-
Front. Phys., 2012, 7(3): 328352
DOI 10.1007/s11467-011-0200-5
REVIEW ARTICLE
Electronic and optical properties of semiconductor and
graphene quantum dots
Wei-dong Sheng1,2,, Marek Korkusinski1, Alev Devrim Guclu1,
Michal Zielinski1,3,
Pawel Potasz1,4, Eugene S. Kadantsev1, Oleksandr Voznyy1, Pawel
Hawrylak1,
1 Institute for Microstructural Sciences, National Research
Council of Canada, Ottawa, Canada
2 Department of Physics, Fudan University, Shanghai 200433,
China
3 Institute of Physics, Nicolaus Copernicus University, Torun,
Poland
4 Institute of Physics, Wroclaw University of Technology,
Wroclaw, Poland
E-mail: [email protected], [email protected]
Received April 21, 2011; accepted July 27, 2011
Our recent work on the electronic and optical properties of
semiconductor and graphene quan-tum dots is reviewed. For strained
self-assembled InAs quantum dots on GaAs or InP substrateatomic
positions and strain distribution are described using valence-force
eld approach and con-tinuous elasticity theory. The strain is
coupled with the eective mass, k p, eective bond-orbitaland
atomistic tight-binding models for the description of the
conduction and valence band states.The single-particle states are
used as input to the calculation of optical properties, with
electron-electron interactions included via conguration interaction
(CI) method. This methodology is usedto describe multiexciton
complexes in quantum dot lasers, and in particular the hidden
symmetryas the underlying principle of multiexciton energy levels,
manipulating emission from biexcitonsfor entangled photon pairs,
and optical control and detection of electron spins using gates.
Theself-assembled quantum dots are compared with graphene quantum
dots, one carbon atom-thicknanostructures. It is shown that the
control of size, shape and character of the edge of graphenedots
allows to manipulate simultaneously the electronic, optical, and
magnetic properties in a singlematerial system.
Keywords quantum dots, electronic structure, multiexciton,
graphene, magnetism
PACS numbers 78.67.Hc, 73.21.La, 73.63.Kv, 73.22.Pr
Contents
1 Introduction 3292 Self-assembled quantum dots 330
2.1 Strain distribution 3302.2 Electronic structure 331
2.2.1 Eective-mass approximation 3312.2.2 Parabolic connement
model 3322.2.3 Eight-band k p approach 3322.2.4 Eective
bond-orbital model 3332.2.5 Empirical tight-binding method 334
2.3 Optical properties 3362.3.1 Photoluminescence:
Polarization
and anisotropy 3362.3.2 Electronelectron interactions and
multiexciton complexes 337
2.3.3 Hidden symmetry 3382.3.4 Fine structure: Electronhole
exchange interaction 3382.4 Quantum dots in magnetic elds
338
2.4.1 Multiexciton FockDarwin spectrum 3392.4.2 Electron g
factors: Distribution and
anisotropy 3392.4.3 Hole g factors: Envelope orbital
momentum 3402.5 Quantum dots in electric elds 341
2.5.1 Quantum-conned Stark eect 3412.5.2 Electrical tuning of
exciton g factors 341
2.6 Single InAs/InP self-assembled quantumdots on nanotemplates
342
3 Graphene quantum dots 3433.1 Introduction 343
c Higher Education Press and Springer-Verlag Berlin Heidelberg
2012
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Wei-dong Sheng, et al., Front. Phys., 2012, 7(3) 329
3.2 Electronic structure Tight-bindingapproach 344
3.3 Dirac fermions 3443.4 Graphene quantum dots 3453.5 Shape and
edge eects 3453.6 Gated quantum dots: Beyond tight-binding
approach 3463.7 Magnetism in triangular quantum dots 3473.8
Excitons in triangular quantum dots 3483.9 Eect of imperfections
349
4 Conclusions 349Acknowledgements 349References 349
1 Introduction
In this article we highlight some of our recent works to-wards
the understanding of electronic and optical prop-erties of
semiconductor [15] and graphene quantum dots[6]. The recent
research on semiconductor quantum dotsfollows naturally the
evolution of semiconductor tech-nology from transistors and lasers
based on bulk sili-con and bulk gallium arsenide to eld eect
transistorsand quantum-well lasers. In these systems, the control
ofmaterial composition in one dimension, e.g., molecularbeam
epitaxy has led to integrated circuits and revolu-tionary changes
in information technology. Semiconduc-tor quantum dots are a
natural step forward in allowingfor the control of material
composition in three dimen-sions and at the nanoscale. Hence
quantum dots are anexample of nanoscience and nanotechnology in
semicon-ductors. There are four major classes of quantum dots:(i)
lateral gated quantum dots, (ii) self-assembled quan-tum dots,
(iii) colloidal nanocrystals, and the most recentaddition, (iv)
graphene quantum dots.
The lateral gated quantum dots are created at a semi-conductor
GaAlAs/GaAs heterojunction containing atwo-dimensional electron gas
as in the eld-eect tran-sistor (FET). On top of the GaAs surface a
pattern ofmetallic gates with nanometer dimensions is
deposited.When negative voltage is applied to the gates,
electronsresiding underneath these gates at the
GaAlAs/GaAsheterojunction feel repulsive potential and are
pushedout from under the gates [1, 3]. By designing the gatesin
such a way that the repulsive potential under the gatesseen by
electrons resembles a volcano, a controlled num-ber of electrons,
down to one, can be trapped in thevolcanos crater. The counting of
electrons in the lateralquantum dot and subsequent isolation of a
single electronhave been demonstrated at the Institute for
Microstruc-tural Sciences [79]. It is now possible to construct
dou-ble [1013] and triple quantum dot molecules [1417]where
individual electrons are isolated, quantum me-chanically coupled,
and manipulated in real time. Some
of the quantum information aspects of lateral quantumdot
molecules can be found in our recent review [4]. Withnanocrystals
being a very active and well-covered eld, inthis review we will
focus on self-assembled and graphenequantum dots.
The appearance of self-assembled quantum dots dur-ing the growth
of InAs layers on GaAs in molecular beamepitaxy was noted as early
as 1985 by Marzin and co-workers [18] and much of the pioneering
work has beencarried out by Petro and co-workers [19] in the
early90s. From early theoretical and experimental
researchself-assembled quantum dots are nding applications
inquantum dot lasers and ampliers [2025] and solar cells[26].
Current research, some theoretical aspects of whichare reviewed in
this paper, is focused on single quantumdots and quantum dot
molecules with potential appli-cations as single photon sources,
sources of entangledphoton pairs, and when charged, quantum bits.
From atheoretical point of view these structures are challengingas
they are neither few-atom molecules nor solids andinvolve
collective behavior of millions of atoms. We willdescribe our
attempts at providing an understanding ofelectronic and optical
properties of million-atom nanos-tructures at dierent levels of
sophistication. In partic-ular, we will discuss to what extent the
single-particlestates in quantum dots can be viewed as states of
twoquantum harmonic oscillators. When the dots are lledwith
electrons, the generalized Hunds rule allows us topredict the spin
of the ground state [27]. When the dotsare populated with electrons
and holes, as in a quan-tum dot laser, the hidden symmetry replaces
the Hundsrule as an underlying principle governing the propertiesof
multiexciton complexes [2832]. The single exciton,controlling the
absorption of photons by a quantum-dot-based solar cell, can be
understood in terms of mixing ofbright and dark congurations by
Coulomb interactions[33]. Manipulating electronic and optical
properties ofsingle self-assembled quantum dots has become
possiblewith the growth of InAs quantum dots on InP
templates,pioneered by Williams and co-workers [34]. This
enabledthe gating of individual dots [35] and embedding of
thesedots in a photonic cavity [36]. These quantum dots
areparticularly interesting because they emit at the
telecomwavelength. We will describe some of the properties
ofInAs/InP quantum dots using the eective bond-orbitalmodel.
Finally, the recent isolation of a single, atomicallythick
carbon graphene layer [37] opened a new eld ofresearch. Since
graphene does not have a gap, size quan-tization opens an energy
gap and turns graphene into asemiconductor. Unlike in semiconductor
quantum dots,the gap in graphene quantum dots can be tuned fromzero
to perhaps even the gap of the benzene ring. How-ever, graphene
dots need to be terminated and the edgesplay a very important role,
with zigzag edges leading
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330 Wei-dong Sheng, et al., Front. Phys., 2012, 7(3)
to energy shells in the middle of the gap and a nitemagnetic
moment. The magnetic moment can in turn berelated to another
property of graphene the sublatticesymmetry. The research on
graphene quantum dots is ata very early stage [38, 39] and we hope
that this shortreview will stimulate its rapid progress.
The research eld covering semiconductor andgraphene quantum dots
has increased tremendously sinceearly 1990s with contributions from
many outstandingscientists. It is not possible to cover the entire
eld andgive credit to everyone in such a short article. We
focushere primarily on the work carried out at the NRC In-stitute
for Microstructural Sciences and refer the readerto existing
reviews for additional coverage of the eld.
2 Self-assembled quantum dots
Self-assembled quantum dots consist of a
low-bandgapsemiconductor A, typically InAs, embedded in a
higher-bandgap semiconductor B, typically GaAs or InP [13]. The two
materials have similar symmetry but dier-ent lattice constants. The
dots are formed during theStranskyKrastanow process of, e.g.,
molecular beamepitaxy of InAs on GaAs. In this process, the
strainbuilding up in the InAs layer is relieved by the formationof
quasi-two-dimensional islands. The islands, typicallypyramidal or
lens-shaped, are capped with GaAs. An ex-ample of a lens-shaped
InAs quantum dot on a wettinglayer is shown in Fig. 1. When
electrons and/or holes areinjected into the sample, they become
conned in InAsquantum dots. The remainder of this section
describesthe electronic properties of self-assembled quantum
dots.Since the two materials are strained, we start with theeect of
strain, followed by a description of electronicproperties of these
systems.
Fig. 1 Indium (red) and Arsenic (blue) atoms in a
lens-shapedInAs quantum dot with diameter of 25 nm and height of
3.5 nmon a 0.6 nm high wetting layer. The GaAs barrier material
atomsare not shown.
2.1 Strain distribution
As aforementioned, self-assembled quantum dots areformed to
relax the strain due to the lattice mismatchbetween two materials
grown on top of each other, likeInAs/GaAs, InAs/InP or CdTe/ZnTe.
