Critical Evaluation of Organic Thin-Film Transistor Models
Markus Krammer, James W. Borchert, Andreas Petritz, Esther Karner-Petritz, Gerburg Schider, Barbara Stadlober, Hagen Klauk, Karin Zojer
AArticle
Critical Evaluation of Organic Thin-Film TransistorModels
Markus Krammer , James W. Borchert , Andreas Petritz , Esther Karner-Petritz , GerburgSchider , Barbara Stadlober , Hagen Klauk and Karin Zojer * Institute of Solid State Physics, NAWI Graz, Graz University of Technology, Petersgasse 16, 8010 Graz,Austria; [email protected] Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany;[email protected] Joanneum Research Materials, Institute for Surface Technologies and Photonics, Franz-Pichler-Straße 30,8160 Weiz, Austria; [email protected] * Correspondence: [email protected]; Tel.: +43-316-873-8974Academic Editor: nameReceived: date; Accepted: date; Published: date
Abstract:
Thin-film transistors (TFTs) represent a wide-spread tool to determine the charge-carriermobility of materials. Mobilities and further transistor parameters like contact resistances arecommonly extracted from the electrical characteristics. However, the trust in such extractedparameters is limited, because their values depend on the extraction technique and on the underlyingtransistor model. We propose a technique to establish whether a chosen model is adequate torepresent the transistor operation. This two-step technique analyzes the electrical measurementsof a series of TFTs with different channel lengths. The first step extracts the parameters for eachindividual transistor by fitting the full output and transfer characteristics to the transistor model. Thesecond step checks whether the channel-length dependence of the extracted parameters is consistentwith the model. We demonstrate the merit of the technique for distinct sets of organic TFTs thatdiffer in the semiconductor, the contacts, and the geometry. Independent of the transistor set, ourtechnique consistently reveals that state-of-the-art transistor models fail to reproduce the correctchannel-length dependence. Our technique suggests that contemporary transistor models requireimprovements in terms of charge-carrier-density dependence of the mobility and/or the considerationof uncompensated charges in the transistor channel.
Keywords: organic thin-film transistor; transistor model evaluation; channel-length dependence;contact resistances; modeling contact effects; equivalent circuit; charge-carrier-mobility extraction
1. Introduction
The fabrication of organic thin-film transistors (TFTs) has reached a level at which devices withexcellent performance, small device-to-device variations, and smooth electrical characteristics withlow hysteresis are routinely available.[1–4] These technological advances are significantly ahead of ourcurrent ability to reliably extract crucial transistor parameters, be that to design circuits, to determinematerial parameters, or to further optimize a device. The most prominent of these transistor parametersare the charge-carrier mobility as a material parameter and the contact resistance as an indicator forthe quality of the metal-semiconductor interfaces. To be able to extract such parameters from theelectrical device characteristics, the transistor operation and, hence, its electric characteristics must beunderstood in terms of these parameters.In general, parameter extraction requires a theoretical model for the transistor operation thatprovides the current-voltage relations on the basis of input parameters that account for the point ofoperation (applied voltages), material properties and the device geometry. While material-relatedtransistor parameters comprise, for example, the charge-carrier mobility and the gate-insulatorpermittivity, the most prominent geometry parameters are the length L and the width W of the a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec of 16 transistor channel and the gate-insulator thickness. Such theoretical models hold the promise of beingable to associate any changes in the current-voltage relation to changes in these parameters. Hence, it isparticularly desirable to utilize a theoretical model that associates the drain current to these transistorparameters, preferably with a closed analytic expression. To obtain reliable and robust associations,it is customary to conceive specific models for each class of TFTs by accounting, for example, for aparticular transport mechanism[5,6] or for particular geometry features, such as short channels.[7] Thepotential success of a theoretical model inherently relies on preliminary assumptions that are guidedby the device geometry and the anticipated transport mechanism. For instance, in the presumablymost prominent model, the gradual channel approximation, it is assumed that all mobile charges areconfined to the interface between the semiconductor and the gate insulator. Despite many effortsto improve the transistor models to better comply with the measured electrical characteristics,[8,9]the development of refined models is hampered as there is no reliable tool to check the consistencybetween the prediction made by a given theoretical model and the experimentally measured electricalcharacteristics.Here we propose a technique to scrutinize the adequateness of the underlying theoretical model.The technique consists of a two-step process that requires a set of TFTs with different channel lengths.The two steps combine the benefits and overcome the drawbacks of the two classes of establishedextraction approaches, namely ’single transistor methods’ and ’channel-length-scaling approaches’.