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Phonon-Limited Mobility in n-Type Single-Layer MoS2 from First Principles

Article in Physical review. B, Condensed matter · March 2012

DOI: 10.1103/PhysRevB.85.115317 · Source: arXiv

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 First-principles study of the phonon-limited mobility in n-type single-layer MoS2
Kristen Kaasbjerg,∗ Kristian S. Thygesen, and Karsten W. Jacobsen
Center 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 approach
for the calculation of the electron-phonon interaction, the deformation potentials and Fröhlich in-
teraction in the isolated MoS2 layer are determined. We find that the calculated room-temperature
arXiv:1201.5284v1 [cond-mat.mtrl-sci] 25 Jan 2012

mobility of ∼ 410 cm2 V−1 s−1 is dominated by optical phonon scattering via deformation potential
couplings and the Fröhlich 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 the
characteristic homopolar mode which is likely to occur in top-gated samples, boosts the mobility
with ∼ 70 cm2 V−1 s−1 and can be observed as a decrease in the exponent to γ = 1.52. Our
findings indicate that the intrinsic phonon-limited mobility is approached in samples where a high-κ
dielectric 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 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 of
Alongside with the rise of graphene and the explo- the phonon-limited mobility in single-layer MoS2 . Since
ration of its unique electronic properties,1–3 the search phonon scattering is an intrinsic scattering mechanism
for other two-dimensional (2D) materials with promis- that often dominates at room temperature, the theo-
ing electronic properties has gained increased interest.4 retically predicted phonon-limited mobility sets an up-
Metal dichalcogenides which are layered materials similar per limit for the experimentally achievable mobilities in
to graphite, provide interesting candidates. Their layered single-layer MoS2 .15 In experimental situations, the tem-
structure with weak interlayer van der Waals bonds al- perature dependence of the mobility can be useful for the
lows for fabrication of single- to few-layer samples using identification of dominating scattering mechanisms and
mechanical peeling/cleavage or chemical exfoliation tech- also help to establish the value of the intrinsic electron-
niques similar to the fabrication of graphene.5–8 How- phonon couplings.16 However, the existence of additional
ever, in contrast to graphene they are semiconductors extrinsic scattering mechanisms often complicates the in-
and hence come with a naturally occurring band gap—a terpretation. For example, in graphene samples impu-
property essential for electronic applications. rity scattering and scattering on surface polar optical
The vibrational and optical properties of single- to few- phonons of the gate oxide are important mobility-limiting
layer samples of MoS2 have recently been studied ex-
tensively using various experimental techniques.9–11 In
contrast to the bulk material which has an indirect-gap,
it has been demonstrated that single-layer MoS2 is a a
Mo 0.6 K K 0.45

direct-gap semiconductor with a gap of 1.8 eV.10,12 To-

ky [2 /a]

a S 0.3
0.30

gether with the excellent electrostatic control inherent
0.0 eV

of two-dimensional materials, the large band gap makes -0.3 0.15

it well-suited for low power applications.13 So far, elec- -0.6

trical characterizations of single-layer MoS2 have shown -0.6 -0.3 0.0 0.3 0.6
0.00

n-type conductivity with room-temperature mobilities in kx [2 /a]

the range 0.5 − 3 cm2 V−1 s−1 .5,6,8 Compared to early
FIG. 1: (Color online) Atomic structure and conduction band
studies of the intralayer mobility of bulk MoS2 where
of single-layer MoS2 . Left: Primitive unit cell and structure
mobilities in the range 100 − 260 cm2 V−1 s−1 were
of an MoS2 layer with the molybdenum and sulfur atoms po-
reported,14 this is rather low. In a recent experiment sitioned in displaced hexagonal layers. Right: Contour plot
the use of a high-κ gate dielectric in a top-gated device showing the lowest lying conduction band valleys as obtained
was shown to boost the carrier mobility to a value of with DFT-LDA in the hexagonal Brillouin zone of single-layer
200 cm2 V−1 s−1 .8 The observed increase in the mobility MoS2 . For n-type MoS2 , the low-field mobility is determined
was attributed to screening of impurities by the high-κ by the carrier properties in the K, K ′ -valleys which are sepa-
dielectric and/or modifications of MoS2 phonons in the rated in energy from the satellite valleys inside the Brillouin
top-gated sample. zone.
 2

factors.17–22 It can therefore be difficult to establish the

value and nature of the intrinsic electron-phonon interac-
tion from experiments alone. Theoretical studies there- 4

fore provide useful information for the interpretation of

experimentally measured mobilities.

eV)
2
In the present work, the electron-phonon interaction in

Energy (
1.6
single-layer MoS2 is calculated from first-principles using
0
a density-functional based approach. From the calcu- 1.4

lated electron-phonon couplings, the acoustic (ADP) and

optical deformation potentials (ODP) and the Fröhlich 2 1.2

interaction in single-layer MoS2 are inferred. Using 1.0

these as inputs in the Boltzmann equation, the phonon- 4

limited mobility is calculated in the high-temperature 
0.8
K M

regime (T > 100 K) where phonon scattering often dom- M  K M

inate the mobility. Our findings demonstrate that the

calculated phonon-limited room-temperature mobility of FIG. 2: (Color online) Band structure of single-layer MoS2 .
∼ 410 cm2 V−1 s−1 is dominated by optical deformation The inset shows a zoom of the conduction band along the path
potential and polar optical scattering via the Fröhlich Γ-K-M. The parabolic band (red dashed line) with effective
interaction. The dominating deformation potentials of mass m∗ = 0.48 me is seen to give a perfect description of
0
DHP = 4.1 × 108 eV/cm and DLO 0
= 2.6 × 108 eV/cm the conduction band valley in the K-point for the range of
originate from the couplings to the homopolar and the energies relevant for the low-field mobility.
intervalley polar LO phonon. This is in contrast to pre-
vious studies for bulk MoS2 , where the mobility has been
assumed to be dominated entirely by scattering on the lattice constant of the hexagonal unit cell shown in Fig. 1
homopolar mode.14,23,24 Furthermore, we show that a was found to be a = 3.14 Å using a 11 × 11 k-point
quenching of the characteristic homopolar mode which is sampling of the Brillouin zone. In order to eliminate in-
polarized in the direction normal to the MoS2 layer can teractions between the layers due to periodic boundary
be observed as a change in the exponent γ of the generic conditions, a large interlayer distance of 10 Å has been
temperature dependence µ ∼ T −γ of the mobility. Such a used in the calculations.
quenching can be expected to occur in top-gated samples
where the MoS2 layer is sandwiched between a substrate
and a gate oxide. A. Band structure
The paper is organized as follows. In Section II the
band structure and phonon dispersion of single-layer In the present study, the band structure of single-layer
MoS2 as obtained with DFT-LDA are presented. The lat- MoS2 is calculated within DFT-LDA.28,29 While DFT-
ter is required for the calculation of the electron-phonon LDA in general underestimates band gaps, the resulting
coupling presented in Section III. From the calculated dispersion of individual bands, i.e. effective masses and
electron-phonon couplings, we extract zero- and first- energy differences between valleys, is less problematic.
order deformation potentials for both intra and inter- We note, however, that recent GW quasi-particle cal-
valley phonons. In addition, the coupling constant for culations which in general give a better description of
the Fröhlich interaction with the polar LO phonon is de- the band structure in standard semiconductors,52 sug-
termined. In Section IV, the calculation of the phonon- gest that the distance between the K, K ′ -valleys and the
limited mobility within the Boltzmann equation is out- satellite valleys located on the Γ-K path inside the Bril-
lined. This includes a detailed treatment of the phonon louin zone not as large as predicted by DFT (see below)
collision integral from which the scattering rates for the and that the valley ordering may even be inverted.53,54
different coupling mechanisms can be extracted. In Sec- This, however, seems to contradict with the experimen-
tion V we present the calculated mobilities as a function tal consensus that single-layer MoS2 is a direct-gap semi-
of carrier density and temperature. Finally, in Section VI conductor10,12 and further studies are needed to clarify
we summarize and discuss our findings. this 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.
II. ELECTRONIC STRUCTURE AND PHONON The calculated DFT-LDA band structure is shown in
DISPERSION Fig. 2. It predicts a direct band gap of 1.8 eV at the
K-poing in the Brillouin zone. The inset shows a zoom
The electronic structure and phonon dispersion of of the bottom of the conduction band along the Γ-K-
single-layer MoS2 have been calculated within DFT in M path in the Brillouin zone. As demonstrated by the
the LDA approximation using a real-space projector- dashed line, the conduction band is perfectly parabolic in
augmented wave (PAW) method.25–27 The equilibrium the K-valley. Furthermore, the satellite valley positioned
 3

on the path between the Γ-point and the K-point lies on

the order of ∼ 200 meV above the conduction band edge 60

and is therefore not relevant for the low-field transport.

