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PHYSICAL REVIEW B 89, 115413 (2014)
Intrinsic optical conductivity of modified Dirac fermion systems
Habib Rostami and Reza Asgari*
School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran
(Received 25 December 2013; revised manuscript received 24 February 2014; published 12 March 2014)
We analytically calculate the intrinsic longitudinal and transverse optical conductivities of electronic systems
which govern by a modified Dirac fermion model Hamiltonian for materials beyond graphene such as monolayer
MoS2and ultrathin film of the topological insulator. We analyze the effect of a topological term in the Hamiltonian
on the optical conductivity and transmittance. We show that the optical response enhances in the nontrivial phase
of the ultrathin film of the topological insulator and the optical Hall conductivity changes sign at transition from
trivial to nontrivial phases which has significant consequences on the circular dichroism property and optical
absorption of the system.
DOI: 10.1103/PhysRevB.89.115413 PACS number(s): 72.20.−i,78.67.−n,78.20.−e
I. INTRODUCTION
Two-dimensional (2D) materials have been one of the most
interesting subjects in condensed matter physics for potential
applications due to the wealth of unusual physical phenomena
that occur when charge, spin, and heat transport are confined
to a 2D plane [1]. These materials can be mainly classified in
different classes which can be prepared as a single atom thick
layer namely, layered van der Waals materials, layered ionic
solids, surface growth of monolayer materials, 2D topological
insulator solids, and finally 2D artificial systems and they
exhibit novel correlated electronic phenomena ranging from
high-temperature superconductivity, quantum valley or spin
Hall effect to other enormously rich physics phenomena. Two-
dimensional materials can be mostly exfoliated into individual
thin layers from stacks of strongly bonded layers with weak
interlayer interaction and a famous example is graphene and
hexagonal boron nitride [2]. The 2D exfoliates versions of
transition metal dichalcogenides that exhibit properties that
are complementary to and distinct from those in graphene [3].
Optical spectroscopy is a broad field and useful to explore
the electronic properties of solids. Optical properties can be
tuned by varying the Fermi energy or the electronic band
structure of 2D systems. Recently, developed 2D systems such
as gapped graphene [4], thin film of the topological insulator
[5,6], and monolayer of transition metal dichalcogenides [3]
provide the electronic structures with direct band gap signa-
tures. The optical response of semiconductors with direct band
gap is strong and easy to explore experimentally since photons
with energy greater than the energy gap can be absorbed or
omitted. The thin film of the topological insulator, on the other
hand, has been fabricated experimentally by using Sb2Te3slab
[7] and has been shown that a direct band gap can be formed
owning to the hybridization of top and bottom surface states.
Furthermore, a nontrivial quantum spin Hall phase has been
realized experimentally which was predicted previously in this
system [8–10]. Although pristine graphene and surface states
of the topological insulator reveal massless Dirac fermion
physics, by opening an energy gap they become formed as
massive Dirac fermions. The thin film of the topological
insulator and monolayer transition metal dichacogenides can
*asgari@ipm.ir
be described by a modified Dirac Hamiltonian. A monolayer
of the molybdenum disulfide (ML-MoS2) is a direct band gap
semiconductor [11], however, its multilayer and bulk show
indirect band gap [3]. This feature causes the optical response
in ML-MoS2to increase in comparison with its bulk and
multilayer structures [12–16].
One of the main properties of ML-MoS2is a circular
dichroism aspect responding to a circular polarized light where
the left- or right-handed polarization of the light couples only
to the Kor Kvalley and it provides an opportunity to
induce a valley polarized excitation which can profoundly
be of interest in the application for valleytronics [17–19].
Another peculiarity of ML-MoS2is the coupled spin valley
in the electronic structure which is owing to the strong
spin-orbit coupling originating from the existence of a heavy
transition metal in the lattice structure and the broken inversion
symmetry, too [20]. These two aspects are captured in a minima
massive Dirac-like Hamiltonian introduced by Xiao et al. [20].
However, it has been shown, based on the tight binding [21,22]
and k.p method [23], that other terms like an effective mass
asymmetry, a trigonal warping, and a diagonal quadratic term
might be included in the massive Dirac-like Hamiltonian. The
effect of the diagonal quadratic term is very important; for
instance, if the system is exposed by a perpendicular magnetic
field, it will induce a valley degeneracy breaking term [21].
The optical properties of ML-MoS2have been evaluated by ab
initio calculations [24] and studied theoretically based on the
simplified massive Dirac-like model Hamiltonian [25], which
is by itself valid only near the main absorbtion edge. A part
of the model Hamiltonian which describes the dynamic of
massive Dirac fermions are known in graphene committee to
have an optical response quite different from that of a standard
2D electron gas. Thus it would be worthwhile to generalize
the optical properties of such systems by using the modified
Dirac fermion model Hamiltonian.
The modified Hamiltonian for ML-MoS2without trigonal
warping effect at Kpoint is very similar to the modified
Dirac equation which has been studied for an ultrathin
film of the topological insulator (UTF-TI) around point
[8,26]. The modified Dirac Hamiltonian reveals nontrivial
quantum spin hall (QSH) and trivial phases corresponding
to the existence and absence of the edge states, respectively.
Those phases have been predicted theoretically [8–10,27] and
recently observed by experiment [7]. An enhancement of the
1098-0121/2014/89(11)/115413(12) 115413-1 ©2014 American Physical Society
HABIB ROSTAMI AND REZA ASGARI PHYSICAL REVIEW B 89, 115413 (2014)
optical response of UTF-TI has been obtained in the nontrivial
phase [28] and a band crossing is observed in the presence
of the structure inversion asymmetry induced by substrate
[29]. Since the modified Dirac Hamiltonian incorporates an
energy gap and a quadratic term in momentum which both
have topological meaning, it is natural to expect that the
topological term of the Hamiltonian plays an important role in
the optical conductivity. In this paper, we analytically calculate
the intrinsic longitudinal and transverse optical conductivities
of the modified Dirac Hamiltonian as a function of photon
energy. This model Hamiltonian covers the main physical
properties of ML-MoS2and UTF-TI systems in the regime
where interband transition plays a main role. We analyze the
effect of the topological term in the Hamiltonian on the optical
conductivity and transmittance. Furthermore, we show that the
UTF-TI system has a nontrivial phase and its optical response
enhances; in addition, the optical Hall conductivity changes
sign at a phase boundary, when the energy gap is zero. This
changing of the sign has a significant consequence on the
circular polarization and the optical absorbtion of the system.
