First-principles theory of field-effect doping in transition-metal dichalcogenides: Structural properties, electronic structure, Hall coefficient, and electrical conductivity

4.2 MB·English

by  Thomas Brumme

First-principles theory of field-effect doping in transition-metal dichalcogenides:
Structural properties, electronic structure, Hall coefficient, and electrical conductivity
Thomas Brumme,∗ Matteo Calandra, and Francesco Mauri
CNRS, UMR 7590 and
Sorbonne Universit´es, UPMC Univ Paris 06, IMPMC - Institut de Min´eralogie,
de Physique des Mat´eriaux, et de Cosmochimie, 4 place Jussieu, F-75005, Paris, France
(Dated: May 5, 2015)
5
1 We investigate how field-effect doping affects the structural properties, the electronic structure
0 and the Hall coefficient of few-layers transition metal dichalcogenides by using density-functional
2 theory. We consider mono-, bi-, and trilayers of the H polytype of MoS2, MoSe2, MoTe2, WS2,
and WSe2 and provide a full database of electronic structures and Hall coefficients for hole and
y
electron doping. We find that, for both electron and hole doping, the electronic structure depends
a
onthenumberoflayersandcannotbedescribedbyarigidbandshift. Furthermore,itisimportant
M
to relax the structure under the asymmetric electric field. Interestingly, while the width of the
conducting channel depends on the doping, the number of occupied bands at each given k point
4
is almost uncorrelated with the thickness of the doping-charge distribution. Finally, we calculate

within the constant-relaxation-time approximation the electrical conductivity and the inverse Hall
]
l coefficient. Wedemonstrate that in some cases the charge determined byHall-effect measurements
l
a candeviatefromtherealchargebyupto50%. Forhole-dopedMoTe2 theHallchargehaseventhe
h wrong polarity at low temperature. We provide the mapping between the doping charge and the
- Hall coefficient. In the appendix we present more than 250 band structures for all doping levels of
s
thetransition-metal dichalcogenides considered within this work.
e
m
t. I. INTRODUCTION A second crucial issue is the determination of the
a
doping charge. Usually the charge is determined via
m
Since the rise of graphene1 and the discovery of Hall-effectmeasurements. However,theinterpretationof
d- topological insulators2 a lot of interesting physics has Hallexperimentsassumesa2Delectron-gasmodel,most
likelyvalidonlyinthelowdopingregime. InTMDs,due
n been found in systems with reduced dimensions. Other
o two-dimensional (2D) material, such as monolayers to the multivalley electronic structure, this assumption
c is highly questionable.
or few-layer systems (nanolayers) of transition-metal
[
dichalcogenides3–7 (TMDs) are gaining importance be- In this paper we solve these issues and provide a

2 cause of their intrinsic band gap. TMDs are MX -type thorough study of structural, electronic and transport
2
v compounds where M is a transition metal (e.g., M = properties and of their changes under field-effect dop-
3
Mo, W) and X representsa chalcogen(S, Se, Te). These ing for TMDs. We use our recently developed first-
2
materials form layered structures in which the different principles theoretical approach to model doping in field-
2
7 X–M–X layers are held together by weak van der Waals effectdevices24. Themethodallowsforcalculationofthe
0 forces. Thus, similar to graphene, one can easily extract electronic structure as well as complete structural relax-
. single or few layers from the bulk compound using the ationinfield-effectconfigurationusingdensity-functional
1
mechanical-exfoliationor other experimental techniques. theory(DFT).WeapplyourapproachtotheHpolytype
0
5 Doping these nanolayers with field-effect transistors of MoS2, MoSe2, MoTe2, WS2, and WSe2.
1 (FETs) is particularly appealing8–21 as it allows for the Thepaperisorganizedasfollows: InsectionIIwesum-
v: exploration the semiconducting, metallic, superconduct- marize the parameters and methods used within the pa-
i ing, and charge-density-wave regimes in reduced dimen- per and the relaxedgeometries of the bulk TMDs. Then
X
sionality. Furthermore, the TMDs are promising mate- wewillfirstshowthe results forthe undoped mono-,bi-,
ar rials to realize valleytronics, i.e., the usage of the valley and trilayer systems (section IIIA). After a brief discus-
index of carriers to process information7,15,22. sion on the quantum capacitance (IIIB), we will investi-
Despite these challenging experimental perspectives, gatethechangesofthegeometryandtheelectronicstruc-
theunderstandingofstructural,electronic,andtransport ture of the different TMDs in sections IIIC and IIID,
properties at high electric field in FET configuration is respectively. Finally, we will focus on how the doping-
stilllimited,particularlyinthephysicallyrelevantcaseof charge concentration can be determined experimentally
multilayer samples. Previous theoretical works20,23 ana- by Hall-effect measurements. We will show in section
lyzedthe highdopinglimitof20nmthickMoS flakes10, IIIE that the results of such a measurement cannot be
2
relevant for ionic-liquid based FETs, by assuming that interpreted within a 2D electron-gas model but that the
only the topmost layer is doped uniformly. However, it specific band structure of the TMDs and its changes in
is unclear to what extent the doping of thick flakes can a field-effect setup need to be taken into account. In
be modeled inthis approximationasthe thicknessofthe section IIIF we will furthermore investigate the density
conductive channel is not experimentally accessible. of states (DOS) at the Fermi energy and the electrical
2
conductivity as function of doping. In the end, we will
summarize our results anddraw some finalconclusionin TABLE I. Comparison of the calculated and experimental30
lattice parameters.
section IV. In the appendix we provide a full database
ofelectronicstructuresforalldopinglevelsconsideredin
a calc. a exp. ccalc. cexp.
this work (in total more than 250 calculations, Figs. 28–
MoS2 3.197˚A 3.160˚A 12.38˚A 12.29˚A
63)andHallcoefficients(Figs.64–66)ofmono-,bi-,and
MoSe2 3.328˚A 3.289˚A 13.07˚A 12.93˚A
trilayer dichalcogenides as a function of doping.
