Modeling of Organic Thin Film Transistors Effect of Contact Resistance

8/6/2019 Modeling of Organic Thin Film Transistors Effect of Contact Resistance

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Modeling of organic thin film transistors: Effect of contact resistances

Dario Natali,a Luca Fumagalli, and Marco SampietroDipartimento di Elettronica e Informazione, Politecnico di Milano, Piaz Leonardo da Vinci 32,

20133 Milano, Italy

Received 17 March 2006; accepted 12 October 2006; published online 2 January 2007

Field effect transistors require an Ohmic source contact and an Ohmic drain contact for ideal

operation. In many real situations, however, and specifically in organic devices, the injection of

charge carriers from metals into semiconductors can be an inefficient process that is non-Ohmic.This has an adverse impact on the performance of thin film transistors and makes the analysis of

electrical measurements a complex task because contact effects need to be disentangled from

transistor properties. This paper deals with the effects of non-Ohmic contacts on the modeling of

organic transistors and gives specific rules on how to extract the real transistor parameters mobility,threshold voltage, and contact resistances using only electrical measurements. The method consistsof a differential analysis of the transfer characteristic curves current versus gate voltage andexploits the different functional dependences of current on gate voltage which is induced by the

presence of contact resistances. This paper fully covers the situations from constant carrier mobility

to power law gate-voltage-dependent mobility, from constant contact resistance to

gate-voltage-dependent contact resistance, and in the linear and in the saturation regime of the

operation of the transistor. It also gives important criteria for the validation of the extracted

parameters to assess whether the conditions for the application of the method are fulfilled. Examples

of application to organic transistors showing various behaviors are given and discussed. 2007

American Institute of Physics. DOI: 10.1063/1.2402349

I. INTRODUCTION

In recent decades organic semiconductors have received

considerable attention, thanks to some of their attractive

properties such as ease of deposition on large areas by means

of low cost and low temperature techniques, possibility of

tuning their chemical and physical properties by means of

chemical tailoring, electroluminescence covering the entire

visible range, and large light absorption coefficient.1

Never-

theless, organic semiconductors have major drawbacks in the

relatively low mobility of charge carriers arising from the

weak intermolecular interaction in the solid state and in the

fact that doping is problematic because it is interstitial and

not substitutional as in inorganic semiconductors.2

One of

the consequences is that the injection of carriers from metal

into organic semiconductors is an issue.3

In fact, low mobil-

ity makes the injection process inefficient: because of their

very small mean free path, injected carriers are likely to be

trapped by the attractive part of the image potential and to

flow back into the metal. In addition, since doping cannot be

easily pursued even though a few examples can be found in

the literature

4

, the strategy of enhancing the current injec-tion by means of heavy local doping of the semiconductorinterfacial regions to provide a tunneling contact, which is

usually adopted with inorganic semiconductors, is not viable.

Therefore one has to rely on proper alignment of metal

Fermi level with highestlowest occupiedunoccupied mo-lecular level in order to achieve an Ohmic contact for holes

electrons. This poses some restrictions to the organic semi-conductors which can be employed because the work func-

tion range of easily processible metals is somewhat limited.

Furthermore, even in the case of a potentially good injecting

contact, nonidealities can interfere: from an energetic point

of view surface dipoles can develop at the interface modify-

ing the expected energy level alignment; from a morphologi-

cal point of view the semiconductor can be more disordered

in the region close to the interface and can display worse

transport properties which give rise to a bad electrical

contact.

5

The result is that the injecting property of metal-organic interfaces is often less than ideal.

This has an adverse impact on organic thin film

transistors6,7 TFTs which require for ideal transistor opera-

tion contacts with the capability of providing any current

with a negligible voltage drop Ohmic contacts. If contactsare non-Ohmic, externally applied voltages partly drop on

the channel and partly drop on the contact regions. This situ-

ation can be modeled by adding contact resistances in series

to the source and to the drain terminals. Because of the volt-

age drops across the contact resistances, the current magni-

tude diminishes and its functional dependence on the exter-

nally applied gate and drain voltages is generally altered. As

a consequence, if one tries to extract carrier mobility fromTFT current measurements without taking into account the

effect of contact resistances, only an apparent mobility is

obtained. It underestimates the real one and does not reflect

the real material properties. This effect is more serious with

TFTs with shorter channels, since their smaller resistivity

makes the voltage drop on the contact resistances larger.

Contact resistances have been heavily studied in inor-

ganic TFTs, but models developed for single-crystalline de-

vices are of limited application since they do not take into

account properly the dependence of mobility on the gateaElectronic mail: [email protected]

JOURNAL OF APPLIED PHYSICS 101, 014501 2007

0021-8979/2007/1011 /014501/12/$23.00 2007 American Institute of Physics101, 014501-1

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voltage which is peculiar to organic and, in general, to amor-

phous and polycrystalline semiconductors. Specific models

have been developed for noncrystalline transistors, but either

the effect of contact resistances is not rigorously treated810

or they are based on the scaling law approach, which exploits

the scaling of current with the TFT channel length.11

In the

literature on organic TFTs various strategies have been de-

veloped to deal with the effect of contact resistances on TFT:

numerical fitting;12,13

the scaling law approach;14,15

the four-probe measurement,

16,17which is an effective and direct

measurement of the potential drop on a portion of the TFT

channel far from the interface regions; and the Kelvin probe

measurement,18,19

which is a powerful method to directly

measure the surface potential of the channel.

The purpose of this paper is to develop a method to

extract the real carrier mobility and the contact resistance

from electrical measurements on TFT structures. These

quantities are essential to the study of the material transport

properties and to the characterization of the injecting prop-

erties of the metal-semiconductor interface. Both the linear

and saturation regimes of transistors are dealt with. This is

important because with low mobility materials the highercurrents that can be obtained in the saturation regime often

make the measurement significantly easier. With respect to

four-probe and Kelvin probe measurements, our method is

based on simple electrical measurements and does not re-

quire either ad hoc lithographies, as in four-probe measure-

ments, or a nontrivial experimental setup as in Kelvin probe

measurements. With respect to numerical fitting, it will be

shown that our method is easier to be applied since no pa-

rameter initialization is needed and the results are easier to

be validated. Only in few cases it will be shown that a scal-

ing law approach, which requires a set of nominal identical

TFTs with different channel lengths, cannot be avoided.