The paper is organized as follows. We introduce the low-
energy model Hamiltonian of ML-MoS2and UTF-TI systems
and then the dynamical conductivity is calculated analytically
by using Kubo formula in Sec. II. The numerical results for
the optical Hall and longitudinal conductivities and optical
transmittance are reported, and we also provide discussions
with circular dichroism in both systems in Sec. III.Abrief
summary of results is given in Sec. IV.
II. THEORY AND METHOD
The low-energy properties of the ML-MoS2and other
transition metal dicalcogenide materials can be described by a
modified Dirac equation [21–23] and the Hamiltonian around
the Kand Kpoints is given by
Hτs =λ
2τs +−λτ s
2σz+t0a0q·στ+
2|q|2
4m0
(α+βσz),
(1)
where the Pauli matrices stand for a pseudospin which indi-
cates the conduction and valence band degrees of freedom, τ=
±denotes the two independent valleys in the first Brillouin
zone, q=(qx,qy) and στ=(τσ
x,σy). The numerical values
of the parameters will be given in Sec. III.
The UTF-TI system, on the other hand, can be described
by a modified Dirac Hamiltonian around the point with two
independent hyperbola (isospin) degrees of freedom [8,26] and
thus the Hamiltonian reads
Hτ=0+τ
2σz+t0a0q·σ+
2|q|2
4m0
(α+τβσz).(2)
Note that the Pauli matrices in this Hamiltonian stand for
the real spin where spin is rotated by operator U=diag[1,i]
which results in U†σxU=−σyand U†σyU=σxand the
isospin index of τ=±indicates two independent solutions of
UTF-TI which are degenerated in the absence of the structure
inversion asymmetry and can be assumed as an internal isospin
(spin, valley or sublattice) degree of freedom. Two mentioned
models, Eqs. (1) and (2), are similar to some extent and
describe similar physical properties.
Generally, the Hamiltonian around the (τ=+) and
K(τ=+) points for UTF-TI and monolayer MoS2systems,
respectively, can be rewritten as
H=a1+b(α+β)q2cq∗
cq a2+b(α−β)q2,(3)
where a1=/2+0,a
2=−/2+0for UTF-TI and
a1=/2,a
2=−/2+λs for ML-MoS2. Note that b=
2/4m0a2
0,c=t0, and we set a0q→q. The eigenvalue and
eigenvector of the Hamiltonian, Eq. (3), can be obtained as
|ψc,v= 1
Dc,v −cq∗
hc,v ,
hc,v =d∓d2+c2q2,d=a1−a2
2+bβ q 2,
(4)
Dc,v =c2q2+h2
c,v,
εc,v =a1+b(α+β)q2−hc,v,
and velocity operators along the xand ydirections are
vx=∂H
∂qx=cσx+2bαqx+2bβ q xσz,
(5)
vy=∂H
∂qy=cσy+2bαqy+2bβ q yσz.
The intrinsic optical conductivity can be calculated by using
the Kubo formula [30–32] in a clean sample and it is given by
σxy (ω)=−ie2
2πh d2qf(εc)−f(εv)
εc−εvψc|vx|ψvψv|vy|ψc
ω+εc−εv+i0++ψv|vx|ψcψc|vy|ψv
ω+εv−εc+i0+,
(6)
σxx(ω)=−ie2
2πh d2qf(εc)−f(εv)
εc−εvψc|vx|ψvψv|vx|ψc
ω+εc−εv+i0++ψv|vx|ψcψc|vx|ψv
ω+εv−εc+i0+,
where f(ω) is the Fermi distribution function. We include only the interband transitions and the contribution of the intraband
transitions, which leads to the fact that the Drude-like term is no longer relevant in this study since the momentum relaxation time
is assumed to be infinite. This approximation is valid at low temperature and a clean sample where defect, impurity, and phonon
scattering mechanisms are ignorable. We also do not consider the bound state of exciton in the systems. After straightforward
calculations (details can be found in Appendix A), the real and imaginary parts of diagonal and off-diagonal components of the
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INTRINSIC OPTICAL CONDUCTIVITY OF MODIFIED . . . PHYSICAL REVIEW B 89, 115413 (2014)
conductivity tensor at τ=+are given by
σRe
xy (ω)=2e2
hqdq(f(εc)−f(εv)) ×c2
d2+c2q2(d−2bβ q 2)P−1
(ω)2−(εc−εv)2,
σIm
xy (ω)=πe2
hqdq f(εc)−f(εv)
εc−εv×c2
d2+c2q2(d−2bβ q 2){δ(ω+εv−εc)−δ(ω+εc−εv)},
(7)
σIm
xx (ω)=−2e2
h
ωqdq f(εc)−f(εv)
εc−εv×c2−c2q2
d2+c2q2c2
2+bβ (a1−a2)P−1
(ω)2−(εc−εv)2,
σRe
xx (ω)=−πe2
hqdq f(εc)−f(εv)
εc−εv×c2−c2q2
d2+c2q2c2
2+bβ (a1−a2){δ(ω+εv−εc)+δ(ω+εc−εv)},
where Re and Im refer to the real and imaginary parts of σand Pdenotes the principle value. It is worthwhile mentioning that the
conductivity for ML-MoS2for τ=−can be found by implementing px→−pxand λ→−λ. Using these transformations, the
velocity matrix elements around the Kpoint can be calculated by taking the complex conjugation of the corresponding results
for the τ=+case. Furthermore, for the UTF-TI case system, we must replace and βby their opposite signs which lead to
the same results in comparison with the ML-MoS2case around Kpoint. More details in this regard are given in Appendix A.
