Solid-S a e Elec onics 197 (2022) 108437
A ailable online 23 Augus 2022
0038-1101/© 2022 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-
nc-nd/4.0/).
Towa ds a DFT-based laye ed model o TCAD simula ions o MoS
2
L. Done i, C. Ma quez, C. Na a o, C. Medina-Bailon, J.L. Padilla, C. Samped o, F. Gamiz
Depa amen o de Elec onica and CITIC, Uni e sidad de G anada, G anada, Spain
ARTICLE INFO
The e iew o his pape was a anged by “S.
C is olo eanu”
Keywo ds:
Densi y unc ional heo y
MoS
2
TCAD
Dielec ic cons an
Densi y o s a es
ABSTRACT
In his wo k, we employ he esul s o a omis ic DFT calcula ion o ex ac use ul pa ame e s o he simula ion o
ew-laye s MoS
2
s uc u es wi h adi ional TCAD ools. In pa icula , we ocus on he cha ge dis ibu ion, which
allows us o ob ain a laye ed model o he dielec ic cons an , and on he e ec i e densi ies o s a es in he
conduc ion and alence bands aking in o accoun he ull 2D densi y o s a es. Using his model, we compu e he
capaci ance o a me al–oxide–semiconduc o s uc u e and compa e i o he one ob ained employing a uni o m
model wi h a e aged e ec i e pa ame e s.
1. In oduc ion
The laye ed s uc u e o 2D semiconduc o s such as MoS
2
p esen s
se e al challenges o hei accu a e simula ion employing s anda d
TCAD ools. Apa om he need o hickness-speci ic ma e ial pa am-
e e s [1–3], desc ip ions based on bulk pa ame e s canno accu a ely
ep oduce he laye ed spa ial cha ge dis ibu ion and he 2D densi y o
s a es. The e o e, e en i he p ope ies o MoS
2
a e being success ully
s udied h ough ab ini io a omis ic me hods [4], i is no s aigh o wa d
o include his knowledge in classical TCAD ools [5].
In his pape , we model he laye ed s uc u e o MoS
2
h ough a s ack
o al e na ing semiconduc o and insula ing laye s ep esen ing he Van
de Waals (VdW) gaps [6], wi h ma e ial pa ame e s ex ac ed di ec ly
om Densi y Func ional Theo y (DFT) calcula ions (Sec ion 2). In
pa icula , in Sec ion 3, we ob ain a laye ed dielec ic model h ough he
analysis o cha ge dis ibu ion in mono- and ew-laye s uc u es wi h an
applied ou -o -plane elec ic ield, and ex ac laye -dependen Nc and
N alues om he calcula ion o he ull Densi y O S a es (DOS). Then,
in Sec ion 4, we implemen ou model and compa e he esul s o he
laye ed model wi h hose o a uni o m model wi h a e aged alues o
he pa ame e s. Finally, we d aw ou conclusions in Sec ion 5.
2. DFT calcula ions
DFT calcula ions a e pe o med wi h Quan umATK ( e sion S-
2021.06) [7], employing i s LCAO calcula o wi h GGA exchange/co -
ela ion, PBE unc ional and G imme DFT-D2 VdW co ec ion.
Spin–o bi in e ac ion is included o ob ain a be e desc ip ion o he
band s uc u e o he alence band a ound i s maximum. We conside
ew-laye MoS
2
s uc u es wi h a numbe o laye s, N, be ween 1 and 10,
keeping he expe imen al alue o he in-plane la ice cons an . Fo each
s uc u e in equilib ium, we compu e he band s uc u e ha will be
employed o ob ain he DOS. Then, we apply an elec ic ield pe pen-
dicula o he semiconduc o laye s by o cing a po en ial di e ence
be ween he bounda ies o he simula ion cell. Fo he biased s uc u es,
we compu e he induced elec on densi y di e ence Δn, and he elec-
os a ic po en ial di e ence ΔV, aking in-plane a e ages as a unc ion
o he ou -o -plane posi ion z [8].
