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A simpli ied app oach o include con inemen
induced band s uc u e changes in o he NsFET
compac model
Aishwa ya Singh
Elec ical Enginee ing
IIT Gandhinaga
Gandhinaga , India
singh [email p o ec ed]
Mohi D. Gane iwala
Elec onics and Compu e Technology
Uni e sidad de G anada
G anada, Spain
mohi @ug .es
Ramandeep Kau
Elec ical Enginee ing
IIT Gandhinaga
Gandhinaga , India
[email p o ec ed]
Niha R. Mohapa a
Elec ical Enginee ing
IIT Gandhinaga
Gandhinaga , India
[email p o ec ed]
Abs ac —This wo k p esen s a simpli ied ma hema ical
me hod o cap u e he k.p-based band s uc u e modi ica ions
wi h con inemen and de ice subs a e/ anspo o ien a ion in
he compac model o quan um con ined Nanoshee FETs. The
change in e ec i e mass wi h con inemen is cap u ed in e ms
o non-pa abolic sub-bands. The es ima ed sub-bands a e used
o compu e in e sion cha ge densi y and ga e capaci ance using
a bo om–up scalable compac model o di e en de ice dimen-
sions and subs a e/channel o ien a ions. The accu acy o he
p oposed me hod is con i med using k.p simula ion in Global
TCAD Solu ions (GTS).
Index Te ms—non-pa abolic subbands, nanoshee FET, k.p
based bands uc u e, quan um con inemen , channel o ien a ion,
e ec i e mass, bo om-up scalable compac model
I. INTRODUCTION
The Silicon Nanoshee FETs (NsFET) ha e been iden i ied
by he semiconduc o indus y as he leading a chi ec u e o
con inue CMOS echnology scaling beyond he 5 nm node
[1]. The NsFETs o e excellen ga e elec os a ics wi h negli-
gible sho -channel e ec s. Fig. 1 shows schema ic diag am
o a NsFET wi h he ex emely hin body (H) and wid h
(W). Fo such a con ined sys em, he Quan um Mechanical
Con inemen (QMC) induced e ec s play a majo ole in
de ice pe o mance. The QMC causes he sepa a ion o ene gy
bands, esul ing in sub-band o ma ion and he eby a ec ing
he densi y o s a es (DOS). Unlike he bulk MOSFETs, ene gy
bands in con ined s uc u e can no longe be conside ed as
a con inuous 3D s ack o ene gies. This a ec s he in e sion
cha ge densi y and his phenomena is mo e p onounced in hin
and na ow NsFETs.
Fig. 2 shows he E-K diag am o a (100)/[110] o ien ed
silicon channel. He e, wo ou o he six alleys (as shown
in Fig. 3), esiding in he con inemen di ec ion (x-y in Fig.
1), a e p ojec ed a he Γpoin (kz= 0). The pai o wo-
old degene a e alleys a e p ojec ed a he o -Γaxis. No e
ha only he lowes ene gy sub-band in each alley is shown.
As shown, he ela i e posi ion o he bands as well as hei
Mohi D. Gane iwala would like o acknowledge he unding om he
Eu opean Union’s Ho izon 2020 esea ch and inno a ion p og am unde he
Ma ie Sclodowska-Cu ie g an ag eemen numbe 101032701
Fig. 1: C oss sec ional iew o a NsFET con ined in he x-y
di ec ion. W and H a e wid h and hickness o NsFET espec-
i ely. The e ical and ho izon al sidewall o he channel has
di e en c ys allog aphic o ien a ion.
Fig. 2: The k.p based E-K diag am o Si channel o ien ed in
(100)/[110] di ec ion o W (nm) x H (nm) = 10x10 and 10x3.
Two old degene a e band (∆2) in con inemen di ec ion a e
p ojec ed a Γ(kz= 0) and pai o wo old degene a e band
(∆4) a o -Γ(nega i e and posi i e kz espec i ely).
cu a u e changes wi h con inemen . This esul s in he change
o he sub-band ene gies, e ec i e mass and DOS. In addi-
ion, hese a ia ions a e di e en o each subs a e/channel
o ien a ion [2]. Mo eo e , due o he non plana na u e o he
NsFET channel, he channel’s e ical and ho izon al side wall
has di e en c ys allog aphic o ien a ions. The e o e, hese
di e en o ien a ions need o be p ope ly conside ed. No e ha
all he o ien a ions men ioned in his wo k a e along he leng h
o he channel.
