Charge Scattering and Mobility in Atomically Thin ... · thinnest possible manifestations of semiconductor materi-als that exhibit an energy band gap. For example, a single-layer

Charge Scattering and Mobility in Atomically Thin Semiconductors

Nan Ma* and Debdeep JenaDepartment of Electrical Engineering, University of Notre Dame, Notre Dame, Indiana 46556, USA

(Received 25 October 2013; revised manuscript received 16 February 2014; published 18 March 2014)

The electron transport properties of atomically thin semiconductors such as MoS2 have attractedsignificant recent scrutiny and controversy. In this work, the scattering mechanisms responsible for limitingthe mobility of single-layer semiconductors are evaluated. The roles of individual scattering rates aretracked as the two-dimensional electron gas density is varied over orders of magnitude at varioustemperatures. From a comparative study of the individual scattering mechanisms, we conclude that allcurrent reported values of mobilities in atomically thin transition-metal dichalcogenide semiconductors arelimited by ionized impurity scattering. When the charged impurity densities are reduced, remote opticalphonon scattering will determine the ceiling of the highest mobilities attainable in these ultrathin materialsat room temperature. The intrinsic mobilities will be accessible only in clean suspended layers, as is also thecase for graphene. Based on the study, we identify the best choices for surrounding dielectrics that will helpattain the highest mobilities.

DOI: 10.1103/PhysRevX.4.011043 Subject Areas: Condensed Matter Physics,Semiconductor Physics

Two-dimensional (2D) layered crystals such as singlelayers of transition-metal dichalcogenides represent thethinnest possible manifestations of semiconductor materi-als that exhibit an energy band gap. For example, a single-layer (SL) MoS2 is around !0.6 nm thick and exhibits anenergy band gap of around !1.8 eV [1]. Such semicon-ductor layers differ fundamentally from ultrathin hetero-structure quantum wells or thin membranes carved out ofthree-dimensional (3D) semiconductor materials becausethere are, in principle, no broken bonds, and no roughnessover the 2D plane. In heterostructure quantum wells, theelectron mobility suffers from variations in the quantum-well thickness. A classic “sixth-power law” from Sakakiet al. [2] shows that since the quantum-mechanical energyeigenvalues in a heterostructure quantum well of thicknessL go as ε ! 1=L2, variations in thickness ΔL lead toperturbations of the energy Δε ! !2ΔL=L3. Since thescattering rate depends on the square of Δε, the rough-ness-limited mobility degrades as μR ! L6. When Lreduces from about !7 to !5 nm for example, μR reducesfrom about 104 to 103 cm2=Vs in GaAs/AlAs quantumwells at 4.2 K [2]. Though low-temperature mobilitiesexceeding 106 cm2=Vs have been achieved in such hetero-structures by scrupulous cleanliness and design to reduceroughness scattering, the statistical variations in the quan-tum-well thickness during the epitaxial growth processpose a fundamental limit to electron mobility.

Because of the absence of intrinsic roughness in atomi-cally thin semiconductors, the expectation is that highermobilities should, in principle, be attainable. However,recent measurements in MoS2 and similar semiconductors[3–5] exhibit rather low mobilities in single layers, whichare, in fact, lower than in their multilayer counterparts.Many-particle transport effects can appear in transition-metal dichalcogenides under special conditions becauseof the contribution of highly localized d-orbitals to theconduction and valence-band-edge eigenstates. Collectiveeffects have been observed in multilayer structures, such ascharge-density waves [6,7] and the appearance of super-conductivity at extremely high metallic carrier densities [8]under extreme conditions. We do not discuss such collec-tive phenomena here. Instead, we focus on single-particletransport in single-layerMoS2; the onlymany-particle effectincluded is free-carrier screening. In this work, we performa comprehensive study of the scattering mechanisms thatlimit electron mobility in atomically thin semiconductors.The mobility is calculated in the relaxation-time approxi-mation (RTA) of the Boltzmann transport equation. Theresults shed light on the experimentally achievable electronmobility by designing the surrounding dielectrics and low-ering the impurity density. The findings thus offer usefulguidelines for future experiments.With the advent of graphene, it was realized that for

ultrathin semiconductors, the dielectric environment plays acrucial role in electron transport. It has now been demon-strated that the dielectric mismatch significantly modifiesthe Coulomb potentials inside a semiconductor thin layer[9–12]. Electrons in the semiconductor can also remotelyexcite polar-optical-phonon modes in the dielectrics[13–19]. Such long-range interactions become stronger asthe thickness of the semiconductor layer decreases. Thus,

