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Effect of Point Defects on the Optical and Transport Properties of MoS2 and WS2

Shengjun Yuan,1, ∗ Rafael Roldan,2, † M. I. Katsnelson,1 and Francisco Guinea2

1Radboud University of Nijmegen, Institute for Molecules and Materials,

Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands2Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco E28049 Madrid, Spain

(Dated: July 7, 2014)

Imperfections in the crystal structure, such as point defects, can strongly modify the optical andtransport properties of materials. Here, we study the effect of point defects on the optical andDC conductivities of single layers of semiconducting transition metal dichalcogenides with the formMS2, where M=Mo or W. The electronic structure is considered within a six bands tight-bindingmodel, which accounts for the relevant combination of d orbitals of the metal M and p orbitalsof the chalcogen S. We use the Kubo formula for the calculation of the conductivity in sampleswith different distributions of disorder. We find that M and/or S defects create mid-gap states thatlocalize charge carriers around the defects and which modify the optical and transport propertiesof the material, in agreement with recent experiments. Furthermore, our results indicate a muchhigher mobility for p-doped WS2 in comparison to MoS2.

PACS numbers:

Introduction— The mobility of current single layercrystals of transition metal dichalcogenides (TMD) ishighly dependent on the screening environment and islimited by the presence of defects in the samples. Theexistence of defects in the chemical and structural com-position of those materials can influence their optical andtransport properties, as revealed by recent experimentalresults. A broad peak at ∼ 700 nm (∼ 1.77 eV) in theoptical spectrum of bilayer MoS2 has been associated toimpurities [1] whereas the mobility of multilayer sam-ples has been shown to highly depend on the substrateand dielectric effects [2]. Vacancy defects in the crys-tal, which can be created by means of thermal annealingand α-particle [3] or electron beam irradiation [4, 5], trapfree charge carriers and localize excitons, leading to newpeaks in the photoluminescence spectra [3]. Recent ex-periments [5] show that the density of sulphur vacanciesin MoS2 is of the order of 10

13 cm−2, corresponding to anaverage defect distance about 1.7 nm. The existence ofline defects, which separate patches or islands where thelayer direction is opposite to its surrounding, can lead tochanges in the carrier mobility [6], and the importanceof short-range disorder has been proposed as the mainlimitation for the mobility of chemical vapor deposition(CVD) grown single-layer MoS2 [7, 8].Therefore, it is necessary to understand the effect of

impurities in the optical and transport properties ofTMD, as a first step to exploit the controlled creationof defects as a route to manipulate their electronic prop-erties. There are several theoretical works which havestudied this problem using ab initio methods [6, 9–15].However, the simulation of realistic disordered samples ofTMD with a random distribution of defects are extremelyexpensive computationally for density functional theory(DFT) methods, since they require a very large unit cellin the calculation. In this paper we follow an alternativeroute and perform a systematic study of the density of

states, optical and DC conductivities of single-layers ofMoS2 and WS2 in the presence of point defects, by meansof a real space tight-binding (TB) model for large sys-tems, containing millions of atoms. In our simulations,Mo/W and S point defects are introduced by eliminationof atoms which are randomly distributed over the sam-ple. This method allows us to study point defects suchas unreconstruced vacancies, chemically bonded atomsor molecules, and strong substitutional defects whichprevent the electronic hopping to the neighbors. Wealso consider clusters of point defects. We use a TBmodel that considers the relevant orbital contributionin the valence and conduction bands, as well as the ef-fect of spin-orbit coupling (SOC) [1, 16]. The opticaland electronic properties are obtained numerically byusing the tight-binding propagation methods (TBPM)[3, 6, 7, 19]. Our results show that point defects cre-ate midgap states whose energy depends on the specificimpurities. We show that optical transitions involvingthe impurity bands lead to a background contribution inthe photoconductivity at low energies, in agreement withrecent experiments [22]. We further calculate the DCconductivity of disordered MoS2 and WS2, finding thatthe impurity states do not contribute to the conductiv-ity within the gap, whereas they lead to a depletion of itoutside the gap.Method— Transition metal dichalcogenides as MoS2

and WS2 are composed, in its bulk configuration, of two-dimensional S-M -S layers (M =Mo,W) stacked on top ofeach other, coupled by weak van der Waals forces. Thetransition metal atoms M are ordered in a triangularlattice, each of them bonded to six S atoms located in thetop and bottom layers, forming a sandwiched material.Similarly as in graphene, the weak interlayer couplingmakes possible to exfoliate this material down to a single-layer [23]. The electronic band structure of MoS2 changesfrom an indirect band gap for multilayer samples, to a

