Experimental determination of the massive Dirac fermion model parameters for MoS$_2$, MoSe$_2$, WS$_2$, and WSe$_2$

0.75 MB·

by  Beom Seo Kim

Experimental determination of the massive Dirac fermion model parameters for
MoS , MoSe , WS , and WSe
2 2 2 2
Beom Seo Kim,1,2,3 Jun-Won Rhim,4,∗ Beomyoung Kim,5,6 Changyoung Kim,1,2 and Seung Ryong Park3,†
1Center for Correlated Electron Systems, Institute for Basic Science, Seoul 151-747, Korea
2Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
3Department of Physics, Incheon National University, Incheon 406-772, Korea
4Max-Planck-Institut fu¨r Physik komplexer Systeme, 01187 Dresden, Germany
5Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea
6 6Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
1
0 Monolayer MX2 (M = Mo, W; X = S, Se) has drawn much attention recently for its possible
application possibilities for optoelectronics, spintronics, and valleytronics. Its exotic optical and
2
electronic properties include a direct band gap, circular polarization dependent optical transitions,

n and valence band (VB) spin band splitting at the K and −K points. These properties can be
a describedwithinaminimalmodel,calledthemassiveDiracfermionmodelforwhichtheparameters
J
needtobeexperimentallydetermined. Weproposethattheparameterscanbeobtainedfromangle

7 resolved photoemission (ARPES) data from bulk 2H-MX2, instead of monolayer MX2. Through
tightbindingcalculations, weshowhowtheelectronicstructureathighsymmetrypointsevolvesas

i] thesystemchangesfrom themonolayertothethreedimensional bulk2H-MX2 . Wefindvanishing
c kz dispersion and almost no change in the direct band gap at the K and −K points, in sharp
s contrasttothestrongkz dispersionattheΓpoint. Thesefactsallowustoextractthegapandspin
l- bandsplittingattheK pointaswellasthehoppingenergyfrom bulkARPESdata. Weperformed
tr ARPESexperimentsonsinglecrystalsofMoS2,MoSe2,WS2,andWSe2 atvariousphotonenergies
m andalsowithpotassiumevaporation. Fromthedata,wedeterminedtheparametersforthemassive
. Dirac fermion model for monolayer MoS2, MoSe2, WS2, and WSe2.
t
a
m PACSnumbers: 71.20.Nr,75.70.Tj,71.15.-m
-
d
I. INTRODUCTION describedinIII-A.TheelectronicstructureofTMDscan
n
be directly measured by ARPES, which has confirmed
o
c The successful exfoliation of graphene1–3 is impor- the direct band gap and the spin band splitting at the
[ K and K points5,24–31. Moreimportantly,theparame-
tant on its own right but also has triggered the in- −
ters for the massive Dirac fermion model can be directly
1 tensive/extensive research on similar two-dimensional
measured by using ARPES. The measured values of ∆
v layered materials. Transition metal dichalcogenides
8 and2λare1.465and0.15eVfortheepitaxialmonolayer
(TMDs) such as NbSe and MoS have strong in-plane
1 2 2 MoS on Au(111), and 1.67 and 0.18 eV for monolayer
covalent and weak out-of-plane van der Waals bonds, 2
4 MoSe grown on graphene5,30.
which reduce the dimensionality from three to two and 2
1
0 allow us to obtain monolayer systems by the exfoliation Thereareacoupleofobstaclesinexperimentallymea-
. method. MonolayerTMDsoftenexhibitqualitativelydif- suring the massive Dirac fermion model parameters by
1
ferent electronic properties compared to the bulk4–6. ARPES. The experiments have been mostly performed
0
6 AmongtheTMDs,the group6TMDs MX2 (M=Mo, on epitaxially grown MX2 monolayer systems due to
1 W; X = S, Se) exhibit interesting electronic properties the difficulty in ARPES experiments on exfoliated MX2
: such as indirect to direct band gap transition from bulk monolayer. Epitaxialstrainandformationofsuperstruc-
v
i tomonolayer4,5,valleydegeneracy7,andspin-orbitinter- ture due to the interaction with the substrate may af-
X action (SOI) induced spin band splitting at the K and fect the parameters whereas other experiments such as
r K points of the hexagonalBrillouinzone8. Fromthese transport measurements have been mostly done on ex-
a f−undamental electronic properties, the valley degeneracy foliated MX2 monolayer systems. It is now possible to
can be lifted by using circularly polarized light9–13 and do ARPES on exfoliated MX2 monolayer with the size
valley Hall effect was observed14–16. These raised the of tens of micrometers with the development of so-called
notion of the valleytronics17–23. micro-ARPES for which the incident light is focused to
sub micrometer size. Unfortunately, however, the qual-
It would be desired to have a simple model that cov-
ityofthedatafromexfoliatedMX monolayerbymicro-
ers these exotic properties of group 6 TMDs for practi- 2
ARPESisstillnotgoodenoughtoextracttheparameters
cal purpose. A minimal model, massive Dirac fermion
quantitatively25,28,29.
