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12

First-principles study of the phonon-limited mobility in n-type single-layer MoS2

Kristen Kaasbjerg,∗ Kristian S. Thygesen, and Karsten W. JacobsenCenter for Atomic-scale Materials Design (CAMD),

Department of Physics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark(Dated: January 26, 2012)

In the present work we calculate the phonon-limited mobility in intrinsic n-type single-layer MoS2

as a function of carrier density and temperature for T > 100 K. Using a first-principles approachfor the calculation of the electron-phonon interaction, the deformation potentials and Frohlich in-teraction in the isolated MoS2 layer are determined. We find that the calculated room-temperaturemobility of ∼ 410 cm2 V−1 s−1 is dominated by optical phonon scattering via deformation potentialcouplings and the Frohlich interaction with the deformation potentials to the intravalley homopo-lar and intervalley longitudinal optical phonons given by 4.1 × 108 eV/cm and 2.6 × 108 eV/cm,respectively. The mobility is weakly dependent on the carrier density and follows a µ ∼ T−γ tem-perature dependence with γ = 1.69 at room temperature. It is shown that a quenching of thecharacteristic homopolar mode which is likely to occur in top-gated samples, boosts the mobilitywith ∼ 70 cm2 V−1 s−1 and can be observed as a decrease in the exponent to γ = 1.52. Ourfindings indicate that the intrinsic phonon-limited mobility is approached in samples where a high-κdielectric that effectively screens charge impurities is used as gate oxide.

PACS numbers: 81.05.Hd, 72.10.-d, 72.20.-i, 72.80.Jc

I. INTRODUCTION

Alongside with the rise of graphene and the explo-ration of its unique electronic properties,1–3 the searchfor other two-dimensional (2D) materials with promis-ing electronic properties has gained increased interest.4

Metal dichalcogenides which are layered materials similarto graphite, provide interesting candidates. Their layeredstructure with weak interlayer van der Waals bonds al-lows for fabrication of single- to few-layer samples usingmechanical peeling/cleavage or chemical exfoliation tech-niques similar to the fabrication of graphene.5–8 How-ever, in contrast to graphene they are semiconductorsand hence come with a naturally occurring band gap—aproperty essential for electronic applications.

The vibrational and optical properties of single- to few-layer samples of MoS2 have recently been studied ex-tensively using various experimental techniques.9–11 Incontrast to the bulk material which has an indirect-gap,it has been demonstrated that single-layer MoS2 is adirect-gap semiconductor with a gap of 1.8 eV.10,12 To-gether with the excellent electrostatic control inherentof two-dimensional materials, the large band gap makesit well-suited for low power applications.13 So far, elec-trical characterizations of single-layer MoS2 have shownn-type conductivity with room-temperature mobilities inthe range 0.5 − 3 cm2 V−1 s−1.5,6,8 Compared to earlystudies of the intralayer mobility of bulk MoS2 wheremobilities in the range 100 − 260 cm2 V−1 s−1 werereported,14 this is rather low. In a recent experimentthe use of a high-κ gate dielectric in a top-gated devicewas shown to boost the carrier mobility to a value of200 cm2 V−1 s−1.8 The observed increase in the mobilitywas attributed to screening of impurities by the high-κdielectric and/or modifications of MoS2 phonons in thetop-gated sample.

In order to shed further light on the measured mobili-ties and the possible role of phonon damping due to a top-gate dielectric,8 we have carried out a detailed study ofthe phonon-limited mobility in single-layer MoS2. Sincephonon scattering is an intrinsic scattering mechanismthat often dominates at room temperature, the theo-retically predicted phonon-limited mobility sets an up-per limit for the experimentally achievable mobilities insingle-layer MoS2.

15 In experimental situations, the tem-perature dependence of the mobility can be useful for theidentification of dominating scattering mechanisms andalso help to establish the value of the intrinsic electron-phonon couplings.16 However, the existence of additionalextrinsic scattering mechanisms often complicates the in-terpretation. For example, in graphene samples impu-rity scattering and scattering on surface polar opticalphonons of the gate oxide are important mobility-limiting

aa

S

Mo

-0.6 -0.3 0.0 0.3 0.6kx [2�/a]

-0.6

-0.3

0.0

0.3

0.6

ky [2�

/a]

K K�

0.00

0.15

0.30

0.45

eV

FIG. 1: (Color online) Atomic structure and conduction bandof single-layer MoS2. Left: Primitive unit cell and structureof an MoS2 layer with the molybdenum and sulfur atoms po-sitioned in displaced hexagonal layers. Right: Contour plotshowing the lowest lying conduction band valleys as obtainedwith DFT-LDA in the hexagonal Brillouin zone of single-layerMoS2. For n-type MoS2, the low-field mobility is determinedby the carrier properties in the K,K′-valleys which are sepa-rated in energy from the satellite valleys inside the Brillouinzone.

2

factors.17–22 It can therefore be difficult to establish thevalue and nature of the intrinsic electron-phonon interac-tion from experiments alone. Theoretical studies there-fore provide useful information for the interpretation ofexperimentally measured mobilities.In the present work, the electron-phonon interaction in

single-layer MoS2 is calculated from first-principles usinga density-functional based approach. From the calcu-lated electron-phonon couplings, the acoustic (ADP) andoptical deformation potentials (ODP) and the Frohlichinteraction in single-layer MoS2 are inferred. Usingthese as inputs in the Boltzmann equation, the phonon-limited mobility is calculated in the high-temperatureregime (T > 100 K) where phonon scattering often dom-inate the mobility. Our findings demonstrate that thecalculated phonon-limited room-temperature mobility of∼ 410 cm2 V−1 s−1 is dominated by optical deformationpotential and polar optical scattering via the Frohlichinteraction. The dominating deformation potentials ofD0

HP = 4.1 × 108 eV/cm and D0LO = 2.6 × 108 eV/cm

originate from the couplings to the homopolar and theintervalley polar LO phonon. This is in contrast to pre-vious studies for bulk MoS2, where the mobility has beenassumed to be dominated entirely by scattering on thehomopolar mode.14,23,24 Furthermore, we show that aquenching of the characteristic homopolar mode which ispolarized in the direction normal to the MoS2 layer canbe observed as a change in the exponent γ of the generictemperature dependence µ ∼ T−γ of the mobility. Such aquenching can be expected to occur in top-gated sampleswhere the MoS2 layer is sandwiched between a substrateand a gate oxide.The paper is organized as follows. In Section II the

band structure and phonon dispersion of single-layerMoS2 as obtained with DFT-LDA are presented. The lat-ter is required for the calculation of the electron-phononcoupling presented in Section III. From the calculatedelectron-phonon couplings, we extract zero- and first-order deformation potentials for both intra and inter-valley phonons. In addition, the coupling constant forthe Frohlich interaction with the polar LO phonon is de-termined. In Section IV, the calculation of the phonon-limited mobility within the Boltzmann equation is out-lined. This includes a detailed treatment of the phononcollision integral from which the scattering rates for thedifferent coupling mechanisms can be extracted. In Sec-tion V we present the calculated mobilities as a functionof carrier density and temperature. Finally, in Section VIwe summarize and discuss our findings.

II. ELECTRONIC STRUCTURE AND PHONON

DISPERSION

The electronic structure and phonon dispersion ofsingle-layer MoS2 have been calculated within DFT inthe LDA approximation using a real-space projector-augmented wave (PAW) method.25–27 The equilibrium

M � K M

�4

�2

0

2

4

Energ

y (eV

)

� K M0.8

1.0

1.2

1.4

1.6

FIG. 2: (Color online) Band structure of single-layer MoS2.The inset shows a zoom of the conduction band along the pathΓ-K-M. The parabolic band (red dashed line) with effectivemass m∗ = 0.48 me is seen to give a perfect description ofthe conduction band valley in the K-point for the range ofenergies relevant for the low-field mobility.

lattice constant of the hexagonal unit cell shown in Fig. 1was found to be a = 3.14 A using a 11 × 11 k-pointsampling of the Brillouin zone. In order to eliminate in-teractions between the layers due to periodic boundaryconditions, a large interlayer distance of 10 A has beenused in the calculations.