*[email protected]

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

PHYSICAL REVIEW X 4, 011043 (2014)

2160-3308=14=4(1)=011043(9) 011043-1 Published by the American Physical Society

one can expect the dielectric environment to significantlyaffect electron transport properties in SL gapped semi-conductors. In this work, we take SL MoS2 as a case studyto investigate such effects. The results and conclusions canbe extended to other SL gapped semiconductors.We first study the effect of the dielectric environment

on Coulomb scattering of carriers from charged impuritieslocated inside the MoS2 single layer. Figure 1(a) shows apoint charge located at the center (z0 ! 0) of a SL MoS2 ofthickness a. Assuming the surrounding dielectric providesa large energy barrier for confining electrons in the MoS2membrane, we consider scattering of electrons within theconduction band minima at the K point, i.e., in the groundstate. The envelope function of mobile electrons is then

ψk⇀"ρ⇀; z# ! χ"z#eik

⇀·ρ⇀

=!!!S

p, where χ"z#!

!!!!!!!!2=a

pcos"πz=a#,

S is the 2D area, k⇀is the in-plane 2D wave vector, and ρ

⇀is

the in-plane location vector of the electron from the pointcharge. The dielectric mismatch between the MoS2 (rela-tive dielectric constant εs) layer and its environment (εe)creates an infinite array of image charges at points zn ! na,where n ! $1;$2… [9,10,20]. The nth point charge has amagnitude of eγjnj, where γ ! "εs ! εe#="εs % εe#. Theseimage charges contribute to the net electric potential seenby the electron, which is given by

VCIunsc"ρ; z# !

X"

n!!"

eγjnj

4πε0εs!!!!!!!!!!!!!!!!!!!!!!!!!!!ρ2 % jz ! znj2

p : (1)

where e is the elementary charge, and ε0 is the vacuumpermittivity. Figure 1 shows the net unscreened Coulombpotential contours in the dielectric=MoS2=dielectric systemwith three different εe. The Coulomb interaction is stronglyenhanced for a low-κ dielectric environment and is dampedfor the high-κ case.When a point charge is located inside a 3D semicond-

uctor, its Coulomb potential is lowered by the dielectric

constant of the semiconductor host alone. For thin semi-conductor layers, the Coulomb potential is determinedby the dielectric constants of both the semiconductor itselfand the surrounding dielectrics. When a high density ofmobile carriers is present in the semiconductor, theCoulomb potential is further screened. For atomically thinsemiconductors, understanding the dielectric mismatcheffect on the free-carrier screening of scattering potentialsis necessary. At zero temperature, static screening by the2D electron gas is captured by the Lindhard function [21]:

ε2d"q;ω ! 0# ! 1% e2

2ε0εsqΠ"q;ω ! 0#"!1 % !2#; (2)

where q is the 2D scattering wave vector, and Π is thepolarizability function at zero temperature [22],

Π"q;ω! 0# ! gsgvm&

2πℏ2

(

1!Θ'q! 2kF(

!!!!!!!!!!!!!!!!!!!!!!!!