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Fig. 3: Six equi alen ene gy ellipsoids o bulk Silicon o ien ed
in kx, kyand kzdi ec ions.
In ea lie li e a u e, i has been shown ha he con en ional
model o bulk MOSFETs needs o be modi ied o accu a ely
ep oduce he NsFET cha ac e is ics [3], [4]. The sub-band
in o ma ion has o be accu a ely calcula ed and included in he
compac model. Since bo h H and W change wi h echnology
o de ice/ci cui design, anywhe e om a ew o se e al
100 sub-band ene gy le els needs o be calcula ed o he
model o seamlessly scale om hin o hick and na ow
o wide NsFETs. The BSIM ex ension o NsFETs uses an
empi ical equa ion o calcula e sub-band ene gy le els ha
equi e se e al i ing pa ame e s [3]. In addi ion, an empi ical
i ing is used o cap u e he e ec i e mass change wi h he
con inemen . Se e al epo s o model he change in e ec i e
mass wi h con inemen a e a ailable. Howe e , almos all o
hem use cu e i ing o expe imen al o simula ion da a [5].
Fu he , as such calcula ions a e equi ed o e e y alley,
which di e s depending on subs a e/channel o ien a ion, his
equi es ex ensi e i ing as well as di e en equa ions o
ma ch he cha ac e is ics. Such ex ensi e use o i ing pa am-
e e s no only make he model p one o con e gence issues bu
also makes i complica ed and empi ical (by losing physical
insigh ).
Recen ly, a bo om-up compac model has been epo ed o
he NsFETs, which modi ies he ene gy calcula ion equa ion
o an in ini e po en ial well o cap u e he wa e- unc ion
pene a ion in an explici manne [4]. This allows o he
accu a e calcula ion o any numbe o sub-bands wi hou he
use o i ing pa ame e , making he model compu a ionally
e icien and scalable. Howe e , he p oposed model in [4]
uses an a e age DOS e ec i e mass and does no include he
e ec o o ien a ion and hickness/wid h scaling.
In his wo k, we ha e shown ha he p oblem o change in
e ec i e mass wi h con inemen can be o mula ed in e ms
o ha ing non-pa abolic sub-bands. The e o e, by modi ying
he widely used non-pa abolic e ec i e mass app oxima ion,
all he e ec s discussed abo e can be included in he compac
model. The model [4] is hen ex ended o calcula e he sub-
band ene gy explici ly using only a couple o addi ional
i ing pa ame e s. This me hod makes he model simple,
compu a ionally e icien and p edic able.
Fig. 4: D ain cu en (IDS) as a unc ion o VGS o a NsFET
wi h h ee shee s and W (nm) x H (nm) = 20x5, ga e
leng h (LG) = 12 nm o VDS = 50 mV and 0.7 V. The
symbols ep esen he TCAD da a and solid lines ep esen
he expe imen al da a [6]. The TCAD deck is well calib a ed
agains he expe imen al da a.
II. SIMULATION SETUP
The simula ed NsFET de ice (shown in Fig. 1) is con ined
in he x-y di ec ion wi h anspo in he z-di ec ion. The
simula ions a e done using he Global TCAD Solu ions (GTS)
amewo k [7]. The TCAD deck is i s calib a ed wi h he
NsFET expe imen al da a ob ained om [6]. Fig. 4 shows he
calib a ed d ain cu en o a 12 nm long n- ype NsFET wi h
h ee shee s and W (nm) x H (nm) = 20x5 o bo h linea (VDS
= 50 mV) and sa u a ion (VDS = 0.7 V) egion. The calib a ed
deck is hen used o simula e he 2D cu su ace o single
shee NsFETs wi h di e en H, W and su ace o ien a ions.