2

-1.0 -0.5 0.0 0.5 1.00.0

0.1

0.2

0.3DOS(1/eV

)

E (eV)

MoS2

Mo defect

Pristine xMo+S=0.2% xMo=0.2% xS=0.2%

S defect

(a)

-1.0 -0.5 0.0 0.5 1.00.0

0.1

0.2

DOS(1/eV

)

E (eV)

WS2

W defect

Pristine xW+S=0.2% xW=0.2% xS=0.2%S defect

(b)

-1.0 -0.5 0.0 0.5 1.00.0

0.1

0.2

0.3

R=a

DOS(1/eV

)

E (eV)

Pristine xCluster_Mo+S=1% xCluster_Mo=1% xCluster_S=1%

MoS2(c)

1.2 1.4 1.6 1.8 2.0 2.2 2.40.0

0.5

1.0

1.5

2.0

2.5

3.0

1.2 1.4 1.6 1.8 2.0 2.2

0.01

0.1

1

/0

(eV)

Pristine xMo+S=0.2% xMo=0.2% xS=0.2% MoS2

(d)

/0

(eV)

1.2 1.4 1.6 1.8 2.0 2.2 2.40.0

0.5

1.0

1.5

2.0

2.5

3.0

1.2 1.4 1.6 1.8 2.0 2.2

0.01

0.1

1

/0

(eV)

Pristine xW+S=0.2% xW=0.2% xS=0.2%

(e)

WS2

/0

(eV)

1.2 1.4 1.6 1.8 2.0 2.2 2.40.0

0.5

1.0

1.5

2.0

2.5

3.0

1.2 1.4 1.6 1.8 2.0 2.2

0.01

0.1

1

/0

(eV)

Pristine xCluster_Mo+S=1% xCluster_Mo=1% xCluster_S=1% MoS2

(f)

/0

(eV)

Figure 1: DOS of MoS2 (a) and WS2 (b) for different types of single point defects, and for MoS2 (c) with clusters of defects withR = a (the labels in the subscript of the concentration x indicate the center of the cluster). The peaks in the DOS associatedto midgap bands due to S or Mo/W defects are marked by arrows. Optical conductivity (in units of σ0 = πe2/2h) for MoS2

(d) and WS2 (e) with single defects, and for MoS2 with clusters of defects, for the same concentration of defects as in (a)-(c).The insets show the same plots in a logarithmic scale.

direct gap semiconductor for single-layers [16, 24]. Weconsider a 6-bands tight-binding model which containsthe proper orbital combination that contributes to thevalence and conduction bands of MS2: 3 d-orbitals ofthe transition metal (dxy , dx2−y2 and d3z2−r2) as well asthe symmetric (antisymmetric) combination of the px, py(pz) orbitals of the top and bottom chalcogen atoms [16,25]. The base vector can be written as

φ†i = (d†

i,3z2−r2, d†

i,x2−y2 , d†i,xy, p

†i,x,A, p

†i,y,A, p

†i,z,S), (1)

where p†i,o,S = (p†i,o,t + p†i,o,b)/√2, p†i,o,A = (p†i,o,t −

p†i,o,b)/√2, o = x, y, z and the subscripts t and b refer

to the top and bottom S layers, respectively.We also consider the intra-atomic spin-orbit coupling