MoTe2 3.536˚A 3.518˚A 14.00˚A 13.97˚A
WS2 3.190˚A 3.153˚A 12.15˚A 12.32˚A
II. COMPUTATIONAL DETAILS WSe2 3.341˚A 3.282˚A 12.87˚A 12.96˚A
All calculations were performed within the framework
of DFT using the Quantum ESPRESSO package25 Packgridof64×64×1kpoints forthe chargedsystems
whichusesaplane-wavebasissettodescribethevalence- and 16×16×1 k points for the neutral ones. In order
electronwavefunction andchargedensity. We employed to correctly determine the Fermi energy in the charged
full-relativistic, projector augmented wave potentials26. systems, we performed a non-self-consistent calculation
While the local-density approximation is known to un- on a denser k-point grid of at least 90×90×1 points
derestimate the lattice parameters, generalized-gradient starting from the converged charge density. All other
approximationsfortheexchange-correlationenergyover- parameters were the same as in the calculations for the
estimate the out-of-plane lattice constant (see, e.g., bulksystems. Forthetotal-energycalculationsofthe1T
Ref. 27 and references therein). In our FET setup a and1T’polytypeofMoS andWSe underFETdoping,
2 2
correct description of the interlayer distance is however wedoubledthe unit-cellsizealongonein-planedirection
very important. Accordingly, we choose the Perdew- andcorrespondinglyhalvedthenumberofkpointsalong
Burke-Ernzerhof functional28 (PBE) for the exchange- this direction.
correlation energy and furthermore included dispersion Severalmethodshavebeendevelopedtostudyelectro-
corrections29 (D2). This also leads to the best agree- staticsinperiodicboundarycondition24,32–34 withdiffer-
ment with both the experimental in-plane and out-of- ent experimental geometries. We used our recently de-
plane lattice parameters (cf., Tab. I). A comparison be- velopedmethod24 asit istailoredforthe FETsetupand
tween PBE and LDA for MoS2 can be found in the ap- allows for structural optimization. The dipole for the
pendix (Fig. 27). dipole correction24,35 was placed at z = d /2 with
dip dip
Using the experimental lattice parameters of the H d =0.01L and L being the unit-cell size in the direc-
dip
polytype of the bulk structures30 as starting geometry, tion perpendicular to the 2D plane – L changed for the
we minimized the total energy as a function of the lat- differentcalculationsandwasbetween34˚Aand48˚A. The
tice parametersuntil it changedby less then 2meV. For charged plane modeling the gate electrode24 was placed
the molybdenum-containing dichalcogenides we used a closetothedipoleatz =0.011L. Apotentialbarrier
mono
cutoffof 50Ryand410Ry (1Ry≈13.6eV) for the wave with a height of V = 2.5Ry and a width of d = 0.1L
0 b
functions and the charge density, respectively, while for was used in order to prevent the ions from moving too
thetungstendichalcogenideswechose55Ry/410Ry. The close to the gate electrode. The final results were found
Brillouinzone (BZ)integrationhasbeenperformedwith to be independent of the separationofthe dipole planes,
aMonkhorst-Packgrid31of16×16×4kpointsandusing as well as the barrier height and width as long as it is
a Gaussian broadening of 0.002Ry ≈27meV. The con- high or thick enough to ensure that the electron den-
vergence with respect to the number of k points as well sity at the position of the dipole and the gate electrode
as the wave-function and charge-density cutoff has been is zero, ρe(z ) = ρe(z ) = 0. As we will often
mono dip
checked. The self-consistent solution of the Kohn-Sham givethedoping-chargeconcentrationperunitcelln(i.e.,
equationswasobtainedwhenthetotalenergychangedby in charge per unit cell, e/unit cell, with the elementary
less than 10−9Ry and the maximum force on all atoms charge e ≈ 1.602×10−19C), we summarized in Tab. II
was less than 5· 10−4 Ry a−01 (a0 ≈ 0.529177˚A is the theconversiontocharge-carrierconcentrationperarean
Bohr radius). The lattice parameters thus determined (in cm−2) for the different dichalcogenides and two typ-
are given in Tab. I and agree within 2% with the experi- ical doping-charge concentrations of n= 0.01e/unit cell
mental values. and n = 0.15e/unit cell. Throughout the paper we will
The finalgeometryofthe bulk systemwasused asthe use n < 0 and n > 0 for electron and hole doping, re-
starting geometry for the relaxation of the layered-2D spectively. Typical charge-carrier concentrations which
systems in an FET setup. To achieve this, the size of can be achieved in experiments using either solid-state
the unit cell was fixed in plane and the perpendicular dielectrics such as SiO or ionic-liquid based FETs can
2
size was increased such that the vacuum region between be found in Tab. III. Note that in principle those max-
the repeated images was at least 23˚A. The layers were imum concentrations could be possible for all TMDs36
stackedasintheHpolytype ofthe bulk compound. The even if we did not find references for, e.g., ionic-liquid-
BZ integration has been performed with a Monkhorst- based field-effect doping of MoSe .
2
3
cus on the changes in the valence-band maximum and
TABLE II. Conversion for the doping-charge concentration
the conduction-band minimum with changing transition
n (in e/unit cell) to charge-carrier concentration n per area
cm−2forthedifferentdichalcogenidesandtwotypicaldoping- metal or chalcogenandcompare them with other results
found in literature27.
charge concentrations of n = 0.01 e/unit cell and n =
0.15e/unit cell. Figure 1 shows the band structure and the projected
density of states (pDOS) for monolayer MoSe with and
2
n=0.01e/unit cell n=0.15e/unit cell without including spin-orbit coupling (SOC). Monolayer
MoS2 n≈0.1127·1014cm−2 n≈1.6911·1014cm−2 molybdenumdiselenideis(asmostTMDs)adirect-band-
MoSe2 n≈0.1042·1014cm−2 n≈1.5636·1014cm−2 gapsemiconductorwith a DFT gapof about1.329eVat
MoTe2 n≈0.0924·1014cm−2 n≈1.3856·1014cm−2 the K point. Our calculated gap is smaller by 83meV
WS2 n≈0.1135·1014cm−2 n≈1.7027·1014cm−2 than the one in Ref. 27 which can be attributed to our
WSe2 n≈0.1035·1014cm−2 n≈1.5521·1014cm−2 slightlylargerin-planelatticeparameterasthesizeofthe
bandgapdecreaseswith increasinglattice constant45–48.
