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

1.7 MB·English

by  Kristen Kaasbjerg

First-principles study of the phonon-limited mobility in n-type single-layer MoS
2
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)
Inthepresentworkwecalculatethephonon-limitedmobilityinintrinsicn-typesingle-layerMoS2
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 Fr¨ohlich in-
2 teractionintheisolated MoS2 layeraredetermined. Wefindthatthecalculated room-temperature
1 mobilityof∼410cm2 V−1 s−1 isdominatedbyopticalphononscatteringviadeformationpotential
0 couplings and the Fr¨ohlich interaction with the deformation potentials to the intravalley homopo-
2 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-
n
perature dependence with γ = 1.69 at room temperature. It is shown that a quenching of the
a
characteristic homopolar mode which is likely to occur in top-gated samples, boosts the mobility
J
with ∼ 70 cm2 V−1 s−1 and can be observed as a decrease in the exponent to γ = 1.52. Our
5
findingsindicatethattheintrinsicphonon-limitedmobilityisapproachedinsampleswhereahigh-κ
2
dielectric that effectively screens charge impurities is used as gate oxide.

]
i PACSnumbers: 81.05.Hd,72.10.-d,72.20.-i,72.80.Jc
c
s
-
l I. INTRODUCTION In order to shed further light on the measured mobili-
r
t tiesandthepossibleroleofphonondampingduetoatop-
m
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 MoS . Since
at ration of its unique electronic properties,1–3 the search phonon scattering is an intrinsic scattering mec2hanism
m 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-
d Metaldichalcogenideswhicharelayeredmaterialssimilar
per limit for the experimentally achievable mobilities in
n tographite,provideinterestingcandidates. Theirlayered
single-layerMoS .15 Inexperimentalsituations,thetem-
o 2
structure with weak interlayer van der Waals bonds al-
c peraturedependenceofthemobilitycanbeusefulforthe
lows for fabrication of single- to few-layer samples using
[ identification of dominating scattering mechanisms and
mechanicalpeeling/cleavageorchemicalexfoliationtech-
also help to establish the value of the intrinsic electron-
1 niques similar to the fabrication of graphene.5–8 How- phononcouplings.16 However,theexistenceofadditional
v
ever, in contrast to graphene they are semiconductors
4 extrinsicscatteringmechanismsoftencomplicatesthein-
and hence come with a naturally occurring band gap—a
8 terpretation. For example, in graphene samples impu-
property essential for electronic applications.
2 rity scattering and scattering on surface polar optical
5 Thevibrationalandopticalpropertiesofsingle-tofew- phononsofthegateoxideareimportantmobility-limiting
.
1 layer samples of MoS2 have recently been studied ex-
0 tensively using various experimental techniques.9–11 In
2 contrast to the bulk material which has an indirect-gap,
:1 it has been demonstrated that single-layer MoS2 is a a Mo 0.6 K K(cid:2) 0.45
v direct-gap semiconductor with a gap of 1.8 eV.10,12 To- a S ]/a 0.3 0.30
Xi gether with the excellent electrostatic control inherent [2(cid:1) 0.0 eV
ar oitfwtwelol--dsuimiteednsfioornallowmaptoewriearlsa,ptphleiclaatrigoensb.a13ndSogafparm, ealkeecs- ky--00..63 0.15
trical characterizations of single-layer MoS2 have shown -0.6-0.30.0 0.3 0.6 0.00
n-typeconductivitywithroom-temperaturemobilitiesin kx [2(cid:0)/a]
the range 0.5 3 cm2 V−1 s−1.5,6,8 Compared to early
− FIG.1: (Coloronline)Atomicstructureandconductionband
studies of the intralayer mobility of bulk MoS where
mobilities in the range 100 260 cm2 V−1 s−21 were of single-layer MoS2. Left: Primitive unit cell and structure
reported,14 this is rather low−. In a recent experiment ofanMoS2 layerwiththemolybdenumandsulfuratomspo-
sitioned in displaced hexagonal layers. Right: Contour plot
the use of a high-κ gate dielectric in a top-gated device
showingthelowestlyingconductionbandvalleysasobtained
was shown to boost the carrier mobility to a value of withDFT-LDAinthehexagonalBrillouinzoneofsingle-layer
200cm2 V−1 s−1.8 Theobservedincreaseinthemobility MoS2. Forn-typeMoS2,thelow-fieldmobility isdetermined
was attributed to screening of impurities by the high-κ bythecarrierproperties intheK,K′-valleyswhich aresepa-
dielectric and/or modifications of MoS phonons in the rated in energy from the satellite valleys inside the Brillouin
2
top-gated sample. zone.