A. Optical conductivity of ML-MoS2
Having obtained the general expressions of the conductivity for the modified Dirac fermion systems, the conductivity of two
examples namely the ML-MoS2and UTF-TI could be obtained. Here, we would like to focus on the ML-MoS2case and explore
its optical properties, although all results can be generalized to the UTF-TI system as well. Therefore, the optical conductivity for
each spin and valley components of ML-MoS2can be obtained by using appropriate substitution in Eq. (7) and results are written as
σRe,τ s
xy (ω)=2e2
h
Pdq(f(εc)−f(εv)) ×τ(
τsq−βq3)
(
τs +βq2)2+q2[4((
τs +βq2)2+q2)−(ω/t0)2],
σIm,τ s
xy (ω)=πe2
2hdq(f(εc)−f(εv)) ×τ(
τsq−βq3)
(
τs +βq2)2+q2δ(ω/t0−2(
τs +βq2)2+q2),
σIm,τ s
xx (ω)=−2e2
h
ωPdq(f(εc)−f(εv)) ×q
(
τs +βq2)2+q2[4((
τs +βq2)2+q2)−(ω/t0)2]
−q31
2+2β
τs
((
τs +βq2)2+q2)3/2[4((
τs +βq2)2+q2)−(ω/t0)2],
σRe,τ s
xx (ω)=−πe2
2hdq(f(εc)−f(εv)) ×q
(
τs +βq2)2+q2−q31
2+2β
τs
((
τs +βq2)2+q2)3/2
×δ(ω/t0−2(
τs +βq2)2+q2),(8)
where
τs =(−λτ s)/2t0,α=bα/t0,β=bβ /t0,στ,s
xy =
σRe,τ s
xy +iσIm,τ s
xy , and στ,s
xx =σRe,τ s
xx +iσIm,τ s
xx .
Note that in the case of UTF-TI, there is no extra spin index
of sas a degree of freedom and
τs might be replaced by
=/2t0; consequently we have στ
xy and στ
xx rather than
στs
xy and στs
xx .Tobemoreprecise,λτ s , which is located out
of the radical in Eq. (8), might be replaced by 0in the c,v
to achieve desirable results corresponding to the UTF-TI. It is
clear that the dynamical charge Hall conductivity vanishes in
both the UTF-TI and ML-MoS2systems due to the presence
of the time reversal symmetry. For the MoS2case, the spin and
valley transverse ac conductivity are given by
σs
xy =
2e
τστ,↑
xy −στ,↓
xy ,σ
v
xy =1
e
sσK,s
xy −σK,s
xy ,
(9)
and for the longitudinal ac-conductivity case, an electric
field can only induce a charge current and corresponding
conductivity is given as
σxx =
τστ,↑
xx +στ,↓
xx .(10)
Moreover, the longitudinal conductivity is the same as ex-
pression given by Eq. (10) for the UTF-TI case, however, the
Hall conductivity is slightly changed. Owing to the coupling
between the isospin and the spin indexes, the hyperbola Hall
conductivity is a spin Hall conductivity [7,8] and it is thus
given by
σhyp
xy =1
eσ+
xy −σ−
xy .(11)
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HABIB ROSTAMI AND REZA ASGARI PHYSICAL REVIEW B 89, 115413 (2014)
B. Intrinsic dc conductivity
To find the static conductivity in a clean sample, we
set ω=0 and thus the interband longitudinal conductivity
vanishes. Consequently, we calculate only the transverse
conductivity in this case. At zero temperature, the Fermi
distribution function is given by a step function, i.e., f(εc,v)=
(εF−εc,v). We derive the optical conductivities for the case
of ML-MoS2and results corresponding to the UTF-TI can
be deduced from those after appropriate substitutions. Most
of the interesting transport properties of ML-MoS2originates
from its spin-splitting band structure for the hole doped case.
Therefore, for the later case, when the upper spin-split band
contributes to the Fermi level state, the dc conductivity is
given by
σK↑
xy =−σK↓
xy =−e2
2hqc
qF
(
K↑q−βq3)dq
((
K↑+βq2)2+q2)3
2
=−e2
2hCK↑+e2
2h
2μ+2b(α−β)q2
F
−λ+2μ+2bαq2
F
,(12)
and for the spin-down component we thus have
σK↓
xy =−σK↑
xy =−e2
2hqc
0
(
K↓q−βq3)dq
((
K↓+βq2)2+q2)3
2
=−e2
2hCK↓,(13)
where qcis the ultraviolate cutoff and μ/t0=
√(
K↑+βq2
F)2+q2
F−
K↑−αq2
Fstands for the chemical
potential and it is easy to show that CKs =sgn(−
λs)−sgn(β) at large cutoff values. In a precise definition,
CKs terms are the Chern numbers for each spin and valley
degrees of freedom and the total Chern number is zero owing
to the time reversal symmetry. Intriguingly, the quadratic
term in Eq. (3), β, leads to a new topological characteristic.
When β > 0, with >λ, the system has a trivial phase
with no edge mode closing the energy gap, however, for
the case that β < 0, the topological phase of the system
is a nontrivial with edge modes closing the energy gap. In
the case of the ML-MoS2, the tight binding model [21,33]
predicts the trivial phase (β>0) with CKs =0. However, a
nontrivial phase is expected by Refs. [22,23] (where β<0)
which leads to CKs =2. In other words, the term proportional
to βhas a topological meaning in Z2symmetry invariant like
the UTF-TI system [8] and the sign of βplays important
role.
The transverse intrinsic dc conductivity for the hole doped
ML-MoS2case, is given by
σs
xy =
eσK↑
xy −σK↓
xy =e
2π
μ+b(α−β)q2
F
−λ+2μ+2bαq2
F
,
σv
xy =2
eσK↑
xy +σK↓
xy =−e
hCK+2
σs
xy ,(14)
where, at large cutoff, CK=[sgn(−λ)+sgn(+λ)]/2−
sgn(β) stands for the valley Chern number and it equals
to zero or 2 corresponding to the nontrivial or trivial band
structure, respectively. In the case of the UTF-TI, the isospin
Hall conductivity is
σhyp
xy =−e
hC+2e
h
μ+b(α−β)q2
F
+2μ+2bαq2
F
,(15)
where μ/t0=√(+βq2
F)2+q2
F−−0−αq2
Fand
C=sgn()−sgn(β) at large cutoff. This result is consistent
with that result obtained by Lu et al. [8]. It should be noted
that in the absence of the diagonal quadratic term, the nonzero
valley Chern number at zero doping predicts a valley Hall
conductivity, which is proportional to sign(). Therefore, the
exitance of edge states, which can carry the valley current, is
anticipated. However, Z2symmetry prevents the edge modes
from existing. Since the Z2topological invariant is zero when
the gap is caused only by the inversion symmetry breaking
[34], thus the topology of the band structure is trivial and there
are no edge states to carry the valley current when the chemical
potential is located inside the energy gap. Therefore, we can
ignore the valley Chern number in σv
xy and thus the results are
consistent with those results reported by Xiao et al. [20]ata
low doping rate where μ−λ.
