Double resonance Raman modes in mono- and few-layer MoTe$_2$

0.95 MB·English

by  Huaihong Guo

Double resonance Raman modes in mono- and few-layer MoTe
2
Huaihong Guo,1,∗ Teng Yang,2,1,† Mahito Yamamoto,3 Lin Zhou,4,5 Ryo Ishikawa,6 Keiji
Ueno,7 Kazuhito Tsukagoshi,3 Zhidong Zhang,2 Mildred S. Dresselhaus,4,5 and Riichiro Saito1
1Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan
2Shenyang National Laboratory for Materials Science, Institute of Metal Research,
Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China
3International Center for Materials Nanoarchitectonics (WPI-MANA),
National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0044, Japan
5 4Department of Electrical Engineering and Computer Science,
1 Massachusetts Institute of Technology,Cambridge, Massachusetts 02139-4037, USA
0 5Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA
2 6Department of Functional Materials Science, Graduate School of
Science and Engineering, Saitama University, Saitama 338-8570, Japan
n
7Department of Chemistry, Graduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan
u
(Dated: June10, 2015)
J

9 Westudythesecond-orderRamanprocessofmono-andfew-layerMoTe2,bycombiningabinitio
densityfunctional perturbation calculations with experimentalRaman spectroscopy using532, 633

] and 785 nm excitation lasers. The calculated electronic band structure and the density of states
i
c showthattheresonanceRamanprocessoccursattheMpointintheBrillouinzone,whereastrong
s optical absorption occurs due to a logarithmic Van-Hovesingularity of electronic density of states.
-
Double resonance Raman process with inter-valley electron-phonon coupling connects two of the
l
r three inequivalent M points in the Brillouin zone, giving rise to second-order Raman peaks due to
t
m the M point phonons. The calculated vibrational frequencies of the second-order Raman spectra
agree with the observed laser-energy dependent Raman shifts in the experiment.
.
t
a
PACSnumbers: 33.40.+f,78.30.Hv,63.20.kd,63.22.Np,63.22.-m,31.15.A-
m
-
d
I. INTRODUCTION present paper.
n
Raman spectra of TMDs have been reported for sev-
o
eral decades17–29. The first-order Raman process in
c
[ Transition metal dichalcogenides (TMDs) have been TMDs has been widely investigated and is now well
intensively studied over past decades, and recently understood16,21,22,24,30–34. However,thestudyofsecond-
2
attracted tremendous attention for a number of orderRamanspectradidnotgiveconsistentassignments
v
applications1–5, since TMDs have a finite energy band for different TMDs18,25,26,35–37. For example, Stacy et
8
7 gap,whichismoreadvantageousthangraphene. Among al.26 and many other groups15,18,29,38–40 assigned some
0 the TMDs, MoTe2 has a stronger spin-orbit coupling of the Raman peaks of 2H-MoS2 to either combination
7 than other Mo related TMD materials (MoX , X = S, orovertonesecond-orderphononmodes occurringat the
2
0 Se), which leads to a longer decoherence time both for Mpoint(centerofhexagonaledge)intheBrillouinzone.
1. valley and spin freedom5 and thus MoTe2 is considered Terrones et al.37 also assigned phonon modes in few-
0 to be used for ideal valleytronic devices6,7. Unlike other layer WSe according to the M point phonon. On the
2
5 TMDs, MoTe has unique structural properties, such other hand, Sourisseau et al.27,28 ascribed the second-
2
1
as a thermally-induced structural phase transition at orderRamanprocessof2H-WS totheKpointphonons.
2
v: around900◦Cfromtheα-phasediamagneticsemiconduc- Berkdemiretal.41 consideredthatelectronsarescattered
i tortotheβ-phaseparamagneticmetal8,9. MoTe hasthe bytheMpointphononsbetweentheKpointandIpoint
X 2
smallestdirectenergybandgap( 1.1eV)attheKpoint (one point along ΓK direction), and that this process
r ∼
a (hexagonal corner) in the Brillouin zone (BZ) among all is responsible for the double resonant Raman process in
semiconducting TMDs, the band gap is similar to the few-layerWS . Furthermore,Sekine et al.25 found a dis-
2
indirect gap ( 1.1 eV) of silicon10,11. MoTe is thus persive Raman peak in 2H-MoS and interpreted this
2 2
suitable for na∼noelectronics devices4,5 and for studying peak in terms of a two-phonon Raman process at the
exciton effects at the K points12. Raman spectroscopy K point. The different assignments for both few-layer
is a useful tool for characterizing TMDs since we get or bulk MoS and WS , either at the M or K points,
2 2
many information13–15 such as number of layers, elec- have not been well investigated, since most of these as-
tronic and phonon properties of TMDs. Recent Raman signmentstosecond-orderRamanpeakshavebeenmade
spectroscopy studies of monolayer and few-layer MoTe merely by comparing phonon frequencies with reference
2
using visible light (1.9 2.5 eV) show that there ex- to some inelastic neutron scattering results19,28. Thus
∼
ists some unassigned Raman spectra which can not be a detailed analysis with use of double resonance Ra-
assigned to a first-order Raman process5,16, and the as- man theory is needed, especially for understanding both
signment of these Raman spectra is the subject of the mono- and few-layer MoTe that are discussed in this
2
2
paper. ters used in this paper are listed in Table S1 and agree
Inthepresentpaper,wemainlystudythesecond-order well with the experiment and other calculations32,45,46.