∑

a λaLa · Sa, where a = M, S accounts for both, thetransition metal M as well as the chalcogen atom S, λa

is the corresponding intra-atomic SO interaction, L isthe atomic angular momentum operator, and S = ~σ isthe spin operator. The optical and electronic properties,such as density of states (DOS), quasi-eigenstates, opticaland DC conductivities, are obtained numerically by usingthe TBPM [3, 6, 7, 19] (more details can be found in theSupplementary Material).Results and discussion— The effect of point defects in

the DOS of MoS2 and WS2 are shown in Fig. 1(a)-(c).The defect concentration (0.2%) for single point defectsin Fig. 1(a) and (b) is chosen to be of the same order

as the intrinsic vacancies observed in recent experiments[5]. For clean samples (black lines), the DOS has a gap ∆which corresponds to the well known direct gap of singlelayer samples at the K points of the Brillouin zone (BZ).Defects in the samples lead to the appearance of a seriesof peaks in the gapped region of the DOS, which are asso-ciated to the creation of midgap states localized aroundthe defects, whose energy and strength depends on thespecific missing atoms, their concentration as well as thespecific arrangement of the point defects as individualmissing atoms [Fig. 1(a) and (b)] or in clusters of pointdefects with variable radius [Fig. 1(c)]. For the sameconcentration of defects, isolated point defects modifymore strongly the DOS than clusters of defects. This isthe reason why we show results for 0.2% of single defects,and 1% of cluster of defects with a radius R = a [35]. Theimpurity states have also an important effect on the opti-cal conductivity [Fig. 1(d)-(e)]. First, let us consider thecase of undoped and clean MoS2 and WS2. Since singlelayers of those TMD are direct gap semiconductors, theonly optical transitions allowed at low energies are twoset of inter-band transitions with ω ≥ ∆ from the edgesof the SOC split valence bands to the conduction bandat the K and K ′ points of the BZ [26]. Those transitionslead to the A and B absorption peaks observed in photo-luminescence experiments [22], and the SOC splitting ofthe valence band manifests itself in the optical conduc-tivity through the step like feature of σ(ω) that can be

3

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.0

0.2

0.4

0.6MoS2

DOS(1/eV

)

E (eV)

Mo defectS defect

(a) Pristine xMo+S=0.1% xMo+S=1%

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.0

0.2

0.4

0.6WS2

DOS(1/eV

)

E (eV)

W defect

S defect

(b)

Pristine xW+S=0.1% xW+S=1%

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.0

0.2

0.4

0.6(c) MoS2

DOS(1/eV

)

E (eV)

Pristine xCluster_Mo+S=0.1% xCluster_Mo+S=1%

R=a

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50

200

400

600

800

1000

1200

1400

1600

MoS2

xMo+S=0.1% xMo+S=1%

(e2 /h

)

(eV)

(d)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50

200

400

600

800

1000

1200

1400

1600

(e2 /h

)

(eV)

WS2

xW+S=0.1% xW+S=1%

(e)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50

200

400

600

800

1000

1200

1400

1600

R=a(e2 /h

)

(eV)

MoS2

xCluster_Mo+S=0.1% xCluster_Mo+S=1%

(f)

Figure 2: Top panels: DOS of (a) MoS2 and (b) WS2 with single defects, and of (c) MoS2 with clusters of defects with radiusR = a. Panels (d)-(f) show the DC conductivity as a function of doping energy for the same concentrations of defects as in(a)-(c).

seen in the black lines (for pristine MoS2 and WS2) ofFig. 1(d)-(f). This feature is especially visible for WS2due to the strong SOC associated to the heavy W atom[1], which lead to a plateau like feature for σ(ω) of ∼ 400meV, in agreement with the energy separation betweenthe spin polarized valence bands. The existence of defectsin the sample lead to flat midgap bands which activatenew optical transitions with ω < ∆ in the optical spec-trum. Most importantly, these new optical transitionslead to a background contribution which appears in theoptical conductivity at low energies, as it can be seen inFig. 1(d)-(f) for different concentration of defects, sug-gesting that resonant impurities, like the defects studiedin this work, could have a relevant contribution to theoptical spectrum of TMDs [22]. Furthermore, it is inter-esting to note that this background contribution due todisorder resembles that observed in the the optical con-ductivity of highly doped graphene [27].

