Download 14a1024 Supplementary Material_Revised-2

Supplementary Material
Exfoliated multilayer MoTe2 field-effect transistors
S. Fathipour,1 N. Ma,1,a W. S. Hwang,2 V. Protasenko,1 S. Vishwanath,1
H. G. Xing,1 H. Xu,1D. Jena,1 J. Appenzeller,3 and A. Seabaugh,1,b
1
Department of Electrical Engineering, University of Notre Dame, Notre Dame, Indiana 46556, USA
2
Department of Materials Engineering, Korea Aerospace University, Seoul 110810, Korea
3
Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA
Fig.*S1(a)*
a) [email protected]
b) [email protected]
(b)
Oxide
(a)
EFG
EFS
qΨ0S
(qΨ0D)
EF
qΦSB
EV
qϕP
qΨ(x)
tS
0 (x)
LCH
x
LD
-qVD
-qVCH
EC
-qVGF
LS
EFD
-qVS
qΦCHS
EV
qΦCHD
qΦ(y)
0 (y)
L
y
Fig. S1. Energy band diagram and model parameters for a Schottky-contacted MoTe2 FET: (a) band diagram in x-direction
for LS< y < LS+LCH and (b) band diagram in y-direction at x = 0.
The MoTe2 transistor is modeled as a long channel FET by two back-to-back metal-semiconductor diodes
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separated by the FET channel. The energy band diagrams for the Schottky-contacted MoTe2 FET in two
directions are shown schematically in Fig. S1, where (a) shows the band diagram in the direction perpendicular to
the TMD surface (x), and (b) shows the band diagram in the direction pointing from source to drain (y). The
transistor channel is divided into three regions in the y-direction consisting of source contact, transistor channel,
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and drain contact. The lengths of each region are indicated LS, LCH and LD, and the voltage drops across each
region are VS, VCH and VD, respectively. Note that L = LS + LCH + LD, and VDS = VS + VD + VCH, where L is the
distance between source and drain contacts and VDS is the applied drain voltage. These three regions are
connected via the electrical potential as well as the current continuity and the output characteristics can be
obtained accordingly, as we will show in the following.
In the channel region between y = LS and y = LS + LCH, the current can be simulated with a drift-diffusion
model. Using a classical gradual-channel approximation for long-channel MOSFETs, one can separate the twodimensional (2D) problem into two coupled one-dimensional (1D) problems
1,2
. For p-channel FETs, the hole
density p in the channel is related to the electrical potential by
é y (x) - V(y) ù
p = N A exp ê ú,
VT
ë
û
(S1)
where NA is the unintentionally doped impurity concentration, Ψ(x) is the potential in the x-direction, V(y) is the
potential in the y-direction along the channel, VT is the thermal voltage kT/q, k is Boltzmann’s constant, T is the
absolute temperature, and q is the electron charge. For a flat-band bias condition along the band-diagram in the xdirection, pFB = NA. Using the gradual-channel approximation, the x-direction electric field in the channel is
assumed to be much larger than the y-direction field. Thus the potential distribution in the x-direction can be
described by the Poisson equation
æ y (x) - V ( y ) ö ù
d 2y (x)
qN é
= - A ê exp ç 2
÷ø -1ú ,
dx
eS ë è
VT
û
(S2)
where εS is the MoTe2 dielectric constant, with the value taken to be the average of the dielectric constants of the
semiconductor channel and the surrounding dielectric environments 3. In the ON-state, the condition of p >> NA
can be achieved, thus the exponential term in Eq. (S2) dominates the Poisson equation
é y (x) - V ( y ) ù
d 2y (x)
qN
= - A exp ê ú.
2
dx
eS
V
ë
û
T
2
(S3)
Eq. (S3) restricts the use of the model to regions of strong accumulation where the accumulation layer thickness is
less than the layer thickness. Multiplying both sides of Eq. (S3) by 2dΨ and integrating from x = 0 to x = ts, where
ts is the thickness of the semiconductor channel, one can thus obtain the surface potential at the source side of the
channel, Ψ0S, and the drain side of the channel region, Ψ0D after Ref. [1],
é 2qN AVT e S
æ V öù
exp ç - GF ÷ ú
è VT ø û
ë 2VT COX
y 0S = VGF + 2VT W ê
y 0D
é 2qN AVT e S
æ V - VCH ö ù
= VGF + 2VT W ê
exp ç - GF
÷ø ú
VT
è
ë 2VT COX
û
,
(S4)
where W is the Lambert function 4,5, which is the solution to W(x) exp[W(x)] = x, COX is the gate capacitance per
unit area. VGF is the shifted Gate voltage given by VGF = VBG − VGFB, where VGFB is the gate flat-band voltage
determined by the gate oxide charges and the work-function difference between gate metal and the channel
semiconductor 6, and is used as a fitting parameter in the current model. To obtain Eq. (S4), we have assumed
that the electric field and the electric potential at the back surface (x = ts) are much smaller compared with the gate
surface (x = 0) 1,5, which is reasonable since most of the carriers are confined very close to the gate surface 6. It is
worth noting that the requirement of p >> NA limits the validity of the current model. If we set the lower limit as
p0 D = 10N A , where p0 D is the hole density at the drain side of the channel region, which is given by
é y -V ù
p0 D = N A exp ê - 0 D CH ú ,
VT
ë
û
(S5)
the upper limit of the shifted gate voltage VGF,MAX can be obtained by combining Eq. (S5) and (S4). Figure S2
shows the VGF,MAX as a function of VCH that satisfy p0 D > 10NA at three different temperatures T= 100 K, 200 K
and 300 K.