2
factors.17–22 It can therefore be difficult to establish the
valueandnatureoftheintrinsicelectron-phononinterac-
tion from experiments alone. Theoretical studies there- 4
fore provide useful information for the interpretation of
experimentally measured mobilities.
) 2
Inthepresentwork,theelectron-phononinteractionin V
e
single-layerMoS2 iscalculatedfromfirst-principlesusing y ( 0 1.6
a density-functional based approach. From the calcu- g
r 1.4
latedelectron-phononcouplings,theacoustic(ADP)and e
n
optical deformation potentials (ODP) and the Fr¨ohlich E 2 1.2
(cid:4)
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 (cid:4) 0.8 K M
(cid:5)
regime(T >100K)wherephononscatteringoftendom- M K M
(cid:3)
inate the mobility. Our findings demonstrate that the
calculatedphonon-limitedroom-temperaturemobility of
FIG. 2: (Color online) Band structure of single-layer MoS2.
410cm2 V−1 s−1 is dominatedby opticaldeformation
Theinsetshowsazoomoftheconductionbandalongthepath
∼
potential and polar optical scattering via the Fr¨ohlich 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
D0 = 4.1 108 eV/cm and D0 = 2.6 108 eV/cm the conduction band valley in the K-point for the range of
HP × LO ×
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-
viousstudiesforbulkMoS ,wherethemobilityhasbeen
2
assumed to be dominated entirely by scattering on the latticeconstantofthehexagonalunitcellshowninFig.1
homopolar mode.14,23,24 Furthermore, we show that a was found to be a = 3.14 ˚A using a 11 11 k-point
×
quenchingofthecharacteristichomopolarmodewhichis sampling of the Brillouin zone. In order to eliminate in-
polarized in the direction normal to the MoS layer can teractions between the layers due to periodic boundary
2
be observedas a changein the exponent γ ofthe generic conditions, a large interlayer distance of 10 ˚A has been
temperaturedependenceµ T−γ ofthemobility. Sucha used in the calculations.
∼
quenchingcanbeexpectedtooccurintop-gatedsamples
where the MoS layeris sandwichedbetweena substrate
2
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
Inthe presentstudy,the bandstructureofsingle-layer
MoS2asobtainedwithDFT-LDAarepresented. Thelat- MoS is calculated within DFT-LDA.28,29 While DFT-
2
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 Fr¨ohlichinteractionwith the polar LO phononis de- the band structure in standard semiconductors,52 sug-
termined. In Section IV, the calculation of the phonon- gestthat the distance between the K,K′-valleysandthe
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
talconsensusthatsingle-layerMoS isadirect-gapsemi-
2
ofcarrierdensityandtemperature. Finally,inSectionVI 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 MoS have been calculated within DFT in M path in the Brillouin zone. As demonstrated by the
2
the LDA approximation using a real-space projector- dashedline,theconductionbandisperfectlyparabolicin
augmented wave (PAW) method.25–27 The equilibrium theK-valley. Furthermore,thesatellitevalleypositioned
3
onthe pathbetweenthe Γ-pointandthe K-pointlies on
the orderof 200meV abovethe conduction bandedge 60
∼
and is therefore not relevant for the low-field transport.
TherightplotinFig.1showsacontourplotofthelow- 50
)
V
estlying conductionbandvalleys inthe two-dimensional
e
Brillouin zone. Here, the K,K′-valleys that are popu- m40
(
lated in n-type MoS2, are positioned at the corners of y
the hexagonalBrillouin zone. Due to their isotropic and nc30
e
parabolic nature, the part of the conduction band rele- u
q20
vant for the low-field mobility can to a good approxima-
e
tion be described by simple parabolic bands Fr
10
~2k2
ε = (1)
k 2m∗ 0 K M
(cid:6) (cid:6)
with an effective electron mass of m∗ = 0.48 m and
e
where k is measured with respect to the K,K′-points
FIG. 3: Phonon dispersion of single-layer MoS2 calculated
inthe Brillouinzone. The two-dimensionalnatureofthe with the small displacement method (see e.g. Ref.30) using
carriersisreflectedintheconstantdensityofstatesgiven a 9×9 supercell. The frequencies of the two optical Raman
by ρ0 = gsgvm∗/2π~2 where gs = 2 and gv = 2 are the active E2g and A1g modes at 48 meV and 50 meV, respec-
spinandK,K′-valleydegeneracy,respectively. Thelarge tively, are in excellent agreement with recent experimental
density of states which follows from the high effective measurements.9
massoftheconductionbandintheK,K′-valleys,results
in non-degenerate carrier distributions except for very
high carrier densities.
