Simultaneous Extraction of Charge Density Dependent Mobility and Variable
Contact Resistance from Thin Film Transistors
Riccardo Di Pietro1*, Deepak Venkateshvaran2, Andreas Klug3, Emil J.W. ListKratochvil3,4, Antonio Facchetti5, Henning Sirringhaus2 and Dieter Neher1
Supporting information
1
Universität Potsdam, Institut für Physik und Astronomie, Karl-Liebknecht-Str. 24-25, 14476 Potsdam,
Germany
2
Cambridge University, Cavendish Laboratory, J. J. Thomson Avenue, CB3 0HE Cambridge, United
Kingdom
3
NanoTecCenter Weiz Forschungsgesellschaft mbH, Franz-Pichler-Straße 32, A-8160 Weiz, Austria
4
Institute of Solid State Physics, Graz University of Technology, Petersgasse 32, A-8010 Graz, Austria
5
Polyera Corporation, 8045 Lamon Ave, STE 140, Skokie, IL 60077-5318, USA
* dipietro@uni-potsdam.de
1 - Full derivation of the saturation mobility equation
In the following derivation the onset voltage for charge accumulation in the channel is assumed to
be zero. The results can be easily extended by substituting 𝑉𝐺 with 𝑉̂𝐺 = 𝑉𝐺 − 𝑉𝑜𝑛 , and taking into
account that all derivatives with respect to the gate voltage are not affected by the constant value of
𝑉𝑜𝑛 (i.e.
̂𝐺 )
𝑑𝑓(𝑉
̂
𝑑𝑉𝐺
=
̂𝐺 )
𝑑𝑓(𝑉
).
𝑑𝑉𝐺
The equations can be used both for p-type (𝑉𝐷 , 𝑉𝐺 − 𝑉𝑜𝑛 < 0) and n-type
(𝑉𝐷 , 𝑉𝐺 − 𝑉𝑜𝑛 > 0) transistors.
The charge sheet model together with the gradual channel approximation allows the charge density
per unit area q at position x along the channel to be defined as:
𝑞(𝑉𝐺 − 𝑉(𝑥)) = 𝐶𝑖 ∙ (𝑉𝐺 − 𝑉(𝑥))
(1)
where |𝑉𝐺 − 𝑉𝑜𝑛 | ≥ |𝑉𝐷 |. In order to consider the effects of gate voltage dependent mobility, we
define an effective mobility (gate voltage dependent) as the average of the charge density
dependent mobility u(V) over the accumulated charge density at position x along the channel:
𝑞
𝜇𝑒𝑓𝑓 (𝑉𝐺 − 𝑉(𝑥)) =
∫0 𝜇(𝜌)𝑑𝜌
𝑞
=
𝑉 −𝑉(𝑥)
𝜇(𝑣)𝑑𝑣
∫0 𝐺
(2)
𝑉𝐺 −𝑉(𝑥)
We note that the function is defined for the whole range where equation (1) is defined. Using
equation (1) the charge accumulated in a strip parallel to the source drain contacts of width dx can
be written as:
𝑑𝑄 = 𝑊𝐶𝑖 ∙ (𝑉𝐺 − 𝑉(𝑥))𝑑𝑥
(3)
Combined with the effective mobility (the average mobility at which charges move) it can be used to