[10]’Single transistor methods’ seek to extract the parameters of an assumed transistor model fromcertain voltage regions in the output or/and transfer characteristics of an individual TFT,[9–13]whereas in ’channel-length-scaling approaches’ parameters are extracted from a series of nominallyequivalent TFTs, that differ only in the channel length, by exploring the scaling of the transistorperformance with the channel length from the perspective of the assumed model.[14–16] Neitherof these two approaches is able to provide a reliable check of the consistency between theoreticalmodel and measured current-voltage characteristics. For ’single transistor methods’ the consistencycan, at most, be checked within the limited region from which the parameters are extracted, and for’channel-length-scaling approaches’, the deviations of model predictions from the measured data isoften hidden by device-to-device variations.The technique we present here combines main aspects of the two classes of extraction methods.This combination allows us to go beyond extraction methods and enables a reliable check of theadequateness of the underlying theoretical model. Our first step analyzes single transistors. Wefit the entire set of measured data points of all output and transfer characteristics at once to theassumed model. As pointed out by Deen et al. [17] and Fischer et al. ,[18] the consideration of allavailable data points guarantees the best possible parameter set describing an individual transistoras a whole and eliminates the aforementioned ambiguity that arises from selecting certain regionsof operation. The extracted parameter set is then used to calculate the corresponding output andtransfer characteristics I D ( V DS ) and I D ( V GS ) . Comparing the calculated electrical characteristics tothe measured ones allows a first check of the validity of the assumed model.[17,18] Furthermore,deviations seen in the characteristics can be analyzed to get an idea of how the model should beimproved. If this check is successful and the characteristics match well, we can proceed to the secondstep and compare the results of the individually extracted parameters of all devices. The secondstep relies on the hypothesis that transistor quantities, such as voltage drops, resistances, and chargemobilities, can be split into contributions from the channel and from the contacts. If the assumed modelcorrectly assigns the contributions to the channel and to the contacts, all channel-length dependenciesare captured explicitly in the model. In turn, all related parameters have to be independent of thechannel length. Hence, if, in a second check, the extracted parameters are found to be independentof the channel length, it can be concluded that the assumed model describes the measured devicesconsistently. The second step is of particular importance, because fitting approaches have the drawbackthat they can produce nice fits even for unreasonable models, provided that a sufficient number ofparameters are considered.[19] As we overcome the drawbacks of both extraction methods and fitting of 16 approaches, our two-step fitting approach (TSFA) is suited for checking complex models and foridentifying problems within those models.We test the merit of our TSFA and scrutinize existing organic TFT models using experimentaldata. We purposefully select five sets of organic TFTs. These sets differ in the semiconductor, thegeometry and the treatment of the semiconductor-contact interface to realize devices with nearly ideal(vanishing contact resistance) to highly non-ideal injection (large, non-linear contact resistances). Inparticular, we fabricate a set of bottom-gate, bottom-contact TFTs and bottom-gate, top-contact TFTswith dinaphtho[2,3-b:2’,3’-f]thieno[3,2-b]thiophene (DNTT) as the semiconductor and Au contacts.In the case of the bottom-gate, bottom-contact TFTs, the Au contact surfaces were functionalizedwith a layer of pentafluorobenzenethiol (PFBT) to reduce the contact resistance.[20] The remainingtransistors are bottom-gate, bottom-contact TFTs; one with pentacene as semiconductor and Aucontacts functionalized with 2-phenylpyrimidine-5-thiol and two with C60 as the semiconductor andAu contacts functionalized with 4-(2-mercaptophenyl)pyrimidine and biphenyl-4-thiol respectively.[21]The DNTT-based bottom-gate, bottom-contact TFT set resembles an ideal transistor behaviour withlow contact resistances very closely. Hence, this set will serve us as a reference and is analyzed indetail. First, we explain the application and interpretation of the most commonly used extractionmethod, the transmission line method (TLM),[14,15] in a step-by-step manner. Second, we illustrateour TSFA on the example of the model assumed in the TLM. Third, we test a more sophisticated modelwith field- and charge-carrier-density-dependent mobility. And finally, we test models with field-and charge-carrier-density-dependent mobility and non-linear contact resistances by analyzing themeasured data of the other four TFT sets.
2. Materials and Methods
This chapter discusses in detail, (i), how to numerically calculate the drain current within theequivalent cirquit model we use, (ii), how we perform the fit of the calculated drain current to themeasured data and, (iii), which transistor technologies we investigated with our TSFA.
The equivalent circuit model employed here is shown in Figure 1. This model containsan ideal transistor in the gradual channel approximation [22] with a field- [7,23] andcharge-carrier-density-dependent mobility [24,25] between the ideal source S’ , drain D’ and gate G’ terminals. At the gate, the threshold voltage V T is considered as an external bias, and source anddrain are connected to ohmic contact resistances R S ,0 and R D ,0 . The experimentally accessible contactsare labeled source S , drain D and gate G . V DSD I D R D,0D ′ G ′ V TG V GS S ′ R S,0S
Figure 1.