3
Fig.*S2
!
!4
300 K
!
V G F ,MA X *(V )
!3
!5
200 K
100 K
!6
NA = 1017 cm-3
!2.0
!1.5
!1.0
!0.5
0.0
V C H *(V )
Fig. S2. Upper limit of VGF as a function of VCH at different temperatures for the Schottky-contacted MoTe2 FET model.
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The drift-diffusion current in the channel is given by1,2
I CH
2
é
ù
w
y 02D - y 0S
=m PCOX ê(VGF + 2VT ) (y 0 D - y 0S ) ú
LCH
2
ë
û
é
æy -V ö
æy ö ù
w
mPVT qN At S ê exp ç 0 D CH ÷ - exp ç 0S ÷ ú
LCH
VT
è
ø
è VT ø û
ë
,
(S6)
where μp is the hole mobility and w is the width of the channel. In Eq. (S6) μp, NA, LCH and VCH are unknowns so
far. As shown in the following, LCH is obtained by solving Poisson’s equation in the source and drain depletion
regions and VCH is then obtained from the current continuity of source and channel. μp and NA are fitting
parameters. In the source and drain region of channel, one should consider the fact that the Schottky barrier
results in LD and LS to be depleted. The potential distribution in above two regions ΦS(x,y) and ΦD(x,y) can only be
described by 2D Poisson equation since they are strongly coupled to both the gate and drain biases,
d 2 F S( D) (x, y) d 2 F S( D) (x, y) qN A
+
=
.
dx 2
dy 2
eS
4
(S7)
Following Young’s method, the potential in the vertical (x) direction is assumed to be a parabolic function of x,
and the 2D Poisson equation can be simplified to a 1D problem 7-9
d 2 F(y) F(y) - VGF qN A
,
=
dy 2
l2
eS
(S8)
where Φ(y)=ΦS(D)(0, y) is the potential distribution along the channel-oxide interface, λ is the screening length,
l = tOX tSe S / e OX , where tox and εox are the thickness and dielectric constant of the gate oxide, respectively.
The boundary conditions considered to solve Eq. (S8) are the following: (1) the source Fermi level is defined as
the reference potential, thus Φ(0)=ΦSB, and Φ(L)=ΦSB+VDS, (2) the potential is continuous at the source/channel
and channel/drain junction, thus ΦCHS=Φ(LS)=ϕP–VS+Ψ0S and ΦCHD=Φ(L−LD)=ϕP–VS+Ψ0D, where ϕP=(EF–EV)/q
and EF is the Fermi energy. Using these boundary conditions, the solution to Eq. (S7) for the surface potential in
the source region, ΦS (0, y), and drain, ΦD (0, y), is
æ L - yö
æ yö
sinh ç S
sinh ç ÷
2
÷
é
è l ø é
è lø
qN A l ù
qN A l ù
qN A l 2
F S (0, y) = ê F SB - VGF +
+
F
V
+
+
V
GF
ê CHS GF
e S úû
e S úû
eS
æL ö
æL ö
ë
ë
sinh ç S ÷
sinh ç S ÷
è lø
è lø
2
æ y - L + LD ö
æ yö
sinh ç
sinh ç ÷
2
÷
é
è
ø é
è lø
qN A l ù
qN A l ù
qN A l 2
l
F D (0, y) = ê F SB + VDS - VGF +
+
F
V
+
+
V
GF
ê CHD GF
e S úû
e S úû
eS
æL ö
æL ö
ë
ë
sinh ç D ÷
sinh ç D ÷
è l ø
è l ø
2
(S9)
To find the length of the source and drain region, i. e., LS and LD, we assume that the electric field at the
source/channel and drain/channel interface is much smaller than that in the source and drain region, which yields
f - V + qN A l 2 / e S
2
LS = l ln éGS (VS ) + GS (VS ) -1 ù , where GS (VS ) = SB GF
úû
ëê
y CHS - VGF + qN A l 2 / e S
f + V - V + qN A l 2 / e S
2
LD = l ln éGD (VS ,VCH ) + GD (VS ,VCH ) - 1 ù , where GD (VS ,VCH ) = SB DS GF
êë
úû
y CHD - VGF + qN A l 2 / e S
.