This is due to the fact that the valence-band maximum
at K is formed by in-plane states of both the transi-
TABLEIII.Maximumexperimentalcharge-carrierconcentra- tion metal and the chalcogen48. On the other hand, the
tion n per area cm−2 for the different dichalcogenides using
conduction-bandminimumatKismainlyformedbyout-
either a solid-state or an ionic-liquid based FET. References
are given after the charge-carrier concentration. of-planeMostates(dz2 withoutSOCandmj =±1/2for
both j = 5/2 and j = 3/2 including SOC) and in-plane
Polarity Solid-state FET Ionic-liquid FET statesofthechalcogen. Thevalence-bandmaximumnear
MoS2 n-type ≈−3.6·1013, [14] ≈−9.8·1014, [37] Γ has basically only out-of-plane states of Mo and Se as
can be seen in Fig. 1. This will become very important
p-type — —
MoSe2 n-type ≈−3.9·1012, [38] — for hole doping of the nanolayers in an FET setup – de-
pendingonwhichvalleyisdoped(KorΓ)onecanexpect
p-type ≈+2.0·1012, [38] —
different doping-charge distributions. Energetically very
MoTe2 n-type ≈−1.3·1013, [39] ≈−1.1·1013, [40] close to the conduction-band minimum at K is a min-
p-type — ≈+1.8·1013, [40] imum half-way between K and Γ. The corresponding
WS2 n-type ≈−1.0·1014, [19] ≈−4.0·1014, [41] pointinkspaceiscalledQinliterature(sometimesΛor
p-type — ≈+3.5·1013, [42] Λmin as it is a minimum along the Λ line, from Γ to K)
WSe2 n-type — ≈−1.4·1014, [16] evenifitis notahighsymmetrypointofthe BZ.Thisis
p-type ≈+9.0·1012, [43] ≈+1.9·1015, [16] also why this minimum does not lie exactly at the same
point for the different TMDs and its position can even
change if the number of layers is increased. The states
close to the Q point have a stronger in-plane character
In order to calculate the Hall tensor R (T;E ), we
ijk F and can thus also lead to a different doping-charge dis-
usedtheBoltzTraPcode44 todeterminetheconductivity
tributionif the doping occursmainly atthis pointin the
tensors in Eqs. (11) and (12) (see section IIIE for more
BZ. The same results for the character of the different
details). We fitted the band structure for eachdoping of
valleys were also obtained in Refs. 47–49.
the different TMDs by using 55-times more plane waves
Thedifferentcharacterofthestatesinthedifferentval-
thanbandsandusedafterwardsthein-planecomponents
leys is even more important for the other TMDs. From
oftheenergy-projectedtensorstocalculatetheHallcoef-
sulfur to tellurium the difference between the minimum
ficient R (T;E ) for temperature T and chemical po-
xyz F at K and Q decreases: for MoS the minimum at K is
tential E . R (T;E ) is the only relevant Hall coef- 2
F xyz F lower by 279meV while it is only 154meV and 72meV
ficient for our 2D systems assuming that the magnetic
lowerfor MoSe and MoTe , respectively. The changein
field is applied perpendicular to the layers. We checked 2 2
the case of the tungsten dichalcogenides is much lower
theconvergencebycalculatingR (T;E )withincreas-
xyz F which however might be due to the stronger spin-orbit
ing number of k points and found that the results for
splitting of the bands near Q compared the splitting at
the 64×64×1 grid and the dense grid of the non-self-
K. The band structures of all undoped TMDs are sum-
consistent calculation are the same.
marized in the appendix, Figs. 19–23.
Increasing the number of layers in TMDs leads to a
well-known change from a direct-band-gap semiconduc-
III. RESULTS
tor to an indirect one27,49,51–54 as shown in Fig. 2. The
change of the direct band gap at K with increasing the
A. Electronic structure of TMDs
number of layers is much smaller than the changes at
Γ or Q. This is due to the small hybridization between
In the following we will first briefly summarize different layers at K as those states have only in-plane
the results for the undoped TMDs before investigat- chalcogen character. On the other hand, both valleys
ing the changes under field-effect doping. We fo- at Γ and Q have contributions from Se p states. Ac-
z
4
MoSe MoSe
2 2
2 2
]
V
e Mo - dz2 V]
(E - E) [v01 no SOC MMSSeeoo ---- ddppxxzx2 z-+ y+ 2p +dyy dzxy E - E) [ev1 BTriillaayyeerr
(
0
2
Γ M K Γ
]
V
E) [ev1 SOC MMMMoooo ---- (((( llll ==== 2222,,,, jjjj ==== 3355////2222,,,, mmmm ==== ±±±±1313////2222 )))) iFdseInGgui.vm2e.ndBriseaelnaldetnisvitdereutcocthutahrneegfeovsarlb(eain-scaean-lbdlatTnrdMilaDmyesarxiMnimvoeuSsmet2ig.EaTtveh.deMiennoeltyrhgbiys-
E - Mo - ( l = 2, j = 5/2, m = ±5/2 ) paper) from a direct-band-gap semiconductor to an indirect
( 0 one when thenumberof layers is increased.
Γ M K Γ 1 2 gap Γ → K. Also for other TMDs one can find different
DOS [states / eV] results46,59,61–63. For a comprehensive review of the the-
oretical papers see also Ref. 27 and references therein.