The strain dis-tribution in the vicinity of a quantum dot can be
de-termined by either the continuum elasticity theory orthe
atomistic valence-force-eld approach [40, 41]. The
domain of strain calculation is typically a
rectangularcomputational box which generally includes the
entirestructure, with characteristic sizes on the micrometerscale.
Depending on the device conguration, xed orfree-standing boundary
conditions are implemented [42].
In the framework of the continuum elasticity theory,the strain
tensor is dened for each unit cell of thestructure as
jk =12
(ujrk
+ukrj
)(1)
where rj and uk are the components of the position vec-tor r and
displacement vector u, respectively. It can beobtained by
minimizing the following elastic energy func-tional:
E =12
[C11
(2xx +
2yy +
2zz
)+C44
(2xy +
2yz +
2zx
)+2C12
(xxyy + yyzz + zzxx
)]d3r (2)
where C11, C44, and C12 are position-dependent elasticconstants,
assuming the value of the quantum dot or thebarrier matrix
materials, respectively.
In the atomistic approach, the elastic energy of eachatom i is a
function of the positions of its nearest neigh-bors,
Ei =12
4j=1
ijd2ij
(R2ij d2ij)2
+3
j=1
4k=j+1
ijik
dijdik
(Rij Rik + 13dijdik
)2(3)
with dij being the bond length, and ij and ij the forceconstants
between atoms i and j. By minimizing thetotal elastic energy which
is a sum over all atoms, theequilibrium positions of all the atoms
are computed andthe corresponding strain tensor can therefore be
obtained[41].
The minimization procedure can proceed either viathe standard
conjugate gradient method, or by a moreintuitive force-eld
approach. As the strain energy is afunction of the positions of all
the unit cells or atoms,i.e., E = E(r1, r2, ), the force exerted on
each unit istherefore given by Fi = E/ri. In one iteration,
eachunit is displaced proportionally to Fi. The iterative
pro-cedure ends when the amplitudes of all the forces
becomesuciently small. In zinc-blende semiconductors such asInAs or
GaAs, shear strain would induce piezoelectriccharge along the
interface between neighboring unit cells[42], as given by
P (r) = 2e14[yz(r)
x+
xz(r)y
+xy(r)
x
](4)
The piezoelectric potential can be obtained by solvingthe
corresponding Poisson equation.
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Wei-dong Sheng, et al., Front. Phys., 2012, 7(3) 331
In Table 1 we list the elastic constants (in unit of 1011
dyne/cm1), piezoelectric modulus (Cm2), ideal bondlengths (A)
and force constants (103 dyne) of InAs, GaAsand InP. Figure 2 shows
the shear strain component xyand piezoelectric potential calculated
for a lens-shapedInAs/GaAs self-assembled quantum dot with a base
di-ameter of 19.8 nm and a height of 2.8 nm placed on
atwo-monolayer wetting layer (ML). Red/blue area corre-sponds to
positive/negative part of the strain and poten-tial.
Table 1 List of material parameters used in the strain
calcula-tion by the continuum or atomistic elasticity theory.
InAs GaAs InP
C11 8.329 11.879 10.22
C12 4.526 5.376 5.76
C44 3.96 5.94 4.6
e14 0.045 0.16 0.035
d 2.622 2.448 2.537
35.18 41.19 43.04
5.50 8.95 6.24
Fig. 2 Isosurface plots of the shear strain component xy
andpiezoelectric potential in a lens-shaped InAs/GaAs
self-assembledquantum dot.
2.2 Electronic structure
Within the framework of continuum elasticity, the eectof strain
on the electronic structure of a self-assembledquantum dot is
described by the BirPikus deformationpotential theory [43]. There
are two major componentsof the strain which modies the band-edge
energies, thehydrostatic (Hs) and the biaxial (Bs), as dened in
thefollowing:
Hs = xx + yy + zz
B2s = (xx yy)2 + (yy zz)2 + (zz xx)2 (5)The band-edge energies
of the conduction bands aremainly aected by the hydrostatic
component of thestrain while the heavy- and light-hole bands are
furtherinuenced by the biaxial strain components, i.e.,
Uc = acHs + Ecbo
Uhh = avHs bvBs + EvboUlh = avHs + bvBs + Evbo (6)
where Ecbo and Evbo are the band osets between the dot
and matrix materials before strain, and ac, av, and bvare the
deformation potential parameters [40].
Figure 3 plots the conning potentials for electrons,heavy- and
light-holes along the growth direction (upperpanel) and along the
diameter of the bottom base of thedot. The height of the dot is
chosen to be the same asthat in Fig. 2 while the diameter is twice
as large for bet-ter visualization. It is seen that the eective
band-gap inthe presence of the strain is increased to about 695
meVfrom 475 meV in the strain free bulk InAs. The strain isalso
seen to enhance the depth of the conning potentialfor the
heavy-holes to about 300 meV from 85 meV. Theheavy- and light-hole
bands which are degenerate in thebulk are now lifted by the biaxial
strain component. Thesplitting in the middle of the dot is about
177 meV.
Fig. 3 Conning potentials for electrons, heavy- and
light-holes(from top to bottom) deformed by the strain InAs/GaAs
self-assembled quantum dot.
Besides the eect on the band-edge energies, the strainalso
modies the eective masses of electrons and holesin the dot.
2.2.1 Eective-mass approximation
The states conned in the quantum dot are often com-posed of
components from both the conduction and va-lence bands. With the
biaxial strain present in the self-assembled quantum dots, the
degeneracy between theheavy and light holes is lifted and the
low-lying statesin the valence band have mostly a heavy hole
character.
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332 Wei-dong Sheng, et al., Front. Phys., 2012, 7(3)
If the eect of band mixing between the conduction andvalence
bands is neglected, the problem could be reducedto two separate
single-band eective-mass equations, onefor electron and the other
for hole, as given by [44]
He = 2
2m//e
(2
x2+
2
y2
)
2
2me
2
z2+ Uc + Vp
Hh =2
2m//h
(2
x2+
2
y2
)+
2
2mh
2
z2+ UhhVp (7)
where Vp is the piezoelectric potential. Although the elec-tron
eective mass is almost isotropic in most IIIV semi-conductors, two
independent components, m//e and me ,are used to reect the
anisotropic geometries of the quan-tum dots, and so are the
analogous masses for holes.
Since there is no a priori rule on how to choose theeective
masses within the model itself, m//e , me , m
//h ,
and mh are treated as adjustable parameters which aredetermined
by tting the calculated energy spectrum tothat obtained by a more
sophisticated approach like themultiband k p or the empirical
tight-binding methoddescribed below. A way to avoid the tting
procedureis to regard the eective masses as
position-dependentvariables [45].
It is found that the electron eective mass in quantumdots is
generally larger than the bulk value and becomesanisotropic in the
dots of large aspect ratio between thevertical and lateral
dimensions. Unlike the bulk mate-rial, the hole eective mass is
seen to be almost isotropicin the dots of small aspect ratio. For
an example of atInAs/GaAs quantum dots, the most appropriate
valuefor the electron and hole eective mass is believed to bethe
electron eective mass in bulk GaAs (0.067 m0) andthe vertical
heavy-hole eective mass in bulk InAs (0.34m0), respectively.
2.2.2 Parabolic connement model
The eective mass model has been applied to lens-shapedquantum
dots [46]. It was shown that, in the adiabaticapproximation, the
radial conning potentials for elec-trons and heavy holes can be
well approximated byparabolas with a given depth and radius of the
dot.The single-electron Hamiltonian is reduced to a Hamilto-nian of
two bosons, two 1D harmonic oscillators (HOs)with creation
(annihilation) operators a+m(am) and en-ergy level spacing e
[7],
Hho = e
(m +
12
)a+mam + e
(n +
12
)a+n an (8)
The corresponding wave functions are those of thetwo-dimensional
harmonic oscillator. A similar modelis applied to heavy holes, and
the analytical solutionfor HO states in a perpendicular magnetic
eld can befound, e.g., in Ref. [7]. It was recently demonstrated
that
InAs/InP and disk-like InAs/GaAs dots grown using theindium-ush
technique developed by Wasilewski and co-workers [47] can be well
described by the HO model.If the HO states are populated by
photoexcited elec-trons and holes with increasing excitation power,
as il-lustrated in Fig. 4(c), the resulting emission
spectrumconsists of several peaks corresponding to electron andhole
shells. The evolution of the emission spectrum withan increasing
magnetic eld is shown in Figs. 4(a) and(b). The spectrum strikingly
resembles that of the 2DHO in a magnetic eld. As already mentioned,
the HOmodel works quite well, but the parameters of the
model,masses, and connement energies e and h are ttingparameters
that must be obtained by other methods likethe multiband k p or
empirical tight-binding methoddescribed below.
Fig. 4 Emission spectrum of an ensemble of InAs/GaAs
self-assembled quantum dots as a function of applied magnetic
eldshows harmonic oscillator states. Reproduced from Ref. [48],
Copy-right c 2004 American Physical Society.
2.2.3 Eight-band k p approach
The eight-band k p method uses eight Bloch functionsat the point
of the Brillouin zone as the basis func-tions to describe electron
states with a nite wave vec-tor. As the lateral size of quantum
dots is usually muchlarger than the lattice constant, the k p
method hasbeen widely used in the calculation of conned
electronstates. In general, the multiband k p Hamiltonian canbe
written as
Hkp = Ebo +Axkxkx +Aykyky +Azkz kz+Bxykxky +Byzkykz
+Bxzkxkz+Cxkx +Cyky +Czkz + VP (9)
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Wei-dong Sheng, et al., Front. Phys., 2012, 7(3) 333
where Ebo is the matrix for the band osets, and A,B, and C are
the matrices of coecients [49]. Withinthe deformation potential
theory, an additional part Hs,which has a similar structure as Hkp,
is added to takeinto account the eects due to strain [50]. The
computa-tional box for the calculation of the electronic
structureof quantum dots is not necessarily as large as that
usedfor the strain since the conned states are mostly local-ized
inside the quantum dots. The Hamiltonian is rstdiscretized on a
reduced three-dimensional mesh, whichresults in a large, sparse
matrix, and then is diagonalizedby utilizing the Lanczos algorithm
[51].
The wave function of a single-particle state, , is thesum of
products of envelope functions k(r) and basisfunctions uk(r), i.e.,
(r) =
n n(r)un(r). The basis
functions are usually taken as |s, |x, |y, and |z, cou-pled with
two eigenspinors | and | . These functionsare referred to as the
uncoupled spinorbital basis. Be-cause of the spinorbit interaction,
only the total angularmomentum is a good quantum number. Therefore,
theeigenfunctions of the total angular momentum operatorare
considered to be a more convenient choice of what isreferred to as
the coupled spinorbital basis. These ba-sis functions are closer to
the band-edge Bloch functionsthan the uncoupled set. Note that the
coecient matri-ces in the k p Hamiltonian take dierent forms in
theuncoupled and coupled spinorbital basis [52].
