Supplementary Online Materials for
Effect of Variations in Diameter and Density on the Statistics
of Aligned Array Carbon-Nanotube Field Effect Transistors
Ahmad E. Islam†, Frank Du†, Xinning Ho, Sung Hun Jin, Simon Dunham, and John A. Rogers*
†
These authors contributed equally
[*] Prof. J. A. Rogers
Departments of Materials Science and Engineering, Chemistry, Mechanical Science and
Engineering, Electrical and Computer Engineering
Beckman Institute for Advanced Science and Technology
and Frederick Seitz Materials Research Laboratory
University of Illinois at Urbana-Champaign
Urbana, Illinois 61801 (USA)
E-mail: jrogers@illinois.edu
Dr. A. E. Islam, F. Du, Dr. X. Ho*, Dr. S. H. Jin
Department of Materials Science and Engineering
and Frederick Seitz Materials Research Laboratory
University of Illinois at Urbana-Champaign
Urbana, Illinois 61801 (USA)
*Current address: Singapore Institute of Manufacturing Technology
71 Nanyang Drive, Singapore 638075
1
120
80
40
0
50
100 150
<N>
2
0
9
40
80
ION/<N> [nA]
40
ION/<N> [nA]
Data for <N>~67
Poisson Fit
3
0
0
6
Counts
20
150
30
60
90
ION/<N> [nA]
3
0
50
100
ION/<N> [nA]
6
Data for <N>~150
Poisson Fit
(e)
0
(d)
120
6
0
2
0
250
Counts
4
Counts
200
Data for <N>~30
Poisson Fit
(c)
0
4
IONION/<N>)
0
Counts
Data for <N>~11
Poisson Fit
(b)
6
Counts
ION [nA]
(a)
60
Data for <N>~197
Poisson Fit
(f)
3
0
60
90
120
150
ION/<N> [nA]
Supplementary Figure 1: (a) Variation of average ION/<N> (µION) for different array-SWNT
FETs, having different nominal number of SWNTs (<N>). (b-f) Histogram of measured ION/<N>
distributions in array-SWNT FETs having (b) <N> = 11 SWNTs; M (i.e., number of samples) =
19, (c) <N> = 30 SWNTs; M = 20, (d) <N> = 67 SWNTs; M = 20, (e) <N> = 150 SWNTs; M =
35 and (f) <N> = 197 SWNTs; M = 17. The distributions are fitted using Poisson statistics.
2
800
(a)
(b)
200
Normalized I
400
ON
<N>
10
50
100
200
500
600
Counts
1.5
M = 5000
0
0
20
40
60
Simulation
1/<N>
1.0
0.5
0.0
0
50
ION/<N> [nA]
100
150
200
<N>
Supplementary Figure 2: Simulation results considering an ideal case (i.e., with no
variation in SWNT density and single, unique diameter distribution across the wafer) for arraySWNT FETs. Simulated (a) ION/<N> distributions and (b) normalized standard deviation
(σION/σION,<N>=1) of ION distributions for array-SWNT FETs. σION/σION,<N>=1 in Fig. b follows the
1/√<N> scaling, as per central limit theorem.
(a)
Aligned SWNTs
(c)
(b)
Drain
Source
Photoresist
Quartz
(d)
(e)
Dielectric
Gate Metal
Dielectric
Drain
(f)
Source
Quartz
Supplementary Figure 3: Steps to fabricate FETs with single SWNT: (a) Grow aligned
array SWNTs on ST-cut quartz substrate. (b) Deposit Ti(2nm)/Pd(60nm) source/drain using
photolithography and lift-off. (c) Deposit photoresist in small area (~1.5μm width) of the
channel. (d) Etch (using O2 plasma: 100 mTorr, 20 sccm, 100 RF power in Plasma-Therm
Reactive Ion Etching system) SWNTs outside the channel (defined using photoresist) and
identify FETs with single SWNT. (e) Deposit SOG(35nm)/HfO2(20nm) dielectric. (f) Deposit
Ti(2nm)/Au(60nm) top-gate using photolithography and lift-off.