This paper is organized as follows. In Sec. II we intro-

duce the effect of constant contact resistances on the transfer

characteristic curves in the simple case of gate-voltage-

independent mobility, whereas the case of gate-voltage-

dependent mobility, which is more realistic and relevant to

organic semiconductors, is considered in Sec. III. In Sec. IV

the method for the extraction of TFT parameters is outlined

and examples of the application of the method to experimen-

tal data are discussed. In Sec. V we remove the hypothesis of

constant contact resistances to comply with those organic

TFTs that show gate-voltage-dependent contact resistances.

Finally in Sec. VI some conclusions are drawn.

II. TRANSFER CHARACTERISTIC CURVESFOR GATE-VOLTAGE-INDEPENDENT MOBILITYAND CONSTANT CONTACT RESISTANCES

We start with the simple case of gate-voltage-

independent mobility 0 and constant source and drain con-

tact resistances, RS and RD. In the following we will consider

a n-type TFT. Because of RS and RD, the source and drain

terminals, S and D, respectively, are not physically directly

accessible. We will name S and D the physically accessible

terminals, as schematically shown in Fig. 1. Generally speak-

ing, the presence of RS and RD has two effects: i the appliedgate voltage VG is not equal to the gate to source voltage VGS

because the source terminal is not grounded but its potential

is raised by the amount RSID by the current ID flowing

through RS so that

VGS = VG RSID; 1

ii the applied drain voltage VD is not equal to the drain tosource voltage VDS because the source terminal is not

grounded and the drain terminal is connected to VD through

RD so that

VDS = VD RSDID , 2where RSD =RS +RD. The situation can be regarded as a non-

linear voltage partition effect between RS, RD, and the TFT,

where the nonlinearity is due to the TFT. We will now derive

analytical expressions for ID, taking into account the effect of

contact resistances by means of Eqs. 1 and 2.

A. Linear regime

We first focus on the regime where VGSVDS. We as-

sume that the applied gate voltage VG is greater than the

applied voltage VD to the extent that VGSVDS. Since VGVD, and VD is larger than the voltage drop on RS, it turns

out that this latter is negligible with respect to VG. Therefore

Eq. 1 can be simplified to VGS VG. Starting from the stan-dard TFT current equation in the linear regime,

ID = KVGS VTVDS, 3

where K=0CoxW/L, W and L are the channel width and

length, respectively, Cox is the oxide capacitance per unit

surface, and VT is the threshold voltage, we substitute VGSwith VG and VDS with Eq. 2 and solve for ID, obtaining

ID =KVDVG VT

1 + KRSDVG VT. 4

In the absence of contact resistances the transfer characteris-tic curves are linear in VG VT, whereas in the presence ofcontact resistances they are linear in VG VT only asymp-totically for VG VT0 and tend to reach a horizontal as-ymptote equal to VD/RSD for VG VT, as shown in Fig.2 left column. This occurs because for large VG the channelresistance becomes negligible with respect to RSD and the

current flow tends to be dominated by the contact resistance

RSD.

Figure 2 also reports ID , the first derivative of current

with respect to VG: instead of being constant and equal to

KVD, ID starts from this value but monotonously tends to 0

for large VG.

FIG. 1. Electrical model for a TFT affected by contact resistances, RS and

RD, in series with the source and drain terminals, respectively.

014501-2 Natali, Fumagalli, and Sampietro J. Appl. Phys. 101, 014501 2007



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B. Saturation regime

In the saturation regime we assume that VD is greater

than VG to the extent that the condition VDSVGS VT is

met. This requires that the voltage drop on RD is not largeenough to drive the device out of saturation. In this situation

only RS has an influence. Starting from the standard TFT

current equation,

ID =1

2KVGS VT

2, 5

we substitute VGS with Eq. 1 and solve for ID, obtaining

ID =VG VT

RS+

1

KRS2

1 1 + 2VG VTKRS. 6

Because of the contact resistance, the quadratic dependence

on VGS VT is lost. To understand the effect of RS on theshape of ID it can be useful to expand in a Taylor series thesquare root term in Eq. 6. A third order expansion leads to

ID 1

2KVGS VT

2 1

2K2RSVGS VT

3. 7

Equation 7 shows that the transfer characteristic curves areasymptotically quadratic in VG VT for VG VT0, but atlarge VG the negative cubic term comes into play and makes

the curve less than quadratic, as can be seen from the simu-

lation reported in Fig. 3 left column. The effect on ID isthat, instead of being a straight line ID = KVGS VT, it isasymptotic to this latter for VG VT0 but for large VGtends to a horizontal asymptote equal to 1/RS Fig. 3. Thisoccurs because at large VG the TFT has a very low resis-

tance; hence any VG variation practically drops entirely on

RS which dominates ID .

III. TRANSFER CHARACTERISTIC CURVESFOR GATE-VOLTAGE-DEPENDENT MOBILITYAND CONSTANT CONTACT RESISTANCES

We now improve the model, introducing a gate-voltage-

dependent mobility but keeping contact resistances constant.

In organic semiconductors it is usually found that mobil-

ity increases upon VGS, that is upon increasing the carrier

concentration. Two possible explanations are reported in the

literature. One lies on the multiple trapping and release trans-

port model,20

where only a fraction of the gate induced

charge contributes to the current flow, the remaining part

being trapped in an exponential tail of trapping states; since

the ratio of free versus trapped carriers is larger at higher

levels of injection, the mobility increases with the gate volt-

age. Alternatively, in the framework of the variable range

hopping model,21

carriers contribute to current flow only

when they are excited to a so-called transport energy level: at

higher carrier concentration the average starting energy is

closer to the transport energy, which reduces the activation

energy and therefore enhances mobility. In both models mo-

bility follows a power law dependence on VGS,

= 0VGS VT, 8

where 0 is an empirical fitting parameter

22

defined as themobility for VG VT =1 V. In the following we analyze theeffect of contact resistances when mobility is in the form of

Eq. 8 in the linear and in the saturation regime of a TFT.