Figure 5 shows the density of states for a lens-shapedInAs/GaAs
self-assembled quantum dot grown on a 2ML wetting layer with a base
diameter of 25.4 nm anda height of 2.8 nm. The conned states in the
conduc-tion bands are seen to form almost equally spaced clus-ters,
with increasing number of states in each one, inanalogy to the HO
model. The density of states in thevalence band is found to
increase steadily as the energyapproaches the continuum. Figure 5
also shows the com-position of the states, i.e., the proportion of
each kind ofcomponents. It is seen that the states in the
conductionband are dominated by the components from the sameband,
however with decreasing proportion as the energyincreases. In the
valence band, the low-lying states arefound to be dominated by the
heavy-hole components,while the proportion of the light-hole
components is non-negligible in high-lying states. Close to the
continuum,the proportion of the heavy- and light-hole componentsis
seen to be saturated at around 60% and 25%, respec-tively.
Since the eight-band k p method keeps a good bal-ance between
handling complicated band mixing ef-fects and computational
complexity, it has becomewidely used in the calculation of the
electronic struc-ture of self-assembled quantum dots, and
successfullyexplained many interesting phenomena, such as
invertedelectron-hole alignment in intermixed single quantumdots
[53] and spontaneous localization of hole states in
Fig. 5 Density of states (lower panel) and composition ofstates
(upper panel) calculated for a lens-shaped InAs/GaAs self-assembled
quantum dot. Small solid dots (blue) are for the compo-nents from
the conduction bands, larger ones (red) for the heavy-hole
components, and open dots (green) for the light-hole
compo-nents.
quantum-dot molecules [54]. The k p method predictedthe
transition between bonding and anti-bonding statesas ground states
of a valence hole in a vertical quantumdot molecule as a function
of the separation between theconstituent quantum dots [55, 56].
This unusual behav-ior related to strong spinorbit coupling was
recently ob-served experimentally by Doty et al. [57].
2.2.4 Eective bond-orbital model
The multiband k p method accounts for the properstructure of the
valence band, including heavy, light, andspin split-o hole
subbands. It is, however, limited tothe vicinity of the point, and
therefore is expected tobreak down as the size of the nanostructure
decreases.The eective bond-orbital model (EBOM) [58] is an
em-pirical sp3 tight-binding method in which indium-arsenicdimers
are replaced by an eective atom. Hence the fullsymmetry of the
zinc-blende lattice is reduced to that ofa fcc lattice. Although
EBOM misses the lack of inversionsymmetry of zinc-blende structures
and the microscopicatomic structure of the unit cell, it can,
however, repro-duce the eective masses of electrons and holes at
the point, as well as conduction and valence band edgesat both the
and X points with the second nearest-neighbor interactions included
[59]. The tight-bindingHamiltonian matrix describing hopping
between eectives and p orbitals of eective atoms on sites R is
given by
H(Rs,Rs) = E000ss R,R + E110ss R, + E
200ss R,
H(Rp,Rp) = R,[E110xx
2p + E
011xx (1 2p )
]+R,
[E200xx
2p + E
002xx (1 2p)
]
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334 Wei-dong Sheng, et al., Front. Phys., 2012, 7(3)
+E000xx R,R
H(Rs,Rp) = E110sx pR,
H(Rp,Rp) = E110xy ppR, (10)
with R = RR, and =
a
2[(1,1, 0), (1, 0,1), (0,1,1)]
= a[(1, 0, 0), (0,1, 0), (0, 0,1)] (11)
being the positions of the nearest and next-nearest neigh-bors,
respectively. The eect of strain is incorporatedthrough the
deformation potential theory and throughthe piezoelectric eect
[60]. Table 2 shows the next-nearest-neighbor parameters for GaAs
and InAs.
Table 2 Next-nearest-neighbor parameters (in eV).
GaAs InAs
E000ss 3.581011 2.698585
E110ss 0.039499 0.124187E200ss 0.264835 0.118389E110sx 0.387077
0.368502
E000xx 3.301012 3.068847E110xx 0.424999 0.412500
E011xx 0.070000 0.112499E200xx 0.186297 0.156340E002xx 0.138401
0.154132
E110xy 0.495000 0.525000
The matrix elements for spin-orbit interaction aregiven by
H(Rx,Ry) = i/3, H(Rx,Ry) = i/3,H(Rx,Rz) = /3, H(Ry,Rz) = i/3, and
theirconjugate terms, with being the spin-orbit splitting inthe
valence band. When the spin-orbit interaction is notvery strong, it
is possible to separate the components ofan envelope function into
two groups. One group consistsof components for spin up basis
functions, |s, |x, |y,and |z, and the other one consists of
components forspin down basis functions. A pseudospin can be
assignedto a state if its polarization, dened as
p =|s(r)|2 + |x(r)|2 + |y(r)|2
+|z(r)|2 dr (12)is either p 1 (a spin up state) or p 0 (a
spindown state). The assignment of pseudospin can be car-ried out
by introducing a small magnetic eld ( 1 mT)along the growth
direction of the quantum dots to liftthe degeneracy induced by the
time-reversal symmetry.
2.2.5 Empirical tight-binding method
In the empirical tight-binding method we rst expandthe wave
function in the basis of atomic orbitals
=R,
cR|R (13)
and next form the Hamiltonian matrix in this atomic ba-sis [61].
The matrix elements are treated as parametersdetermined by
comparison with ab-initio and/or experi-mental results for bulk
materials. Signicant amount ofwork has been devoted to ab-initio
determination of va-lence and conduction bands, eective masses and
bandosets for strained materials of quantum dot and
barriermaterials [62, 63].
By comparison, in the pseudopotential approach [64,67], the
self-consistent potential seen by an electron ina bulk material is
replaced by a sum of eective atomicpotentials. These atomic
potentials are next used to gen-erate the one-electron potential of
the nanostructure.
In our tight-binding approach [61] the wave functionon each atom
is described by ten valence orbitals foreach spin: one of type s,
three of type p, ve of type d,and an additional s orbital
accounting for higher-lyingstates. Each orbital is doubly
spin-degenerate, thus re-sulting in a total of 20 bands. The
resulting Hamiltonianof a quasiparticle in an N -atom quantum dot,
written inthe language of second quantization, is
HTB =N
i=1
20=1
ic+ici +
Ni=1
20=1,=1
i,c+ici
+N
i=1
4j=1
20,=1
ti,jc+icj (14)
where c+i (ci) is the creation (annihilation) operatorof a
carrier on the orbital localized on the site i, iis the
corresponding on-site energy; and ti,j describesthe hopping of the
particle between orbitals on neigh-boring sites. Coupling to
farther neighbors is neglected.Finally, i, accounts for the
spinorbit interaction byintroducing nite matrix elements connecting
p or-bitals of opposite spin, residing on the same atom, fol-lowing
the description given by Chadi [68]. For example,py, |H | , pz =
i/3. Spinorbit-type coupling be-tween d orbitals is neglected. Here
we assume that eachsite holds 20 orbitals and is surrounded by 4
neighbors.
Hopping, i.e., o-diagonal matrix elements of ourHamiltonian are
calculated according to the recipe givenby Slater and Koster [69].
In this approach, the hop-ping matrix elements ti,j are expressed
as geometricfunctions of two-center integrals and depend only on
therelative positions of the atoms i and j. Contributionsfrom
three-center and higher integrals are neglected. Forexample, if the
two atoms are connected by a bond alongthe x axis, then orbitals s
and pz create a bond andthe matrix element ts,pz = Vs,pz = 0
vanishes becauseof the symmetry. On the other hand, if the
direction ofthe bond is along y axis, then the bond is of a typeand
ts,pz = Vs,pz is nite. In the general case the near-est neighbors
are connected by bonds of any directiond = |d| (lx+ my + nz), with
d being the bond length
-
Wei-dong Sheng, et al., Front. Phys., 2012, 7(3) 335
and l, m, n the direction cosines. Then the tunnelingts,pz
element can be expressed in terms of projecting thepz orbital onto
the bond and in the direction perpendicu-lar to it. Since the
perpendicular projections give -typebonds, their contribution is
zero. The Hamiltonian ma-trix element is thus ti,s,j,pz = nVsp .
Similar sets of rulesare dened for all other t matrix elements
[69]. This ap-proach reduces the number of unknown matrix
elementsas they can be related via SlaterKoster rules to a
rel-atively small subset of two-center integrals V, . Thisis
particularly useful within the framework of empiri-cal tight
binding, where E, , and V, parametersare not directly calculated,
but rather obtained by t-ting the TB bulk model results to
experimentally knownband gaps and eective masses at high symmetry
pointsof the Brillouin zone [70]. We stress here that we havebeen
tting the TB model not only to bulk propertiesat point, but also at
X and L points to account formultivalley couplings.
The TB parametrization used so far are given, e.g., inRefs.
[7173], where it was demonstrated that the inclu-sion of d orbitals
in the basis allows to obtain much betterts of the masses and
energy gaps to the target materialvalues. In particular, the
treatment of the conductionband edge is signicantly improved, which
is importantfor small nanostructures [74]. In this work we use
ourown parametrization, analogous to work by Klimeck etal. [72],
but giving a better agreement with target bulkproperties. More
details will be presented in our futurework.
In order to address the treatment of the interface be-tween InAs
and GaAs we note that these two materialsshare the same anion
(Arsenic). Thus during the ttingprocedure the diagonal matrix
elements on arsenic arekept the same in both materials. This
approach removesthe necessity of averaging on-site matrix elements
for in-terface atoms. Additionally, to account for the band o-set
(BO) between the materials forming the interface,the t for InAs is
performed in such a way that the topof the valence band of InAs is
set to be equal to the BOvalue relative to the GaAs. This removes
the necessityof shifting values of diagonal matrix element for
inter-face atoms, which would result in two dierent sets
ofparameters for Arsenic: one for InAs and another forGaAs.