Equivalent circuit model with an ideal transistor in the gradual channel approximation and afield- and charge-carrier-density-dependent mobility in the channel, connected to ohmic source anddrain resistances R S ,0 and R D ,0 . The threshold voltage V T is included as an external bias. The contactsof the ideal transistor are labeled source S’ , drain D’ and gate G’ and the experimentally accessiblecontacts are labeled source S , drain D and gate G . of 16 The mobility µ at a certain position x in the channel can be written as: µ ( x ) = µ exp β s L L (cid:12)(cid:12)(cid:12)(cid:12) V D S V (cid:12)(cid:12)(cid:12)(cid:12)! (cid:18) V GS − V T − V ChS ( x ) V (cid:19) γ (1)with the channel potential with respect to the source V ChS ( x ) at this position x , the gate-sourcevoltage V G S = V GS − V T , the drain-source voltage V D S , the mobility prefactor µ , the channellength L , the exponent of the field sensitivity β , the charge-carrier density sensitivity γ , and a constantlength scale L = µ m. To conveniently address both hole and electron conduction, we introducea constant potential scaling factor V with V = V = − L and theconstant potential scale V are chosen arbitrarily and are necessary only to avoid inconsistenciesregarding the units within the corresponding power functions. The exponential term mimics asimplified Poole-Frenkel field-dependence[7,23] and the right term describes the charge-carrier-densitydependence with a power law behavior.[24,25]Incorporating the gradual channel approximation (for details see [8,22]) leads to an implicit systemof equations that determines the drain current I D for given applied gate-source and drain-sourcevoltages V GS and V DS : v G S = V (cid:16) V GS − V T − I D r S ,0 W (cid:17) v G D = V (cid:16) V GS − V T − V DS + I D r D ,0 W (cid:17) I D = V | V | WC I µ L ( γ + ) exp β r L L | v G S − v G D | ! h v γ + G S Θ ( v G S ) − v γ + G D Θ ( v G D ) i (2)The reduced voltages v G S and v G D are the voltages at the ideal gate G’ to source S’ and gate G’ to drain D’ contacts divided by V . The heaviside-function Θ ( x ) is 1 for x ≥ x < r = RW reduced by the channel width W , i.e., the source-sided r S ,0 = R S ,0 W and the drain-sided r D ,0 = R D ,0 W resistances, as well as the gate capacitance per unitarea, C I , are introduced.In summary, the drain current I D as output quantity is implicitly determined by two inputquantities V GS and V DS , six fit parameters V T , µ , r S ,0 , r D ,0 , β and γ , two constants L and V and threegeometry factors L , W and C I . The gate capacitance per unit area C I is a geometry factor, because it isapproximately calculated from the thickness and the dielectric constant of the gate oxide.The implicit system of equations (2) can be numerically solved with the bisection methodincorporating knowledge of the desired fixed point. We start by setting I ( ) D = v ( ) G S and v ( ) G D , and then substituting the latter in the right-handside of the third equation. This gives I ( ) D and defines the search interval [ I D , min , I D , max ] =[ min ( I ( ) D , I ( ) D ) , max ( I ( ) D , I ( ) D )] . Now the recurrent series starts by taking the midpoint I D , MP =( I D , min + I D , max ) /2 and plugging it into the first two equations and the right side of the third equationof (2) to get I D , calc . If I D , MP < I D , calc , the new search interval is [ I D , MP , min ( I D , max , I D , calc )] and if I D , MP > I D , calc , the new search interval is [ max ( I D , min , I D , calc ) , I D , MP ] . Calculating I D , MP and I D , calc iscontinued until the desired accuracy is reached. Fitting measured characteristics to this model is performed with a Gauß-Newton algorithmincluding the variation of Marquardt.[26] The algorithm has been modified slightly to be able to handleminimum and maximum values of parameters. In our case, µ , r S ,0 , r D ,0 and β have to be positive and γ > − of 16 The Gauß-Newton-Marquardt algorithm calculates the difference ∆ a = a − a ( ) between thecurrent model parameters a ( ) and the suggested new model parameters a by solving the linearequation system ( A + λ D ) ∆ a = b (3)with matrices A and D , the convergence parameter λ introduced by Marquardt and a vector b . Thematrix A is given by ( A ) ij = n ∑ k = σ k ∂ I D ( V ( k ) DS , V ( k ) GS ; a ( ) ) ∂ a i ∂ I D ( V ( k ) DS , V ( k ) GS ; a ( ) ) ∂ a j , (4)containing the sum over all n measured values k , the standard deviation σ k and the partial derivatives ∂ I D ( V ( k ) DS , V ( k ) GS ; a ( ) ) / ∂ a i / j of the calculated drain current I D at the measured data values V ( k ) DS and V ( k ) GS and the current model parameters a ( ) with respect to the model parameter a i and a j , respectively. Thematrix D is a diagonal matrix consisting of the diagonal elements of A , ( D ) ij = δ ij ( A ) ij with δ ij beingthe Kronecker delta returning 1 if i = j and 0 if i = j . The vector b is given by b i = n ∑ k = I ( k ) D − I D ( V ( k ) DS , V ( k ) GS ; a ( ) ) σ k ∂ I D ( V ( k ) DS , V ( k ) GS ; a ( ) ) ∂ a i (5)involving the measured drain current I ( k ) D corresponding to the measured voltages V ( k ) DS and V ( k ) GS .To consider minimum and maximum values of model parameters, the matrices A and D , thevector b and the convergence parameter λ are evaluated as in Ref. [26] and the linear equation system(3) is solved to receive ∆ a . Before going on with this calculated value for ∆ a , it is checked if any of thesuggested parameters a = a ( ) + ∆ a are out of bounds. If this is the case, the corresponding value for ∆ a j of the entry j that is allowed to stay within the boundaries is calculated (e.g., ∆ a j = a maxj − a ( ) j ifthe upper boundary is exceeded) and plugged into the linear equation system (3) by eliminating thecorresponding equation j and transferring ( A ) ij ∆ a j to the right side b i → b i − ( A ) ij ∆ a j . The new linearequation system is solved and the model parameters are checked again. This procedure is iterativelycontinued until all model parameters are in bounds. Following this, the Gauß-Newton algorithm iscontinued.To calculate the required derivatives of the model function with respect to the model parameters,a few definitions are useful: T = β r L L v γ + G S Θ ( v G S ) − v γ + G D Θ ( v G D ) ( γ + ) p | v G S − v G D | sgn ( v G S − v G D ) , (6) T G S = v γ + G S Θ ( v G S ) + T , (7) T G D = v γ + G D Θ ( v G D ) + T , (8)˜ µ = µ exp β r L L | v G S − v G D | ! , (9) D I D = + | V | C I ˜ µ L ( T G S r S ,0 + T G D r D ,0 ) (10) of 16 The sign function sgn ( x ) is -1 if x <
