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Applied Surface Science 325 (2015) 27–32

Contents lists available at ScienceDirect

Applied Surface Science

jou rn al h om ep age: www.elsev ier .com/ locate /apsusc

hase transition, effective mass and carrier mobility of MoS2

onolayer under tensile strain

heng Yua,∗, Hao D. Xiongb,c,∗∗, Kwesi Eshuna, Hui Yuana,c, Qiliang Lia,∗

Department of Electrical and Computer Engineering, George Mason University, Fairfax, VA 22030, USADepartment of Electrical and Electronic Engineering, South University of Science and Technology of China, Shenzhen, ChinaSemiconductor and Dimensional Metrology Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

r t i c l e i n f o

rticle history:eceived 7 October 2014eceived in revised form5 November 2014ccepted 16 November 2014vailable online 22 November 2014

a b s t r a c t

We report a computational study on the impact of tensile strain on MoS2 monolayer. The transitionbetween direct and indirect bandgap structure and the transition between semiconductor and metalphases in the monolayer have been investigated with tensile strain along all direction configurationswith both x-axis and y-axis components εxy (εx and εy). Electron effective mass and the hole effectivemass are isotropic for biaxial strain εxy = εx = εy and anisotropic for εxy with εx /= εy. The carrier effectivemass behaves differently along different directions in response to the tensile strain. In addition, the

eywords:train effectwo-dimensional materialsoS2 monolayerobility enhancement

hase transition

impact of strain on carrier mobility has been studied by using the deformation potential theory. Theelectron mobility increases over 10 times with the biaxial strain: εx = εy = 9.5%. Also, the mobility decreasesmonotonically with the increasing temperature as � ∼ T−1. These results are very important for futurenanotechnology based on two-dimensional materials.

© 2014 Elsevier B.V. All rights reserved.

. Introduction

Recently, MoS2 monolayer, with layered honeycomb structuren which the transition metal atoms (Mo) are sandwiched betweenulfur (S) atoms, is of great interest because of its extraordinarylectronic, optical and mechanical properties [1,2]. These prop-rties benefit from the direct energy bandgap (Eg) in monolayeroS2 (Eg ≈ 1.8 eV). In comparison, bulk (or multilayer) MoS2 is

eld together by relatively weak van der Waals forces betweendjacent layers and has indirect energy bandgap (Eg ≈ 1.3 eV) [3].lso, two-dimensional (2D) MoS2 monolayer can be obtained fromulk MoS2 by using exfoliation technique which has been used tobtain graphene [2,4]. It has also been successfully demonstratedhat chemical vapor deposition (CVD) method can produce 2D

oS2 monolayer [5,6]. In addition, compared to the popular 2D

raphene, the existence of the native bandgap renders MoS2onolayer a promising candidate for field-effect transistors with

∗ Corresponding authors.∗∗ Corresponding author at: Department of Electrical and Electronic Engineering,outh University of Science and Technology of China, Shenzhen, China.

E-mail addresses: syu12@masonlive.gmu.edu (S. Yu),aoxiong941805@gmail.com (H.D. Xiong), qli6@gmu.edu (Q. Li).

ttp://dx.doi.org/10.1016/j.apsusc.2014.11.079169-4332/© 2014 Elsevier B.V. All rights reserved.

excellent current on/off ratio (≈108) [7], which may open a suiteof applications in logic and memory integrated circuits.

In experiment, tensile strain in MoS2 monolayer can be achievedby stretching or bending elastomeric substrate with external stress.Strain in MoS2 monolayer can be detected by Raman spectroscopyin which the E2g mode splits and shifts due to the strain [5–8]. Ithas been also demonstrated that strain can reduce photoluminesce-nce and tune the polarization associated with K–K direct bandgaptransition [8,9]. To date, many works have been reported on thesimulations and modeling about the transformation of energyband structure activated by strain [10,11]. The strain effects onelectronic, optical and magnetic properties have also been wellinvestigated and predicted by Mahapatra et al. [12,13]. It is widelyrecognized that tensile strain will cause reduction in Eg. Also, theconduction band minimum (CBM) and valence band maximum(VBM) will be overlapping each other at Fermi level under suffi-ciently large strain [10,14]. However, the distinct transition fromdirect to indirect bandgap structure and the transition from semi-conductor to metal phase have not yet been found. The dependenceof carrier effective mass on crystal direction has not been stud-ied. In addition, the low carrier mobility (�) is probably one of

the major barriers for the application of MoS2 based field-effecttransistor (FET) [15,16]. There have been found that the value of� can be enhanced to about 200 cm2 V−1 S−1 by the depositionof high-dielectric HfO2 layer [7]. In this work, we have studied

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8 S. Yu et al. / Applied Surf

he effect of tensile strain on the phase transitions, carrier effec-ive mass and mobility in MoS2 monolayer at room temperatureRT). The temperature depended mobility of electrons and holes arelso investigated. The results have quantitatively demonstrated thearrier mobility enhancement by using tensile strain, this suggest-ng an alternative to improve the electrical performance of MoS2

onolayer for device application.

