SUPPLEMENTAL MATERIAL FOR:
The role of insulator charging in transport across ultra-thin metalinsulator-semiconductor junctions
By Ayelet Vilan.
Department of Materials and Interfaces, Weizmann Institute of Science, POB 26, Rehovot,
76100, Israel.
I.
Details of numerical simulation
The numerical simulation follows closely the method proposed by Tarr et al. (Ref. 11 of
main text). This procedure defines two unknown variables:
 𝐵𝐵, the band-bending in the semiconductor, and
 𝑄𝐹, the position of the quasi-Fermi level of minority carriers with respect to the bulk
Fermi level (see Figure 1 of main text).
At each applied bias, these parameters are computed by solving two equations:
 Charge neutrality (i.e., Equation 2 of main text) and
 Continuity of current by minority carriers.
The second balance means that tunneling of minority carriers across the insulator equals
the supply of minority carriers by the semiconductor by mechanism of diffusion and generationrecombination. After the 𝐵𝐵 and 𝑄𝐹 are known the net-current is computed as the sum of
electrons and holes currents, using Equation 1 of main text. The net bias is the sum of change in
𝐵𝐵 and Ψ (Equation 4 of main text) rather than an independent parameter. This yields a varying
spacing in 𝑉, which can be latter smoothed by spline interpolation. Setting 𝑉 as an output of the
computation rather than an input parameter, fasten the computation, because the more influential
parameter in each regime is fixed. Thus for depletion and inversion regimes, the 𝐵𝐵 is taken as
an independent fixed parameter while Ψ and 𝑄𝐹 are computed by iteratively nulling the charge
balance minority current balance. For accumulation regime, the order is inversed, and Ψ is taken
as the arbitrary input parameter and 𝐵𝐵 and 𝑄𝐹 are computed numerically. Initially, the
1
transition between the two sub-regimes is found by computing the flat-band potential form the
interface energy alignment.
The following Table SI list the parameters used in the simulations.
TABLE SI. List of materials constants used in the simulation
Insulator
Silicon
parameters
Property
Value
Units
Permittivity
3.5
ε0
Effective massa)
0.4
me
Barrier Height electronsb)
2.4
eV
Barrier Height holesb)
3.6
eV
Area
1
cm2
Type
n
Doping level of Si
Energy of dopants from
gap edge
temperature
1.00E+15
cm-3
0.05
eV
298
K
Forbidden energy gap
1.12
eV, @RT
Nc
2.80E+19
cm-3 , @RT
Nv
2.65E+19
cm-3 , @RT
Permittivity
11.7
ε0
Richardson
120.1735
A∙cm-2K-2
effective mass holes
0.65
me
effective mass electrons
0.37
me
electrons' lifetime
10
seconds
electrons' mobility
1450
cm2∙V-1sec-1
holes' lifetime
10
seconds
hole' mobility
500
cm2∙V-1sec-1
a)
This empirical factor is added to adjust the computed tunneling decay length (β) to
experimentally known values (β∼1Å-1); b) Based on Ref. 16.
2
II.
Derivation of Cheung-Werner method for a general potential drop in series
to a diode
Here I briefly reproduce the mathematical derivation of Eq. 10 of main text following Ref. 27.
Starting from a generic diode equation describing the current density, 𝐽, as a function of the
applied bias, 𝑉, and including a potential drop, 𝑈𝑆 , in series to the diode:
V−U
J = J0 ∙ [exp ( nkTS) − 1]
(S1
Notice that the parameters 𝐽0 and 𝑛 could get different interpretations depending on the details of
the diode. Differentiating Eq. S1 with respect to bias yields:
dJ
V−US
dV
= J0 ∙ exp (
nkT
)∙
1−US ′
nkT
=
J+J0
nkT
(1 −
dUS
dJ
∙
dJ
dV
)
(S2
The second equality uses Eq. S1 and further differentiates 𝑈𝑆 with respect to current instead of
voltage. Collecting arguments and neglecting 𝐽0 ≪ 𝐽, gives:
dJ
1 = dV ∙ [
nkT
J
+
dUS
dJ
]
(S3
𝑑𝑉
𝑑𝑉
Finally multiplying both sides by the inversed derivative, 𝐽 ∙ 𝑑𝐽 ≡ 𝑑 ln 𝐽, yields:
dV
= nkT + J ∙
d ln J
dUS
(S4
dJ
Eq. S4 is identical to Eq. 10 of main text, and it is strait forward to show that expressing it in
terms of current (𝐼, as in Eq. 10) or current density (𝐽, as in Eqs. S1-S4) does not matter as long
as all terms are consistent.
