Table Of ContentCorrection of exciton binding energy in monolayer transition metal dichalcogenides
Zi-Wu Wang,1 Yao Xiao,1 Wei-Ping Li,2 Run-Ze Li,1 and Zhi-Qing Li1
1Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology,
Department of Physics, Tianjin University, Tianjin 300354, China∗
2Department of Basic Courses, Tianjin Sino-German University
of Applied Sciences, Tianjin, 300350 People’s Republic of China
Abstract: We theoretically investigate the corrections of exciton binding energy in monolayer
7 transitionmetaldichalcogenides(TMDs)duetotheexciton-opticalphononcouplingintheFro¨hlich
1 interaction model by using the linear operator combined Lee-Low-Pines variational method. We
0 not only consider the excitons couple with the intrinsic longitudinal optical (LO) phonon modes,
2 but also the surface optical phonon modes that induced by the polar substrates underneath the
TMDs. We find that exciton binding energies are corrected in a large scale due to these exciton-
n
optical phonon couplings. We discuss the dependences of exciton binding energy on the cut-off
a
wave vector of optical phonon modes, the polarization parameters of materials and the interlayer
J
distance between thepolar substrates and TMDs. These results provide potential explanations for
3
thedivergence of theexciton bindingenergy between experiment and theory in TMDs.
] Keywords: Transitionmetaldichalcogenides, Excitonbindingenergy,Surfaceopticalphonon.
l
l
a
h I. INTRODUCTION
-
s
e
m Monolayer transition metal dichalcogenides MX
2
(TMDs) (Transition metal M=Mo, W; Chalcogen X=S,
.
at Se) have recently attracted great attention because of
m their distinctive electronic and optical properties differ-
ingfromtheircorrespondingbulkmaterials. Theseprop-
-
d erties present a vast arena of potential applications[1, 2]
n such as optoelectronics and valleytronics, quantum in-
o
formatics and spintronics. One of the most significantly
c
properties of these monolayer materials arising from the
[
reduced dimensionality and dielectric screening is the
1 strongCoulombinteractionbetweenelectronsandholes.
FIG. 1: Schematic diagram of the monolayer TMDs/polar
v
Thisleadstotheformationoftightlyboundexcitonsand substrates system. The monolayer material lies in the x−y
9
5 large exciton binding energies, which implies a rich exci- plane and z0 is the distance between monolayer TMDs and
5 tonic physics in these monolayer materials[2, 3]. polar substrate in thevertical direction z.
0 There were a number of works have been focused
0
on the excitonic properties of these monolayer materi-
.
1 als in both experiment and theory, particularly on the ever,thepronounceddeviationfromtheusualhydrogenic
0 exciton binding energies. In theoretical aspect, first- model in the several lower excitonic levels were also ob-
7 principle calculation based on GW and Bethe-Salpeter served in experiments[17, 18]. The modified Coulomb
1
equation (GW-BSE) approach were used extensively to potential[17, 18] and effective two-dimensional dielec-
:
v investigate the excitonic states and binding energy[4–8]. tric constant[19] models based on the nonlocal nature
i The very large exciton binding energies in the range of of dielectric screening are proposed to explain these de-
X
0.5 1 eV are predicted and confirmed by many conse- viations effectively. Recently, an analytically solvable
r ∼
a quent experiments such as the photoluminescence ex- model that the interband coupling between electron and
citation spectroscopy[9], femtosecond transient absorp- hole states are considered to qualitatively explain non-
tion spectroscopy and microscopy[10], the optical re- hydrogenic exciton spectra observed in these monolayer
flectionspectra[11, 12],time-resolvedphotoluminescence TMDs[20]. In addition, several exciton complexes and
spectroscopy[13]. Within the frame ofWannier-Mottex- the corresponding binding energies are investigated by
citon, the simple hydrogenic model is also employed[14, using the fractional dimensional space approach[21], in
15], in which the usualRydberg seriesof energylevels of which the exciton binding energy depends on the modu-
the excitonic states are obtained and applied to explain lationofthedimensionalityparameterα,whichunderlies
severalexperimentalresults successfully[9,15, 16]. How- theconfinementeffectsofvariousgeometricalstructures.