results in the so-calledLO-TOsplitting between the two
modes in the long-wavelength limit. The inclusion of
this effect from first-principles requires knowledge of the
B. Phonon dispersion
Borneffectivecharges.32–34Intwo-dimensionalmaterials,
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-basedsmall-displacementmethod30 usinga9 couplingtothemacroscopicpolarizationoftheLOmode
×
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-calledhomopolarmodewhichischara∼cteristicforlay-
With three atoms in the unit cell, single-layer MoS
2 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-wavelengthlimit, the frequency
→ stationary. The change in the potential associated with
ofthe remaining transverseacoustic (TA) and longitudi-
this lattice vibration has previously been demonstrated
nalacoustic(LA) modes aregivenbythe in-plane sound
toresultinalargedeformationpotentialinbulkMoS .14
velocity c as 2
λ
ω =c q. (2)
qλ λ
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 andopticalbranches evenatthe high-
In this section, we present the electron-phonon cou-
symmetry points at the zone-boundarywhere the acous-
plinginsingle-layerMoS obtainedwithafirst-principles
tic and optical modes become similar. The two lowest 2
based DFT approach. The method is based on a su-
optical branches belong to the non-polar optical modes.
percell approach analogous to that used for the calcula-
Due to an insignificant coupling to the charge carriers,
tion of the phonon dispersion and is outlined in App. A.
they are not relevant for the present study.
The calculated electron-phonon couplings are discussed
The next two branches with a phonon energy of
∼ inthecontextofdeformationpotentialcouplingsandthe
48 meV at the Γ-point are the transverse (TO) and lon-
Fr¨ohlich interaction well-known from the semiconductor
gitudinal (LO) polar optical modes where the Mo and S
literature.
atoms vibrate in counterphase. In bulk polar materials,
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 phononinteraction,thecouplingstrengthforthephonon
4
mode with wavevectorq andbranchindex λ is givenby for the effective mass, MN = Aρ, and the expression in
Eq. (5) is obtained.
~ For scattering on acoustic phonons, the coupling ma-
gλ = Mλ , (3)
kq s2MNωqλ kq trix element is linear in q in the long-wavelengthlimit,
M =Ξ q, (6)
whereω isthephononfrequency,M isanappropriately qλ λ
qλ
defined effective mass, N is the number of unit cells in
where Ξ is the acoustic deformation potential.
the crystal, and λ
Inthecaseofopticalphononscattering,bothcoupling
to zero- and first-order in q must be considered. The in-
Mλ = k+qδV (r)k (4)
kq h | qλ | i teractionviatheconstantzero-orderopticaldeformation
isthecouplingmatrixelementwherekisthewavevector potential Dλ0 is given by
of the carrier being scattered and δV is the change in
qλ M =D0. (7)
the effective potential per unit displacement along the qλ λ
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-
bothintraandintervalleyphononscatteringofthecarri-
ments. Therefore, only symmetry-allowed phonons can
ers in the K,K′-valleys need to be considered. Here, the
coupleto the carriersviathe zero-orderinteraction. The
coupling constants for these scattering processes are ap-
couplingviathefirst-orderinteractionisgivenbyEq.(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 D1. Both the
proach,theintraandintervalleyscatteringfortheK,K′- λ
zero- and first-order deformation potential coupling can
valleysarethusassumedindependentonthewavevector
give rise to intra and intervalley scattering.
of the carriers.