C. Intrinsic dynamical conductivity
In this section, we analytically calculate the dynamical
conductivity of the modified Dirac Hamiltonian which results
in the trivial and nontrivial phases. Using the two-band
Hamiltonian, including the quadratic term in momentum, the
optical Hall conductivity for each spin and valley components
are given by
σRe,τ s
xy (ω)=τe2
h[Gτs(ω,qF)−Gτs(ω,qc)],
σIm,τ s
xy (ω)=τπe2
2h
τs −βq2
0,τ s
ωn(ω)2ε
F−λτs
−2αq2
0,τ s −ω−(ω→−ω)
×(n(ω)−(1 +2β
τs)),(16)
where Re and Im indicate to the real and imaginary parts,
respectively, and Gτs(ω,q) reads as below (details are given
in Appendix B):
Gτs(ω,q)
=
τs
ωn(ω)ln
ωm(q)
n(ω)−2(
τs +βq2)2+q2
ωm(q)
n(ω)+2(
τs +βq2)2+q2
+1
4βωn(ω)ln
ωm(q)
n(ω)−2(
τs +βq2)2+q2
ωm(q)
n(ω)+2(
τs +βq2)2+q2
−1
4βωln
ω−2(
τs +βq2)2+q2
ω+2(
τs +βq2)2+q2
,(17)
where m(q)=1+2β
τs +2β2q2,n(ω)=
1+4β
τs +β2(ω)2,ω=ω/t0,ε
F=εF/t0,
and λ=λ/t0.Thevalueofq0,τ s can be evaluated from
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INTRINSIC OPTICAL CONDUCTIVITY OF MODIFIED . . . PHYSICAL REVIEW B 89, 115413 (2014)
m(q0,τ s )=n(ω). Note that qc, the ultraviolate cutoff, is
assumed to be equal to 1/a0. Some special attentions might
be taken for the situation in which there is no intersection
between the Fermi energy and the band energy, for instance,
in a low doping hole case of the ML-MoS2in which the Fermi
energy lies in the spin-orbit splitting interval. In this case,
the Fermi wave vector (qF, which has no contribution to the
Fermi level) vanishes.
The quadratic terms can also affect profoundly on the
longitudinal dynamical conductivity which plays main role
in the optical response when the time reversal symmetry is
preserved. In this case, one can find
σRe,τ s
xx (ω)=−πe2
4h
1
n(ω)1−1+4β
τs
22q0,τ s
ω2
×2ε
F−λτs −2αq2
0,τ s −ω
−(ω→−ω)(n(ω)−(1 +2β
τs))
σIm,τ s
xx (ω)=−e2
h[Hτs(ω,qF)−Hτs(ω,qc)],(18)
where Hτs(ω,q) is given by (details are given in Appendix B)
Hτs(ω,q)
=(1 +2β
τs)m(q)−(1 +4β
τs)
2β2ω(
τs +βq2)2+q2
+1+4β
τs
2β2(ω)2ln
ω
2−(
τs +βq2)2+q2
ω
2+(
τs +βq2)2+q2
+(1 +2β
τs)(1 +4β
τs)+β2(ω)2
2β2(ω)2n(ω)
×ln
ω
2
m(q)
n(ω)−(
τs +βq2)2+q2
ω
2
m(q)
n(ω)+(
τs +βq2)2+q2
.(19)
It is worthwhile mentioning that the Gand Hfunctions do not
depend on the αterm given in Eq. (3). For β=0inEq.(3),
we have m(q)/n(ω)→1, 1/n(ω)→1−2β
τs, therefore
Gτs(ω,q) reduces to gτs(ω,q) and in a similar way, Hτs reduces
to hτs.Heregτs and hτs read as below:
gτs(ω,q)=−λτ s
4ωln
ω−(−λτ s)2+4t2
0q2
ω+(−λτ s)2+4t2
0q2
,
hτs(ω,q)=−λτ s
2ω
−λτ s
(−λτ s)2+4t2
0q2
+1
41+−λτ s
ω2
×ln
ω−(−λτ s)2+4t2
0q2
ω+(−λτ s)2+4t2
0q2
.(20)
Using Eqs. (8) and (20), the conductivity simplifies when β=
0 and the results are
σRe,τ s
xy (ω)=τe2
h[gτs(ω,qF)−gτs(ω,qc)],
σIm,τ s
xy (ω)=τπe2
4h
−λτ s
ω[(2εF−λτ s −ω)
−(ω→−ω)](ω−(−λτ s)).(21)
The longitudinal conductivity for the case of β=0isgiven
by the following relations for the electron doped case:
σRe,τ s
xx (ω)=−πe2
8h1+−λτ s
ω2(ω−(−λτ s))
×[(2εF−λτ s −ω)−(ω→−ω)]
σIm,τ s
xx (ω)=−e2
h[hτs(ω,qF)−hτs(ω,qc)].(22)
These relations are consistent with those results reported in
Ref. [25]. Furthermore, dropping the λterm gives rise to the
optical conductivity of gapped graphene and the result is in
good agreement with the universal conductivity of graphene
[35]for=λ=α=β=0.
III. NUMERICAL RESULTS
In most numerical results, we use set0:λ=
0.08 eV, =1.9eV,t0=1.68 eV,α =m0/m+=0.43,β =
m0/m−−4m0v2/(−λ)=2.21 where m±=memh/
(mh±me) and v=t0a0/. These values have been obtained
in Ref. [21]. Moreover, for the sake of completeness,
we introduce two other sets of the parameters as
t0=1.51 eV,β =1.77, and another set t0=2.02 eV; β=0
corresponding to the same effective masses (α=0for
me=−mh=0.5m0) for electron and hole bands. These
parameters are calculated by using the procedure reported
in Ref. [21]. The later comparison helps us to perceive the
validity of the effective mass approximation for the ML-MoS2
system and for this purpose, we assume the same effective
masses for electron and hole bands to compare the spin Hall
conductivity resulted from the Dirac-like and modified Dirac
Hamiltonians. Notice that all energies are measured from the
center of the energy gap.