3
L ~ 10m
VDS= -0.05V
30
-2
-2
-1
0
1
VG [V]
-1
(b)
0
-ID,SM [nA]
5
0
-2
10
10
10
40
(a)
-ID,SS [nA]
-ID,SS [nA]
L ~ 10m
15 VDS= -0.05V
0
1
2
20
10
0
-2
-1
VG [V]
0
1
2
VG [V]
Supplementary Figure 4: (a) Measured drain current (ID,SS) vs. gate voltage characteristics
(inset in semilog scale) of three FETs with single semiconducting SWNT (SS-SWNT FETs). (b)
Measured drain current (ID,SM) vs. gate voltage characteristics of three FETs with single metallic
SWNT (SM-SWNT FETs).
2
10
- ID,SS [nA]
0
10
-2
10
Simulation
-4
10
-1.0
-0.5
0.0
VG-VT1 [V]
d [nm]
0.60
0.72
0.86
1.02
1.47
1.75
2.10
2.51
3.00
0.5
Supplementary Figure 5: Simulated ID,SS vs. VG-VT1 characteristics (L = 10μm, VDS = 0.05V) for SS-SWNT FETs with different diameter. Here, VT1  VG@(ID=IMAX/100).
4
(a)
ON
ON
Normalized I I
σION ≡ σ(ION/<N>)
μION ≡ μ(ION/<N>)
Simulation
1/<N>
1.0
0.5
0.0
0
50
100
150
1.5
Normalized IONION
1.5
1.0
0.5
0.0
200
σION ≡ σ(ION/N)
μION ≡ μ(ION/N)
Simulation
1/<N>
(b)
0
50
<N>
100
150
200
<N>
Supplementary Figure 6: Eliminating the effect of SWNT density variation: Simulated
σION/√μION (normalized by its value for <N>=1) vs. <N>, where only the variation of SWNT
density (Fig. 2a) is considered in calculating ION, and σION and μION are defined as the standard
deviation and average for (a) ION/<N> and (b) ION/N. Counting the number of SWNTs in arraySWNT FET and then calculating σION and μION using ION/N, rather than ION/<N>, eliminates the
deviation from 1/√<N>, thereby subtracts the effect of SWNT density variation from the
performance statistics.
1.5
1.5
(b)
Normalized I
Normalized V
T
MIN
(a)
1.0
0.5
0.0
0
50
100
<N>
150
200
1.0
0.5
0.0
0
50
100
150
200
<N>
Supplementary Figure 7: Variation of normalized standard deviation (with respect to the
value at <N> ~ 11) with <N> for (a) VT (defined as VG@IMIN) and (b) IMIN (minimum of drain
current ID,ARRAY) in array-SWNT FETs.
5
-1
10
-3
(a)
10
-1.5
VDS = -0.1nm; d = 1.1nm
Symbol: Data from
Franklin et al.
Line: Simulation
-1.0
-0.5
VG-VT1 [V]
5
PDF [x10 ]
3
2
1
0 -6
10
EOT
1nm
2nm
5nm
10nm
20nm
6
L = 300nm
ION [A]
- ID,SS [A]
10
SS-SWNT FET (N=1) simulation
L = 300nm
8
L = 15nm
1
4
2
0
0.0
(b)
0
1
2
d [nm]
3
SS-SWNT FET (N=1) simulation
L = 300nm
EOT
1nm
5nm
20nm
2nm
10nm
(c)
-4
10
-2
10
ION [A]
0
10
2
10
Supplementary Figure 8: (a) Simulated ID,SS vs. VG-VT1 of short-channel SS-SWNT FETs
are calibrated with measurements of Ref. 1 to extract the device parameters α = 3m/K-s, Gc0 =
1/5kΩ, and VA = 16q/3πd. The multiple sub-band approximation for VA is obtained by
multiplying 2 with the single sub-band expression of VA (as used in Ref. 2). These device
parameters are used to simulate scaled array-SWNT FETs in section 5. (b) Simulated ION
(|ID,SS|@VG-VT1= - 1V, VDS = -0.05V) vs. diameter (d) for L = 300nm SS-SWNT FETs at
different equivalent oxide thickness (EOT). (c) Simulated ION distributions for a given diameter
distribution (Fig. 6a) narrows down at smaller EOT.