write the drift current equation:
𝐼=
𝑑𝑄
𝑑𝑡
= 𝑊𝐶𝑖 ∙ (𝑉𝐺 − 𝑉(𝑥)) ∙ 𝜇𝑒𝑓𝑓 (𝑉𝐺 − 𝑉(𝑥))
𝑑𝑉(𝑥)
𝑑𝑥
𝑉 −𝑉(𝑥)
= 𝑊𝐶𝑖 (∫0 𝐺
𝜇(𝑣)𝑑𝑣 )
𝑑𝑉(𝑥)
𝑑𝑥
(4)
By separating the variables in equation (4) it is possible to write the general equation for charge
transport in field-effect transistors:
𝐼=
𝑉 ∗
𝑉 −𝑉(𝑥)
𝑊
𝐶 ∫ 𝐷 (∫0 𝐺
𝜇(𝑣)𝑑𝑣 ) 𝑑𝑉(𝑥)
𝐿 𝑖 𝑉𝑆∗
(5)
In the saturation regime it is possible to assume 𝑉𝐷∗ = 𝑉𝐺 , therefore equation (5) can be written as:
𝐼=
𝑉
𝑊
𝐶 ∫ 𝐺
𝐿 𝑖 𝑉𝑆∗
𝑉 −𝑉(𝑥)
(∫0 𝐺
𝜇(𝑣)𝑑𝑣 ) 𝑑𝑉(𝑥)
(6)
In order to extract the mobility from equation (6) we consider the following function:
𝑉 −𝑉(𝑥)
𝜎(𝑉𝐺 − 𝑉(𝑥)) = ∫0 𝐺
𝜇(𝑣)𝑑𝑣
(7)
And expand it around 𝑉𝐺 in a Taylor series:
𝜎(𝑉𝐺 − 𝑉(𝑥)) = ∑∞
𝑖=0
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉(𝑥))𝑖
𝑖!
(8)
2
From which it is possible to write the generalized integral of 𝜎(𝑉𝐺 − 𝑉(𝑥)) as:
∫ 𝜎(𝑉𝐺 − 𝑉(𝑥))𝑑𝑉(𝑥) = − ∑∞
𝑖=0
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉(𝑥))𝑖+1
(𝑖+1)!
(9)
The simplest way to obtain 𝜇(𝑉) is to perform a second derivative of the saturation current:
𝑑2 𝐼
𝑑𝑉𝐺 2
≅
𝑊
𝐶 𝜇(𝑉)
𝐿 𝑖
(10)
where the terms containing 𝑉𝑆 ∗ have been neglected. However, performing a second derivative on a
transfer curve introduces a high noise in the result, thus requiring longer integration times and/or
more data points to obtain a reasonable signal to noise ratio.
In order to perform a single derivation also in the saturation regime, it is sufficient to calculate the
following derivative:
2
𝑑 √𝐼
𝑑𝐼 2
)
√𝐼 𝑑𝑉𝐺
1
(𝑑𝑉 ) = (2
𝐺
1
2
𝑑𝐼
= 4|𝐼| (𝑑𝑉 )
(11)
𝐺
Replacing the integral in equation (6) with equation (8), the first derivative can be expressed as:
𝑑𝐼
𝑑𝑉𝐺
𝑑
𝑊
= 𝑑𝑉 { 𝐿 𝐶𝑖 [− ∑∞
𝑖=0
𝐺
𝑉𝐺
𝜎 (𝑖) (𝑉𝐺 )
𝑖+1
(−𝑉(𝑥))
]
}
(𝑖+1)!
𝑉𝑆∗
∑∞
𝑖=0
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉𝑆 ∗ )𝑖+1 ]
(𝑖+1)!
∑∞
𝑖=0
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉𝐺 )𝑖
(𝑖)!
+ ∑∞
𝑖=1
∑∞
𝑖=1
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉𝑆 ∗ )𝑖
(𝑖)!
− ∑∞
𝑖=0
𝑊
𝐶
𝐿 𝑖
=
[− ∑∞
𝑖=1
=
𝑊
𝑑
𝜎 (𝑖) (𝑉𝐺 )
∑∞
(−𝑉𝐺 )𝑖+1
𝐶
[−
𝑖
𝑖=0
(𝑖+1)!
𝐿
𝑑𝑉𝐺
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉𝐺 )𝑖
(𝑖)!
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉𝑆 ∗ )𝑖
(𝑖)!
− ∑∞
𝑖=0
+
+
𝑑𝑉 ∗
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉𝑆 ∗ )𝑖 𝑆 ]
(𝑖)!