The real part of the optical Hall and longitudinal
conductivities for the two sets of parameters, with and without
quadratic terms, are illustrated in Figs. 1and 2where top
and bottom panels indicate electron and hole doped systems,
respectively. The effect of the mass asymmetry between
the effective masses of the electron and hole (α) bands
is neglected and it will be discussed later. It is clear that
the quadratic term βcauses a reduction of the intensity
of the optical Hall conductivity with no changing of the
position of peaks for both electron and hole doped cases. The
position of peaks in the real part of Hall conductivity is given
by ω=√(−λτ s)2+4t2
0qFs2for the β=0 case and
ωm(qFs)n(ω)−1−2(
τs +βqFs2)2+qFs 2=0 and
ω−2(
τs +βqFs2)2+qFs 2=0 for each spin
component with corresponding Fermi wave vector qFs
and for the case that β= 0. Surprisingly, the last two
equations for the latter case simultaneously fulfilled the
115413-5
HABIB ROSTAMI AND REZA ASGARI PHYSICAL REVIEW B 89, 115413 (2014)
0 1 2 3 4
¯hω(eV)
−0.05
0.00
0.05
0.10
0.15
0.20
−σxy(e2/¯h)
(a)
s=1,β =0
s=1,β =1.77
s=−1,β =0
s=−1,β =1.77
0 1 2 3 4
¯hω
(
eV
)
−0.05
0.00
0.05
0.10
0.15
0.20
−σxy(e2/¯h)
(b)
s=1,β =0
s=1,β =1.77
s=−1,β =0
s=−1,β =1.77
FIG. 1. (Color online) Real part of the Hall conductivity (in units
of e2/)for(a)electronwithεF=1 eV and (b) hole with εF=
−1eV+λdoped cases as a function of photon energy (in units
of eV) around the Kpoint. Electron and hole masses are set to
be 0.5m0and for two sets of parameters, β=0,t0=2.02 eV and
β=1.77,t0=1.51.
equation m(qFs)=n(ω) in frequency. In the energy range
shown in the figures, the numerical value of the peak position
for both cases are approximately equal and it indicates that
the position of peaks and steplike shape do not change due
to the βterm in a certain Fermi energy. It should be noticed
that the intensity of the real part of σxx decreases with the
quadratic term. Consequently, it indicates that the effective
mass approximation of the Hamiltonian for the ML-MoS2is
not completely valid because two sets of parameters with the
same effective masses are showing distinct results.
A. Mass asymmetry between electron and hole
In this subsection, we consider the mass asymmetry
between electron and hole bands and then the conductivity
of the ML-MoS2is calculated for the Hamiltonian given in
Eq. (3). The results are illustrated in Figs. 3and 4around the
Kpoint. Due to the mass asymmetry, a small splitting between
0 1 2 3 4
¯hω(eV)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
σ
xx
(e
2
/¯h)
(a)
s=1,β =0
s=1,β =1.77
s=−1,β =0
s=−1,β =1.77
0 1 2 3 4
¯hω(eV)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
σ
xx
(e
2
/¯h)
(b)
s=1,β =0
s=1,β =1.77
s=−1,β =0
s=−1,β =1.77
FIG. 2. (Color online) Real part of the longitudinal conductivity
(in units of e2/) for (a) electron with εF=1 eV and (b) hole with
εF=−1eV+λdoped cases as a function of photon energy (in units
of eV) around the Kpoint. Electron and hole masses are set to
be 0.5m0and for two sets of parameters, β=0,t0=2.02 eV and
β=1.77,t0=1.51.
electron and hole doped cases takes place in the spin-up
component. On the other hand, there is considerable splitting
between electron and hole doped cases due to both spin-orbit
coupling and mass asymmetry for the spin-down case. We also
note a sharp onset in the imaginary part of the conductivity,
minimum energy associated with the possible interband optical
transition. Moreover, corresponding to the onset in σIm
xy (σRe
xx )
where there is a peak in its real (imaginary) part at the same
energy as they are related by the Kramers-Kroning relations.
The position of peaks or steplike configuration of the
dynamical conductivity, can be controlled by the doping rate.
Figure 5shows the difference between the position of those
peaks, δω =ω↑−ω↓, around the Kpoint for electron and
hole doped cases corresponding to the real part of the Hall
conductivity for each spin component. As it is clearly shown
in this figure, δω increases linearly from a negative value to a
positive one up to a saturation value (2λ) for the hole doped
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INTRINSIC OPTICAL CONDUCTIVITY OF MODIFIED . . . PHYSICAL REVIEW B 89, 115413 (2014)
1.8 1.9 2.0 2.1 2.2
¯hω(eV)
0.00
0.05
0.10
0.15
0.20
0.25
−σxy(e2/¯h)
(a)
s= +1, electron doped
s=−1, electron doped
s= +1, hole doped
s=−1, hole doped
1.8 2.0 2.2 2.4 2.6 2.8 3.0
¯hω(eV)
0.00
0.02
0.04
0.06
0.08
0.10
−σxy(e2/¯h)
(b)
s= +1, electron doped
s=−1, electron doped
s= +1, hole doped
s=−1, hole doped
FIG. 3. (Color online) (a) Real and (b) imaginary parts of the
optical Hall conductivity (in units of e2/) as a function of photon
energy (in units of eV) around the Kpoint. Red (blue) color stands
for the electron (hole) doped case with εF=1eV(εF=−1eV+λ)
and solid (dashed) line indicates the spin-up (-down).
case. The linear part of the result originates from the spin
splitting in the valence band and the fact that there are two
Fermi wave vectors in which one component spin has zero
Fermi wave vector and does not change by increasing the
doping rate. Finally, by increasing the Fermi energy, two Fermi
wave vectors contribute to the calculations and the position of
both peaks move in the same way and lead to a saturation value
for δω.