A. Linear regime

In case of ideal Ohmic contacts, the TFT current equa-

tion with mobility according to Eq. 8 reads

ID = KVGS VT+1VDS, 9

where a power law dependence on VGS VT is obtained,which is the fingerprint of mobility in the form of Eq. 8.

The effect of RS and RD is analogous to that outlined inSec. II. Substituting Eq. 2 in Eq. 9 and letting VGS = VG,we obtain

ID =KVDVG VT

+1

1 + KRSDVG VT+1

. 10

The power law dependence on VG VT is lost: ID isasymptotic to the power law of Eq. 9 only for VG VT0 and tends to reach a horizontal asymptote equal to

VD/RSD for large VG, as can be seen in Fig. 2 right column.The first derivative ID is not monotonous, as it is in the

case of VG-independent mobility Sec. II: for VG VT0it tends to a power law as it would be in case of Ohmic

FIG. 2. Simulation of the transfer characteristic curves ID and of its first

derivative ID in the linear regime in the case of constant mobility leftcolumn and VG-dependent mobility right column. The curves in solidlines are affected by constant contact resistances; the dashed lines are for

Ohmic contacts. Parameters: 0 =1 cm2/V s, =0.2, Cox = 10

4 F/m, W/L

=1000, VT=0 V, VD =1 V, and RSD = 5 k.

FIG. 3. Simulation of the transfer characteristic curves ID and of its first

derivative ID in the saturation regime in the case of constant mobility leftcolumn and VG-dependent mobility right column. The curves in solidlines are affected by constant contact resistances; the dashed lines are for

Ohmic contacts. Parameters: 0 =1 cm2/V s, =0.2, Cox = 10

4 F/m, W/L

=1000, VT=0 V, VD =30 V, and RS = 5 k.




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contacts, then reaches a maximum, and finally tends to 0 forlarge VG Fig. 2. In correspondence of the ID maximum,

23

the transfer characteristic curve shows an upward to down-

ward concavity change. The concavity change in the transfer

characteristic curve and a nonmonotonous ID are the finger-

prints for the presence of constant contact resistances when

mobility is in the form of Eq. 8.

B. Saturation regime

In the saturation regime with ideal Ohmic contacts, the

TFT current equation with a mobility according to Eq. 8reads

ID =K

+ 2VGS VT

+2 . 11

A power law dependence on VGS VT is obtained, which isagain the fingerprint of mobility in the form of Eq. 8.

To take into account the effect of RS we substitute Eq.

1 in Eq. 11, and we obtain

ID =

K

+ 2 VG RSID VT+2

. 12

The power law dependence on VG VT is lost because ofcontact resistances, but in general it is not possible to derive

an explicit expression for ID, and therefore very little can be

said about the nature of the deviation of Eq. 12 from thepower law of Eq. 11. For the same reason, regarding ID it isonly possible to say that it tends to 1/RS for large VG, as in

the case of a VG-independent mobility seen in Sec. II. With

respect to the linear regime, no clear fingerprints for the pres-

ence of RS can be found, and it is difficult to assess the

presence of contact resistances from the shape of the transfer

characteristic curve: a deviation from the power law in itself

is not representative since in principle it might be due to

other phenomena, different from the presence of contact re-

sistances. Simulated plots are reported in Fig. 3 right col-umn.

IV. EXTRACTION OF TRANSISTOR PARAMETERSWITH THE DIFFERENTIAL METHOD FOR CONSTANTCONTACT RESISTANCES

For the case of mobility in the form of Eq. 8 and ofconstant contact resistances, we develop a method for the

extraction of the TFT parameters, 0, , VT, and the contact

resistances RS and RD. The method simply requires an elec-

trical measurement on a single device, and it can be appliedboth in the linear and in saturation regimes.

A. Parameter extraction in the linear regime

Equation 10 can be rewritten as follows:

fx =gx

1 + gx, 13

if we let = KVD, = KRSD, gx = VG VT+1, and fx

=ID. Functions in the form of Eq. 13 have the relevantproperty that the ratio z between fx2 and the first derivativeof fx does not depend on ,

z =fx2

fx=

gx2

gx. 14

If we apply this property to the expression of transfer char-

acteristic curves in the linear regime, we obtain a quantity

which does not depend on RSD,

z =ID

2

ID

=K

+ 1

VG VT+2VD , 15

and we are therefore left with three unknowns: 0, , and VT.

This property, although already demonstrated to hold true for

a VG-independent mobility by Jain24

and for specific models

of mobility including degradation and saturation effects by

Fikry et al.,25

is indeed general: it arises only because ID can

be written in the form of Eq. 14 and it holds true for anydependence of mobility on VG.

At this point one might directly fit Eq. 15 to extract theremaining unknowns, but it is possible to further reduce the

number of unknowns by noting that z is in the general form,

fx =

x x0

, x

x00, xx0, 16

if we let x = VG, x0 = VT, = KVD/+ 1, =+2, and fx=z. Functions in the form of Eq. 16 have the relevantproperty

8that the ratio w between the integral function of

fx and fx itself is linear in x x0 and does not depend onthe multiplying factor ,

w =x0

xfxdx

fx=

1

+ 1x x0 . 17

If we apply this property to z in Eq. 15, we obtain

w =V

T

VGzdVG

z

0VGzdVG

z=

1

+ 3VG VT, 18

where the lower limit of integration has been extended from

VT to 0, under the hypothesis that ID is negligible for VGVT. The quantity w plotted as a function of VG is a straight

line and contains only two unknowns, and VT, which can

be easily extracted from its slope and from its abscissa inter-

section, respectively. With these two parameters, we can now

solve Eq. 15 for K and then extract 0. In this way threeTFT parameters are known, and it is possible to extract the

last unknown parameter RSD. If we solve Eq. 10 for RSD,

we obtain

RSD =VD

ID

1

KVG VT+1

= Rtot RTFT. 19

The series of the contact resistances RSD is calculated as the

difference between the overall device resistance Rtot and the

TFT resistance RTFT.

To ascertain whether the conditions for the applicability

of the method are met namely, mobility in the form of apower law of VG VT as in Eq. 8 and constant contactresistances and, consequently, to validate the extracted pa-rameters, w has to be a straight line with a slope between 0




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and 1/3, provided that can range from infinity to 0, and the

extracted contact resistance when plotted against VG has to

be a horizontal line.