Finally, there is the second type of interface that ariseson the
edges of the computational box. Here, the ap-pearance of free
surfaces leads to the existence of dan-gling bonds. Their presence
results in spurious surfacestates, with energy inside the gap of
the barrier ma-terial, making it dicult to distinguish spurious
statesfrom the single-particle states of the quantum dot. Anenergy
shift for dangling bonds that mimics the passiva-tion procedure,
described in Ref. [75], is performed in or-der to move the energies
of surface-localized states away
from the energies corresponding to conned quantum-dot
states.
Then, a parallel Lanczos diagonalizer is used to resolvethe
Kramers-degenerate doublets. Because the computa-tional domain
necessary for the converged tight-bindingcalculation involves the
order of 1 million atoms, theresulting tight-binding matrices are
very large, i.e., 20million by 20 million. This presents a
signicant numer-ical problem, but utilizing matrix sparsity,
parallel com-putation paradigm, and the fact that we need only
sev-eral lowest electron and hole states rather than the
entireeigenspectrum of the Hamiltonian, we achieve linear scal-ing
of the computational resources as a function of thenumber of
atoms.
To illustrate the application of the dierent methods,we show in
Fig. 6 the results of calculation for the samedot obtained with the
eective bond-orbital method (la-beled as EBOM), the tight-binding
(TB) model, and theempirical pseudopotential method (EMP). The
inputparameters in EBOM and TB calculations are chosen tobe similar
to the ones used in empirical pseudopotentialcalculations (EMP1)
from Ref. [64]. Another empiricalpseudopotential calculation (EMP2)
is shown for com-parison [65]. As expected, the structure of
electron statesis similar in all cases, however, the hole states
dier sig-nicantly. EBOM maintains approximately a
shell-likestructure of hole levels. This diers from both TB andEMP
and can be attributed to the replacement of zinc-blende with cubic
symmetry in EBOM. There is a goodagreement between TB and EMP1
calculation, with acharacteristic large/large/small level spacing
betweensubsequent hole levels h1 h2 h2 h3 h3 h4.Surprisingly, two
pseudopotential (EMP1/EMP2) cal-culations predict dierent details
of hole levels, most
Fig. 6 Electron (blue/upper) and hole (red/lower)
single-particleenergies calculated for eective bond-orbital method
(EBOM) andtight-binding (TB) model. The parameters of the TB model
werechosen to be similar to empirical pseudopotential calculation
from[64]; results of this calculation are shown as EMP1. Another
em-pirical pseudopotential calculation (EMP2) is shown for
compari-son [65]. Reproduced from Ref. [61], Copyright c 2010
AmericanPhysical Society.
-
336 Wei-dong Sheng, et al., Front. Phys., 2012, 7(3)
likely due to a slightly dierent choice of tting parame-ters or
tted results in pseudopotential tting procedure.More specically,
due to the insucient number of ttingparameters, the
pseudopotentials used in the EMP cal-culations give eective masses
of GaAs and InAs o fromthe experimental values by more than 30%
[60]. Conse-quently, the separations among the energy levels in
thevalence bands calculated by both EBOM and TB are dif-ferent from
those by EMP. Overall there is a good agree-ment between EBOM, the
tight-binding and empiricalpseudopotential models.
2.3 Optical properties
2.3.1 Photoluminescence: Polarization and anisotropy
Let us begin our discussion with the EBOM model andrst
temporarily neglect the electronelectron interac-tion. The momentum
matrix element between an elec-tron state e =
enun and a hole state h =
hnun
along the polarization direction e is given by
h|e p|e =mn
un|e p|umhn|em
+m
hm|e p|em (15)
If we neglect the contribution from the envelope-functionpart of
the wave function [76], this element can be furthersimplied as
h|px|e = iP0 [hx|es+ hx|es
hs|ex hs|ex]
h|py|e = iP0 [hy|es+ hy|es
hs|ey hs|ey]
(16)
where iP0 = s|px|x = s|py|y denotes the couplingbetween the
conduction and valence bands. For circularpolarization + or , the
momentum matrix element isthen given by phe = [h|px|e ih|py|e]
/
2. It is
straightforward to show that phe = p+he in the absence of
the magnetic eld.Figure 7 gives a schematic view of interband
transition
in an elongated quantum dot. The emission is found tobe
partially polarized, which is regarded to be relatedto the
structural anisotropy [77]. The degree of linearpolarization of
interband transitions is then dened by
Peh =|e|px|h|2 |e|py|h|2|e|px|h|2 + |e|py|h|2 (17)
which is found to be closely related to the polarizationof
envelope functions as given by [78]
Pef =|hx|hx|2 |hy |hy|2|hx|hx|2 + |hy |hy|2
(18)
For the ground hole state in a quantum dot elongatedalong the x
direction, the probability of the electron inthe px orbital is
found to be larger than py [78]. Hence,it is this selective
occupancy that leads to the opticalanisotropy.
Fig. 7 Schematic view of the interband transition between
theground electronic state e and hole state h and the
intersubbandtransitions from e to the rst two excited states xe
and
ye in a
quantum dot elongated along the x = [110] direction.
Figure 7 also shows two intersubband transitions fromthe ground
electronic state. These transitions are polar-ized along the long
and short axes of the structure. In atraditional single-band
picture, it is trivial to nd thatthe intersubband transitions from
the ground state e toxe and
ye are respectively polarized along the x and y
directions due to the spatial symmetry of the states. Inthe
presence of the mixing between the conduction andvalence bands,
however, the polarization of intersubbandtransitions becomes a
non-trivial issue.
In the multiband formalism, the electronic states con-sist not
only of components from the conduction band,but also those from the
valence bands. Let us considerthe expansions, e =
kS |k, xe =
kX |k, and
ye =
kY |k where the summation is over the basis{s, x, y, z}. The
momentum matrix elements for the tran-sitions e xe and e ye are
then given by [79]xe |px|e = iP0
[sX |xS xX |sS]xe |py|e = iP0
[sX |yS yX |sS]ye |px|e = iP0
[sY |xS xY |sS]ye |py|e = iP0
[sY |yS yY |sS] (19)Here the momentum matrix elements among the
enve-lope functions have been shown to be much smaller thanP0 and
hence are neglected. It is therefore seen that thepolarization of
the intersubband transitions now dependson those minor components
of the electronic states, suchas yS and
yX from the valence bands in the presence of
the mixing between the conduction and valence bands.Further
study has shown that the linearly polarized
intersubband transitions are due to the simultaneous
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Wei-dong Sheng, et al., Front. Phys., 2012, 7(3) 337
vanishing of the four coupling terms, sX |yS, yX |sS,sY |xS, and
xY |sS. The physics behind this rule canbe understood in terms of
directional interactions be-tween the local atomistic orbitals
[79].
Figure 8 shows the linear polarization of the primaryinterband
transition against the aspect ratio for the dotswith various
lateral sizes and heights. It is seen that thelinear polarization
of the primary interband transitionexhibits a quadratic dependence
on the lateral aspectratio (). This dependence is also aected by
other struc-tural parameters such as the lateral size and height.
Moreimportantly, it is found that Peh() exhibits almost thesame
behavior for dots of similar aspect ratio betweenthe lateral size
and height [80]. This reveals the possibil-ity of optical
characterization of structural properties ofself-assembled quantum
dots.
Fig. 8 Linear polarization of the primary interband
transition,Peh in dotted lines, calculated as a function of the
aspect ratioof the quantum dots with various lateral sizes and
heights. Theshape and size of the symbols correspond to the denoted
struc-tures. Quadratic t is shown in solid lines. Reproduced from
Ref.[80], Copyright c 2008 American Physical Society.
2.3.2 Electronelectron interactions and
multiexcitoncomplexes
The absorption and emission spectra of a quantum dotare
determined by an exciton, an interacting electron-hole pair. There
are four exciton states, two dark andtwo bright. The electronhole
exchange interaction, tobe discussed later, leads to the splitting
of the four ex-citon states into a dark doublet and a bright
doublet[81], and modies the polarization of emitted photons.The
emission spectra from a quantum dot with dierentnumber of excitons
are dierent due to electron-electroninteractions and this allows us
to identify the emissionfrom a single exciton, and hence the
emission of a singlephoton. This is the principle behind the
quantum dot asthe single photon emitter. An emission cascade from
abiexciton through two potentially indistinguishable exci-ton
states to the quantum dot ground state leads to theemission of a
pair of entangled photons [82, 83].
In a quantum-dot laser, the number of electronholepairs in a
quantum dot increases with external excitationpower. The electrons
and holes interact via Coulomb in-teraction, and these electronhole
pairs form multiexci-
ton complexes. A given N -exciton complex is equivalentto a
specic articial excitonic atom. An understandingof the electronic
properties such as the ground state en-ergy, total spin, total
angular momentum, and emissionand absorption spectra requires an
understanding of therole of electronelectron and electronhole
interactionsin electronhole complexes occupying electronic shells
ofa quantum dot. For a given number of electronhole pairsN the
interacting Hamiltonian reads,
Hex =
i
Eei c+i ci
i
Ehi h+i hi
ijkl
V heijklh+i c
+j ckhl
+12
ijkl
V eeijklc+i c
+j ckcl +
12
ijkl
V hhijklh+i h
+j hkhl
(20)
where Eei and Ehi are the energy levels of the conned
states in the conduction and valence bands, respectively.A
general Coulomb matrix element is dened by
Vij,kl =e2
40
i (r1)
j (r2)
1|r1 r2|k(r2)l(r1)
(21)
and we have V eeij,kl = Vij,kl. Note here that a connedstate in
the valence band h obtained from any multi-band methods can be
regarded as a hole state only aftera conjugate transformation, h h.
Hence we ndthat V hhij,kl = Vkl,ij and V
heij,kl = Vik,jl. Once the single-
particle states are obtained, the Hamiltonian for multi-excitons
can be solved by the congurationinteractionmethod [30, 46, 61,
84].