The right plot in Fig. 1 shows a contour plot of the low- 50

meV)
est lying conduction band valleys in the two-dimensional
Brillouin zone. Here, the K, K ′ -valleys that are popu- 40

Frequency (
lated in n-type MoS2 , are positioned at the corners of
30
the hexagonal Brillouin zone. Due to their isotropic and
parabolic nature, the part of the conduction band rele-
20
vant for the low-field mobility can to a good approxima-
tion be described by simple parabolic bands
10

~2 k 2
εk = (1) 0
2m∗  K M 
∗
with an effective electron mass of m = 0.48 me and
where k is measured with respect to the K, K ′ -points FIG. 3: Phonon dispersion of single-layer MoS2 calculated
in the Brillouin zone. The two-dimensional nature of the with the small displacement method (see e.g. Ref.30) using
carriers is reflected in the constant density of states given a 9 × 9 supercell. The frequencies of the two optical Raman
by ρ0 = gs gv m∗ /2π~2 where gs = 2 and gv = 2 are the active E2g and A1g modes at 48 meV and 50 meV, respec-
spin and K, K ′ -valley degeneracy, respectively. The large tively, are in excellent agreement with recent experimental
density of states which follows from the high effective measurements.9
mass of the conduction band in the K, K ′ -valleys, results
in non-degenerate carrier distributions except for very
high carrier densities. results in the so-called LO-TO splitting between the two
modes in the long-wavelength limit. The inclusion of
this effect from first-principles requires knowledge of the
B. Phonon dispersion
Born effective charges.32–34 In two-dimensional materials,
however, the lack of periodicity in the direction perpen-
The phonon dispersion has been obtained with the dicular to the layer removes the LO-TO splitting.35 The
supercell-based small-displacement method30 using a 9 × coupling to the macroscopic polarization of the LO mode
9 supercell. The resulting phonon dispersion shown in will therefore be neglected here.
Fig. 3 is in excellent agreement with recent calculations
The almost dispersionless phonon at ∼ 50 meV is the
of the lattice dynamics in two-dimensional MoS2 .31
so-called homopolar mode which is characteristic for lay-
With three atoms in the unit cell, single-layer MoS2
ered structures. The lattice vibration of this mode cor-
has nine phonon branches—three acoustic and six opti-
responds to a change in the layer thickness and has the
cal branches. Of the three acoustic branches, the fre-
sulfur layers vibrating in counterphase in the direction
quency of the out-of-plane flexural mode is quadratic in
normal to the layer plane while the Mo layer remains
q for q → 0. In the long-wavelength limit, the frequency
stationary. The change in the potential associated with
of the remaining transverse acoustic (TA) and longitudi-
this lattice vibration has previously been demonstrated
nal acoustic (LA) modes are given by the in-plane sound
to result in a large deformation potential in bulk MoS2 .14
velocity cλ as
ωqλ = cλ q. (2)
Here, the sound velocity is found to be 4.4 × 103 and
6.5 × 103 m/s for the TA and LA mode, respectively. III. ELECTRON-PHONON COUPLING
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- In this section, we present the electron-phonon cou-
tic and optical modes become similar. The two lowest pling in single-layer MoS2 obtained with a first-principles
optical branches belong to the non-polar optical modes. based DFT approach. The method is based on a su-
Due to an insignificant coupling to the charge carriers, percell approach analogous to that used for the calcula-
they are not relevant for the present study. tion of the phonon dispersion and is outlined in App. A.
The next two branches with a phonon energy of ∼ The calculated electron-phonon couplings are discussed
48 meV at the Γ-point are the transverse (TO) and lon- in the context of deformation potential couplings and the
gitudinal (LO) polar optical modes where the Mo and S Fröhlich interaction well-known from the semiconductor
atoms vibrate in counterphase. In bulk polar materials, literature.
the coupling of the lattice to the macroscopic polariza- Within the adiabatic approximation for the electron-
tion setup by the lattice vibration of the polar LO mode phonon interaction, the coupling strength for the phonon
 4

mode with wave vector q and branch index λ is given by for the effective mass, M N = Aρ, and the expression in
s Eq. (5) is obtained.
λ ~ For scattering on acoustic phonons, the coupling ma-
gkq = Mλ , (3) trix element is linear in q in the long-wavelength limit,
2M N ωqλ kq
Mqλ = Ξλ q, (6)
where ωqλ is the phonon frequency, M is an appropriately
defined effective mass, N is the number of unit cells in
where Ξλ is the acoustic deformation potential.
the crystal, and
In the case of optical phonon scattering, both coupling
λ
Mkq = hk + q|δVqλ (r)|ki (4) to zero- and first-order in q must be considered. The in-
teraction via the constant zero-order optical deformation
is the coupling matrix element where k is the wave vector potential Dλ0 is given by
of the carrier being scattered and δVqλ is the change in
the effective potential per unit displacement along the Mqλ = Dλ0 . (7)
vibrational normal mode.
The coupling via the zero-order deformation potential is
Due to the valley degeneracy in the conduction band,
dictated by selection rules for the coupling matrix ele-
both intra and intervalley phonon scattering of the carri-
ments. Therefore, only symmetry-allowed phonons can
ers in the K, K ′ -valleys need to be considered. Here, the
couple to the carriers via the zero-order interaction. The
coupling constants for these scattering processes are ap-
coupling via the first-order interaction is given by Eq. (6)
proximated by the electron-phonon coupling at the bot-
with the acoustic deformation potential replaced by the
tom of the valleys, i.e. with k = K, K′ . With this ap-
first-order optical deformation potential Dλ1 . Both the
proach, the intra and intervalley scattering for the K, K ′ -
zero- and first-order deformation potential coupling can
valleys are thus assumed independent on the wave vector
give rise to intra and intervalley scattering.
of the carriers.
The absolute value of the calculated coupling matrix el-
In the following two sections, the calculated electron-
ements |Mqλ | are shown in Fig. 4 for phonons which cou-
phonon couplings are presented.36 The different deforma-
ple to the carriers via deformation potential interactions.
tion potential couplings are discussed and the functional
They are plotted in the range of phonon wave vectors rel-
form of the Fröhlich interaction in 2D materials is es-
evant for intra and intervalley scattering processes,39 and
tablished. Piezoelectric coupling to the acoustic phonons
only phonon modes with significant coupling strengths
which occurs in materials without inversion symmetry,
have been included. Although the coupling matrix ele-
is most important at low temperatures37 and will not
ments are shown only for k = K, the matrix elements
be considered here. As deformation potentials are of-
for k = K′ are ′related through time-reversal symmetry
ten extracted as empirical parameters from experimental K K
as Mqλ = M−qλ . The three-fold rotational symmetry
mobilities, it can be difficult to disentangle contributions
of the coupling matrix elements in q-space stems from
from different phonons. Here, the first-principles calcula-
the symmetry of the conduction band in the vicinity of
tion of the electron-phonon coupling allows for a detailed
the K, K ′ -valleys (see Fig. 1). Since the symmetry of the
analysis of the couplings and to assign deformation po-
matrix elements has not been imposed by hand, slight
tentials to the individual intra and intervalley phonons.
deviations from the three-fold rotational symmetry can
be observed.
A. Deformation potentials
The acoustic deformation potential couplings for the
TA and LA modes are shown in the two top plots of
Fig. 4. Due to the inclusion of Umklapp processes here,
The deformation potential interaction describes how the coupling to the TA mode does not vanish as is most
carriers interact with the local changes in the crystal po- often assumed.40 Only along high-symmetry directions in
tential associated with a lattice vibration. Within the de- the Brillouin zone is this the case. This results in a highly
formation potential approximation, the electron-phonon anisotropic coupling to the TA mode. On the other hand,
coupling is expressed as38 the deformation potential coupling for the LA mode is
s perfectly isotropic in the long-wavelength limit but also
~ becomes anisotropic at shorter wavelengths. In agree-
gqλ = Mqλ , (5)
2Aρωqλ ment with Eq. (6), both the TA and LA coupling matrix
elements are linear in q in the long-wavelength limit.
where A is the area of the sample, ρ is the atomic mass The next two plots show the couplings to the intraval-
density per area and Mqλ is the coupling matrix element ley polar TO and homopolar optical modes. While the
for a given valley which is assumed independent on the interaction with the polar TO phonon corresponds to
k-vector of the carriers. This expression follows from a first-order optical deformation potential, the interac-
the general definition of the electron-phonon coupling in tion with the homopolar mode acquires a finite value of
Eq. 3 by setting the effective mass M equal to the sum of ∼ 4.8 eV/Å in the Γ-point and corresponds to a strong
the atomic masses in the unit cell. With this convention zero-order deformation potential coupling. The large
 5