A. Band structure

In the present study, the band structure of single-layerMoS2 is calculated within DFT-LDA.28,29 While DFT-LDA in general underestimates band gaps, the resultingdispersion of individual bands, i.e. effective masses andenergy differences between valleys, is less problematic.We note, however, that recent GW quasi-particle cal-culations which in general give a better description ofthe band structure in standard semiconductors,52 sug-gest that the distance between the K,K ′-valleys and thesatellite valleys located on the Γ-K path inside the Bril-louin zone not as large as predicted by DFT (see below)and that the valley ordering may even be inverted.53,54

This, however, seems to contradict with the experimen-tal consensus that single-layer MoS2 is a direct-gap semi-conductor10,12 and further studies are needed to clarifythis issue. In the following we therefore use the DFT-LDA band structure and comment on the possible con-sequences of a valley inversion in Section VI.The calculated DFT-LDA band structure is shown in

Fig. 2. It predicts a direct band gap of 1.8 eV at theK-poing in the Brillouin zone. The inset shows a zoomof the bottom of the conduction band along the Γ-K-M path in the Brillouin zone. As demonstrated by thedashed line, the conduction band is perfectly parabolic inthe K-valley. Furthermore, the satellite valley positioned

3

on the path between the Γ-point and the K-point lies onthe order of ∼ 200 meV above the conduction band edgeand is therefore not relevant for the low-field transport.The right plot in Fig. 1 shows a contour plot of the low-

est lying conduction band valleys in the two-dimensionalBrillouin zone. Here, the K,K ′-valleys that are popu-lated in n-type MoS2, are positioned at the corners ofthe hexagonal Brillouin zone. Due to their isotropic andparabolic nature, the part of the conduction band rele-vant for the low-field mobility can to a good approxima-tion be described by simple parabolic bands

εk =~2k2

2m∗(1)

with an effective electron mass of m∗ = 0.48 me andwhere k is measured with respect to the K,K ′-pointsin the Brillouin zone. The two-dimensional nature of thecarriers is reflected in the constant density of states givenby ρ0 = gsgvm

∗/2π~2 where gs = 2 and gv = 2 are thespin andK,K ′-valley degeneracy, respectively. The largedensity of states which follows from the high effectivemass of the conduction band in the K,K ′-valleys, resultsin non-degenerate carrier distributions except for veryhigh carrier densities.

B. Phonon dispersion

The phonon dispersion has been obtained with thesupercell-based small-displacement method30 using a 9×9 supercell. The resulting phonon dispersion shown inFig. 3 is in excellent agreement with recent calculationsof the lattice dynamics in two-dimensional MoS2.

31

With three atoms in the unit cell, single-layer MoS2has nine phonon branches—three acoustic and six opti-cal branches. Of the three acoustic branches, the fre-quency of the out-of-plane flexural mode is quadratic inq for q → 0. In the long-wavelength limit, the frequencyof the remaining transverse acoustic (TA) and longitudi-nal acoustic (LA) modes are given by the in-plane soundvelocity cλ as

ωqλ = cλq. (2)

Here, the sound velocity is found to be 4.4 × 103 and6.5× 103 m/s for the TA and LA mode, respectively.The gap in the phonon dispersion completely sepa-

rates the acoustic and optical branches even at the high-symmetry points at the zone-boundary where the acous-tic and optical modes become similar. The two lowestoptical branches belong to the non-polar optical modes.Due to an insignificant coupling to the charge carriers,they are not relevant for the present study.The next two branches with a phonon energy of ∼

48 meV at the Γ-point are the transverse (TO) and lon-gitudinal (LO) polar optical modes where the Mo and Satoms vibrate in counterphase. In bulk polar materials,the coupling of the lattice to the macroscopic polariza-tion setup by the lattice vibration of the polar LO mode

� K M �0

10

20

30

40

50

60

Frequency

(meV

)

FIG. 3: Phonon dispersion of single-layer MoS2 calculatedwith the small displacement method (see e.g. Ref.30) usinga 9 × 9 supercell. The frequencies of the two optical Ramanactive E2g and A1g modes at 48 meV and 50 meV, respec-tively, are in excellent agreement with recent experimentalmeasurements.9

results in the so-called LO-TO splitting between the twomodes in the long-wavelength limit. The inclusion ofthis effect from first-principles requires knowledge of theBorn effective charges.32–34 In two-dimensional materials,however, the lack of periodicity in the direction perpen-dicular to the layer removes the LO-TO splitting.35 Thecoupling to the macroscopic polarization of the LO modewill therefore be neglected here.

The almost dispersionless phonon at ∼ 50 meV is theso-called homopolar mode which is characteristic for lay-ered structures. The lattice vibration of this mode cor-responds to a change in the layer thickness and has thesulfur layers vibrating in counterphase in the directionnormal to the layer plane while the Mo layer remainsstationary. The change in the potential associated withthis lattice vibration has previously been demonstratedto result in a large deformation potential in bulk MoS2.

14

III. ELECTRON-PHONON COUPLING

In this section, we present the electron-phonon cou-pling in single-layer MoS2 obtained with a first-principlesbased DFT approach. The method is based on a su-percell approach analogous to that used for the calcula-tion of the phonon dispersion and is outlined in App. A.The calculated electron-phonon couplings are discussedin the context of deformation potential couplings and theFrohlich interaction well-known from the semiconductorliterature.

Within the adiabatic approximation for the electron-phonon interaction, the coupling strength for the phonon

4

mode with wave vector q and branch index λ is given by

gλkq =

√

~

2MNωqλMλ

kq, (3)

where ωqλ is the phonon frequency,M is an appropriatelydefined effective mass, N is the number of unit cells inthe crystal, and

Mλkq = 〈k+ q|δVqλ(r)|k〉 (4)

is the coupling matrix element where k is the wave vectorof the carrier being scattered and δVqλ is the change inthe effective potential per unit displacement along thevibrational normal mode.Due to the valley degeneracy in the conduction band,

both intra and intervalley phonon scattering of the carri-ers in the K,K ′-valleys need to be considered. Here, thecoupling constants for these scattering processes are ap-proximated by the electron-phonon coupling at the bot-tom of the valleys, i.e. with k = K,K′. With this ap-proach, the intra and intervalley scattering for the K,K ′-valleys are thus assumed independent on the wave vectorof the carriers.In the following two sections, the calculated electron-

phonon couplings are presented.36 The different deforma-tion potential couplings are discussed and the functionalform of the Frohlich interaction in 2D materials is es-tablished. Piezoelectric coupling to the acoustic phononswhich occurs in materials without inversion symmetry,is most important at low temperatures37 and will notbe considered here. As deformation potentials are of-ten extracted as empirical parameters from experimentalmobilities, it can be difficult to disentangle contributionsfrom different phonons. Here, the first-principles calcula-tion of the electron-phonon coupling allows for a detailedanalysis of the couplings and to assign deformation po-tentials to the individual intra and intervalley phonons.

A. Deformation potentials

The deformation potential interaction describes howcarriers interact with the local changes in the crystal po-tential associated with a lattice vibration. Within the de-formation potential approximation, the electron-phononcoupling is expressed as38

gqλ =

√

~

2AρωqλMqλ, (5)

where A is the area of the sample, ρ is the atomic massdensity per area and Mqλ is the coupling matrix elementfor a given valley which is assumed independent on thek-vector of the carriers. This expression follows fromthe general definition of the electron-phonon coupling inEq. 3 by setting the effective massM equal to the sum ofthe atomic masses in the unit cell. With this convention

for the effective mass, MN = Aρ, and the expression inEq. (5) is obtained.For scattering on acoustic phonons, the coupling ma-

trix element is linear in q in the long-wavelength limit,

Mqλ = Ξλq, (6)

where Ξλ is the acoustic deformation potential.In the case of optical phonon scattering, both coupling

to zero- and first-order in q must be considered. The in-teraction via the constant zero-order optical deformationpotential D0

λ is given by

Mqλ = D0λ. (7)

The coupling via the zero-order deformation potential isdictated by selection rules for the coupling matrix ele-ments. Therefore, only symmetry-allowed phonons cancouple to the carriers via the zero-order interaction. Thecoupling via the first-order interaction is given by Eq. (6)with the acoustic deformation potential replaced by thefirst-order optical deformation potential D1

λ. Both thezero- and first-order deformation potential coupling cangive rise to intra and intervalley scattering.The absolute value of the calculated coupling matrix el-

ements |Mqλ| are shown in Fig. 4 for phonons which cou-ple to the carriers via deformation potential interactions.They are plotted in the range of phonon wave vectors rel-evant for intra and intervalley scattering processes,39 andonly phonon modes with significant coupling strengthshave been included. Although the coupling matrix ele-ments are shown only for k = K, the matrix elementsfor k = K′ are related through time-reversal symmetryas MK

qλ = MK′

−qλ. The three-fold rotational symmetryof the coupling matrix elements in q-space stems fromthe symmetry of the conduction band in the vicinity ofthe K,K ′-valleys (see Fig. 1). Since the symmetry of thematrix elements has not been imposed by hand, slightdeviations from the three-fold rotational symmetry canbe observed.The acoustic deformation potential couplings for the