Infactthattheagreementsbetweenexperimentsandthe-
ories are obtained as above mentioned works. However,
from the comparison of the obtained binding energies in
∗Electronicaddress: [email protected] experiments and theoretical models, it can be concluded
2
that the causes regarding exciton binding energies can
be variedinalargescalearestillnotclearlyandthe val- 840
ues of exciton binding energies for the same monolayer
(a)
TMDsinvariousexperimentsandtheoreticalmodelscan 820
V)
not get inconsistent. Therefore, the determining ingre- e
m 800
dients of exciton binding energies need to be explored y (
further. g
er 780
In the presentpaper, we mainly consider the influence n
e
ofexciton-opticalphononscouplingsontheexcitonbind- ng 760 MoS2
ing energies in monolayer TMDs by using the linear op- ndi MoSe2
ecroantsoidrearntdhueneixtacrityotnracnosufporlemsawtiiotnhmtheethiondtr.inWsiecnloontgointuly- n bi 580 WS2
o
dinal optical (LO) phonon modes, but also the surface cit 560 WSe2
x
optical phonon modes that induced by the polar sub- E
strates under the TMDs. We find that the exciton bind- 540
ing energy can be varied from tens of meV to several
520
hundreds of meV, depending on the cut-off wave vector
0.0 0.8 1.6 2.4 3.2 4.0
ofphononmodes,the polarizationstrengthofsubstrates
9 -1
and the distance between the monolayer materials and kC (10 m )
polar substrates. These theoretical results highlight and 840
enrich the understanding of the exciton states and exci- (b)
ton binding energy in these monolayer materials as well V) 800
e
as the layered heterostructures. m
y ( MoS2
g 760
er MoSe2
II. THEORETICAL MODEL n
We consider the structure that the monolayer TMDs nding e 720 WWSSe22
areplaced onthe polarsubstrate as schemedin Fig.1,in bi
n 560
whichthesurfaceoptical(SO)phononmodesareinduced o
bythesubstrates. TheexcitonHamiltonianofthisstruc- xcit
E 520
ture taking into account the LO and SO phonons and
the couplingsbetweenexcitonandthese opticalphonons
modes can be written as 480
0.01 0.1
H =H +H +H , (1a)
ex ph ex−ph
0
P2 p2 e2 FIG. 2: (a) Exciton binding energies of different TMDs as a
H = + , (1b)
ex 2M 2µ − 4πεrr functionofthecut-offwavevectorkc ofLOphononmodesat
η =0.05. (b) Exciton binding energies of different TMDs as
0
a function of thepolarization parameter η .
0
H = ~ω a† a , (1c)
ph ν k,ν k,ν
k,ν
X istheopticalphononenergy,ν=LO,SO ,SO standsfor
1 2
the LO mode and SO mode with two brabches, respec-
H = [M ξ (r)a exp(ik R) tively. a (a†)isannihilation(creation)aopticalphonon
ex−ph k,ν k k,ν · k k
with wave vector k. The third term H denotes the
Xk,ν e−ph
+M∗ ξ∗(r)a† exp( ik R)]. (1d) exciton-optical phonon coupling in the Fro¨hlich interac-
k,ν k k,ν − · tion including two parts: one is the exciton-LO phonon
coupling and the corresponding coupling element[22–24]
The term H denotes the exciton Hamiltonian con-
ex
sists of an electron and a hole interacting with each
other through a dynamically screened Coulomb poten- (e2η0Lm~ωLO) kσ
M = erfc( ), (2)
tial e2/(4πεrr), M =me+mh, µ=m−e1+m−h1 are the k,LO s (2Aε0k) 2
center of mass, reduced mass, respectively. m (m ) is
e h
the electron (hole) effective mass, r is the coordinate of inwhichη isthepolarizationparameteranddetermined
0
relative motion, P is the momentum of center of mass by the polarization properties of the monolayer TMDs,
and p is the momentum of relative motion. The second L is the monolayer thickness, erfc is the complemen-
m
termH describestheopticalphononHamiltonian,~ω tary error function and σ denotes the effective width of
ph ν
3
~
tthereaeclteicotnroonfiLcOBlpohcohnsotnatwesitihsbchaasergdeocnarthrieercsoinnstmraoinnoeldayiner- Rj =i(2Mλ)21(Bj −Bj†), (5b)
materials, A is the quantization area in the monolayer
plane, ε is the permittivity of vacuum, R is the center
0
of mass coordinate; Another is the exciton-SO phonon
coupling with the coupling element[25–27] (Bj′,Bj†)=δj′j, (5c)
e2η′~ω
M = SOγ exp( kz ), (3) and the well-knownLee-Low-Pines(LLP) unitary trans-
k,SO 0
s 2Aε0k − formation by[27–29]
γ=1,2
X
where the interlayer parameter η′ = (κ κ )/[(κ +
0 ∞ ∞
−
1)(κ +1)] is a combination of the known dielectric con-
0 U =exp i a† a k R , (6a)
stants of the substrates and denotes the polarization 1 − k,ν k,ν ·
k,ν
strength of polar substrate, κ (κ ) is the high (low) X
∞ 0
frequency dielectric constant. z denotes the distance
0
between the polar substrate and monolayer TMDs. The
element ξ (r) is expressed as
k
ξk(r)=exp(−iβ1k·r)−exp(−iβ2k·r), (4) U2 =exp (Fk,νa†k,ν −Fk∗,νak,ν), (6b)
k,ν
X
in which β1 = me/M and β2 = mh/M is the fraction of
electron and hole, respectively.