B. The differential method in the linear regime:

An example

We analyze the application of the method to poly-3-

hexylthiophene-based transistors.

Transistors have been developed in bottom contact con-

figuration. An oxide layer, 130 nm thick acting as a gate

dielectric, has been grown by thermal oxidation on a highly

p-doped silicon layer, acting as the gate. Platinum-titanium

drain and source contacts, with titanium acting as an adhe-

sion layer, have been photolithographically defined on the

oxide surface. Devices with channel lengths ranging from

3 to 12 m and channel widths of 15 mm have been devel-

oped. Silicon dioxide has been plasma polished for 10 min

Hummer Sputter Coater 6.2 Anatech LTD, functionalizedby immersion for 10 min in a 10% by volume solution ofdimethyldichlorosilane in chloroform and rinsed in pure

chloroform for 5 min. Poly-3-hexylthiophene Aldrich, re-gioregularity 97% was purified by soxhlet extraction inmethnanol and hexane. A 3 mg/ml chloroform solution was

prepared, filtered through a 0.45 m pore size PTFE mem-

brane and spin coated at 2000 rpm. Electrical measurements

have been performed in vacuum, at about 105 mbar, with a

HP4142B modular dc source/monitor.

In Fig. 4 the measured transfer characteristic curve in the

linear regime of a TFT with L = 3 m, recorded at VD=1 V, and its first derivative ID , obtained by means of nu-

merical differentiation, are shown. From the fact that IDtends to saturate for high VG and that ID is nonmonotonous

comparable to Fig. 2, right column, it can be expected thatconstant contact resistances and VG-dependent mobility in

the form of Eq. 8 are present.Starting from the transfer characteristic curve and from

ID , it is possible to compute the quantities z and w according

to Eqs. 15 and 18, respectively, which are shown in Fig.5. From the plot of w we extract =0.057 and VT=5.47 V. From z we compute K and hence 0 =0.783

102 cm2 /V s, and finally it is possible to extract RSD=34.9 k. This latter is plotted in Fig. 6 top together withthe VG-dependent mobility bottom. The initial guess sug-

gested by the shape of ID and ID is confirmed through the

application of the method.26

The effect of contact resistances

can now be assessed in a quantitative way and turns out to be

considerable: at VG =30 V half of the applied voltage VD=1 V drops on the contact resistance and half drops on the

transistor channel.It has to be noted that the conditions for the application

of the method are met only for VG14 V: only in thisrange is w linear in VG VT, as evidenced in Fig. 5, and RSDVG independent, as stated in Fig. 6 top.

1. Differential method versus fitting procedure

Let us now compare the results obtained with the differ-

ential method with the ones that might be obtained with a

simple fitting procedure on the same experimental data of

Fig. 4.

Firstly, the experimental curve has been fitted by means

of Eq. 10 in the range of validity of the differential method,that is, for VG14 V. Extracted parameters are summa-rized in Table I and the mobility obtained by fitting is also

plotted in Fig. 6 fit 1, bottom. The two procedures producethe same results and are indeed equivalent because they are

based on the same equation for the current and they are ap-

plied on the same data range.

Secondly, we apply the fitting procedure without using

the information gained with the differential method, that is,

FIG. 4. Experimental transfer characteristic curve ID vs VG in the linear

regime of a poly-3-hexylthiophene TFT with L = 3 m measured at VD= 1 V ID, top. The first derivative ID with respect to VG has been obtainedby numerical differentiation bottom.

FIG. 5. Quantities z top and w bottom calculated with the differentialmethod presented in Sec. IV A starting from ID and ID of Fig. 4. The dashed

line bottom represents a linear interpolation of w in the range VG14 V: intersection with the w=0 axis gives VT, from the line slope

=0.057 is calculated.

FIG. 6. Top: contact resistance RSD extracted with the differential method of

Sec. IV A starting from ID and ID of Fig. 4. The dashed line is a linear fitting

to RSD in the range VG14 V. Bottom: comparison between the mobilityextracted with the differential method of Sec. IV A solid line and themobility obtained by means of fitting with Eq. 10 in two ranges: VG14 V fit 1 and VG7 V fit 2.




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by starting from VG =7 V based on the fact that the transfercharacteristic curve extends nicely for VG5 V. A sizeabledifference between the differential method and the fitting

method starts to display, as can be appreciated from Table I

and from Fig. 6 fit 2, bottom. The fitting procedure returnsa mobility which is not correct in the range 7 V VG14 V, as it overestimates the real mobility even in the

range VG14 V, which was correctly fitted when the fittingrange was VG14 V. This highlights the important advan-tage of the differential method over the fitting procedure. The

parameters of the fitting curve are chosen to minimize themean square error between the experimental data and the

fitting curve over a chosen data range, which is tentatively

adjusted to exclude the region where the fitting is not good

because of model inadequacy. On the contrary, the differen-

tial method presented here is applied pointwise: the self-

consistency tests indicated in Sec. IV A permit us to identify

directly and unambiguously the data ranges where results are

not correct because the method cannot be applied, without

compromising the extraction of parameters in the data ranges

where the method can be correctly applied.

C. The differential method in the saturation regime

We now turn to the problem of extracting TFT param-

eters from transfer characteristic curves in the saturation re-

gion. This can be advantageous with respect to the extraction

in the linear regime because of the flowing of higher cur-

rents.

It is useful to start from the expression of ID which reads

ID =dID

dVG=

KVG VT RSI+1

1 + KRSVG VT RSI+2

. 20

If we let = K, = KRS, and gx = VG VTRSI+1

, ID

hasthe same form of Eq. 13, so we can apply again the prop-erty of Eq. 14. The quantity z in this case reads

z =ID

2

ID= + 1

1 RSID

KVG VT RSI+2

. 21

With respect to the linear regime, we now have to compute

both the first and the second derivative with respect to VG of

the experimental transfer characteristic curve. In the quantity

z all the four unknowns are still present, but if we multiply

z1 by ID, we obtain the following relevant quantity:

z1ID =ID

ID2ID =

+ 1

+ 21 RSID , 22

where only two unknowns are left, and RS. Equation 22is a straight line if plotted against ID , making it easy to ex-

tract and RS from the ordinate intersection and from the

slope of the curve, respectively. To extract the other twounknowns we can calculate VGS = VG RSID and plot IDagainst VGS. This latter plot is a power law in VGS VTaccording to Eq. 12 and therefore can be fitted directly toextract VT and 0, or we can apply the property of Eq. 17to obtain the quantity w,

w =VT

VGSIDdVGS

ID

0VGSIDdVGS

ID=

1

+ 3VGS VT. 23

From the abscissa intersection of w it is possible to extract

VT, and finally it is possible to solve ID to extract K and

hence 0.