0, 1 if x > x =
0. With these definitions, the derivativescan be written in a compact way: ∂ I D ∂ V T = − | V | WC I ˜ µ LD I D ( T G S − T G D ) (11) ∂ I D ∂µ = I D µ D I D (12) ∂ I D ∂ r S ,0 = − | V | C I ˜ µ T G S I D LD I D (13) ∂ I D ∂ r D ,0 = − | V | C I ˜ µ T G D I D LD I D (14) ∂ I D ∂γ = − I D D I D ( γ + ) − V | V | WC I ˜ µ L ( γ + ) D I D h ln ( v G S ) v γ + G S Θ ( v G S ) − ln ( v G D ) v γ + G D Θ ( v G D ) i (15) ∂ I D ∂β = I D D I D r L L | v G S − v G D | (16)In addition to these derivatives, starting values for the fitting procedure are required. Initially, we canset all parameters to zero except the mobility prefactor µ and the threshold voltage V T . These twoparameters can be estimated from the saturation regime of the output characteristics. In this regimewith only µ and V T being non-zero, the drain current I D is calculated by I D , sat = WC I µ ( V GS − V T ) /2 L . Performing a linear fit of p I D , sat ( V GS ) provides starting values for µ and V T . With thesestarting values, the first fit is performed by optimizing only µ and V T . Starting from these optimizedparameters, more and more parameters are included in the fitting procedure. The next fit, e.g., isoptimizing µ , V T , r S ,0 and r D ,0 followed by a fit of µ , V T , r S ,0 , r D ,0 and γ and a final fit of µ , V T , r S ,0 , r D ,0 , γ and β . When changing the order of included fit parameters (e.g. β before γ ), the optimizedparameters should converge to the same solution within the chosen numerical accuracy. All TFTs were fabricated on flexible plastic substrates and share aluminum oxide as gate dielectriclayer. The TFTs investigated in particular detail are bottom-gate, bottom-contact TFTs with a 30 nmthick layer of DNTT as the semiconductor and Au contacts that are treated with PFBT to increase thework function of the contacts[27] and to improve the semiconductor morphology across the contactinterface[20]. The ultrathin 5.3 nm aluminum oxide gate dielectric layer enables operation voltagesbelow 3 V.[28] This set of TFTs was chosen because it appears to closely resemble an ideal transistor, asdemonstrated by a nearly perfect linear behavior in the linear regime of the output characteristics, lowcontact resistances, and good reproducibility. This nearly ideal behavior is maintained even for thesmallest channel length of L = µ m.The remaining sets of TFTs, that were analyzed for comparison, are a series of bottom-gate,top-contact TFTs[29] and series of bottom-gate, bottom-contact TFT with either pentacene or C60 as thesemiconductor and Au contacts decorated with biphenyl-based SAMs containing embedded dipoles(one phenyl ring exchanged by pyrimidine) to adjust the work function of the contacts.[21]
3. Results
Before our TSFA is applied, we analyze the data measured for our set of DNTT-based bottom-gate,bottom-contact TFTs with the widely used transmission line method (TLM). This analysis is performed of 16 to (i) put the measured data into a perspective commonly shared in our field of research and (ii)highlight the benefits and drawbacks of the TLM.In principle, the TLM is able to take into account non-idealities like non-ohmic contact resistances.However, when applying the most common TLM extraction procedure, the model assumptions arerather strict, as it assumes ideal transistors that satisfy the gradual channel approximation [22] andhave a constant mobility and ohmic source and drain resistances.[14,15] With these model assumptions,the drain current I D in the linear regime of the output characteristics can be written as I D = V WC I µ | V | ( L + L T ) (cid:20)(cid:16) V GS − V T − I D r S ,0 W (cid:17) − (cid:16) V GS − V T − V DS + I D r D ,0 W (cid:17) (cid:21) . (17)The transfer length L T accounts for a channel-length-independent extension of the channel in thecontact regions. In bottom-gate, top-contact TFTs, L T can be interpreted as the additional distancethat charge carriers have to travel through the semiconductor to reach the channel (see e.g. [18]). Forbottom-gate, bottom-contact TFTs, charges are injected very close to the channel and travel significantlya shorter distance through the semiconductor before reaching the channel. This implies that L T by itsown is not a physically interpretable parameter but rather has to be seen as a weighting factor for anon-ohmic contribution to the contact resistance.The parameter extraction procedure consists of three parts. In the first part, the ON-state resistance r ON is calculated from the slope of the measured output characteristics: r ON = lim V DS → W ∂ V DS ∂ I D = | V | L + L T V C I µ ( V GS − V T ) + r C ,0 (18)with r C ,0 = r S ,0 + r D ,0 . Note that it is important to extract r ON for V DS → S1 ). To determine r ON , we performed a linear fit for the four smallest measured drain-sourcevoltages and forced this fit to go through the origin V DS = I D = r ON as afunction of the channel length L for different V GS is shown in Figure 2( a ). The measured r ON behaveslinearly with respect to L and the intercept of all curves for different gate-source voltages at the bottomleft is approximately at L ≈ − µ m and r ON ≈ Ω cm.In the second part, the inverse slope ∆ L / ∆ r ON = C I µ ( V GS − V T ) V / | V | is extracted fromFigure 2( a ) and plotted versus V GS (see Figure 2( b )). The slope of this graph yields the intrinsic channelmobility µ = /Vs and the x-axis intercept gives the threshold voltage V T = − r ON ( L = ) = r Sh L T + r C ,0 is plotted as a function of the sheet resistance r Sh = | V | [ V C I µ ( V GS − V T )] − (seeFigure 2( c )). The slope from the linear fit of this data is the transfer length L T = µ m and the y-axisintercept yields the ohmic contact resistance of r C ,0 = Ω cm. of 16 −
10 0 10 20 30 40 50 60 70 800246810 V GS = -1.67 V-2.00 V-2.33 V-2.67 V-3.00 V (a) ∆L ∆r ON r ON (L = 0) channel length L / µ m O N - r e s i s t a n c e r ON / k Ω c m − − − − − − − − (b) slope = − C I µ V T gate source voltage V GS / V i n v e r s e s l o p e ∆ L ∆ r ON / ( M Ω ) − (c) slope = L T r C,0 sheet resistance r Sh / M Ω r ON ( L = ) / k Ω c m Figure 2.