III.
Computing the insulator charging coefficient
Eq. 8b of main text:
d
2ε kNC
R ins = ε D √ πAsc∗ε
i
0T
∙ eβd⁄2
(8b, main text
Plunging physical constants:
d[Å]
R ins = ε D[μm] 10−4 [
i
μm
Å
2∙1.38∙10−23 [J⁄K]
εsc NC [cm−3 ] βd⁄2
e
T[K]
] √π∙110[A⁄K2 cm2 ]∙8.85∙10−14 [F/cm] ∙ √
(S5
The following term is for any semiconductor, where the insulator thickness, 𝑑 is in Å, the contact
diameter, 𝐷 is in μm, the dielectric constants, 𝜀𝑖 , 𝜀𝑠𝑐 are dimensionless and the density of states in
the conduction band, 𝑁𝐶 is in cm-3:
d
εsc NC
i
T
R ins = 0.95 ∙ 10−10 ∙ ε D eβd⁄2 ∙ √
∙ [√J⁄AF]
(S6
3
Finally, the following silicon-specific parameters (see Ref. 25 and §) are substituted into
Equation S6:
𝑁𝐶,𝑆𝑖 = 6.2 · 1015 · T1.5 [𝑐𝑚−3 ]
𝜀𝑠𝑖 = 11.7;
or:
𝑁𝐶,𝑆𝑖,𝑅𝑇 = 2.8 ∙ 1019 [𝑐𝑚−3 ]
d
R ins |Si = 0.0256 ∙ T 0.25 ∙ ε D ∙ eβd⁄2
(S7a
i
and at room temperature:
d
R ins |Si = 10∙ε D ∙ eβd⁄2
(S7b
i
Using 𝑅𝑖𝑛𝑠 = 𝑝0 ⁄𝐷, 𝑝0 is extracted by fitting 𝑅𝑖𝑛𝑠 to 1⁄𝐷 , and can be further used to extract the
tunneling decay coefficient, 𝛽:
d
p0 = 10∙ε ∙ eβd⁄2
(S8)
i
Which explain the factor ‘10’ in Eq. 13b.
IV.
Additional data for Me-Sty and C16
Original diode curves for Pb/C16-Si(111)
Current Density [Acm-2]
A.
1
0.1
0.01
3 m
10 m
30 m
100 m
1E-3
1E-4
1E-5
1E-6
-0.5
0.0
0.5
1.0
Bias to Pb [V]
§
http://www.ioffe.rssi.ru/SVA/NSM/Semicond/Si
4
FIGURE S1: Diode curves for of the C16 alkyl-chain tMIS, showing the current density against
applied bias on a semi-log plot. The data is generally highly reproducible (cf. Fig. 3 of main text
for Me-Sty), except for the smallest diameter on reverse bias where perimeter shunt increases the
current. A slight deviation from area scaling starts appearing toward 1V but it is much weaker
than for Me-Sty. The figure shows averaged curves, using the same data set used to compute
Figure 5b of main text.
FIGURE S2: The ideality
factor extracted from the
constant
parameter
(“intercept”) of the parabolic
fit (Eq. 12, main text) of
inversed derivative against
square root of the current
(Fig. 5 of main text), for
tMIS with a) Me-Sty (data of
Fig. 5a) and b) C16 (data of
Fig. 5b). The diameter (Xaxis) is shown on a log-scale
for clarity. As can be seen,
except
for
the
two
exceptionally large diameters
of Me-Sty, the extracted
ideality factor values are
independent of the diameter
as predicted by the model.
This is in a marked
difference to the reciprocal
dependence on diameter of
the linear coefficient of the
fit (𝑅𝑖𝑛𝑠 ), shown in Fig. 5c,d
of main text.