TA and LA modes are shown in the two top plots ofFig. 4. Due to the inclusion of Umklapp processes here,the coupling to the TA mode does not vanish as is mostoften assumed.40 Only along high-symmetry directions inthe Brillouin zone is this the case. This results in a highlyanisotropic coupling to the TA mode. On the other hand,the deformation potential coupling for the LA mode isperfectly isotropic in the long-wavelength limit but alsobecomes anisotropic at shorter wavelengths. In agree-ment with Eq. (6), both the TA and LA coupling matrixelements are linear in q in the long-wavelength limit.The next two plots show the couplings to the intraval-

ley polar TO and homopolar optical modes. While theinteraction with the polar TO phonon corresponds toa first-order optical deformation potential, the interac-tion with the homopolar mode acquires a finite value of∼ 4.8 eV/A in the Γ-point and corresponds to a strongzero-order deformation potential coupling. The large

5

K K�Intravalley

K K�Intervalley

FIG. 4: (Color online) Deformation potential couplings insingle-layer MoS2. Only phonon modes with significant cou-pling strength are shown. The contour plots show the absolutevalue of the coupling matrix elements |Mqλ| in Eq. (4) withk = K and k′ = K+ q as a function of the two-dimensionalphonon wave vector q (note the different color scales in theplots). The upper four plots correspond to intravalley scat-tering and the two bottom plots to intervalley scattering onphonons with wave vector q = 2K. The two zero-order defor-mation potential couplings to the intravalley homopolar andintervalley LO phonon play an important role for the room-temperature mobility.

deformation potential coupling to the homopolar modestems from its characteristic lattice vibration polarizedin the direction perpendicular to the layer correspondingto a change in the layer thickness. The associated changein the effective potential from the counterphase oscilla-tion of the two negatively charged sulfur layers resultsin a significant change of the potential towards the cen-ter of the MoS2 layer. As the electronic Bloch functions

have significant weight here, the homopolar lattice vi-bration gives rise to a significant shift of the K,K ′-valleystates. The large deformation potential associated withthe coupling to the homopolar mode is also present inbulk MoS2.

14

The two bottom plots in Fig. 4 show the couplingsfor the intervalley TA and polar LO phonons. Due tothe nearly constant frequency of the intervalley acousticphonons, their couplings are classified as optical defor-mation potentials in the following. Both the shown cou-pling to the intervalley TA phonon and the coupling tothe intervalley LA phonon result in first-order deforma-tion potentials. The intervalley coupling for the polarLO phonon gives rise to a zero-order deformation poten-tial which results from an identical in-plane motion of thetwo sulfur layers in the lattice vibration of the intervalleyphonon.In general, the deformation potential approximation,

i.e. the assumption of isotropic and constant/linear cou-pling matrix elements, in Eqs. (6) and (7) is seen tohold in the long-wavelength limit only. At shorter wave-lengths, the first-principles couplings become anisotropicand have a more complicated q-dependence. When de-termined experimentally from e.g. the temperature de-pendence of the mobility,16 the deformation potentialsin Eqs. (6) and (7) implicitly account for the more com-plex q-dependence of the true coupling matrix element.In Section V, the theoretical deformation potentials forsingle-layer MoS2 are determined from the first-principleselectron-phonon couplings. This is done by fitting theirassociated scattering rates to the scattering rates ob-tained with the first-principles coupling matrix elements.The resulting the deformation potentials, can be used inpractical transport calculations based on e.g. the Boltz-mann equation or Monte Carlo simulations. It shouldbe noted that similar routes for first-principles calcula-tions of deformation potentials have been given in theliterature.41–43

B. Frohlich interaction

The lattice vibration of the polar LO phonon givesrise to a macroscopic electric field that couples to thecharge carriers. For bulk three-dimensional systems, thecoupling to the field is given by the Frohlich interactionwhich diverges as 1/q in the long-wavelength limit,40

gLO(q) =1

q

√

e2~ωLO

2ǫ0V

(

1

ε∞− 1

ε0

)1/2

, (8)

where ǫ0 is the vacuum permittivity, V is the volume ofthe sample , and ε∞ and ε0 are the high-frequency opticaland static dielectric constant, respectively.In atomically thin materials, the two-dimensional na-

ture of both the LO phonon and the charge carriers leadsto a qualitatively different q-dependence of the Frohlich

6

K0

25

50

75

100

g LO(q)

(meV

)

First-principlesAnalytic

FIG. 5: (Color online) Frohlich interaction for the polar opti-cal LO mode. The dashed line shows the analytical couplingin Eq. (9) with the coupling constant gFr = 98 meV andthe effective layer thickness σ = 4.41 A fitted to the long-wavelength limit of the calculated coupling (dots).

interaction. The situation is similar to 2D semiconduc-tor heterostructures where the Frohlich interaction hasbeen studied using dielectric continuum models and mi-croscopic based approaches.44,45 From the microscopicconsiderations presented in App. B, we derive the fol-lowing functional form for the Frohlich interaction to thepolar LO phonon in 2D materials,

gLO(q) = gFr × erfc(qσ/2). (9)

Here, the coupling constant gFr is the equivalent of thesquare root factors in Eq. (8), σ is the effective width ofthe electronic Bloch states and erfc is the complementaryerror function.Figure 5 shows the first-principles coupling gLO to the

polar LO phonon in single-layer MoS2 (dots) as a func-tion of the phonon wave vector along the Γ-K path. Thedashed line shows a fit of the coupling in Eq. (9) to thelong-wavelength limit of the calculated coupling. With acoupling constant of gFr = 98 meV and an effective widthof σ = 4.41 A, the analytic form gives a perfect descrip-tion of the calculated coupling in the long-wavelengthlimit. For shorter wavelengths, the zero-order deforma-tion potential coupling to the intervalley LO phonon fromFig. 4 becomes dominant, and a deviation away from thebehavior in Eq. (9) is observed.

IV. BOLTZMANN EQUATION

In the regime of diffuse transport dominated by phononscattering, the mobility can be obtained with semiclas-sical Boltzmann transport theory. In the absence ofspatial gradients the Boltzmann equation for the out-of-equilibrium distribution function fk of the charge carriersreads46

∂fk∂t

+ k · ∇kfk =∂fk∂t

∣

∣

∣

∣

coll

, (10)

where the time evolution of the crystal momentum is gov-erned by ~k = qE and q is the charge of the carriers. Inthe case of a time-independent uniform electric field, thesteady-state version of the linearized Boltzmann equationtakes the form

q

~E · ∇kf

0k = − q

kBTE · vkf

0k(1− f0

k)

=∂fk∂t

∣

∣

∣

∣

coll

, (11)

where f0k = f0(εk) is the equilibrium Fermi-Dirac dis-

tribution function, ∂f0k/∂εk = −f0

k(1 − f0k)/kBT , and

vk = ∇kεk/~ is the band velocity. In linear response,the out-of-equilibrium distribution function can be writ-ten as the equilibrium function plus a small deviation δfaway from its equilibrium value,46 i.e.

δfk = fk − f0k = f0

k(1− f0k)ψk. (12)

Here, the last equality has defined the deviation func-tion ψk. Furthermore, following the spirit of the itera-tive method of Rode,47 the angular dependence of thedeviation function is separated out as

ψk = φk cos θk, (13)

where θk is the angle between the k-vector and the forceexerted on the carriers by the applied field E. In gen-eral, φk is a function of the k-vector and not only itsmagnitude k. However, for isotropic bands as consideredhere, the angular dependence of the deviation function isentirely accounted for by the cosine factor in Eq. (13).38

Considering the phonon scattering, the collision inte-gral describes how accelerated carriers are driven backtowards their equilibrium distribution by scattering onacoustic and optical phonons. In the high-temperatureregime of interest here, the collision integral can be splitup in two contributions. The first, accounting for thequasielastic scattering on acoustic phonons, can be ex-pressed in the form of a relaxation time τel. The second,describing the inelastic scattering on optical phonons,will in general be an integral operator Iinel. The colli-sion integral can thus be written

∂fk∂t

∣

∣

∣

∣

coll

= −δfkτelk

+ Iinel [ψk] , (14)

where the explicit forms of the relaxation time and theintegral operator will be considered in detail below.With the above form of the collision integral, the Boltz-

mann equation can be solved by iterating the equation

φk =

[

eEvkkBT

+ I ′inel [φk]

]

×(

1

τelk

)−1

(15)

for the deviation function φk. Here, the angular depen-dence cos θk and a f0

k(1 − f0k) factor have been divided

out and the new integral operator I ′inel is defined by

Iinel = f0k(1− f0

k) cos θkI′inel. (16)

7

From the solution of the Boltzmann equation the cur-rent and drift mobility of the carriers can be obtained.Taking the electric field to be oriented along the x-direction, the current density is given by

jx ≡ σxxEx = −4e

A

∑

k

vxδfk, (17)

where σxx is the conductivity, A is the area of the sample,and vi = ~ki/m

∗ is the band velocity for parabolic bandswith i = x, y, z. From the definition µxx = σxx/ne of thedrift mobility, it follows that

µxx =e〈φk〉m∗

(18)

where the modified deviation function φk having units oftime is defined by

φk =m∗kBT

eEx~kφk, (19)

and the energy-weighted average 〈·〉 is defined by

〈A〉 = 1

n

∫

dεk ρ(εk)εkAk

(

−∂f0

∂εk

)

. (20)

Here, n is the two-dimensional carrier density. In the re-laxation time approximation φk = τk and the well-knownDrude expression for the mobility µ = e〈τk〉/m∗ is recov-ered.

