)vanishes. Inthesecond
Wearenowinapositiontoenumerateapproximations termthegapcancelsout,seeEq.(14),andthedispersion
andrestrictionsnecessarytousethesemirelativisticanal- becomes
ogy of Eqs. (17) – (19). As to approximations, we have
neglecteddistantbandskeepingonlyones-likeandonep- ε(k)= Cℏk (20)
±
like level. This results in the two-bandmodel, the heavy
holes also appearing but not being a part of the picture. i.e. it is linear in k. In terms of the relativistic analogy
WehavealsoneglectedthefreeelectronterminEq.(15). thismeansthatonedealswiththe “extremerelativistic”
Astorestrictions,thetwo-bandmodelisvalidfornottoo situation. The transition from straight band ordering to
large k values, approximatelyup the inflection points on the inverted one in the HgCdTe system is illustrated in
the complete ε(k) curve. Second, the sphericity of the Fig. 2.
bands described in Eq. (15) is related to the fact that in In narrow gap lead chalcogenides PbTe, PbSe, PbS
III-V InSb-like and II-VI HgTe-like materials the band andtheirternaryalloystheconductionbandminimaand
extrema of our interest occur in the center of the Bril- valencebandmaximaareatthe Lpointsofthe Brillouin
louinzone (the Γ point). Inother materials,for example zone. Asaconsequence,the ε(k)dispersionrelationsare
in IV-VI lead chalcogenides, this may not be the case. not spherical but spheroidal, the complete bands having
Finally, as we show below, in semiconductors one does cubicsymmetry. However,thetwo-bandk p description
·
not deal in reality with relativistic electron velocities, so is still valid for not too high electron energies. Choosing
5
the zero of energy in the middle of the gap one has, see oftheenergywithrespecttoquasimomentum,seebelow.
Dimmock and Wright (1964) However, we show that a considerably more useful defi-
nitionofthe massm∗ comesfromarelationbetweenthe
ε 2 ℏ2k2 ℏ2k2 21 electronvelocityv andits quasimomentumℏk. Thus we
ε= g +ε ⊥ + k , (21) define the “velocity” effective mass
±" 2 g 2m∗⊥0 2m∗k0!#
(cid:16) (cid:17) mˆ∗v=ℏk, (23)
where m∗ and m∗ effective masses describe the band
⊥0 k0
ellipsoids. One can obtain the spherical description in where mˆ∗ is generally a tensor. There is
k by performing a transformation of k and k compo-
⊥ k ∂ε 1 dεk 1 dε
nents, see Aronov and Pikus (1966). v = = i = k δ , (24)
i ∂ℏk ℏdk k ℏkdk j ij
TernaryalloysofPbSnSeandPbSnTealsoofferapos- i
j
X
sibility of going through zero energy gap, see Strauss
(1967), Nimtz and Schlicht (1983). As follows from wherethe summationisoverthe coordinates: j =1,2,3.
Eq. (21), in this situation one also deals with the linear On the other hand, there is
dispersionε(k). Since in leadsalts the band structure in
each material depends strongly on the temperature, one vi =ℏ kj(m∗)−ij1 (25)
can reach vanishing gaps as functions of T. j
X
We conclude this section by remarking that the band
descriptions of Eqs. (15) and (20) offer better approxi- in which (m∗)−ij1 is the inverse mass tensor. Comparing
mationstorealε(k)dispersionsthanmayappearatfirst Eqs. (24) and (25) one obtains
glance,seeZawadzki and Kol odziejczak(1964). Suppose
1 dε
one deals with a spherical energy band of an almost ar- (m∗)−1 = δ , (26)
bitrary form ε(k) having a minimum at the k=0 point. ij ℏ2kdk ij
This occurs, for example, when one takes into account
where δ is the Kronecker delta. This means that the
ij
more bands in the k p theory. In this situation one can
inversetensorofthe effective massfora sphericalenergy
expand quite general·ly the k2(ε) dependence in a power
band is a scalar. As a consequence, Eq. (26) becomes
series of energy
1 1 dε
k2(ε)=λ0+λ1ε+λ2ε2+λ3ε3+... . (22) m∗ = ℏ2kdk. (27)
It is seen that, taking the first two terms of this expan- The constitutive equality (23) with the scalareffective
sion, one obtains the standard parabolic description of mass m∗ takes the form
an energy band. However, taking the first three terms
and solving the resulting quadratic equation for the en- m∗v=ℏk. (28)
ergy one obtains an ε(k) relation of the type given by
Bywritingdownthethreecomponentsoftheaboveequa-
Eq. (15). The conclusion of this reasoning is that the
tion, squaring them , adding and taking the square root
firstnonparabolicapproximationtoanyε(k)dependence
ofboth sides, one shows thatthe relation(28) holds also
is always of the relativistic type. Also the linear ε(k)
for the absolute values of velocity and quasimomentum,
dispersions are accounted for by taking λ = λ = 0
0 1 i.e.