𝑑𝑉𝐺
=
𝑊
𝐶
𝐿 𝑖
𝑑𝑉 ∗
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉𝑆 ∗ )𝑖 𝑆 ]
(𝑖)!
𝑑𝑉𝐺
[𝜎(𝑉𝐺 ) +
(12)
While the current can be written as:
𝐼=
𝑉
𝑊
𝐶 ∫ 𝐺
𝐿 𝑖 𝑉𝑆∗
𝜎(𝑉𝐺 − 𝑉(𝑥))𝑑𝑉(𝑥) =
𝑊
𝐶
𝐿 𝑖
[− ∑∞
𝑖=0
𝜎 (𝑖) (𝑉𝐺 )
((−𝑉𝐺 )𝑖+1
(𝑖+1)!
𝑊
𝐶
𝐿 𝑖
[− ∑∞
𝑖=0
𝑉𝐺
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉(𝑥))𝑖+1 ]
(𝑖+1)!
𝑉𝑆∗
=
− (−𝑉𝑆 ∗ )𝑖+1 )]
(13)
Combining equations (11), (12) and (13), the following equation is obtained:
2
𝑑 √𝐼
)
𝑑𝑉𝐺
(
=
𝑊
𝐶
4𝐿 𝑖
2
(𝑖)
𝑖
𝑖 𝑑𝑉𝑆∗
𝜎(𝑖) (𝑉𝐺 )
∞ 𝜎 (𝑉𝐺 )
∗
−∑
∗
∙
(−𝑉
)
(−𝑉
)
]
𝑖=0 (𝑖)!
𝑆
𝑆
(𝑖)!
𝑑𝑉𝐺
[𝜎(𝑉𝐺 )+∑∞
𝑖=1
𝜎(𝑖) (𝑉 )
𝐺
𝑖+1 −(−𝑉 ∗ )𝑖+1 )|
|− ∑∞
𝑆
𝑖=0 (𝑖+1)! ((−𝑉𝐺 )
(14)
In order to write equation (13) in a more useful form, it is necessary to perform some simplifications.
First it is assumed that
𝑑𝑉𝑆∗
𝑑𝑉𝐺
≪ 1 so that we can remove it from the numerator of the fraction.
Furthermore, keeping in mind that the conductivity of the depleted channel is negligible, i.e. 𝜎(0) =
0, it is possible to write:
𝜎(0) = ∑∞
𝑖=0
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉𝐺 )𝑖
𝑖!
⇒ 𝜎(𝑉𝐺 ) = − ∑∞
𝑖=1
𝜎 (𝑖) (𝑉𝐺 )
(−𝑉𝐺 )𝑖
𝑖!
(15)
3
Then equation (14) can be written as:
𝑑 √𝐼
2
(𝑑𝑉 )
𝑊
𝐶
4𝐿 𝑖
=
𝐺
𝑊
4𝐿
=
2
𝜎(𝑖) (𝑉 )
𝐺
𝑖+1 −(−𝑉 ∗ )𝑖+1 )|
|− ∑∞
𝑆
𝑖=0 (𝑖+1)! ((−𝑉𝐺 )
𝑖
𝜎(𝑖) (𝑉𝐺 )
𝜎(𝑖) (𝑉𝐺 )
(−𝑉𝐺 )𝑖 +∑∞
(−𝑉𝑆∗ ) ]
𝑖=1
𝑖!
𝑖!
2
[− ∑∞
𝑖=1
𝑊
4𝐿
=
𝑖
𝜎(𝑖) (𝑉𝐺 )
𝜎(𝑖) (𝑉𝐺 )
(−𝑉𝐺 )𝑖 +∑∞
(−𝑉𝑆∗ ) ]
𝑖=1
𝑖!
𝑖!
[− ∑∞
𝑖=1
(16)
𝜎(𝑖) (𝑉 )
𝐺
𝑖+1 −(−𝑉 ∗ )𝑖+1 )|
|𝜎(𝑉𝐺 )𝑉𝐺𝑆∗ −∑∞
𝑆
𝑖=1 (𝑖+1)! ((−𝑉𝐺 )
𝑖
𝜎(𝑖) (𝑉𝐺 )
𝜎(𝑖) (𝑉𝐺 )
(−𝑉𝐺 )𝑖 +∑∞
(−𝑉𝑆∗ ) ]
𝑖=1
𝑖!