Theabsolutevalueofthecalculatedcouplingmatrixel-
In the following two sections, the calculated electron-
ements M areshowninFig.4forphononswhichcou-
phononcouplingsarepresented.36Thedifferentdeforma- | qλ|
pletothecarriersviadeformationpotentialinteractions.
tion potential couplings are discussed and the functional
Theyareplottedintherangeofphononwavevectorsrel-
form of the Fr¨ohlich interaction in 2D materials is es-
evantforintraandintervalleyscatteringprocesses,39and
tablished. Piezoelectriccouplingtotheacousticphonons
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
tenextractedasempiricalparametersfromexperimental as MK = MK′ . The three-fold rotational symmetry
mobilities,itcanbe difficulttodisentanglecontributions qλ −qλ
of the coupling matrix elements in q-space stems from
fromdifferentphonons. Here,thefirst-principlescalcula-
the symmetry of the conduction band in the vicinity of
tionoftheelectron-phononcouplingallowsforadetailed
theK,K′-valleys(seeFig.1). Sincethesymmetryofthe
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.
The acoustic deformation potential couplings for the
A. Deformation potentials
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
carriersinteractwiththe localchangesin the crystalpo- oftenassumed.40Onlyalonghigh-symmetrydirectionsin
tentialassociatedwithalatticevibration. Withinthede-
theBrillouinzoneisthisthecase. Thisresultsinahighly
formation potential approximation, the electron-phonon
anisotropiccouplingtotheTAmode. Ontheotherhand,
coupling is expressed as38
the deformation potential coupling for the LA mode is
perfectly isotropic in the long-wavelength limit but also
~
becomes anisotropic at shorter wavelengths. In agree-
g = M , (5)
qλ qλ
s2Aρωqλ mentwith Eq.(6), both the TA andLAcoupling matrix
elements are linear in q in the long-wavelengthlimit.
where A is the area of the sample, ρ is the atomic mass The next two plots show the couplingsto the intraval-
densityperareaandM isthe couplingmatrixelement ley polar TO and homopolar optical modes. While the
qλ
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.3bysettingtheeffectivemassM equaltothesumof 4.8 eV/˚A 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
have significant weight here, the homopolar lattice vi-
Intravalley
brationgivesrisetoasignificantshiftoftheK,K′-valley
K K(cid:7)
states. The large deformation potential associated with
the coupling to the homopolar mode is also present in
bulk MoS .14
2
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 givesrise to a zero-orderdeformationpoten-
tialwhichresultsfromanidenticalin-planemotionofthe
twosulfurlayersinthelatticevibrationoftheintervalley
phonon.
In general, the deformation potential approximation,
i.e. the assumption of isotropic and constant/linearcou-
pling matrix elements, in Eqs. (6) and (7) is seen to
hold in the long-wavelengthlimit only. At shorter wave-
lengths,thefirst-principlescouplingsbecome anisotropic
Intervalley and have a more complicated q-dependence. When de-
K K(cid:8) 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-layerMoS aredeterminedfromthefirst-principles
2
electron-phonon couplings. This is done by fitting their
associated scattering rates to the scattering rates ob-
tainedwiththefirst-principlescouplingmatrixelements.
The resulting the deformationpotentials, 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-
plingstrengthareshown. Thecontourplotsshowtheabsolute
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. Fro¨hlich interaction
phonon wave vector q (note the different color scales in the
plots). The upper four plots correspond to intravalley scat-
The lattice vibration of the polar LO phonon gives
tering and the two bottom plots to intervalley scattering on
rise to a macroscopic electric field that couples to the
phononswithwavevectorq=2K. Thetwozero-orderdefor-
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 Fr¨ohlich interaction
temperature mobility. which diverges as 1/q in the long-wavelengthlimit,40
1 e2~ω 1 1 1/2
LO
g (q)= , (8)
deformation potential coupling to the homopolar mode LO qs 2ǫ0V (cid:18)ε∞ − ε0(cid:19)
stems from its characteristic lattice vibration polarized
inthe directionperpendicularto the layercorresponding where ǫ0 is the vacuum permittivity, V is the volume of
toachangeinthelayerthickness. Theassociatedchange thesample,andε∞andε0arethehigh-frequencyoptical
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- tureofboththeLOphononandthechargecarriersleads
ter of the MoS layer. As the electronic Bloch functions to a qualitatively different q-dependence of the Fr¨ohlich
2