The k.p along wi h he Poisson equa ion is used o simula e
he elec os a ic cha ac e is ics. This me hodology accu a ely
cap u es he QMC e ec s such as sub-band ene gy change,
alley spli ing, and e ec i e mass a ia ions. The compac
model p oposed in [4] is used as he baseline model, which is
hen modi ied using he app oxima ion p oposed o ma ch he
k.p simula ion da a.
III. METHODOLOGY
The ene gy alue o he sub-band can be calcula ed as,
Ei,j =ℏ2
2m∗"iπ
Wb2
+jπ
Hb2#(1)
He e, he subsc ip i, j indica es i h and j h sub-bands, ℏ
is he educed Planck’s cons an , m*is he a e age DOS
e ec i e mass, Wband Hba e he modi ied wid h and heigh
espec i ely as epo ed in [4]. Fig. 5 shows he simula ed
ga e capaci ance (Cgg) o (100)/[110] o ien ed silicon channel
NsFETs wi h W (nm) x H (nm) = 10x3 and 3x3 using bo h
k.p and he model de eloped in [4]. The model in [4] uses
(1) o sub-band ene gy calcula ion. The obse ed de ia ion
om k.p simula ion is due o he assump ion o a e age DOS
e ec i e mass in (1). The a e age DOS e ec i e mass a ec s
bo h he lowes ene gy subband le el and DOS, hus di ec ly
a ec ing he h eshold ol age o he de ice.
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TABLE I: Con ined (x-y di ec ion) and T anspo (z di ec ion) e ec i e mass o Silicon ellipsoidal alleys in di e en
subs a e/ anspo o ien a ion
O ien a ion P incipal axis Valley mx* my* mz*
(100)/[110]
Along kxE - 1,2 2mlm /(ml+m ) m (ml+m )/2
Along kyE - 3,4 2mlm /(ml+m ) m (ml+m )/2
Along kzE - 5,6 m mlm
(100)/[100]
Along kxE - 1,2 m m ml
Along kyE - 3,4 mlm m
Along kzE - 5,6 m mlm
(110)/[100]
Along kxE - 1,2 (ml+m )/2 2mlm /(ml+m ) m
Along kyE - 3,4 (ml+m )/2 2mlm /(ml+m ) m
Along kzE - 5,6 m m ml
(110)/[110]
Along kxE - 1,2 m 2mlm /(ml+m ) (ml+m )/2
Along kyE - 3,4 m 2mlm /(ml+m ) (ml+m )/2
Along kzE - 5,6 mlm m
(a) (b)
Fig. 5: Ga e capaci ance (Cgg) o (100)/[110] o ien ed NsFET
wi h W(nm) x H(nm), (a) 10x3 and (b) 3x3 using k.p (open
ci cles) and model [4] (solid lines). The de ia ion in Cgg is due
o he assump ion o a e age DOS e ec i e mass in model [4].
The e o e, o cap u e he e ec o de ice o ien a ion on sub-
band ene gy le els, (1) needs o be modi ied by in oducing
o ien a ion dependen e ec i e mass. The modi ied equa ion
can be w i en as below.
Ei,j=i2ℏ2
2m∗
xπ
Wb2
+j2ℏ2
2m∗
yπ
Hb2
(2)
He e, m∗
x, m∗
ya e con ined e ec i e masses in x and y
di ec ions espec i ely. I can be calcula ed by using he
e ec i e masses associa ed wi h he p incipal axis o ellip-
soids ( ans e se (m ) and longi udinal (ml) e ec i e mass).