Raman spectra of monolayer MoTe , based on ab initio TheMonkhorst-Packscheme47isusedtosampletheBril-
2
electronic band calculations, density functional pertur- louinzone(BZ) overa 17 17 1k-meshforMoTe2. The
× ×
bation theory and associated experimental Raman spec- phonon energy dispersion relations of MoTe2 are calcu-
troscopy. From our electronic band structure calcula- lated based on density functional perturbation theory48.
tion, we have found a saddle point of the energy disper- In order to visualize the vibrational modes at the M
sion at the M points in the BZ, which gives rise to two- point, we use a 2 2 1 supercell in which the M point
× ×
dimensionalVanHovesingularity(VHS)intheelectronic is zone-folded to the Γ point in the folded BZ.
density of states. The energy separation between two The optical absorption spectrum is calculated by the
VHSs(2.30and2.07eV)arematchedtothe2.33eV(532 real (ǫ′) and imaginary (ǫ′′) parts of the dielectric func-
nm) and 1.96 eV (633 nm) laser excitations in our ex- tionasafunctionofphotonenergy,respectively,basedon
periment in which resonant optical absorption occurs at the PAW methodology49 and the conventionalKramers-
theMpoints. Consequently,inter-valleyelectron-phonon Kronig transformation. The absorption coefficient α is
scattering process occurs between two of the three in- described by α = 4πκE /(hc), where E is the incident
γ γ
equivalentM(M,M′, M′′)points,givingriseto adouble laserexcitationenergy,hthePlanckconstant,cthespeed
resonance Raman process. We propose that this double oflight,andκ is the extinctioncoefficient50, thatis, κ =
resonance process occurring at the M points is essential
q(√ǫ′2+ǫ′′2 ǫ′)/2.
for the interpretation of the observed second-order Ra- −
man peaks of MoTe . The experimental second-order
2
Raman frequencies as a function of laser excitation en-
ergy confirm our interpretation of the double resonance III. EXPERIMENTAL METHOD
Raman spectra.
Organization of the paper is as follows. In II and III, Bulk crystals of α-MoTe were prepared through a
2
we briefly show computational details and experimental chemical vapor transport method51. Atomically thin
methods,respectively. InIV,bothexperimentalandthe- crystals of MoTe were mechanically exfoliated from the
2
oretical results for mono- and few-layer MoTe Raman bulk crystals onto silicon substrates with a 90 nm-thick
2
spectra are given. We assign second-order Raman peaks oxidelayerontop. Thethicknessesoftheatomiccrystals
by comparing the calculated results with the observed were identified using optical microscopy and first-order
Raman spectra. In V summary of the paper is given. Raman spectroscopy16. Raman spectroscopy measure-
Further information such as layer-number dependence is ments for monolayer MoTe were performed using 532,
2
discussed in Supplemental Material. 633 and 785 nm excitation lasers for discussing the ob-
served phonon dispersion due to double resonance Ra-
?
man theory . Raman spectroscopy studies of few-layer
II. COMPUTATIONAL DETAILS MoTe2under532nmlaserexcitationwerealsoperformed
andtheresultsarediscussedinthesupportingmaterials.
The grating sizes were 1200, 1800 and 2400 lines/mm
We performed the electronic and phonon energy dis-
for the 785, 633 and 532 nm laser excitation measure-
persion calculations on monolayer and few-layer MoTe
2 ments, respectively. The magnification of the objective
by using first-principles density functional theory within
lens was 100x. The accumulation times were 150-200
the local density approximation (LDA) as implemented
seconds. The laser power was kept below 0.1 mW, 0.2
in the Quantum-Espresso code42. The structures of
mW and 3.0 mW for the 532, 633 and 785 nm excita-
monolayer and few-layer MoTe that are used in the
2 tion lasers, respectively, in order to avoid damage to the
present experiment are α-MoTe 16. The AA′ and AA′A
2 samples. Allmeasurementswereperformedatroomtem-
stacking geometries are used in the computationalstudy
perature in the backscattering configuration. Since this
for bi-layerandtri-layerMoTe , respectively,since these
2 papergivesgreateremphasistoatheoreticalunderstand-
stacking geometries are more stable than the other
ing of the 2nd-order Raman scattering processes, we do
possibilities43. AA′stackingreferstothegeometrywhich
not present many experimental details. However, more
hasaMo(Te)atomofonelayerontopofaTe(Mo)atom
details about the characterization of the number of lay-
oftheotherlayer. Thefew-layerMoTe areseparatedby
2 ers in our MoTe samples can be obtained in one of the
25 ˚A from one another in a unit cell of the calculation 2
previousworksonMoTe synthesisandcharacterization,
toeliminatetheinter-few-layerinteraction. Weusedpro- 2
based on the first-order Raman spectra16.