Intravalley have significant weight here, the homopolar lattice vi-

K K bration gives rise to a significant shift of the K, K ′ -valley
states. The large deformation potential associated with
the coupling to the homopolar mode is also present in
bulk MoS2 .14
The two bottom plots in Fig. 4 show the couplings
for the intervalley TA and polar LO phonons. Due to
the nearly constant frequency of the intervalley acoustic
phonons, 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 to
the intervalley LA phonon result in first-order deforma-
tion potentials. The intervalley coupling for the polar
LO phonon gives rise to a zero-order deformation poten-
tial which results from an identical in-plane motion of the
two sulfur layers in the lattice vibration of the intervalley
phonon.
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 to
hold in the long-wavelength limit only. At shorter wave-
lengths, the first-principles couplings become anisotropic
Intervalley and have a more complicated q-dependence. When de-
K K termined experimentally from e.g. the temperature de-
pendence of the mobility,16 the deformation potentials
in 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 for
single-layer MoS2 are determined from the first-principles
electron-phonon couplings. This is done by fitting their
associated scattering rates to the scattering rates ob-
tained with the first-principles coupling matrix elements.
The resulting the deformation potentials, can be used in
practical transport calculations based on e.g. the Boltz-
mann equation or Monte Carlo simulations. It should
be noted that similar routes for first-principles calcula-
tions of deformation potentials have been given in the
FIG. 4: (Color online) Deformation potential couplings in
literature.41–43
single-layer MoS2 . Only phonon modes with significant cou-
pling strength are shown. The contour plots show the absolute
value of the coupling matrix elements |Mqλ | in Eq. (4) with
k = K and k′ = K + q as a function of the two-dimensional B. Fröhlich interaction
phonon wave vector q (note the different color scales in the
plots). The upper four plots correspond to intravalley scat-
tering and the two bottom plots to intervalley scattering on The lattice vibration of the polar LO phonon gives
phonons with wave vector q = 2K. The two zero-order defor- rise to a macroscopic electric field that couples to the
mation potential couplings to the intravalley homopolar and charge carriers. For bulk three-dimensional systems, the
intervalley LO phonon play an important role for the room- coupling to the field is given by the Fröhlich interaction
temperature mobility. which diverges as 1/q in the long-wavelength limit,40
s 1/2
e2 ~ωLO

1 1 1
gLO (q) = − 0 , (8)
deformation potential coupling to the homopolar mode q 2ǫ0 V ε∞ ε
stems from its characteristic lattice vibration polarized
in the direction perpendicular to the layer corresponding where ǫ0 is the vacuum permittivity, V is the volume of
to a change in the layer thickness. The associated change the sample , and ε∞ and ε0 are the high-frequency optical
in the effective potential from the counterphase oscilla- and static dielectric constant, respectively.
tion of the two negatively charged sulfur layers results In atomically thin materials, the two-dimensional na-
in a significant change of the potential towards the cen- ture of both the LO phonon and the charge carriers leads
ter of the MoS2 layer. As the electronic Bloch functions to a qualitatively different q-dependence of the Fröhlich
 6

100 where the time evolution of the crystal momentum is gov-

First-principles erned by ~k̇ = qE and q is the charge of the carriers. In
Analytic the case of a time-independent uniform electric field, the
75 steady-state version of the linearized Boltzmann equation
(meV)
takes the form
50
q q
E · ∇k fk0 = − E · vk fk0 (1 − fk0 )
gLO(q)

~ kB T
∂fk
25 = , (11)
∂t coll

0 where fk0 = f 0 (εk ) is the equilibrium Fermi-Dirac dis-

K tribution function, ∂fk0 /∂εk = −fk0 (1 − fk0 )/kB T , and
vk = ∇k εk /~ is the band velocity. In linear response,
FIG. 5: (Color online) Fröhlich interaction for the polar opti- the out-of-equilibrium distribution function can be writ-
cal LO mode. The dashed line shows the analytical coupling ten as the equilibrium function plus a small deviation δf
in Eq. (9) with the coupling constant gFr = 98 meV and away from its equilibrium value,46 i.e.
the effective layer thickness σ = 4.41 Å fitted to the long-
wavelength limit of the calculated coupling (dots). δfk = fk − fk0 = fk0 (1 − fk0 )ψk . (12)

Here, the last equality has defined the deviation func-

interaction. The situation is similar to 2D semiconduc- tion ψk . Furthermore, following the spirit of the itera-
tor heterostructures where the Fröhlich interaction has tive method of Rode,47 the angular dependence of the
been studied using dielectric continuum models and mi- deviation function is separated out as
croscopic based approaches.44,45 From the microscopic ψk = φk cos θk , (13)
considerations presented in App. B, we derive the fol-
lowing functional form for the Fröhlich interaction to the where θk is the angle between the k-vector and the force
polar LO phonon in 2D materials, exerted on the carriers by the applied field E. In gen-
eral, φk is a function of the k-vector and not only its
gLO (q) = gFr × erfc(qσ/2). (9) magnitude k. However, for isotropic bands as considered
Here, the coupling constant gFr is the equivalent of the here, the angular dependence of the deviation function is
square root factors in Eq. (8), σ is the effective width of entirely accounted for by the cosine factor in Eq. (13).38
the electronic Bloch states and erfc is the complementary Considering the phonon scattering, the collision inte-
error function. gral describes how accelerated carriers are driven back
Figure 5 shows the first-principles coupling gLO to the towards their equilibrium distribution by scattering on
polar LO phonon in single-layer MoS2 (dots) as a func- acoustic and optical phonons. In the high-temperature
tion of the phonon wave vector along the Γ-K path. The regime of interest here, the collision integral can be split
dashed line shows a fit of the coupling in Eq. (9) to the up in two contributions. The first, accounting for the
long-wavelength limit of the calculated coupling. With a quasielastic scattering on acoustic phonons, can be ex-
coupling constant of gFr = 98 meV and an effective width pressed in the form of a relaxation time τel . The second,
of σ = 4.41 Å, the analytic form gives a perfect descrip- describing the inelastic scattering on optical phonons,
tion of the calculated coupling in the long-wavelength will in general be an integral operator Iinel . The colli-
limit. For shorter wavelengths, the zero-order deforma- sion integral can thus be written
tion potential coupling to the intervalley LO phonon from ∂fk δfk
Fig. 4 becomes dominant, and a deviation away from the =− + Iinel [ψk ] , (14)
∂t coll τkel
behavior in Eq. (9) is observed.
where the explicit forms of the relaxation time and the
integral operator will be considered in detail below.
IV. BOLTZMANN EQUATION With the above form of the collision integral, the Boltz-
mann equation can be solved by iterating the equation
In the regime of diffuse transport dominated by phonon    −1
scattering, the mobility can be obtained with semiclas- eEvk ′ 1
sical Boltzmann transport theory. In the absence of φk = + Iinel [φk ] × (15)
kB T τkel
spatial gradients the Boltzmann equation for the out-of-
equilibrium distribution function fk of the charge carriers for the deviation function φk . Here, the angular depen-
reads46 dence cos θk and a fk0 (1 − fk0 ) factor have been divided
′
out and the new integral operator Iinel is defined by
∂fk ∂fk
+ k̇ · ∇k fk = , (10)
∂t ∂t coll
Iinel = fk0 (1 − fk0 ) cos θk Iinel
′
. (16)
 7

From the solution of the Boltzmann equation the cur- Here, n is the two-dimensional carrier density. In the re-
rent and drift mobility of the carriers can be obtained. laxation time approximation φ̃k = τk and the well-known
Taking the electric field to be oriented along the x- Drude expression for the mobility µ = ehτk i/m∗ is recov-
direction, the current density is given by ered.
4e X
jx ≡ σxx Ex = − vx δfk , (17)
A
k

where σxx is the conductivity, A is the area of the sample,

and vi = ~ki /m∗ is the band velocity for parabolic bands A. Phonon collision integral
with i = x, y, z. From the definition µxx = σxx /ne of the
drift mobility, it follows that
The phonon collision integral has been considered in
great detail in the literature for two-dimensional elec-
ehφ̃k i
µxx = (18) tron gases in semiconductor heterostructure (see e.g.
m∗ Ref.37). In general, these treatments consider scattering
where the modified deviation function φ̃k having units of on three-dimensional or quasi two-dimensional phonons.
time is defined by In atomically thin materials the phonons are strictly two-
dimensional which results in a slightly different treat-
m∗ kB T ment. In the following, a full account of the two-
φ̃k = φk , (19) dimensional phonon collision integral is given for scat-
eEx ~k
tering on acoustic and optical phonons.
and the energy-weighted average h·i is defined by
With the distribution function written on the form
1
Z 
∂f 0
 in Eq. (12), the linearized collision integral for electron-
hAi = dεk ρ(εk )εk Ak − . (20) phonon scattering takes the form46
n ∂εk