and λ = 0. Thus, the above two-band formulas are
2
6
just special cases of the general rule. It is clear that the m∗v =ℏk (29)
above phenomenological expansion is valid also for two-
dimensional and one-dimensional systems. The above velocity mass appears in the cyclotron reso-
nance and, most important, in the transport theory. On
the other hand, the mass relating force to acceleration:
III. CONSEQUENCES OF BAND STRUCTURE F=Mˆ∗a, is given by
We now follow the first physical consequences of the 1 ∂2ε
(M∗)−1 = . (30)
semirelativistic ε(k) relation given by Eq. (15). ij ℏ2∂k ∂k
i j
The mass M∗ is less useful than the mass m∗ defined
A. Effective Mass above. In particular, it is not a scalar for a spherical
energy band.
We begin with the important concept of the effective We cannowdetermine the maximumelectronvelocity
mass of charge carriers. In standard textbooks on semi- in energy bands described by the two-level model. One
conductors,the effective mass of electrons or holes is de- can use Eq. (15), calculate the velocity from Eq. (24)
fined by a relation between force and acceleration. This andtakeitslimitforlargek values. Butasimplerwayis
leads to the inverse mass given by the second derivative to use directly the correspondence of Eqs. (17) and (18)
6
and obtain the same result. According to relativity, the
maximum electron velocity is c. We can write
c= 2m0c2/2m0 21 →(εg/2m∗0)21 =u. (31)
Clearly, one (cid:0)obtains the s(cid:1)ame for the light holes. The
maximum velocity u in a material can be estimated by
using experimentally determined values of ε and m∗. It
g 0
turns out that the resulting value is u 108cm/s. Fur-
≃
ther,thisvalueisverysimilarforvarioussemiconducting
materials. Thus u is about 300 times smaller than the
light velocity c indicating that we do not deal in semi-
conductorswithtrulyrelativisticvelocities. Thevalueof
u is reasonable since typical electron velocities in semi-
conductors are of the order of 107cm/s. By using u one
can write the dispersion relation (15) in the form
1
ε= (m∗u2)2+u2(ℏk)2 2 , (32) FIG. 3 Electron effective mass in InSb versus free electron
± 0 density,asmeasuredatroomtemperaturebyvariousauthors.
whichisevenmorer(cid:2)eminiscentofthere(cid:3)lativisticformula Solidline,calculatedforthetwo-bandmodel,representsmass
(16), with u replacing c and ℏk replacing p. It is known values at the Fermi energy. The latter, counted from the
that in special relativity one also defines the mass by bandedge,isindicatedontheupperabscissa. AfterZawadzki
the relation between velocity and momentum: p = mv. (1974).
This velocity mass is a scalar and it depends on the ve-
locity (or energy). The relativistic increase of the elec-
Combingthis withEq.(34)andusingthe definition (29)
tron mass with growing velocity has been observed in
for the effective mass m∗, we obtain
many experiments, see e.g. Guye and Lavanchy (1915),
Rogers et al.(1940),Kaplan(1955). Objectionsconcern-
m∗v(1 v2/u2)=ℏk, (37)
ing the use of velocity-dependent mass in the theory of 0 −
relativityhavebeenexpressedintheliterature,assucha
from which it follows
massdoesnothavepropertransformationproperties,see
e.g. Ginzburg (1979). However, in semiconductors the m∗ =m∗(1 v2/u2)−21, (38)
energy-dependent effective mass is an important physi- 0 −
cal quantity. which is in direct analogy with the relativistic formula
One can determine an explicit form of the velocity quoted above.
mass within the two-band model. Calculating dε/dk =
Now we give a few experimental examples for the
ε ℏ2k/(ε2m∗) and using Eq. (31) for u, we get
g 0 energy-dependent effective masses of electrons in semi-
conductors which illustrate their semirelativistic behav-
m∗(ε)=m∗(2ε/ε )=ε/u2. (33)
0 g ior. At low temperatures the mass is usually measured
Thusthemassisproportionaltotheenergycountedfrom at the Fermi energy within the band. As a consequence,
the middle of the gap. It then follows oneoftenplotsthemassasafunctionoftheelectronden-
sity which determines the Fermi energy. Figure 3 shows
ε=m∗u2. (34) the effective masses of electrons in InSb, as measured
byvariousauthors,plottedversustheelectrondensityn.