Our results for the DC conductivities are shown in Fig.2, which demonstrate a significant asymmetry betweenelectrons and holes, in reasonable agreement with exper-iments [2] (note that the observations are done in multi-layered samples). The fact that Mo and W defects leadto localized states well inside the gap, as illustrated bythe densities of states also shown in Fig. 2, combinedwith the fact that the bands at the K and K ′ points ofthe BZ can be approximated by an effective gapped Diracequation [28, 29], suggests that these defects can act asresonant scatterers [30], which give rise to a mobility al-

most independent of the carrier density (see below andthe T-matrix analysis in the Supplementary Material).

The localization of these midgap states are clearly seenby their local density of states (LDOS) plotted in Fig.3. The amplitude of each orbital in real space is ob-tained from the average of quasiegenstates with differentinitial states. The profiles of the localized states showeither hexagonal symmetry or mirror symmetry, depend-ing on the type of the orbital and the energy of themidgap states. For S defects, the impurity state at theenergy−0.86 eV is localized mainly (∼ 65.6%) on d3z2−r2

orbitials of neighboring Mo atoms, with small amount(∼ 9.6%) on pz,S orbitals of neighboring S atoms. ForMo defects, on the other hand, there are two midgapstates, one centered at 0.18 eV, with mainly localizedpx(y),A (∼ 23.4%) and d3z2−r2 (∼ 21.2%) orbitals, andanother centered at 0.4 eV, with mainly localized d3z2−r2

(∼ 46.4%) and px(y),A (∼ 21.6%) orbitals. The detailedDOS of each orbital as a function of energy is shown inthe Supplementary Material.

Results for the carrier mobility, defined as µ (E) =σ (E) /ene (E), where the charge density ne (E) is ob-tained from the integral of density of states via ne (E) =∫ E

0 ρ (ε) dε are shown in Fig. 4. We notice that n-dopingcorresponds to Mo(W) point defects, whereas p-dopingcorresponds to S point defects. We observe that forn-doped samples, MoS2 and WS2 show similar mobili-ties, whereas for p-doped samples, the mobility of WS2is larger than for MoS2. Our results show that in gen-

4

-10 -5 0 5 10

-10

-5

0

5

10 (a)

S DefectsMo-3d3z2-r2

E=-0.84eV

Y (n

m)

X (nm)

1.000E-10

3.466E-09

1.201E-07

4.163E-06

1.443E-04

0.005000

-10 -5 0 5 10

-10

-5

0

5

10 (c)

Mo DefectsMo-3d3z2-r2

E=0.188eV

Y (n

m)

X (nm)

1.000E-10

2.885E-09

8.326E-08

2.402E-06

6.931E-05

0.002000

-10 -5 0 5 10

-10

-5

0

5

10 (e)

Mo DefectsMo-3d3z2-r2

E=0.40eV

Y (n

m)

X (nm)

1.000E-10

2.885E-09

8.326E-08

2.402E-06

6.931E-05

0.002000

-10 -5 0 5 10

-10

-5

0

5

10 (b)

Y (n

m)

X (nm)

1.000E-10

2.187E-09

4.782E-08

1.046E-06

2.287E-05

5.000E-04

S DefectsS-3Pz

E=-0.84eV

-10 -5 0 5 10

-10

-5

0

5

10 (d)

Mo DefectsS-3Px

E=0.188eV

Y (n

m)

X (nm)

1.000E-10

2.885E-09

8.326E-08

2.402E-06

6.931E-05

0.002000

-10 -5 0 5 10

-10

-5

0

5

10 (f)

Mo DefectsS-3Px

E=0.40eV

Y (n

m)

X (nm)

1.000E-10

2.885E-09

8.326E-08

2.402E-06

6.931E-05

0.002000

Figure 3: LDOS of for the corresponding dominant orbitals in the impurity states of MoS2 with single point defects. The whitepoints indicate the position of the defects. The concentration of the defects is 0.1% for all panels. Results are obtained fromthe average of quasi-eigenstates for one hundred samples with fixed distribution of point defects but different initial states.Please notice the different color scale in each panel.

eral, the mobilities of TMDs are low, but they are largerfor holes than for electrons, in agreement with previousexperimental results [2, 7]. The results for the mobilitysuggest that it is independent of carrier concentration,except at the edge of the valence band. As we have dis-cussed before, this is consistent with the expected fea-tures of resonant scatterers. A more detailed analysis ofresonant scatterers in gapped Dirac systems is requiredin order to make this statement more quantitative, how-ever. Note, finally, that our analysis leaves out the effectof the missing charge at the defect, which can lead to along range potential, and to intravalley scattering [31].