𝑖!
2
[− ∑∞
𝑖=1
|− ∑∞
𝑖=1
𝜎(𝑖) (𝑉𝐺 )
𝜎(𝑖) (𝑉𝐺 )
𝑖+1 −(−𝑉 ∗ )𝑖+1 )|
(−𝑉𝐺 )𝑖 𝑉𝐺𝑆∗ −∑∞
𝑆
𝑖=1 (𝑖+1)! ((−𝑉𝐺 )
𝑖!
By separating the contribution of the first derivative and including the rest of the terms of the Taylor
series expansion in a parameter 𝑘(𝑉𝐺 ), it is possible to rewrite equation (16) as:
2
𝑑 √𝐼
)
𝑑𝑉𝐺
(
𝜎 ′ (𝑉𝐺 )2 𝑉𝐺𝑆∗ 2
𝑊
∙
′
4𝐿 |𝜎 (𝑉𝐺 )𝑉𝐺𝑆∗ 𝑉𝐺 −1𝜎 ′ (𝑉𝐺 )((𝑉𝐺 )2 −(𝑉𝑆∗ )2 )|
=
𝑘(𝑉𝐺 ) =
2
𝑊 𝜎 ′ (𝑉𝐺 )
4𝐿 |1(𝑉𝐺 −𝑉𝑆∗ )|
=
2
𝑉𝐺𝑆∗
𝑊
𝜎 ′ (𝑉𝐺 )
𝑉
4𝐿 | 𝐺 −1(𝑉𝐺 +𝑉𝑆∗ )|
𝑉𝐺𝑆∗ 2
𝑊
𝑉𝐺𝑆∗
∙ 𝑘(𝑉𝐺 )
(17)
∙ 𝑘(𝑉𝐺 ) = 2𝐿 𝜎′(𝑉𝐺 ) ∙ 𝑘(𝑉𝐺 )
where 𝑘(𝑉𝐺 ) depends on the particular kind of charge density dependence of the mobility. In the
simpler case of a power law dependence of the mobility:
𝜇(𝑣) = 𝛼(𝑣)𝛽
(18)
it can be easily seen how equation (17) is reduced to:
2
𝑑 √𝐼
)
𝑑𝑉𝐺
(
=
𝑊
𝐶 𝜇(𝑉𝐺 ) ∙
2𝐿 𝑖
𝑘,
𝑘=
𝛽+2
2(𝛽+1)
(19)
and the value of 𝑘 becomes gate voltage independent (for the full derivation see below). Most
importantly, it can be seen that:
0 ≤ 𝛽 ≤ +∞ ⇒
−1 < 𝛽 < 0
1 ≥ 𝑘 ≥ 0.5
(20)
⇒ +∞ > 𝑘 > 1
In the physically relevant case of a mobility increasing with charge density, the correction factor
𝑘(𝑉𝐺 ) will therefore be limited between 1 and 0.5 although it will not be constant throughout the
𝑑 √𝐼
2
curve. If it is possible to fit the function (𝑑𝑉 ) with the power law dependence in the range used for
𝐺
the measurement, the extracted value of 𝛽 can then be used to approximate the value of k to a
constant value, and the mobility in the saturation regime can be obtained as:
2𝐿
𝑑 √𝐼
𝑖
𝐺
2
𝜇(𝑉𝐺 ) = 𝑊𝐶 ∙𝑘 (𝑑𝑉 )
(21)
without the need to perform a second derivative of the transfer characteristics. This approximation
holds also when the approximation with a power law dependence is not perfect due to the relatively
little dependence of the value of k with respect to the value of 𝛽. The value of 𝐶(𝑉𝐺 ) can be easily
4
obtained either by calculating the geometric capacitance of the dielectric layer or by measuring the
capacitance-voltage characteristics of the actual device.