This formula is in a direct analogy with the famous rela-
tion of special relativity: E = mc2. In both expressions ThecorrespondingvaluesoftheFermienergy,ascounted
fromthe conductionband edge, areindicated onthe up-
theenergyisgivenbythevelocitymassmultipliedbythe
per abscissa. It is seen that the experimental values are
square of maximum velocity.
in good agreement with the predictions of the two-band
In the theory of relativity one often quotes the depen-
model. Somewhat too high experimental values at low
denceofthemassonvelocity: m(v)=m (1 v2/c2)−1/2.
0 − densities result from the fact that, at the room temper-
Wecanderiveananalogousformulafortheeffectivemass
ature, the electron gas is not degenerate, so the average
ofcarriersinasphericalenergybandwithinthetwo-band
model. Equation (15) gives: v =(u2/ε)ℏk , from which measured masses are higher than those at the Fermi en-
i i ergy. At the highest densities n = 1019cm−3 the Fermi
one easily gets
energy is about 0.5eV above the band edge while the
v2ε2/u4 =(ℏk)2. (35) gap value at 300K is around 0.18eV, so that one is well
within the nonparabolic (or semirelativistic) regime.
On the other hand
The situation is qualitatively similar for electrons in
v2ε2 (ℏk)2v2 GaAs. This material is a medium-gap semiconductor,
u2 =m∗02v2+ u2 . (36) having εg 1.5eV. As the electron density n increases
≃
7
FIG. 4 Electron effective mass in HgSe versus free electron
density. The solid line is calculated for the “inverted” band
model. AfterKonczykowski(1974),seealsoZawadzki(1974).
from1016 to1020cm−3,theelectroneffectivemassm∗ at
the Fermienergyincreasesfrom0.065m to 0.13m , see
0 0
Pfeffer and Zawadzki (1990). This example illustrates
the fact that, what matters for the semirelativity is not
FIG.5 Band-edgeelectroneffectivemassatT =4.2Kversus
so much an absolute value of the gap, but rather the ra- the interaction gap in Hg1−xCdxTe alloy system. At x =
tio of electron energy above the band edge to the gap 0.165 the mass goes to zero. The value of ε0 = 300meV
−
value. Electrons in any material will exhibit semirel- corresponds to HgTe. After Guldner et al. (1977), see also
ativistic properties if excited sufficiently high into the Rigaux (1980).
band.
The situation is quite similar for the “inverted” band
structure of HgTe-type semiconductors, as discussed
above. Figure 4 illustrates the increase of electron mass
in HgSe. In this materialone cannot reach electron den-
sities below 2 1017cm−3.
× It is seen that, indeed, in agreement with Eq. (39), the
As we mentioned above, in HgCdTe one can have a
mass is linear in n1/3, i.e. it is proportional to k.