2 
∂fk 2π X |gqλ | 0
fk0 1 − fk+q

0
=− 2
× Nqλ (ψk − ψk+q ) δ(εk+q − εk − ~ωqλ )
∂t coll ~ ǫ (q, T )
qλ

fk0 0 0



+ 1− fk−q 1+ Nqλ (ψk − ψk−q ) δ(εk−q − εk + ~ωqλ ) (21)

0
where Nqλ = N 0 (~ωqλ ) is the equilibrium distribution for the proper screened mobility.
of the phonons given by the Bose-Einstein distribution
0
function and fk±q = f 0 (εk ± ~ωq ) is understood. The
different terms inside the square brackets account for 1. Quasielastic scattering on acoustic phonons
scattering out of (ψk ) and into (ψk±q ) the state with
wave vector k via absorption and emission of phonons. For quasielastic scattering on acoustic phonons at high
Screening of the electron-phonon interaction by the car- temperatures, the energy of the acoustic phonon can be
riers themselves is accounted for by the static dielectric neglected in the collision integral in Eq. (21). As a con-
function ǫ, which is both wave vector q and tempera- sequence, the collision integral can be recast in the form
ture T dependent. Due to the large effective mass of of the following relaxation time37
the conduction band, the charge carriers in single-layer 1 X
MoS2 are non-degenerate except at very high carrier con- el
= (1 − cos θkk′ ) Pkk′ (22)
τk
centrations. Semiclassical screening where the screening k′
length is given by the inverse of the Debye-Hückel wave where the summation variable has been changed to k′ =
vector kD = e2 n/2kB T therefore applies.48 For the con- k ± q and the transition matrix element for quasielastic
sidered values of carrier densities and temperatures, this scattering on acoustic phonons is given by
corresponds to a small fraction of the size of the Bril- 
2π X
louin zone, i.e. kD ≪ 2π/a. As scattering on phonons Pkk′ = |gqλ |2 Nqλ
0
δ(εk′ − εk )
in general involves larger wave vectors, screening by the ~ q
carriers can to a good approximation be neglected here. 
0


The calculated mobilities therefore provide a lower limit + 1 + Nqλ δ(εk − εk ) . (23)
′
 8

For isotropic scattering as assumed here in Eq. (6), the scattering is given by
square of the electron-phonon coupling can be expressed 
as 2π X
Pkk′ = |gqλ |2 × Nλ0 δ(εk′ − εk − ~ωλ )
~ q
2 Ξ2λ ~q
|gqλ | = , (24) 
2Aρcλ + 1 + Nλ0 δ(εk′ − εk + ~ωλ ) .

where the acoustic phonon frequency has been expressed (28)

in terms of the sound velocity cλ . Except at very low tem-
peratures ~ωq ≪ kB T implying that the equipartition ap- For the in-scattering part, the desired cos θk factor in
proximation Nq0 ∼ kB T /~ωq ≫ 1 for the Bose-Einstein Eq. (16) can be extracted from ψk′ using the rela-
distribution applies. With the resulting q-factors in tion cos θk′ = cos θk cos θkk′ − sin θk sin θkk′ . Since the
the transition matrix element in Eq. (23) canceling, the sine term vanishes from symmetry consideration, the in-
cos θkk′ in the k′ -sum in Eq. (22) vanishes and the first scattering part of the inelastic collision integral reduces
term yields a factor density of states divided by the spin to
and valley degeneracies ρ0 /gs gv . The relaxation time for
acoustic phonon scattering becomes in
X 1 − fk0′
I ′ inel [φk ] = φk′ cos θkk′ Pkk′ . (29)
1 − fk0
k′
1 m∗ Ξ2 kB T
= . (25)
el
τk ~3 ρc2λ The evaluation of the k′ -sum in Eqs. (27) and (29) is
outlined in App. C. Here, the assumption of dispersion-
The independence on the carrier energy and the τ −1 ∼ T less optical phonons allows for an semianalytical treat-
temperature dependence of the acoustic scattering rate ment. For zero- and first-order coupling within the defor-
are characteristic for charge carriers in two-dimensional mation potential approximation, the sum can be carried
heterostructures and layered materials.14,37 out analytically. The resulting expressions for the colli-
sion integral are given in Eqs. (C14), (C15), (C16), (C17)
and (C18). In the case of the Fröhlich interaction, the
2. Inelastic scattering on dispersionless optical phonons angular part of the q-integral must be done numerically.

For inelastic scattering on optical phonons the phonon

energy can no longer be neglected and the collision in- 3. Optical deformation potential scattering rates
tegral in Eq. (21) must be considered in full detail. Un-
der the reasonable assumption of dispersionless optical In spite of the fact that the collision integral for in-
phonons ωqλ = ωλ , the inelastic collision integral can be elastic scattering on optical phonons cannot be recast in
treated semianalytically. The overall procedure for the the form of a (momentum) relaxation time,49 a scatter-
evaluation of the collision integral is given below, while ing rate related to the inverse carrier lifetime can still
the calculational details have been collected in App. C. be defined from the out-scattering part of the inelastic
The resulting expressions for the collision integral apply collision integral alone. The scattering rate so defined is
to both intravalley and intervalley optical and intervalley given by
acoustic phonons.
In the following, the integral operator for inelastic scat- 1 X 1 − fk0′
= Pkk′ (30)
tering in Eq. (16) is split up in separate out- and in- τkinel 1 − fk0
k′
scattering contributions,
out in
and corresponds to the imaginary part of the elec-
I ′ inel [φk ] = I ′ inel [φk ] + I ′ inel [φk ], (26) tronic self-energy in the Born approximation.46 Below,
the resulting scattering rates for zero-order and first-
which include the terms in Eq. (21) involving ψk and order deformation potential scattering are given for non-
ψk±q , respectively. With the contributions from differ- degenerate carriers, i.e. with the Fermi factors in Eq. (30)
ent phonon branches λ adding up, scattering on a single neglected. They follow straight-forwardly from the ex-
phonon with branch index λ is considered in the follow- pressions for the out-scattering part of the collision in-
ing. tegral derived in App. C. It should be noted that the
From Eq. (21), the out-scattering part of the collision scattering rate for the zero-order deformation poten-
integral follows directly as tial interaction given below in Eq. (31), in fact defines
a proper momentum relaxation time because the in-
out
X 1 − fk0′ scattering part of the collision integral vanishes in this
I ′ inel [φk ] = −φk Pkk′ , (27)
1 − fk0 case (see App. C).
k′
For zero-order deformation potential scattering, the
where the transition matrix element for optical phonon scattering rate is independent of the carrier energy and
 9

given by the total scattering rate from the various electron-phonon

couplings to the intra and intervalley phonons which have
m∗ (Dλ0 )2
 
1 ~ωλ /kB T been grouped according to their coupling type, i.e. acous-
= 1+e Θ(εk − ~ωλ ) Nλ0 . (31)
τkinel 2~2 ρωλ tic deformation potentials (ADPs), zero/first-order opti-
cal deformation potentials (ODPs) and the Fröhlich in-
Here, Θ(x) is the Heavyside step function which assures teraction. The acoustic deformation potential scatter-
that only electrons with sufficient energy can emit a ing includes the quasielastic intravalley scattering on the
phonon. TA and LA phonons with linear dispersions. Scatter-
The scattering rate for coupling via the first-order de- ing on intervalley acoustic phonons is considered as op-
formation potential is found to be tical deformation potential scattering. Both the total
scattering rate and the contributions from the different
m∗2 (Dλ1 )2

1 coupling types have been obtained using Matthiessen’s
= (2εk + ~ωλ ) + e~ωλ /kB T
τkinel ~4 ρωλ rule by summing the scattering rates from the individual
 phonons, i.e.
× Θ(εk − ~ωλ ) (2εk − ~ωλ ) Nλ0 . X
−1
τtot = τλ−1 . (33)
(32)
λ