. Methodology

In this study, first principle calculations were carried out bysing the Virtual Nanolab Atomistix ToolKit (ATK) package withensity functional theory (DFT) [17]. The localized density approx-

mation (LDA) exchange correlation with a double zeta polarizedDZP) basis is used with a mesh cut-off energy of 150 Ry [18]. Theemperature is set to 300 K for our simulations on the bandstruc-ure, charge carriers’ effective mass and their mobility under strainffect at RT. For the investigation on the temperature dependedobility, we set this value to 200 K, 300 K and 400 K, respec-

ively. A unit cell containing two Mo atoms and four S atoms washosen with a periodic boundary condition. We used 11 × 11 × 1onkhorst-Pack k-grid mesh in this simulation [19]. All atomic

ositions and lattice constants were optimized by using the gen-ralized gradient approximations (GGA) [20] with the maximumellmann-Feynman forces of 0.05 eV/Å. The Pulay-mixer algorithm

s employed as iteration control parameter with tolerance valuef 10−5. The maximum number of fully self-consistent field (SCF)teration steps is set to 100. The result of structural optimizationndicated that the simple orthorhombic lattice constant a0 = 5.47 A,0 = 3.16 A, and the thickness of MoS2 monolayer = 3.17 A. a0 and0 represent the lattice constant in armchair and zigzag directions,espectively. This optimized MoS2 monolayer structure is in a goodgreement with other publications [11,21].

. Results and discussion

At first, we studied the bandstructure of MoS2 monolayer foromparison. Fig. 1(a) shows the lattice structure of MoS2 mono-ayer where the orthorhombic supercell, the smallest rectangulartructure used for elastic constants calculation, is enclosed withinhe solid lines. The supercell is built in the orthorhombic way toemonstrate the carrier conduction along the [1,0] (armchair) and

0,1] (zigzag) directions. This has been shown to be more intuitivehan the hexagonal lattice [22]. The K point, which is defined ashe fractional reciprocal coordinates (1/3, 1/3) in hexagonal lat-ice, is equivalent to (1/3, 0), the midpoint between � and Y pointsor orthorhombic supercell [22] in this study. The bandstructure of

ig. 1. (a) Atomic structure of MoS2 monolayer: the rectangle supercell selected for the nub) The energy band structure of unstrained MoS2 monolayer. (For interpretation of the rhis article.)

ience 325 (2015) 27–32

unstrained MoS2 monolayer is shown in Fig. 1(b) with a bandgap of1.82 eV at RT, which is in a good agreement with the previous stud-ies [10,23] In addition, the valence band energy at � (0, 0) is slightlylower than the VBM at K(1/3, 0), indicating a direct bandgap at Kpoint.

Then, we investigated the electronic properties of MoS2monolayer under tensile strain for different in-plane directionconfigurations. The components of strain along [1,0] and [0,1] direc-tions are noted as εx and εy, respectively. In this work, the strainsare evaluated as the lattice stretching percentage. We definedεx �a0/a0 and εy �b0/b0, where �a0 and �b0 are the increaseof lattice constants a0 and b0, respectively, due to the tensile strain.A wide range of strain configurations (magnitude from 0 to 18%,in all directions) has been employed in the study. For the pur-pose of precision and computational efficiency, a small incrementof strain, �ε = 0.1%, has been used to study the direct–indirectbandgap transition, while a large �ε = 2% has been used to studythe semiconductor–metal transition. The exact transition bound-aries have been found by using Newton’s method with a precisionof 0.0075% in strain and 0.001 eV in Eg.