FIG. 1. (color online) Band structure and density of states Molybdenum ditelluride is especially peculiar, since the
projected onto atomic orbitals for monolayer MoSe2 with- calculations show that its valence-band maximum is lo-
out (upper panel) and with (lower panel) including spin-
cated at K even in the trilayer case. Furthermore, we
orbit coupling. The energy is given relative to the valence-
find that the difference between the maximum at Γ and
band maximum E . As apparent in the case without SOC,
v K is only 26meV in bulk MoTe which is in agreement
mainly in-plane states contribute to the valence-band maxi- 2
with the experimental results in Ref. 64.
mum near K (Mo dx2−y2 and dxy, Se px/y), while the max-
imum near Γ is formed by out-of-plane states (Mo dz2, Se In section IIID we will see that, e.g., the varying dif-
p ). This also holds in theSOCcase wherethevalence-band ferencebetweentheconduction-bandminimumatKand
z
states with mainly in-plane character can be found near K Q for the different TMDs will also lead to a different
(j =5/2,mj =±5/2,±3/2 and j =3/2,mj =±3/2) while thickness of the conductive channel for electron doping
those with more out-of-plane character can be found near Γ inanFETsetup,whilenearlyallTMDswillbehavesim-
(mj =±1/2forbothj =5/2andj =3/2,seeRef.50forthe ilarly under hole doping. However, before investigating
states in terms of spherical harmonics). On the other hand,
thechangesunderfield-effectdopingwewanttofocuson
the conduction-band minimum near K has mainly contribu-
another problem that can make it difficult to dope a 2D
tions from out-of-plane Mo and in-plane Se states (without
SOC: Mo dz2 and Se px/y, with SOC: Mo mj = ±1/2 for system – the quantum capacitance.
both j =5/2 and j =3/2).
B. Quantum capacitance
cordingly, the maximum (minimum) at Γ (Q) shift up
(down) in energy which eventually leads to an indirect A prominent example in which the quantum ca-
band gap between Γ and Q (see also Ref. 49 for an in- pacitance hinders the doping via field-effect setup is
depthanalyses). Thetransitionbetweentheindirectgap graphene. Due to its linear dispersion relation at the
Γ→KandΓ→Qhappensatdifferentnumberoflayers K points, the charge that can be induced in an FET
and occurs either in the bilayer case (MoSe ), in the tri- setup is much smaller than the corresponding charge at
2
layer case (WS and WSe ), or in the bulk limit (MoS , the gate electrode65,66. Thus, doping concentrations ex-
2 2 2
MoTe ). Dependingontheleveloftheorythatwasused, ceeding 1013cm−2 are hardly achievable using common
2
one canalsofind verydifferent results in literaturewhen dieletrics such as SiO or HfO . Similarly, the quan-
2 2
this transition occurs. Most calculations were done for tum capacitance in nanolayers of TMDs could also re-
MoS for which the transition either already occurs for duce the amount of induced charge. In the following we
2
thebilayer46,55,56 orwithlargernumberoflayers57–59. In wanttoshowthatTMDsarehoweverquitedifferentfrom
fact,Ramasubramaniametal. haveshowninRef.60that graphene and that the quantum capacitance is not rele-
using the experimental bulk distance between the layers vant in their case as soon as the Fermi energy is within
inthebilayercasealsoleadstoanindirectbandgapΓ→ the conduction or valence band. In fact, experimen-
Q while relaxation using PBE+D2 leads to an indirect tallydopingconcentrationsinthe orderof1014cm−2 are
5
TABLE IV. Geometrical capacitances for parallel-plate ca-
pacitors with 100nm SiO2 (relative permittivity εr = 3.9),
10nm HfO2 (εr ≈ 20), and an 1nm thick ionic-liquid (IL)
FET (ε ≈ 15), quantum capacitance for electron doping of
r
the K valley (m ≈ 0.5m0) and hole doping of the Γ valley
(m ≈ m0) of MoS2 (ν = 4), and the overall capacitance of
a setup as shown in Fig. 3(b). All capacitances are given in
unitsof µFcm−2.
FIG. 3. (a) Schematic illustration of an FET setup in which CK =133.9 CΓ =267.7
the 2D metallic system is separated from the gate electrode Q Q
by a dielectric with dielectric constant εox of thickness dox. CSiO2 =0.035 C =0.035 C =0.035
(b) Equivalent circuit for the overall capacitance seen at the CHfO2 =1.771 C =1.748 C =1.759
gate electrode. CIL =13.28 C =12.08 C =12.65
possible using ionic-liquid based FETs10,16,41,67,68 (cf.,
onset potential of the conduction band (valence band) is
Tab. III).
given by
The term “quantum capacitance” was first used by
Serge Luryi69 in order to develop an equivalent circuit g g mq2
s v
C = , (5)
modelto describethe incomplete screeningofanelectric Q π¯h2
field by a 2D electron gas. When a 2D metallic system
whereg andg , m,andq arethe spinandvalleydegen-
is contacted by a gate electrode (separated by a dielec- s v
eracies, the effective mass, and the charge, respectively.
tric as shown in Fig. 3(a)), the electric field generated
Depending on (i) the dielectric thickness d , (ii) the ef-
by the charges on the dielectric surface leads to a shift ox
fectivemassofthe2Dmetallicsystem,and(iii)thenum-
of the Fermi level of the 2D metal. This effect results in
berofvalleysν =g g ,thequantumcapacitanceC can
a modified capacitance with respect to the geometrical s v Q
be relevant or not.