1!"2kFq

#2

s )

;

(3)

where gs, gv are the spin and valley degeneracy factors,respectively, m& is the electron mass, kF is the Fermi wavevector, and Θ'…( is the Heaviside unit-step function. Thefunction !1 is the form factor, and !2 is the dielectricmismatch factor, which are defined by the equations [23]

!1 !Z

χ2"z#dzZ

χ2"z0# exp"!qjz ! z0j#dz0; (4)

!2 !2χ%χ! exp"!qa#"εe ! εs#2 ! "χ2! % χ2%#"ε2e ! ε2s#

exp"qa#"εe % εs#2 ! exp"!qa#"εe ! εs#2;

(5)

where χ$ !Rdz exp"$qz#χ2"z#. The free-carrier screen-

ing is taken into account by dividing the unscreened

(a) (b) (c)

e s e

-a/2 0 a/2

!

zC

harg

ed i

mpu

rity

e="s e=100e=1

MoS2

Cou

lom

b P

oten

tial i

n Li

near

Sca

le

z

FIG. 1. Coulomb potential contours due to an on-center point charge for three different dielectric environments: εe ! 1, 7.6 "!εs#, 100.

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scattering matrix elements by ε2d. Equation (2) can berecast as the Thomas-Fermi formula: ε2d ! 1% qeffTF=q, inanalogy to the case in the absence of a dielectric mismatch.Here, qeffTF corresponds to the Thomas-Fermi screeningwave vector q0TF without a dielectric mismatch. Figure 2(a)shows the ratio qeffTF=q

0TF that captures the effect of the

dielectric mismatch on screening at zero temperature. The2D electron density is ns ! 1012 cm!2 in this figure. Ascan be seen, the free-carrier screening is weakened by ahigh-κ dielectric, and it is enhanced in the low-κ case. Thisdependence is opposite to the effect of the dielectricenvironment on the net unscreened Coulomb interaction.The momentum relaxation rate "τm#!1 due to elastic

scattering mechanisms is evaluated using Fermi’s goldenrule in the form

1

τm! 2π

ℏ

Zd2k0

"2π#2jMkk0 j2

ε22d"1 ! cos θ#δ"Ek ! Ek0#; (6)

whereMkk0 is the matrix element for scattering from state kto k0, θ is the scattering angle, and Ek and Ek0 are theelectron energies for states k and k0, respectively. For thecharged impurity scattering momentum relaxation rate"τcm#!1, the scattering matrix element is evaluated as

Mkk0 !e2

2ε0εsS1

q! 4

$γ

exp"qa# ! γ

4π2 sinh"qa2 #4π2"qa# % "qa#3

%2'1 ! exp"! qa

2 #(π2 % "qa#2

4π2"qa# % "qa#3

%: (7)

Figure 2(b) shows "τcm#!1 with the impurity density ofNI ! 1012 cm!2. εe and ns are varied over 2 orders ofmagnitude to map out the parameter space. Evidently,"τcm#!1 still reduces monotonically with increasing εe

because the weakening of the unscreened Coulomb poten-tial is stronger.The reduction of "τcm#!1 for a high-κ environment is

much enhanced for high ns0 , as indicated in Fig. 2(b).When εe varies from 1 to 100, "τcm#!1 decreases about!1.4 times for ns ! 1011 cm!2, and about !2.6 times forns ! 1013 cm!2. From the perspective of screening, noticefrom Fig. 2(a) that in a low-κ environment, qeffTF is higher forsmall-angle scattering events. This means the smaller thescattering angle, the stronger is the screening. Thus screen-ing favors randomizing the electron momentum. A high-κenvironment reverses this process: small angle scatteringevents are weakly screened, and thus such scattering eventsare favored. Thus, as εe increases, the electron transportbecome more directional. Though qeffTF decreases, the netscreening efficiency increases. These tendencies areenhanced as ns increases. From the scattering potentialpoint of view, a higher ns leads to a larger Fermi wavevector kF. As shown schematically in the inset of Fig. 2(a),the same q ! jki ! kfj with high ns corresponds to asmaller scattering angle than a lower-ns case, leading to areduced "τcm#!1. This effect on the Coulomb scatteringmatrix element is multiplied by the dielectric mismatchfactor; thus, a high-ns system shows stronger εe depend-ence at zero temperature.For finite temperatures, following Maldague [22,24,25],

the static polarizability function is

Π"q; T; EF# !Z"