jectoraugmented-wave(PAW)pseudopotentials44 witha
plane-wave cutoff energy of 65 Ry to describe the inter-
action between electrons and ions. The spin-orbit split
IV. RESULTS AND DISCUSSION
electronic band structures were calculated by relativistic
pseudopotentialsderivedfromanatomicDirac-likeequa-
tion. Theatomiccoordinatesarerelaxeduntiltheatomic Figure 1(a) shows Raman spectra of mono- to tri-
force is less than 10−5 Ry/Bohr. The lattice parame- layerMoTe underthe532nm(2.33eV)laserexcitation.
2
3
(a) (b)

8000 8800000000
A1g E12g B12g 1L A 2.33 eV
⑧1 2L 1g 1.96 eV
sity (a. u.) 4000 !2 ✁3 ✠4 Si ✂5 ✄6 3L sity (a. u.) nsity (a.u.) 4400000000 ☎1 ✆2 ✝3 E12g Si ✞5 1✟.56 8 eV
en ennte ✡4
nt ntI 22000000
I E I
1g
0 00
100 150 200 250 300 350 400 100 115500 220000 225500 330000 335500 440000
Raman shift (cm-1) RaRmamaann sShhiifftt ( c(mcm-1)-1)
Figure1: (Coloronline) (a)Ramanspectraofmono-,bi-,andtri-layerMoTe2 under2.33 eV(532 nm)laserexcitation energy,
showing the small intensity peaks ωi (i = 1, 2, ..., 6) and an additional calibration peak from Si besides the typical Raman
peaks of A1g, E12g and B12g that are frequently observed in MoTe25,16. (b) Raman spectra of monolayer MoTe2 under2.33 eV
(532 nm),1.96 eV (633 nm), and 1.58 eV (785 nm) laser excitation energies.
The Raman spectra show three strong peaks, including Fig. 2(a), we show the calculated electronic energy band
the in-plane E1 mode and the out-of-plane A mode andDOS.FromtheparabolicbandsaroundtheKpoint,
2g 1g
for monolayer and few-layer MoTe , as well as the E two-dimensional (2D) DOS gives constant values, while
2 1g
modeandtheB1 modeforbi-andtri-layerMoTe ,that attheMpointwehavefoundsaddlepointsoftheenergy
2g 2
were observed previously5,16,52. These spectra are as- dispersionintheelectronicband,thecorrespondingDOS
signed to the first-order Raman spectra of the Γ point gives logarithmic logE - E0 divergence as is known as
| |
phonons16. However, we observe additional peaks with Van Hove Singularity (VHS) in 2D systems.
relatively small intensities, which can not be assigned as InFig.2(b),weshowthecalculatedreal(ǫ′)andimag-
first-order Raman spectra16. In Fig. 1(a), we denote the inary(ǫ′′)partsofthedielectricconstantandthe optical
weak Raman spectra as ω (i = 1, 2, ..., 6).
i absorption coefficient α as a function of photon energy.
Since the second-order Raman spectra are dispersive ǫ′′ hasanonzerovaluefromthephotonenergyEγ =1.10
as a function of excitation energy? , we performed Ra- eV which corresponds to the energy gap, and a peak at
man spectroscopy measurements using different excita- Eγ 1.80 eV. A strong optical absorption at 1.80 eV
∼
tion laser energies. In Fig. 1(b) we show Raman spectra from our calculation agrees well with the large peak at
of monolayer MoTe under 2.33 eV (532 nm), 1.96 eV around1.83eVfromtheopticalabsorptionmeasurement
2
(633nm)and1.58eV(785nm)laserexcitationenergies. by Ruppert et al.11 on monolayerMoTe2. Moreover,the
HeretheRamanintensitiesarenormalizedusingthenon- electronicexcitationenergyof2.30eV(2.07eV)fromthe
resonant Si peak intensity at 520 cm−1 (not shown in third(top)valencebandtothelowest(thirdlowest)con-
Fig.1(b)). We findthat the peak positionsofeachω ei- duction band at the M point matches very well with the
i
therupshiftordownshiftbyuptoseveralcm−1fordiffer- laserenergy2.33eV(1.96eV)usedinourexperiment,as
ent laserexcitationenergies,suggestingthat these peaks indicated by pink (orange) arrows in the band structure
are associated with a second-order Raman process? . of Fig. 2(a). Thus resonant optical absorption at these
laser lines is expected.