capacitance C per unit surface
ox
Table IV shows the total capacitance, geometrical ca-
C =ε ε d−1, (1) pacitance for typical gate dielectrics, and the quantum
ox ox 0 ox
capacitance for electron doping of the K valley or hole
obtained for a capacitor having a dielectric constant ε doping of the Γ valley of MoS (the effective masses of
ox 2
and thickness d . the different valleys were taken from Ref. 46). As the
ox
As showninFig.3(b), the quantumcapacitanceC is geometrical capacitance of a parallel-plate capacitor in-
Q
in series with that of the dielectric, namely creases with decreasing thickness of the dielectric, an
ionic-liquid (IL) FET with an 1nm thick electric double
1 1 1
layer(innerHelmholtzplane)showsthelargestdeviation
= + . (2)
C Cox CQ of the total capacitance C from the geometrical capaci-
tanceC –evenforathindielectriclayerof10nmHfO ,
IL 2
Under the simplified assumption that both C and C
ox Q C ≈ 0.986C . Yet, for TMDs the number of valleys
are independent of the applied gate voltage V , the HfO2
G increases with increasing doping (valleys at K and Q or
charge induced in the 2D system n can be written as:
Q KandΓ for electronorhole doping,respectively),which
also leads to a considerably larger DOS at the Fermi en-
0 ∀VG ∈(Vv,Vc) ergythanforasinglequadraticband. Thus,thequantum
nQ = CC((VVvc−−VVGG)) ∀∀VVGG <>VVvc (3) ctoapCac≈itaCnocxe.CSQimisilafurrrtehseurltinschraevaesebde7e0nwfhoiucnhdinbytuNrnanleaMdas
andDebdeep JenainRef.71whoalsoprovideadetailed

where V and V are the onset potentials to fill the description of the low-doping regime n<1013cm−2
v c
valence-band minimum or conduction-band maximum,
respectively. Thus, when C ≫C , we have
Q ox
C. Structural changes under field-effect doping
−1
C
ox
C =C 1+ ≈C . (4)
ox (cid:18) C (cid:19) ox The doping via FET setup has only a minor influence
Q
on the structure of the TMDs – the changes are much
and we regain the classical result, i.e., the charge that smaller than for, e.g., ZrNCl24. This is due to the weak
can be induced in the 2D system depends only on the polarity of the bond between the transition-metal and
applied gate voltage and the capacitance between gate the chalcogen. The largest change can be found for the
and sample. layerthickness(z componentofthe chalcogen–chalcogen
As shown in Refs. 69,70, the quantum capacitance in vector) of the layer closest to the charged plane repre-
this case and for gate voltages larger (smaller) than the senting the gate electrode – the layer thickness increases
6
ing of WS nanotubes85 and monolayer MoS 86, and by
1H-MoS2 0.8 n ≈ −0.44 e / unit cell transfer of2hot electrons generated in gold na2noparticles
V] n≈ −4.96 ⋅ 1014 cm-2 to monolayer MoS 87. In order to determine if the FET
e 2
11TT’--MMooSS22 tottot) [ - EE1H-MoSx-MoS22000...246 11TT-’M-MooSS22 wfisilnaeniygttdue,hrpwtthchesaatectrnuottolmchettaeapudlar1eerTtseno’eisosnrutfgrcFyuh1icoTgatf.-uptM4rhheoteahbSs1ee2eHcttoaropmantonadellsysiett1myniTopeon’re-rgMfe(yoEsroftS1toeaoH2lrbte-Mlct(ehtEorefSooxt2mno-r)Mt.deoolnoWSepo2ce)---
( tron doping larger than n = −0.44e/unit cell in close
0.0 agreementwiththe resultsofRefs.77,78. Thus,itseems
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 that the interaction between the H/Li atoms and the
n [e / unit cell]
MoS layer has only a minor influence on the phase sta-
2
bility as the transition occurs in our FET setup at the
FIG. 4. (left) Structure of 3 different structural phases of
samedoping. Wealsocalculatedtheenergydifferencebe-
monolayer MoS2. The H polytype is the one found in the tween 1T’-WSe and 1H-WSe , (Etot −Etot ),
bulk compound where the coordination of the molybdenum 2 2 1T’-WSe2 1H-WSe2
(gray) is trigonal prismatic. The T polytype with octahe- for a few electron-doping concentrations and found that
dralcoordination canchangetotheT’polytypeinwhichthe foraconcentrationofn=−0.35e/unitcellthe1T’poly-
molybdenum atoms form zig-zag chains. The sulfur atoms type becomes more stable by 63meV. In the following,
are shown in yellow. (right) With increasing electron doping wewillthus onlyconsiderdopingofthe Hpolytype with
the difference between the total energy of 1T/1T’ structure electron concentrations n≥−0.35e/unit cell as it is the
(Exto-Mt oS2)andthetotalenergyofthe1Hpolytype(E1toHt-MoS2) moststablestructurefoundinnatureandisoftenusedto
decreases. For a doping larger than n = −0.44e/unit cell
preparethe samplesby the mechanical-cleavagemethod.
(n≈−4.96·1014cm−2)1T’-MoS2 isthelowest-energy struc-
ture.
D. Band structure in FET setup
by≈0.06˚Aforalargeelectrondopingofn=−0.3e/unit
cell(n≈−3.16·1014cm−2)anddecreasesby≈0.02˚Afor In the following section, we want to investigate the
influence of field-effect doping on the electronic proper-
a large hole doping of n=+0.3e/unit cell. This change
ties of the TMDs. The doping via FET setup changes
is mainly due to the increase/decreaseof the chalcogen–
the bandstructure considerablyasexemplified inFigs.5
transition-metalbondlengthofthosebeingclosesttothe
gate – we find ≈ +0.04˚A for n = −0.3e/unit cell and and 6, which show the band structures for different elec-
≈−0.02˚A for n=+0.3e/unitcell. Accordingly,there is tronandhole dopinglevels formono-andtrilayerMoS2,
respectively. Intheappendixwealsodemonstratethatit
alsoasmallchangeinthe anglebetweenthe firstchalco-
is important to correctly model the FET setup by com-
gen,thetransition-metal,andthesecondchalcogenofup
to +0.9◦ (−0.4◦) for large electron (hole) doping. Note paring the band structures of mono- and trilayer MoSe2
calculated with a compensating jellium background to
that even if the structural changes seem to be small it
those calculated with our method (Figs. 25, 26). Fur-
is still important to relax the system in the FET setup.