Gene ally, due o he non-alignmen o he p incipal axis o
ellipsoids wi h subs a e/channel o ien a ion, hese e ec i e
masses a e enso quan i ies [8]. To ela e he e ec i e mass
in ol ed in he de ice coo dina e sys em wi h he e ec i e
mass associa ed wi h ellipsoids, a o a ion ma ix is used in
his wo k. This ans o ms he ellipsoid coo dina e sys em in o
he de ice coo dina e sys em and esul s in he o ma ion o he
in e se e ec i e mass enso symme ic ma ix in he de ice
Fig. 6: Va ia ion o conduc ion band minima Ecwi h espec o
W using k.p (symbols), (2) (solid lines) and (1) (do ed line)
o H = 3 nm o bo h Γand o -Γ alleys o (100)/[110]
o ien a ion.
coo dina e sys em. Using his ma ix, e ec i e mass can be
compu ed o any a bi a ily o ien ed de ice in bo h con ined
di ec ions and anspo di ec ions. Table I shows he e ec i e
mass o Silicon ellipsoidal alleys shown in Fig. 3 in e ms o
bulk ml(0.89m0, m0is he es mass o an elec on) and m
(0.19m0) o di e en subs a e/ anspo o ien a ions.
Fig. 6 shows he a ia ion o g ound s a e ene gy (conduc-
ion band minima (Ec)) o NsFET wi h H = 3 nm and a ying
W. I co esponds o NsFET o dimension W (nm) x H (nm)
= 10x3 wi h quasi 2D channel o NsFETs o dimension 3x3
wi h quasi 1D channel. He e he symbols and he do ed line
shows Ecob ained om he k.p simula ions and he o iginal
equa ion (1) as p oposed in [4], which uses he a e age DOS
e ec i e mass. I can be seen ha he a e age DOS e ec i e
mass no only ails o cap u e he a ia ion o e ec i e mass
wi h hickness, bu i is also e oneous o W = 10 nm. The
modi ica ion o [4] as gi en in (2) wi h he use o m∗
xand m∗
y
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p esen ed in Table I is plo ed using solid lines in Fig. 6. The
modi ied model accu a ely p edic s he alue o Ec o W =
10 nm o bo h Γand o -Γ alleys. Howe e , i s ill does no
cap u e he accu a e a ia ion o e ec i e mass wi h W.
A. Modi ied E ec i e Mass App oach
Due o con inemen , he e ec i e mass is no cons an
wi h hickness [2]. This beha io can be mimicked using
he non-pa abolic co ec ion o E-K dispe sion. The non-
pa abolic co ec ion inc eases he e ec i e mass wi h ene gy
[9], whe eas in he con ined sys em, he e ec i e mass changes
wi h he educ ion in hickness. Fu he , wi h he con inemen ,
he ene gy o he sub-band also inc eases along wi h he
change in he cu a u e o E-K. The e o e, we a e looking a
sub-bands wi h highe e ec i e mass, which a e also highe
in ene gy wi h espec o he uncon ined s uc u e. Hence, i
he non-pa abolic co ec ion equa ion is modi ied in such a
way ha i changes he e ec i e mass wi h he same ene gy
which co esponds o he desi ed ene gy o E-K o educed
hickness, he change in e ec i e mass wi h he educ ion in
hickness could also be cap u ed. The ene gy calcula ion can
hus be modi ied using he me hod p oposed in [10] as,
Emod
i,j =ENP
i,j +
−1 + q1+4α1
ℏ2k2
z
2m∗
z
2α1
(3)
He e, kzand m∗
za e he wa e ec o and e ec i e mass a he
bo om o sub-band in he anspo di ec ion (z). ENP
i,j deno es
he sub-band ene gy minima a kz= 0 and can be w i en as,
ENP
i,j =Ui,j +−1 + p1+4α2(Ei,j −Ui,j)
2α2
(4)
He e, Ei,j can be calcula ed using (2), Ui,j is he expec a ion
alue o po en ial ene gy wi h espec o wa e unc ion o i h,
j h sub-bands (Ψi,j) and α1, α2a e used as i ing pa ame e s.
The Ui,j o NsFET is calcula ed as,
Ui,j =DΨ∗
i,j|˜
Φ|Ψi,jE(5)
whe e, ˜
Φis he pe u bing po en ial as explained in [4]. The
inal exp ession a e sol ing (5) is,
Ui,j =qQin
4Cc
(6)
He e, Qin is in e sion cha ge densi y and Ccis cen oid
capaci ance [4]. The Qin in ol ed in (6) can be calcula ed
using (3) and u ilizing he app oach explained in [10]. I is
gi en as,
Qin =−q"Cq1X
i,j
F−1
2 qΦc−Egmod
i,j −Eemod
i,j
kBT!