We introduce the creation B† and annihilation B op- where the index j refers to the x, y, λ is the variational
j j parameter and F (F∗) is the variational function. B†
erator for the center of mass momentum P and coordi- k k j
and B are boson operators satisfying the boson com-
nate R by j
mutative relation. Substituting Eq. (5a) and (5b) into
M~λ21 the first term of Eq. (1b), and then performing the LLP
P =( (B†+B ), (5a) transformation, we get
j 2 j j
Hex = U2−1U1−1HexU1U2 =He(xr)+ ~2λ (Bj†Bj +1)+ ~4λ (Bj†Bj†+BjBj)+~(2~Mλ )21 (a†k,ν +Fk∗,ν)
j j k,ν
X X X
e ~2
(ak,ν +Fk,ν) kj(Bj†+Bj)+ 2M k·k′(a†k,ν +Fk∗,ν)(a†k′,ν +Fk∗′,ν)(ak,ν +Fk,ν)(ak′,ν +Fk′,ν)
j k,k′,ν
X X
~2 ~2
− 2µ (∇rFk,ν ·∇rFk′,νa†k,νa†k′,ν +∇rFk∗,ν ·∇rFk∗′,νak,νak′,ν)+ µ ∇rFk,ν ·∇rFk∗′,νa†k,νak′,ν
k,k′,ν k,k′,ν
X X
~2 ~2 ~2
+ F 2 (a† F a F∗ ) (a† 2F a 2F∗ ), (7)
2µ |∇r k,ν| − 2µ k,ν∇r k,ν − k,ν∇r k,ν ·∇r − 2µ k,ν∇r k,ν − k,ν∇r k,ν
k,ν k,ν k,ν
X X X
with the Hamiltonian H(r) = ~2 2/2µ e2/4πε r de- where ζ = 2 , and a = 4πεr~2 is the Bohr
scribes the relative motieoxn of−the∇elerctron−-hole pairr and n a0(n−21) 0 µe2
radius, F(a,b,z) is the confluent hypergeometric func-
be solved exactly in the hydrogenic model[14, 18, 19].
tion, and the quantum numbers l takes the values
Thecorrespondingeigenfunctioninpolercoordinate(r,θ)
0, 1, 2e,..., (n 1). The obtained binding energy
can be written as ± ± ± −
of the nth excitonic state in this hydrogenic model
is[14, 18, 19]
1
ζ (n+ l 1)! 2 1 1 e2
φ = n | |− exp(ilθ) E(n) = . (9)
n,l (2l )! (2n 1)(n l 1)! √2π b 2(n 1/2)24πεra0
| | (cid:20) − −| |− (cid:21) −
(ζ r)|l|exp( ζnr)F( n+ l +1,2l 1,ζ r), This equation is extensively used to describe the exciton
n n
− 2 − | | | |− bindingenergyofnthexcitonicstate,inwhichthedielec-
(8) tricconstantε playsakeyroletodeterminethebinding
e r
4
energy and related to many elements, e.g. the extension values of ε in Ref.30.