model, is simple but can cover all the interesting low
energyelectronicstructurepropertiesmentionedabove7. Our idea is to extract the parameters from ARPES
The model has only three independent parameters: the datafrombulksystemsinsteadofmonolayerMX . Even
2
effective hopping (t), band gap without SOI (∆), and though the massive Dirac fermion model is applicable
spin band splitting (2λ). The details of the model are only for monolayer MX , we show that we can extract
2
2
III. RESULTS AND DISCUSSION
A. Tight binding calculations for
electronic-structure evolution from monolayer to
bulk MX2
Figure 1 is a schematic sketch of the massive Dirac-
fermion model. Two cases are illustrated in the figure,
one without SOI and the other with SOI. The Hamilto-
nian of the massive Dirac Fermion model including SOI
reads
∆ σˆ 1
Hˆ =at(τk σˆ +k σˆ )+ σˆ λτ z − sˆ (3.1)
x x y y z z
2 − 2
FIG.1: SchematicsketchofthemassiveDiracfermionmodel. where a is the lattice constant, t the effective hopping
Gray VB at the front-left K point is for the case without
parameter, τ the valley index, σˆ the Pauli matrices for
SOIwhilered/blueVBedgescorrespondtothespinup/down
the basis functions, ∆ the direct band gap size without
states for thecase with SOI.
SOI, 2λ the SOI induced spin band splitting size, and
sˆ the Pauli matrix for spin (see Ref.[7] for more de-
z
tails). Note that there are only three free-parameters in
this model, ∆, 2λ, and t. As stated earlier, the goal of
themassiveDiracfermionparametersfromtheelectronic ourresearchistodeterminetheseparametersexperimen-
structure of bulk materials. Tight binding calculation tally for MoS , MoSe , WS , and WSe monolayer. On
2 2 2 2
result shows how the electronic structure at high sym- the other hand, ARPES experiments were performed on
metrypoints evolvesasthe systemchangesfromthe two bulk MX for which the low-energyelectronic properties
2
dimensionalmonolayertothethreedimensionalbulk2H- are not governed by the massive Dirac Fermion model.
MX2. It also reveals that the direct band gap at K and For example, the VB maximum is not located at the K
K points for the monolayer is identical to that for the point but at the Γ point in MX . We performed tight
− 2
bulkduetolackofthe kz dispersion. Ourphotonenergy binding calculations with a focus on how the electronic
dependent ARPES data indeed confirms the vanishing structure at the K and Γ points evolves from monolayer
kz dispersion near the K and −K points. Accordingly, MX2 tobulkMX2. Ourcalculationsshowthatelectronic
all the appropriate parameters (t, ∆ and 2λ) for MoS2, structure evolution at K point is small enough that we
MoSe2, WS2, and WSe2 systems were successfully ex- can extract the massive Dirac fermion parameters from
tracted from the experimental ARPES data. From now the electronic structure of MX .
2
on, we omit ”bulk 2H-” for the bulk sample. First, we consider the k dependent VB dispersion
z
at the in-plane Γ point. The conduction band (CB) is
not treated here since it is not easy to find an effective
modelforthisbandduetothemultiplemixingwithother
bands. Meanwhile, the VB is well separated from other
bands and the mixing could be negligible. In this paper,
weneglectthespindegreeofthefreedomwhichdoesnot
II. METHODS affect the band broadening. At Γ, the orbital composi-
tion of the VB is known to be
ARPES measurements were done at the beam line |ψΓVBi = c˜1|d(0e)i−c1|p(0e)i (3.2)
4.0.3.2 (MERLIN) of the Advanced Light Source
equipped with a VG-SCIENTA R8000 analyzer. The where|d(0e)i=|dz2iand|p(0e)i=(|pAzi−|pBz i)/√2. Here,
total energy resolution was better than 20 meV. Four AandB representthechalcogenatomsattheupperand
high quality single-crystal samples were purchased from lower side of the MX slab. From now on, we omit the
2
2D Semiconductors and HQGraphene. All the data superscript (e) of the p orbital. c˜ = 1 c2 and its
were taken under 40 K in a base pressure better than value for various TMDszhas been o1btained−by1Fang et.