Considering that the second-order process is usually
connected to the electronic properties by double reso- In Fig. 2(c), we plot equi-energy contour for the con-
?
nance Raman theory , we calculate the electronic band duction band energies E which correspond to the
band
structure,thedensityofstates(DOS)andtheopticalab- laserexcitationenergiesE . E =1.52and1.57eVfor
γ band
sorptionspectra ofmonolayerMoTe as shownin Fig.2. c1 and 1.72 eV for c2 correspond to the E = 1.58, 2.33
2 γ
Ourcalculatedenergydispersionreproducesthemainob- and1.96eV,respectively,asindicatedbyblue,pink,and
served features11,32,54 such as the direct electronic band orange arrowsin Fig. 2(a). The conduction bands c1, c2
gap (1.00 eV) and the spin-orbit splitting of the valence and the valence bands which have the same spin6,53 are
band ( 230meV) at the K point whichare close to 1.10 highlighted in red solid lines in the band structure. The
eV and∼250 meV, respectively, obtained from the op- equi-energy contour lines Ec1 and Ec2 in Fig. 2(c)
band band
tical absorption measurements11. It is noted that the bothshowthesaddlepointshapeoftheenergydispersion
energy band gap is underestimated by the local den- in the vicinity of the M point. That is, the energy band
sity functional method. In order to fit the experimental Ec1 (Ec2 ) starting from the M point monotonically
band band
data, the conduction bands are upshifted by 0.10 eV, decreases (increases) along the M-K direction, whereas
so are the DOS and optical absorption spectrum. In this band energy increases (decreases)along the M-Γ di-
4
(a) (b)
12
ε'
2
1.72 eV 1.72 eV 8 α 10
1.57 eV band c2 1.57 eV
)
V 1.52 eV 1.52 eV 6 8
(end 1 Eγ = 2.33 eV 1.80 eV band c1 E1.γ9 =6 2e.V3 3 eV ΄ε΄΄, 4 ε'' 6 (aα
Eba 1.96 eV 1.58 eV 1.80 eV ε .u
1.58 eV 2 4 .)
0
0 2
-1 -2 0
Γ M K I Γ 0 5 10 15 0 1 2 3 4 5
DOS (states/eV/unit cell) E (eV)
γ
(c)
E (eV) E (eV)
band band
3.40 3.40
band c1 band c2
M' K' M' K'

M M'' 2.20 M 2.20
k k
y Γ K y Γ K
1.20 1.20
equi-energy 1.96
equi-energy
contour lines
contour lines
2.33 Eband Eγ Eband Eγ
1.57 2.33
1.72 1.96
1.52 1.58
k k
x x
Figure 2: (Color online) (a) Electronic energy band structure E(k) and the density of states (DOS), (b) the real (ǫ′), and
imaginary (ǫ′′) parts of thedielectric function and the optical absorption coefficient α as a function of photon energy Eγ, and
(c) an equi-energy contour plot of the two conduction bands (c1 and c2) with the same spin for monolayer MoTe2. Energy
bands are split by spin-orbit interaction, the conduction bands c1, c2 and the valence bands which have the same spin6,53
are highlighted in red solid lines in (a). When we consider energy separation between two Van-Hove singularities (VHSs)
of the DOS, the peak at 1.80 eV in the absorption spectrum corresponds to optical transition at the M point (black double
headed arrow in (a)). (c) The equi-energy contour plot for VHSEc1 = 1.52 and 1.57 eV (left) and Ec2 = 1.72 eV (right).
band band
All equi-energy contours appear around the M point. The E energies of 1.52, 1.57 and 1.72 eV (horizontal lines in (a))
band
correspond to theexperimental laser energies (1.58, 2.33 and 1.96 eV) in blue,pink and orange colors.