The comparison of corrected and uncorrected mobility with the second derivative method and the
experimental methods is shown in figure S1. The same approach is followed in the main text,
showing in that case an even better agreement with the four point probe measurements.
Mobility (cm2/Vs)
0.3
Uncorrected mobility
Corrected mobility
Saturation mobility (II derivative)
power law fit (=0.76, k=0.78)
0.2
0.1
N2200
0.0
0
20
40
60
Gate Voltage
fpp mobility VD=5V
0.3
fpp mobility VD=10V
Mobility (cm2/Vs)
Saturation mobility
Saturation mobility (corrected)
0.2
0.1
N2200
0.0
0
20
40
60
Gate Voltage
Figure S1. (a) Comparison of the different mobility extraction methods, the uncorrected mobility
(k=1), the corrected mobility (equation (18)) obtained with a power law approximation (red trace)
and the second derivative method (equation (9)) with a 15 point Savitzky-Golay filtering (third order
polynomial). (b) Four point probe mobility (for VDS=5V and VDS=10V) and the corrected and
uncorrected mobility showing the higher agreement with the correction.
5
2 - Full derivation of the saturation mobility equation for a power law
dependent mobility
Starting from the following function (where the onset voltage 𝑉𝑜𝑛 is explicitly taken into account):
𝜇(𝑉) = 𝛼(𝑉 − 𝑉𝑜𝑛 )𝛽
{
0
𝑉 ≥ 𝑉𝑜𝑛
𝑉 < 𝑉𝑜𝑛
(22)
It is possible to calculate the current-voltage characteristics of the device using the following
equation (the inner integral lower limit needs to be shifted to 𝑉𝑜𝑛 , as function (22) is only defined for
𝑉𝐺 ≥ 𝑉𝑜𝑛 and is identically 0 below that value):
𝑉
𝑉 −𝑉(𝑥)
𝐼𝐿 = 𝑊𝐶𝑖 ∫0 𝐺 ∫𝑉 𝐺
𝑜𝑛
𝛼(𝑣 − 𝑉𝑜𝑛 )𝛽 𝑑𝑣 𝑑𝑉(𝑥)
(23)
Keeping 𝑉𝐺 as a parameter, the double integration can be performed by taking into account that the
generalized integral is zero for lower integration parameter:
𝑉
𝑉 −𝑉(𝑥)
∫0 𝐺 ∫𝑉 𝐺
𝑜𝑛
𝑉 −𝑉𝑜𝑛 1
𝐶 𝛼(𝑉𝐺
𝛽+1 𝑖
𝐶𝑖 𝛼(𝑣 − 𝑉𝑜𝑛 )𝛽 𝑑𝑣 𝑑𝑉(𝑥) = ∫0 𝐺
1
𝐶 𝛼(𝑉𝐺
(𝛽+1)(𝛽+2) 𝑖
− 𝑉(𝑥) − 𝑉𝑜𝑛 )𝛽+1 𝑑𝑉(𝑥) =
1
− 𝑉𝑜𝑛 )𝛽+2 = (𝛽+1)(𝛽+2) 𝐶𝑖 𝛼(𝑉𝐺 − 𝑉𝑜𝑛 )𝛽+2
(24)
Where the limit on the outer integral is shifted always to keep into account the presence of the
onset voltage (channel is pinched-off, that is charge density at the upper voltage limit is 0, if onset
voltage is not taken into account it would become negative)
Therefore the current can be written as:
1
𝐼 = (𝛽+1)(𝛽+2)
𝑊
𝐶 𝛼(𝑉𝐺
𝐿 𝑖
− 𝑉𝑜𝑛 )𝛽+2
(25)
And the saturation mobility can be extracted from the following derivation:
2
𝑑 √𝐼
(𝑑𝑉 )
𝐺
𝛽+2
=
𝑊
1
𝑑(𝑉 −𝑉 ) 2
𝐶 𝛼 ( 𝐺 𝑑𝑉𝑜𝑛
𝐿 (𝛽+1)(𝛽+2) 𝑖
𝐺
(𝛽+2)
𝑊
2
) =
(𝛽+2)2
𝑊
𝐶 𝛼 ((𝑉𝐺
𝐿 4(𝛽+1)(𝛽+2) 𝑖
𝛽
2
𝑊
− 𝑉𝑜𝑛 ) 2 ) = 2𝐿 𝐶𝑖 𝛼(𝑉𝐺 −
(𝛽+2)
𝑉𝑜𝑛 )𝛽 2(𝛽+1) = 2𝐿 𝐶𝑖 𝜇(𝑉𝐺 ) 2(𝛽+1)
(26)
(𝛽+2)
With a correction factor now gate voltage independent and equal to 2(𝛽+1) .