3. Laye ed model
Some examples o DFT calcula ions wi h an applied bias a e shown in
Fig. 1. The beha io o he o al cha ge densi y and he dipole momen
densi y in each laye as a unc ion o he ex e nal elec ic ield sugges s a
laye ed dielec ic model, whe e MoS
2
laye s a e sepa a ed by VdW gaps
wi h a di e en dielec ic cons an , as he one shown in Fig. 2. By i ing
he DFT esul s, he wo dielec ic cons an s and he hickness o he
di e en laye s can be ex ac ed [8]. The esul ing pa ame e alues (see
Table 1) a e sligh ly di e en om hose o Re . [8] because he e he
spin–o bi in e ac ion is included in he DFT calcula ion and a sligh ly
di e en la ice cons an is employed. The esul s sugges ha he
hickness o he ex e nal laye s is di e en om ha o he in e nal ones
and also ha an emp y gap is needed o “ ill” he hickness o he mul i-
laye s uc u e, as shown by he emp y ec angles a he sides o he
E-mail add ess: [email p o ec ed] (L. Done i).
Con en s lis s a ailable a ScienceDi ec
Solid S a e Elec onics
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h ps://doi.o g/10.1016/j.sse.2022.108437
Recei ed 4 July 2022; Recei ed in e ised o m 12 Augus 2022; Accep ed 15 Augus 2022
Solid S a e Elec onics 197 (2022) 108437
2
laye ed s uc u e in Fig. 2: he hickness o such gaps is gi en by
d/2− s/2, whe e d=6.15 Å is he in e laye dis ance, equal o hal he c
la ice cons an o bulk MoS
2
.
I is no s aigh o wa d o model he DOS o 2D ma e ials h ough
he usual 3D exp essions in ol ing he e ec i e mass because o se e al
easons. Fi s o all, i mus be aken in o accoun ha he posi ion o he
Conduc ion Band Minimum (CBM) and Valence Band Maximum (VBM)
in ecip ocal space a y wi h he numbe o laye s, N: he alley mul-
iplici ies a e modi ied and also he co esponding band cu a u es
which de ine he e ec i e masses change. Fo example, as we can see in
Fig. 3, he gap is di ec in he mono-laye case, while i becomes indi ec
o N>1. Then, e en assuming ha he ma e ial pa ame e s can a y
wi h N, he DOS o 2D bands p esen s a unc ional dependence on ene gy
which is di e en om he one o 3D ma e ials. As an example, in Fig. 4,
we show he DOS, g(E), o mono- and some ew- laye s s uc u e. In each
case, he DOS is app oxima ely cons an nea he ex ema o each band
and disc e e s eps can be obse ed when he minimum (o maximum) o
di e en bands a e eached. The na ow peaks nea he jumps a e a i-
ac s o he nume ical p ocedu e ( o imp o e he smoo hness o he
esul ing DOS, we employ he e ahed on me hod [9] o B illouin zone
in eg a ion, bu some small luc ua ions a e s ill p esen ).
Fo non-degene a e bulk ma e ials, he elec on densi y, n, can be
ob ained by:
n=Ncexp(−Ec−EF
kT )(1)
whe e Nc is an e ec i e densi y o s a es in he conduc ion band, Ec is he
CBM, EF is he Fe mi le el, k is Bol zmann cons an , T is he absolu e
empe a u e. In his case, Nc is p opo ional o m3/2
DOS (whe e mDOS is he
DOS e ec i e mass), because o he ene gy dependence o
Fig. 1. Excess elec on densi y Δn (a, c) and elec ic ield E (b, d) o mono-
laye (a and b) and 4-laye (c and d) MoS
2
, wi h 2 V bias. In (a) and (c),
closed and open ci cles ep esen he z posi ions o Mo and S a oms,
espec i ely.
Fig. 2. Laye ed model o MoS
2
, showing he meaning o he di e en hickness
pa ame e s o Table 1 and he in e laye dis ance, d.
Table 1
Pa ame e s o he laye ed dielec ic model o Fig. 2.
∊s ∊b s (Å) i (Å)
22.8 2.2 5.15 4.53
Fig. 3. Band s uc u e o MoS
2
om mono-laye o 8 laye s. The ligh blue
a eas deno e he band gap and i is shown o help isualize he band ex ema
(bo h CBM and VBM a e a K poin o mono-laye , CBM a Q and VBM a Γ
poin s o mul i-laye s uc u es). Ene gies a e e e enced o he Fe mi le el.