0

Π"q;ω ! 0#4kBTcosh2'"EF ! E#=2kBT(

dE; (8)

where EF is the Fermi energy and kB is the Boltzmannconstant. Figure 3(a) shows the calculated temperature-dependent polarizability normalized to the zero-temperature value at different ns. The electron gas is lesspolarizable at higher temperatures and lower ns.Polarizability is caused by the spatial redistribution ofthe electron gas induced by the Coulomb potential; thus, itis proportional to ns. As temperature increases, the thermalenergy randomizes the electron momenta, accelerating thetransition of the electron system back into an equilibriumdistribution, consequently weakening the polarization.The decrease of polarizability reduces the free-carrierscreening. Figure 3(b) shows the temperature-dependentCoulomb-scattering-limited mobility (μimp) at two differentns. The dielectric mismatch effect is more significant forlow ns because of the fast decrease of the polarizabilitywith increasing temperature. For high ns, on the other hand,the dielectric mismatch effect is not as drastic. The shape ofthe temperature-dependent μimp curve is highly dependenton the polarizability and ns. Consequently, if the electrontransport is dominated by impurity scattering, one can inferns from the shape of the temperature dependence of theelectron mobility.

FIG. 2. Effect of dielectric mismatch on the (a) free-carrierscreening and (b) Coulomb momentum relaxation rate at zerotemperature. The inset of (a) shows schematically the scatteringangle for different electron densities.

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Much interest exists in using atomically thin semi-conductors as possible channel materials for electronicdevices, in which such layers are in close proximity todielectrics. To that end, we investigate both the intrinsic andextrinsic phonon scattering in SL MoS2. Kaasbjerg et al.[26] have predicted the theoretical intrinsic phonon-limitedmobility (μi!ph) of SL MoS2 from first principles using adensity-functional-based approach. They estimated a room-temperature upper limit for the experimentally achievablemobility of about 410 cm2=Vs, which weakly depended onns. Their estimate did not include the effects of free-carrierscreening and dielectric mismatch. In light of the strongeffect of these factors on the Coulomb scattering, weevaluate μi!ph in MoS2 in the Boltzmann transport formal-ism with the modified free-carrier screening. The materialparameters for SL MoS2 were obtained from Ref. [27]. Themomentum relaxation rate due to quasielastic scattering byan acoustic phonon is given by

1

τacm! "2

ackBTm&

2πℏ3ρsv2s

Zπ

!π

"1 ! cos θ#dθε22d

; (9)

where ρs is the areal mass density of SL MoS2, vs is thesound velocity, and "ac is the acoustic deformation poten-tial. For inelastic electron-optical phonon interactions, themomentum relaxation rate in the RTA is obtained bysumming the emission and absorption processes,

1

τopm!

Θ'Ek ! ℏωνop(

τ%op% 1

τ!op; (10)

where ωνop is the frequency of the νth optical-phonon

mode. The momentum relaxation rates with superscripts

“%” and “!” are associated with phonon emission andabsorption, respectively. For optical deformation potentials(ODP) [26],

1

τ$0!ODP!D2

0m&"Nq% 1

2$12#

4πℏ2ρsω

Zπ

!π

"1! "k0=k#cosθ#dθε22d

; (11)

1

τ$1!ODP!

D21m

&"Nq % 12 $

12#

4πℏ2ρsω

Zπ

!π

q2"1 ! "k0=k# cos θ#dθε22d

;

(12)

where D is the optical deformation potential, Nq !1='exp"ℏω=kBT# ! 1( is the Bose-Einstein distributionfor optical phonons of energy ℏω, and the subscripts 0and 1 denote the zero- and first-order ODP, respectively.The scattering rate by polar-optical (LO) phonons is

given by the Fröhlich interaction [28],

1

τ$LO! e2ωm&

8πℏ2

1

ε0

"1

ε"! 1

εs

#"Nq %

1

2$ 1

2

#

!Zπ

!π

1

q!1

"1 ! "k0=k# cos θ#dθε22d

; (13)

where ε" is the high-frequency relative dielectric constant,and !1 is the form factor defined by Eq. (4).Figure 4(a) shows the ns-dependent screened μi!ph at

room temperature. For comparison, the unscreened μi!ph is

FIG. 3. (a) The normalized polarizability and (b) impurity-limited mobility at different electron densities as a function oftemperature.