6
3 – Transconductance estimation in the linear regime
In the linear regime 𝑉𝑆 ∗ , 𝑉𝐷∗ ≪ 𝑉𝐺 , therefore Equation (5) can therefore be rewritten as:
𝐼=
𝑊
𝐶
𝐿 𝑖
[− ∑∞
𝑖=0
𝑉 𝐷∗
𝜎 (𝑖) (𝑉𝐺 )
𝑖+1
(−𝑉(𝑥))
]
(𝑖+1)!
𝑉𝑆∗
(27)
And now the derivative with respect to the gate voltage can be easily calculated, and if the series is
limited to the first order (which is not a strong assumption since 𝑉𝑆 ∗ , 𝑉𝐷∗ ≪ 𝑉𝐺 , therefore the Taylor
series is always evaluated close to 𝑉𝐺 ), and neglect terms containing
𝑑𝑉𝐷∗𝑆∗
𝑑𝑉𝐺
for the same reasons, the
first derivative of the current in the linear regime can be written as:
𝑑𝐼
𝑑𝑉𝐺
≅
𝑊
𝐶 𝜇(𝑉𝐺 )𝑉𝐷∗ 𝑆 ∗
𝐿 𝑖
(28)
7
4 - Example of correction factor approximation
As an example of how the correction factor can be estimated for an arbitrary function, the transfer
characteristics have been simulated starting from the following polynomial function:
𝐶𝑖 𝜇(𝑉) = 1.17 ∙ 𝜃(𝑉) ∙ (0.1 ∙ 𝑉 0.5 + 2 ∙ 10−5 ∙ 𝑉 2.5 )
(29)
In figure S2(a) the chosen mobility function is shown alongside the two power law components,
responsible for the low voltage and high voltage part of the plot. A simple power law fit is shown in a
dashed line.
Using equation (19) mobility is extracted from the calculated transfer characteristics (figure S2(b)),
without any correction factor (in red) and using the approximated correction factor obtained from
the power law fit (in blue). The procedure returns a much more accurate mobility value despite the
rather poor agreement between the original curve and the fit.
The reason for this is shown in figure S2(c), where the mobility ratio (equal to 1/𝑘) is calculated for
the full curve (black trace), for the two power components of equation (29) (red and green traces)
and for the power law fit of the chosen function. As shown in the mathematical derivation, the
correction factor for an arbitrary function is in general gate voltage dependent, but in any case is
within the values of the exponents of the different monomers which are used in the polynomial
function. The rough approximation resulting from the power law fit returns a value that is an
average of the complete gate voltage dependent correction factor. Although it does not fully recover
the initial mobility, it greatly improves the accuracy of the mobility extraction, which is in turn
fundamental for the contact resistance extraction.
8
2.5
V0.5+V2.5
power law fit (V1.08)
V0.5
V2.5
Mobility (a.u.)
2.0
1.5
1.0
0.5
0.0
0
20
40
60
80
Gate voltage
2.5
Simulated
Extracted
Corrected (k=0.74)
Mobility (a.u.)