6
Supplementary Section 1: Conductance Simulation for SS-SWNT FETs
S1.1 Calculation of GSS (conductance of semiconducting SWNT)
Calculation of GDS using eqn. (2) of the main manuscript requires estimation of GSS for both
electron and hole. As we know EFi at any gate bias VG from eqn. (3) of the main manuscript, we
can calculate the electron density (QSS,e) and hole density (QSS,h) for semiconducting SWNT
using –

QSS,e = - q  dE*sign(E)*v(E)*f  sign(E)*(E-E Fi ) ,
(1)
0
0
QSS,h = - q  dE*sign(E)*v(E)*f  sign(E)*(E-E Fi ) ,
(2)
-
and hence the respective conductances GSS,e and GSS,h using 2 –
4q 2 λ e(h)
G SS,e(h) =
,
h λ e(h) +L
where
v(E)=
4 E u(E-E Fi )
πhvF E 2 -E Fi 2
SWNT, λ e(h) =τ F v F
[ QSS,e(h) /VA ]2
1+[ QSS,e(h) /VA ]2
is
the
density
(3)
of
states
of
semiconducting
is the mean free path for electron (hole), EFi = Ei – EF, Ei is
the intrinsic Fermi level of the semiconducting SWNT or the mid-gap energy level of graphene,
EF is the Fermi energy level of the semiconducting SWNT, F(E) is the Fermi distribution,
sign(E) is the sign of energy level E, u(E) is the unit step function, VA = 8q/3πd (single-sub-band
approximation) is a diameter dependent quantity, h is Planck’s constant, τF-1 = αT/d is the
scattering rate within the channel at Fermi velocity vF, T is the temperature in °K, and α is the
scattering coefficient.
7
S1.2 Calculation of GC (contact conductance of semiconducting SWNT with Pd)
Calculation of GDS at different VG using eqn. (2) of the main manuscript also requires an
estimation of contact conductance GC, which is a product of VG-independent quantity GC0 and
VG-dependent transmission probability TC for the contact, i.e.,
G C =G C0TC .
(4)
We calculate TC = Ttherm + TSB + TBTBT using the following three components: (i) thermionic
emission component: Ttherm = exp(-Ebarrier/kT); where Ebarrier is the thermal barrier for carrier
injection from contact into the channel, (ii) Schottky barrier tunneling component: TSB = exp(2∫kzdz);3 where kz is carrier’s parallel momentum (extracted from the E-k relationship) at
distance z away from the contact into the channel, (iii) band to band tunneling component: TBTBT
= exp(-2πEG2/qhvFFz),4 where EG is the bandgap, and Fz is the electric field at the location of
band to band tunneling. For large diameter and at large gate bias (VG >> VT), there is no
Schottky barrier at the contact
5, 6
, hence TC ~ 1 and GC ~ GC0; whereas for small diameter, the
presence of Schottky barrier makes TC ~ Ttherm + TSB < 1. TBTBT only exists during electron
conduction for large diameter SWNTs at large positive gate bias.
References:
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2. X. J. Zhou, J. Y. Park, S. M. Huang, J. Liu and P. L. McEuen, Physical Review Letters
95 (14), - (2005).
3. X. Yang, G. Fiori, G. Iannaccone and K. Mohanram, presented at the Great Lake
Symposium on VLSI, 2010 (unpublished).
4. A. Liao, Y. Zhao and E. Pop, Physical Review Letters 101 (25), - (2008).
5. Z. H. Chen, J. Appenzeller, J. Knoch, Y. M. Lin and P. Avouris, Nano Letters 5 (7),
1497-1502 (2005).
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6. W. Kim, A. Javey, R. Tu, J. Cao, Q. Wang and H. J. Dai, Applied Physics Letters 87
(17), - (2005).
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