4.5 10−11 Torr. For the photon energy dependence, al..32 p
×
we used the photon energy between 50 and 100 eV. Al- One can construct a Bloch wave function with the
kali Metal Dispensers from SAES Getters were used for translational symmetry along z-axis as
potassiumevaporationexperimentsandevaporationwas
conducted in situ with the samples at the measurement Ψl(u) = 1 ψn,l(u) einkzc (3.3)
position. | ΓVB,kzi √N | ΓVB i
n
X
3
tion.
t(LL) = p′(r )H′ p (r ) (3.7)
p′i,pj h i i | | j j i
r r
i j
= (V V ) +V δ (3.8)
ppσ − ppπ r2 ppπ ij
where V is an exponentially decaying function of
ppσ(π)
the distance between p orbitals.32 For the case of ∆ ,
Γ,kz
only the p orbital is involved, so that r r /r2 =(δ /δ)2
z i j z
(see Fig. 2). As a result, we have
c2
∆ = 1 pB,n,u H′ pA,n,l(δ ) 1+eikzc
Γ,kz − 2 h z | | z i i
i
FIG. 2: Crystal structure of MX2. Inversion symmetry is D X (cid:0) (cid:1)
broken within an MX2 monolayer. In MX2, the layers are = Γ 1+eikzc (3.9)
− 2
stacked in a way that inversion symmetry is restored. The
figure on the right is the zoom-in of the part marked by the where (cid:0) (cid:1)
circle in the crystal structure. It shows the definitions of the 2
δ
parameters used in thecalculation. D =3c2 (V V ) z +V . (3.10)
Γ 1( ppσ − ppπ (cid:18) δ (cid:19) ppπ)
Then, the energy spectrum at the Γ point becomes
where n is the layer index, and l and u represent lower
and upper MX slab in the unit cell. c is the lattice E± =ǫ D (1+cosk c) (3.11)
2 ΓVB ΓVB ± Γ z
constant along z direction and we set the gauge so that
forwhichtheenergydifferenceisthemaximumatk =0
there is no k dependence in the same unit cell. Here, z
z
and vanishes at the zone boundary.
ψn,l(u) is a function of k and k and is constructed to
| ΓVB i x y If one experimentally measures the bandwidth at Γ
satisfy the Bloch condition in the xy-plane.
along the k direction, one can extract the relation be-
z
Ifweassumethatǫ istheVBenergyofMX mono-
ΓVB 2 tween two fundamental interlayer hopping parameters
layer, the effective Hamiltonian for the 3D bulk system
V and V from the Eq. (3.10) and Eq. (3.11).
ppσ ppπ
at the point is given by
As an example, for MoS , we obtain V = 0.6344
2 ppσ
and V = 0.0592 in eV, assuming δ = 3.4261 (S-
ppπ
HΓ ≈(cid:18)∆ǫΓ∗ΓV,kBz ∆ǫΓΓV,kBz(cid:19) (3.4) SDΓdis=tan0.c4e2)8a4n−edVδ,zso=t2h.a9t.3t2h,3e4 bAasnadwriedstuhlt,atweΓepstoiimntaties
about 1.7 eV which is comparable to the experimental
where ∆Γ,kz = hΨuΓVB,kz|H′|ΨlΓVB,kzi. Here, H′ is the results. Theseparametersareobtainedfromasimpleex-
interlayer hopping term which will be explained below. ponential form of the overlapintegraland may be tuned
Its eigenvalues are evaluated to be for realistic systems.
Now,weconsidertheVBandCBstatesattheKpoint.
EΓ±VB = ǫΓVB ±|∆Γ,kz| (3.5) For these states, the orbital composition is completely
different from that of the states at the Γ point. At K,
The off-diagonal component ∆Γ,kz, which is dependent states have equal contribution from px and py orbitals
on kz, corresponds to the band broadening and can be while pz orbital contribution is almost absent. The or-
calculated as follows. bital composition is given by
c2 ψ =c˜ d(e) +c p(e) (3.12)
∆ 1 pB,n,u H′ pA,n,l(δ ) | KVBi 6| 2 i 6| 1 i
Γ,kz ≈ − 2 h z | | z i i
i and
X
c2 (e) (e)
− 21 hpAz,n,u|H′|pBz,n+1,l(δ˜i)ieikzc(3.6) |ψKCBi=c˜5|d0 i+c5|p−1i (3.13)
Xi where |d(0e)i = |dz2i, |d(2e)i = (|d(xe2)−y2i + i|d(xey)i)/√2
where the vector δi represents the nearest neighbor sites and p(e) = (pA + pB ) i(pA + pB ). Here,
betweenMX layers,andδ˜ = δ . Notethatthenearest | ±1i | xi | xi ± | yi | yi
neighbor vec2tors between silabs−ini the same unit cell are c˜n = 1−c2n. In this case, we consider following ef-
fective 4 4 Hamiltonian for CB and VB of the 3D bulk
in opposite direction to those in different unit cells due p×
system.