rection. Whenwedecreaselaserexcitationenergiesfrom Ramanprocess. Selectingoneprocessfromthreepossible
2.33to 1.58 eV,the contourlines in Fig.2(c) left change inter-valleyscatteringofphoto-excitedcarriers(M M′
the shapes aroundthe M point, suggesting that the area or M M′′ or M′ M′′), we show the schemat←ic→view
←→ ←→
insideofthecontourlineforeachMpointdecreaseswith ofthedoubleresonanceprocessinFig.3(b). InFig.3(b),
increasing laser excitation energy. This situation is op- a photo-excited electron will be scattered from M to M′
posite to the case of graphene in which the equi-energy by emitting a phonon with energy ~ω and momentum
p1
contour is almost circular and the area increases with ~q, which conserves the energy and momentum of both
−
increasing energy. the electron and the phonon. Then the electron is scat-
tered back to M by emitting the second phonon with
Ifthelaserexcitationenergymatchesthe opticaltran-
energy ~ω and momentum +~q. As analyzed in the
sitionenergyattheMpoint,weexpectthatmanyphoto- p2
Fig.3(a),themomentum~qofthephononcanonlyhavea
excited electrons at the M point are scattered to the in-
equivalent M′ or M′′ points by an inter-valley resonant magnitudeof −M−−M→′ or −M−M−→′′ or −M−′−M→′′ . Whenwemea-
| | | | | |
electron-phonon interaction, as shown in Fig. 3(a). The sure the value of q from the Γ point, the ~q corresponds
| |
inter-valley electron-phonon process, together with the to −Γ−M→′′ or −Γ−K−M→′ or −Γ→M, respectively. Since −Γ−K−M→′ =
electron-photonprocess,gives rise to a double resonance
(cid:76)(cid:81)(cid:87)(cid:72)(cid:85)(cid:16)(cid:89)(cid:68)(cid:79)(cid:79)(cid:72)(cid:92)(cid:3)(cid:72)(cid:79)(cid:68)(cid:86)(cid:87)(cid:76)(cid:70)(cid:3)(cid:86)(cid:70)(cid:68)(cid:87)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)
5
M K transverseandout-of-planeacousticmodes,respectively.
(a)
Weassignthesecond-orderRamanmodesattheMpoint
M
for the laser energy of 2.33 eV. The strategy for the as-

signment is to choose any two phonon modes at the M
M K M
point as one overtone or (difference) combination mode
M
and to list in Table I those that have frequency values

withinthe experimentalω 6cm−1. Theseassignments
i
±
are marked in Fig. 4(b) by using color arrows. The red,
green, blue, purple, magenta, and cyan arrows represent
(b) k k+q overtoneorcombinationRamanmodesofω (i=1,2,...,
i
q 6), respectively. Downwardarrowswith an a ” ” super-
−
p1 script represent a two-phonon process with one phonon
p2 -q annihilated and one phonon created to generate ”differ-
ence combination mode”, while upward arrows with the
M M ”+”or”2”superscriptsareusedforcombinationorover-
tonemodesinwhichtwophononsarecreated. Forexam-
ple, ω is assignedto either the ω− (= E1 (M) - LA(M))
1 1 2g
or ω2 (= 2TA(M)), ω to ω− (= E1 (M) - TA(M)), ω
Figure 3: (Color online) (a) In the double-resonance pro- 1 2 2 2g 3
cess, photon-excited electrons are scattered inelastically by to either ω32 (= 2LA(M)) or ω3+ (= E1g(M) + TA(M)),
emitting phonons and make inter-valley (i.e., M←→M′, ω4 to ω4+ (= A1g(M) +LA(M)), ω5 to either the ω52 (=
M′←→M′′, and M←→M′′) transitions, as shown by thesolid 2A1g(M)) orω5+ (=E12g(M)+ TA(M))andω6 to theω6+
double-endarrowsinthefirstBZ.Therelatedphononvectors (= E1 (M)+ LA(M)) mode. Usingthe samephononas-
2g
~q measuredfrom theΓpointarealsogivenbythesix dashed signment,weobtainthesecond-orderRamanfrequencies
arrows. (b) The schematics of the double resonance process
for Eγ = 1.96 eV both at the M point and near the M
(electron-photon and electron-phonon processes). This pro- pointwithalengthof 1MK andfor1.58eVatonepoint
cess generates two phonons (~ωp1, -~q) and (~ωp2, ~q). 3
away from the M point with 10%M′M′′.
InTableI,welistphononfrequenciesofthefirst-order
−Γ−M→′ + −Γ→Γ −Γ−M→′, all phonon modes related to double (upper) and the second-order(lower) Raman spectra for
≡ both the experiment and the calculation and also the
resonanceprocesscorrespondtophononmodesatthe M
phonon assignments. The first-order experimental Ra-
point.
man spectra (A and E1 ) do not change the frequency
In order to specify the phonons contributing to the 1g 2g
forthelaserenergies(2.33and1.96eV)withintheexper-
second-orderRamanprocess,weanalyzedpossibleinter-
imental error (1 cm−1) and agree with the calculations.
valley phonon vectors ~qMM′ for the three laser energies The small frequency downshift of the first order spectra
and calculated the phonon spectra of monolayer MoTe ,
2 (1.7 cm−1 atmost) from2.33eV to 1.58eV laserexcita-
as shown in Fig. 4. For the laser with an energy of 2.33
tion is checked to be not from the laser-induced thermal
eV, we analyze possible ~qMM′ wave vectors between two effect as discussed in the Supplemental Material. Al-
elliptical equi-energy contour lines (red lines) in the left
though we did not specify the origin of the downshift
panel of Fig. 4(a). The dark area for ~qMM′ shows that precisely, the origin of the change for the first order can
themostprobablephononvectorsappearattheMpoint.