To ascertain whether the conditions for the applicabilityof the method are met, namely, the mobility in the form of a

power law of VG VT as in Eq. 8 and constant contactresistances, the plot of the quantity z1 ID vs ID has to be a

straight line with a negative slope and its intersection with

the y axis has to occur at an ordinate between 1/2 and 1,

provided that can range from 0 to infinity; w has to be a

straight line with a slope between 0 and 1/3.

D. The differential method in the saturation regime:An example

We now apply the method to the experimental transfer

characteristic curve reported in Fig. 7, measured on a TFTwith L = 3 m, recorded at VD =30 V TFT realization as inSec. IV B.

From experimental data we have computed by means of

numerical differentiation the first and second derivatives of

the current with respect to VG, ID and ID respectively, also

shown in Fig. 7: these are the starting quantities to extract

transistor parameters with the differential method.

In Fig. 8 we plot z1ID vs ID : in the range 5 V VG11.5 V, z1ID is a straight line with a negative slope and

with an ordinate intersection occurring between 0.5 and 1,

meaning that the TFT is affected by constant contact resis-

tances. From the ordinate intersection we extract =1.05 and

TABLE I. Comparison of TFT parameters for the example of Sec. IV B

extracted in the linear regime by means of the differential method valid forVG 14 V, by means of fitting with Eq. 10 for VG 14 V, and bymeans of fitting with Eq. 10 for VG 7 V.

0 cm2/V s VT V RSD k

Different ial method 0.78102 5.47 0.057 34.9

Fitting Eq. 10 for VG 14 V 0.82102 5.62 0.045 34.9

Fitting Eq. 10 for VG 7 V 0.61102 5.02 0.166 37.6

FIG. 7. Transfer characteristic curve in the saturation regime of a poly-3-

hexylthiophene TFT with L = 3 m measured at VD =30 V ID, top. Thefirst and second derivatives with respect to VG have been obtained by nu-

merical differentiation ID and ID , bottom.




7/12

from the slope we extract RS =8.35 k. Outside this range,

z1ID does not behave clearly as a straight line and the

method cannot be applied, which means that mobility is not

in the form of a power law and/or contact resistances are not

constant. The parameter extraction is completed by calculat-

ing VGS and by fitting the ID vs VGS plot. Results are sum-

marized in Table II. Although the effect of contact resistanceis minor in this case as the voltage drop on RS is only 0.6 V

for VG =11.5 V, the method is able to identify the presenceof the contact resistance.

1. Differential method versus fitting procedure

Let us compare these results with those which can be

obtained by means of a simple fitting approach directly on

the transfer characteristic curve of Fig. 7. We have seen that

in the saturation regime explicit expressions for the transfer

characteristic curves are available only in two special cases:

constant contact resistances with = 0 Eq. 6 or arbitrary

but in the absence of contact resistances Eq. 11.The fitting in the framework of the first model, which is

based on VG-independent mobility, is not meaningful: by us-

ing Eq. 7, which is the third order Taylor expansion of Eq.6, the cubic term has a positive coefficient instead of anegative one, which means that the contact resistance should

be negative. This occurs because the transfer characteristic

curve is more than quadratic, instead of being less than qua-

dratic as requested by Eq. 6. On the contrary, the fitting inthe framework of the second model, VG-dependent mobility

and absence of contact resistances, is possible, as it predicts

transfer characteristic curves to be more than quadratic,

which is experimentally found. By fitting the curve in the

range 5 V VG27.5 V, the difference between the fittingcurve and the experimental measurement is in the range of a

few percent only, but using the method this difference is less

than 0.1%, as it is shown in Fig. 9 top. Also the mobility

obtained by fitting is different from the one extracted by

means of the differential method, as shown in Fig. 9 bot-tom: it has a higher 0 but a lower factor extracted pa-rameters are reported in Table II. Indeed, we know from themethod that contact resistances play a role in the range

5 V

VG

11.5 V and therefore the fitting, which does nottake into account the contact resistance, is less accurate than

the method.

V. METHODS FOR VG-DEPENDENT CONTACTRESISTANCES

It is often reported in the literature that contact resis-

tances can be VG dependent.12,1417,19,2731

We investigate

how to extract TFT parameters when contact resistances

have an arbitrary dependence on VG.

To solve the problem, which has one more degree of

freedom with respect to the case of VG-independent contactresistances analyzed in Sec. IV, let us bias the TFT in the

linear regime and reconsider z, the key quantity of the dif-

ferential method presented in Sec. IV A. In the case of

VG-dependent contact resistances the reciprocal of z reads

z1 =ID

ID2 =

+ 1

KVDVG VT+2

RSD

VD. 24

With respect to the case of VG-independent contact resis-

tances, there is an additional term which contains the first

derivative of RSD with respect to VG. Note that only the first

addendum in Eq. 24 depends on K and therefore on theTFT channel length L; the second addendum does not de-

pend on K under the reasonable assumption that contact re-sistances are interface related phenomena and do not depend

on the transistor length. We can exploit this feature to extract

TFT parameters.

FIG. 8. Quantity z1ID calculated with the differential method presented in

Sec. IV C starting from ID, ID , and ID of Fig. 7. The ordinate intersection of

the linear fit in the range 5 V VG11.5 V gives , and its slope gives RS.

TABLE II. Comparison of TFT parameters for the example of Sec. IV D extracted in the saturation regime by

means of the differential method of Sec. IV C valid for 5 V VG 11.5 V and by means of fitting with Eq.11 for 5 V VG 27.5 V.