B. Circular dichroism and optical transmittance
One of the main optical properties of the monolayer tran-
sition metal dichalcogenide system is the circular dichroism
when it is exposed by a circularly polarized light in which left-
or right-handed light can be absorbed only by the Kor K
valley and it makes the material promising for the valleytronic
field. This effect originates from the broken inversion sym-
metry and it can be understood by calculating the interband
optical selection rule P±=m0ψc|vx±ivy|ψvfor incident
1.8 2.0 2.2 2.4 2.6 2.8 3.0
¯hω(eV)
0.00
0.02
0.04
0.06
0.08
0.10
σxx(e2/¯h)
(a)
s= +1, electron doped
s=−1, electron doped
s= +1, hole doped
s=−1, hole doped
1.8 1.9 2.0 2.1 2.2
¯hω(eV)
0.0
0.1
0.2
0.3
0.4
0.5
σxx(e2/¯h)
(b)
s= +1, electron doped
s=−1, electron doped
s= +1, hole doped
s=−1, hole doped
FIG. 4. (Color online) (a) Real and (b) imaginary parts of the
optical longitudinal conductivity (in units of e2/) as a function
of photon energy (in units of eV) around the Kpoint. Red(blue)
color stands for the electron (hole) doped case with εF=1eV
(εF=−1eV +λ) and the solid (dashed) line indicates the spin-up
(-down).
right-(+) and left-(−)handed light. The photoluminescence
probability for the modified Dirac fermion Hamiltonian is
|P±|=m0t0a0
1±τd−2bβ q 2
d2+c2q2,(23)
where q2=q2
x+q2
y. Notice that the mass asymmetry term α
has no effect on the optical selection rule. The selection rule
can simply prove the circular dichroism in the ML-MoS2.
Another approach which helps us to understand this effect is
to calculate the optical conductivity around the Kpoint of two
kinds of light polarizations as σ±=s{σKs
xx ±σKs
xy }, which
has been calculated by using the Dirac-like model [19,25], and
now we modify that by using the modified Dirac Hamiltonian.
Figure 6shows the coupling of the light and valleys. Note
that Re[σ−] is large and comparable in size for either spin-up
or -down while Re[σ+] is small in comparison. The valley
around the Kpoint can couple only to the left-handed light
115413-7
HABIB ROSTAMI AND REZA ASGARI PHYSICAL REVIEW B 89, 115413 (2014)
0.0 0.1 0.2 0.3 0.4 0.5
|μ−μ0|(eV)
−0.2
−0.1
0.0
0.1
0.2
¯hδω(eV)
hole doped
electron doped
FIG. 5. (Color online) Difference between the position of the
peak in the real part of the Hall conductivity, δω =ω↑−ω↓,for
both spin components for the electron doped case including mass
asymmetry as a function of the chemical potential. Note that μ0,
which is the band edge in the conduction and valence bands, is 0.95 eV
and −0.87 eV for the electron and hole doped case, respectively.
and this effect is washed up by increasing the frequency
of the light and the result is in good agreement with recent
experimental measurements [12].
Furthermore, the optical transmittance is an important
physical quantity and it can be evaluated stemming from the
conductivity. The optical transmittance of a free standing thin
film exposed by a linear polarized light is given by [36]
T(ω)=1
2
2
2+Z0σ+(ω)
2
+
2
2+Z0σ−(ω)
2,(24)
1.0 1.5 2.0 2.5 3.0 3.5 4.0
¯hω
(
eV
)
0.00
0.05
0.10
0.15
0.20
0.25
Re[σ]
Re[σ−]
Re[σ+]
FIG. 6. (Color online) Real part of the optical conductivity
around Kpoint, for left-(solid) hand and right-(dashed) handed
light. It indicates the appearance of the circular dichroism effect for
the modified Dirac equation. The electron (εF=1 eV) doped case
including mass asymmetry.
1.50 1.75 2.00 2.25 2.50
¯hω(eV)
0.90
0.92
0.94
0.96
0.98
1.00
Transmittance
hole doped
electron doped
FIG. 7. (Color online) Optical transmittance in a finite frequency
for the electron (εF=1 eV) and hole (εF=−1eV+λ) doped cases
including mass asymmetry.
where Z0=376.73and σ±(ω)=σxx(ω)±iσxy are the
vacuum impedance and the optical conductivity of the thin
film, respectively. For the ML-MoS2case, the total Hall
conductivity in the presence of the time reversal symmetry is
zero and the total longitudinal conductivity is given by σxx =
2(σK↑
xx +σK↓
xx ). The optical transmittance of the multilayer
of MoS2systems has been recently measured [33] and it is
about 94.5% for each layer in the optical frequency range. The
optical transmittance of the ML-MoS2is displayed in Fig. 7for
both electron and hole doped cases using the numerical value
defined as set0. The result shows that the optical transmittance
is about 98% for the frequency range in which both spin
components are active for giving response to the incident
light. Importantly, for the electron dope case, there are two
minimums with distance about 0.16 eV/in frequency which
mostly indicates the spin-orbit splitting (2λ) in the valence
band and it is consistent with the results illustrated in Fig. 5.
The optical transmittance for the electron doped case is about
98% in all frequency ranges. Moreover, for the hole dope case
as it is shown in Fig. 5, the optical transmittance changes
by the tuning doping rate. Interestingly, at μ=−0.942 eV
the difference between the position of peaks of two spin
components, δω, is approximately zero. Consequently, the
total optical conductivity enhances in this resonating doping
rate which has significant effect on the optical transmittance of
the system where the transmittance decreases and particularly
reaches to a value less than 90% at the resonance frequency
when δω 0. Our numerical calculations show that the
hole doped ML-MoS2is darker than the electron doped one
especially close to the resonance frequency. Furthermore, this
feature provides an opportunity for measuring the spin-orbit
coupling by an optical transmittance measurement.
C. Optical response in the nontrivial phase
The modified Dirac Hamiltonian shows a nontrivial phase
when β < 0 and it has been numerically shown that in this
phase a light matter interaction enhances due to the change
115413-8
INTRINSIC OPTICAL CONDUCTIVITY OF MODIFIED . . . PHYSICAL REVIEW B 89, 115413 (2014)
TABLE I. Numerical parameter for the ultrathin film of a
topological insulator [8].
L(˚
A) (eV) t0(eV) αβ
20 0.14 −2.22 −1.05 23.67
25 0.0 −2.21 −2.37 18.41
32 −0.04 −2.20 −3.94 6.31
of the parabolic band dispersion into the shape of a Mexican
hat with two extrema [28]. To fulfill such a band dispersion, a
negative value β with a large absolute value is required and it
is accessible for an ultrathin film of the topological insulator.