FIG. 4. Electron mobility in MoS2 due to intrinsic phononscattering at room temperature with the electron-phonon interaction(a) fully screened and (b) partially screened. The dashed lines showmobilities limited by unscreened phonon modes, and the solid linesshow the mobilities limited by fully screened modes.

NAN MA AND DEBDEEP JENA PHYS. REV. X 4, 011043 (2014)

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also shown as a reference (blue line). The unscreenedvalues remain effectively constant (about 380 cm2=Vs) overthe range of ns of interest (1011–1013 cm!2). This is inagreementwith the previous predictions (320–410 cm2=Vs)[26,29]. However, the screened increases sharply withincreasing ns. As can be seen in Fig. 4(a), introducing ahigh-κ dielectric leads to a reduction of μi!ph; the highestvalues of μi!ph reduce from 3100 to 1500 cm2=Vs as εeincreases from about !7.6 to about !20. The strongdependence of μi!ph on the dielectric environment is entirelydue to the dielectric-mismatch effect on free-carrier screen-ing since the unscreened phonon-scattering matrix elementis not affected by εe. Over the entire range of ns, longitudinaloptical phonon scattering is dominant. This finding isdifferent from previous works on multilayer MoS2 transportwhere the room-temperature μi!ph was determined byhomopolar phonon scattering [30–32].We have used the static dielectric function for calculat-

ing the screened interactions due to different modes ofphonons in the limit ω ! 0. Scattering mechanisms vialong-range Coulomb interactions, such as charged impu-rities, polar-optical phonons, and piezoelectric acousticphonons, can be effectively screened by free carriers.However, free carriers may not respond to rapidly chang-ing scattering potentials originating from short-rangeinteractions. There are arguments about to what extentthe short-range deformation potentials induced by acoustic(ADP) and optical phonons (ODP) are screened by freecarriers. Boguslawski and Mycielski [33] argue that in asingle-valley conduction band, the deformation potentials(both ADP and ODP) are screened in the same way as themacroscopic (long-range) phonon potentials. But formultivalley semiconductors (Ge), only the longitudinalacoustic (LA) mode of the ADP can be effectivelyscreened by free carriers. The free-carrier screening ofthe transverse acoustic (TA) mode ADP and ODP can, to agood approximation, be neglected. [34]. In SL MoS2,Kaasbjerg et al. [27] have argued that the LA mode of theADP can be treated as screened by the long-wavelengthdielectric function, while the screening of the TA modeADP by free carriers can be neglected.Figure 4(b) highlights the effect of the partially screened

electron-phonon interaction compared to the fully screenedversion in Fig. 4(a). For the plot in Fig. 4(b), we havescreened the polar-optical and LA phonon scattering asin Fig. 4(a), and we leave the TA and ODP interactionsunscreened. The highest μi!ph reached by free-carrierscreening effects is reduced to about 750 cm2=Vs by notscreening the DP modes. The mobility is dominated bythe polar-optical phonon interaction at low carrier densityand by TA and ODP at moderate and high densities. Thescattering of electrons due to piezoelectric phonons is notconsidered because it is relevant only at very low temper-atures and because there are still uncertainties in thepiezoelectric coefficients of SL MoS2 [27,35].

In both cases, the calculated room-temperature μi!ph aremuch higher than reported experimental values, implyingthat there is still much room for improvement of mobilitiesin atomically thin semiconductors. For the rest of this work,we use the fully screened intrinsic phonon scattering, asshown in Fig. 4(a). To pinpoint the most severe scatteringmechanisms limiting the mobility in current samples, wediscuss an extrinsic phonon-scattering mechanism at playin these materials, again motivated by similar processes ingraphene.Electrons in semiconductor nanoscale membranes can

excite phonons in the surrounding dielectrics via long-range Coulomb interactions, if the dielectrics support polarvibrational modes. Such “remote phonon” or “surface-optical” (SO) phonon scattering has been investigatedrecently for graphene and found to be far from negligible[15–17]. SO phonon scattering can severely degradeelectron mobility; however, this process has not beenstudied systematically in atomically thin semiconductors.The electron-SO phonon interaction Hamiltonian is[15,17,18]

He!SO ! eFν

X

q

&e!qz

!!!q

p "eiq⇀·ρ⇀

aν%q % e!iq⇀·ρ⇀

aνq#'; (14)

where aν%q (aνq) represents the creation (annihilation) oper-ator for the νth SO phonon mode. Neglecting the dielectricresponse of the atomically thin MoS2 layer in lieu of thesurrounding media, the electron-SO phonon couplingparameter Fν is

F2ν !