A multiexciton complex, such as a biexciton, is com-posed of
more than one electronhole pair. If CiN is thei-th eigenstate of an
N -exciton complex, its wave func-tion can be generally written as
iN =
ciN |C iN . Re-
combination of an electronhole pair from the N -excitoncomplex
would reduce the number of excitons to N 1.By taking into account
every possible recombination, onecan obtain the radiative lifetime,
dened by
1k
=ne2
20m0c3
[f
+k () + f
k ()
]2d (22)
with fk () being the oscillator strength,
fk () =
2m0
i
f(EiN )|ciN |2f
|cfN1|2
|CfN1|P |CiN |2(EiN EfN1 )(23)
The radiative lifetime of a single exciton is typicallyaround 1
ns. Note that the nal state of the (N 1)-exciton complex, fN1, may
not be its ground state afterthe recombination of an electronhole
pair. The emissionintensity is given by
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338 Wei-dong Sheng, et al., Front. Phys., 2012, 7(3)
IN () =
i
f(EiN )|ciN |2f
|cfN1|2
|CfN1|P |CiN |2 (EiN EfN1 )(24)
where the probability function f(EiN ) is given byexp(EiN/T
)/
j exp(EjN/T ) and CiN is the i-th
eigenstate of the N -exciton system. The operator Pdescribes all
possible ways of electronhole recombina-tion, i.e., P =
nm p
nmhncm. In the absence of
magnetic eld we have I+N (E) = IN (E). The emis-
sion spectrum observed in experiments is a sum of
thecontribution from individual multiple exciton complexesweighed
by the corresponding occupation number nk,i.e., I() =
nkIk(). Recent photon correlation ex-
periments allow to untangle complicated emission spec-tra and
extract spectra corresponding to xed excitonnumbers [85].
2.3.3 Hidden symmetry
The main diculty with determining the ground stateof the
multiexciton complex exists for partially lled de-generate quantum
dot states. All congurations havethe same energy and there is no
single congurationwhich dominates the ground state. Fortunately, it
wasshown that the fully interacting electronhole Hamil-tonian and
exciton creation operator on any degener-ate shell satises the
following commutation relation:[Hex, P+] = EXP+ where P+ creates an
exciton andEX is a single exciton binding energy [28]. This
commu-tation relation allows to construct exact eigenstates ofthe
fully interacting Hamiltonian, called multiplicativestates. These
states are the ground states of multiex-citon complexes and, as a
consequence, emission from adegenerate shell takes place with
energy EX independentof the population of this shell, i.e., the
number of mul-tiexciton complexes N . This property of quantum
dotsis called hidden symmetry. For more information on thehidden
symmetry in quantum dots we refer the reader toRefs. [28, 30, 32,
61, 84].
2.3.4 Fine structure: Electronhole exchangeinteraction
Let us denote the total energy of an ideal ground state ofa
semiconductor, i.e., a fully occupied valence band plusan empty
conduction band, as E0. If one removes an elec-tron from a state i
in the valence band to a state j inthe conduction band, the energy
of the system changesto Eij . If one adds an electron back to the
same statein the valence band, the dierence between Eij and E0would
just be the interaction energy between two elec-trons, one in state
i and the other one in state j, i.e,E0 +C = Eij +Vij,ij Vij,ji.
Here C is the energy of in-
teractions between the electron in the conduction band,and each
electron in the valence band, and can be re-garded as a constant.
Hence, the energy of an electronhole pair is given by Vij,ji +
Vij,ij with the rst termbeing the electronhole direct attraction
and the otherone the electronhole exchange repulsion. In
general,the matrix element for the electronhole exchange
inter-action is given by
Vij,kl =e2
40
i(r1)j (r2)
1|r1 r2|
k(r2)l(r1)
(25)
In bulk, the electronhole exchange interaction arisesfrom the
overlap between the s orbitals in cations, wheremost of the
electron wave function is localized, and p or-bitals in anions,
accounting for most of the wave functionof the hole. This also
accounts for the mixing betweenconduction and valence bands.
The electronhole exchange interaction is responsiblefor the ne
structure in the optical spectra of an exciton.Without the exchange
interaction, the exciton composedof an electron with spin s = 1/2
and a heavy holewith spin = 3/2 would be four-fold degenerate.
WithL = s+ as the total angular momentum, the four statesform a
bright doublet with L = 1 and a dark doubletwith L = 2. The
electronhole exchange interactionsplits the dark and bright
doublets, with dark excitons atlower energy. Even more importantly,
the bright doubletis also split into two linearly polarized exciton
states bythe long-range electronhole exchange interaction.
Thesplitting of the two exciton doublets is a function ofthe
anisotropy of the quantum dot. Since the splittingprevents the
emission of entangled photon pairs in thebiexciton cascade, the
theory of the ne structure of ex-citon has been extensively
studied, e.g., by Takagahara[86, 87], Ivchenko and co-workers [88],
and Kadantsevet al. [89] and analyzed in atomistic approaches [61,
90].Since the ne structure splitting is on the order of tensto
hundreds of eV, cautions must be taken in extract-ing numbers from
multi-million atom simulations. Sig-nicant eort is also devoted to
controlling the excitonne structure [9193].
2.4 Quantum dots in magnetic elds
The eect of magnetic eld has been incorporated intothe
EBOMHamiltonian by introducing Peierls phase fac-tors as
follows:
H(R,R) ei eRRR
A(r)dr H(R,R) (26)where A(r) is the vector potential. The Zeeman
eect isincluded by adding the spin terms to the diagonal
matrixelements,
H(R(),R()) H(R(),R()) 12g0BBz (27)
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Wei-dong Sheng, et al., Front. Phys., 2012, 7(3) 339
where g0 is the g factor of a bare electron. The mag-netic
properties of quantum dots such as the eectiveelectron g factors
depend sensitively on the parametersused in the calculation. In the
EBOM Hamiltonian allthe tting parameters are established in a
closed form asa function of band edges and eective masses, and
thuscan be uniquely determined.
2.4.1 Multiexciton FockDarwin spectrum
Multiexciton complexes in quantum dots have been in-vestigated
by us in Refs. [48, 94]. As mentioned previ-ously, the connement
for the electrons in the conductionbands in a at quantum dot can be
well approximated asa two-dimensional parabolic potential. The
energy spec-trum of conned electronic states is therefore the same
asthat of a harmonic oscillator, which is also known as
theFockDarwin spectrum in the presence of applied mag-netic eld. In
experiments, shown in Fig. 4, one observesthe emission spectrum
from multiple exciton complexesinstead of the energy spectrum of
single-particle states.Nevertheless, there is a correlation between
the spectraof single-particle states and emission of multiple
excitons.According to Koopmans theorem for a few-particle sys-tem,
it is a good approximation to treat the additionenergy, EN EN1,
i.e., emission energy of multipleexcitons as the corresponding
single-particle excitationenergy, Een Ehn . Hence, it is not
surprising to observeFockDarwin-like spectra in experiments on
ensemble ofquantum dots [48] as well as in single dots [95].
Fig. 9 Contour plot of the multiexciton emission spectrum of
alens-shaped InAs/GaAs self-assembled quantum dot.
Numerical simulation of multiexciton emission in-volves several
steps from determination of strain dis-tribution, calculation of
the single-particle energy spec-trum, computing multiexciton states
by using theconguration-interaction method, solving rate
equationsto determine occupation of multiple exciton states tonally
obtaining the emission spectrum [94]. Figure 9shows the
multiexciton emission spectrum calculated fora lens-shaped
InAs/GaAs self-assembled quantum dotwhich is grown on a 2 ML
wetting layer and has a base
diameter of 19.8 nm and height of 3.4 nm. For a dot ofsuch small
lateral dimensions, the diamagnetic shift ofthe single exciton line
is barely noticeable. However, thesplitting of the two p-like
states and crossing of the p andd orbitals is observed.
2.4.2 Electron g factors: Distribution and anisotropy
In bulk semiconductors such as GaAs and InAs, the elec-tron g
factor can deviate substantially from the bare elec-tron g factor
due to strong band mixing eects and thespinorbit interaction. In
turn, in self-assembled quan-tum dots the complicated environment
like the strongquantum connement and long-ranged strain eld
makeselectron g factors greatly dierent from those in the bulk[96].
Interestingly, in InAs/GaAs self-assembled quan-tum dots the
ensemble average value of the electron gfactor does not deviate too
much from that in single dots[97]. It implies that the g factors
for individual islandsare similar, and therefore the inhomogeneous
broaden-ing of the quantum dot size and composition does notinuence
the measured g factors signicantly.
Fig. 10 Electron g factors as a function of the single
excitonemission energy calculated for InAs/GaAs quantum dots with
var-ious sizes, shapes and composition proles. Circles and
diamondsstand for lens-shaped and pyramidal dots, respectively. The
size ofthe symbols corresponds to the denoted structures.
Figure 10 shows the electron g factors calculated asa function
of the emission energy for an ensemble ofInAs/GaAs dots with
various sizes, shapes and com-position proles [98]. For all the
samples, the electrong factors are found to carry a negative sign
and havemagnitudes smaller than 2.0. Except for some extremelylarge
or small samples, the electron g factors fall between0.5 and 1.0.
To understand this interesting behavior,we need to recall that an
electronic state e in quantumdots, in the coupled spinorbital
basis, is composed ofcomponents from the conduction bands (s),
heavy (hh),light (lh), and split-o (sh) hole subbands,
e = s|s+ hh|hh+ lh|lh+ sh|sh (28)The electronic state is
dominated by the s componentfrom the conduction bands followed by
the light and
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340 Wei-dong Sheng, et al., Front. Phys., 2012, 7(3)
heavy hole components from the valence bands. In thebulk, the
holes in dierent bands have distinctive g fac-tors, i.e., gshh = 6,
gslh = gssh = 2 where is anempirical parameter. In quantum dots we
propose thatthe longitudinal electron g factor can be expressed
as
ge = 2.0 |s|s|2 6 |hh|hh|22 |lh|lh|2 2 |sh|sh|2 (29)
with now being a tting parameter. By tting this em-pirical
formulation with the result from numerical calcu-lation, we nd a
weak dependence of on the dimensionsof the dots. Typically, it
falls between 9.4 and 10.1. Thetheory therefore well explains why
the uctuation of size,shape, and even composition prole does not
have muchinuence on the electron g factor [99].
Electron g factors are almost isotropic in bulk semi-conductors
such as InAs or GaAs. However, like the elec-tron eective mass
[44], they become anisotropic in self-assembled quantum dots. The
in-plane electron g factoris given by
g//e = 2|s|s|2+g//lh|lh|lh|2+g//sh|sh|sh|2(30)and further we
have an anisotropic electron g factor [100]
ge g//e = 6 |hh|hh|2 + 2 |lh|lh|2 (31)As g//lh = 4 and g//sh = 2
are isotropic within thegrowth plane of the dots, the in-plane
electron g factorremains isotropic within the lateral
dimensions.
2.4.3 Hole g factors: Envelope orbital momentum
In contrast with electronic states, the valence-bandstates have
more complicated composition structuredue to strong mixing between
heavy- and light-holesubbands. As these two bands have distinctive
g fac-tor tensors [see Eq. (29)], the hole states in quantumdots
thus exhibit interesting behavior. Figure 11 showsthe hole g
factors calculated for the same set of thequantum dots as in Fig.