The tensile strain will induce direct to indirect bandgap transi-tion in MoS2 monolayer [24]. However, very few studies reportedthe exact transition boundary on this issue with detailed informa-tion, and there is a large difference among them. For example, thestudy by Li [25] reported that 1% uniaxial strain is insufficient forthis transition, while the study by Yun et al. [10] showed that thecritical point for the transition from direct to indirect bandgap is0.3% biaxial strain. In order to find out this transition, we carriedout an explicit study: tensile strain along different combinations of[1,0] (εx) and [0,1] (εy) directions were applied on a MoS2 mono-layer. Both εx and εy range from 0 to 0.7% with a step �ε = 0.1%, thistotally including 64 combinational configurations. In addition, theexact transition boundary is found by using Newton’s method witha smaller step. Fig. 2(a) shows the bandgap as a function of tensilestrain εxy (εx, εy). A clear boundary for the direct to indirect bandgaptransition is found to be a straight line: εxy with εx + εy ≈ 0.66%, wellcovering the result of the aforementioned study by Yun et al. whoreported that εx = εy ≈ 0.3% is the critical point for this transition[10].

For the study of semiconductor to metal transition, a widerange of strain configurations εxy (0 ≤ εx, εy ≤ 18%) with a step�ε = 2% along both [1,0] and [0,1] directions has been studied,this generating a 10 × 10 mesh grids. Similar to the study of directto indirect bandgap transition, the exact semiconductor to metal

transition is found by using Newton’s method with a smaller step.Fig. 2(b) shows the bandgap of MoS2 monolayer as a function ofεx and εy. The bandgap decreases monotonically with increasingapplied strain, and becomes zero when the edges of valence

merical simulation is within the red line. The arrows represent the lattice directions.eferences to color in this figure legend, the reader is referred to the web version of

S. Yu et al. / Applied Surface Sc

Fig. 2. The bandgap of MoS2 monolayer decreases with increasing strain: εx

(=�a0/a0) and εy (=�b0/b0): (a) under small values of strain εxy (εx ≤ 0.7%, εy ≤ 0.7%),(t(

aftttthttvr

b) under a large range of strain εxy (0 ≤ εx , εy ≤ 18%), and (c) under the rela-ively small values of uniaxial strain along x axis (εx ≤ 1.2%, εy = 0) and along y axisεy ≤ 1.2%, εx = 0).

nd conduction bands overlap the Fermi level at a straight lineollowing a formula εx + εy ≈ 19.06%. For the strain beyond this line,he MoS2 monolayer becomes metallic. This result also indicateshat the strains along [1,0] and [0,1] directions contribute equallyo the change in bandgap. Yun et al. obtained εx = εy ≈ 9.8% ashe point for semiconductor–metal transition by considering theexagonal lattice and biaxial stretch [10], which is approximately

he middle point (εx = εy ≈ 9.53%) of the straight line derived inhis study. In addition, the bandgap is reduced monotonically andery linearly with increasing strain in both the direct bandgapegion and the indirect bandgap regions at relatively small strain

ience 325 (2015) 27–32 29

(<1.2%) As shown in Fig. 2(c), the Eg decreasing rate in directbandgap region is ≈50.86 meV per 1% strain (meV/%) which is veryclose to the experimental result: 48 meV/% reported by Wang etal. [26]. However, the Eg decreasing rate in the indirect bandgapregion is 89.25 meV/%, indicating a faster decreasing rate thanthe rate in the direct bandgap region. Strong enhancement ofballistic conductance in 2D MoS2 structure can be achieved bysemiconductor–metal transition, which is advantageous for highlyefficient carrier transport in these semiconducting monolayers.Also, bandgap closing (i.e., semiconductor–metal transition) canbe realized by hydrogenation on the MoS2 monolayer surface,which introduces in-gap hybridized states [27].

Besides the bandgap, strain also changes the energy band cur-vature which is directly related to the carrier effective mass (m*):m* = �

2/(∂2E/∂k2), where � is the reduced Planck constant, E is theenergy and k is momentum. In this work, we have studied the elec-tron effective mass (m∗

e) at K(1/3, 0), the hole effective mass m∗h

atK(1/3, 0) and � (0, 0) of MoS2 monolayers under different strainsεxy (εx, εy) at RT. Fig. 3 shows the electron or hole effective mass atvarious tensile strain configurations: εx = 0, 4%, 8% and 12% with εy

varying from 0 to 12%. With the Atomistix ToolKit package in VirtualNanolab [17], we are able to study the effective mass along differ-ent directions in the 2D lattice. In this work, effective mass along[1,0] (armchair), [0,1] (zigzag) or [1,1] (armchair) direction has beencalculated. The m∗

e at K point and m∗h

at K point are ≈0.47 me and≈0.60 me at zero strain, respectively, agreeing with the previousstudies by other groups [10–12]. The m∗

hat � point is ≈2.70 me at

zero strain and decreases with strain, agreeing with Cai et al. [22].As shown in Fig. 3, the values of carrier effective mass along thesethree directions decrease or increase with tensile strain εxy (εx, εy)at a different rate. The anisotropic effective mass under strain willlead to anisotropic carrier mobility, which may lead to direction-dependent conductivity. In addition, we found that the values ofeffective mass along these three direction converges at a crossoverpoint at biaxial strain εx = εy.