A. Phonon collision integral

The phonon collision integral has been considered ingreat detail in the literature for two-dimensional elec-tron gases in semiconductor heterostructure (see e.g.Ref.37). In general, these treatments consider scatteringon three-dimensional or quasi two-dimensional phonons.In atomically thin materials the phonons are strictly two-dimensional which results in a slightly different treat-ment. In the following, a full account of the two-dimensional phonon collision integral is given for scat-tering on acoustic and optical phonons.

With the distribution function written on the formin Eq. (12), the linearized collision integral for electron-phonon scattering takes the form46

∂fk∂t

∣

∣

∣

∣

coll

= −2π

~

∑

qλ

|gqλ|2ǫ2(q, T )

×[

f0k

(

1− f0k+q

)

N0qλ (ψk − ψk+q) δ(εk+q − εk − ~ωqλ)

+ f0k

(

1− f0k−q

) (

1 +N0qλ

)

(ψk − ψk−q) δ(εk−q − εk + ~ωqλ)

]

(21)

where N0qλ = N0(~ωqλ) is the equilibrium distribution

of the phonons given by the Bose-Einstein distributionfunction and f0

k±q = f0(εk ± ~ωq) is understood. Thedifferent terms inside the square brackets account forscattering out of (ψk) and into (ψk±q) the state withwave vector k via absorption and emission of phonons.Screening of the electron-phonon interaction by the car-riers themselves is accounted for by the static dielectricfunction ǫ, which is both wave vector q and tempera-ture T dependent. Due to the large effective mass ofthe conduction band, the charge carriers in single-layerMoS2 are non-degenerate except at very high carrier con-centrations. Semiclassical screening where the screeninglength is given by the inverse of the Debye-Huckel wavevector kD = e2n/2kBT therefore applies.48 For the con-sidered values of carrier densities and temperatures, thiscorresponds to a small fraction of the size of the Bril-louin zone, i.e. kD ≪ 2π/a. As scattering on phononsin general involves larger wave vectors, screening by thecarriers can to a good approximation be neglected here.The calculated mobilities therefore provide a lower limit

for the proper screened mobility.

1. Quasielastic scattering on acoustic phonons

For quasielastic scattering on acoustic phonons at hightemperatures, the energy of the acoustic phonon can beneglected in the collision integral in Eq. (21). As a con-sequence, the collision integral can be recast in the formof the following relaxation time37

1

τelk=∑

k′

(1− cos θkk′)Pkk′ (22)

where the summation variable has been changed to k′ =k ± q and the transition matrix element for quasielasticscattering on acoustic phonons is given by

Pkk′ =2π

~

∑

q

|gqλ|2[

N0qλδ(εk′ − εk)

+(

1 +N0qλ

)

δ(εk′ − εk)

]

. (23)

8

For isotropic scattering as assumed here in Eq. (6), thesquare of the electron-phonon coupling can be expressedas

|gqλ|2 =Ξ2λ~q

2Aρcλ, (24)

where the acoustic phonon frequency has been expressedin terms of the sound velocity cλ. Except at very low tem-peratures ~ωq ≪ kBT implying that the equipartition ap-proximation N0

q ∼ kBT/~ωq ≫ 1 for the Bose-Einsteindistribution applies. With the resulting q-factors inthe transition matrix element in Eq. (23) canceling, thecos θkk′ in the k′-sum in Eq. (22) vanishes and the firstterm yields a factor density of states divided by the spinand valley degeneracies ρ0/gsgv. The relaxation time foracoustic phonon scattering becomes

1

τelk=m∗Ξ2kBT

~3ρc2λ. (25)

The independence on the carrier energy and the τ−1 ∼ Ttemperature dependence of the acoustic scattering rateare characteristic for charge carriers in two-dimensionalheterostructures and layered materials.14,37

2. Inelastic scattering on dispersionless optical phonons

For inelastic scattering on optical phonons the phononenergy can no longer be neglected and the collision in-tegral in Eq. (21) must be considered in full detail. Un-der the reasonable assumption of dispersionless opticalphonons ωqλ = ωλ, the inelastic collision integral can betreated semianalytically. The overall procedure for theevaluation of the collision integral is given below, whilethe calculational details have been collected in App. C.The resulting expressions for the collision integral applyto both intravalley and intervalley optical and intervalleyacoustic phonons.In the following, the integral operator for inelastic scat-

tering in Eq. (16) is split up in separate out- and in-scattering contributions,

I ′inel[φk] = I ′outinel[φk] + I ′

ininel[φk], (26)

which include the terms in Eq. (21) involving ψk andψk±q, respectively. With the contributions from differ-ent phonon branches λ adding up, scattering on a singlephonon with branch index λ is considered in the follow-ing.From Eq. (21), the out-scattering part of the collision

integral follows directly as

I ′outinel[φk] = −φk

∑

k′

Pkk′

1− f0k′

1− f0k

, (27)

where the transition matrix element for optical phonon

scattering is given by

Pkk′ =2π

~

∑

q

|gqλ|2 ×[

N0λδ(εk′ − εk − ~ωλ)

+(

1 +N0λ

)

δ(εk′ − εk + ~ωλ)

]

.

(28)

For the in-scattering part, the desired cos θk factor inEq. (16) can be extracted from ψk′ using the rela-tion cos θk′ = cos θk cos θkk′ − sin θk sin θkk′ . Since thesine term vanishes from symmetry consideration, the in-scattering part of the inelastic collision integral reducesto

I ′ininel[φk] =

∑

k′

φk′ cos θkk′Pkk′

1− f0k′

1− f0k

. (29)

The evaluation of the k′-sum in Eqs. (27) and (29) isoutlined in App. C. Here, the assumption of dispersion-less optical phonons allows for an semianalytical treat-ment. For zero- and first-order coupling within the defor-mation potential approximation, the sum can be carriedout analytically. The resulting expressions for the colli-sion integral are given in Eqs. (C14), (C15), (C16), (C17)and (C18). In the case of the Frohlich interaction, theangular part of the q-integral must be done numerically.

3. Optical deformation potential scattering rates

In spite of the fact that the collision integral for in-elastic scattering on optical phonons cannot be recast inthe form of a (momentum) relaxation time,49 a scatter-ing rate related to the inverse carrier lifetime can stillbe defined from the out-scattering part of the inelasticcollision integral alone. The scattering rate so defined isgiven by

1

τ inelk

=∑

k′

Pkk′

1− f0k′

1− f0k

(30)

and corresponds to the imaginary part of the elec-tronic self-energy in the Born approximation.46 Below,the resulting scattering rates for zero-order and first-order deformation potential scattering are given for non-degenerate carriers, i.e. with the Fermi factors in Eq. (30)neglected. They follow straight-forwardly from the ex-pressions for the out-scattering part of the collision in-tegral derived in App. C. It should be noted that thescattering rate for the zero-order deformation poten-tial interaction given below in Eq. (31), in fact definesa proper momentum relaxation time because the in-scattering part of the collision integral vanishes in thiscase (see App. C).For zero-order deformation potential scattering, the

scattering rate is independent of the carrier energy and

9

given by

1

τ inelk

=m∗(D0

λ)2

2~2ρωλ

[

1 + e~ωλ/kBTΘ(εk − ~ωλ)

]

N0λ. (31)

Here, Θ(x) is the Heavyside step function which assuresthat only electrons with sufficient energy can emit aphonon.The scattering rate for coupling via the first-order de-

formation potential is found to be

1

τ inelk

=m∗2(D1

λ)2

~4ρωλ

[

(2εk + ~ωλ) + e~ωλ/kBT

×Θ(εk − ~ωλ) (2εk − ~ωλ)

]

N0λ.

(32)

Due to the linear dependence on the carrier energy, zero-order scattering processes dominate first-order processesat low energies. Only under high-field conditions wherethe carriers are accelerated to high velocities will first-order scattering become significant.The expressions for the scattering rates in Eqs. (31)

and (32) above apply to scattering on dispersionless in-tervalley acoustic phonons and intra/intervalley opticalphonons. Except for a factor of

√εk ± ~ωλ originat-

ing from the density of states, the energy dependenceof the scattering rates is identical to that of their three-dimensional analogs.38

V. RESULTS

In the following, the scattering rate and phonon-limited mobility in single-layer MoS2 are studied as afunction of carrier energy, temperature T and carrier den-sity n using the material parameters collected in Tab. I.Here, the reported deformation potentials represent ef-fective coupling parameters for the deformation potentialapproximation in Eqs. (6) and (7) (see below).