Parameter extraction in the framework of the transmission line method (TLM), performedhere on bottom-gate, bottom-contact TFTs based on the small-molecule semiconductor DNTT. In ( a ),the ON-state resistance r ON = W ∂ V DS / ∂ I D for V DS → V GS .From a linear fit of these data points, the inverse slope ∆ L / ∆ r ON and the y-axis intercept r ON ( L = ) are extracted. The insert shows a magnification of the intercept of all fit lines and the extracted r ON values for the smallest channel length (symbols). In ( b ), ∆ L / ∆ r ON plotted versus V GS yields thethreshold voltage V T = µ = /Vs. In ( c ), r ON ( L = ) = r Sh L T + r C ,0 is plotted versus the sheet resistance r Sh = | V | [ V C I µ ( V GS − V T )] − to obtain thetransfer length L T = µ m and the total ohmic contact resistance r C ,0 = Ω cm. To check the reliability of the parameters extracted by the TLM, the following requirements mustbe fulfilled: • Looking at V DS → V DS .An S-shape of the curves in this region is a clear indicator for a non-ohmic contact resistance. • The measured data must be represented by the linear fits for all three cases r ON versus L , ∆ L / ∆ r ON versus V GS and r ON ( L = ) versus r Sh . • The transfer length L T and the total contact resistance r C ,0 must be equal to the intercept of the r ON ( L ) curves for different V GS .For the set of TFTs analyzed in Figure 2, all of the above requirements are indeed met. Smalldeviations of the extracted r ON values for different channel lengths from the linear fit (see Figure 2( a ))can be attributed to device-to-device variations. A closer look, however, reveals inconsistencies. Theinset in Figure 2( a ) shows a magnification of r ON versus L close to L = r ON valuesfor the smallest channel length L = µ m (crosses). As can be seen, the fit lines do not cross all in onepoint. In addition, r ON of the TFT with the smallest channel length L = µ m is always a factor ofapproximately two below the linear fit. Both inconsistencies do not prevent a further analysis, becausethe deviation of the L = µ m TFT might be due to short-channel effects, while the fact that the fit lines of 16 do not cross in one point could be a consequence of the drain-source voltage being too large to extractthe ON-resistance in a reliable manner (cf. supplementary materials Figure S1 ). These explanations donot necessarily affect the validity of the model system.As we are able to calculate characteristics for given parameters, we can compare outputcharacteristics calculated with the parameters extracted using the TLM to the measured outputcharacteristics. This comparison is shown in the first row of Figure 3 for different channel lengths. Ascan be seen, the calculated curves (black lines) deviate substantially from the measured curves (graysymbols), regardless of the channel length. These deviations indicate a problem within the TLM thatwas not spotted by the reliability check performed above. Upon closer inspection, it can be noticedthat the calculations match the experimental data better for longer channel lengths. The curves forthe devices with the largest channel length L = µ m (see Figure 3( d )) and also for the intermediatechannel lengths L = µ m (see Figure 3( c )) and L = µ m (see Figure 3( b )) show at least a reasonablygood match, whereas in Figure 3( a ) the drain current is by far too small for the device with the smallestchannel length L = µ m. For the three longer channel lengths, the slope at the beginning of thelinear regime is captured quite well, while the match becomes increasingly worse upon increase of V DS into the saturation regime. The better agreement in the linear regime is related to the fact that theparameters in the TLM are extracted from the slope at V DS → − − − (a) L = 2 µ m d r a i n c u rr e n t I D / µ A − − (b) L = 8 µ m − − − (c) L = 40 µ m − − − − (d) L = 80 µ m T L M p a r a m e t e r s − − − − − − V GS = − V − V − V − V − V − V − V (e) drain source voltage V DS / V d r a i n c u rr e n t I D / µ A − − − − − (f) drain source voltage V DS / V − − − − − − (g) drain source voltage V DS / V − − − − − − − (h) T S F A c o n s t . m o b ili t y drain source voltage V DS / V Figure 3.