+Cq2X
i,j
F1
2 qΦc−Egmod
i,j −Eemod
i,j
kBT!# (7)
Fig. 7: Va ia ion o conduc ion band minima (Ec) wi h espec
o W using k.p (symbols) and (3) (solid lines) o H = 3 nm
o Γ alley o (100)/[110] o ien a ion. The modi ied sub-band
ene gy (3) i s he simula ion da a e y well.
Fig. 8: Roo mean squa e (RMS) e o in conduc ion band
minima (Ec) calcula ion o di e en de ice o ien a ion o W
(nm) x H (nm) = 10x3 o 3x3 wi hou i ing (2) and wi h
i ing (3). The signi ican educ ion in e o can be seen on
using he p oposed model.
He e, Cq1 and Cq2 a e he cons an s [10], Φcis he po en ial a
he cen e o NsFET (x,y = 0), Egmod
i,j is he sub-band ene gy
le el conside ing only geome ical con inemen gi en by (4),
Eemod
i,j co esponds o elec ical con inemen [10], F-1/2, F1/2 is
he Fe mi-Di ac in eg al o o de -1/2 and 1/2, kBis Bol zmann
cons an and T is empe a u e.
Fig. 7 shows he a ia ion o Ecwi h W ob ained om he
k.p simula ion and om (3). I can be seen ha he modeled Ec
ma ches wi h he k.p simula ion esul s and accu a ely cap u es
he a ia ion o e ec i e mass wi h con inemen . Fig. 8 shows
he oo mean squa e (RMS) e o o di e en o ien a ions o
NsFET anging om W (nm) x H (nm) = 10x3 o 3x3. I can
be seen ha he e o is signi ican ly educed using (3) o all
he di e en combina ions o subs a e/channel o ien a ion.
IV. RESULTS AND DISCUSSION
The modi ied ene gy exp ession (3) and cha ge exp ession
(7) in eg a ed wi h compac model in [4] is used o es i-
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(a) (b)
Fig. 9: (a) In e sion cha ge densi y (Qin ) and (b) ga e capac-
i ance (Cgg) as a unc ion o VGS o NsFET wi h di e en
dimensions in (100)/[110] o ien a ion. The model is in good
ag eemen wi h k.p simula ion da a.
(a) (b)
Fig. 10: (a) In e sion cha ge densi y (Qin ) and (b) ga e
capaci ance (Cgg) as a unc ion o VGS o NsFET wi h
di e en dimensions in (110)/[100] o ien a ion. The model is
in good ag eemen wi h k.p simula ion da a.
ma ing he elec os a ics beha io o NsFETs. Fig. 9 shows
he Qin and Cgg o NsFETs wi h H = 3 nm and a ying
W o (100)/[110] de ice o ien a ion. The model is in good
ag eemen wi h he k.p simula ion da a. The e o e, he me hod
p oposed he e can cap u e he a ia ion o e ec i e mass using
simpli ied equa ion wi h only wo i ing pa ame e s. Fig. 10
shows he Qin and Cgg o (110)/[100] de ice o ien a ion. The
good ag eemen wi h simula ion da a con i ms ha he p o-
posed model is equally accu a e o di e en subs a e/channel
o ien a ions.
V. CONCLUSION
To summa ize, a modi ied app oach was p oposed o com-
pu e sub-band ene gy le els in quan um con ined sys em. I
was shown ha he p oposed app oach can be used o di e -
en de ice dimensions and subs a e/ anspo o ien a ions wi h
only wo i ing pa ame e s. The p oposed model cap u ed he
e ec o e ec i e mass change wi h con inemen on ene gies
while ensu ing he simplici y o compac model. The modi ied
sub-bands we e used o compu e he NsFET elec os a ics
using bo om-up scalable compac model [4]. The da a om
he modi ied model ag eed e y well wi h he da a om k.p
simula ions he eby con i ming he accu acy o he model.
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