A. Scattering rates

With access to the first-principles electron-phononcouplings, the scattering rate taking into account theanisotropy and more complex q-dependence of the first-principles coupling matrix elements can be evaluated.They are obtained using the expression for the scatteringrate in Eq. (30) with the respective transition matrix el-ements for acoustic and optical phonon scattering givenin Eqs. (23) and (28).50 In order to account for the dif-ferent coupling matrix elements in the K,K ′-valleys, thescattering rates have been averaged over different high-symmetry directions of the carrier wave vector k. Theresulting scattering rates are shown in Fig. 6 as a func-tion of the carrier energy for non-degenerate carriers atT = 300 K. The different lines show the contributions to

the total scattering rate from the various electron-phononcouplings to the intra and intervalley phonons which havebeen grouped according to their coupling type, i.e. acous-tic deformation potentials (ADPs), zero/first-order opti-cal deformation potentials (ODPs) and the Frohlich in-teraction. The acoustic deformation potential scatter-ing includes the quasielastic intravalley scattering on theTA and LA phonons with linear dispersions. Scatter-ing on intervalley acoustic phonons is considered as op-tical deformation potential scattering. Both the totalscattering rate and the contributions from the differentcoupling types have been obtained using Matthiessen’srule by summing the scattering rates from the individualphonons, i.e.

τ−1tot =

∑

λ

τ−1λ . (33)

For carrier energies below the optical phonon frequen-cies, the total scattering rate is dominated by acoustic de-formation potential scattering. At higher energies zero-order deformation potential scattering and polar opticalscattering via the Frohlich interaction become dominant.Due to the linear dependence on the carrier energy, thefirst-order deformation potential scattering on the inter-valley acoustic phonons and the optical phonons is in

Parameter Symbol Value

Lattice constant a 3.14 A

Ion mass density ρ 3.1× 10−7 g/cm2

Effective electron mass m∗ 0.48 me

Transverse sound velocity cTA 4.2 × 103 m/s

Longitudinal sound velocity cLA 6.7 × 103 m/s

Acoustic deformation potentials

TA ΞTA 1.6 eV

LA ΞLA 2.8 eV

Optical deformation potentials

TA D1

K,TA 5.9 eV

LA D1

K,LA 3.9 eV

TO D1

Γ,TO 4.0 eV

TO D1

K,TO 1.9 eV

LO D0

K,LO 2.6× 108 eV/cm

Homopolar D0

Γ,HP 4.1× 108 eV/cm

Frohlich interaction (LO)

Effective layer thickness σ 4.41 A

Coupling constant gFr 98 meV

Optical phonon energies

Polar LO ~ωLO 48 meV

Homopolar ~ωHP 50 meV

TABLE I: Material parameters for single-layer MoS2. Un-less otherwise stated, the parameters have been calculatedfrom first-principles as described in the Secs. II and III. TheΓ/K-subscript on the optical deformation potentials, indicatecouplings to the intra/intervalley phonons.

10

0.00 0.05 0.10 0.15 0.20 0.25Energy (eV)

1010

1011

1012

1013

1014

Sca

tteri

ng r

ate

(s

1)

T=300 K

ADPZero-order ODPFirst-order ODPFr�hlichTotalFitted

FIG. 6: (Color online) Scattering rates for the differentelectron-phonon couplings as a function of carrier energy atT = 300 K. The scattering rates have been calculated fromthe first-principles electron-phonon couplings in Figs. 4 and 5.The (black) dashed line shows the scattering rate obtainedusing the fitted deformation potential parameters defined inEqs. (6) and (7). The kinks in the curves for the optical scat-tering rates mark the onset of optical phonon emissions.

general only of minor importance for the low-field mo-bility. This is also the case here, where it is an orderof magnitude smaller than the other scattering rates foralmost the entire plotted energy range. The jumps inthe curves for the optical scattering rates at the opticalphonon energies ~ωλ are associated with the thresholdfor optical phonon emission where the carriers have suf-ficient energy to emit optical phonons

B. Deformation potentials

In this section, we determine the deformation potentialparameters in single-layer MoS2. They can be used inthe following study of the low-field mobility within theBoltzmann approach outlined in Section IV.The energy dependence of the first-principles based

scattering rates in Fig. 6 to a high degree resembles thatof the analytic expressions for the deformation poten-tial scattering rates in Eqs. (25), (31) and (32). Forexample, the acoustic and zero-order deformation po-tential scattering rates are almost constant in the plot-ted energy range. The first-principles electron-phononcouplings can therefore to a good approximation be de-scribed by the simpler isotropic deformation potentialsin Eqs. (6) and (7). The deformation potentials are ob-tained by fitting the associated scattering rates for eachof the intra and intervalley phonons separately to thefirst-principles scattering rates. The resulting deforma-tion potential values are summarized in Tab. I. In analogywith deformation potentials extracted from experimentalmobilities, the theoretical deformation potentials repre-sent effective coupling parameters that implicitly account

for the anisotropy and the full q-dependence of the first-principles electron-phonon couplings. However, as mo-mentum and energy conservation limit phonon scatteringto involve phonons in the vicinity of the Γ/K-point,39

the fitting procedure yields deformation potentials closeto the direction averaged q → Γ/K limiting behaviorof the first-principles coupling matrix elements. For thezero-order deformation potentials, the sampling of thecoupling matrix elements away from the Γ-point in thek′ sum in Eq. (30), leads to a deformation potentialswhich is slightly smaller than the Γ-point value of thecoupling matrix element.The total scattering rate resulting from the fitted de-

formation potentials is shown in Fig. 6 to give an al-most excellent description of the first-principles scatter-ing rate. With the mobility in the relaxation time ap-proximation given by the energy-weighted average of therelaxation time (see Eq. (18)), the difference between theassociated mobilities will be negligible. Hence, the defor-mation potentials provide well-founded electron-phononcoupling parameters for low-field studies of the mobility.

C. Mobility

The high density of states in the K,K ′-valleys of theconduction band in general results in non-degenerate car-rier distributions in single-layer MoS2. This is illustratedin the left plot of Fig. 7 which shows the carrier densityversus the position of the Fermi level for different temper-atures. At room temperature, carrier densities in excessof ∼ 8 × 1012 cm−2 are needed to introduce the Fermilevel into the conduction band and probe the Fermi-Diracstatistics of the carriers. Thus, only at the highest re-ported carrier densities of n ∼ 1013 cm−2,5 are the carri-ers degenerate at room temperature. For the lowest tem-perature T = 100 K, the transition to degenerate carriersoccurs at a carrier density of ∼ 2×1012 cm−2. The tran-sition from non-degenerate to degenerate distributions isalso illustrated by the discrepancy between the full anddashed lines which shows the Fermi level obtained withBoltzmann statistics.The right plot of Fig. 7 shows the phonon-limited drift

mobility calculated with the full collision integral as afunction of carrier density for the same set of tempera-tures. The dashed lines represent the results obtainedwith Boltzmann statistics and the relaxation time ap-proximation using the expressions in Section IV for opti-cal phonon scattering and Matthiessen’s rule for the to-tal relaxation time. The strong drop in the mobility from∼ 2450 cm2 V−1 s−1 at T = 100 K to ∼ 400 cm2 V−1 s−1

at T = 300 K, is a consequence of the increased phononscattering at higher temperatures due to larger phononpopulation of, in particular, optical phonons. The rela-tively low intrinsic room-temperature mobility of single-layer MoS2 can be attributed to both the significantphonon scattering and the large effective mass of 0.48me in the conduction band. While the mobility decreases

11

�0.10 �0.05 0.00 0.05 0.10 F�Ec (eV)

1011

1012

1013

1014C

arr

ier

densi

ty (cm

�

2)

T=100 KT=200 KT=300 K

1011 1012 1013

Carrier density (cm�2 )

0

500

1000

1500

2000

2500

3000

Mobili

ty (cm

2V

�

1s�

1)

T=100 KT=200 KT=300 K

FIG. 7: (Color online) Left: Carrier density as a functionof the Fermi energy for different temperatures. The Fermienergy is measured relative to the conduction band edge Ec

Right: Mobility as a function of carrier density for the sametemperatures. In both plots, the (black) dashed lines show theresults obtained with Boltzmann statistics and the relaxationtime approximation (only right plot).