ℏωνSO

2Sε0

"1

ε"ox % ε"ox0! 1

ε0ox % ε"ox0

#; (15)

where ε"ox (ε0ox) is the high- (low-) frequency dielectricconstant of the dielectric hosting the SO phonon, and ε"ox0 isthe high-frequency dielectric constant from the dielectricon the other side of the membrane. The frequency of the SOphonon ων

SO is [17,36]

ωνSO ! ων

TO

"ε0ox % ε"ox0ε"ox % ε"ox0

#1=2

; (16)

where ωνTO is the νth bulk transverse optical-phonon

frequency in the dielectric. The scattering rate due to theSO phonon is then given by

1

τ$SO! 32π3e2F2

vm&Sℏ3a2

"Nq %

1

2$ 1

2

#

!Zπ

!π

1

qsinh2"aq2 #

"4π2q% a2q3#2"1 ! "k0=k# cos θ#dθ

ε22d: (17)

CHARGE SCATTERING AND MOBILITY IN ATOMICALLY … PHYS. REV. X 4, 011043 (2014)

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Table I summarizes the parameters for some commonlyused dielectrics.Figure 5 shows the room-temperature electron mobility

for various dielectric environments for two representativetemperatures, 100 K and 300 K. NI and ns are both about1013 cm!2. The solid lines show the net mobility bycombining the scattering from charged impurities, andintrinsic and SO phonons, whereas the dashed lines showthe cases neglecting the SO phonons. When SO phononscattering is absent, the electron mobility is limited almostentirely by μimp, which increases with εe because of thereduction of Coulomb scattering by dielectric screening.The addition of the SO phonon scattering does not changethings much at 100 K, except for the highest εe case(HfO2=ZrO2). But it drastically reduces the electron mobil-ity at room temperature, as is evident in Fig. 5. For instance,neglecting SO phonon scattering, one may expect thatby using HfO2=ZrO2 as the dielectrics instead of SiO2=air,the RT mobility μimp should improve from about !45 to80 cm2=Vs. However, when the SO phonon scattering isin action, the mobility in the HfO2=MoS2=ZrO2 structure isactually degraded to around 25 cm2=Vs, even lower thanthe SiO2=air case. Thus, SL MoS2 layers suffer fromenhanced SO phonon scattering if they are in closeproximity to high-κ dielectrics that allow low-energy polarvibrational modes.To calibrate our calculations, we study the temperature-

dependent electron mobility for SL MoS2 embeddedbetween SiO2 and HfO2 and compare the calculationswith reported experimental results. This structure is oftenused in top-gated MoS2 field effect transistors (FETs); thus,understanding the transport in it provides a pathway tounderstanding the device characteristics. In Fig. 6(a), theblue curves indicate calculated values of μimp with differentNI , and the red line shows the SO phonon-scatteringlimited mobility (μSO), with ns ! 1013 cm!2. The temper-ature-dependent μSO of each SO phonon mode followsthe Arrhenius rule: μSO # exp"ℏω0=kBT#, and the net μSOis dominated by the softest phonon mode with the lowestenergy. The black curves indicate the net mobilitiesconsidering all scattering mechanisms discussed in thiswork. The open squares are the experimental resultsmeasured by the Hall effect on SL MoS2 FETs fromRef. [4]. The NI and ns necessary to fit the data areindicated in Fig. 6(a). At low temperatures, the

experimental electron mobility in SL MoS2 is entirelylimited by μimp. This is really not unexpected; it tookseveral decades of careful epitaxial growth and ultracleancontrol to achieve the high mobilities in III-V semicon-ductors at low temperatures. Based on this study, we predict

TABLE I. SO phonon modes for different dielectrics.