0 cm2/V s VT V RS k

Differential method 8.3104 1.8 1.05 8.35

Fitting Eq. 11 for 5 V VG 27.5 V 5.98103 1.03 0.305 0

FIG. 9. Top: percent difference between the experimental transfer charac-

teristic curve of Fig. 7 and the curves reconstructed by means of the differ-

ential method solid line and by means of the fitting dashed line. Bottom:mobility extracted by means of the differential method solid line and bymeans of the fitting dashed line.




8/12

Let us consider two nominally identical TFTs with dif-

ferent channel lengths L1 and L2 and let us calculate z1 and z2according to Eq. 24. Since in these quantities the secondaddendum RSD /VD is the same, they can be combined to

eliminate it,

z*1 = z11 z2

1 =+ 1

K*VDVG VT+2

, 25

where K* =0CoxW/L1 L2. As the quantity z* does not de-pend on RSD, we can now simply follow the recipe for the

VG-independent contact resistances in the linear regime cal-

culating w* 0VGz*dVG/z

* = VG VT/+ 3, extractingVT and from it, calculating K

* from z*, and then solving for

RSD according to Eq. 19. The reliability of obtained resultsis assured by checking that w* plotted against VG is a straight

line with slopes between 0 and 1/3.

Attempts to solve the problem in the saturation regime

proved unsuccessful: as a consequence of the implicit nature

of Eq. 12, it was not possible to write the relevant quantityz as the sum of L-dependent and L-independent addenda and

then to exploit a scaling law approach as in the linear regime.

A. Contact resistances with a power law dependenceon VG

We now investigate a particular VG dependence of con-

tact resistances. Let us assume that contact resistances are

due to accumulated regions characterized by a low mobility:

in top contact TFTs these could be identified as the regions

under the source and drain electrodes,31

partially damaged by

the contact deposition; in bottom contact TFTs they could be

identified with the regions whose morphology is disturbed by

the presence of a triple interface formed by the dielectric, the

contact metal, and the semiconductor. In both these cases

contact resistances would depend on the reciprocal of the

number of accumulated carriers na times their VG-dependent

mobility, RnaVG1. If we now assume that the mobil-

ity in the channel region and the mobility in the contact

region are characterized by the same but by different mul-

tiplying factors, 0 and cont, respectively, contact resis-

tances have a power law dependence on VG VT and can bewritten in the general form

R =

WVG VT+1

, 26

with =Lcont/Coxcont, where Lcont is the length of the con-tact accumulation channel.

32It is easy to see by direct sub-

stitution of Eq. 26 into Eqs. 10 and 12 that such a de-pendence has an unexpected impact on the transfer

characteristic curves: unlike the case of constant contact re-

sistances presented in Sec. III, transfer characteristic curves

retain the power law functional dependence on VG encoun-

tered in the case of Ohmic contacts, and contact resistances

affect only the multiplying constant. In the linear regime we

have

ID = KlinVG VT+1VD, 27

where

Klin =K

1 + K/W.

In the saturation regime we have

ID = KsatVG VT+2 , 28

where

Ksat =

K

+ 2 1 Ksat

W +2

.

Whereas with constant contact resistances the departing

from the power law arises because of the nonlinear partition

effect between the TFT and the contact resistances, with the

dependence on VG as in Eq. 26, a compensation occursbetween the nonlinearities of the TFT and those of the con-

tact resistances. In fact, the higher the VG, the higher the IDbut the lower the contact resistances, and the net result is that

the partition between these two nonlinear elements remains

linear. Therefore in the linear regime VDS is linearly propor-

tional to the applied VD and in the saturation regime VGS is

linearly proportional to the applied VG.

Since in this case the dependence of ID on VG is notaltered by contact resistances, a differential approach cannot

be used and one must resort to a scaling law approach to

extract TFT parameters, exploiting the dependence of the

multiplying factors on L. In other words, this means that

even if ID shows a power law dependence on VG VT, thisdoes not imply the absence of contact resistances, but only

the absence of VG-independent contact resistances: if the

methods outlined in Sec. IV return contact resistances being

equal to 0 , this means that constant contact resistances are

equal to 0 , but nothing can be said about the presence of

VG-dependent contact resistances in the form of Eq. 26.Therefore in the case of ID following the power law form of

Eqs. 9 and 11, one should always perform a scaling lawtest. Methods to deal with such situations are outlined below.

1. Linear regime

The parameters and VT can be easily obtained from the

transfer characteristic curves. To extract the VG-dependent

contact resistance, we observe that

1

Klin=

L

0CoxW+

W. 29

Therefore plotting 1/Klin vs L, a straight line should be ob-

tained: from its intersection with the y axis can be calcu-

lated, and from its slope 0 can be extracted.

2. The linear regime with constant and VG-dependentresistances: An example

Let us consider now a contact resistance given by the

sum of constant term and of a VG-dependent term, RSD=RSD0

+/WVG VT+1. In this case, firstly one should

apply the method of Sec. IV A to extract the VG-independent

term RSD0and the parameters VT, , and Klin, and secondly

1 /Klin should be plotted versus L to extract 0 and .

As an example we consider the case of bottom contact

pentacene based transistors33

having a 230 nm thick SiO2




9/12

layer, Gold-indium tin oxide ITO drain and source contactsITO acting as an adhesion layer, channel lengths rangingfrom 2.5 to 20 m, and channel widths of 10 mm. Silicon

dioxide had been functionalized with 1,1,1,3,3,3-

hexamethyldilisazane HMDS Merck applied in gas phaseat 110 C, and pentacene was synthesized according to Ref.

34, purified in a Craphys sublimator and deposited without

heating the substrates with a rate of 0.010.02 nm/s at a

pressure of 1.1107 mbar. The final pentacene thickness

was of 30 nm. Electrical measurements have been performed

under ambient conditions by means of a Credence M3650test system.

Transfer characteristic curves in the linear regime, mea-

sured at VD =1 V, do not behave according to power laws.Therefore, firstly they have been analyzed by means of the

method of Sec. IV A to extract RSD0, the constant part of the

the contact resistance. Results are reported in Table III in

terms ofRSD0, VT and and show that TFTs were affected by

constant contact resistances RSD0in the range of 500 k.

Secondly, a scaling law test has been performed on the

extracted Klin by plotting 1/Klin versus L. As shown in Fig.

10 top, the intersection with the y axis does not occur in theorigin, which means that VG-dependent contact resistances in

the form of Eq. 26 are present, in addition to the alreadyextracted constant term RSD0

. The extracted parameters and

0 are reported in Table III.