r
ofexcitonicradius,thereducedmassofexciton,the sub-
strate environment. The nonhydrogenic Rydberg Series Substituting Eq. (5a) and (5b) into the term H and
ph
ofexcitonicstatescanalsobe wellfittedviathe modula- H , and then performing the LLP unitary transfor-
e−ph
tion of parameter ε [19]. In this paper, we adopted the mation, we have
r
~2k2
H = U−1U−1H U U = (~ω + )(a† a +a† F +F∗ a +F∗ F ), (10)
ph 2 1 ph 1 2 ν 2M k,ν k,ν k,ν k,ν k,ν k,ν k,ν k,ν
k,ν
X
e
~k2
H = U−1U−1H U U = M∗ ξ∗(r)(a† +F∗ )+M ξ (r)(a +F ) exp( )
ex−ph 2 1 e−ph 1 2 k,ν k k,ν k,ν k,ν k k,ν k,ν −4Mλ
Xk,ν h i
e ~ ~
exp[−(2Mλ)12 kjBj†]exp[−(2Mλ)12 kjBj] (11)
j j
X X
Inordertoobtainthebindingenergyofexcitonicground vector of exciton describing the center of mass motion,
state, we choose Φ = φ 0 0 as the ground 0 is the zero phonon state. The exciton binding en-
1,0 ex ph ph
| i | i| i | i | i
state wave-function of system. φ is the ground state ergytakingintoaccountthe exciton-opticalphononcou-
1,0
| i
of exciton as given in Eq. (8), 0 is the vacuum state plings can be obtained via
ex
| i
E(λ,F ,F∗ ) = ΦH +H +H Φ
k,ν k,ν h | ex ph ex−ph| i
~λ ~2k2 ~k2
= Eb(0)e+ 2 e+ (~eων + 2M )Fk∗,νFk,ν + Mk∗,νFk∗,νexp(−4Mλ)hφ1,0|ξk∗(r)|φ1,0i
k,ν k,ν
X X
~k2 ~2
+ M F exp( ) φ ξ (r)φ + F 2, (12)
k,ν k,ν 1,0 k 1,0 r k,ν
−4Mλ h | | i 2µ |∇ |
k,ν k,ν
X X
in which the relations B† 0 = 1 , B 0 = 0 and phonons coupling on the exciton binding energy, we set
j| iex | iex j| iex
a 0 =0 are applied. thecaseλ=0. Convertingthesummationofwavevector
k ph
M| iinimizing Eq. (12)withrespecttof∗ andf , weget into the integral in Eq. (14), the exciton binding energy
k k
can be expressed as
M φ ξ∗(r)φ
f = k,νh 1,0| k | 1,0i, (13a)
k − ~2k2/2M +~ων E = E(0) A kc[ |Mk,ν|2ℜ21
b b − (2π)2 ~2k2/2M +~ω
fk∗ =−Mk~,2νkhφ2/1,20M|ξk+(r)~|ωφν1,0i (13b) f ~2k2 Xν|Mk,ν|2ℜZ022 ]kdk 2πdθ,ν (15)
− 2µ (~2k2/2M +~ω )2
Substituting Eq. (13a) and (13b) into Eq. (12), we get ν Z0
with
~λ M 2 φ ξ (r)φ 2
Eb = Eb(0)+ 2 −Xk,ν | k,ν~|2k|h2/12,M0| k+~ω|ν1,0i| ℜ1 = (ζ2+βζ032k2)3/2 − (ζ2+βζ032k2)3/2,
~2 M 2 φ ξ (r)φ 2 0 1 0 2
k,ν 1,0 r k 1,0
| | |h |∇ | i| , (14)
− 2µXk,ν (~2k2/2M +~ων)2 ℜ2 = (ζ2+βζ032k2)3/2 + (ζ2+βζ032k2)3/2,
inwhicheachtermisindependentofλexceptthesecond 0 1 0 2
term describing the center of mass movement. In or- where k is the cut-off wave vector of optical phonon
c
der to clearly distinguish the influence of exciton-optical mode. The first term in integral denotes the motion of
5
TABLEI:ParametersusedtoobtainresultsforFigs.2,3and
840
4. The effective masses of electron me and hole mh (in units
(a) of the free electron mass m0) are obtained from the Ref.31.