Conclusions— We have studied the effect of point de-fects in the DOS, optical and DC conductivity of sin-gle layers of TMDs like MoS2 and WS2. The existenceof point defects in the sample creates flat midgap bandswhich activate new optical transitions in the optical spec-trum, leading to a background contribution which ap-pears in the optical conductivity at low energies, in agree-ment with photoconductivity measurements. Our re-sults show a significant asymmetry between electrons andholes. The DC conductivities and mobilities are larger forholes, in agreement with experiments, and we find highermobilities for p-doped WS2 than for MoS2. Mo and Wdefects induce localized states well inside the gap, sug-gesting a behavior similar to that of resonant scatterersin graphene.

-5 0 5 10 150

200

400

600

800

1000

1200

1400

random defects x=0.1%

(cm

2 V-1s-1)

ne (1013cm2)

MoS2 WS2

Figure 4: Mobility for MoS2 and WS2 for the same concentra-tion (0.1%) of point defects. n-doping corresponds to Mo(W)point defects and p-doping corresponds to S point defects.

Acknowledgments— We thank the European UnionSeventh Framework Programme under grant agreementn604391 Graphene Flagship. The support by the Sticht-ing Fundamenteel Onderzoek der Materie (FOM) andthe Netherlands National Computing Facilities founda-tion (NCF) are acknowledged. S.Y. and M.I.K. thankfinancial support from the European Research CouncilAdvanced Grant program (contract 338957). RR ac-

5

knowledges financial support from the Juan de la CiervaProgram (MEC, Spain). R.R. and F.G. thank finan-cial support from MINECO, Spain, through Grant No.FIS2011-23713, and the European Research Council Ad-vanced Grant program (contract 290846).

∗ Electronic address: s.yuan@science.ru.nl† Electronic address: rroldan@icmm.csic.es

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the arrangement of the defects in our calculation.

6

Supplementary Material

Tight-binding band structure and distribution of

defects— In Fig. 5 we show the band structure of MoS2and WS2 obtained from the TB model used in the calcu-lations. The TB parameters are given in Ref. [1].

(a)

MoS2

G M GK-2

-1

0

1

2

G M GK

-2

-1

0

1

2

k

Ene

rgyHe

VL

(b)

WS2

G M GK-2

-1

0

1

2

G M GK

-2

-1

0

1

2

k

Ene

rgyHe

VL

Figure 5: Band structure of MoS2 (a) and WS2 (b) obtainedby the TB model used in the text. The Slater-Koster TBparameters are those given in Ref. [1]

In Fig. 6 we show an sketch of the distribution ofdefects considered in the main text.

Tight-binding Propagation Method— Our method isbased on the numerical solution of time-dependentSchodinger equation (TDSE) in the TB model. The ini-tial state |ϕ〉 is considered as a random superposition ofall orbitals over the whole space which covers all the en-ergy eigenstates [2, 3]

|ϕ〉 =∑

i,o,σ

ai,o,σ |i, o, σ〉 , (2)

where ai,o,σ are random complex numbers normalized as∑

i,o,σ |ai,o,σ|2= 1, and |i, o, σ〉 represents the o orbital

with spin σ at site i. The density of states can be ob-tained by the Fourier transformation of the overlap be-tween the time-evolved state |ϕ(t)〉 ≡ e−iHt |ϕ〉 and theinitial state |ϕ〉 as [2, 3]

ρ (ε) =1

2π

∫ ∞

−∞

eiεt 〈ϕ|ϕ(t)〉 dt. (3)

Here we use units such that ~ = 1. The time-evolutionoperator e−iHt is calcualted numerically by using Cheby-shev polynomial algorithm, extremly efficient for a TBHamitlonian H which is a sparse matrix. Within theTBPM, the optical conductivity (omitting the Drudecontribution at ω = 0) is calculated by using the Kubo