to the way of the stacking.33 There are no phase factors
that are dependent on k and k in the above since we ǫ 0 ∆ α
x y KVB KVB,kz K,kz
are considering only the Γ point. Details are presented 0 ǫ β ∆
in Appendix A. HK ≈∆∗KVB,kz βKK∗C,kBz ǫKKV,kBz KC0B,kz(3.14)
Now, we use the following Slater-Koster approxima-  α∗K,kz ∆∗KCB,kz 0 ǫKCB 
 
4
2.0 conclusion that ∆KVB,kz = 0 at the K point35. By the
same procedure,one caneasily find that ∆ is also
KCB,kz
vanishing. As a result, the matrix representation of the
effective Hamiltonian reduces to
1.0
ǫ 0 0 α
KVB K,kz
0 ǫ β 0
E-E(eV)F 0.0 BMuolnkolayer ∆ HK ≈α∗K0,kz βKK∗0C,kBz ǫKK0V,kBz ǫK0CB. (3.20)
 
Without SOI α and β in the Hamiltonian are calculated to be
K,kz K,kz
-1.0 2λ
α = Ψu H′ Ψl (3.21)
K,kz h KVB,kz| | KCB,kzi
D (3.22)
K
≈
Γ Κ
and
FIG. 3: Monolayer MX2 bands (solid black line) and pro- β = Ψu H′ Ψl (3.23)
jected bulk bands (gray region) along the Γ - K direction, K,kz h KCB,kz| | KVB,kzi
extracted from Ref. [35]. Blue dash-dot line is the band DKeikzc (3.24)
≈
structurearound theK point when there is noSOI.
where
3c c δ 2
where ∆KVB,kz and ∆KCB,kz are mixing between same DK = 5 6 ⊥ (Vppσ Vppπ). (3.25)
orbitals and α and β are between different ones. 4 δ −
K,kz K,kz (cid:18) (cid:19)
The upper (lower) 2 2 diagonal block is for the upper
× Note that α = β∗ due to the layer index. Here,
(lower) slab. K,kz 6 K,kz
As in the previous case, one can approximately eval- δ1,⊥ = (a/2,a/2√3,0), δ2,⊥ = ( a/2,a/2√3,0), and
−
uate ∆KVB,kz as follows. Neglecting terms involving d δ3,⊥ = (0,−a/√3,0) (details are given in Appendix
orbitals, we have A). For the case of MoS as an example, we have
2
D 0.0263 eV from the parameters given by δ =
K ⊥
∆KVB,kz = hΨuKVB,kz|H′|ΨlKVB,kzi (3.15) 1.824≈4˚A, δ = 3.4261˚A, Vppσ = 0.6344eV, and Vppπ =
0.0592eV.32,34
= c426 ((Vppσ −Vppπ)δδi2,2⊥ +2Vppπ)eiK·δi c−omFiensally, the effective Hamiltonian at the K point be-
i
X
+×ec4i26KX·δ˜iiei(kz(cVppσ −Vppπ)δ˜δ˜i2,2⊥ +2Vppπ)(3.16) HK ≈ǫDK00VKB DǫKKe00C−Bikc DǫKK00VeiBkc ǫDK00CKB. (3.26)
where δ2 = δ2 +δ2 and K represents the position  
i,⊥ i,x i,y Its eigenvalues are evaluated to be
of a K point in the Brillouin zone of monolayer MX .
2
Sthinecseaδmi2,e⊥,voanleuecsanforseatllδt⊥2he=nδei2a,⊥re.stTnheeingh∆boKrVhBo,kpzpicnagnabree EK± = ǫVB +ǫCB±p(ǫV2B −ǫCB)2+4DK2 (3.27)
further simplified to
which is independent of k . Since ǫ ǫ 2D ,
z VB CB K
∆ =R f(K)+f(K)∗eikzc (3.17) as an approximation, we just have| − | ≫
KVB,kz KVB
where (cid:0) (cid:1) D2
ǫ ǫ K (3.28)
KVB → KVB − ǫ ǫ
c2 δ2 CB− VB
R = 6 (V V ) ⊥ +2V (3.18)
KVB 4 ppσ − ppπ δ2 ppπ and
(cid:26) (cid:27)
D2
and ǫ ǫ + K . (3.29)
KCB → KCB ǫ ǫ
CB VB
f(k)= eik⊥·δi. (3.19) −
Thismeans thatthe CB andVB energiesatthe K point
i
X are k -independent and that they are subject to tiny en-
z
Refer to Appendix A for details. As was in the case of ergy shifts as we go from the monolayer to bulk cases.
the grapheneDirac point, f(K)=0andwe arriveatthe In obtaining the results, two factors were crucial. First,
5
FIG. 4: (a) Intensity plot of WSe2 ARPES data in energy and momentum (kz, kk) space. kz dependent ARPES is taken by
using different photon energies. kz of 9.0 and 11.0 correspond to the incident photon energies of 58 and 94 eV, respectively.