be neglected for simplicity. The experimental second-
Otherphononvectors,suchas~qM′M′′ and~qMM′′,aresim- order peak positions of ω are either upshifted or down-
ilar to ~qMM′ (not shown). For the 1.96 and 1.58 eV exci- shifted with the laser eneirgy E by a small magnitude
tationlasers,asalreadyseeninFig.2(c),theequi-energy γ
of about 1-3 cm−1. It is pointed out that the frequency
contours have a larger area than that for 2.33 eV. Then
of ω is upshifted from 2.33 eV to 1.58 eV even though
we expect that ~qMM′ for Eγ = 1.96 and 1.58 eV arises the 1both the first-order spectra are downshifted. This
from the vicinity of the Brillouin zone near the M point,
can be understood by double resonance Raman theory
as shown in the central and right panels of Fig. 4(a).
in which the resonance phonon frequency changes along
The relatively dark region indicates that: (1) the ~qMM′ the phonon dispersion near the M point. In the right of
for Eγ = 1.96 eV appear both at the M point and in the
TableI,thesecond-orderphononfrequenciesfor2.33and
vicinity of the M point (approximately with lengths of
=31M1.K58) aeVlonagpptheaerMinKthdeirveicctiniointyaonfdth(e2)Mthpeoi~qnMtMa′lofnogrtEhγe 1M.5)8aenVdlas1e0r%exMcMita′ti(odnesnoatreedobbtyai”noeffdaMt”~q)M,Mre′s=pe−Γct→Mive(layt.
∼
M′M′′ (orMM′′)directionapproximatelywithlengthsof For Eγ = 1.96 eV, the phonon assignments are obtained
10%M′M′′ (or 6.5%MM′′) away from the M′′ point. either at ~qMM′ = −Γ→M (at M) or ∼ 13MK (off M).
InFig.4(b),weshowthecalculatedphonondispersion Theseassignmentscanbefurtherconfirmedbytheob-
relations of monolayer MoTe . The symmetry for each served relative Raman intensity and the phonon disper-
2
opticalphononband52,suchasE1 andE ,islabeledin sionrelation. First,inFig.1(b),alargerrelativeintensity
2g 1g
Fig.4(b). HereLA,TAandZArefertothelongitudinal, ofRamanspectra ω canbe found atEγ = 2.33eV than
i
6
(a)
q
K M΄ M΄ M΄ MM΄
q qMM΄ M q (cid:2778)(cid:2780)(cid:1787)(cid:1785)qMM΄
M MM΄ MM΄ M
q
MM΄
(cid:705) (cid:705) (cid:705)
E = 2.33 eV 1.96 eV 1.58 eV
γ
(b)
2.33(1.96) eV 1.58 eV 1.96 eV
300
E1
2g
E12g ω +
ω - 6
200 2
ω +
5
)
-1m A1g ω1-
c
ω ( E1g
ω 2

5
E
1g
100

ω +
3
LA

TA ω 2 ω +
3 4

ω 2
1
ZA
0

Γ M΄ M K Γ
Figure 4: (Color online) (a) All possible phonon vectors qMM′ coming from the double resonance Raman process due to the
laser energies of 2.33, 1.96 and 1.58 eV. The ~qMM′ is equal to either ΓM (for the laser energy Eγ = 2.33 and 1.96 eV) or
∼ 13MK(forEγ =1.96eV)or∼10%MM′ (forEγ =1.58eV),asindicatedbyanarrowawayfromtheΓpoint. (b)Thephonon
dispersion relations of a monolayer MoTe2. The second-order phonon modes are analyzed at the M′ point for thelaser energy
Eγ = 2.33 eV, and then at two k points marked by two vertical dashed lines which are shifted leftward and rightward from
the M point for the laser energy Eγ = 1.58 eV and 1.96 eV, respectively. The red, green, blue, purple, magenta, and cyan
vertical arrows are used for modes ω1, ω2, ω3, ω4, ω5, and ω6, respectively. The superscripts ”+”, ”-”, and ”2” for ω refer to
possible ”combination modes” (i.e. ω4+ = A1g(M) + LA(M), ω6+ = E12g(M) + LA(M)),”difference modes” (i.e. ω1− = E12g(M)
− LA(M), ω− = E1 (M) − TA(M)) and ”overtones” (i.e. ω2 = 2TA(M), ω2 = 2LA(M)), respectively.