V) The values of dielectric constant εr and LO phonon energy
me are taken from Ref.30 and 32, respectively. Estimations for
770
y ( the exciton binding energies Eb(0) are obtained by using Eq.
rg (12) in these monolayer TMDs.
e
en MoS2 MoSe2 WS2 WSe2
ding 700 me(m0) 0.51 0.64 0.31 0.39
n mh(m0) 0.58 0.71 0.42 0.51
citon bi 630 MMooSS22//Sh-iOBN2 ~Eω(Lε0O)r(((mεm0e)eVV)) 484.12766 483.37414 454.81303 453.66033
x b
E MoS2/HfO2
MoS2/ZrO2
560 TABLE II: Surface-optical phonon modes and the dielectric
0.0 0.8 1.6 2.4 3.2 4.0 constants of polar substrates used in this paper. Parameters
9 -1 havebeen taken from Refs.25 and 26.
kC (10 m ) h-BN SiO HfO ZrO
2 2 2
800 κ (ε ) 5.1 3.9 22.0 24.0
0 0
eV) (b) ~ωSκO∞,1((εm0)eV) 140.11 26.05 51.90 42.50
y (m 640 ~ωSO,2(meV) 195 146 53 71
g
r
e
n 480
e III. DISCUSSION AND RESULTS
ng MoS2/h-BN
ndi 320 MoS2/SiO2 The influence of the exciton-LO phonon coupling on
bi
n MoS2/HfO2 the exciton binding energies are presented in Fig. 1 for
o different TMDs materials. We set the monolayer thick-
cit 160 MoS2/ZrO2 ness of TMDs L =0.4 nm andthe effective width of the
x m
E electronic Block state σ=0.5 nm throughout the numer-
ical calculation[21, 23]. Fig. 1(a) presents the depen-
0
dence of exciton binding energies on the cut-off wave
0.1 1
vector k at η = 0.05. As can be seen that exci-
c 0
z0 (nm) ton binding energies decrease with increasing the cut-
off wave vector. When the k reaches above the value
c
FIG. 3: In the MoS2/different polar substrates systems, (a) of 4.0 109m−1, the effect of kc on exciton binding
×
excitonbindingenergiesasafunctionofthecut-offwavevec- energies becomes insignificant, which means that only
torkc ofSOphononmodesatz0=0.5nm;(b)excitonbinding the phonons with small wave-vector, corresponding to
energies as a function of the distance z0. the long-wavelengthphonons, arestronglycoupled with
the charge carriers, which is consistent with the results
that small-momentum optical phonons give the mainly
contributions to the Fro¨hlich interaction based on first-
principle calculation[33]. Indeed, the influence of cut-off
wave vector (k ) of LO and SO phonon mode on the ex-
c
citon binding energies are similar in following section, so
center of mass couples to the optical phonons. The cor- we choose k =4.0 109m−1 as the limitation value for
c
×
rection of which to the exciton binding energy is very all numerical calculations presented in this paper. On
small, since the effective masses of electron and hole are the other hand, the strength of Fro¨hlich interaction de-
very closer in these monolayer TMDs as listed in table pends on the polarization properties of material, which
I, leading to the polarization clouds of them cancel each is described by the polarization parameter η in Eq. (2)
0
other. The second term arises from the relative motion and related to several ingredients, such as the dielectric
of exciton couples to the optical phonons, which gives tensor[33], screening of substrate material[7], thus lead-
the dominate contribution to the correction of the ex- ing to its precise estimation is a complex problem. We
citon binding energy and was assigned to the modifi- only qualitative analysis the influence of η on exciton
0
cation to the Coulomb potential in traditional exciton binding energies in Fig. 2(b) for different monolayer
problems[29]. TMDs. One can see that the exciton binding energies
6
can change a hundred of meV at η = 0.1. Hence, the
0
exciton binding energies are corrected remarkably if the
larger values of η obtained.