10. Compared with electrong factors, the hole g factors are seen to
be distributed
Fig. 11 Hole g factors against the single exciton emission
en-ergy calculated for the same set of InAs/GaAs quantum dots as
inFig. 10.
in more wider region, ranged from 2.5 to 1.5. Further-more,
their dependence on the emission energy is foundto be more
irregular.
In bulk InAs and GaAs both electron and hole g fac-tors are
negative. In InAs/GaAs quantum dots, electrong factors are found to
always carry a negative sign. Thereason why some dots can have
positive hole g factors liesin the nonzero envelope orbital momenta
(NEOM) car-ried by the components of the hole states [101].
Keepingin mind the expansion of any single-particle wave func-tion
as in Eq. (28), in general, one only needs to considerthe
contribution from the Bloch functions to the overallg factor
because envelope functions usually do not carryany NEOM, and
therefore have no eect on the g factor.
However, we nd that the light-hole component of theground hole
states in a quantum dot does carry NEOMdespite the fact that the
heavy-hole part does not. Thereason why the heavy hole component of
a zero angularmomentum can mix with the light-hole parts of
nonzeroangular momenta lies in the fact that the total
angularmomentum, i.e., that of the envelope part plus that ofthe
Bloch part, is a good quantum number in a systemwith cylindrical
symmetry. Even when the symmetry isbroken by the shear strain, we
nd that the conservationof the total angular momentum is still a
good approx-imation. Including the contribution from the
envelopefunctions, we have the overall hole g factor
gh = gsh + goh (32)
where goh denotes the contribution from NEOM,
goh = 2
nhh,lh,sh,sn|Lz|n+ n|Lz|n (33)
Although the light-hole component carries NEOM, itscontribution
to the overall g factor is small because ofits small projection in
the hole state. It is found thatthe heavy-hole part gains more and
more NEOM as theheight of the dot increases while the lateral
dimensionis xed. If the contribution from NEOM carried by
theheavy-hole component exceeds that from the Bloch partin absolute
amplitude, the overall hole g factor is seen tochange its sign and
becomes positive. If the positive holeg factor and the negative
electron g factor are similar inamplitude, the overall exciton g
factor can even vanish[101].
Apart from the sign change, the hole g factor also ex-hibits an
interesting anisotropy. As the biaxial strain inquantum dots splits
the heavy and light hole apart, thelow-lying states in the valence
bands are dominated bytheir heavy-hole components (see Fig. 5).
Recall that thebasis functions for the heavy-hole band at the
pointare given by |x + i|y and |x i|y , with orbitalangular momenta
oriented along the growth (z) direc-tion. The spinorbit interaction
couples the spin to theorbital angular momentum and results in the
z direction
-
Wei-dong Sheng, et al., Front. Phys., 2012, 7(3) 341
being the easy-axis of polarization of total angular mo-mentum
(quasi-spin). If a hole state is composed of onlythe heavy-hole
component, it would lead to the hole spinfrozen along the growth
direction and thus a zero g factorin the presence of a magnetic eld
applied in the growthplane [102].
For a hole state in a realistic quantum dot, its mi-nor
light-hole and split-hole components help to retain asmall g factor
in the Voigt conguration. Unlike an al-most isotropic in-plane
electron g factor [99], the in-planehole g factor can be highly
anisotropic due to NEOMcarried by mainly the light-hole parts
[103].
2.5 Quantum dots in electric elds
As in bulk materials and quantum wells, electronic andoptical
properties of self-assembled quantum dots canbe probed and then
tuned by applying external electricelds. The electric eld can be
applied either along thegrowth direction or in the growth
plane.
2.5.1 Quantum-conned Stark eect
In the presence of an external electric eld F appliedalong the z
(growth) direction, the Hamiltonian for anelectron conned in
quantum dots is given by H + qeFz.For at quantum dots with strong
connement along thevertical direction, the additional term for the
electriceld qeFz can usually be regarded as a perturbation.Hence
the energy of the electron ground state 0 in theelectric eld
becomes
E0 0|H|0+ qeF 0|z|0
+q2eF2n=1
0|z|nn|z|0E0 En (34)
with E0 = 0|H |0 being the energy in the ab-sence of the eld.
The coecients of the linear andquadratic terms are given by = qez
and =q2e
n=1 |z0n|2/E0n, respectively. Combining the for-mulations for
electrons and holes together, we have thetransition energy for an
electronhole pair given by
Ee Eh = (Ee Eh) + pF + F 2 (35)where p = qe(ze zh) is the
built-in dipole moment,and = e h measures the polarization of the
elec-tron and hole states. As the eect of electric eld on
theexciton binding energy can be approximated by a
similarexpansion, the above expression may also be used for
theemission energy of single excitons. If there is a
non-zerobuilt-in dipole moment in the dot, the Stark shift wouldbe
asymmetric [104].
Since in the Stark shift the linear term in electriceld is the
electronhole dipole moment, the informa-tion about the relative
positions of the electron and hole
states in the quantum dots can be determined by experi-ments.
For a quantum dot with a homogeneous composi-tion prole, the ground
hole state is found to be localizedcloser to the bottom of the dot
than the electron state[40]. This is due to the fact that the
electron state is af-fected by only the hydrostatic strain while
hole states areheavily inuenced by the biaxial strain component.
Sur-prisingly, an inverted electronhole alignment was foundin an
experiment on the Stark eect in self-assembledquantum dots [105],
which was later attributed to theinter-diusion eect [53].
The lateral electric eld has also been applied to exci-ton
complexes in quantum dots. The two main reasonsare i) attempt to
modify the anisotropy of the quantumdot and hence the exciton ne
structure [86, 106, 107]and ii) modication of the biexciton binding
energy [35,114]. It was shown that the electric eld alone
cannotremove the exciton ne structure splitting. However,
theremoval of the biexciton binding energy provides an al-ternative
route to generation of entangled photon pairswithout the need for
the modication of quantum dotstructural parameters such as
anisotropy [35, 114].
2.5.2 Electrical tuning of exciton g factors
The eect of an applied electric eld on a quantum dotis not
limited to the Stark shift observed in the opticaltransitions.
Other properties such as polarization of thephotoluminescence
emission and even exciton eectiveg factor may also be aected.
Electrical tuning of exci-ton g factors in single [108] and stacked
[109] InAs/GaAsquantum dots has already been demonstrated in
recentexperiments.
The spin splitting of an exciton state consists of
thecontributions from both electron and hole states, i.e.,gx =
ge+gh. Although it is still very dicult to measureelectron and hole
g factors independently in experiment,ge and gh can be calculated
separately. Our model sys-tem here is a pyramidal dot with a base
length of b=20 nm. Figure 12 plots the hole g factor as a function
ofthe applied eld for the dot of various height. For all the
Fig. 12 Hole g factor as a function of the applied electric
eldfor a pyramidal InAs/GaAs quantum dot of various height whichis
illustrated schematically in the inset.
-
342 Wei-dong Sheng, et al., Front. Phys., 2012, 7(3)
dots, we nd that electron g factor is strongly resistive tothe
applied electric eld and exhibits very little changeover a broad
range of the eld strength, and thereforedo not show its result in
the gure. On the contrary, thehole g factor is seen quite sensitive
to the applied eldand even changes its sign in the dots of high
aspect ratio.
The g factors of electronic states are known to be in-sensitive
to the size, geometry, and even the compositionprole of the quantum
dots [99]. The independence ofge on the applied electric eld can be
attributed to thesame physical mechanism. The hole g factor has
beenshown to increase with the height of the dot [101].
Thisdimensional dependence of the hole g factor is explainedin
terms of NEOM carried in the ground state of holes.As the eect of
an applied electric eld is equivalent tothe change in the eective
connement, the electric de-pendence of the hole g factor can be
understood in thesimilar way [110].
For eective tuning of exciton g factors in single quan-tum dots
a relatively strong electric eld is usually re-quired. It has
already been shown that, with the sameapplied electric eld, a
double dot often exhibit a largerStark shift compared with the
single dot of the samedimension. The model system adopted here
involves twocoupled disk-like InAs quantum dots, each having a
di-ameter of 15.3 nm, separated by a GaAs barrier witha thickness
of 4.5 nm. Figure 13 plots the g factors ofthe ground and rst
excited hole states calculated as afunction of the electric eld for
the coupled dots with aseparation of d = 4.5 nm. A resonance
structure can beseen in the spectra of hole g factors, which is
very dier-ent from the monotonic dependence of their counterpartsin
the single quantum dots. The probability density ofthe ground and
rst hole excited states at F = 0, 10kV/cm (o resonance) and F = 5.4
kV/cm (on reso-nance) is plotted to highlight the origin of the g
factor
Fig. 13 g factors of the ground and rst excited hole states in
alens-shaped double InAs/GaAs quantum dot, calculated as a
func-tion of the applied electric eld. Probability density of the
statesat the various electric elds are illustrated. Reproduced from
Ref.[111], Copyright c 2009 Institute of Physics.
resonances [111]. It is noted that recent theoretical work[112]
by using the eight-band k p method reports onthe exciton g factors
of the opposite sign to our resultand the experiment [109].
Finally, we would like to pointout that a magnetic impurity may
lead to dramatic vari-ation in electron g factors [113] though
electrical tuninghas been shown to be nearly ineective.
2.6 Single InAs/InP self-assembled quantum dots
onnanotemplates
Formation of self-assembled quantum dots during
theStranskiKrastanow growth of epitaxial layers is a veryuseful way
of controlling matter in three dimensions inthe one-dimensional
growth. The disadvantage of this ap-proach is the random in-plane
nucleation of quantumdots and the variation of their sizes.
Williams and co-workers [34, 115] proposed a dierent approach,
involv-ing growth of single InAs quantum dots on InP
nan-otemplates. The fabrication of nanotemplates starts
withlithographically dened patterns for the initialization ofthe
growth of InP pyramid. The starting area of the pyra-mid is dened
with the accuracy of tens of nanometers.The growth of the pyramid
is governed by the stability ofcrystallographic facets and, when
interrupted, results ina formation of nanotemplates dened with
atomic preci-sion. InAs dots are grown on such nanotemplates and
arecovered with InP to complete the pyramid. Such struc-ture, shown
in Fig. 14(a), can be functionalized by eithermetallic gates or by
building photonic crystals aroundthem for a fully controlled
light-matter system [36]. Therst steps toward the theory of InAs
quantum dots onInP templates have been made by some of us in Ref.