The black dashed lines indicate the expected kz dispersion of the bands with DΓ=0.3 eV [Eq. 3.11]. Three selected cuts on
theright handsidealong thebrown dashed lines areARPESintensity mapsat constant energies in themomentum space(kz,
kk). Also shown are ARPESintensity maps of (b) MoS2, (c) MoSe2, (d) WS2, (e) WSe2 at a constant bindingenergy of −1.7
eV. The dashed lines are guides to eyefor the electronic states near theK point. These lines are straight along the kz.
there are no p orbital components in both the CB and cancellationsfromthenearestneighborhoppingsarenot
z
VB states at the K point. Second, we have diminishing possible at the Γ point (k = k = 0), the resulting en-
x y
sums of phase factors due to the C3 symmetry. One can ergyspectra of the 3DMX become dispersivealong the
2
obtain the same results for K′ since the basis wave vec- k direction.
z
tors are just complex conjugates of the wave vectors at
At the K point, on the other hand, we have both the
K, namely, Eq. (3.12) and (3.13).
out-of-plane (dz2) and in-plane orbital (px and py) com-
Summarizing the theoretical part, upon stacking of ponents for M and X atoms, respectively. Among them,
MX slabs, we find that the k dispersions at two high only the p and p orbitals are responsible for the in-
2 z x y
symmetry points Γ andK arecompletely different. This terlayercoupling because the distance between M atoms
canbeinterpretedintermsoftheorbitalcompositionand in the neighboring slabs is much larger than that of the
the discrete rotational symmetry of the system at those nearest neighbor X atoms and thus the overlap between
points. At the Γ point, the eigenstates mainly consist of dz2 orbitalsis negligible. One may immediately expect a
the out-of-plane orbitalssuchas the dz2 orbitalofthe M small dispersion along the kz direction due to the small
atom and the p orbital of the X atom. As a result, the inter-planehoppingbetweenp andp orbitalscompared
z x y
overlap integral between different layers is expected to to the p orbitals. However, we have shown that even
z
be large compared to the in-plane orbitals. Since phase thissmalldispersionissuppressedandthebandbecomes
6
FIG. 5: (a)-(d) ARPES data along the Γ to K from MoS2, MoSe2, WS2, and WSe2. Green dotted lines indicate the band
dispersions near the K point. (e)-(h) ARPES data after potassium evaporation. The concentration of the doped electrons by
potassium evaporation can be estimated from the Fermi surface volume. The estimated electron doping concentrations are
4.6×1013 cm−2, 7.2×1013 cm−2, 11×1013 cm−2, and 7.6×1013 cm−2 for MoS2, MoSe2,WS2, and WSe2, respectively.
almost dispersionless along the k direction due to the lation results and shows a strong k dispersion. The
z z
graphene-likephase cancellationamongthe nearesthop- breadth in the ARPES data in the energy direction is
ping processes stemming from the C3 symmetry of the due to the finite escape depth of the ARPES process (fi-
system. nite k resolution). k dispersions in MoS , MoSe , and
z z 2 2
The experimentalperspectivesoftheaboveresultsare WS near the in-plane Γ point are as strong as that in
2
as follows. The direct band gap at the K-point in MX WSe [Fig. 4(b),(c),(d)].
2 2
remains almost the same with the gap of the monolayer. On the other hand, photon energy dependent ARPES
Spin band splitting is expected to depend only on the data show no k dispersion near the K point as seen in
z
atomicspin-orbitcouplingofMatominMX2 andshould Fig. 4(b)-(e), consistent with our calculation results in
be independent of the number of layers. These results Eq. (3.27). Dashed lines in Fig. 4(b)-(e) are guides to
tell us that we can extract the massive Dirac Fermion eye which are straight (that is, no k dispersion). Since
z
parametersfromthe electronicstructureofMX2. Figure the energy of the band at a specific in-plane momentum
3showstheexpectedARPESdatafromMX2 asARPES is the same regardless of kz, ARPES spectra near the K
captures a range of kz due to the finite photoelectron point are very sharp in comparison to the Γ point data,
escape-depth. both in the energy and in-plane momentum directions.
This fact can be seen in Fig. 4(b)-(e) as well as in Fig.
5(a)-(d).
B. ARPES measurements on bulk 2H-MX2 In order to extract the massive Dirac fermion param-
eters, we need ARPES data along the in-plane Γ to K
As our theoreticalworkshowsthat we canextractthe (seeFig. 5). 2λofMoS ,MoSe ,WS ,andWSe canbe
2 2 2 2
appropriate parameters from MX data, we performed clearly observed in the data shown in Fig. 5(a)-(d). 2λ
2
photon energy dependent ARPES to obtained the k is drastically increased as the transition metal changes
z
dispersive electronic structure. Figure 4(a) shows the fromMo to W since 2λ mostly relies onthe atomic spin-
ARPES data taken with incident photon energies be- orbitcouplingofthetransitionmetalatom. Theeffective
tween 50 and 100 eV near the in-plane Γ point. Black hopping integral, t, can also be estimated by fitting the
dashedlinesindicatebanddispersionsexpectedfromEq. datainFig. 5(a)-(d)becausetislinearlyproportionalto
(3.11). The data is in good agreement with the calcu- the curvature of valence band dispersionat the K point.