Fig. 4. DOS o MoS
2
s uc u es wi h di e en numbe o laye s.
L. Done i e al.
Solid S a e Elec onics 197 (2022) 108437
3
g(E)∝(E−Ec)1/2. Howe e , as we can obse e in Fig. 4, g(E)shows
nei he a bulk no a pu e 2D beha io (whose DOS is a single s ep
unc ion). The e o e, we employ a nume ical app oach o compu e he
e ec i e 2D DOS in he conduc ion band, Nc,2D, aking in o accoun he
whole DOS, wi h he ollowing exp ession:
Nc,2D=∫∞
Ec
g(E)exp((E−Ec)/kT)dE (2)
which is ela ed o he 2D elec on densi y by n2D=
Nc,2Dexp( − (Ec−EF)/kT). The esul s o Eq. (2) a oom empe a u e
(T=300k) and an analogous one o N ,2D a e shown in Fig. 5(a). The
jump be ween mono-laye and bi-laye is due o he ac ha he posi ion
o he conduc ion band minima and alence band maxima change: he
DOS e lec s he change in he e ec i e mass and in he degene acies o
he ex ema posi ion in he ecip ocal space. Fo example, ega ding
Nc,2D, in Fig. 4 we can see ha he heigh o he i s s ep o he DOS o
he conduc ion band is much smalle in he mono-laye case han in he
mul i-laye case. The densi y o s a es is la ge o N>1 because he
CBM is a Q poin which has a la ge mul iplici y han he K poin (6 s.
2) and also because o a di e en band cu a u e. Then, he inc ease o
la ge N is caused by he ac ha he subsequen bands ge close in
ene gy o he i s one. To ob ain 3D e ec i e densi ies o s a es Nc and
N , we no malize Nc,2D and N ,2D by he co esponding semiconduc o
hickness:
Nc/ =Nc/ ,2D
s+ (N−1) i
(laye ed model)(3)
These bulk pa ame e s show a p onounced dec ease as N g ows, as
depic ed in Fig. 5(b), because he inc ease o he 2D quan i ies (Fig. 5(a))
is much smalle han he inc ease o he hickness.
Finally, we compu e he empe a u e dependence o he e ec i e
densi ies o s a es. In Fig. 6(a), we ep esen no malized Nc and N as a
unc ion o T: o N=2 we can obse e a linea dependence on T, as
expec ed o 2D ma e ials, and when N inc eases he beha io becomes
mo e simila o he bulk one, wi h a T3/2 dependence (especially a
la ge empe a u es). Indeed, in hick 2D sys ems, he sum o many s ep
unc ions co esponding o he di e en sub-bands can gi e ise o an
o e all shape simila o he 3D case. This ac should no be su p ising,
because also o he p ope ies, such as he size o he band gap Eg,
app oach he bulk alue al eady o N=8 [2]. A beha io co e-
sponding o a ela i ely la ge exponen is also obse ed o he mono-
laye case: his can be explained aking in o accoun he wo close
conduc ion band minima which gi e ise o qui e close s eps in he DOS
(Fig. 4).
4. TCAD simula ions
We implemen he laye ed model o MoS
2
s uc u es o di e en
hickness in Sen au us TCAD [5], wi h pa ame e s aken om Table 1.
Fo compa ison, we also in oduce a uni o m model wi h an e ec i e
a e aged dielec ic cons an [8]. The elec on and hole densi ies a e
compu ed employing he alues o Nc and N ex ac ed in he p e ious
sec ion, which depend on he sample hickness. In he case o he uni-
o m model, hese alues ha e been co ec ed o ake in o accoun he
di e en e ec i e semiconduc o olume, ha is:
Nc/ =Nc/ ,2D
Nd (uni o m model)(4)
As shown in he p e ious sec ion, he DOS beha io does no ully
co espond o ei he 2D o 3D model, so ha he alues o Nc and N a e
di ec ly se h ough he pa ame e s Nc300 and N 300, in eDOSMass
and hDOSMass, espec i ely (wi h Fo mula =2). We i s simula e
isola ed MoS
2
s uc u es wi h an applied bias, in o de o compa e he
elec os a ic po en ial p o ile and elec ic ield wi h he ou pu o he
DFT calcula ions o Sec ion 2. The esul s o he case o a 4-laye
s uc u e and an applied bias o 2 V a e shown in Fig. 7, exhibi ing
good ag eemen , a leas o he laye ed model.