0 810
Inferred from Eqs. (3) and (15), it can be known that
the corrections of exciton-SO phonon coupling to exci- MoS /SiO
) 2 2
V
ton binding energies depend on the cut-off wave vector e
k , polarization parameter of substrate η′ and distance m 780
c (
z . These dependences are presented in Fig. 3. Surface y
0 g
optical phonon modes and dielectric constants of polar er
n
substrates are given in Table II, which have been taken e 750
from Refs. 25 and 26. In Fig. 3(a), exciton binding ng
energies in monolayer MoS for different polar substrate di
2 n
are presented at distance z0=0.5 nm. Comparing with bi 720
Fig. 2(a), it can be seen that the correction of exciton n
o
binding energies are enhanced more obviously with in- cit
creasing kc due to two branches of SO phonon modes Ex 690 LO phonon
included. Moreover, the corrections increase from h-BN SO phonon
to ZrO substrates, which because of the polarization LO+SO phonon
2
parameter η′ that denotes the polarization strength of
660
substratesincreasesfromh-BNtoZrO substratesasta-
2 0.0 0.8 1.6 2.4 3.2 4.0
ble II listed. The increasing of this parameter denotes
9 -1
the enhanced strength of exciton-SO phonons coupling, kC (10 m )
andthus resulting inthe correctioneffect morestronger.
Therefore, the polarization of substrate plays a signifi- FIG. 4: Exciton binding energy as functions of the cut-off
cant role on the exciton binding energies in these mono- wave number of optical phonon including LO and SO modes
layer materials. However,the coupling strength between in MoS2/SiO2 system at thedistance z0=0.5 nm andthepa-
rameter η =0.05.
exciton and SO phonons can be modulated by the dis- 0
tance z as plotted in Fig. 3(b). From it, we can see
0
that the corrections of exciton binding energies decrease
sharplyandcanbevariedinrangeofseveralhundredsof been assigned to result in these discrepancies in several
meVfordifferentpolarsubstratewithincreasingthedis- works[6, 7, 14]. The present model proved that exciton-
tance z . Moreover, the exciton binding energies change optical phonons coupling is an effective many-body in-
0
hardly when this distance exceeding z =3.0 nm, which teractionaffectingexcitonbindingenergiesremarkablyas
0
means that the contribution of SO phonons to correc- plottedinFig. 4. Especially,theSOphononsinducedby
tion are very minor or negligible above this distance and thepolarsubstrateshavepotentialinfluenceontheprop-
LO phonons dominate the correction of exciton binding ertiesofexcitonicstatesatcertaindistancez0. TheseSO
energy. This also provides the highlight for how to eval- phonons modes generate a series of novel quantum phe-
uateandcontrolthe influence ofSOphononsinducedby nomena in the van der Waals heterostructures based on
the substrates on the exciton states in experiments. In these monolayer TMDs materials and arouse more and
addition, the screening effect of substrate also leads to more interesting in recent years[34].
the large reduction in exciton binding energy has been In summary, we theoretically investigate the correc-
calculated based on the first-principle[7]. Although both tions of exciton binding energies in monolayer TMDs by
this screening effect and SO phonons are related to the considering the exciton couples with LO phonons and
polar substrates and give potential correction to exciton SO phonons that induced by polar substrates. We find
binding energy, the correlations between them can not that (1) exciton binding energies are correctedin a large
discuss quantitative in the present study. scale due to exciton-optical phonon coupling and those
The total corrections of exciton-LO and SO phonons long-wave length phonons give the dominate contribu-
coupling on exciton binding energies for the monolayer tions to the correction;(2)the polarizationparameterof
MoS /SiO system are presented in Fig. 4. It can be LOphononandthe polarizationofsubstratesplaya sig-
2 2
seen that the magnitude of exciton binding energy can nificant role in determining the magnitude of correction;
be tuned in a very large scale due to this two kinds of (3) the influence ofSO phononsbecome veryweak when
exciton-optical phonons coupling, which not only pro- the distance between monolayer TMDs and substrates
vides an potential explanation for the divergence of the exceedsthe rangeof 3 nm. These resultshighlightthe
∼
exciton binding energy between experiment and theory, discrepancies between theory and experiment regarding
but also proposes a way to identify the variation of ex- exciton binding energies in these monolayer TMDs.
citon binding energy by controlling the polarization of This work was supported by National Natural Science
substrate, the distance z and dielectric environment Foundation of China (NO. 11304355and NO. 11674241)
0
in experiment. In fact, many-body mechanisms have and Young Core Instructor of Peiyang (NO.11303267).
7
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