strongly with increasing temperature, it is relatively inde-pendent on the carrier density. The weak density depen-dence of the mobility originates from the energy-weightedaverage in Eq. (20) where the derivative of the Fermi-Dirac distribution changes slowly as a function of carrierdensity for non-degenerate carriers. As the Fermi energyis introduced into the band with increasing carrier den-sity, the derivative of the Fermi-Dirac distribution to alarger extent probes the scattering rate at higher energiesleading to a decrease of the mobility. This effect is mostprominent at T = 100 K where the level of degeneracy islarger compared to higher temperatures.Surprisingly, the relaxation time approximation is seen

to work extremely well. The deviation from the full treat-ment at high carrier densities stems from the assumptionof non-degenerate carriers and not the relaxation timeapproximation. The reason for the good performanceof the relaxation time approximation shall be found inthe in-scattering part of the collision integral in the fulltreatment. As the in-scattering part of the collision in-tegral vanishes for zero-order deformation potential cou-pling and is small compared to the out-scattering partotherwise, the full collision integral does not differ signif-icantly from the corresponding scattering rate as definedin Eq. (30) of Section IV, i.e. with the in-scattering partneglected. The good performance of the relaxation timeapproximation therefore seems to be of general validityfor the phonon collision integral even in the presence in-elastic scattering on optical phonons.Finally, we study the temperature dependence of the

mobility in more detail. In general, room temperaturemobilities are to a large extent dominated by opticalphonon scattering. This is manifested in the temperaturedependence of the mobility which follows a µ ∼ T−γ lawwhere the exponent γ depends on the dominating scat-tering mechanism. For acoustic phonon scattering abovethe Bloch-Gruneisen temperature, the temperature de-pendence of the scattering rate in Eq. (25) results inγ = 1. At higher temperatures where optical phononscattering starts to dominate, the mobility acquires a

100 200 300 400 500Temperature (K)

102

103

104

Mobili

ty (cm

2V

�

1s�

1)

n=1011 cm�2

AcousticTotalQuenched

FIG. 8: (Color online) Mobility vs temperature. For compari-son, the mobility in the presence of only acoustic deformationpotential scattering with the µ ∼ T−1 temperature depen-dence is shown. The (gray) shaded area shows the variationin the mobility associated with a 10% uncertainty in the cal-culated deformation potentials.

stronger temperature dependence with γ > 1. In thisregime, the exponent depends on the optical phonon fre-quencies and the electron-phonon coupling strength. Inan early study of the in-plane mobility of bulk MoS2,

14

the measured room-temperature exponent of γ ∼ 2.6 wasfound to be consistent with scattering on the homopolarmode via a zero-order deformation potential.In Fig. 8, we show the temperature dependence of the

mobility at n = 1011 cm−2 calculated with the full col-lision integral. For comparison, the mobility limited byacoustic phonon scattering with the µ ∼ T−1 tempera-ture dependence is also included. To illustrate the effectof an uncertainty in the calculated deformation poten-tials, the shaded area shows the variation of the mo-bility with a change in the deformation potentials of±10%. The calculated room-temperature mobility of∼ 410 cm2 V−1 s−1 is in fair agreement with the recentlyreported experimental value of ∼ 200 cm2 V−1 s−1 inthe top-gated sample of Ref.8 where additional scatteringmechanisms as e.g. impurity and surface-optical phononscattering must be expected to exist. Over the plottedtemperature range, the mobility undergoes a transitionfrom being dominated by acoustic phonon scattering atT = 100 K to being dominated by optical phonon scat-tering at higher temperatures with a characteristic ex-ponent of γ > 1. At room temperature, the mobilityfollows a temperature dependence with γ = 1.69. Theincrease in the exponent is almost exclusively due to theoptical zero-order deformation potential couplings andthe Frohlich interaction while the first-order deformationpotential couplings contribute only marginally. The ex-ponent found here is considerably lower than the above-mentioned exponent of γ ∼ 2.6 for bulk MoS2, indicat-ing that the electron-phonon coupling in bulk and single-layer MoS2 differ. Indeed, the transition from an indirect

12

band gap in bulk MoS2 to a direct band gap in single-layer MoS2 which shifts the bottom of the conductionband from the valley located along the Γ-K path to theK,K ′-valleys, could result in a change in the electron-phonon coupling.In top-gated samples as the one studied in Ref.8, the

sandwiched structure with the MoS2 layer located be-tween substrate and gate dielectric, is likely to result in aquenching of the characteristic homopolar mode which ispolarized in the direction normal to the layer. To addressthe consequence of such a quenching, the mobility in theabsence of the zero-order deformation potential originat-ing from the coupling to homopolar mode is also shownin Fig. 8. Here, the curve for the quenched case shows adecrease in the characteristic exponent to γ = 1.52 andan increase in the mobility of ∼ 70 cm2 V−1 s−1 at roomtemperature. Despite the significant deformation poten-tial of the homopolar mode, the effect of the quenchingon the mobility is minor.

VI. DISCUSSIONS AND CONCLUSIONS

Based on our finding for the phonon-limited mobility insingle-layer MoS2, it seems likely that the low experimen-tal mobilities of ∼ 1 cm2 V−1 s−1 reported in Refs.5,6,8are dominated by other scattering mechanisms as e.g.charged impurity scattering. The main reason for the in-crease in the mobility to 200 cm2 V−1 s−1 observed whendepositing a high-κ dielectric on top of the MoS2 layerin Ref.8 is therefore likely to be impurity screening. Thequenching of the out-of-plane homopolar phonon only ledto a minor increase in the mobility and cannot alone ex-plain the above-mentioned increase in the mobility. Itmay, however, also contribute. A comparison of the tem-perature dependence of the mobility in samples with andwithout the top-gate structure can help to clarify the ex-tent to which a quenching of optical phonons contributesto the experimentally observed mobility increase. Withour theoretically predicted room-temperature mobility of410 cm2 V−1 s−1, the observed enhancement in the mo-bility to 200 cm2 V−1 s−1 induced by the high-κ dielec-tric, suggests that dielectric engineering51 is an effectiveroute towards phonon-limited mobilities in 2D materialsvia efficient screening of charge impurities.A rough estimate of the impurity concentrations re-

quired to dominate phonon scattering can be inferredfrom the phonon scattering rate of τ−1 ∼ 1013 s−1 inFig. 6. The scattering rate is related to the mean-freepath λ of the carriers via λ = τ〈v〉 where 〈v〉 is their meanvelocity. Using the velocity of the mean energy carriersfor a non-degenerate distribution where 〈εk〉 = kBT , wefind a mean-free path of λ ∼ 14 nm at T = 300 K. Inorder for impurity scattering to dominate, the impurityspacing must be on the order of the phonon mean-freepath or smaller. This results in a minimum impurityconcentration of ∼ 5 × 1011 cm−2 in order to dominatephonon scattering. The high value of the estimated im-

purity concentration needed to dominate phonon scat-tering is in agreement with the experimental observationthat low-mobility single-layer MoS2 samples are heavilydoped semiconductors.5

As discussed in Section II, independent GW quasi-particle calculations53,54 suggest that the ordering of thevalleys at the bottom of the conduction band might notbe as clear as predicted by DFT. If this is the case, thesatellite valleys inside the Brillouin zone must also betaken into account in the solution of Boltzmann equa-tion. This gives rise to additional intervalley scatteringchannels that together with the larger average effectiveelectron mass m∗ ∼ 0.6 me of the satellite valley willresult in mobilities below the values predicted here.

In conclusion, we have used a first-principles basedapproach to establish the strength and nature of theelectron-phonon coupling and calculate the intrinsicphonon-limited mobility in single-layer MoS2. The cal-culated room-temperature mobility of 410 cm2 V−1 s−1

is to a large extent dominated by optical deformationpotential scattering on the intravalley homopolar andintervalley LO phonons as well as polar optical scat-tering on the intravalley LO phonon via the Frohlichinteraction. The mobility follows a µ ∼ T−1.69 tem-perature dependence at room temperature characteris-tic of optical phonon scattering. A quenching of thehomopolar mode likely to occur in top-gated samples,results in a change of the exponent in the tempera-ture dependence of the mobility to 1.52. With effectivemasses, phonons and measured mobilities in other semi-conducting metal dichalcogenides being similar to thoseof MoS2,

14,23 room-temperature mobilities of the sameorder of magnitude can be expected in their single-layerforms.

Acknowledgments

The authors would like to thank O. Hansen and J.J. Mortensen for illuminating discussions. KK has beenpartially supported by the Center on Nanostructuringfor Efficient Energy Conversion (CNEEC) at StanfordUniversity, an Energy Frontier Research Center fundedby the U.S. Department of Energy, Office of Science,Office of Basic Energy Sciences under Award Numberde-sc0001060. CAMD is supported by the LundbeckFoundation.