Measured (gray symbols) and calculated (black lines) output characteristics fordifferent channel lengths of DNTT-based bottom-gate, bottom-contact TFTs (corresponding transfercharacteristics, see supplementary materials Figure S2 ). Note that the symbols appear as an apparentthick line due to the close spacing of the voltage points. The calculated curves in the first row wereobtained using the parameters extracted using the TLM, whereas the results in the second row werecalculated using the TSFA for the model used within the TLM. One weakness of using the TLM to extract parameters is that all parameters have to be thesame for all TFTs within the set of different channel lengths. However, those parameters can varyconsiderably even for nominally equivalent TFTs. Then, device-to-device variations would potentiallybe able to explain the deviations of the measured and calculated output characteristics. So the questionarises, whether the deviations can be attributed to the extraction method (TLM) or to the underlyingtransistor model. To answer this question, we analyzed the measured TFT data with our TSFA. Weextract an effective mobility µ e f f , threshold voltage V T , source resistance r S ,0 and drain resistance r D ,0 for each TFT individually. The calculated output characteristics of these fits can be seen in the secondrow in Figure 3. For all channel lengths, the calculated curves have notably improved compared to theones referring to the TLM. As only minor deviations can be spotted, the important information takenfrom those curves is that the first step of our TSFA is conditionally passed. The details of the deviations between the measured and calculated output characteristics are discussed after the completion of thesecond step below.For the second step, we have to plot the extracted parameters versus the channel length, as shownin Figure 4. To be consistent with the model assumptions, these parameters need to be independentof L . In Figure 4( a ), the threshold voltage V T exhibits a minor dependence on the channel length L with an increase of about 100 mV for the smallest channel length. In Figure 4( b ), a clear L dependenceof the effective mobility µ e f f (symbols) can be seen. If we strictly stick to the model underlying theTLM, we could surmise that this dependence could be related to the transfer length L T . To checkwhether the introduction of a transfer length conceptually lifts the L dependence, we can incorporate L T into the second step by replacing µ e f f by µ intr LL + L T . Then, the value of the intrinsic mobility µ intr should be constant.[30] A fit of µ e f f = µ intr LL + L T is shown as a solid line in Figure 4( b ). The shape ofthis fit does not represent the extracted parameter µ e f f well because it systematically overestimatesthe extracted parameters for intermediate channel lengths and underestimates them for high channellengths. This poor match of the shapes indicates a problem with the model system. The right panelFigure 4( c ) displays the combined contact resistance r C ,0 = r S ,0 + r D ,0 . Rather than being independentof the channel length, the contact resistance r C ,0 grows by more than a factor of three with increasing L .This is a clear indicator for an inadequate transistor model. − − − − − (a) channel length L / µ m t h r e s h o l d v o l t a g e V T / V (b) channel length L / µ m m o b ili t y µ e ff / c m V s (c) channel length L / µ m c o n t a c t r e s i s t a n c e r C , / k Ω c m Figure 4.
Channel-length dependence of the parameters extracted with the TSFA for the model usedwithin the TLM. The variation of the threshold voltage V T in ( a ) shows only minor L dependence.For the mobility in ( b ), the appearing L dependence (crosses) can not be consistently described by atransfer length L T with the corresponding fit µ e f f = µ intr LL + L T (solid line) and for the contact resistance r C ,0 = r S ,0 + r D ,0 in ( c ), the distinct linear increase with L can not be explained at all. As a consequence,the model does not pass the second step. To find the reason for the failure of the model, a closer look at the deviations of the calculatedoutput characteristics from the measured ones can give an idea (see second row in Figure 3). Thedeviations occur as two distinct symptoms. First, the shape of the calculated curves at the transitionbetween linear and saturation regime does not really fit to the measured data and second, the measureddata shows a linear trend in the saturation regime which is not captured by the calculated curves.The first symptom appears regardless of the channel length and can be diminished by assuming acharge-carrier-density-dependent mobility of the form µ = µ ( V G − V Ch ) γ as suggested by percolationtheory [24] or multiple trapping and release [25]. The second symptom is more pronounced for shorterchannels indicating a field-dependence of the mobility. As a first attempt, we assume a simplifiedPoole-Frenkel behavior of the form exp ( β √ V DS / L ) .[7,23] Incorporating a field- and charge-carrier-density-dependent mobility in the model leads to aclear improvement of the deviations between the measured and calculated output characteristics (seeFigure 5( a ) to ( d )). Especially the TFT with the smallest channel length shows a much better agreementdue to the improved description of the saturation regime with the Poole-Frenkel behavior. For allchannel lengths, the curves of the more positive gate-source voltages V GS > − We again move on to examine the L dependence of the extracted parameters. The most relevantparameters are the mobility prefactor µ and the combined contact resistance r C ,0 = r S ,0 + r D ,0 shownin Figure 5( e ) and ( f ). The mobility prefactor µ exhibits a slightly lower L dependence comparedto the effective mobility µ e f f examined earlier (cf. Figure 4( b )). The L dependence of r C ,0 is evenmore pronounced with approximately one order of magnitude between smallest and largest channellength (see Figure 5( f )), provoking a failure of this model. To illustrate the significant influence ofthe length-dependence of the contact resistance, Figure S3 (in the supplementary material) shows thedisagreement of measured and calculated output characteristics when taking the contact resistanceof the device with the smallest channel length (shown in Figure S3 ( a ) to ( d )) and the largest channellength (shown in Figure S3 ( e ) to ( h )). The remaining parameters, V T , γ and β do not have such apronounced L dependence (not shown). − − − − − − V GS = − V − V − V − V − V − V − V (a) L = 2 µ m drain source voltage V DS / V d r a i n c u rr e n t I D / µ A − − − − − (b) L = 8 µ m drain source voltage V DS / V − − − − − − (c) L = 40 µ m drain source voltage V DS / V − − − − − − − (d) L = 80 µ m T S F A v a r . m o b ili t y drain source voltage V DS / V (e) channel length L / µ m m o b ili t y µ / c m V s (f) channel length L / µ m c o n t a c t r e s i s t a n c e r C , / k Ω c m Figure 5.