Due to the linear dependence on the carrier energy, zero- For carrier energies below the optical phonon frequen-
order scattering processes dominate first-order processes cies, the total scattering rate is dominated by acoustic de-
at low energies. Only under high-field conditions where formation potential scattering. At higher energies zero-
the carriers are accelerated to high velocities will first- order deformation potential scattering and polar optical
order scattering become significant. scattering via the Fröhlich interaction become dominant.
The expressions for the scattering rates in Eqs. (31) Due to the linear dependence on the carrier energy, the
and (32) above apply to scattering on dispersionless in- first-order deformation potential scattering on the inter-
tervalley acoustic phonons and intra/intervalley
√ optical valley acoustic phonons and the optical phonons is in
phonons. Except for a factor of εk ± ~ωλ originat-
ing from the density of states, the energy dependence
of the scattering rates is identical to that of their three- Parameter Symbol Value
dimensional analogs.38 Lattice constant a 3.14 Å
Ion mass density ρ 3.1 × 10−7 g/cm2
V. RESULTS Effective electron mass m∗ 0.48 me
Transverse sound velocity cTA 4.2 × 103 m/s
In the following, the scattering rate and phonon- Longitudinal sound velocity cLA 6.7 × 103 m/s
limited mobility in single-layer MoS2 are studied as a Acoustic deformation potentials
function of carrier energy, temperature T and carrier den- TA ΞTA 1.6 eV
sity n using the material parameters collected in Tab. I. LA ΞLA 2.8 eV
Here, the reported deformation potentials represent ef- Optical deformation potentials
fective coupling parameters for the deformation potential TA 1
DK,TA 5.9 eV
approximation in Eqs. (6) and (7) (see below). LA 1
DK,LA 3.9 eV
1
TO DΓ,TO 4.0 eV
1
A. Scattering rates TO DK,TO 1.9 eV
0
LO DK,LO 2.6 × 108 eV/cm
0
With access to the first-principles electron-phonon Homopolar DΓ,HP 4.1 × 108 eV/cm
couplings, the scattering rate taking into account the Fröhlich interaction (LO)
anisotropy and more complex q-dependence of the first- Effective layer thickness σ 4.41 Å
principles coupling matrix elements can be evaluated. Coupling constant gFr 98 meV
They are obtained using the expression for the scattering Optical phonon energies
rate in Eq. (30) with the respective transition matrix el- Polar LO ~ωLO 48 meV
ements for acoustic and optical phonon scattering given
Homopolar ~ωHP 50 meV
in Eqs. (23) and (28).50 In order to account for the dif-
ferent coupling matrix elements in the K, K ′ -valleys, the
scattering rates have been averaged over different high- TABLE I: Material parameters for single-layer MoS2 . Un-
symmetry directions of the carrier wave vector k. The less otherwise stated, the parameters have been calculated
resulting scattering rates are shown in Fig. 6 as a func- from first-principles as described in the Secs. II and III. The
tion of the carrier energy for non-degenerate carriers at Γ/K-subscript on the optical deformation potentials, indicate
T = 300 K. The different lines show the contributions to couplings to the intra/intervalley phonons.
 10

for the anisotropy and the full q-dependence of the first-

14 T =300 K principles electron-phonon couplings. However, as mo-
10
mentum and energy conservation limit phonon scattering
Scattering rate (s 1 ) to involve phonons in the vicinity of the Γ/K-point,39
1013 the fitting procedure yields deformation potentials close
to the direction averaged q → Γ/K limiting behavior
of the first-principles coupling matrix elements. For the
1012 ADP zero-order deformation potentials, the sampling of the
Zero-order ODP coupling matrix elements away from the Γ-point in the
First-order ODP k′ sum in Eq. (30), leads to a deformation potentials
1011 Fr hlich which is slightly smaller than the Γ-point value of the
Total coupling matrix element.
Fitted The total scattering rate resulting from the fitted de-
10100.00 0.05 0.10 0.15 0.20 0.25
Energy (eV) 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-
FIG. 6: (Color online) Scattering rates for the different proximation given by the energy-weighted average of the
electron-phonon couplings as a function of carrier energy at relaxation time (see Eq. (18)), the difference between the
T = 300 K. The scattering rates have been calculated from
associated mobilities will be negligible. Hence, the defor-
the first-principles electron-phonon couplings in Figs. 4 and 5.
The (black) dashed line shows the scattering rate obtained
mation potentials provide well-founded electron-phonon
using the fitted deformation potential parameters defined in coupling parameters for low-field studies of the mobility.
Eqs. (6) and (7). The kinks in the curves for the optical scat-
tering rates mark the onset of optical phonon emissions.
C. Mobility

general only of minor importance for the low-field mo- The high density of states in the K, K ′ -valleys of the
bility. This is also the case here, where it is an order conduction band in general results in non-degenerate car-
of magnitude smaller than the other scattering rates for rier distributions in single-layer MoS2 . This is illustrated
almost the entire plotted energy range. The jumps in in the left plot of Fig. 7 which shows the carrier density
the curves for the optical scattering rates at the optical versus the position of the Fermi level for different temper-
phonon energies ~ωλ are associated with the threshold atures. At room temperature, carrier densities in excess
for optical phonon emission where the carriers have suf- of ∼ 8 × 1012 cm−2 are needed to introduce the Fermi
ficient energy to emit optical phonons level into the conduction band and probe the Fermi-Dirac
statistics of the carriers. Thus, only at the highest re-
ported carrier densities of n ∼ 1013 cm−2 ,5 are the carri-
B. Deformation potentials ers degenerate at room temperature. For the lowest tem-
perature T = 100 K, the transition to degenerate carriers
In this section, we determine the deformation potential occurs at a carrier density of ∼ 2 × 1012 cm−2 . The tran-
parameters in single-layer MoS2 . They can be used in sition from non-degenerate to degenerate distributions is
the following study of the low-field mobility within the also illustrated by the discrepancy between the full and
Boltzmann approach outlined in Section IV. dashed lines which shows the Fermi level obtained with
The energy dependence of the first-principles based Boltzmann statistics.
scattering rates in Fig. 6 to a high degree resembles that The right plot of Fig. 7 shows the phonon-limited drift
of the analytic expressions for the deformation poten- mobility calculated with the full collision integral as a
tial scattering rates in Eqs. (25), (31) and (32). For function of carrier density for the same set of tempera-
example, the acoustic and zero-order deformation po- tures. The dashed lines represent the results obtained
tential scattering rates are almost constant in the plot- with Boltzmann statistics and the relaxation time ap-
ted energy range. The first-principles electron-phonon proximation using the expressions in Section IV for opti-
couplings can therefore to a good approximation be de- cal phonon scattering and Matthiessen’s rule for the to-
scribed by the simpler isotropic deformation potentials tal relaxation time. The strong drop in the mobility from
in Eqs. (6) and (7). The deformation potentials are ob- ∼ 2450 cm2 V−1 s−1 at T = 100 K to ∼ 400 cm2 V−1 s−1
tained by fitting the associated scattering rates for each at T = 300 K, is a consequence of the increased phonon
of the intra and intervalley phonons separately to the scattering at higher temperatures due to larger phonon
first-principles scattering rates. The resulting deforma- population of, in particular, optical phonons. The rela-
tion potential values are summarized in Tab. I. In analogy tively low intrinsic room-temperature mobility of single-
with deformation potentials extracted from experimental layer MoS2 can be attributed to both the significant
mobilities, the theoretical deformation potentials repre- phonon scattering and the large effective mass of 0.48
sent effective coupling parameters that implicitly account me in the conduction band. While the mobility decreases
 11