thermore, we also provide the band structures for more
Otherwise, the band structure can be quite different for
doping levels and all the other TMDs in the appendix
doping |n| >0.15e/unit cell as exemplified in Fig. 24 in
(in total more than 250 calculations, Figs. 28–63). We
the appendix.
summarized the evolution of the band structure with in-
Eveniftheinternalstructurechangesonlyslightlyun-
creasingdopingintheleftpanelofFigs.7–11whichshow
der FET doping, electron doping could induce a phase
thepositionofthedifferentbandextremawithrespectto
transition where the structure of the full nanolayer sys-
tem is altered. It is well known that lithium72,73 or the Fermi energy. Additionally, the right panel in those
potassium74 intercalated MoS can undergo a phase figures shows the relative amount of doping charge per
2
valley given by
transition in which the Mo coordination changes from
a direct-band-gap, semiconducting, trigonal-prismatic
e ε1≤εi,k≤ε2
structure (labeled 2H, “2” as there are two layers in the 2
nα = |ψi,k| . (6)
ufonuintdceellx)pteoriammenettaallllyic7,5,o76ctaahneddrsahloownnet(h1eTo)r.eIttichaallsy7b7e,7e8n, Nα kX∈Ωα Xi
thatalsomonolayerMoS canundergothisphasetransi- Here α = {Γ,K,Q}, Ω defines the subset of k points
2 α
tion. Inthe calculations,forahighelectrondopingbyH which are closer to, e.g., α = Γ than to any α 6= Γ,
(Ref. 77) or Li (Ref. 78) adsorption of n ≈ −0.35e/unit Nα is the total number of those k points, and εi,k is the
cell or n=−0.44e/unit cell, respectively,the octahedral eigenenergyforbandiatk. Theinterval[ε ,ε ]isalways
1 2
phases suchas the 1T or 1T’phases become more stable chosensuchthattheprobabilitydensityisintegratedbe-
than the 1H phase. In the 1T’ phase the molybdenum tweenanenergywithintheformerbandgapE andthe
m
forms zig-zag chains like tungsten in WTe 79–84. Such a FermienergyE ofthedopedsystem,i.e., [E ,E ]and
2 F F m
transitionwasalsofoundexperimentallybyrheniumdop- [E ,E ] for hole and electron doping, respectively.
m F
7
Monolayer MoS
2
Electron doping Hole doping
1.0 0.2
n = 0.00
0.0
0.8
V] 0.6 V] -0.2
e e
E) [F 0.4 E) [F-0.4
E - E - -0.6
( (
0.2
n = 0.00 -0.8
0.0 -1.0
Γ M K Γ Γ M K Γ
0.2
n≈ +2.25 ⋅ 1013 cm-2
0.8
0.0
V] 0.6 V] -0.2
e e
) [F 0.4 ) [F-0.4
E E
E - 0.2 E - -0.6
( (
n≈ −2.25 ⋅ 1013 cm-2 -0.8
0.0
-1.0
Γ M K Γ Γ M K Γ
0.4
n≈ +1.69 ⋅ 1014 cm-2
0.6 0.2
V] 0.4 V] 0.0
e e
E) [F 0.2 E) [F-0.2
E - E - -0.4
( 0.0 (
n≈ −1.69 ⋅ 1014 cm-2 -0.6
-0.2
-0.8
Γ M K Γ Γ M K Γ
FIG. 5. Band structure for different FET induced doping of monolayer MoS2. The figures in the left column are for electron
dopingwhiletherightcolumn shows theholedopingcase. FormonolayerMoS2 mainly thevalleysat Kare filledand onlyfor
ahighdopingofn≈±1.69·1014cm−2 (n=±0.15e/unitcell) asmallamountofchargeisinthemaximumatΓ(holedoping)
or in the minimum at Q (electron doping). The band structures for more doping levels and all the other TMDs can be found
in theappendix (Figs. 28–63).
Electron doping andthedopingchargeissolelylocalizedaroundQ.Com-
paringour results ofthe changes in the conductionband
for electron doping of monolayer MoS (Figs. 5, 7) with
2
Forn-typedopingofmonolayerMoS2 (asforallmono- literature shows that it is important to correctly model
layerTMDs)thedopingchargefirstoccupiestheextrema thesystem–whiletheauthorsofRef.88findanupshift
at K. For small doping (as long as only one valley is of the Q valley with increasing electron doping, we see a
doped)thebandsarerigidlyshifted. However,assoonas down shift. The opposite shift in Ref. 88 might be due
a second valley is close to the Fermi energy, the doping to the free-electron states at Γ (i.e., the states in the
cannot be described by a rigid shift of the bands any- vacuum between the repeated images, cf., also Ref. 89)
more. For electron doping, the down shift of the bands which approach the Fermi energy with increasing dop-
at K slows down and, as the valley at Q starts to get ing. Also the authors of Refs. 20,23,89 find a down shift
occupied,is eventuallyreversedintoanupshift. Finally, of the Q valley further supporting our results even if in
forhighelectrondoping,theKvalleyisunoccupiedagain
8
Trilayer MoS
2
Electron doping Hole doping
1.0 0.2
n = 0.00
0.0
0.8
V] 0.6 V] -0.2
e e
E) [F 0.4 E) [F-0.4
E - E - -0.6
( (
0.2
-0.8
n = 0.00
0.0 -1.0
Γ M K Γ Γ M K Γ
0.2
n≈ +2.25 ⋅ 1013 cm-2
0.8
0.0
V] 0.6 V] -0.2
e e
) [F 0.4 ) [F-0.4
E E
(E - 0.2 (E - -0.6
n≈ −2.25 ⋅ 1013 cm-2 -0.8
0.0
-1.0
Γ M K Γ Γ M K Γ
0.8
n≈ +1.69 ⋅ 1014 cm-2
0.4
0.6
V] V] 0.2
) [eF 0.4 ) [eF 0.0
E E
E - 0.2 E - -0.2
( (
0.0 -0.4
n≈ −1.69 ⋅ 1014 cm-2
-0.6
-0.2
Γ M K Γ Γ M K Γ
FIG. 6. Band structure for different FET induced doping of trilayer MoS2. The figures in the left column are for electron
dopingwhiletherightcolumnshowstheholedopingcase. Incontrasttothemonolayercase,thedopingatΓ/Q(hole/electron
doping) ismoreimportant in trilayerMoS2. Thebandstructuresformore dopinglevelsandall theotherTMDs can befound
in theappendix (Figs. 28–63).