The sign and the absolute values of the parameters can be
manipulated by the thickness of the thin film, while in the case
of the ML-MoS2, to the best of our knowledge, it is barely
possible to create a Mexican-hat-like dispersion relation even
for the model Hamiltonian with a nontrivial topology phase
[22,23]. In this case, we plot the optical Hall and longitudinal
conductivities of the UTF-TI in its trivial and nontrivial phases.
In the UTF-TI [37] system, in which only in-plan components
of momentum are relevant, one can find the Hamiltonian given
by Eq. (2) where the numerical value of the model parameters
depends on the thickness of the thin films [8,26]. We consider
three different thicknesses for which three sets of parameters
[8] are listed in Table I. We also neglect the value of 0which
is just a constant shift in the energy.
As can be seen from Table I, a sample with L=20 ˚
Aor
L=32 ˚
A indicates the trivial or nontrivial phases, respec-
tively. However, for a sample with L=25 ˚
A the energy gap
vanishes and thus at critical thickness, L=25 ˚
A, the trivial to
nontrivial phase transition takes place. Hereafter, we call that
a phase boundary.
Now, we calculate the real part of the Hall and longitudinal
conductivities for τ=+ and the results are illustrated in
Fig. 8. It shows that the conductivity enhances in the nontrivial
phase which is consistent with previous numerical work
[28]. More interestingly, we are now showing that the Hall
conductivity changes sign through changing the thickness and
it is very important in the circular dichroism effect. This
changing of the sign means a different helicity of the light
can be coupled to the system. It is worth mentioning that the
circular dichroism effect on the electronic system governing
modified Dirac Hamiltonian is also possible when energy
gap is zero [19,25]. The selection rule equation reads as
|P±|=m0t0a0
(1 ∓τbβq/
(bβ q )2+c2) for the case of zero
gap. This expression indicates that the circular polarization is
achievable away from the point even in the absence of the
energy gap. It might be emphasized that the peak in the optical
conductivity at zero energy gap originates from a nonzero
Fermi energy in which the low energy part of phase space
is no longer available for a photon absorbtion process [38]
based on the Pauli exclusion principle. More precisely, there
is a peak at energy point ω≈2εFin the topological insulator
case and it can be seen from Eqs. (16) and (18). Therefore, the
peak disappears at zero Fermi energy for a gapless system. In
Fig. 9, we show the optical conductivity for the two helicities of
light for τ=+. The results show that the circular polarization
0.0 0.1 0.2 0.3 0.4 0.5
¯hω(eV)
−0.05
0.00
0.05
0.10
0.15
0.20
σxy(e2/¯h)
(a)
×(−1)
L=20˚
A
L=25˚
A
L=32˚
A
0.0 0.1 0.2 0.3 0.4 0.5
¯hω
(
eV
)
0.00
0.02
0.04
0.06
0.08
0.10
σxx(e2/¯h)
(b) L=20˚
A
L=25˚
A
L=32˚
A
FIG. 8. (Color online) Real part of the Hall (a) and longitudinal
(b) conductivity for τ=1 and different values of film thickness.
It is clear that in the nontrivial phase the optical response of the
system is stronger than that of its trivial one. The Fermi energy is
εF=||/2+0.03 eV.
changes sign for the negative value of the gap and it gets more
strength in the nontrivial phase rather than the trivial phase.
IV. SUMMARY
We have analytically calculated the intrinsic conductivity of
the electronic systems which govern a modified Dirac Hamil-
tonian by using the Kubo formula. We have studied the effect
of the quadratic term in momentum β, which has been recently
predicted, and found the different optical responses. This
discrepancy originates from the different topological structures
of the systems. Our calculations show that the βterm has no
effect on the position of the peak of the optical conductivity
but it has considerable effect on its magnitude. Therefore, it
shows that the same effective mass approximation for electron
and hole bands for monolayer MoS2cannot fully describe
the optical properties. The effect of the strong spin-orbit
interaction can be traced by the difference of the energy interval
between the position of the peak in the optical conductivity for
115413-9
HABIB ROSTAMI AND REZA ASGARI PHYSICAL REVIEW B 89, 115413 (2014)
0.0 0.1 0.2 0.3 0.4
¯hω(eV)
0.00
0.05
0.10
0.15
0.20
Re[σ]
(a)
L=20˚
A, Re[σ−]
L=20˚
A, Re[σ+]
0.0 0.1 0.2 0.3 0.4
¯hω
(
eV
)
0.00
0.05
0.10
0.15
0.20
0.25
Re[σ]
(b) L=25˚
A, Re[σ−]
L=25˚
A, Re[σ+]
L=32˚
A, Re[σ−]
L=32˚
A, Re[σ+]
FIG. 9. (Color online) Circular dichroism effect for different
values of the thickness. The real part of the optical conductivity
around the Kpoint is shown for (a) L=20 ˚
Aand(b)L=25 ˚
Aand
31 ˚
A. The Fermi energy is εF=||/2+0.03 eV.
the two spin components in electron and hole doped cases.
We have shown that this interval for the electron doped case
is approximately constant while for the hole doped case, it
increases from a negative value to a positive one, and then it
increases linearly up to a saturation value. The effect of the
mass asymmetry in monolayer MoS2induces a small splitting
between the conductivity spectrum for the electron and hole
doped cases. The circular dichroism effect is investigated for
the modified Dirac Hamiltonian of the monolayer MoS2by
calculating the selection rule and the optical conductivity. We
have also obtained the optical transmittance of the monolayer
MoS2for the hole and electron doped cases and the results
show that the valence band spin splitting has considerable
effect on the intensity of the transmittance.
We have also studied the effect of the quantum phase
transition, which occurs owing to the reducing of the thickness,
on the optical conductivity of the thin film of the topological
insulator. We have shown that at the phase boundary, when
the energy gap is zero, the diagonal quadratic term plays
a significant role on the optical conductivity and selection
rule. Moreover, we have illustrated that the optical response
enhances and the optical Hall conductivity changes sign in the
nontrivial phase (QSH) and the phase boundary.
ACKNOWLEDGMENTS
R.A. would like to thank the Institute for Material Research
in Tohoku University for its hospitality during the period when
the last part of this work was carried out.