vanishing interaction gap, see Fig. 1. For the result-
ing “ultrarelativistic” linear ε = uℏk dependence, see
Eq. (20), one obtains from Eq. (29) the effective mass
As we mentioned before, in lead chalcogenides and
m∗ =ε/u2 =ℏk/u. (39) their mixed crystals the bands are ellipsoidal and one
deals with transverse and parallel effective masses, see
It follows that for ε = 0 the mass vanishes. Thus, at Eq. (21). It follows from formula (21) that both masses
thebandedgeonedealswiththree-dimensional“massless dependonthe energythesameway. Itisimportantthat
fermions”. However, as the energy increases the mass is alsointhis caseonecandefinethe effectivemassesrelat-
nonzero. From Eq. (39) we obtain again the relation ing electron velocity to its quasimomentum (Zawadzki,
ε = m∗u2. For the vanishing gap the mass is a linear 1974; Z´ukotyn´ski and Kolodziejczak, 1963). In order to
function of quasimomentum ℏk. This relation can be follow the energy dependence of the masses one defines
directly verified. the density-of-states effective mass m∗ =42/3(m2m )1/3,
d ⊥ k
Equation (39) indicates that the effective mass at the see Zawadzki (1974). Figure 7 shows the concentration
band edge is proportional to the interaction gap ε (or dependence of m∗ masses in PbS and PbTe, as mea-
g d
ε ). Figure5showstheelectronmassm∗inHg Cd Te sured by various authors. The straight lines are cal-
0 0 1−x x
at T = 4.2K versus the interaction gap going through culated for the spheroidal two-band model. It is seen
zero for chemical composition x 0.165. On the other that the latter describes the data very well. Similar
≃
hand, Fig. 6 illustrates dependence of the mass on the agreement is obtained for PbSe (Zawadzki, 1974). This
wave vector k for HgCdTe in the zero-gap situation, as demonstrates that the semirelativity generalized to the
given by Eq. (39). For the degenerate electron gas the spheroidal bands works for the narrow-gap lead chalco-
electron density is n=(1/3π2)k3, so that k =(3π2n)1/3. genides.
8
∗2
FIG. 7 Density-of-states mass m in PbS and PbTe at
d
T = 77K versus free electron density n2/3. Experimental
data taken by various authors, straight lines calculated for
thegeneralized two-band model. After Zawadzki (1974).
FIG. 6 Electron effective mass in Hg0.9Cd0.1Te at T =
300Kversuselectrondensityn1/3 demonstratinglinearvaria-
tion of the mass with quasimomentum, see Eq. (39). This
corresponds to the “extreme relativistic” regime. After
Galazka and Sosnowski (1967), see also Zawadzki (1974).
B. Dispersion in the Gap
One of important physical quantities in the rela-
tivistic quantum mechanics for electrons is the Comp-
ton wavelength λ = ℏ/m c = 3.8 10−3˚A, see e.g.
c 0
×
Bjorken and Drell (1964). There exists a corresponding
quantityλ inthe two-bandk p descriptionofelectrons
z
·
in NGS whichis defined with the use of relativisticanal-
FIG. 8 Energy versus imaginary wave vector squared in
ogy (Zawadzki, 1997, 2005b): the forbidden gap of InAs. Experimental data are after
Parker and Mead (1968), solid line is a theoretical fit after
λz = mℏ∗0u =ℏ(cid:18)m∗02εg(cid:19)21 . (40) aEnqd. (u41=).1.3T3h×e a1d08jucsmte/ds.pAarfatemreZtearwsaadrzekiλ(z20=054b1)..5angstrom
In NGS one can have m∗ 5 10−2m and since u
csi/z3a0b0le, olennegtohbtfoarinssemλzico≃n0d2≃u×cto1×r0n4λacno≃str05u0ct˚Au.resT.his is≃a LEeqt. (u4s1)codnessicdreibretshtehecacsoenkdxuc=tionkyan=d l0ig.htFohrolekz2ba≥nd0s.,
The dispersion relation(15) can be rewritten in terms For k2 < 0, that is for imaginary values of k = iκ ,
z z z
of u and λz in the form, see Zawadzki (2005b) this equation describes the dispersion in the forbidden
energy gap. The length λ can be determined experi-
1 z
ε(k)= ℏu λ−2+k2 2 . (41) mentally. The gap region is classically forbidden, but
± z
(cid:0) (cid:1)
9
it can become accessible through one-dimensional quan-
tum tunneling. Figure 8 shows the dispersion in the
gap of InAs obtained from tunneling experiments with
double Schottky barriers. The solid line in Fig. 8 in-
dicates the fit obtained using Eq. (41). The value of
λ is determined directly by k for which the energy
z z0
vanishes: λ−2 = k2 . This gives λ = 41.5˚A and
z z0 z
u = 1.33 108cm/s, in good agreement with the above
×
estimations. Similar experiments were carried in GaAs
and general picture of the dispersion in the gap is sim-
ilar to that shown in Fig. 8, see Padovaniand Stratton
(1966), Conley and Mahan (1967), Pfeffer and Zawadzki
(1990).