Appendix A: First-principles calculation of the

electron-phonon interaction

The first-principles scheme for the calculation of theelectron-phonon interaction used in this work is outlinedin the following. Contrary to other first-principles ap-proaches for the calculation of the electron-phonon inter-action which are based on pseudo potentials,41–43,55 the

13

present approach is based on the PAW method56 and isimplemented in the GPAW DFT package.25–27

Within the adiabatic approximation for the electron-phonon interaction, the electron-phonon coupling matrixelements for the phonon qλ and Bloch state k can beexpressed as

gλkq =

√

~

2MNωqλ

∑

α

〈k+ q|eqλα · ∇qαV (r)|k〉, (A1)

where N is the number of unit cells, M is an appropri-ately defined effective mass, ωqλ is the phonon frequency,and the q-dependent derivative of the effective potentialfor a given atom α is defined by

∇qαV (r) =

∑

l

eiq·Rl∇αlV (r), (A2)

where the sum is over unit cells in the system and∇αlV isthe gradient of the potential with respect to the positionof atom α in cell Rl. The polarization vectors eqλ of thephonons appearing in Eq. (A1) are normalized accordingto∑

α(Mα/M)|eqλα |2 = 1.In order to evaluate the matrix element in Eq. (A1),

the Bloch states are expanded in LCAO basis orbitals|iRn〉 where i = (α, µ) is a composite index for atomicsite and orbital index and Rn is the lattice vector to then’th unit cell. In the LCAO basis, the Bloch states areexpanded as |k〉 =

∑

i cki |ik〉 where

|ik〉 = 1

N

∑

n

eik·Rn |iRn〉. (A3)

are Bloch sums of the localized orbitals. Inserting theBloch sum expansion of the Bloch states in the matrixelement in Eq. (A1), the matrix element can be written

〈k+ q|eqλ·∇qV (r)|k〉 =∑

ij

c∗i cj〈ik+ q|eqλ · ∇qV (r)|jk〉

=1

N2

∑

ij

c∗i cj∑

lmn

ei(k+G)·(Rn−Rm)−iq·(Rm−Rl)

× 〈iRm|eqλ · ∇lV (r)|jRn〉 (A4)

where the k-labels on the expansion coefficients havebeen discarded for brevity. The phase factor from the re-ciprocal lattice vector G can be neglected since Rn ·G =2πN . By exploiting the translational invariance of thecrystal the matrix elements can be obtained from thegradient in the primitive cell as

〈iRm|eqλ · ∇lV (r)|jRn〉= 〈iRm −Rl|eqλ · ∇0V (r)|jRn −Rl〉 (A5)

by performing the change of variables r′ = r − Rl inthe integration. Inserting in Eq. (A4) and changing thesumming variables to Rm −Rl and Rn −Rl we find forthe matrix element

〈k+ q|eqλ · ∇qV (r)|k〉 = 1

N2

∑

ij

c∗i cj∑

lmn

eik·(Rn−Rm)−iq·Rm〈iRm|eqλ · ∇0V (r)|jRn〉

=1

N

∑

ij

c∗i cj∑

mn

eik·(Rn−Rm)−iq·Rm〈iRm|eqλ · ∇0V (r)|jRn〉 (A6)

Here, the sum over k in the first equality produces afactor ofN . The result for the matrix element in Eq. (A6)is similar to the matrix element reported in Eq. (22) ofRef.55 where a Wannier basis was used instead of theLCAO basis.

From the last equality in Eq. (A6), the procedure fora supercell-based evaluation of the matrix element hasemerged. The matrix element in the last equality in-volves the gradients of the effective potential ∇0V withrespect to atomic displacement in the reference cell atR0.These can be obtained using a finite-difference approxi-mation for the gradient where the individual components

are obtained as

∂V

∂Rαi=V +αi(r)− V −

αi(r)

2δ. (A7)

Here, V ±αi denotes the effective potential with atom α

located at Rα in the reference cell displaced by ±δ indirection i = x, y, z. The calculation of the gradient thusamounts to carrying out self-consistent calculations forsix displacements of each atom in the primitive unit cell.Having obtained the gradients of the effective potential,the matrix elements in the LCAO basis of the super-cell must be calculated and the sums over unit cells andatomic orbitals in Eq. (A6) can be evaluated.In general, the matrix elements of the electron-phonon

interaction must be converged with respect to the super-

14

cell size and the LCAO basis. In particular, since thesupercell approach relies on the gradient of the effectivepotential going to zero at the supercell boundaries, thesupercell must be chosen large enough that the potentialat the boundaries is negligible. For polar materials wherethe coupling to the polar LO phonon is long-ranged innature, a correct description of the coupling can only beobtained for phonon wave vectors corresponding to wave-lengths smaller than the supercell size. Large supercellsare therefore required to capture the long-wavelengthlimit of the coupling constant (see e.g. main text).

1. PAW details

In the PAW method, the effective single-particle DFTHamiltonian is given by

H = −1

2∇2 + veff(r) +

∑

α

∑

i1i2

|pαi1〉∆Hαi1i2〈pαi2 |, (A8)

where the first term is the kinetic energy, veff is the effec-tive potential containing contributions from the atomicpotentials and the Hartree and exchange-correlation po-tentials and the last term is the non-local part includ-ing the atom-centered projector functions pαi (r) andthe atomic coefficients ∆Hα

i1i2. In contrast to pseudo-

potential methods where the atomic coefficients are con-stants, they depend on the density and thereby also onthe atomic positions in the PAW method.

The diagonal matrix elements (i.e. q = 0) in Eq. (A4)correspond to the first-order variation in the band en-ergy εk upon an atomic displacement in the normal modedirection eqλ. Together with the matrix elements forq 6= 0, these can be obtained from the gradient of thePAW Hamiltonian with respect to atomic displacements.The derivative of (A8) with respect to a displacement ofatom α in direction i results in the following four terms

∂H

∂Rαi=∂veff∂Rαi

+∑

i1,i2

[

|∂pαi1∂Rαi

〉∆Hαi1i2〈p

αi2 |

+ |pαi1〉∆Hαi1i2〈

∂pαi2∂Rαi

|

+ |pαi1〉∂∆Hα

i1i2

∂Rαi〈pαi2 |

]

. (A9)

While the gradient of the projector functions in thefirst and second terms inside the square brackets canbe evaluated analytically, the gradients of the effectivepotential and the projector coefficients in the last termare obtained using the finite-difference approximation inEq. (A7).

Once the gradient of the PAW Hamiltonian has beenobtained, the matrix elements in Eq. (A4) can be ob-tained. Under the assumption that the smooth pseudoBloch wave functions ψk from the PAW formalism is a

good approximation to the true quasi particle wave func-tion, the matrix element follows as

〈k+ q|eqλ · ∇qV (r)|k〉 = 〈ψk+q|eqλ · ∇qH |ψk〉, (A10)

where ∇qH is given in Eq. (A2). With the pseudo wavefunction expanded in an LCAO basis, the matrix elementis evaluated following Eq. (A6).In order to verify the calculated matrix elements, we

have carried out self-consistent calculations of the vari-ation in the band energies with respect to atomic dis-placements along the phonon normal mode. For thecoupling to the Γ-point homopolar phonon in single-layer MoS2, we find that the calculated matrix elementof 4.8 eV/A agrees with the self-consistently calculatedvalue to within 0.1 eV/A.As the matrix elements of the electron-phonon inter-

action are sensitive to the behavior of the potential andwave functions in the vicinity of the atomic cores, vari-ations in electron-phonon couplings obtained with dif-ferent pseudo-potential approximations can be expected.We have confirmed this via self-consistent calculations ofthe band energy variations, which showed that the cou-pling to the homopolar Γ-point phonon increases with0.3 eV/A compared to the PAW value when using norm-conserving HGH pseudo potentials.

Appendix B: Frohlich interaction in 2D materials

In the long-wavelength limit, the lattice vibration ofthe polar LO mode gives rise to a macroscopic polar-ization that couples to the charge carriers. In three-dimensional bulk systems, the coupling strength is givenby Eq. 8 which diverges as 1/q. Using dielectric contin-uum models, the coupling has been studied for confinedcarriers and LO phonons in semiconductor heterostruc-tures44,45 where a non-diverging interaction is found. Foratomically thin materials, however, macroscopic dielec-tric models are inappropriate. Using an approach basedon the atomic Born effective charges Z∗, a functionalform that fits the calculated values of the Frohlich inter-action in Fig. 5 is here derived.