6
wherethetimeevolutionofthecrystalmomentumisgov-
100
First-principles erned by ~k˙ =qE and q is the charge of the carriers. In
Analytic the caseof a time-independent uniformelectric field, the
75 steady-stateversionofthelinearizedBoltzmannequation
)
V takes the form
e
m q q
() 50 ~E·∇kfk0 =−k TE·vkfk0(1−fk0)
q B
(
gLO = ∂fk , (11)
25 ∂t
(cid:12)coll
(cid:12)
0 where fk0 = f0(εk) is the e(cid:12)(cid:12)quilibrium Fermi-Dirac dis-
K tribution function, ∂f0/∂ε = f0(1 f0)/k T, and
(cid:9) v = ε /~ is the bkandkveloc−ity.k In−linekar rBesponse,
k k k
∇
the out-of-equilibrium distribution function can be writ-
FIG.5: (Coloronline)Fr¨ohlichinteractionforthepolaropti-
cal LO mode. The dashed line shows the analytical coupling tenasthe equilibriumfunction plusa smalldeviationδf
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 ˚A fitted to the long-
wavelength limit of thecalculated coupling (dots). δfk =fk−fk0 =fk0(1−fk0)ψk. (12)
Here, the last equality has defined the deviation func-
tion ψ . Furthermore, following the spirit of the itera-
interaction. The situation is similar to 2D semiconduc- k
tive method of Rode,47 the angular dependence of the
tor heterostructures where the Fr¨ohlich interaction has
deviation function is separated out as
been studied using dielectric continuum models and mi-
croscopic based approaches.44,45 From the microscopic
ψ =φ cosθ , (13)
considerations presented in App. B, we derive the fol- k k k
lowingfunctionalformfortheFr¨ohlichinteractiontothe where θ is the angle betweenthe k-vector andthe force
k
polar LO phonon in 2D materials, exerted on the carriers by the applied field E. In gen-
eral, φ is a function of the k-vector and not only its
k
g (q)=g erfc(qσ/2). (9)
LO Fr× magnitudek. However,forisotropicbandsasconsidered
here,theangulardependence ofthe deviationfunctionis
Here, the coupling constant g is the equivalent of the
Fr entirely accounted for by the cosine factor in Eq. (13).38
square root factors in Eq. (8), σ is the effective width of
Considering the phonon scattering, the collision inte-
theelectronicBlochstatesanderfcisthecomplementary
gral describes how accelerated carriers are driven back
error function.
towards their equilibrium distribution by scattering on
Figure 5 shows the first-principles coupling g to the
LO
acoustic and optical phonons. In the high-temperature
polar LO phonon in single-layer MoS (dots) as a func-
2
regime of interest here, the collision integralcan be split
tionofthephononwavevectoralongthe Γ-K path. The
up in two contributions. The first, accounting for the
dashed line shows a fit of the coupling in Eq. (9) to the
quasielastic scattering on acoustic phonons, can be ex-
long-wavelengthlimit of the calculatedcoupling. With a
pressedin the form of a relaxationtime τ . The second,
couplingconstantofg =98meVandaneffectivewidth el
Fr
of σ =4.41 ˚A, the analytic form gives a perfect descrip- describing the inelastic scattering on optical phonons,
will in general be an integral operator I . The colli-
tion of the calculated coupling in the long-wavelength inel
sion integral can thus be written
limit. For shorter wavelengths, the zero-order deforma-
tionpotentialcouplingtotheintervalleyLOphononfrom
∂f δf
k k
Fig.4becomesdominant,andadeviationawayfromthe ∂t =−τel +Iinel[ψk], (14)
behavior in Eq. (9) is observed. (cid:12)coll k
(cid:12)
where the explicit(cid:12)forms of the relaxation time and the
(cid:12)
integral operator will be considered in detail below.
IV. BOLTZMANN EQUATION
Withtheaboveformofthecollisionintegral,theBoltz-
mann equation can be solved by iterating the equation
Intheregimeofdiffusetransportdominatedbyphonon
scattering, the mobility can be obtained with semiclas- φ = eEvk +I′ [φ ] 1 −1 (15)
sical Boltzmann transport theory. In the absence of k k T inel k × τel
(cid:20) B (cid:21) (cid:18) k (cid:19)
spatial gradients the Boltzmann equation for the out-of-
equilibriumdistributionfunctionfkofthechargecarriers for the deviation function φk. Here, the angular depen-
reads46 douentcaencdotshθeknaenwdiantfekg0r(a1l−opfek0r)atfoarctIo′r haisvedebfieneenddbiyvided
∂f ∂f inel
k +k˙ f = k , (10)
∂t ·∇k k ∂t (cid:12)coll Iinel =fk0(1−fk0)cosθkIi′nel. (16)
(cid:12)
(cid:12)
(cid:12)
7
From the solution of the Boltzmann equation the cur- Here, n is the two-dimensionalcarrierdensity. In the re-
rent and drift mobility of the carriers can be obtained. laxationtimeapproximationφ˜ =τ andthewell-known
k k
Taking the electric field to be oriented along the x- Drudeexpressionforthemobilityµ=e τ /m∗ isrecov-
k
h i
direction, the current density is given by ered.