7
MoS MoSe WS WSe
2 2 2 2
TABLE I: Parameters for the massive Dirac fermion model
determinedfromthebulkARPESdata. Alsogivenintheta-
blearethevaluesfrompublishedARPESdataonmonolayers 1.0
grown on various substrates. The parameters are expressed
inunitofeV.Notethattvalueswith*markareobtainedby eV) ∆−λ
fittingthe dispersions of thepublished data. E(F 0.0
E-
∆ 2λ t ∆-λ
MoS2 1.90 0.16 1.01 1.82 -1.0 2λ
MoSe2 1.67 0.20 0.90 1.57
WS2 1.86 0.44 1.25 1.82
WSe2 2.04 0.48 1.13 1.62
FIG.6: TheelectronicstructuresofmonolayerMoS2,MoSe2,
MoS2/Au(111) [30] 1.465 0.15 1.10* 1.39
WS2, and WSe2 near the K point predicted by the massive
MoSe2/bilayer graphene [5] 1.67 0.18 0.90* 1.58
DiracfermionmodelwiththevaluesinTable1. Blueandred
WS2/Au(111) [31] 0.42
lines indicate the spin lifted valenceband.
t of MoS is, therefore, larger than that of MoSe as the The latter case demonstrates the effect of the substrate
2 2
curvatureislargerinMoS thaninMoSe ascanbeseen on the electronic structures of the monolayer.
2 2
from the data in Fig. 5(a),(b). The extracted t values t and 2λ determined by ARPES are in quantitative
for MoS2, MoSe2, WS2, and WSe2 are given in table I. agreements with the results from the first principles
In order to observe the direct band-gap size, ∆ λ, calculations7. Variationof∆inMX qualitativelyagrees
− 2
at the K point, it is necessary to see the bottom of the with the results from the first principles calculations,
CB.The problemis thatthe states arenotoccupiedand but ∆ observed by ARPES is consistently 0.2 eV larger
thus cannot be observed by ARPES. One way to cir- than that from the first principles calculations7. Since
cumvent the problem is to populate the CB bottom by potassium evaporation concentration dependent experi-
potassium evaporation. Potassiumhas very low electron ments on WSe recently revealed that ∆ decreases from
2
affinity and, when dosed on the sample surface, provides 1.6 to 1.45 eV27, ∆ of pristine MX is expected to be
2
electrons. ARPESexperimentsafterthepotassiumevap- evenlarger. Therefore,thefirstprinciplescalculationson
oration reveal the CB minimum (CBM) from which we MX clearly underestimate the true ∆. This along with
2
can determine ∆ [Fig. 5(e)-(h)]. The CBM is found to the substrate effect discussed above justify our study.
be located at the K point in MoS2 and MoSe2, while it In summary,we performedsystematic ARPESstudies
is located at the Σ point in WS2 and WSe2. We note onMX2 (M=Mo,W;X=S,Se)familyanddetermined
thattheCBMofmonolayerWS2 andWSe2 is locatedat themassiveDiracfermionparametersofmonolayerMX2
the K point. This is because the kz dispersion at the Σ withtheassistancefromtightbindingcalculationswhich
pointforWS2 andWSe2 causesthe CBMatthe Σpoint clearly shows how to determine the parameters from the
to be locatedevenlowerthanthatatthe K point. Here, electronic structure of MX . Our work provides the fun-
2
we emphasize that CB and VB nearthe K point arenot damentalinformationonthe quantitative understanding
affectedwhenlayersarestackedandthat,asaresult,the of the electrical and optical properties of this material
massive Dirac fermion parameters including ∆ could be family.
correctly observed.
All the parameters are summarized in the upper part
of Table I. These values can be regarded as those for
Acknowledgments
freestandingmonolayers. Usingtheseexperimentallyob-
tainedparameters,wesketchtheexpectedminimalband
We thank Yeongkwan Kim, Jonathan D. Denlinger,
structures of MoS , MoSe , WS , and WSe near the K
2 2 2 2 Jongkeun Jung, and Soohyun Cho for assistance in the
point[Fig. 6]. Notethatthetvalueaffectsthecurvature
experiments. We also thank Wonshik Kyung for helpful
of the VB and CB, and ∆ λ indicates the estimated
− discussions. This work was supported by the Incheon
direct band-gap size.
NationalUniversityResearchGrantin20130817. B.S.K.