1. Polarization field and potential

In two-dimensional materials, the polarization fromthe lattice vibration of the polar LO phonon is orientedalong the plane of the layer. It can be expressed in termsof the relative displacement uq of the unit cell atoms as

Pq(z) =Z∗

ε∞Auqfq(z), (B1)

where q is the two-dimensional phonon wave vector, ε∞is the optical dielectric constant, Z∗ is the Born effectivecharge of the atoms (here assumed to be the same for allatoms) and fq describes the profile of the polarization in

15

the direction perpendicular (here the z-direction) to thelayer. The associated polarization charge ρ = −∇ · Pis given by ρq = −iq · Pq. The resulting scalar poten-tial which couples to the carriers follows from Poisson’sequation. Fourier transforming in all three directions,Poisson’s equation takes the form

(q2 + k2)φq(k) = −i Z∗

ε∞Aq · uqfq(k), (B2)

where k is the Fourier variable in the direction perpen-dicular to the plane of the layer and q ‖ uq for the LOphonon.In 2D materials, the z-profile of the polarization field

can to a good approximation be described by a δ-function, i.e.

fq(z) = δ(z). (B3)

Inserting the Fourier transform fq(k) = 1 in Eq. (B2),we find for the potential

φq(z) = −iZ∗uqε∞A

∫

dk eikzfq(k)q

q2 + k2

= −iZ∗uqε∞A

e−q|z| (B4)

which in agreement with the findings of Refs.44,45 doesnot diverge in the long-wavelength limit.

2. Electron-phonon interaction

In three-dimensional bulk systems, the q-dependenceof the Frohlich interaction is given entirely by the 1/q-divergence of the potential associated with the latticevibration of the polar LO phonon. However, in two di-mensions, the interaction follows by integrating the po-tential with the square of the envelope function χ(z) ofthe electronic Bloch state. Hence, the Frohlich interac-tion is given by the matrix element

gLO(q) =

∫

dz χ∗(z)φq(z)χ(z). (B5)

For simplicity, we here assume a Gaussian profile for theenvelope function

χ(z) =1

π1/4√σe−z2/2σ2

, (B6)

where σ denotes the effective width of the electronicBloch function. With this approximation for the enve-lope function, the Frohlich interaction becomes

gLO(q) = gFr × eq2σ2/4erfc(qσ/2)

≈ gFr × erfc(qσ/2) (B7)

where gFr is the Frohlich coupling constant, erfc is thecomplementary error function and the last equality holdsin the long-wavelength limit where q−1 ≫ σ. As shownin Fig. 5 in the main text, this functional form for theFrohlich interaction gives a perfect fit to the calculatedelectron-phonon coupling for the polar LO mode.

Appendix C: Evaluation of the inelastic collision

integral

Following Eq. (26) in the main text, the inelastic colli-sion integral for scattering on optical phonons is split upin separate out- and in-scattering contributions,

I ′inel[φk] = I ′outinel[φk] + I ′

ininel[φk], (C1)

which are given by Eq. (27) and (29), respectively. Withthe assumption of dispersionless optical phonons, theFermi-Dirac and Bose-Einstein distribution functions donot depend on the phonon wave vector and can thus betaken outside the q-sum.The out-scattering part of the collision integral then

takes the form

I ′outinel[φk] = −2π

~

φk1− f0

k

×[

N0λ(1− f0

k+q)∑

q

|gq|2 δ(εk+q − εk − ~ωλ)

+ (1 +N0λ)(1 − f0

k−q)∑

q

|gq|2 δ(εk−q − εk + ~ωλ)

]

.

(C2)

For the in-scattering part, the additional cos θk,k±q fac-tor is rewritten as

cos θk,k±q =k ± q cos θk,q

k√

1± ~ωλ

εk

, (C3)

leading to the following general form of the in-scatteringpart of the collision integral

I ′ininel[φk] =

2π

~

1

1− f0k

×[

N0λ(1− f0

k+q)φk+q

k√

1 + ~ωλ

εk

×∑

q

|gq|2 (k + q cos θk,q)δ(εk+q − εk − ~ωλ)

+ (1 +N0λ)(1 − f0

k−q)φk−q

k√

1− ~ωλ

εk

×∑

q

|gq|2 (k − q cos θk,q)δ(εk−q − εk + ~ωλ)

]

.

(C4)

This depends on the deviation function in φk±q ≡ φ(εk±~ωλ) and thus couples the deviation function at the initialenergy εk of the carrier to that at εk ± ~ωλ. The twoterms inside the square brackets account for emission andabsorption out of the state k ± q and into the state k,respectively.

16

1. Integration over q

The in- and out-scattering contributions to the inelas-tic part of the collision integral in Eqs. (C2) and (C4)can be written on the general form

I ′in/outinel (k) =

∑

q

f(q)δ(εk±q − εk ∓ ~ωqλ), (C5)

where the function f accounts for the q-dependent func-tions inside the q-sum. The conservation of energy andmomentum in a scattering event is secured by the δ-function entering the collision integral.Following the procedure outlined in Ref.14, the integral

over q resulting from the q-sum in the collision integral isevaluated using the following property of the δ-function,

∫

dq F (q)δ [g(q)] =∑

n

F (qn)

|g′(qn)|. (C6)

Here, F (q) = qf(q) and qn are the roots of g. Rewritingthe argument of the δ-function in Eq. (C5) as

εk±q − εk ∓ ~ωqλ =~2(k± q)2

2m∗− ~

2k2

2m∗∓ ~ωqλ

=~2

2m∗

(

q2 ± 2kq cos θk,q ∓ 2m∗ωqλ

~

)

≡ ~2

2m∗g(q) (C7)

we get

I ′(k) =A

(2π)2

∫

dθk,q2m∗

~2

∑

n

qnf(qn)

|g′(qn)|. (C8)

Depending on the q-dependence of the phonon frequency,different solutions for the roots qn result.

a. Dispersionless optical phonons

With the assumption of dispersionless optical phonons,i.e. ωqλ = ωλ, the roots of g in Eq. (C7) become

q/k = ∓ cos θk,q ±√

cos2 θk,q ± ~ωλ

εk(C9)

for absorption (upper sign in the first and last term) andemission (lower sign in the first and last term), respec-tively.For absorption there is only one positive root qa+ given

by the plus sign in front of the square root,

qa+ = −k cos θk,q + k

√

cos2 θk,q +~ωλ

εk(C10)

where all values of θk,q are allowed. The absorption termsin the collision integral then becomes

I ′a(k) =A

(2π)2

∫ 2π

0

dθk,qm∗

k~2qa+f(q

a+)

√

cos2 θk,q + ~ωλ

εk

. (C11)

In the case of phonon emission, both possible roots inEq. (C9)

qe± = k cos θk,q ± k

√

cos2 θk,q − ~ωλ

εk(C12)

can take on positive values. However, the minussign inside the square root restricts the allowed val-ues of the integration angle to the range θk,q ≤ θ0 =

arccos (~ωλ/εk)1/2. Furthermore, in order to secure a

positive argument to the square root, the electron en-ergy must be larger than the phonon energy, εk > ~ωλ.Hence, the emission terms can be obtained as

I ′e(k) =A

(2π)2Θ(εk − ~ωλ)

×∫ θ0

−θ0

dθk,qm∗

k~2qe+f(q

e+) + qe−f(q

e−)

√

cos2 θk,q − ~ωλ

εk

. (C13)

2. Collision integral for optical phonon scattering

In the following two sections, analytic expressions forcollision integral in the case of zero- and first-order op-tical deformation potential coupling are given. For theFrohlich interaction the integration over θ can not be re-duced to a simple analytic form and is therefore evaluatednumerically.

a. Zero-order deformation potential

For the zero-order deformation potential coupling inEq. (7), the θ-integration in the out-scattering part ofthe collision integral yields a factor of 2π resulting in thefollowing form

I ′outinel[φk] = − m∗D2

λ

2~2ρωλ

φk1− f0

k

×[

(1− f0k+q) + (1− f0

k−q)e~ωλ/kBTΘ(εk − ~ωλ)

]

N0λ.

(C14)

For the in-scattering part, the q-independence of the de-formation potential results in a vanishing q-sum and thein-scattering contribution is zero, i.e.

I ′ininel[φk] = 0. (C15)

17

b. First-order deformation potential

For first-order deformation potentials, we find for theout-scattering part

I ′outinel[φk] = −m

∗2Ξ2λ

~4ρωλ

φk1− f0

k

×[

(1− f0k+q)(2εk + ~ωλ)

+ (1− f0k−q)(2εk − ~ωλ)e

~ωλ/kBTΘ(εk − ~ωλ)

]

N0λ.

(C16)

For the in-scattering part, the result for the first andsecond term inside the square brackets in Eq. (C4) is

I ′ininel[φk] = −m

∗2Ξ2λ

~4ρωλ

1− f0k+q

1− f0k

φk+q√

1 + ~ωλ

εk

N0λ(εk + ~ωλ)

(C17)

and

I ′ininel[φk] = −m

∗2Ξ2λ

~4ρωλ

1− f0k−q

1− f0k

φk−q√

1− ~ωλ

εk

× (1 +N0λ) (εk − ~ωλ)Θ(εk − ~ωλ)

× 1

π

(

π

2+ arccos

√

~ωλ

εk+ arcsin

√

~ωλ

εk

)

, (C18)

which accounts for absorption and emission, respectively.

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18

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