Results of the TSFA for the model with field- and charge-carrier-density-dependent mobility.In ( a ) to ( d ), output characteristics for different channel lengths indicate a good agreement of themeasured data (gray symbols) and the fit (black lines). Corresponding transfer characteristics are foundin the supplementary materials Figure S2 . In ( e ) and ( f ), the channel-length dependence of the mobilityprefactor µ and the contact resistance r C ,0 = r S ,0 + r D ,0 indicate a failure of the model to properlyrepresent the TFTs. To identify the problem of the model, we can have a look at the output characteristics of all channellengths, Figure 5( a ) to Figure 5( d ). In the saturation regime, the calculated curve for V GS = − V GS = − V GS .The spacing of the curves in the saturation regime is not only determined by thecharge-carrier-density dependence of the mobility, but also by the contact resistances (explainedin more detail in the supplementary material, Figure S4 ). Assuming a constant mobility and nocontact resistance, the saturation current I D , sat increases quadratically with the gate-source voltage, ( V GS − V T ) . On the other hand, assuming a constant mobility and a very high contact resistance, thesaturation current would increase linearly with the gate-source voltage. This means that increasingboth, mobility and contact resistance, can lead to similar I D ( V DS ) curves for the highest V GS anddifferent spacing for lower V GS (see Figure S4 ).This effect could possibly explain the increase of r C ,0 with L in the following way. If thecharge-carrier-density dependence of the mobility is captured incorrectly, the spacing of the outputcharacteristics for different gate-source voltages V GS will be wrong as well. The spacing is corrected by way of a compensating, though incorrect, change in the contact resistance. As the error of the mobilityscales with L in the calculation of the drain current because it is a channel property, and the contactresistance has no L scaling effect, the extracted value of the contact resistance is forced to scale with L to compensate the mobility.The over- and underestimation of I D for the second lowest and lowest V GS , respectively, suggeststhat the contact resistance tries to reduce the spacing for higher V GS and, hence, is too high. This changein spacing could be achieved as well if the mobility would decrease with increasing charge-carrierdensity. This decrease should only happen for high charge-carrier densities, because for lowcharge-carrier densities, related to low V GS , the increasing mobility of the improved TFT modeldescribes the measured curves much better than the constant mobility model. So the evaluation of ourTSFA suggests that the mobility should first increase and later decrease with increasing charge-carrierdensity. Experimental hints indicating such a behavior of the mobility were recently found by Bittle et al. [31] and Uemura et al. [32]; Fishchuk et al. [33] suggested such a behavior from a theoretical pointof view.Besides improving the mobility, a potential alternative problem in the transistor model is thatthe gradual channel approximation disregards the fact that organic semiconductors are in principleinsulators. As a consequence, all mobile charge carriers have to be brought externally into thechannel. This charge accumulation is not compensated by charges of opposite polarity, in contrast toconventional semiconductors. This uncompensated charge accumulation affects the electric field atthe contact with increasing impact for increasing channel length. Including this charge cloud in thetransistor model might also be able to diminish the L dependence of the contact resistance. We note that the failure of the transistor model illustrated above is not a peculiarity of thechosen experimental TFT technology. Neither changing the geometry, nor the organic semiconductor,helps to improve the applicability of this transistor model. To confirm this claim, four moretransistor technologies are investigated. These other technologies include a similar TFT set as above,only the Au contacts were changed from bottom-contact to top-contact while the thickness of theDNTT layer was kept at 30 nm (called DNTT - TC in the following).[34] In addition, three otherbottom-gate, bottom-contact TFT series were examined: pentacene on Au contacts coated with a SAMof 2-phenylpyrimidine-5-thiol (Pentacene - BP0-down), C60 with 4-(2-mercaptophenyl)pyrimidine(C60 - BP0-up) and C60 with biphenyl-4-thiol (C60 - BP0) (for detail, see [21]). For DNTT - TC andC60 - BP0, non-linearities in the linear regime of the output characteristics were modeled with agate-voltage-dependent Schottky diode at the source side to get a reasonable agreement of measuredand fitted characteristics (for details about the Schottky diode, see [21]).Figure 6 shows the ohmic part of the contact resistance r C ,0 as a function of the channel length L for all of the additional four TFT series. The approximately linear dependence of r C ,0 on L results in asimilar failure of the transistor model in the second step of our TSFA for each device series. (a) channel length L / µ m c o n t a c t r e s i s t a n c e r C , / k Ω c m DNTT - TC (b) channel length L / µ m Pentacene - BP0-down (c) channel length L / µ m C60 - BP0-up (d) channel length L / µ m C60 - BP0
Figure 6.
Ohmic part of the contact resistance r C ,0 versus the channel length for different deviceseries, i.e., a DNTT-based bottom-gate, top-contact TFT ( a ) and bottom-gate, bottom-contact TFTs withPentacene on Au/BP0-down, C60 on Au/BP0-up and C60 on Au/BP0 contacts, respectively ( b ) to( d ). Despite different geometries, organic semiconductors, and contact preparations, all series exhibita clear channel-length dependence of the ohmic part of the contact resistance r C ,0 . This leads to afailure of the transistor model in all instances. The substantial fluctuations of r C ,0 for low L -values forthe two C60 series (including transistors with r C ,0 = Ω cm) reflects the fact that the uncertaintiesof the ohmic contact resistance for those channel lengths is in the order of the actual value. This highuncertainty does not obscure the clear increase of r C ,0 with L .