4e
j σ E = v δf , (17)
x xx x x k
≡ −A
k
X
whereσ istheconductivity,Aistheareaofthesample,
xx
andv =~k /m∗ isthebandvelocityforparabolicbands A. Phonon collision integral
i i
withi=x,y,z. Fromthe definitionµ =σ /neofthe
xx xx
drift mobility, it follows that
The phonon collision integral has been considered in
e φ˜ great detail in the literature for two-dimensional elec-
µ = h ki (18) tron gases in semiconductor heterostructure (see e.g.
xx m∗
Ref.37). In general,these treatments consider scattering
wherethemodifieddeviationfunctionφ˜ havingunitsof on three-dimensional or quasi two-dimensional phonons.
k
Inatomicallythinmaterialsthephononsarestrictlytwo-
time is defined by
dimensional which results in a slightly different treat-
m∗k T ment. In the following, a full account of the two-
φ˜ = B φ , (19)
k eE ~k k dimensional phonon collision integral is given for scat-
x
tering on acoustic and optical phonons.
and the energy-weighted average is defined by
h·i With the distribution function written on the form
in Eq. (12), the linearized collision integral for electron-
1 ∂f0
A = dε ρ(ε )ε A . (20) phonon scattering takes the form46
k k k k
h i n −∂ε
Z (cid:18) k(cid:19)
∂f 2π g 2
∂tk =− ~ ǫ|2(qqλ,|T)× fk0 1−fk0+q Nq0λ(ψk−ψk+q)δ(εk+q−εk−~ωqλ)
(cid:12)coll qλ (cid:20)
(cid:12) X (cid:0) (cid:1)
(cid:12)
(cid:12) +f0 1 f0 1+N0 (ψ ψ )δ(ε ε +~ω ) (21)
k − k−q qλ k− k−q k−q− k qλ
(cid:21)
(cid:0) (cid:1)(cid:0) (cid:1)
where N0 = N0(~ω ) is the equilibrium distribution for the proper screened mobility.
qλ qλ
of the phonons given by the Bose-Einstein distribution
function and f0 = f0(ε ~ω ) is understood. The
k±q k ± q
different terms inside the square brackets account for 1. Quasielastic scattering on acoustic phonons
scattering out of (ψ ) and into (ψ ) the state with
k k±q
wave vector k via absorption and emission of phonons. Forquasielasticscatteringonacousticphononsathigh
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
MoS2 arenon-degenerateexceptatveryhighcarriercon- τel = (1−cosθkk′)Pkk′ (22)
centrations. Semiclassical screening where the screening k k′
X
length is given by the inverse of the Debye-Hu¨ckel wave where the summation variable has been changedto k′ =
vector kD = e2n/2kBT therefore applies.48 For the con- k q and the transition matrix element for quasielastic
sidered values of carrierdensities and temperatures, this sca±ttering on acoustic phonons is given by
corresponds to a small fraction of the size of the Bril-
2π
ilnouginenzeornale,iniv.eo.lvkesDla≪rge2rπw/aa.veAvsecstcoartst,ersicnrgeeonningphboynothnes Pkk′ = ~ |gqλ|2 Nq0λδ(εk′ −εk)
q (cid:20)
X
carriers can to a good approximation be neglected here.
The calculated mobilities therefore provide a lower limit + 1+Nq0λ δ(εk′ −εk) . (23)
(cid:21)
(cid:0) (cid:1)
8
For isotropic scattering as assumed here in Eq. (6), the scattering is given by
square of the electron-phononcoupling can be expressed
as Pkk′ = 2~π |gqλ|2× Nλ0δ(εk′ −εk−~ωλ)
g 2 = Ξ2λ~q , (24) Xq (cid:20)
qλ
| | 2Aρcλ + 1+Nλ0 δ(εk′ −εk+~ωλ) .