2 2g 1 3
at1.58eV.ThissupportstheassignmentsattheMpoint large(small)PhDOSattheMpoint(offtheMpoint). It
and at ~qMM′ 10%MM′ for Eγ = 2.33 eV and 1.58 eV, is therefore reasonable to assign the ω2, ω3 and ω6 (ω1,
respectively, s∼ince phonon density of states (PhDOS) at ω4 and ω5) at the M point (off the M point), as under-
the M point is larger than that at ~qMM′ 10%MM′ as linedinTableI.Second,itisnotedthattheexperimental
∼ spectra of ω (at the three laser energies) are too weak
shown in Fig. 4(b). Similarly, for Eγ = 1.96 eV, a rela- 4
to determine the frequency values. The ω is assigned
tivelylarge(small)intensityoftheRamanspectraω ,ω 4
2 3
as a combination mode related to the A (M) as listed
and ω (ω , ω and ω ) in Fig. 1(b) may arise from the 1g
6 1 4 5
7
Table I: Phonon frequencies of the first- and second-order Raman spectra of monolayer MoTe2 are listed for three different
laser energies Eγ. A comparison for first and second-order Raman spectra is given between the experimentally unassigned
modesωi (i=1,2,..., 6)andthecalculated phononfrequenciesbothat theMpointandneartheMpointdenotedby(offM)
whose~q liesaround 1MKand10%MM′ forEγ =1.96eVand1.58eV,respectively. Theconsistentdispersionbehaviorforthe
3
second-order Raman spectra between the experiment and the theory are underlined. The frequency is given in units of cm−1.
NA:not available.
Experiment Calculation Phonon assignments
Eγ 2.33 eV 1.96 eV 1.58 eV 2.33 eV 1.96 eV 1.58 eV
E1g NA NA NA 116.9 first-order
A1g 171.1 171.3 170.0 172.2 first-order
E1 235.8 236.2 234.1 236.9 first-order
2g
(at M) (at M/off M) (off M)
ω1 138.2 140.7 139.0 137.4 137.4/146.6 140.1 2TA(M)
132.8 132.8/133.1 133.2 or E1 (M)-LA(M)
2g
ω2 183.7 183.6 181.5 178.3 178.3/177.2 177.8 E12g(M)-TA(M)
ω3 205.5 205.3 205.9 197.9 197.9/208.2 200.5 E1g(M)+TA(M)
202.7 202.7/196.0 200.4 or 2LA(M)
ω4 ∼264.0 260-270 260-270 263.0 263.0/256.6 261.0 A1g(M) + LA(M)
ω5 329.0 333.2 ∼328.0 323.3 323.3/317.2 321.6 2A1g(M)
315.7 315.7/323.8 317.9 or E1 (M)+TA(M)
2g
ω6 347.4 348.4 346.5 348.3 348.3/348.5 348.0 E12g(M)+LA(M)
in Table I. Then it is interesting to discuss why the ω of the LA modes, suggestingthat the TA modes may in-
4
becomesweakerandbroaderwhenthefirst-orderA (Γ) duce as large deformation potential as the LA modes in
1g
mode becomes stronger by changing the laser excitation the electron-phonon interaction. Usually, the TA modes
energyfrom2.33eV to 1.96eV/1.58eV. We expect that around the Γ point do not contribute to the electron-
the photo-excited electron may choose either an intra- phonon interaction. Nevertheless, this seems not to be
valleyscatteringprocess(A (Γ))oraninter-valleyscat- the case at the M point.
1g
tering process (A (M)), so that A (M) competes with
1g 1g So far we have only considered the electron-involved
A (Γ), resulting in the alternative Raman intensity of
1g double resonance process for MoTe and did not dis-
A (Γ) and ω in the experiment. The ω (A (M)) has 2
1g 4 4 1g cuss the hole-involved double resonance process. As
no VHS of PhDOS compared with all the other ω’s at
Venezuela et al.55 pointed out, the enhancement by the
the M point as seen from Fig. 4(b) which is relevant to
double resonance is strong in graphene for one-electron
the weak intensity of ω , too.
4 and one-hole scattering, since the energy denominators
of the Raman intensity formula becomes zero for the
In Fig. 5, we show the normal modes of the phonons
symmetric π and π∗ bands of graphene (triple reso-
at the M point in which the phonon vibration ampli-
nance). In the case of TMDs, however, a similar effect
tude is visualized in the 2 2 supercell. As seen from
Fig. 4(b), the two groups o×f optical phonon modes (E1 is not expected since the conduction band and the va-
2g lencebandarenotsymmetricaroundtheFermienergy53.
and (E or A )) are well separated by about 50 cm−1.