The carrier effective mass at K and � points under the biax-ial strain is shown in Fig. 4(a). The biaxial strain applied on MoS2monolayer causes different responses from the m∗

hat � (0, 0), m∗

hand m∗

e at K(1/3, 0). The m∗h

at � (0, 0) decreases significantlywith increasing biaxial strain, while the m∗

e and m∗h

at K(1/3, 0)only change slightly. Our result shows that m∗

e ≈ 0.47 me and m∗h

≈0.60 me at zero strain. The localized atomic basis may have someunexpected inconsistency with the forces and structural relaxation.Thus we compare our results by ATK with other approaches: theeffective mass and bandgap transition simulated in our work atvarious values of strain are very close to the report using VASP [11]and the report using FLAPW [10]. Also, the study on the elastic mod-ulus shows that it does not exhibit significant change with strainor temperature, conforming to our assumption in the next section[28]. Therefore, the results of direct–indirect band gap transition inthis work are accurate [29].

The carrier mobility is a very important parameter to evaluatethe electrical properties of materials. Since the carrier effectivemass is affected by tensile strain, it is expected that the carriermobility will also be affected. Kaasbjerg et al. have clearly shownthat the carrier mobility in these 2D monolayers is dominated byacoustic phonon scattering via intra- and intervalley deformationpotential couplings at room temperature [30,31]. The mobility canbe figured out by the deformation potential (DP) theory raised byBardeen and Shockley [32]. Based on the effective mass approxi-mation, the DP theory determines the mobility as [33]:

� = 2e�3C

3KBT |m∗|2E21

(1)

30 S. Yu et al. / Applied Surface Science 325 (2015) 27–32

F % (from nsidet

wtctttCtmlAedsftia

F

ig. 3. The carrier effective mass as a function of strain εy for εx = 0%, 4%, 8% and 12ass at K(1/3, 0) (in the middle row) and � (0, 0) (in the bottom row) have been co

he MoS2 becomes metallic for the strain εxy with εy + εx > 19.06%.

here e is electron charge, m* is the carrier effective mass, KB ishe Boltzmann constant and T is the temperature, E1 is the DPonstant denoting the shift of the band edges (Eedge) (CBM for elec-rons and VBM for holes) induced by strain (ε): dEedge/dε, and C ishe elastic modulus of a uniformly deformed crystal activated byhe strain. For a 2D crystal, the in-plane elastic modulus (stiffness)

= [∂E2/∂ε2]/S0, where E is the total energy of supercell and S0 ishe area of the supercell. Yue et al. [11] determined that C of MoS2

onolayer is a constant: 123 N/m, which we have used in the calcu-ation of carrier mobility as a function of biaxial strain εxy = εx = εy.s displayed in Fig. 3, in the case of isotropic strain εxy (εx = εy), theffective mass will show the same value along different in-planeirections. The tensorial value of effective mass is degenerate intocalar. Therefore the calculation of mobility is the scalar product of

our variables (the effective mass of charge carriers, temperature,he DP constant and the elastic modulus) and accordingly obtain-ng a scalar result. In other words, when the MoS2 monolayer ispplied by the isotropic strain εxy (εx = εy), the mobility of charge

ig. 4. (a) The electron and hole effective mass at K and � points as a function of biaxial st

m left to right). The electron effective mass at K(1/3, 0) (in top row), hole effectivered. It should be noted that the simulated values of effective mass is precise when

carriers will be the same along various in-plane directions. In thisstudy, we only focus on this degenerate mobility under isotropicstrain εxy (εx = εy).

As shown in Fig. 4(b), electron mobility (�e) is significantlyimproved by the strain. The enhancement of electron mobilityis mainly attributed to the reduced value of DP constant as theincreasing strain. Larger strain slows down the changing rate ofconduction band minimum (CBM), thus E1 will reduce by theexpression: E1 = dEedge/dε. The hole mobility (�h) exhibits a largereduction from 194 cm2 V−1 s−1 to 18.4 cm2 V−1 s−1 when the biax-ial strain increases from 0% to 2%. This reduction is caused bythe strain-induced direct to indirect bandgap transition whereVB shifts from K(1/3, 0) to � (0, 0). Since the hole effective massabruptly jumps from 0.6 me to 1.95 me, accordingly the inversely

proportional mobility undergoes deep reduction from 0% to 2%strain. As the biaxial strain further increases, �h begins to increaseand reach 155 cm2 V−1 s−1 at εxy = εx = εy = 9.5%. In comparison,�e increases monotonically from 79 cm2 V−1 s−1 (unstrained) to

rain (εxy = εx = εy); (b) the electron and hole mobility at CBM and VBM, respectively.