Ideality Factor, n
Extracted Ideality factors
1.5
1.0
Me-Styrene
0.5
a
1
10
100
1,000 10,000
Contact Diameter [m]
Ideality Factor, n
B.
2.5
2.0
1.5
1.0
C16 alkyl
b
3
10
30
100
Contact Diameter [m]
5
Extracted parabolic coefficients for Me-Sty
FIGURE S3: An apparent ‘series resistance’
factor extracted from fitting the inversed derivative
of Me-Sty to √𝐼 . The figure plots of the
coefficient multiplying 𝑥 2 = 𝐼 (see Eq. 12 of main
text) against the contact diameter, on a log-log
scale. Dots are fitting-extracted coefficients and
line is a fit to a reciprocal relation: 𝑅𝑆 =
23.6𝑘Ω⁄𝐷.
Square Parameter, RS []
C.
1000
100
10
1
10
100
1,000
Contact Diameter [m]
D.
Inversed derivative plots for C16 as a function of temperature
Inverse Derivative [V]
FIGURE S4: Inversed derivative
plots for Pb/C16-Si(111) tMIS with a
0.25
diameter of 100 μm for varying
300K
273K
temperatures. Symbols are raw data
0.20
250 K
of typical curves (not averaged) and
230 K
lines are linear fits. The parameters
200 K
0.15
extracted from these fits are shown in
Fig. 6 of main text. The inset is a
0.10
blown-up of the near 0 region,
showing that the variation in the
0.05
0.05
intercept (reported in Fig. 6a of main
text) are directly observable and not a
0.0m
0.2m
0.4m
0.6m
0.8m
1.0m
0.00
fitting artefact. As the temperature
0
1m 2m 3m 4m 5m 6m
reduces, the behavior becomes less
sqrt(Current) [A0.5]
linear. However, increasing the order
of the fit to a parabola, as done for
Me-Sty (Fig. 5a) is physically unrealistic because it would yield a negative ‘series resistance’,
because the curves bend downward. This is another indication that the explanation of the
inversed derivative (Eq. 12) should be further refined to reveal a more accurate explanation for
deviation from linearity.
6
FIGURE
S5:
Inversed
derivative analysis for tMIS
made of Hg/Cn-Si(111) showing
a) averaged current-voltage
curves; b) inversed derivative
against square-root of the
current; and c) Polynomial
coefficients plotted against the
molecular length. Data in (a) is
taken from Ref. 33 and was
numerically differentiated to
yield the symbols shown in (b).
Lines in (b) are fits to a thirdorder polynomial:
𝑑𝑉
𝑑 ln 𝐽
= 𝑝0 + 𝑝1 𝑥 + 𝑝2 𝑥 2 + 𝑝3 𝑥 3 ,
Current Density [Acm-2]
INVERSED DERIVATIVE PLOTS FOR VARYING LENGTH ALKYL CHAINS
Inversed Derivative [V]
V.
1
0.1
0.01
1E-3
1E-4
1E-5
1E-6
1E-7
1E-8
0.0
C12
C14
C16
C18
a
0.2
0.4
0.6
0.8
1.0
Bias to Hg [V]
0.4
0.3
0.2
C12
C14
C16
C18
Polynomial Coefficients
where 𝑝0 , 𝑝1 , 𝑝2 & 𝑝3 are the
0.1
constant
(intercept),
linear,
square and cubic parameters,
0.0
0.2
0.4
0.6
0.5
respectively. These are the
sqrt(current) [A ]
coefficients shown in (c) except
1000
for 𝑝0 because it didn’t vary
Linear
100
much between the different
Square
Cubic
curves, as expected (𝑝0 = 𝑛𝑘𝑇)
10
with extracted 𝑝0 values yielding
1
𝑛 = 1.4 𝑡𝑜 2.2. The log-scaled
0.1
Y-axis in (c) reveals that all
parabolic
coefficients
were
0.01
length-dependent and especially
12 13 14 15 16 17 18
the highest one, 𝑝3 has a clear
Number of Carbons
exponential dependence. This is
a clear indication that these
higher coefficients originate in the insulator (of varying thickness) and not due to an external
resistance of the system (thickness independent).
b
c
7