[42].
As an example of a deterministically functionalizedsingle
quantum dot we show the result of the applicationof gates to the
single quantum dot, resulting in a fullytunable optical structure.
Figure 14(b) shows the emis-sion spectrum of a single photo-excited
quantum dot as afunction of the bottom gate voltage. We see a
number ofemission lines, which correspond to charged exciton
com-plexes X0, X, X2, or dierent number of electrons Nin the
initial states (right-hand axis). The black circlesand lines create
a phase diagram of the emission spec-trum as predicted in Ref.
[27]. The emission for X2
consists of two lines, corresponding to the singlet andtriplet
two-electron complexes in the nal state, respec-tively. Hence, the
electron spin can be detected optically.The eect of spin can also
be seen in the emission fromother multiexciton complexes.
Because the nanotemplate can be elongated, it al-lows for the
control of the shape of the quantum dot.One would expect that the
resulting splitting of the twop-shell states would be a measure of
the elongation.Unfortunately this is not so straightforward.
Figure15(a) shows the InAs quantum dot on an elongated
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Wei-dong Sheng, et al., Front. Phys., 2012, 7(3) 343
Fig. 14 (a) Single InAs quantum dot in an InP pyramid
withmetallic gates on the sides of the pyramid. (b) Emission
spectrumas a function of the bottom gate voltage. Dierent emission
linescorrespond to dierent charged exciton states, X0,X, X2,
ordierent number of electrons in the initial states (right-hand
axis).The black circles and lines create a phase diagram of the
emissionspectrum as predicted in Ref. [27]. Reproduced from Ref.
[116],Copyright c 2009 Institute of Physics.
template. Figure 15(b) shows the emission from the ve-exciton
complex in which three electrons and three holespopulate the two
p-shell states. We see that even fora fully symmetric quantum dot
there are two emissionpeaks. These two peaks correspond to two
dierent four-exciton nal states, singletsinglet and
triplettriplet,which dier by exchange and correlation energy. As
thestructure becomes highly asymmetric, we can attemptto extract
the p-shell splitting from the emission spec-tra. Much work is
needed for the full understanding ofInAs/InP quantum dots on
patterned substrates to real-ize this promising technology.
3 Graphene quantum dots
3.1 Introduction
Carbon atom, a basis of organic chemistry, gives rise toa rich
variety of chemical structures, allotropes, due tothe exibility of
its -bonds. One of the allotropes isgraphene, a two-dimensional
sheet of carbon atoms ar-ranged in a hexagonal honeycomb lattice.
The theory ofgraphene has been developed at the National
ResearchCouncil of Canada by Wallace as early as 1947 [117].Since
then graphene was considered as a starting pointin the
understanding of other allotropes such as graphite
Fig. 15 (a) Schematic view of a single InAs quantum dot ona InP
pyramid template. (b) Emission spectrum from the ve-exciton to the
four-exciton complex as a function of the elongationof the
template. The two emission peaks correspond to two four-exciton nal
states, determined by spin for symmetric structureand splitting of
the p shell for asymmetric structure. Reproducedfrom Ref. [42],
Copyright c 2005 American Physical Society.
(stack of graphene layers), carbon nanotubes (rolled-upcylinders
of graphene), and fullerenes (wrapped grapheneby the introduction
of pentagones on the honeycomblattice). Graphene has been
investigated extensively inGraphite Intercalation Compounds (GIC)
[118]. Inter-calation of graphite with, e.g., Lithium or
Potassiumleads to increased separation of graphene sheets and
in-troduction of electrons or holes. GIC were equivalent todoped
semiconductors. The theory of optical propertiesof graphene in GIC
was developed by Blinowski and co-workers [119] and by one of us
[120] and has been com-pared with reectivity experiments. The
theory of elec-tronic screening and plasmons in GIC was also
developedat that time [121]. Recently, following the rst
experi-mental isolation of a single graphene sheet in 2004 byGeim,
Novoselov and co-workers [37], both experimen-tal and theoretical
research on graphene has increasedexponentially due to the unique
physical properties andpromising potential for applications
[122].
One of the most interesting electronic properties ofgraphene is
the zero-energy gap and relativistic natureof quasi-particle
dispersion close to the Fermi level pre-dicted by Wallace [117]:
low energy excitations are mass-less Dirac fermions, mimicking the
physics of quantumelectrodynamics at much lower velocities than
light. Onthe other hand, with ongoing improvements in
nanofabri-cation techniques [123], the zero-energy gap of the
Diracquasi-particles can be opened by engineering the size,
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344 Wei-dong Sheng, et al., Front. Phys., 2012, 7(3)
shape, edge, and carrier density. This in turn oers
pos-sibilities to control electronic [38, 124128, 130132],magnetic
[39, 123, 130138] and optical [138142] prop-erties of a
single-material nanostructure simultaneously.As a result, there is
a growing interest in studying lower-dimensional structures such as
graphene ribbons [144147], and more recently graphene quantum dots
[126129, 130]. In the following we will concentrate on
theelectronic structure, optical properties and magnetism
ofgraphene quantum dots. For transport properties, pleaserefer to
Ref. [143] for more detail review.
3.2 Electronic structure Tight-binding approach
The sp2 hybridization of carbon atom 2s and px, py or-bitals
leads to trigonal -bonds responsible for a robusthexagonal lattice
of carbon atoms. The remaining pz or-bitals from each carbon atom
form the -band, whichis well separated from the lled -band. Hence,
the elec-tronic states of graphene in the vicinity of the Fermi
levelcan be understood in terms of pz electrons on a hexag-onal
lattice. The hexagonal honeycomb lattice consistsof two triangular
sublattices as shown in Fig. 16. Redatoms form the sublattice A,
and blue atoms form thesublattice B. There are two carbon atoms, of
type A andB, in a unit cell. Following Wallace [117], the
wavefunc-tion of graphene can be written as a linear combinationof
pz orbitals localized on sublattices A and B:
k(r) =RA
A(RA)z(r RA)
+RB
B(RB)z(r RB) (36)
where z is a localized pz orbital. R represents the po-sition of
an atom on sublattice = A or B, and is afunction of the lattice
vectors a1,a2, and b shown in Fig.16. Within the tight-binding
formalism, the Hamiltoniancan be written as:
H = i,j,
tijcicj (37)
Fig. 16 Honeycomb lattice structure of graphene. The two
inter-penetrating triangular sublattices are illustrated by blue
and redcarbon atoms.
where the electrons can hop between sites i and j via
thetunneling matrix element ti,j . For graphene, the
nearestneighbor hopping energy is about t 2.8 eV and
thenext-nearest neighbor hopping energy t 0.1 eV [122].For
simplicity, let us consider only the rst nearest neigh-bor
interaction.
For bulk graphene the wave function that satises theperiodicity
of both sublattices can be classied by thewave vector k and written
as:
k(r) = AkRA
eikRAz(r RA)
+BkRB
eikRB+bz(r RB) (38)
Solving the Schrodinger equation for graphene involvesnding the
solution for the coecients of the two sublat-tices for each wave
vector k:(
0 tf(k)
tf(k) 0
)(Ak
Bk
)= E
(Ak
Bk
)(39)
where
f(k) = 1 + eika1 + eika2 (40)
The energy spectrum Ek and the wave function are thengiven
by
E(k) = t|f(k)| (41)Ak = eikBk (42)
with k being the phase of f(k). The energy spectrumE(k) is shown
in Fig. 17. For the charge neutral system,each carbon atom gives
one electron to the pz orbital. Asa result, the Fermi level is at
E(k) = 0, and the signcorresponds to the electron and hole
branches, respec-tively. Although the electron-hole symmetry is
conservedhere, if one includes second neighbor interaction t,
theelectron and hole branches become asymmetric.
Fig. 17 Energy spectrum of graphene obtained from the
nearest-neighbor tight-binding model.
3.3 Dirac fermions
One of the most striking properties of graphene is re-
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Wei-dong Sheng, et al., Front. Phys., 2012, 7(3) 345
vealed if we expand the energy spectrum close to Fermilevel Ek =
0, around the K and K points of the Bril-louin zone. In contrast
with semiconductor quantumdots, where the energy spectrum E(q) =
q2/(2m) isquadratic, in graphene we nd a linear dispersion
E(q) vFq (43)where q is the momentum relative to one of the K
orK points, called Dirac points, and vF = 3tb/2 106m/s is the
velocity of Dirac fermions. We thus obtainrelativistic-like
dispersion relation with a velocity 300times smaller than that of
light. We can also expandEq. (39) around one of the K points which
gives thetwo-dimensional Dirac equation:
ivF (r) E(r) (44)The wave functions around K and K points are
givenby
(k) 12
(eik
1
)around K (45)
12
(e+ik
1
)around K (46)
As discussed above, the wave function for Dirac Fermionshas two
components corresponding to the two sublat-tices. The two-component
character is often referred to aspseudo-spin. In analogy with real
spin, the Dirac Fermionwave function acquires Berrys phase if we
adiabaticallychange the wave vector k along a closed loop
enclosingthe Dirac point. Hence the sublattice structure adds
anontrivial topological eect to the physics of graphene.
3.4 Graphene quantum dots
As we can see from Eq. (43) and Fig. 17, graphene is agapless
material. As a result, in analogy with the phe-nomenon of Klein
tunneling for massless particles in rel-ativistic quantum
mechanics, it is not possible to conneDirac electrons in graphene
electrostatically using metal-lic gates as in semiconductor quantum
dots. Various indi-rect ways were proposed for opening a gap to
conne theelectrons, such as size quantization in a graphene
ribbon[144147] or using bilayer graphene [148155]. In this re-view
we will focus on graphene islands with size quantiza-tion in the
two dimensions, i.e., graphene quantum dotswith edges created by,
e.g., etching [123]. Quantum con-nement in such systems was
experimentally observed[38, 124, 125] and there is increasing
interest in creat-ing quantum dots with well controlled shapes and
edges[123, 129].
3.5 Shape and edge eects
Graphene sheet can be cut along dierent crystallo-
graphic directions as seen in the top panel of Fig. 18,resulting
in dierent types of edges. The most stableedge types are armchair
and zigzag edges [156, 157]. To-gether with the shape of the
quantum dot, the edge typeplays an important role in determining
the electronic,magnetic, and optical properties. In this section,
we willcompare the properties of quantum dots of various shapeand
edge, and study their properties as a function of theirsize. We
consider three dierent quantum dots as illus-trated in Fig. 18: (a)
hexagonal dot with armchair edges,(b) hexagonal dot with zigzag
edges, and (c) triangulardot with zigzag edges. Their energy
spectra, shown onthe lower panel of Fig. 18, were calculated by
numeri-cally solving the tight-binding Hamiltonian of Eq. (37)in
the nearest neighbors approximation. Clearly the en-ergy spectrum
around the Fermi level (E = 0) stronglydepends on the structure.