(cid:21)
wherethe acousticphononfrequencyhasbeen expressed (cid:0) (cid:1) (28)
intermsofthesoundvelocityc . Exceptatverylowtem-
λ
ppreoraxtimuraetsio~nωqN≪q0 ∼kBkTBiTm/p~lωyqing≫th1atfotrhetheqeuBipoasret-iEtiionnstaepin- FEoqr. t(h1e6)inc-ascnatbteerinexgtrpaacrtte,dthferomdesiψrekd′ ucosisnθgk tfhacetorrelain-
distribution applies. With the resulting q-factors in tion cosθk′ = cosθkcosθkk′ sinθksinθkk′. Since the
−
the transition matrix element in Eq. (23) canceling, the sine term vanishes from symmetry consideration, the in-
cosθkk′ 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
andvalley degeneraciesρ /g g . The relaxationtime for
0 s v
acoustic phonon scattering becomes I′iinnel[φk]= φk′cosθkk′Pkk′11−ffk00′. (29)
1 m∗Ξ2k T Xk′ − k
B
= . (25)
τel ~3ρc2 The evaluation of the k′-sum in Eqs. (27) and (29) is
k λ
outlined in App. C. Here, the assumption of dispersion-
Theindependenceonthecarrierenergyandtheτ−1 T less optical phonons allows for an semianalytical treat-
∼
temperature dependence of the acoustic scattering rate ment. Forzero-andfirst-ordercouplingwithinthedefor-
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-
sionintegralaregiveninEqs.(C14),(C15),(C16),(C17)
and (C18). In the case of the Fr¨ohlich interaction, the
2. Inelastic scattering on dispersionless optical phonons angular part of the q-integralmust be done numerically.
Forinelasticscatteringonopticalphononsthe 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
tobothintravalleyandintervalleyopticalandintervalley given by
acoustic phonons.
terIinngthienfoEllqo.w(in1g6,)tihsesinptlietgruaploipnersaetpoarrafotreinoeulta-staincdscaint-- τi1nel = Pkk′11−ffk00′ (30)
scattering contributions, k Xk′ − k
and corresponds to the imaginary part of the elec-
I′inel[φk]=I′oinuetl[φk]+I′iinnel[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 ψ and
k orderdeformationpotentialscatteringare givenfor non-
ψ , respectively. With the contributions from differ-
k±q degeneratecarriers,i.e. withtheFermifactorsinEq.(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-
1 f0
I′oinuetl[φk]=−φk Pkk′ 1− fk0′, (27) sccaastet(esreinegAppapr.tCo)f.the collision integral vanishes in this
k′ − k
X 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 thetotalscatteringratefromthevariouselectron-phonon
couplingstotheintraandintervalleyphononswhichhave
1 = m∗(Dλ0)2 1+e~ωλ/kBTΘ(ε ~ω ) N0. (31) beengroupedaccordingtotheircouplingtype,i.e. acous-
τinel 2~2ρω k− λ λ tic deformation potentials (ADPs), zero/first-orderopti-
k λ (cid:20) (cid:21)
cal deformation potentials (ODPs) and the Fr¨ohlich 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 ingincludes the quasielasticintravalleyscatteringonthe
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
1 = m∗2(Dλ1)2 (2ε +~ω )+e~ωλ/kBT coupling types have been obtained using Matthiessen’s
τkinel ~4ρωλ (cid:20) k λ rulebysummingthe scatteringratesfromthe individual
phonons, i.e.
Θ(ε ~ω )(2ε ~ω ) N0.
× k− λ k− λ λ
(cid:21) τ−1 = τ−1. (33)
(32) tot λ
λ
X
Due tothelineardependence onthecarrierenergy,zero- For carrierenergies below the optical phonon frequen-
order scattering processes dominate first-order processes cies,thetotalscatteringrateisdominatedbyacousticde-
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. scatteringviatheFr¨ohlichinteractionbecomedominant.