n =1011 cm2
14
3000
T =100 K
10
2)
 104

1)
T =200 K 
Acoustic
cm
2500

cm2 V 1 s
T =300 K 
Total
Carrier density (

13
2000
T =100 K
10

Mobility (cm2 V 1 s 1 )
1500 T =200 K  Quenched
T =300 K

Mobility (
10
12
1000 
500

10
11
0 11 12 13
103
Ec (eV)
0.10 0.05 0.00 0.05 0.10

cm2 )
10 10 10

F Carrier density (

FIG. 7: (Color online) Left: Carrier density as a function

of the Fermi energy for different temperatures. The Fermi
energy is measured relative to the conduction band edge Ec 2
Right: Mobility as a function of carrier density for the same 10100 200 300 400 500
temperatures. In both plots, the (black) dashed lines show the Temperature (K)
results obtained with Boltzmann statistics and the relaxation
time approximation (only right plot). FIG. 8: (Color online) Mobility vs temperature. For compari-
son, the mobility in the presence of only acoustic deformation
potential scattering with the µ ∼ T −1 temperature depen-
strongly with increasing temperature, it is relatively inde- dence is shown. The (gray) shaded area shows the variation
pendent on the carrier density. The weak density depen- in the mobility associated with a 10% uncertainty in the cal-
dence of the mobility originates from the energy-weighted culated deformation potentials.
average in Eq. (20) where the derivative of the Fermi-
Dirac distribution changes slowly as a function of carrier
density for non-degenerate carriers. As the Fermi energy stronger temperature dependence with γ > 1. In this
is introduced into the band with increasing carrier den- regime, the exponent depends on the optical phonon fre-
sity, the derivative of the Fermi-Dirac distribution to a quencies and the electron-phonon coupling strength. In
larger extent probes the scattering rate at higher energies an early study of the in-plane mobility of bulk MoS2 ,14
leading to a decrease of the mobility. This effect is most the measured room-temperature exponent of γ ∼ 2.6 was
prominent at T = 100 K where the level of degeneracy is found to be consistent with scattering on the homopolar
larger compared to higher temperatures. mode via a zero-order deformation potential.
Surprisingly, the relaxation time approximation is seen In Fig. 8, we show the temperature dependence of the
to work extremely well. The deviation from the full treat- mobility at n = 1011 cm−2 calculated with the full col-
ment at high carrier densities stems from the assumption lision integral. For comparison, the mobility limited by
of non-degenerate carriers and not the relaxation time acoustic phonon scattering with the µ ∼ T −1 tempera-
approximation. The reason for the good performance ture dependence is also included. To illustrate the effect
of the relaxation time approximation shall be found in of an uncertainty in the calculated deformation poten-
the in-scattering part of the collision integral in the full tials, the shaded area shows the variation of the mo-
treatment. As the in-scattering part of the collision in- bility with a change in the deformation potentials of
tegral vanishes for zero-order deformation potential cou- ±10%. The calculated room-temperature mobility of
pling and is small compared to the out-scattering part ∼ 410 cm2 V−1 s−1 is in fair agreement with the recently
otherwise, the full collision integral does not differ signif- reported experimental value of ∼ 200 cm2 V−1 s−1 in
icantly from the corresponding scattering rate as defined the top-gated sample of Ref.8 where additional scattering
in Eq. (30) of Section IV, i.e. with the in-scattering part mechanisms as e.g. impurity and surface-optical phonon
neglected. The good performance of the relaxation time scattering must be expected to exist. Over the plotted
approximation therefore seems to be of general validity temperature range, the mobility undergoes a transition
for the phonon collision integral even in the presence in- from being dominated by acoustic phonon scattering at
elastic scattering on optical phonons. T = 100 K to being dominated by optical phonon scat-
Finally, we study the temperature dependence of the tering at higher temperatures with a characteristic ex-
mobility in more detail. In general, room temperature ponent of γ > 1. At room temperature, the mobility
mobilities are to a large extent dominated by optical follows a temperature dependence with γ = 1.69. The
phonon scattering. This is manifested in the temperature increase in the exponent is almost exclusively due to the
dependence of the mobility which follows a µ ∼ T −γ law optical zero-order deformation potential couplings and
where the exponent γ depends on the dominating scat- the Fröhlich interaction while the first-order deformation
tering mechanism. For acoustic phonon scattering above potential couplings contribute only marginally. The ex-
the Bloch-Grüneisen temperature, the temperature de- ponent found here is considerably lower than the above-
pendence of the scattering rate in Eq. (25) results in mentioned exponent of γ ∼ 2.6 for bulk MoS2 , indicat-
γ = 1. At higher temperatures where optical phonon ing that the electron-phonon coupling in bulk and single-
scattering starts to dominate, the mobility acquires a layer MoS2 differ. Indeed, the transition from an indirect
 12

band gap in bulk MoS2 to a direct band gap in single- purity concentration needed to dominate phonon scat-
layer MoS2 which shifts the bottom of the conduction tering is in agreement with the experimental observation
band from the valley located along the Γ-K path to the that low-mobility single-layer MoS2 samples are heavily
K, K ′ -valleys, could result in a change in the electron- doped semiconductors.5
phonon coupling. As discussed in Section II, independent GW quasi-
In top-gated samples as the one studied in Ref.8, the particle calculations53,54 suggest that the ordering of the
sandwiched structure with the MoS2 layer located be- valleys at the bottom of the conduction band might not
tween substrate and gate dielectric, is likely to result in a be as clear as predicted by DFT. If this is the case, the
quenching of the characteristic homopolar mode which is satellite valleys inside the Brillouin zone must also be
polarized in the direction normal to the layer. To address taken into account in the solution of Boltzmann equa-
the consequence of such a quenching, the mobility in the tion. This gives rise to additional intervalley scattering
absence of the zero-order deformation potential originat- channels that together with the larger average effective
ing from the coupling to homopolar mode is also shown electron mass m∗ ∼ 0.6 me of the satellite valley will
in Fig. 8. Here, the curve for the quenched case shows a result in mobilities below the values predicted here.
decrease in the characteristic exponent to γ = 1.52 and In conclusion, we have used a first-principles based
an increase in the mobility of ∼ 70 cm2 V−1 s−1 at room approach to establish the strength and nature of the
temperature. Despite the significant deformation poten- electron-phonon coupling and calculate the intrinsic
tial of the homopolar mode, the effect of the quenching phonon-limited mobility in single-layer MoS2 . The cal-
on the mobility is minor. culated room-temperature mobility of 410 cm2 V−1 s−1
is to a large extent dominated by optical deformation
potential scattering on the intravalley homopolar and
VI. DISCUSSIONS AND CONCLUSIONS intervalley LO phonons as well as polar optical scat-
tering on the intravalley LO phonon via the Fröhlich
Based on our finding for the phonon-limited mobility in interaction. The mobility follows a µ ∼ T −1.69 tem-
single-layer MoS2 , it seems likely that the low experimen- perature dependence at room temperature characteris-
tal mobilities of ∼ 1 cm2 V−1 s−1 reported in Refs.5,6,8 tic of optical phonon scattering. A quenching of the
are dominated by other scattering mechanisms as e.g. homopolar mode likely to occur in top-gated samples,
charged impurity scattering. The main reason for the in- results in a change of the exponent in the tempera-
crease in the mobility to 200 cm2 V−1 s−1 observed when ture dependence of the mobility to 1.52. With effective
depositing a high-κ dielectric on top of the MoS2 layer masses, phonons and measured mobilities in other semi-
in Ref.8 is therefore likely to be impurity screening. The conducting metal dichalcogenides being similar to those
quenching of the out-of-plane homopolar phonon only led of MoS2 ,14,23 room-temperature mobilities of the same
to a minor increase in the mobility and cannot alone ex- order of magnitude can be expected in their single-layer
plain the above-mentioned increase in the mobility. It forms.
may, however, also contribute. A comparison of the tem-
perature dependence of the mobility in samples with and
without the top-gate structure can help to clarify the ex- Acknowledgments
tent to which a quenching of optical phonons contributes
to the experimentally observed mobility increase. With The authors would like to thank O. Hansen and J.
our theoretically predicted room-temperature mobility of J. Mortensen for illuminating discussions. KK has been
410 cm2 V−1 s−1 , the observed enhancement in the mo- partially supported by the Center on Nanostructuring
bility to 200 cm2 V−1 s−1 induced by the high-κ dielec- for Efficient Energy Conversion (CNEEC) at Stanford
tric, suggests that dielectric engineering51 is an effective University, an Energy Frontier Research Center funded
route towards phonon-limited mobilities in 2D materials by the U.S. Department of Energy, Office of Science,
via efficient screening of charge impurities. Office of Basic Energy Sciences under Award Number
A rough estimate of the impurity concentrations re- de-sc0001060. CAMD is supported by the Lundbeck
quired to dominate phonon scattering can be inferred Foundation.
from the phonon scattering rate of τ −1 ∼ 1013 s−1 in
Fig. 6. The scattering rate is related to the mean-free
path λ of the carriers via λ = τ hvi where hvi is their mean
velocity. Using the velocity of the mean energy carriers Appendix A: First-principles calculation of the
for a non-degenerate distribution where hεk i = kB T , we electron-phonon interaction
find a mean-free path of λ ∼ 14 nm at T = 300 K. In
order for impurity scattering to dominate, the impurity The first-principles scheme for the calculation of the
spacing must be on the order of the phonon mean-free electron-phonon interaction used in this work is outlined
path or smaller. This results in a minimum impurity in the following. Contrary to other first-principles ap-
concentration of ∼ 5 × 1011 cm−2 in order to dominate proaches for the calculation of the electron-phonon inter-
phonon scattering. The high value of the estimated im- action which are based on pseudo potentials,41–43,55 the
 13