1g 1g Since we don’t have a program to calculate electron-
The visualization of these M point phonon modes shows
phonon matrix element by first-principles calculations,
thattheLOandTOE (M)modes(230 300cm−1)and
the LA(M) andTA(M)2gmodes (60 100c∼m−1)arebond- we can’t calculate Ramanintensity for eachdouble reso-
∼ nance process,55 which will be a future work.
stretching modes of the Mo-Te bond. The LO A (M)
1g
and TO E (M) modes (120 180 cm−1) arises from the Finally,webrieflycommentontheexcitoneffectonthe
1g
∼
relativemotionoftwoTeatomsoccupyingthesamesub- doubleresonanceRamanspectraofMoTe . Sincetheex-
2
lattice. The six vibrational modes in Fig. 5 are Raman citonic states appear at the direct energy gap at the K
active, since the volume or the area of the supercell un- pointofthe BZ56,57,we expect thatthe excitonicenergy
dervibrationgivesnon-zerodeformationpotential. Thus is around 1.1 eV for MoTe , which is much smaller than
2
all the modes can contribute to the 2nd-order overtone, thethreelaserenergiesweused(1.58-2.33eV).Asfaras
combination and difference modes. Furthermore, the vi- weconsideronlyMoTe ,wecansafelyneglecttheexciton
2
brationalmagnitudeoftheTAmodesisnolessthanthat effectattheKpoint. However,inthecaseofotherTMDs
8
LO, E1 (M) TO, E1 (M) less, when we discuss the M-point phonon of the other
2g 2g
246.9 cm-1 234.3 cm-1 TMDs, we expect that the second-order Raman spectra
donotappear,sincetheopticaltransitionenergiesatthe
M point is larger than 3.0 eV. It would be interesting if
top
ultra-violetlasercouldbeusedfordetectingtheM-point
view
phonon modes for other TMDs, which will be a future
work. In the case of MoTe , on the other hand, since
2
side
E (M) 1.8eV,weexpectthedoubleresonanceRaman
view g ∼
process for conventional laser energies. Although we did
not consider the exciton effect at the M point explicitly
LO, A1g(M) TO, E1g(M) asK.F.Maket al.59 didingraphene,webelievethatthe
161.8 cm-1 129.2 cm-1
present discussion of the double resonance Raman effect
should work well, because only optical transition matrix
elements that are common in the resonant Raman pro-
cess are enhanced by the exciton effect, and the relative
Raman intensity thus obtained do not change much for
the M point.60,61
LA(M) TA(M)
101.4 cm-1 68.6 cm-1
V. CONCLUSION
Insummary,wehaveinvestigatedthesecond-orderRa-
man spectra of MoTe systems, based on an ab initio
2
density functional calculation, and compared with the
experimental Raman spectroscopy. The calculated elec-
tronicbands,theVanHovesingularitiesinthedensityof
states,andtheabsorptionspectraindicateastrongopti-
Figure5: (Coloronline)VisualizationofthepossibleRaman- calabsorptionattheMpointintheBZofMoTe2. Using
activemodesattheMpointforthesecond-orderprocesssug- double resonance Raman theory, combination/difference
gested in Figure 4. Both the top and side views are given orovertonemodeofphononsattheMpointareassigned
and red arrows represent the vibrational directions and am- to identify the experimental Raman modes of ω (i = 1,
i
plitudes of themotion of individual atoms. 2, ..., 6). This study facilitates a better understanding
of the electron-phonon interaction and the second-order
Raman process in TMD systems.
TableII:LaserenergiesthatareusedintheRamanspectraof
TMDs. Eg(K),Eg(M)are,respectively,theopticaltransition
energies at the K and M point, which are obtained by LDA
calculations. Detail is discussed in thetext. Acknowledgments
TMDs laser(eV) Eg(K) Eg(M)
MoS218,26 1.83-2.41 1.7 >3.0 M.Y. and K.T. sincerely acknowledge Prof. Meiyan
WS227,28,41 1.83-2.41 1.9 >3.0 Ni and Dr. Katsunori Wakabayashi for critical sugges-
WSe236,37 2.41 1.7 >3.0 tions on experiments. T.Y. and Z.D.Z acknowledge the
MoTe2thiswork 1.96, 2.33 1.1 ∼1.8 NSFC under Grant Nos. 11004201, 51331006 and the
IMR SYNL-Young Merit Scholars for financial support.
R.S. acknowledges MEXT Grants Nos. 25286005 and
(MoS 26,WS 27,28,41,58,WSe 36,37)energygapsattheK 25107005. M.S.D. acknowledges NSF-DMR Grant No.
2 2 2
point (E (K)) lie in the range of 1.7-1.9 eV as shown in 10-04147. K.U. and K.T. acknowledge MEXT Grant
g
Table II. The laser energies used in the experiments lie No. 25107004. Calculations were performed in the HPC
in a range from 1.83 to 2.41 eV. Thus the exciton effect cluster at the Institute of Metal Research in Shenyang,
needstobeconsideredforsomelaserenergies. Neverthe- China.
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