2.0
1.5
1.0
0.5
0.0
0
20
40
60
80
Gate voltage
1/k
1.5
1.0
V0.5+V2.5
power law fit (k=0.74)
V0.5 (k=0.83)
V2.5 (k=0.64)
0.5
0
20
40
60
80
Gate voltage
Figure S2. (a) A model polynomial mobility curve, used for the calculation of the transfer
characteristics (black trace) is shown together with the two power law components (red and green
trace) and a power law fit (dashed trace). (b) The model mobility curve (black trace), together with
the uncorrected mobility extracted from the calculated transfer characteristics (red trace) and the
saturation mobility corrected using the scaling factor extracted from the power law fit (blue trace).
(c) Calculated correction factors for the full polynomial mobility (black trace, gate voltage
dependent), for the two different components of the polynomial mobility (red and green)
demonstrating how the real correction factor is always in between the two extreme values, and the
correction factor obtained from the power law fit (averaging the variable correction factor of the
initial function.
9
5 - Extended experimental details
All the devices were prepared in a bottom-contact top-gate geometry. A hall bar structure with
patterned semiconductor [1] has been used for the four point probe measurement (W=200µm,
L=100µm, probe at ¼ and ¾ of the channel length), while a pre-patterned substrate with different
channel length transistors (W=2.85mm, L=5/12/50µm) has been used for the TLM measurements.
Devices for the OSC thickness dependent measurement have been performed on interdigitated
source-drain electrodes prepared by shadow mask evaporation (W=149mm, L=100µm, contact
thickness 20nm). The effective contact length, required for the extraction of contact resistivity for
the interdigitated pattern is defined as half the width of the contact finger (𝐿𝐶 = 50 𝜇𝑚).
P(NDI2OD-T2) and PBTTT, an n- and a p-type organic semiconductor respectively, were dissolved in
dichlorobenzene for the four point probe devices, while P(NDI2OD-T2) dissolved in a 1:1 mixture of
chloronaphthalene:xylene was used for the TLM devices. All solutions had an 8 g/l concentration
(except for the variable thickness measurements where different concentrations from 5 to 20 g/l
were used) and were deposited via spin-coating at 1000 rpm for 60 seconds and annealed at 200°C
and 180°C for P(NDI2OD-T2) and PBTTT, respectively. A dielectric layer made of EG-PMMA (Polymer
Source P8509P-MMA) with a concentration of 60 g/l was subsequently spin-cast at 1500 rpm for 90
seconds and dried overnight at 80°C. Aluminum gate electrodes were deposited through shadow
mask evaporation. All measurements have been performed at room temperature in inert
atmosphere using an Agilent 4155C semiconductor parameter analyzer and a needle probe station,
in all cases performing forward and backwards voltage sweeps. The capacitance was determined
from the measured thickness of the PMMA layer (roughly 550 nm for all devices). All thickness
measurements were performed using a Dektak3ST stylus profilometer.
[1]
J.-F. Chang, M. C. Gwinner, M. Caironi, T. Sakanoue, and H. Sirringhaus, Adv. Funct. Mater. 20,
2825 (2010).
10
6 - N2200 four-point probe mobility extraction
0.30
N2200
Mobility (cm2/Vs)
0.25
0.20
0.15
0.10
linear (Vd=5V)
0.05
saturation (Vd=60V)
0.00
0
20
40
60
Gate Voltage (V)
Figure S3. Extracted mobility from the linear and saturation transfer curves of an N2200 four point
probe transistor, using equation (9) and (18) respectively.
Figure S3 shows the extracted mobility of an N2200 transistor. At low gate voltages the saturation
mobility becomes lower than the linear one, thus preventing the calculation of any contact
resistance. The unexpected behavior is possibly due to the additional preparation steps required for
pattering the semiconductor, as the same effect is not present in any other N2200 device. This could
lead to an invalidation of the assumption
𝑑𝑉𝑆∗
𝑑𝑉𝐺
≪ 1, necessary for the extraction of accurate values of
the mobility in the saturation regime. However, at higher gate voltages the correct dependence is
recovered and the model can be used to correctly estimate the contact resistance (as reported in the
main text).