While the hexagonal armchairand trigonal zigzag dots have a well
dened gap of theorder of 0.2t (shown by a red arrow), for the
hexagonalzigzag dot the gap is much smaller. Moreover, in
additionto valence and conduction bands, the trigonal zigzag
dotspectrum shows a shell of degenerate levels at the Fermilevel
[39, 130, 132136, 138]. As we will see in the fol-lowing section,
this degenerate band is responsible forinteresting electronic and
magnetic properties.
Fig. 18 Single-particle tight-binding spectrum of (a) arm-chair
hexagonal, (b) zigzag hexagonal, and (c) zigzag triangulargraphene
quantum dot structures consisting of similar number ofcarbon atoms.
Top panel shows the atomic positions. Reproducedfrom Ref. [133],
Copyright c 2010 American Physical Society.
In order to understand further the electronic proper-ties near
the Fermi level, in Fig. 19 we plot the energy gapas a function of
the number of atoms. For the hexagonalarmchair dot (red dots), the
gap decays as the inverseof the square root of number of atoms N ,
from hundredatom to million atom nanostructures. This is
expectedfor conned Dirac fermions with photon-like linear en-ergy
dispersion (Egap kmin 2/x 1/
N), as
pointed out in Refs. [130, 139, 158]. However, the re-placement
of the edge from armchair to the zigzag has
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346 Wei-dong Sheng, et al., Front. Phys., 2012, 7(3)
a signicant eect on the energy gap. The energy gap ofhexagonal
structure with zigzag edges decreases rapidlyas the number of atoms
increases. This is due to thezigzag edges leading to localized
states at the edge ofthe quantum dot, similar to whispering gallery
modesof photons localized at the edge of photonic microdisk[159].
Figure 19 also shows the eect on the energy gapof deforming the
hexagonal structure into a triangle whilekeeping zigzag edges.
Unlike the hexagonal zigzag struc-ture, all three edges of the
triangle are composed of atomsof the same sublattice. As a result,
they do not hybridizeand form a degenerate band at Fermi level. The
energygap shown in Fig. 19 corresponds to transitions from
thetopmost valence to the lowest conduction band state.The energy
gap in the triangular zigzag structure followsthe power law
Egap
N since it is due to connement
of bulk states, whereas the gap for the hexagonal arm-chair
structure is due to the hybridization of the edgestates. We note
that the energy gap changes from 2.5eV (green light) for a quantum
dot with 100 atomsto 30 meV (8 THz) for a quantum dot with a
millionatoms and a diameter of 0.4 micrometer. The pres-ence of a
partially occupied band of degenerate statesin the middle of a well
dened energy gap oers uniqueopportunity to control magnetic and
optical propertiesof triangular graphene nanostructures
simultaneously.
Fig. 19 Energy band gap as a function of number of carbonatoms
for triangular zigzag, hexagonal armchair and hexagonalzigzag
structures. Reproduced from Ref. [133], Copyright c 2010American
Physical Society.
3.6 Gated quantum dots: Beyond tight-bindingapproach
While the basic tight-binding model of Eq. (37) ac-curately
describes the electronic properties of bulkgraphene, the situation
can be more complicated forgated nite-size systems. The
experimental values usedfor rst and second nearest neighbors
hopping terms tand t are based on the assumption that electronic
oc-cupation on every site is constant (equal to one), whichmay not
be true at the edges of a quantum dot. More-
over, for a gated system, the average occupation per sitecan be
higher or lower than one. Thus, one must takeinto account the
deviation of the electronic density fromthe bulk values due to edge
eects and doping via theelectrostatic gate [39].
Interaction eects due to additional charge density oneach site
can be investigated within a mean-eld ap-proach using a combination
of tight-binding and self-consistent HartreeFock methods (TBHF).
For pz elec-trons, the interacting many-body Hamiltonian can
bewritten as
H = il
tilcicl
+12
ijkl
ij|V |klcicjckcl (47)
and the corresponding mean-eld HartreeFock Hamil-tonian is
HMF = il
tilcicl +
il
jk
jk (ij|V |kl
ij|V |lk)cicl (48)where the operator ci creates a pz electron on
site i withspin . Note that at this stage the unknown hoppingterms
til do not include the eect of electronelectroninteractions. We
want to express the Hamiltonian for oursystem as a function of
experimentally measured bulktight-binding parameters il . For the
graphene sheet,the mean-eld Hamiltonian is written as
HbulkMF = il
tilcicl +
il
jk
bulkjk (ij|V |kl
ij|V |lk)cicl (49)
il
ilcicl (50)
We can now re-express our mean-eld Hamiltonian as
HMF = HMF HbulkMF + HbulkMF = il
ilcicl
+il
jk
(jk bulkjk )(ij|V |kl
ij|V |lk)cicl (51)which must be solved self-consistently to
obtain Hartree-Fock quasi-particle states.
In order to take into account the eect of induced gatecharge
away from charge neutrality, we assume that elec-trons in the
graphene island interact with the electronstransferred to the gate
via the term vgii given by
vgii(qind) =Nsitej=1
qind/Nsite
(xi xj)2 + (yi yj)2 + d2gate(52)
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Wei-dong Sheng, et al., Front. Phys., 2012, 7(3) 347
where (xi, yi) are the coordinates of the atoms. Thismodel
assumes that the induced charge qind = N issmeared out at positions
(xi, yi) at a distance dgate fromthe quantum dot. Our nal
HartreeFock Hamiltonian isthen given by
HMF = il
ilcicl
+il
jk
(jk bulkjk )(ij|V |kl
ij|V |lk)cicl +i
vgii(qind)cici (53)
After self-consistent diagonalization of the Hamilto-nian as in
Eq. (53) for a TBHF reference level dened byN = Nref , we obtain
TB+HF quasi-particles denoted bythe creation operator bp, with
eigenvalues p and eigen-functions |p. We can then start lling the
conductionstates above Fermi level one by one to investigate
cor-relation eects. The general Hamiltonian for the wholesystem can
be written as
H = H + HMF HMFfrom which, after extensive algebra, it can be
shown thatin the rotated basis of bp quasi-particles and by
neglect-ing scatterings from/to the degenerate shell, the
Hamil-tonian for Nadd + Nref electrons reduces to the congu-ration
interaction problem for the Nadd added electronsgiven by
H =p
pbpbp +
12
pqrs
pq|V |rsbpbqbrbs
+pq
p|vg(Nadd)|qbpbq
+2pp|vg(Nadd)|p (54)
where the indices without the prime sign (p, q, r, s) runover
states above the TBHF reference level, while theindex with the
prime sign p runs over valence states(below the TBHF reference
level). Here the rst termon the right-hand side represents the
energies of quasi-particles; the second term describes the
interaction be-tween the added quasiparticles, the third term
describesthe interaction between the quasiparticles and the
gate;and the last term is a constant giving the interactionenergy
between the Nref electrons and the charge onthe gate. We then build
all possible many-body con-gurations within the degenerate shell
for a given elec-tron number Nadd, for which Hamiltonian matrices
cor-responding to dierent Sz subspaces are constructed
anddiagonalized.
3.7 Magnetism in triangular quantum dots
As discussed in the previous section, when an electron is
conned to a triangular atomic thick layer of graphenewith zigzag
edges, its energy spectrum collapses to a shellof degenerate states
at the Fermi level (the Dirac point).This is similar to the edge
states in graphene ribbons[144147], but the shell is isolated from
the other statesby a gap. Indeed, the zigzag edge breaks the
symmetrybetween the two sublattices of the honeycomb lattice,
be-having like a defect. Therefore, electronic states localizedon
the zigzag edges appear with energy in the vicinity ofthe Fermi
level. The degeneracy Nedge is proportional tothe edge size and can
be made macroscopic. A non-trivialquestion addressed here is the
specic spin and orbitalconguration of the electrons as a function
of the sizeand the fractional lling of the degenerate shell of
edgestates. Due to the strong degeneracy, many-body eectscan be
expected to be as important as in the fractionalquantum Hall eect,
but without the need for a magneticeld. Calculations based on the
Hubbard approximation[134, 135] and local spin density functional
theory [135,136] showed that the neutral system (i.e., at
half-lling)has its edge states polarized.
In order to study many-body eects within thecharged degenerate
shell using the conguration inter-action method, we rst perform a
HartreeFock calcula-tion for the charged system of N Nedge
electrons, withempty degenerate shell and Nedge electrons
transferredto the gate, as shown in Fig. 20 (a). The spectrum of
HFquasi-particles is shown in Fig. 20 (b) with black lines.Due to
the mean-eld interaction with the valence elec-trons and charged
gate, a group of three states is nowseparated from the rest by a
small gap of 0.2 eV. Thethree states correspond to HF
quasiparticles localized inthe three corners of the triangle. The
same physics oc-curs in density functional calculation within local
densityapproximation (LDA), shown in the inset of Fig. 20 (b).Hence
we see that the shell of almost degenerate stateswith a well dened
gap separating them from the valenceand conduction bands exists in
the three approaches.
The wave functions corresponding to the shell
ofnearly-degenerate zero-energy states obtained from TB-HF
calculations are used as a basis set in our congura-tion
interaction calculations where we add Nadd electronsfrom the gate
to the shell of degenerate states. In Fig.21, total spin S of the
ground state as a function ofthe lling of the degenerate shell is
shown for dierentsizes of quantum dots. Three aspects of these
resultsare particularly interesting: (i) for the charge neutralcase
(Nadd Nedge = 0), for all the island sizes studied(Nedge = 3 7),
the half-lled shell is maximally spinpolarized as indicated by red
arrows, in agreement withDFT calculations [135, 136]. The
polarization of the half-lled shell is also consistent with the
Lieb theorem forthe Hubbard model for bipartite lattice [160]. (ii)
Thespin polarization is fragile away from half-lling. If weadd one
extra electron (NaddNedge = 1), magnetization
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348 Wei-dong Sheng, et al., Front. Phys., 2012, 7(3)
Fig. 20 (a) Electronic density in a triangular graphene islandof
97 carbon atoms where 7 electrons were moved to the metal-lic gate
at a distance of dgate. (b) S