In Fig. 10 bottom the apparent mobility, no more af-fected by the constant contact resistance RSD0

but still af-

fected by the VG-dependent contact resistance, and the real

mobility, are shown as functions of L for VG =10 V. Theapparent mobility underestimates the real mobility: the dif-

ference between the two can be as high as an order of mag-

nitude for L =2.5 m. This is consistent with the fact that

after the extraction of the only RSD0, the voltage drop on the

TFT, calculated as VD VRSD0, would appear to be of the

order of 0.50.6 V, whereas in reality, after the extraction of

the VG-dependent term also, it is only in the range of 60 mV.

In Fig. 11 it is shown how the amount of the voltage drop on

the VG-dependent part of the contact resistance gets progres-

sively smaller for the longer channels. As a consequence, the

difference between the apparent and real mobilities dimin-

ishes. The residual dependence of the mobility on L is due tothe different factors of the TFTs.

3. Comparison with the scaling law approach

Now we compare the results of our method, in which we

combined a differential approach and a scaling law approach,

with the results that would have been obtained on the same

data through a pure scaling law approach.

To this extent we solve Eq. 27 for the total deviceresistance Rtot = VD/ID to highlight the dependence on L,

Rtot = VDID

= L0CoxW + WVG VT+1 + RSD0. 30

Equation 30 states that if we plot Rtot vs L for different VG,we obtain a bundle of straight lines whose center lies in the

second quadrant at the coordinates 0Cox,RSD0. Theslope of the lines decreases if VG increases. It is interesting to

note that in the case of constant contact resistances the center

of the bundle is located on the y axis at Rtot =RSD, and in the

case of pure VG-dependent contact resistances, that is RSD0= 0 , it is located on the x axis at L = 0Cox.

TABLE III. Parameters of TFTs of Sec. V A 2. To extract , VT, and the constant part of the contact resistances

RSD0, the method of Sec. IV A has been applied the range of validity is also reported in the table. Subsequently

the method of Sec. V A 1 has been applied to extract the VG-dependent part of the contact resistance in terms

of and 0.

L m VT V RSD0 k VG range V kV m 0 cm2/V s

2.5 +1.8 0.22 535 417 109 1.6102

10 +1.6 0.32 493 412 109 1.6102

20 +1.72 0.38 529 420 109 1.6102

FIG. 10. Top: extraction ofand 0 for the example of Sec. V A 2 exploit-

ing the dependence of Klin on L. Bottom: app is the mobility at VG=10 V extracted applying only the method of Sec. IV A and consequently

still affected by VG-dependent contact resistances; real is obtained combin-

ing the differential method of Sec. IV A with the scaling law method of Sec.

V A 1 and reflects the real material property.

FIG. 11. For the example of Sec. V A 2, percent voltage drop on the tran-

sistor channel VTFT, on the constant contact resistance VRSD0, and on

the VG-dependent part of the contact resistances VRVG. Percent voltagedrops are shown as a function of VG for L =2.5, 10, and 20 m.




10/12

The Rtot vs L plot is shown in Fig. 12. Straight lines

which approximatively converge to a point in the second

quadrant can be drawn. From its coordinates we extract

RSD0 900 k and 0 36610

3 m3/s. These values

should be compared with RSD0 500 k and 0=183

103 m3/s obtained with our method. To understand thereasons for these differences, note that the pure scaling law

approach needs the TFTs with different L to be identical in

terms of0, , VT, and . But this is not true in our case as

evidenced by the results in Table III: the factor varies with

L, and in particular it is higher for shorter channels. There-

fore, in the Rtot vs L plot, the points relative to shorter chan-

nel lengths have an ordinate higher than expected, and the

extracted parameters are not correct. With the differential

method instead, RSD0, , and VT are extracted separately on

each device and are therefore allowed to vary. When the

dependence of Klin on L is exploited in Eq. 29, only and0 are required to be identical among the devices with dif-

ferent L, thus relaxing the requisite of identical parameters

among the TFTs with different channel lengths.

4. Saturation regime

Due to the difficulty to find an explicit expression for

Ksat in Eq. 28, let us expand Ksat in a Taylor series. To firstorder it gives

Ksat 1+ 2

K1 + K/W

, 31where the second multiplying factor arises because of contact

resistances.

It is useful in this case to calculate 1/Ksat,

1

Ksat= + 2 L

0CoxW+

W . 32

The parameters and 0 can be obtained by plotting 1/Ksatvs L, in analogy to the procedure for the linear regime of

Sec. V A 1

In the special case of= 0, Ksat can be solved in closed

form,

Ksat =K

2 2W

K+

2W2

K221 1 + 2K

W , 33

with the second multiplying factor arising because of contact

resistances. The only two unknowns in this case are and 0which can be extracted measuring KsatL from devices withdifferent L, and fitting it with Eq. 33. It can be useful tosimplify the fitting and to consider a Taylor series expansion

of Ksat. If we call app the apparent mobility extracted with-out taking into account the presence of contact resistances,

app =2ID

CoxW/LVG VT

2, 34

expanding Ksat to third order, it turns out that

app = 01 0CoxL

. 35In Eq. 35 the term 0Cox/L can be interpreted as thepercent error made extracting the mobility without taking

into account contact resistances: the higher the 0 and the

shorter the L, the larger the percent error because the tran-

sistor is more conductive and the relative incidence of con-

tact resistances is larger. The recipe to extract contact resis-

tances goes as follows: from a set of TFTs with different L,

one has to extract app according to Eq. 34. If plotted ver-sus 1/L, the values ofapp should fit on a straight line with

a negative slope: the extrapolated intersection with the ordi-

nate axis gives 0, while from the slope it is possible to

extract . In the following an example of application is

given.

5. The saturation regime with= 0: An example

The TFT active material used in this case is a co-oligomer based on fluorenone and thiophene moieties,35

2,7-

bis5-n-hexyl-2,2-bithiophene-5-yl-fluoren-9-one, and ithas been vacuum deposited on substrates held at different

temperatures Tsub during deposition 100, 130, 150, and190 C. On each substrate channel lengths of 3, 6, and12 m were present substrates as in Sec. IV B.