(S10)
The values GS and GD are very close to unity, therefore, LS and LD are much smaller than the screening length, λ.
For example, when VS = VD = 1 V, NA = 1017 cm-3, VG = -60 V, we have LS ~ 14 nm and LD ~ 5 nm. For long-
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channel devices, the channel region length LCH thus can be approximated as LCH ≈ L which in our case is
approximately 1 μm. It is worth noting that when a negative drain bias VDS is applied, the source Schottky
junction is reversed biased and the drain Schottky junction is forward biased, we assume VD << VS+VCH and thus
the voltage drop on the drain region can be ignored. The current through the reverse-biased source Schottky
junction IS has three components: the thermionic emission current (JTE) over the Schottky barrier, the field
emission current (JFE) near the Fermi level, and the thermionic-field emission current (JTFE) which is composed of
thermally excited carriers tunneling through a thinner barrier than the field emission carriers. These current
components are given by 10,11
æ f öé
æV öù
JTE = A*T 2 exp ç - SB ÷ ê1- exp ç S ÷ ú
è VT ø ë
è VT ø û
J FE
2
æ
2qfSB 3 2 ö
æ E00 ö æ fSB - VS ö
=A ç
exp
ç 3E f - V ÷
è k ÷ø çè fSB ÷ø
è
00
SB
S ø
*
JTFE = A*
(S11)
é
ù
æ qfSB ö
T
fSB
æ qV ö
p E00 q ê -VS +
exp ç - S ÷
ú exp ç 2
÷
è e' ø
k
cosh ( E00 kT ) û
è E0 ø
ë
where A* = 4p qm*k 2 h 3 , E00 =
qh
4p
NA
éE
æE ö
æ E öù
, E0 = E00 coth ç 00 ÷ , and e ' = E00 ê 00 - tanh ç 00 ÷ ú .
*
è kT ø û
è kT ø
m es
ë kT
Now the total source current is given by
I S = wt S ( JTE + J FE + JTFE ) .
(S12)
By equating Eq. (S12) and Eq. (S6) to preserve the current continuity, and combining with the assumption that
VDS ≈ VS+VCH, the unknowns VCH and VS can be defined and we can obtain the device characteristics, as shown by
black lines in Fig. 3(d). The gate flat band voltage and the doping in the channel are fitting parameters in the
calculation with values of VGFB = 0.5 V and NA = 1017 cm-3 with ΦSB = 0.12 V and μP = 6.4 cm2/Vs. As can be seen
from Fig. 3(d), a good fit to the experimental data is obtained.
To find out the importance of the Schottky barrier at the source region, we calculate the proportion voltage
drops in the source region in VDS as a function of VGF, as shown in Fig. S3. As expected, the Schottky barrier
plays a more important role at higher gate voltage and lower temperature. The calculated room temperature
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transfer characteristics and the temperature-dependent current density at fixed gate and drain voltage: VBG = -60V
and VDS = -1V in comparison to the experimental results are shown in Fig. S4. The discrepancy between the
calculated and measured temperature-dependent current density is probably caused by the assumption that both
carrier mobility and impurity density in the channel are constant while temperature is changing. Precise fitting of
Fig.*S3
the temperature-dependence of the device characteristics need further transport modeling of the channel material
as well as with other temperature-dependent parameters taken into consideration.
!
VDS=-1V
ΦSB= 0.12 V
NA = 1017 cm-3
µp= 6.4 cm2/Vs
200 K
!
VSS //V
VDS
D (V)
175 K
125 K
150 K
225 K
250 K
275 K
300 K
Fig. S3. Proportion of voltage drop in source region VS in VDS as a function of VGF at different temperature.
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Fig. S4. (a) Room temperature transfer characteristic for the 30-layer Schottky-contacted MoTe2 FET. (b) Temperature-
dependent current density at VBG = -60V and VDS = -1V. Open square symbols indicate the experimental
measured results, and solid black lines show the calculated results based on the model described above.
ACKNOWLEDGMENTS
This work was supported in part by the Office of Naval Research (ONR), the National Science Foundation (NSF), and
the Center for Low Energy Systems Technology (LEAST), a STARnet Semiconductor Research Corporation program
sponsored by MARCO and DARPA.
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