present approach is based on the PAW method56 and is are Bloch sums of the localized orbitals. Inserting the
implemented in the GPAW DFT package.25–27 Bloch sum expansion of the Bloch states in the matrix
Within the adiabatic approximation for the electron- element in Eq. (A1), the matrix element can be written
phonon interaction, the electron-phonon coupling matrix
elements for the phonon qλ and Bloch state k can be
expressed as X
hk + q|êqλ ·∇q V (r)|ki = c∗i cj hik + q|êqλ · ∇q V (r)|jki
s
ij
λ ~ X
gkq = hk + q|êqλ q
α · ∇α V (r)|ki, (A1) 1 X ∗ X i(k+G)·(Rn −Rm )−iq·(Rm −Rl )
2M N ωqλ α = 2 c cj e
N ij i
lmn
where N is the number of unit cells, M is an appropri- × hiRm |êqλ · ∇l V (r)|jRn i (A4)
ately defined effective mass, ωqλ is the phonon frequency,
and the q-dependent derivative of the effective potential
for a given atom α is defined by
X where the k-labels on the expansion coefficients have
∇qα V (r) = eiq·Rl ∇αl V (r), (A2) been discarded for brevity. The phase factor from the re-
l ciprocal lattice vector G can be neglected since Rn · G =
2πN . By exploiting the translational invariance of the
where the sum is over unit cells in the system and ∇αl V is crystal the matrix elements can be obtained from the
the gradient of the potential with respect to the position gradient in the primitive cell as
of atom α in cell Rl . The polarization vectors êqλ of the
phonons appearing in Eq. (A1) are normalized according
2
to α (Mα /M )|êqλ
P
α | = 1.
In order to evaluate the matrix element in Eq. (A1), hiRm |êqλ · ∇l V (r)|jRn i
the Bloch states are expanded in LCAO basis orbitals = hiRm − Rl |êqλ · ∇0 V (r)|jRn − Rl i (A5)
|iRn i where i = (α, µ) is a composite index for atomic
site and orbital index and Rn is the lattice vector to the
n’th unit cell. In the
P LCAO basis, the Bloch states are by performing the change of variables r′ = r − Rl in
expanded as |ki = i cki |iki where the integration. Inserting in Eq. (A4) and changing the
1 X ik·Rn summing variables to Rm − Rl and Rn − Rl we find for
|iki = e |iRn i. (A3) the matrix element
N n

1 X ∗ X ik·(Rn −Rm )−iq·Rm

hk + q|êqλ · ∇q V (r)|ki = c cj e hiRm |êqλ · ∇0 V (r)|jRn i
N 2 ij i
lmn
1 X ∗ X ik·(Rn −Rm )−iq·Rm
= c cj e hiRm |êqλ · ∇0 V (r)|jRn i (A6)
N ij i mn

Here, the sum over k in the first equality produces a are obtained as
factor of N . The result for the matrix element in Eq. (A6) ∂V V + (r) − Vαi
−
(r)
is similar to the matrix element reported in Eq. (22) of = αi . (A7)
Ref.55 where a Wannier basis was used instead of the ∂Rαi 2δ
LCAO basis. Here, Vαi±
denotes the effective potential with atom α
located at Rα in the reference cell displaced by ±δ in
direction i = x, y, z. The calculation of the gradient thus
amounts to carrying out self-consistent calculations for
From the last equality in Eq. (A6), the procedure for six displacements of each atom in the primitive unit cell.
a supercell-based evaluation of the matrix element has Having obtained the gradients of the effective potential,
emerged. The matrix element in the last equality in- the matrix elements in the LCAO basis of the super-
volves the gradients of the effective potential ∇0 V with cell must be calculated and the sums over unit cells and
respect to atomic displacement in the reference cell at R0 . atomic orbitals in Eq. (A6) can be evaluated.
These can be obtained using a finite-difference approxi- In general, the matrix elements of the electron-phonon
mation for the gradient where the individual components interaction must be converged with respect to the super-
 14

cell size and the LCAO basis. In particular, since the good approximation to the true quasi particle wave func-
supercell approach relies on the gradient of the effective tion, the matrix element follows as
potential going to zero at the supercell boundaries, the
supercell must be chosen large enough that the potential hk + q|êqλ · ∇q V (r)|ki = hψ̃k+q |êqλ · ∇q H|ψ̃k i, (A10)
at the boundaries is negligible. For polar materials where
the coupling to the polar LO phonon is long-ranged in where ∇q H is given in Eq. (A2). With the pseudo wave
nature, a correct description of the coupling can only be function expanded in an LCAO basis, the matrix element
obtained for phonon wave vectors corresponding to wave- is evaluated following Eq. (A6).
lengths smaller than the supercell size. Large supercells In order to verify the calculated matrix elements, we
are therefore required to capture the long-wavelength have carried out self-consistent calculations of the vari-
limit of the coupling constant (see e.g. main text). ation in the band energies with respect to atomic dis-
placements along the phonon normal mode. For the
coupling to the Γ-point homopolar phonon in single-
layer MoS2 , we find that the calculated matrix element
1. PAW details of 4.8 eV/Å agrees with the self-consistently calculated
value to within 0.1 eV/Å.
In the PAW method, the effective single-particle DFT As the matrix elements of the electron-phonon inter-
Hamiltonian is given by action are sensitive to the behavior of the potential and
wave functions in the vicinity of the atomic cores, vari-
1 XX ations in electron-phonon couplings obtained with dif-
H = − ∇2 + veff (r) + |p̃α α α
i1 i∆Hi1 i2 hp̃i2 |, (A8)
2 ferent pseudo-potential approximations can be expected.
α i i 1 2
We have confirmed this via self-consistent calculations of
where the first term is the kinetic energy, veff is the effec- the band energy variations, which showed that the cou-
tive potential containing contributions from the atomic pling to the homopolar Γ-point phonon increases with
potentials and the Hartree and exchange-correlation po- 0.3 eV/Å compared to the PAW value when using norm-
tentials and the last term is the non-local part includ- conserving HGH pseudo potentials.
ing the atom-centered projector functions p̃α i (r) and
the atomic coefficients ∆Hiα1 i2 . In contrast to pseudo-
Appendix B: Fröhlich interaction in 2D materials
potential methods where the atomic coefficients are con-
stants, they depend on the density and thereby also on
the atomic positions in the PAW method. In the long-wavelength limit, the lattice vibration of
The diagonal matrix elements (i.e. q = 0) in Eq. (A4) the polar LO mode gives rise to a macroscopic polar-
correspond to the first-order variation in the band en- ization that couples to the charge carriers. In three-
ergy εk upon an atomic displacement in the normal mode dimensional bulk systems, the coupling strength is given
direction êqλ . Together with the matrix elements for by Eq. 8 which diverges as 1/q. Using dielectric contin-
q 6= 0, these can be obtained from the gradient of the uum models, the coupling has been studied for confined
PAW Hamiltonian with respect to atomic displacements. carriers and LO phonons in semiconductor heterostruc-
The derivative of (A8) with respect to a displacement of tures44,45 where a non-diverging interaction is found. For
atom α in direction i results in the following four terms atomically thin materials, however, macroscopic dielec-
tric models are inappropriate. Using an approach based
∂H ∂veff X  ∂ p̃α on the atomic Born effective charges Z ∗ , a functional
= + | i1 i∆Hiα1 i2 hp̃α
i2 | form that fits the calculated values of the Fröhlich inter-
∂Rαi ∂Rαi i ,i ∂Rαi
1 2 action in Fig. 5 is here derived.
∂ p̃α
i2
+ |p̃α α
i1 i∆Hi1 i2 h |
∂Rαi
1. Polarization field and potential
∂∆Hiα1 i2 α

+ |p̃α
i1 i hp̃i2 | . (A9)
∂Rαi In two-dimensional materials, the polarization from
the lattice vibration of the polar LO phonon is oriented
While the gradient of the projector functions in the
along the plane of the layer. It can be expressed in terms
first and second terms inside the square brackets can
of the relative displacement uq of the unit cell atoms as
be evaluated analytically, the gradients of the effective
potential and the projector coefficients in the last term Z∗
are obtained using the finite-difference approximation in Pq (z) = uq fq (z), (B1)
ε∞ A
Eq. (A7).
Once the gradient of the PAW Hamiltonian has been where q is the two-dimensional phonon wave vector, ε∞
obtained, the matrix elements in Eq. (A4) can be ob- is the optical dielectric constant, Z ∗ is the Born effective
tained. Under the assumption that the smooth pseudo charge of the atoms (here assumed to be the same for all
Bloch wave functions ψ̃k from the PAW formalism is a atoms) and fq describes the profile of the polarization in
 15