FIG. 9 Mean internal energy and mean specific heat of the
Coming back to the relativistic electrons in vacuum,
nondegenerate electron gas calculated for a nonparabolic en-
one obtains from Eq. (16) the dispersion analogous to
ergy band versus β =k0T/εg. For β =0 the “classical” val-
Eq. (41) ues of 3/2 are obtained. After Zawadzki and Ko lodziejczak
(1964), see also Zawadzki (1974).
1
E = c ℏ2λ−2+p2 2 . (42)
± c
For energies in the gap,(cid:0) i.e. for p2(cid:1) < 0, it takes the computed numerically without much difficulty. In par-
z
one-dimensional form similar to that shown in Fig. 8, ticular,atT =0onedealswiththe completedegeneracy
properly scaled. To our knowledge, there have been no of the electron gas, so that ∂f0/∂ε=δ(ε ζ), where δ
− −
attempts to investigate this relation for relativistic elec- is the Dirac delta function. The integral (44) is then re-
trons in vacuum. ducedtoonepointatε=ζ. Thisgiveskζ =(3π2)1/3n1/3.
The above relation was used in Fig. 6 to show that the
effectivemassinHgCdTeforvanishingenergygapispro-
IV. STATISTICAL PROPERTIES OF ELECTRON GAS portional to k.
However, we shall be interested here in the nondegen-
Description of statistical and thermodynamic prop- erateelectrongasobeyingtheMaxwell-Boltzmann(MB)
erties of carriers in the relativistic-like energy bands statistics. In this limit there is f0 = ∂f0/∂z. Let us
−
of narrow-gap and middle-gap semiconductors requires calculateinternalenergyoftheelectrongasUMB forthis
a specialized mathematical formalism, as reviewed by situation (Zawadzki, 1974; Zawadzkiand Kol odziejczak,
Zawadzki (1974). Here we present limited results which 1964). Carrying integration by parts one obtains
can be directly compared to those for the relativistic
Juttner gas, see Ju¨ttner (1911), Synge (1957), Arzeli`es U = 2 f εd3k =nk T z d , (45)
(1968). We begin with the simplest calculation of a MB (2π)3 0 0 dz
Z (cid:18) (cid:19)
three dimensional electron concentration n in an arbi-
trary spherical energy band ε(k). It is assumed that where, in general, an average value is defined as A¯ =
the electrons are governed by the Fermi-Dirac statistics A / 1 and the differenciation acts on the k3(ε) func-
h i h i
f =[exp(z η)+1]−1,wherez =ε/k T andη =ζ/k T tion. For nonparabolic bands described by Eq. (15) the
0 0 0
are the redu−ced energy and the reduced Fermi energy, internal energy becomes
respectively. We have
U 3 1D1/2(β)
MB = k T 1 , (46)
n=(1/3π2) 1 , (43) n 2 0 0D3/2
h i 0
where in general where nDm are the generalized Fermi integrals, see
l
Zawadzki et al. (1965)
∞
∂f
A = 0 Ak3(ε)dε. (44) ∞
h i − ∂ε
Z (cid:18) (cid:19) nDm(β)= exp( z)zn(z+βz2)m(1+2βz)ldz. (47)
0 l −
Z
0
In order to obtain Eq. (43) one writes d3k =
k2dksinθdθdϕ, integrates over the angles θ and ϕ, and Hereβ =k T/ε characterizesthedegreeofnonparabol-
0 g
integratesoverdkbypartsnoticingthatatk =0thereis icity. For β = 0 one deals with a parabolic band and in
ε=0 and at k = there is f =0. It turns out that all this case D integrals can be expressed by the Γ special
0
∞
statisticalpropertiesoftheelectrongascanbeexpressed function. It can then be easily verified that one obtains
bytheintegrals(44)withreasonablybehavingA(ε)func- U /n = (3/2)k T, which is a well known result for a
MB 0
tions. Since ∂f /∂ε is nonzero only in a limited range nondegenerate electron gas. However, for a nondegen-
0
−
of energies around the Fermi level, integrals (44) can be erate gas in which electrons have a nonparabolic ε(k)
10
dispersion this is not the case. Figure 9 shows the cal- in which ε is the energy and x denote the classical
±
culated mean internal energy as a function of β. With turning points for which q(x ) = 0. The transverse
±
increasing β, i.e. growing nonparabolicity, the mean in- momentum components p and p have been omitted.
y z
ternalenergyincreasesfrom(3/2)k T to 3k T. The cor- Changing the variables s = (ε eEx)/(ε /2) and using
0 0 g
−
responding mean specific heat at a constant volume is ε /2=m∗u2 one obtains
g 0
C = ∂U/∂T and it also increases from (3/2)k to 3k .
v 0 0
It is known that the relativistic nondegenerate electron 2 1 ε 2 +1 πm∗2u3
gasinvacuumhasidenticalstatisticalpropertiestothose P exp g 1 s2ds = 0 .