Figure 6: Sketch of a MoS2 or WS2 sheet with defect-clusters.Top left: R = a with cluster centers on Mo (MoS6 defects);top right: R = a with cluster centers on S (Mo3S2 defects);bottom left: R = a with cluster centers on Mo and S (MoS6and Mo3S2 defects); bottom right: random defect-clusters(R < 3.5a), as described in the text. For illustrative pur-poses, the size of the sample shown in this sketch is 50× 100(considerably smaller than the sizes used in our simulations),and the concentration of defects is approximately equal to1%.

formula [3, 4]

σαβ (ω) = limǫ→0+

e−βω − 1

ωΩ

∫ ∞

0

e−ǫt sinωt

×2 Im 〈ϕ|f (H)Jα (t) [1− f (H)] Jβ |ϕ〉 dt,(4)

where β = 1/kBT is the inverse temperature, Ω is the

sample area, f (H) = 1/[

eβ(H−µ) + 1]

is the Fermi-

Dirac distribution operator, and the time-dependent cur-rent operator in the α (= x or y) direction is Jα (t) =eiHtJαe

−iHt.The DC conductivity at zero temperature is calculated

by using the Kubo formula at ω → 0 [3, 4]

σαα =ρ (ε)

Ω

∫ ∞

0

dt Re[

e−iǫt 〈ϕ| JαeiHtJα |ε〉]

,

(5)

where |ε〉 is the normalized quasi-eigensate [8]. The accu-racy of TBPM is mainly determined by the time intervaland total time steps used in the Fourier transformation.The main limitation of the numerical calculations is thesize of the physical memory that can be used to store thequasi-eigenstates |Φ (E)〉 in the calculation of DC con-ductivity. In the present work, we have fixed the tem-perature to T = 300 K for the optical conductivity and

7

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.00

0.01

0.02

0.03

0.04

0.05

xS=0.1%

DO

S (1

/eV

)

E (eV)

Mo 3d3z2-r2

Mo 3dx2-y2

Mo 3dxy

S 3px,A

S 3py,A

S 3pz,S

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.00

0.01

0.02

0.03

0.04

0.05

DO

S (1

/eV

)

E (eV)

Mo 3d3z2-r2

Mo 3dx2-y2

Mo 3dxy

S 3px,A

S 3py,A

S 3pz,S

xMo=0.1%

Figure 7: Orbital resolved DOS for MoS2 with 0.1% of Sdefects (top panel) or Mo defects (bottom panel). See alsoFig. 3 of the main text.

to T = 0 for the DC conductivity. We study systemscontaining 2400 × 2400 atoms, with periodic boundaryconditions.

Orbital DOS— Here we complement the informationfor the localized states around the impurities shown inFig. 3 of the main text, and we show in Fig. 7 thecontribution of each orbital to the DOS of MoS2 with0.1% of S or Mo defects.

Dependence of the conductivity on the concentration of

defects— In Fig. 8 we show results similar to Fig. 1 ofthe main text, but comparing different concentrations ofdefects. The height of the peaks in the DOS in the mid-dle of the gap, associated to localized states around theimpurities, increases with disorder, resulting in a largerbackground contribution to the optical conductivity forenergies lower than the gap.The mobility of the samples decreases with the con-

centration of defects, as it is shown in Fig. 9 for sampleswith clusters of defects, as stated in the figure.

-1.0 -0.5 0.0 0.5 1.00.0

0.1

0.2

0.3

0.4

0.5

DOS(1/eV

)

E (eV)

MoS2

Mo defect

Pristine xMo+S=0.1% xMo+S=0.2% xMo+S=0.5% xMo+S=1%S defect

(a)

1.2 1.4 1.6 1.8 2.0 2.2 2.40.0

0.5

1.0

1.5

2.0

2.5

3.0

/0

(eV)

Pristine xMo+S=0.1% xMo+S=0.2% xMo+S=0.5% xMo+S=1%

MoS2

(b)

-1.0 -0.5 0.0 0.5 1.00.0

0.1

0.2

0.3

DOS(1/eV

)

E (eV)

WS2

W defect

Pristine xW+S=0.1% xW+S=0.2% xW+S=0.5% xW+S=1%

S defect

(c)

1.2 1.4 1.6 1.8 2.0 2.2 2.40.0

0.5

1.0

1.5

2.0

2.5

3.0

/0

(eV)

Pristine xW+S=0.1% xW+S=0.2% xW+S=0.5% xW+S=1%

(d)

WS2

Figure 8: Comparison of DOS and optical conductivity ofMoS2 (a) and WS2 (b) for different concentrations of defects.