11
7 - Uncertainty estimations
The measurement accuracy for the used equipment (Agilent 4155C) is dependent on many factors
including measurement integration time, applied voltage and measurement range. For the voltages
and ranges of interest in this work, it’s safe to assume that the relative error on a current
measurement is constant and equal to 𝜉𝑖 = 10−3.
The numerical derivation for a generic data point n is performed according to the following formula:
𝑑𝐼
𝐼
1
−𝐼
𝐼
−𝐼
𝛾𝑛 = 𝑑𝑉𝐷 | = 2 (𝑉𝐷 𝑛 −𝑉𝐷 𝑛−1 + 𝑉𝐷 𝑛+1 −𝑉𝐷 𝑛 )
𝐺
𝐺𝑛
𝑛
𝐺 𝑛−1
𝐺 𝑛+1
(29)
𝐺𝑛
With the relative error that can be calculated as:
𝜉𝛾𝑛
𝛾𝑛
≅𝐼
𝜉𝑖 ∙𝐼𝐷 𝑛
(30)
𝐷 𝑛 −𝐼𝐷 𝑛−1
And for the saturation regime it is possible to derive an analogous formula:
𝜉𝜑𝑛
𝜑𝑛
≅ 2(𝐼
𝜉𝑖 ∙𝐼𝐷 𝑛
(31)
𝐷 𝑛 −𝐼𝐷 𝑛−1 )
with 𝜑𝑛 =
𝑑 √𝐼𝐷
𝑑𝑉𝐺
| .
𝑛
Starting from the equations above, it is possible to determine the uncertainty for mobility and
contact resistance using a simple error propagation rule for uncorrelated values:
𝜉𝜇𝑠𝑎𝑡
| =
𝜇𝑠𝑎𝑡 𝑛
𝜉𝑅𝐶
| ≅
𝑅𝐶 𝑛
𝜉𝜇𝑙𝑖𝑛
| ≅𝐼
𝜇𝑠𝑎𝑡 𝑛
𝜉𝑉𝐷′𝑆′
𝜉𝑖 ∙𝐼𝐷 𝑛
(32)
𝐷 𝑛 −𝐼𝐷 𝑛−1
𝜉 ∙𝐼𝑠𝑎𝑡
𝑖
| ≅ √(𝐼𝑠𝑎𝑡 −𝐼𝐷𝑠𝑎𝑡𝑛
𝑉𝐷′𝑆′ 𝑛
𝐷
𝑛
𝐷
𝑛−1
2
) +(
𝑙𝑖𝑛
𝜉𝑖 ∙𝐼𝐷
𝑛
𝑙𝑖𝑛 −𝐼 𝑙𝑖𝑛
𝐼𝐷
𝑛 𝐷 𝑛−1
2
)
(33)
The result of this analysis is shown in figure S4 for the four point probe measurement, where the
difference between the measured and calculated values (relative to the calculated values) is
compared against the accuracy of the calculated values (taken as twice the standard deviation).
12
Figure S4. The difference between the calculated and experimental values reported in figure 2 (main
article) relative to the calculated values is plotted: (a) pBTTT mobility, (b) P(NDI2OD-T2) mobility, (c)
pBTTT contact resistance, (d) P(NDI2OD-T2) contact resistance. These are compared against the
accuracy of the calculated values, displayed as the black lines. Values in the gray shaded area are not
significant since differences are below the accuracy of the calculated values. Only the calculation on
the backwards traces are shown for clarity.
13
8 - N2200 four-point probe mobility extraction
Figure S5 shows the total device resistance at different source-gate voltages, obtained by fitting the
output curve between 𝑉𝐷𝑆 = 1𝑉 and 𝑉𝐷𝑆 = 4𝑉.
Figure S5. Total device resistance of the TLM devices plotted against channel length, for different
gate voltages. The linear fits used for the calculation of contact and channel resistance are also
shown in the plot for each data set.
14