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 √ε ~ω originat-
k λ
±
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 ˚A
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 MoS are studied as a Acoustic deformation potentials
2
functionofcarrierenergy,temperatureT andcarrierden- 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
fectivecouplingparametersforthedeformationpotential TA DK1,TA 5.9 eV
approximation in Eqs. (6) and (7) (see below). LA DK1,LA 3.9 eV
TO DΓ1,TO 4.0 eV
A. Scattering rates TO DK1,TO 1.9 eV
LO DK0,LO 2.6×108 eV/cm
With access to the first-principles electron-phonon Homopolar DΓ0,HP 4.1×108 eV/cm
couplings, the scattering rate taking into account the Fr¨ohlich interaction (LO)
anisotropy and more complex q-dependence of the first- Effective layer thickness σ 4.41 ˚A
principles coupling matrix elements can be evaluated. Coupling constant gFr 98 meV
Theyareobtainedusingtheexpressionforthescattering
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
in Eqs. (23) and (28).50 In order to account for the dif- Homopolar ~ωHP 50 meV
ferentcoupling matrixelements inthe 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-subscriptontheopticaldeformationpotentials,indicate
couplings to theintra/intervalley phonons.
T =300K. The different lines show the contributions to
10
for the anisotropy and the full q-dependence of the first-
1014 T=300 K principles electron-phonon couplings. However, as mo-
mentumandenergyconservationlimitphononscattering
to involve phonons in the vicinity of the Γ/K-point,39
)
1s(cid:10) 1013 the fitting procedure yields deformation potentials close
e ( to the direction averaged q Γ/K limiting behavior
at of the first-principles coupling→matrix elements. For the
g r1012 ADP zero-order deformation potentials, the sampling of the
n
ri Zero-order ODP coupling matrix elements away from the Γ-point in the
e
t First-order ODP k′ sum in Eq. (30), leads to a deformation potentials
cat1011 Fr hlich which is slightly smaller than the Γ-point value of the
S (cid:11)
Total coupling matrix element.
Fitted
The total scattering rate resulting from the fitted de-
1010
0.00 0.05 0.10 0.15 0.20 0.25 formation potentials is shown in Fig. 6 to give an al-
Energy (eV)
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 proximationgivenbythe energy-weightedaverageofthe
electron-phonon couplings as a function of carrier energy at relaxationtime(seeEq.(18)),thedifferencebetweenthe
T = 300 K. The scattering rates have been calculated from
associatedmobilitieswillbenegligible. Hence,thedefor-
thefirst-principleselectron-phononcouplingsinFigs.4and5.
mation potentials provide well-founded electron-phonon
The (black) dashed line shows the scattering rate obtained
coupling parameters for low-fieldstudies of the mobility.
using the fitted deformation potential parameters defined in
Eqs.(6)and(7). Thekinksinthecurvesfortheopticalscat-
tering rates mark theonset 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
conductionbandingeneralresultsinnon-degeneratecar-
of magnitude smaller than the other scattering rates for
rierdistributionsinsingle-layerMoS . Thisisillustrated
2
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
versusthepositionoftheFermilevelfordifferenttemper-
phonon energies ~ω are associated with the threshold
λ 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 ∼ ×
levelintotheconductionbandandprobetheFermi-Dirac
statistics of the carriers. Thus, only at the highest re-
portedcarrierdensities ofn 1013 cm−2,5 arethe carri-
B. Deformation potentials ersdegenerateatroomtemp∼erature. Forthelowesttem-
peratureT =100K,thetransitiontodegeneratecarriers
Inthissection,wedeterminethedeformationpotential occursatacarrierdensityof 2 1012 cm−2. Thetran-
∼ ×
parameters in single-layer MoS . They can be used in sitionfromnon-degenerateto degeneratedistributions is
2
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 ratesin Fig. 6 to a high degree resemblesthat TherightplotofFig.7showsthephonon-limiteddrift
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 proximationusing the expressionsinSectionIVforopti-
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 talrelaxationtime. Thestrongdropinthemobilityfrom
in Eqs. (6) and (7). The deformation potentials are ob- 2450cm2 V−1 s−1atT =100Kto 400cm2V−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-
tionpotentialvaluesaresummarizedinTab.I.Inanalogy tively low intrinsic room-temperature mobility of single-
withdeformationpotentialsextractedfromexperimental layer MoS can be attributed to both the significant
2
mobilities, the theoretical deformation potentials repre- phonon scattering and the large effective mass of 0.48
senteffectivecouplingparametersthatimplicitlyaccount m intheconductionband. Whilethemobilitydecreases
e