S. Yu et al. / Applied Surface Science 325 (2015) 27–32 31

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Fig. 5. The comparison of the electron and hole m

25 cm2 V−1 s−1 at biaxial strain = 9.5%, about 10-time enhance-ent. This result indicates that the performance of monolayeroS2 transistors can be enhanced with tensile strain. In addi-

ion, our results, �e ≈ 79 cm2 V−1 s−1 and �h ≈ 194 cm2 V−1 s−1 atero strain, are very close to the results obtain by Cai et al.:e ≈ 72.16 cm2 V−1 s−1 and �h ≈ 200.52 cm2 V−1 s−1 [22]. It shoulde noted that the MoS2 monolayer remains stable under strain up to.5% (in our cases) because negative frequencies mode in phononispersion will not be created by the biaxial tensile strain <13%.owever, higher strain can still lead to an abrupt deformation in

he phonon band structure as well as a sudden change in lattice34].

We also give a comparison of the electron and hole mobility ofoS2 monolayer at various temperatures (200 K, 300 K and 400 K).

ince the electron velocity with energy kBT around room tempera-ure (from 200 K to 400 K) is in the order of magnitude of 107 cm s−1,nd according to � = h/mv, the corresponding wavelength possesses

value of several nanometers, which is much larger than the lat-ice constant, thus the electron is scattered mainly by the acoustichonons [33]. Therefore the aforementioned mathematical expres-ion of mobility, which takes the temperature, the elastic modulus,he DP constant and the effective mass of charge carriers intoccount, remains good validity for our investigated temperatures.ig. 5 displays this temperature depended carriers’ mobility undertrain. As shown they decrease monotonically as the temperaturesises up. Their altering trend under applied strain shows great sim-larity between different temperatures: the electron mobility isnhanced monotonically as the increasing strain strength (shownn Fig. 5(a)) whereas the hole mobility reaches minimal value at 2%train (shown in Fig. 5(b)). The electron mobility of MoS2 mono-ayer under 9.5% strain rises up over 1200 cm2 V−1 s−1 at 200 K.ur study indicates � ∼ T−1 of temperature dependence. The relax-tion time for acoustic phonon scattering is independent on thearrier energy and they exhibit temperature dependence �−1 ∼ T.herefore the mobility acquires a � ∼ T−1 dependence characteris-ic for two-dimensional systems and layered materials dominatedy acoustic phonon scattering [35,36]. It should be noted that theffective mass, deformation potential constant and elastic modulusn Eq. (1) has negligible temperature dependence. As reported byhao et al. [28], the stress vs. strain lines almost overlap each othernder strain less than 10% for 200 K, 300 K and 400 K. Therefore, thelastic modulus, which is determined by the mathematic expres-ion: C = d6/d∈, where 6 is the stress and ∈ is the strain, is almosthe same at different temperatures in our study.

. Conclusions

In summary, we have explicitly investigated the effect of tensiletrain on 2D MoS2 monolayer using first-principle calculation.

e have found that the bandgap is reduced monotonically and

y at different temperatures (200 K, 300 K, 400 K).

harmoniously by the increasing tensile strain. Two straight lines:εxy (εx, εy) with εx + εy = 0.66% and εx + εy = 19.06% were found asthe transition boundaries between the direct and indirect bandgapstructure and between the semiconductor and metal phases,respectively. The strains along [1,0] (armchair) and [0,1] (zigzag)directions have an equal contribution to bandgap modulation.Moreover, the values of carrier effective mass along differentdirections behave differently under tensile strain, although theyare equal to each other for biaxial strain εxy = εx = εy. In addition,we have found that carrier mobility can be enhanced significantlyby strain and lowering temperature. Overall, this study has made avery interesting observation with detailed information on the effectof tensile strain on carrier effective mass and mobility in MoS2monolayer, which is very helpful for future research on monolayerMoS2 electronics. The method and conclusion developed in thiswork can also benefit the research on other 2D TMDs and devices.

Acknowledgements

This work was supported in part by the U.S. NSF Grant ECCS-1407807 and in part by Virginia Microelectronics Consortiumresearch grant.

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