In the lower part of Table I, we also list the parame-
and C.K. were supported by IBS-R009-D2, Korea.
ters determined from the published data. In comparing
the values, one finds the values for free standing mono-
layerMoSe (predicted) andmonolayerMoSe grownon
2 2
Appendix A: Details on the tight binding analysis
bilayer-grapheneareverysimilar. Thiscanbeattributed
to the fact that the lattice mismatch between bilayer-
grapheneandMoSe isonly 0.3%. Ontheotherhand, In this section, we show detailed derivation of the
2
≈
free standing monolayer MoS and monolayer MoS on Hamiltonian matrix elements of the 3D bulk MX sys-
2 2 2
Au(111) have quite different parameters because of the tems at the Γ and K points.
large lattice mismatch between MoS and the substrate. The band splitting at Γ point induced by the stacking
2
8
of MX layers is evaluated to be where δ represents three nearest neighbor sites between
2 i
two planes in the same unit cell, and δ˜ = δ is for an-
i i
∆Γ,kz = hΨuΓVB,kz|H′|ΨlΓVB,kzi (A1) other pair of planes in different unit cells−. From (A1)
to (A2), all overlap integrals between d and p are ne-
c2 pn,u H′ pn,l(δ )
≈ 1h z | | z i i glected. In addition, in obtaining (A4), we assume that
i
X the nearest neighbor hoppings are dominant.
+c21hpnz,u|H′ |pnz+1,l(δ˜i)ieikzc (A2)
i
X
c2
= 1 pA,n,u pB,n,u H′ pA,n,l(δ )
2 h z |−h z | | z i i
(cid:16) (cid:17) Xi (cid:16)
c2
pB,n,l(δ ) + 1 pA,n,u pB,n,u (A3)
− z i i 2 h z |−h z |
(cid:17) (cid:16) (cid:17)
×H′ |pzA,n+1,l(δ˜i)i−|pBz,n+1,l(δ˜i)i eikzc
Xi (cid:16) (cid:17)
c2
1 pB,n,u H′ pA,n,l(δ )
≈ − 2 h z | | z i i
i
X Derivationofthematrixelementsof(3.14)isasfollows.
c2
− 21hpAz,n,u|H′ |pBz,n+1,l(δ˜i)ieikzc (A4) First, ∆KVB,kz is
i
X
∆ = Ψu H′ Ψl (A5)
KVB,kz h KVB,kz| | KVB,kzi
≈ c26hp1(e),n,u|H′ |p1(e),n,l(δi)ieiK·δi +c26hp1(e),n,u|H′ |p1(e),n+1,l(δ˜i)ieiK·δ˜ieikzc (A6)
i i
X X
c2
= 6 pA,n,u + pB,n,u i pA,n,u + pB,n,u H′ pA,n,l(δ ) + pB,n,l(δ )
4 h x | h x | − h y | h y | | x i i | x i i
i
X(cid:8)(cid:0) (cid:1) (cid:0) (cid:1)(cid:9) (cid:8)(cid:0) (cid:1)
c2
+i pA,n,l(δ ) + pB,n,l(δ ) eiK·δi + 6 pA,n,u + pB,n,u i pA,n,u + pB,n,u H′
| y i i | y i i 4 h x | h x | − h y | h y |
i
(cid:0) (cid:1)(cid:9) X(cid:8)(cid:0) (cid:1) (cid:0) (cid:1)(cid:9)
pA,n+1,l(δ˜) + pB,n+1,l(δ˜) +i pA,n+1,l(δ˜) + pB,n+1,l(δ˜) eiK·δ˜ieikzc (A7)
× | x i i | x i i | y i i | y i i
c26(cid:8)(cid:16) pB,n,u i pB,n,u H′ p(cid:17)A,n,l((cid:0)δ ) +ipA,n,l(δ ) eiK·δi (cid:1)(cid:9)
≈ 4 h x |− h y | | x i i | y i i
i
X(cid:0) (cid:1) (cid:0) (cid:1)
+c426 hpAx,n,u|−ihpAy,n,u| H′ |pBx,n,l(δ˜i)i+i|pBy,n,l(δ˜i)i eiK·δ˜ieikzc (A8)
Xi (cid:0) (cid:1) (cid:16) (cid:17)
where K is the position of a K point in the momen- involvingp andp simultaneouslycanceleachother. As
x y
tum space of MX monolayer. We also retained only the a result, ∆ becomes
2 KVB,kz
dominant nearest neighbor hoppings in obtaining (A6)
and(A8). Sincet(LL) isinvariantunderi j,the terms
p′ipj ↔
9
c2
∆KVB,kz = 46 hpBx,n,u|H′|pAx,n,l(δi)i+hpBy,n,u|H′|pAy,n,l(δi)i eiK·δi
i
X(cid:0) (cid:1)
c2
+ 46 hpAx,n,u|H′|pBx,n,l(δ˜i)i+hpAy,n,u|H′|pBy,n,l(δ˜i)i eiK·δ˜ieikzc (A9)
Xi (cid:16) (cid:17)
= c426 ((Vppσ −Vppπ)δδi2,2⊥ +2Vppπ)eiK·δi + c426 ((Vppσ −Vppπ)δ˜δ˜i2,2⊥ +2Vppπ)eiK·δ˜ieikzc (A10)
i i
X X
= c26 (V V )δi2,⊥ +2V eiK·δi +eiK·δ˜ieikzc (A11)
4 ( ppσ − ppπ δ2 ppπ)
Xi (cid:16) (cid:17)
c2 δ2
= 46 (Vppσ −Vppπ) δ⊥2 +2Vppπ eiK·δi +e−iK·δieikzc (A12)
(cid:26) (cid:27) i
X(cid:0) (cid:1)
where we use the fact that δ2 δ2 = δ2 + δ2 is show that eiK·δi = e−iK·δi =0.