SiO2a AlNa BNb Al2O3

a HfO2a ZrO2

a

ε"ox 3.9 9.14 5.09 12.53 23 24ε"ox 2.5 4.8 4.1 3.2 5.03 4ω1SO 55.6 81.4 93.07 48.18 12.4 16.67

ω2SO 138.1 88.5 179.1 71.41 48.35 57.7aRef. [15]bRef. [37]

FIG. 5. Electron mobility as a function of an environmentdielectric constant. Dashed lines show the mobility withoutconsidering the SO phonons.

FIG. 6. (a) Temperature-dependent electron mobility (blacklines) in SiO2=MoS2=HfO2 structure. The blue lines indicate μimpand the red lines show μSO. Open squares show experimentalresults from single-layer MoS2 FETs from Ref. [4]. (b) Room-temperature phonon-determined electron mobilitiesμph and(c) the critical impurity densities Ncr corresponding to μimp !μph in SL MoS2 surrounded by different dielectrics. Dashed linesshow the fitted μph and Ncr.

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that the low-temperature mobilities in atomically thinsemiconductors can be significantly improved by loweringthe impurity density. The room-temperature mobility inIII-V semiconductors is limited by intrinsic polar-opticalphonon scattering. For comparison, we find that for SLMoS2, the room-temperature mobility is considerablydegraded by SO phonon scattering, even with NI as highas 6 ! 1012 cm!2, as shown in Fig. 6. When SO phononscattering is absent, the room-temperature mobility isexpected to be about 130 cm2=Vs with NI!6!1012 cm!2,but the measured values are typically lower (about50 cm2=Vs). Consequently, using HfO2 as gate dielectricscan modestly improve μimp. However, the strong SOphonon scattering that comes with HfO2 can severelydecrease the high-temperature electron mobility in cleanMoS2 with low charged impurity densities.An important question then is, which dielectric can help

improve the room-temperature electron mobility in SLMoS2? To answer that question, in Fig. 6(b), we plot theroom-temperature (intrinsic% SO) phonon-limited electronmobility (μph) in SL MoS2 surrounded by different dielec-trics. From the overall trend, μph decreases with increasingεe, and suspended SL MoS2 shows the highest potentialelectron mobility (over 10; 000 cm2=Vs). It is worth notingthat if the scattering of electrons by intrinsic phonons isonly partially screened, as shown in Fig. 4(b), the highestachievable mobility in SL MoS2 will be an order lower(around 1000 cm2=Vs). However, these high values areattainable in suspended SL MoS2. Because μph for MoS2surrounded by high-κ materials is dominated by SO phononscattering, the values do not vary much. The criticalimpurity densities (Ncr) corresponding to μimp ! μph areshown in Fig. 6(c). As long as NI $ Ncr, μimp completelymasks μph. When NI < Ncr, the electron mobility becomesdominated by phonons and moves towards the upper limit.High μph indicates a greater potential for attaining higherelectron mobilities. However, we also need the sample to behighly pure. In high-κ environments that support low-energy polar vibrational modes, there is not as much roomfor improving the electron mobility as in low-κ structures.A compromise is seen for Aluminum Nitride (AlN)- andBoron Nitride (BN)-based dielectrics, which by virtue ofthe light atom N, allows high-energy optical modes inspite of their polar nature. From Figs. 6(b) and 6(c), onecan obtain two useful relationships for single-layer MoS2:μph ! 35000=ε2.2e cm2=Vs and Ncr ! 1010ε2.5e cm!2, withns set at a typical on-state carrier density of 1013 cm!2, asshown by dashed lines. These empirical relations shouldguide the proper choice of dielectrics and the maximumallowed impurity densities.To further illustrate the relative importance of SO

phonon and charged impurity scattering in SL MoS2, wevary NI and ns in different dielectric environments andcheck the changing trends of electron mobilities at roomtemperature. Figure 7(a) shows the net electron mobilities