≃ −ℏu e E 2 −  e Eℏ
presented above: its mean internal energy per electron | | (cid:16) (cid:17) −Z1 p | |
at high temperatures is not (3/2)k0T but 3k0T and its   (50)
mean specific heat per electron at a constant volume is IfwedefinethecriticalelectricfieldE bythecondition
cr
not(3/2)k0but3k0,seeSynge(1957),Kubo(1965). The P 1, we obtain eEcrλz (π/2)εg, where, as before,
reasonforthis similarityisthe correspondenceofdisper- λ ≃= ℏ/m∗u. The physica≈l meaning of E is that it
z 0 cr
sion relations indicated above. gives the potential drop equal to the gap over the dis-
tance λ . Taking ε =0.2eV and λ =20˚A one obtains
z g z
E 106V/cm. This is close to operating fields for
cr
V. INTERBAND TUNELLING semic≃onductor tunnel diodes (Calawa et al., 1960).
The above reasoning can be transposed to the tun-
The phenomenon of tunneling between valence and
neling in the frame of the Dirac equation for electrons
conductionbands canalso be interpretedinterms ofthe
andpositrons, see e.g. Akhiezer and Berestetskii(1981),
relativistic analogy. We begin by reproducing the sim-
Greiner and Reinhardt(1994). Figure10isstillvalidfor
plestsemiclassicalcalculationforaconstantelectricfield
this case. Using the relativistic analogy, i.e. replacing
within the framework of the two-band model of a semi- ε /2bym c2, m∗ bym andubyc,the transitionprob-
g 0 0 0
conductor. Figure 10 shows the space dependence of the
ability is
potential energy V(x)=eEx and the corresponding en-
ergy gap between +(εg/2)+V(x) and −(εg/2)+V(x). P πm02c3 . (51)
The electron-hole pair creation results from the electron ≃ − e Eℏ
tunnelingfromthefilledvalencebandthroughtheclassi- (cid:18) | | (cid:19)
cally forbidden region. The probability of such a tunnel The condition for the critical field is now eE λ
cr c
process is described semiclassically by (π/2)2m c2, which means that E should cause the po≃-
0 cr
tential drop of 2m c2 over the distance λ . This gives
0 c
2 x+ E (relat) 1016V/cm, which is not accessible in ter-
P ≃exp −ℏZx− q(x)dx!, (48) recsrtrial con≃ditions. This shows the power of relativistic
analogy: thephysicsissimilarforrelativisticelectronsin
where the imaginary momentum is, cf. Eq. (41) vacuum and two-band electrons in semiconductors but
the parameters involved are much more “user friendly”
1
1 ε 2 2 in semiconductors.
q(x)= g (ε eEx)2 , (49)
u 2 − −
(cid:20)(cid:16) (cid:17) (cid:21)
VI. ELECTRONS IN MAGNETIC FIELD
We begin this section by considering properties of rel-
ativistic electrons in vacuum in the presence of a mag-
netic field andthen proceedwith the relativistic analogy
describingmagneticbehaviorofelectronsinsemiconduc-
tors. In order to find eigenenergies of relativistic elec-
trons in a uniform magnetic field H z one has to solve
k
the Dirac equation for the problem. This can be done
analytically taking the vector potential A=[ Hy,0,0],
−
seeRabi(1928), Johnson and Lippmann(1949),Strange
(1998). The result is
E(n,p , )=
z
±
= m c2 2+2m c2 D(n,p ) 1ℏω 21 , (52)
0 0 z c
± ± 2
(cid:26) (cid:20) (cid:21)(cid:27)
FIG. 10 Interband electron tunneling in a constant electric (cid:0) (cid:1)
fieldwithinthetwo-bandmodelofasemiconductor(schemat- D(n,p )=ℏω n+ 1 +p2/2m , (53)
ically). z c 2 z 0
(cid:18) (cid:19)