APPENDIX A
In this Appendix, the details of the calculations deriving
Eq. (8) are presented. Since ψc|ψv=0, we get
ψc|vx|ψv=cψc|σx|ψv+2bβ qxψc|σz|ψv,
(A1)
ψv|vy|ψc=cψv|σy|ψc+2bβ qyψv|σz|ψc,
owing to the fact that the mass asymmetry parameter αplays
no role in the velocity matrix elements. Using hchv=−c2q2
we have
ψc|σx|ψv= −c
DcDv
[qhv+q∗hc],
ψv|σy|ψc= ic
DcDv
[qhc−q∗hv],(A2)
ψc|σz|ψv=ψv|σz|ψc= 2c2q2
DcDv
.
In this case
ψc|vx|ψv= c2
DcDv{−[qhv+q∗hc]+4bβq xq2},
(A3)
ψv|vy|ψc= c2
DcDv{i[qhc−q∗hv]+4bβq yq2}.
Consequently, the product of the velocity matrix elements are
ψc|vx|ψvψv|vy|ψc
=c4
(DcDv)2{−i(qhv+q∗hc)(qhc−q∗hv)+(4bβ q 2)2qxqy
+4bβ q 2(−qy(qhv+q∗hc)+iqx(qhc−q∗hv))},
ψc|vx|ψvψv|vx|ψc
=c4
(DcDv)2{|qhv+q∗hc|2+(4bqxβq2)2
−4bqxβq2(qhv+q∗hc+q∗hv+qhc)}.(A4)
Using tan φ=qy/qx, one can find
(qhv+q∗hc)(qhc−q∗hv)
=−2ic2q4sin 2φ−4q2dd2+c2q2
−qy(qhv+q∗hc)+iqx(qhc−q∗hv)
=2q2[−id2+c2q2+dsin 2φ],
qhv+q∗hc+q∗hv+qhc=4qd cos φ,
|qhv+q∗hc|2=4q2(d2+c2q2sin φ2),
(DcDv)2=4c2q2[d2+c2q2].(A5)
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INTRINSIC OPTICAL CONDUCTIVITY OF MODIFIED . . . PHYSICAL REVIEW B 89, 115413 (2014)
After substituting Eq. (A5) into Eq. (A4), we get
ψc|vx|ψvψv|vy|ψc=c2q2sin 2φ
d2+c2q2−c2
2+2bβ (bβ q 2+d)+ic2
d2+c2q2{d−2bβ q 2},
(A6)
ψc|vx|ψvψv|vx|ψc=c2−c2q2cos φ2
d2+c2q2{c2+2bβ (a1−a2)}.
Using dφ sin 2φ=0,dφcos φ2=π, one can find
σxy =e2
hqdq f(εc)−f(εv)
εc−εv×c2
d2+c2q2(d−2bβ q 2)1
ω+εc−εv+i0+−1
ω+εv−εc+i0+,
σxx =−ie2
hqdq f(εc)−f(εv)
εc−εv×c2−c2q2
d2+c2q2c2
2+bβ (a1−a2) 1
ω+εc−εv+i0++1
ω+εv−εc+i0+.
(A7)
Using (x+i0+)−1=Px−1−iπδ(x) where Pstands for principal value, it is easy to show that the real and imaginary parts of
diagonal and off-diagonal components of the conductivity tensor read as below:
σRe
xy =2e2
hqdq(f(εc)−f(εv)) ×c2
d2+c2q2(d−2bβ q 2)P−1
(ω)2−(εc−εv)2,
σIm
xy =πe2
hqdq f(εc)−f(εv)
εc−εv×c2
d2+c2q2(d−2bβ q 2){δ(ω+εv−εc)−δ(ω+εc−εv)},
(A8)
σIm
xx =−2e2
h
ωqdq f(εc)−f(εv)
εc−εv×c2−c2q2
d2+c2q2c2
2+bβ (a1−a2)P−1
(ω)2−(εc−εv)2,
σRe
xx =−πe2
hqdq f(εc)−f(εv)
εc−εv×c2−c2q2
d2+c2q2c2
2+bβ (a1−a2){δ(ω+εv−εc)+δ(ω+εc−εv)}.
To find the conductivity around the Kpoint we must
implement the following changes: px→−pxand λ→−λ.
Using these transformations, the velocity matrix elements
around the Kpoint can be calculated by taking complex
conjugation of the corresponding results around the Kpoint.
Moreover, according to the following dimensionless parame-
ters, εc−εv=2d2+c2q2, and thus δ(ω+εc−εv)→0
for positive frequency in the absorbtion process. Thus Eq. (8)
for the dynamical transverse and longitudinal conductivity is
obtained.
APPENDIX B
In this Appendix, the details of calculations for some
integrals which appear in our model are presented. Using
new variables y=βq2+
τs +(2β)−1and a2=
τs/β+
(4β2)−1, it is easy to show that (
τs +βq2)2+q2=y2−a2
and we have
Gτs(ω,q)=1
β2
τs +1
2βI1−I2,
Hτs(ω,q)=
ω
βI1−2
τs +1
2βI3
+2
τs +1
2β
τs +1
2βI4,(B1)
where I1,I2,I3, and I4are given by
I1=Pdy
y2−a2[4(y2−a2)−(ω)2],
I2=Pydy
y2−a2[4(y2−a2)−(ω)2],
(B2)
I3=Pydy
(y2−a2)3
2[4(y2−a2)−(ω)2],
I4=Pdy
(y2−a2)3
2[4(y2−a2)−(ω)2].
I1and I4can be calculated by defining uas a new variable
where y=a
√1−u2and it leads to
I1=1
2ω4a2+(ω)2ln
u−ω
√4a2+(ω)2
u+ω
√4a2+(ω)2
,
I4=1
a2−I1+1
(ω)2u+4a2+(ω)2
(ω)3
×ln
u−ω
√4a2+(ω)2
u+ω
√4a2+(ω)2
⎫
⎬
⎭
.(B3)
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HABIB ROSTAMI AND REZA ASGARI PHYSICAL REVIEW B 89, 115413 (2014)
By defining y2=u2+a2,I2and I3are obtained as
I2=1
4ωln
u−ω
2
u+ω
2
,I
3=1
(ω)2u+1
(ω)3ln
u−ω
2
u+ω
2
.(B4)
Using the above expressions for I1,I2,I3, and I4,itiseasytoproveEqs.(17) and (19).
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