in SL MoS2 as a function of NI with ns ! 1013 cm!2.Figures 7(b) and 7(c) show the electron mobility as afunction of ns for NI ! 1011 and 1013 cm!2. The electronmobility is weakly dependent on the dielectric environmentat highNI (>1013 cm!2), as shown in the dashed box in thebottom right corner of Fig. 7(a). Within this window, high-κdielectrics can improve the mobility, but only very nomi-nally because the unscreened mobilities are already quitelow. When NI is lowered below about 1012 cm!2, a low-κenvironment shows higher electron mobility. For most ofthe dielectric environments, when NI > 1012 cm!2, themobility fits the following empirical impurity-scattering-dominated relationship: μ% 4200='NI=1011 cm!2(cm2=Vs,as shown by the dashed line in Fig. 7(a). Using thisexpression, one can estimate NI from measured electronmobility for high ns. As ns decreases, electron mobilitiesin different dielectric environments start to separatefrom each other, as shown in Fig. 7(c). In this case, theelectron mobilities can fit the following relationship:μ% 3500

NI=1011 cm!2 'A"εe#%" ns1013 cm!2#1.2(cm2=Vs for ns <

1013 cm!2, shown as dashed lines in Fig. 7(c). A"εe# isa fitting constant depending on εe, and some values arelisted in the inset table of Fig. 7(c). High-κ dielectrics withlow-energy phonons (HfO2, ZrO2) severely degrade theelectron mobility over the entire NI range because ofthe dominant effect of SO phonon scattering. Note that thedielectric mismatch effect can be slightly overestimatedhere since we have assumed the thickness of the dielectricto be infinite [25]. In top-gated FETs, the top dielectriccould be very thin. Thus, the capability of improvingelectron mobility by high-κ dielectrics can be even lesssignificant. Since most applications require high mobilities,

FIG. 7. The room-temperature net electron mobilities in SLMoS2, considering all kinds of scattering mechanisms as afunction of (a) NI with fixed ns at 1013 cm!2; (b) and (c) nswith NI fixing at 1011 and 1013 cm!2, respectively. The numberson the curves show the average dielectric constant of thesurrounding dielectrics. Dashed lines show the fitted electronmobilities.

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high ns, and high εe to be present simultaneously in thesame structure for achieving the highest conductivities,AIN=Al2O3 or BN/BN encapsulation emerges as the bestcompromise among the dielectric choices considered here.One can also conceive of dielectric heterostructures, with afew BN layers closest to MoS2 to damp out the SO phononscattering, followed by higher-κ dielectrics to enhancethe gate capacitance for achieving high carrier densities.All this, however, requires ultraclean MoS2 to start with,with NI well below 1012 cm!2 to attain the high room-temperature mobilities, about 1000 cm2=Vs. The presenceof high impurity densities will always mask the intrinsicpotential of the materials, and this is the most importantchallenge moving forward.In conclusion, carrier transport properties in atomically

thin semiconductors are found to be highly dependent onthe dielectric environment and on the impurity density. Forcurrent 2D crystal materials, electron mobilities are mostlydominated by charged impurity scattering. Remote pho-nons play a secondary role at high temperature dependingon the surrounding dielectrics. The major point is thatthe mobilities achieved to date are far below the intrinsicpotential in these materials. High-κ gate dielectrics canincrease the electron mobility only for samples infectedwith very high impurity densities. Clean samples withlow-κ dielectrics show much higher electron mobilities.AlN- and BN-based dielectrics offer the best compromiseif a high-mobility and high-gate capacitance are simulta-neously desired, as is the case in field-effect transistors. Thetruly intrinsic mobility limited by the atomically thinsemiconductor itself can only be achieved in ultracleansuspended samples, as is the case for graphene.

The authors thank Dr. Andras Kis, Dr. KristenKaasbjerg, and Deep Jariwala for useful discussions andfor sharing experimental data. The research is supported inpart by a NSF ECCS grant monitored by Dr. AnupamaKaul, AFOSR, and the Center for Low Energy SystemsTechnology (LEAST), one of the six centers supported bythe STARnet phase of the Focus Center Research Program(FCRP), a Semiconductor Research Corporation programsponsored by MARCO and DARPA.

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