⊥ ≡ i,⊥ i,x i,y i i
independent of i. With δ1,⊥ = (a/2,a/2√3,0), δ2,⊥ = The othePr matrix elePments related to the slight shifts
( a/2,a/2√3,0), and δ = (0, a/√3,0), one can of the band edges are evaluated as follows.
3,⊥
− −
α = Ψu H′ Ψl (A13)
K,kz h KVB,kz| | KCB,kzi
≈ c5c6hp1(e),n,u|H′ |p−(e1),n,l(δi)ieiK·δi +c5c6hp1(e),n,u|H′ |p−(e1),n+1,l(δ˜i)ieiK·δ˜ieikzc (A14)
i i
X X
c c
5 6 pB,n,u +i pB,n,u H′ pA,n,l(δ ) +ipA,n,l(δ ) eiK·δi
≈ − 4 h x | h y | | x i i | y i i
i
c5c6 X(cid:0) pA,n,u +i pA,n,u (cid:1)H′(cid:0) pB,n,l(δ˜) +ipB,n,l(δ˜)(cid:1) eiK·δ˜ieikzc (A15)
− 4 h x | h y | | x i i | y i i
c c Xi (cid:0) (cid:1) (cid:16) (cid:17)
= − 546 hpBx,n,u|H′|pAx,n,l(δi)i+hpBy,n,u|H′|pAy,n,l(δi)i+2ihpBx,n,u|H′|pAy,n,l(δi)i eiK·δi
i
−c54c6 X(cid:0)hpAx,n,u|H′|pBx,n,l(δ˜i)i+hpAy,n,u|H′|pBy,n,l(δ˜i)i+2ihpBx,n,u|H′|pAy,n,l(δ˜i)i(cid:1) eiK·δ˜ieikzc (A16)
Xi (cid:16) (cid:17)
c c (δ +iδ )2
= − 546 (Vppσ −Vppπ) i,x δ2 i,y +2Vppπ eiK·δi
i (cid:26) i (cid:27)
X
−c54c6 i ((Vppσ −Vppπ)(δ˜i,x+δ˜i2iδ˜i,y)2 +2Vppπ)eiK·δ˜ieikzc (A17)
X
= c5c6 (V V ) δ⊥ 2 (δi,x+iδi,y)2eiK·δi + (δ˜i,x+iδ˜i,y)2eiK·δ˜ieikzc (A18)
− 4 ppσ − ppπ (cid:18) δ (cid:19) i ( δ⊥2 δ˜⊥2 )
X
and, in the same way,
β = Ψu H′ Ψl (A19)
K,kz h KCB,kz| | KVB,kzi
≈ −c54c6 (Vppσ −Vppπ)(cid:18)δδ⊥(cid:19)2 i ((δi,x−δ⊥2iδi,y)2eiK·δi + (δ˜i,x−δ˜⊥2iδ˜i,y)2eiK·δ˜ieikzc). (A20)
X
10
At K =(4π/3a,0,0), one can show that as
3 = (δi,x+iδi,y)2eiK·δi = (δ˜i,x−iδ˜i,y)2eiK·δ˜i 3c5c6 δ⊥ 2
− i δ⊥2 i δ˜⊥2 αK,kz = 4 δ (Vppσ −Vppπ) (A23)
X X (cid:18) (cid:19)
(A21)
and
and
(δ˜ +iδ˜ )2 (δ iδ )2 3c c δ 2
0 = i,x δ˜2 i,y eiK·δ˜i = i,x−δ2 i,y eiK·δi. βK,kz = 45 6 δ⊥ (Vppσ −Vppπ)eikzc. (A24)
i ⊥ i ⊥ (cid:18) (cid:19)
X X
(A22)
OnecanfindthatwegetthesameresultattheK′ point,
Asaresult,wehavesimple formulaeforα andβ ( 4π/3a,0,0).
K,kz K,kz −
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