Nex , we simula e he capaci ance o me al/oxide/semiconduc o
s uc u es composed by a 1 nm SiO
2
laye and MoS
2
wi h di e en
numbe o laye s (inse in Fig. 8). The semiconduc o is conside ed
sligh ly doped (wi h ND equi alen o a 2D densi y o 1 ×1010 cm2), he
me al wo k unc ion co esponds o MoS
2
midgap, he bo om elec ode
is kep a Vbo =0V, while he op elec ode bias, V op, is a ied. In
Fig. 8, we show he simula ed capaci ance compa ed o he one ob ained
o simila s uc u es whe e MoS
2
laye s a e subs i u ed by a uni o m
Fig. 5. 2D (a) and 3D (b) e ec i e DOS o MoS
2
s uc u es as a unc ion o he
numbe o laye s N.
Fig. 6. Tempe a u e dependence o N
c (a) and N (b), no malized o 1 a T=
150 K in log–log scale. The expec ed beha io o bulk ma e ials (∝T3/2) and o
s ic ly 2D ma e ials (∝T) a e also shown as do ed lines.
L. Done i e al.
Solid S a e Elec onics 197 (2022) 108437
4
ma e ial wi h a e aged pa ame e s [8]. A ze o bias, he esul s ob ained
wi h he laye ed and uni o m models a e he same, while he esul s
di e o la ge posi i e and nega i e alues o V op . The main di e ence
o all alues o N a e he smalle capaci ance o he laye ed models o
la ge absolu e alues o V op and he la ge alues o V op needed o each
he in e sion/accumula ion egions. These esul s can be explained by
no ing ha he laye ed model o Fig. 2 includes small dielec ic egions
be o e he i s semiconduc o laye and a e he las one.
5. Conclusions
Employing DFT simula ions, we ha e ex ac ed dielec ic cons an s
and e ec i e DOS in he conduc ion and alence bands o he imple-
men a ion o a laye ed MoS
2
model o be employed in TCAD simula-
ions. We ha e shown he ele ance o aking in o accoun he laye ed
s uc u e o ew-laye MoS
2
by compa ing he TCAD esul s wi h hose
ob ained wi h a uni o m model.
Decla a ion o Compe ing In e es
The au ho s decla e ha hey ha e no known compe ing inancial
in e es s o pe sonal ela ionships ha could ha e appea ed o in luence
he wo k epo ed in his pape .
Acknowledgmen s
The au ho s would like o hank he inancial suppo o p ojec s
H2020-MSCA-IF-2019 Re . 895322 (EU Ho izon 2020 p og amme),
TEC2017-89800-R (Spanish S a e Resea ch Agency, AEI), Juan de la
Cie a Inco po aci´
on Fellowship scheme 307 unde g an ag eemen No.
IJC2019-040003-I (MICINN/AEI). P18-RT-4826 (Regional Go e nmen
o Andalusia) and B-TIC-515-UGR18 (Uni e si y o G anada). Funding
o open access cha ge: CBUA/Uni e sidad de G anada.
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Fig. 7. Elec os a ic po en ial (a) and elec ic ield (b) compa ed be ween DFT
calcula ion and TCAD simula ions employing he laye ed model (solid cu e)
and he uni o m model (dashed cu e), o he 4-laye MoS
2
wi h an applied
bias o 2V.
Fig. 8. Capaci ance o he s uc u e depic ed in he inse ob ained wi h TCAD
simula ions, o he laye ed (solid lines) wi h di e en numbe o laye s N and
uni o m (dashed lines) co esponding models. The oxide capaci ance C
ox is
shown as a e e ence.
L. Done i e al.