-5 0 5 10 150

200

400

600

800

1000

1200

(cm

2 V-1s-1)

ne (1013cm2)

xCluster_Mo+S=0.1% xCluster_Mo+S=1%

R=aMoS2

-5 0 5 10 150

100

200

300

400

500

600

700

(cm

2 V-1s-1)

ne (1013cm2)

xCluster=1% xCluster=5%

R<3.5aMoS2

Figure 9: Comparison of mobility of MoS2 for different con-centrations of cluster defects: (a) cluster centers located onMo or S with R = a, and (b) random clusters with R < 3.5a.

Low energy model for the DC conductivity— The nu-merical results shown in Fig. 2 for the DC conductivitycan be complemented with a low energy approximationin which we can calculate the conductivity using the Tmatrix [5], which accounts for the scattering of electronsby resonant impurities

T (E) =V 2

E − ǫd − V 2g0(E)(6)

8

where V is the potential accounting for the impurity, andg0(E) is the local unperturbed Green’s function, whichfor a semiconductor with electron-hole asymmetry can beobtained from a density of states of the form N0(E) =DcΘ(E−∆/2)+DvΘ(−∆/2−E), from which we obtain

g0(E) = Dc log

∣

∣

∣

∣

E −∆/2

E −Wc

∣

∣

∣

∣

+Dv log

∣

∣

∣

∣

E +Wv

E +∆/2

∣

∣

∣

∣

(7)

where Wc(v) accounts for the width of the conduction(valence) bands. The case of interest here, which is de-fects, can be considered by the limit V → ∞ whichleads to T → −1/g0(E). From this the conductivitycan be calculated from σ = (2e2/h)EF τ where τ−1 =(2π/~)ni|T (E)|2N0(EF ) is the scattering rate in termsof the concentration of impurities ni.

∗ Electronic address: s.yuan@science.ru.nl† Electronic address: rroldan@icmm.csic.es

[1] R. Roldan, M. P. Lopez-Sancho, E. Cappelluti, J. A. Silva-Guillen, P. Ordejon, and F. Guinea, ArXiv e-prints (2014),

1401.5009.[2] A. Hams and H. De Raedt, Phys. Rev. E 62, 4365 (2000).[3] S. Yuan, H. De Raedt, and M. I. Katsnelson, Phys. Rev.

B 82, 115448 (2010).[4] A. Ishihara, Statistical Physics (Academic Press, New

York, 1971).[5] T. Wehling, M. Katsnelson, and A. Lichtenstein, Chemical

Physics Letters 476, 125 (2009).[6] S. Yuan, R. Roldan, H. De Raedt, and M. I. Katsnelson,

Phys. Rev. B 84, 195418 (2011).[7] S. Yuan, T. O. Wehling, A. I. Lichtenstein, and M. I. Kat-

snelson, Phys. Rev. Lett. 109, 156601 (2012).[8] Following Ref. [3], the normalized quasi-eigenstate |ε〉 =

|Φ (ε)〉 / |〈ϕ|Φ (ε)〉| is defined from the quasi-eigenstate|Φ (E)〉, which is a superposition of the degenerateeigenstates with the same eigenenergy E, obtainedas the Fourier transform of |ϕ(t)〉, i.e. |Φ(E)〉 =1

2π

∫∞

−∞dteiEt |ϕ (t)〉 . The quasi-eigenstate is not exactly

an energy eigenstate, unless the corresponding eigenstateis not degenerate at energy E. The average of the realspace distribution of the amplitude for different realizationof intial states can be used to examine the localization ofthe modes [3, 6, 7].

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

DOS(1/eV

)

E (eV)

Pristine xCluster=1% xCluster=5%

MoS2

R<3.5a