those works the asymmetric electric field in an FET has at K and Q45,46,48 and thus decrease the doping-charge
not been taken into account. The amount of doped elec- concentration needed to solely dope the valley at Q (cf.,
tronsneededtohavethechargecompletelylocalizedatQ Fig. 27).
depends on the TMD (i.e., the initial energy difference
InmultilayerMoS ,WS ,andWSe firstthe valleyat
between the minimum at K and Q) and is larger than 2 2 2
n=−2.2·1014cm−2(n≈−0.2e/unitcell): thetransition KisdopedandbothvalleysatKandQareoccupieduntil
occurs for MoS at n≈−3.83·1014cm−2, for MoSe at n≈−3.3·1014cm−2 while in bi- andtrilayerMoSe2 and
2 2
n≈−2.5·1014cm−2,forMoTe atn≈−2.22·1014cm−2, MoTe2 the order is reversed: electrons first occupy the
2
for WS at n ≈ −3.41 · 1014 cm−2, and for WSe at valley at Q and the doping at K is always smaller. This
2 2
n ≈ −3.31· 1014 cm−2. Please note, that using LDA is due to the minimum at Q being lower in energy than
the one at K in the undoped system (see Figs. 8, 9). For
couldslightlychangetheseresultsasitleadstoasmaller
unit cell. The compressivein-plain strain wouldthen re- a doping of n<∼−2.1·1014cm−2 for MoSe2 the K valley
ducethedifferencebetweentheconduction-bandminima is even unoccupied. Yet, one can expect that for small
electrondoping(|n|<1013cm−2)ofthicksamples(more
9
than3layers)ofMoS ,WS ,andWSe theelectronswill culated the planar-averaged doping-charge distribution
2 2 2
also first occupy the Q valley as this valley is lowered in along z,
energy with increasing number of layers.
e ε1≤εi,k≤ε2
Hole doping ρd||op(z)= Ω2DNk Z dA Xi,k |ψi,k(r)|2 (7)
HereΩ istheunitcellareaandN isthetotalnumber
2D k
Forp-typedopingofthemonolayerTMDs,thedoping of k points. The interval [ε ,ε ] is defined as above, i.e.,
1 2
charge first occupies the extrema at K. However,in con-
[E ,E ] and [E ,E ] for hole and electron doping, re-
F m m F
trasttotheelectron-dopingcase,inthehigh-hole-doping
spectively. This property not only reveals the thickness
limit (n > +0.2e/unit cell, n >∼ +2.1· 1014 cm−2) of of the conductive channel but also the relative distribu-
the monolayer TMDs both valleys at Γ and K are oc-
tion among the different layers.
cupied – the relative amount of doping charge in the Γ
In our calculations for low hole doping the charge is
valleyisevenhigherthanthatinK.Thetransitionwhen
nearly evenly distributed between the first two layers
the Γ valley is more occupied than the K valley again
withonlysmallcontributionsatthethirdlayerascanbe
depends on the TMD, i.e., on the initial energy differ-
seeninthe upper panelofFig.13. Increasingthe doping
ence between the valence-band maxima. In the case of
(i.e., increasing the gate voltage, lower panel of Fig. 13)
MoS forexampleevenasmalldopingis enoughtodope
2 thechargeismoreandmorelocalizedonthelayerclosest
more the Γ valley. The only exception from this picture
tothegatewithanegligibleamountofholesonthethird
is MoTe for which K is always more occupied even for
2 layer. In the case of electron doping of multilayer MoS ,
n=+0.35e/unitcell(n≈+3.23·1014cm−2, cf., Fig.9). 2
thechargeislocalizedonthefirstlayerasonlytheKval-
Hole doping of the bilayer and trilayer TMDs is again
leyisdoped(smallcontributionoftheout-of-planestates
very similar: in all investigated compounds (except
of sulfur). Only for higher doping when the minimum at
MoTe ) first the conduction-band maximum at Γ is oc-
2 Q is occupied a small amount of charge can be found on
cupied,whileforhigherdopingalsoKstartstogetfilled. the second layer (cf., lower panel of Fig. 13 and right
Most interestingly, in some cases the second band at K
panel of Fig. 7). One can also see that the asymmetry
is never occupied. For trilayer WS and WSe it is even
2 2 inthedoping-chargedistributionismorepronouncedfor
pushed down in energy, effectively increasing the split-
holedopingandthatthesysteminthiscasemovescloser
tingofthespin-orbit-splitbands. ForWSe itispossible
2 to the dielectric. Furthermore, for low hole doping the
to achieve even higher hole-doping concentrations than
doping charge on the second layer is even slightly larger
shown in Fig. 11 by using an IL-FET16. We thus calcu-
than the one on the first.