the direction perpendicular (here the z-direction) to the Appendix C: Evaluation of the inelastic collision
layer. The associated polarization charge ρ = −∇ · P integral
is given by ρq = −iq · Pq . The resulting scalar poten-
tial which couples to the carriers follows from Poisson’s Following Eq. (26) in the main text, the inelastic colli-
equation. Fourier transforming in all three directions, sion integral for scattering on optical phonons is split up
Poisson’s equation takes the form in separate out- and in-scattering contributions,
Z∗
(q 2 + k 2 )φq (k) = −i q · uq fq (k), (B2) out in
I ′ inel [φk ] = I ′ inel [φk ] + I ′ inel [φk ], (C1)
ε∞ A
where k is the Fourier variable in the direction perpen- which are given by Eq. (27) and (29), respectively. With
dicular to the plane of the layer and q k uq for the LO the assumption of dispersionless optical phonons, the
phonon. Fermi-Dirac and Bose-Einstein distribution functions do
In 2D materials, the z-profile of the polarization field not depend on the phonon wave vector and can thus be
can to a good approximation be described by a δ- taken outside the q-sum.
function, i.e. The out-scattering part of the collision integral then
fq (z) = δ(z). (B3) takes the form
Inserting the Fourier transform fq (k) = 1 in Eq. (B2), out 2π φk
I ′ inel [φk ] = −
we find for the potential ~ 1 − fk0
Z ∗ uq q
Z "
φq (z) = −i dk eikz fq (k) 2 2
X
ε∞ A q + k2 × Nλ0 (1 − fk+q 0
) |gq | δ(εk+q − εk − ~ωλ )
∗ q
Z uq −q|z|
= −i e (B4) #
ε∞ A X 2
+ (1 + Nλ0 )(1 − 0
fk−q ) |gq | δ(εk−q − εk + ~ωλ ) .
which in agreement with the findings of Refs.44,45 does q
not diverge in the long-wavelength limit. (C2)

For the in-scattering part, the additional cos θk,k±q fac-

2. Electron-phonon interaction tor is rewritten as

In three-dimensional bulk systems, the q-dependence k ± q cos θk,q

cos θk,k±q = q , (C3)
of the Fröhlich interaction is given entirely by the 1/q- k 1 ± ~ω λ
εk
divergence of the potential associated with the lattice
vibration of the polar LO phonon. However, in two di- leading to the following general form of the in-scattering
mensions, the interaction follows by integrating the po- part of the collision integral
tential with the square of the envelope function χ(z) of
the electronic Bloch state. Hence, the Fröhlich interac- in 2π 1
tion is given by the matrix element I ′ inel [φk ] =
Z ~ 1 − fk0
"
gLO (q) = dz χ∗ (z)φq (z)χ(z). (B5) φk+q
× Nλ0 (1 − fk+q
0
) q
k 1 + ~ω λ
For simplicity, we here assume a Gaussian profile for the εk
envelope function |gq |2 (k + q cos θk,q )δ(εk+q − εk − ~ωλ )
X
×
1 2 2 q
χ(z) = √ e−z /2σ , (B6)
π 1/4σ φk−q
+ (1 + Nλ0 )(1 − fk−q
0
) q
where σ denotes the effective width of the electronic k 1 − ~ω
εk
λ

Bloch function. With this approximation for the enve- #

lope function, the Fröhlich interaction becomes X 2
× |gq | (k − q cos θk,q )δ(εk−q − εk + ~ωλ ) .
q2 σ2 /4
gLO (q) = gFr × e erfc(qσ/2) q
(C4)
≈ gFr × erfc(qσ/2) (B7)
where gFr is the Fröhlich coupling constant, erfc is the This depends on the deviation function in φk±q ≡ φ(εk ±
complementary error function and the last equality holds ~ωλ ) and thus couples the deviation function at the initial
in the long-wavelength limit where q −1 ≫ σ. As shown energy εk of the carrier to that at εk ± ~ωλ . The two
in Fig. 5 in the main text, this functional form for the terms inside the square brackets account for emission and
Fröhlich interaction gives a perfect fit to the calculated absorption out of the state k ± q and into the state k,
electron-phonon coupling for the polar LO mode. respectively.
 16

1. Integration over q In the case of phonon emission, both possible roots in

Eq. (C9)
The in- and out-scattering contributions to the inelas-
tic part of the collision integral in Eqs. (C2) and (C4) r
e ~ωλ
can be written on the general form q± = k cos θk,q ± k cos2 θk,q − (C12)
εk
in/out
X
I ′ inel (k) = f (q)δ(εk±q − εk ∓ ~ωqλ ), (C5)
q can take on positive values. However, the minus
where the function f accounts for the q-dependent func- sign inside the square root restricts the allowed val-
tions inside the q-sum. The conservation of energy and ues of the integration angle to the range θk,q ≤ θ0 =
momentum in a scattering event is secured by the δ- arccos (~ωλ /εk )1/2 . Furthermore, in order to secure a
function entering the collision integral. positive argument to the square root, the electron en-
Following the procedure outlined in Ref.14, the integral ergy must be larger than the phonon energy, εk > ~ωλ .
over q resulting from the q-sum in the collision integral is Hence, the emission terms can be obtained as
evaluated using the following property of the δ-function,
Z X F (qn ) A
dq F (q)δ [g(q)] = . (C6) I ′ e (k) = Θ(εk − ~ωλ )
|g ′ (qn )| (2π)2
n Z θ0 e
m∗ q+ f (q e ) + q−
e
f (q−e
)
Here, F (q) = qf (q) and qn are the roots of g. Rewriting × dθk,q 2
q + . (C13)
−θ0 k~ cos2 θk,q − ~ω λ
the argument of the δ-function in Eq. (C5) as εk

~2 (k ± q)2 ~2 k 2
εk±q − εk ∓ ~ωqλ = ∗
− ∓ ~ωqλ
2m 2m∗
~2 2m∗ ωqλ
 
2
= q ± 2kq cos θ k,q ∓ 2. Collision integral for optical phonon scattering
2m∗ ~
2
~
≡ g(q) (C7)
2m∗ In the following two sections, analytic expressions for
we get collision integral in the case of zero- and first-order op-
tical deformation potential coupling are given. For the
A 2m∗ X qn f (qn )
Z
′ Fröhlich interaction the integration over θ can not be re-
I (k) = dθk,q . (C8)
(2π)2 ~2 n |g ′ (qn )| duced to a simple analytic form and is therefore evaluated
numerically.
Depending on the q-dependence of the phonon frequency,
different solutions for the roots qn result.

a. Dispersionless optical phonons a. Zero-order deformation potential

With the assumption of dispersionless optical phonons, For the zero-order deformation potential coupling in
i.e. ωqλ = ωλ , the roots of g in Eq. (C7) become Eq. (7), the θ-integration in the out-scattering part of
r
~ωλ the collision integral yields a factor of 2π resulting in the
q/k = ∓ cos θk,q ± cos2 θk,q ± (C9) following form
εk
for absorption (upper sign in the first and last term) and
out m∗ D2 φk
emission (lower sign in the first and last term), respec- I ′ inel [φk ] = − 2 λ
tively. 2~ ρωλ 1 − fk0
a
 
For absorption there is only one positive root q+ given 0 0 ~ωλ /kB T
by the plus sign in front of the square root, × (1 − fk+q ) + (1 − fk−q )e Θ(εk − ~ωλ ) Nλ0 .
r (C14)
a ~ωλ
q+ = −k cos θk,q + k cos2 θk,q + (C10)
εk
For the in-scattering part, the q-independence of the de-
where all values of θk,q are allowed. The absorption terms
formation potential results in a vanishing q-sum and the
in the collision integral then becomes
in-scattering contribution is zero, i.e.
Z 2π a a
′ A m∗ q+ f (q+ )
I a (k) = dθ k,q . (C11)
(2π)2 0 k~2 cos2 θ + ~ωλ
q
in
k,q εk
I ′ inel [φk ] = 0. (C15)
 17

b. First-order deformation potential and

For first-order deformation potentials, we find for the

out-scattering part

m∗2 Ξ2 φk

out 0
I ′ inel [φk ] = − 4 λ × 0
(1 − fk+q )(2εk + ~ωλ ) in m∗2 Ξ2λ 1 − fk−q φk−q
~ ρωλ 1 − fk0 I ′ inel[φk ] = −
~4 ρωλ 1 − fk0
q
 1 − ~ωλ
εk
0
+ (1 − fk−q )(2εk − ~ωλ )e~ωλ /kB T Θ(εk − ~ωλ ) Nλ0 .
× (1 + Nλ0 ) (εk
− ~ωλ ) Θ(εk − ~ωλ )
(C16) r r !
1 π ~ωλ ~ωλ
× + arccos + arcsin , (C18)
For the in-scattering part, the result for the first and π 2 εk εk
second term inside the square brackets in Eq. (C4) is
0
in m∗2 Ξ2λ 1 − fk+q φk+q
I ′ inel [φk ] = − Nλ0 (εk + ~ωλ )
~4 ρωλ 1 − fk0
q
~ωλ
1+ εk
(C17) which accounts for absorption and emission, respectively.

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