Transfer characteristic curves can be well fitted by the

quadratic law of Eq. 6, but app, the mobility extractedaccording to Eq. 34, is L dependent, and in particular it ishigher for the longer channel lengths, as shown in Fig. 13.

Therefore it is to be concluded that VG-dependent contact

resistances in the form of Eq. 26 are present. An example

of mobility extraction according to the scaling law approachof Sec. V A 4 is given for Tsub =190 C in Fig. 14, where the

app vs 1/L plot is reported. The overall results in terms of

0 are reported in Fig. 13: the apparent mobility underesti-

mates the real one by a factor which can be as high as 3 for

the 3 m channels, which are the most severely affected by

contact resistances. The values of the contact resistances as a

function of VG are plotted in Fig. 15. For VG close to VT, RScan be as high as a few megaohms; for large VG, RS de-

creases to a value which can range from tens of kilohms for

Tsub =130 C to almost 1 M for Tsub =190C. Interestingly

enough, the lowest contact resistance corresponds to the set

of TFT grown at Tsub =130C, which gave the highest mo-

FIG. 12. Pure scaling law approach applied to the same experimental data

used for Figs. 10 and 11. The position of the center of the bundle is different

from the one which is expected applying our method solid circle andprovides a constant contact resistance RSD0

900 k instead of RSD0 500 k.




11/12

bility. This can be more easily appreciated by looking at Fig.

16, where the factor , which sets the magnitude for contact

resistances, is plotted versus 0: the higher the 0, the lower

the and the contact resistance. This agrees with recently

published experiments on TFTs Refs. 29 and 30 and ondiodes36 and can be explained on the basis of the theory of

metal-organic semiconductor interface developed by Scott

and Malliaras,37

which predicts the rate of carrier injection to

be proportional to the carrier mobility.

VI. CONCLUSIONS

This paper has shown how powerful the differential

method can be in extracting from simple electrical measure-

ments the relevant parameters of organic TFTs when in the

presence of contact resistances and of a mobility dependent

on the gate voltage as a power law. The large spectrum of

cases that has been analyzed reflects the many real physical

situations encountered in the realization of organic transis-

tors and can be synthetically summarized in the following.

Constant contact resistances modify the dependence of

current on the gate voltage. In the linear regime of transistor

operation, the method requires the transfer characteristic

curve and its first derivative as inputs; in the saturation re-

gime also the second derivative of current with respect to

gate voltage is needed. The method returns the mobility, the

threshold voltage and the value of contact resistances. The

method also gives important criteria for the validation of the

extracted parameters to assess whether the conditions for its

application are fulfilled. With respect to the fitting procedure,

the method has three advantages: i it can be applied even in

the absence of an explicit expression for current, as in thesaturation regime, ii it does not require any parameter ini-tialization, and iii it is applied pointwise. This latter meansthat data ranges where the underlying device model is not

applicable to the experimental measurements can be easily

identified, and they do not compromise the parameter extrac-

tion in the remaining data ranges.

In the case of contact resistances with an arbitrary de-

pendence on the gate voltage, we propose a method which

applies only in the linear regime and combines a differential

approach together with a scaling law approach: the transfer

characteristic curves, together with their first derivatives, of

two nominally identical TFTs with different channel lengths

are needed as inputs of the method.Finally the special case of contact resistances with a

power law dependence on the gate voltage has been consid-

ered. Interestingly enough, the dependence of current on the

gate voltage is not altered by contact resistances, since the

nonlinearities of the TFT and of the contact combine to give

a linear partition effect. Therefore this case would be indis-

tinguishable from a transistor with Ohmic contacts, except

for the dependence of current on the channel length. Conse-

quently, a scaling law test should always be performed to

discriminate between the two cases and to assess the correct

TFT parameters.

FIG. 14. Example of extraction of mobility with the scaling law method of

Sec. V A 4 for a TFT held at Tsub = 190 C during vacuum deposition of the

active material see Sec. V A 5. For each channel length nominal values:L =3, 6, and 12 m, values obtained by means of optical microscopy L = 2,

5, and 11.5 m, maximum down triangles, minimum down triangles,and mean measured apparent mobilities squares are reported. The dashedline is a linear interpolation of the mean apparent mobilities: from its y axis

intersection 0 has been obtained; from its slope has been extracted.

FIG. 13. Apparent mobility app extracted according to Eq. 34 in thesaturation regime from the set of TFTs of Sec. V A 5 hollow stars L= 3 m, hollow circles L = 6 m, and hollow squares L = 12 m. Dashedlines are only a guide to the eye. Also plotted is the mobility 0, solidtriangles extracted according to the scaling law method of Sec. V A 4.

FIG. 15. Gate-voltage-dependent contact resistances RS extracted from the

set of TFTs of Sec. V A 5 according to the scaling law method of Sec.

V A 4. Data refers to the TFTs with L = 6 m.

FIG. 16. Parameter of the contact resistances extracted from the set of

TFTs of Sec. V A 5 according to the scaling law method of Sec. V A 4.




12/12

Combining the methods above, also the general case of

contact resistances given by the sum of a constant term and

of a gate-voltage-dependent term can be treated: the former

is extracted by means of the differential method, the latter

exploiting the dependence of the current on the channel

length. With respect to the pure scaling law approach, our

method proves to be more robust, thanks to the reduced num-

ber of parameters required to be identical among transistors

with different channel lengthsThe given examples have shown that the effect of con-

tact resistances on real organic TFT can be significant and

that their correct extraction can be managed with the help of

the methods presented in the paper.

ACKNOWLEDGMENTS

The authors are grateful to A. Bolognesi and P. Di Gian-

vincenzo ISMAC-CNR, Milano, Italy for poly-3-hexylthiophene purification and deposition and to S. Masci

for careful bonding of devices. Data on pentacene TFTs are

courtesy of Dr. Olaf R. Hild, Fraunhofer Institut fr Photo-

nische Mikrosysteme IPMS, Dresden Germany. The fi-nancial support of Project Teseo Fondazione Cariplo and ofProject MIUR-FIRB RBNE033KMA are also gratefully ac-

knowledged.

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the following expression: =0VG VT /1 V, where the term VG

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Modeling of Organic Thin Film Transistors Effect of Contact Resistance

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