latedfortrilayerWSe alsosomehigherdopingcases(up
2 Figure 14 summarizes the doping-charge distribution
to n = +1e/unit cell, or n ≈ +1·1015 cm−2) in order
forallbi- andtrilayerTMDs. Inthe hole-dopingcaseall
to understand what might happen for such a high dop-
TMDsbehavesimilarlyexceptMoTe : forlowdopingthe
2
ing. Ourcalculatedbandstructure is verydifferent from
holes are delocalized over the first two layers with only
the one in Ref. 16 which was calculated without proper
small contributions in the third layer and, thus, the bi-
treatmentoftheFETsetup. Foradopingofoneholeper
and trilayer systems are nearly the same. Increasing the
unit cellthe secondbandatKis pushed downbelow the
dopingleadstostrongerlocalizationofthechargewithin
FermienergyandalsothefirstbandatKislowered. The
the first layer and an effective narrowing of the conduc-
bandstructureinFig.12showsthattheformerbandgap
tivechannel. OnecanalsoeasilyunderstandwhyMoTe
2
is closed and the first two bands are nearly unoccupied.
behaves differently: in all multilayer TMDs first the val-
However,sincethesearebandslocalizedonthefirstlayer
ley at Γ is doped, since it is the valence-bandmaximum,
close to the gate, we believe that for such a high doping
while in MoTe the K valley is the maximum. As we
2
the ions of the ionic liquid might start to interact with
haveseeninsectionIIIAthe statesclose to Γ havelarge
WSe . In our simplified model without inclusion of the
2 out-of-plane contributions of both the transition metal
fulldielectricitisdifficulttoprovethisstatementandwe
and the chalcogen atom, while the valley at K is com-
willthusleavethisinterestingproblemforfutureinvesti-
posedonly of in-plane states. Increasing the hole doping
gationsandwillconcentratehereonlowerdopingvalues.
however also leads for MoTe to a small doping of the
2
Γ valley. Thus, the amount of charge within the second
layer increases slightly in the beginning. The small kink
for trilayer MoTe close to n =+0.46·1014cm−2 is due
2
Conductive channel to both bands at Γ and K being close to the Fermi en-
ergy. Inthislowdopinglimit(pervalley)thecalculation
Another important property in order to understand wouldrequireaninfinitenumberofkpointstofullycon-
different experiments on FET doping on TMDs is the verge the results. Further increase of the doping leads
doping-charge layer thickness, i.e., the size and shape as for all TMDs to a larger screening of the electric field
of the conductive channel created and influenced by the andthereforetothe strongerlocalizationwithinthe first
gatevoltage. Tovisualizethe conductivechannelwecal- layer as can also be seen in the lower panel Fig. 13.
10
Electron doping Hole doping Electron doping Hole doping
MoS
2 n [1014 cm-2] n [1014 cm-2]
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
Γ 1.0
Monolayer Electron filledempty --00000.....02442 KKQ filledHole empty 00000.....02468 nnnΓKQ
1.0
V] 0.4
e 0.8
Bilayer E - E) [F--0000....0242 n [n]α 000...246
( 0.0
1.0
0.4
r 0.8
e 0.2
y 0.6
a 0.0
Tril -0.2 00..24
-0.4
0.0
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
n [e / unit cell] n [e / unit cell]
FIG. 7. Position of the different band minima/maxima E (left panel) with respect to the Fermi level and relative amount of
i
dopingchargepervalleynα (rightpanel)asafunctionofdopingformono-,bi-,andtrilayerMoS2. “Q”labelstheconduction-
band minimum half-way between K and Γ. In each graph, the scale and the units for the lower x axis are given in the lowest
graph, while those of the upper x axis are given in the uppermost graph. Two different line styles for the two spin-orbit-split
conduction-bandminima at K were used in order to enhancethe readability. Lines are guides for theeye.
For electron doping we can divide the different TMDs MoS ishowevertwiceaslargeasinthetungstensystems
2
into two different classes: (i) those in which the con- (≈ 300meV for MoS and ≈ 150meV for WS /WSe ).
2 2 2
ductive channel for low doping (n ≈ −1013cm−2) has a Thus, the valley at the Q point is doped much earlier in
thickness of just one layer (MoS , WS , WSe ) and (ii) WS /WSe than in MoS .
2 2 2 2 2 2
those with a three-layer-thick channel (MoSe , MoTe ).
2 2
Using the results of Figs. 7–11 one can see that in the
TMDs of class (i) initially the K valley is doped while Number of occupied bands
in (ii) the Q valley is occupied. Since the chalcogen
states close to the conduction-band minimum at K have
It is important to note, that the thickness of the
mainly in-plane character (in contrast to the transition-
doping-charge distribution, the number of occupied
metalstateswhichhavedz2 character),thehybridization bandsatagivenkpoint,andthe numberofTMDlayers
betweenthelayersissmallandtheelectronsaremorelo- (i.e., the system size) are uncorrelated. Indeed, as can
calized within the first layer. The chalcogen states close
be seeninFig.15,the totalspin-valleydegeneracyν can
to Q on the other hand have a large out-of-plane contri-
be quite similar for different number of layers, whereas
bution which leads to a stronger hybridization between
the doping charge is localized on one or two layers as
thelayers. Withincreasingdopingtheelectricfieldofthe
seen in the previous section. Here the total spin-valley
gate is more and more screened and the size of the con-
degeneracy has been calculated by counting the number
ductive channel reduces to one layer. Furthermore, one
of valleys within the interval [ε ,ε ] as defined above:
1 2
can also understand why the tungsten dichalcogenides
have a steep increase of the channel thickness in the be- ν = να = gαgα. (8)
ginning while this is not the case for MoS : first, the s v
2 Xα X
difference between the conduction-band minimum at K
and Q is much smaller in multilayer WS2/WSe2 than in Heregsαandgvαarethespinandvalleydegeneraciesofthe
MoS2 and second, a small electron doping can also re- valley at k point α = {Γ,K,Q}. For electron doping ν
sultsinaneffectiveseparationofthesingle(doped)layer is muchhigher thanin the hole doping caseas the valley
from the multilayer system. The difference between the degeneracyforthe conduction-bandminimumcloseto Q
conduction-band minimum at K and Q for monolayer is g = 6 (spin degeneracy g = 1). Thus, as soon as
v s