4. Summary and Conclusions
In this paper, we propose a two-step fitting approach (TSFA) to check whether a transistor modelis capable of describing the experimental characteristic of TFT devices. Only a valid transistor model,that correctly discriminates between contact and channel properties, enables one to reliably extract,interpret, and compare contact resistances and channel mobilities of TFTs. The TSFA relies on a seriesof transistors with varying channel length and consists of two steps. First, the chosen transistor modelis fitted to all measured data points of output and transfer characteristics of each TFT separately toextract the transistor parameters of each device. Second, one checks whether the extracted parametersdepend on the channel length. The latter consistency check is successful if (i) the measured data isrepresented well by the current-voltage curves calculated with the model and the transistor parametersand (ii) the extracted parameters are independent of the channel length. Our approach offers a clearbenefit compared to currently used extraction methods, i.e., the reliability of the tested model can beeasily checked. Due to the investigation of each individual TFT as a whole, the reason for a failure ofthe transistor model can be identified from the nature of the deviations between the measured dataand the curves calculated with the extracted parameters.We line out the indicators that are available to judge consistency within the TSFA by using thetransistor model underlying the transmission line method (TLM) as an illustrative example. TFTswith particularly small contact resistances served as test set, i.e., TFTs whose operation resemble theideal transistor behavior as closely as possible. This test set readily exemplifies, that inconsistenciescannot be necessarily spotted within the parameter extraction step, but rather require a secondstep for validity checking. An analysis with the TLM of the test set gave, at the first glance, anapparently consistent picture comprising (i) a linear onset of the output characteristics for zerodrain-source voltage and (ii) a high quality of all performed linear fits to the corresponding datapoints. However, the transistor characteristics calculated with the extracted parameters failed toreproduce the measured curves. The subsequent validity check of the TSFA for the model assumed inthe TLM was not passed, because the extracted contact resistances retained a pronounced dependenceon the channel length. Such inconsistencies ought to be removed or, at least diminished, by improvedtransistors models. For example, the model underlying the TLM can be improved by accountingfor a field- and charge-carrier-density-dependent mobility.[7,23–25] Even though the TSFA attestsbetter agreement between measured and calculated characteristics, also this improved model failsthe subsequent validity check of the TSFA due to a marked remnant channel-length dependenceof the contact resistance. The failure of the advanced transistor model featuring a field- and charge-carrier-density-dependent mobility was demonstrated for a broad selection of transistors,i.e., TFTs in a top-contact architecture, with different organic semiconductors, and high injectionbarriers that resulted in profound non-linear contributions to the contact resistance.To improve the currently available transistor models, we need to face two aspects. On the onehand, the analysis of the deviations of the measured and calculated characteristics suggests thatthe charge-carrier-density dependence of the mobility is not captured correctly. Hence, a mobilitymodel that is particularly suitable for the predominantly two-dimensional charge transport throughthe channel of a thin-film transistor has to be developed. On the other hand, the gradual channelapproximation should be reconsidered by accounting for the charge accumulation in the channel,whose effect on the electric field distribution is not compensated by charges of opposite polarity withinthe organic semiconductor. Our TSFA can be used to check each stage of model improvement.
Author Contributions:
Data Analysis and Writing—Original Draft Preparation, M.K. and K.Z.; Fabrication andMeasurement of DNTT TFTs, J.B. and H.K.; Fabrication and Measurement of Pentacene and C60 TFTs, A.P., E.K-P.,G.S and B.S.; Editing J.B., H.K., M.K. and K.Z.
Funding:
This research was funded by FWF grant number I 2081-N20.
Conflicts of Interest:
The authors declare no conflict of interest.1. Guo, X.; Xu, Y.; Ogier, S.; Ng, T.N.; Caironi, M.; Perinot, A.; Li, L.; Zhao, J.; Tang, W.; Sporea, R.A.; Nejim,A.; Carrabina, J.; Cain, P.; Yan, F. Current Status and Opportunities of Organic Thin-Film TransistorTechnologies.
IEEE Transactions on Electron Devices , , 1906–1921. doi:10.1109/TED.2017.2677086.2. Paterson, A.F.; Singh, S.; Fallon, K.J.; Hodsden, T.; Han, Y.; Schroeder, B.C.; Bronstein, H.; Heeney, M.;McCulloch, I.; Anthopoulos, T.D. Recent Progress in High-Mobility Organic Transistors: A Reality Check. Advanced Materials , , 1801079. doi:10.1002/adma.201801079.3. Yamamura, A.; Watanabe, S.; Uno, M.; Mitani, M.; Mitsui, C.; Tsurumi, J.; Isahaya, N.; Kanaoka, Y.;Okamoto, T.; Takeya, J. Wafer-scale, layer-controlled organic single crystals for high-speed circuit operation. Science Advances , , eaao5758. doi:10.1126/sciadv.aao5758.4. Ogier, S.D.; Matsui, H.; Feng, L.; Simms, M.; Mashayekhi, M.; Carrabina, J.; Terés, L.; Tokito, S. Uniform,high performance, solution processed organic thin-film transistors integrated in 1 MHz frequency ringoscillators. Organic Electronics , , 40–47. doi:10.1016/j.orgel.2017.12.005.5. Pasveer, W.F.; Cottaar, J.; Tanase, C.; Coehoorn, R.; Bobbert, P.A.; Blom, P.W.M.; de Leeuw, D.M.; Michels,M.A.J. Unified Description of Charge-Carrier Mobilities in Disordered Semiconducting Polymers. PhysicalReview Letters , . doi:10.1103/PhysRevLett.94.206601.6. Li, J.; Ou-Yang, W.; Weis, M. Electric-field enhanced thermionic emission model for carrier injectionmechanism of organic field-effect transistors: understanding of contact resistance. Journal of Physics D:Applied Physics , , 035101. doi:10.1088/1361-6463/aa4e95.7. Locci, S.; Morana, M.; Orgiu, E.; Bonfiglio, A.; Lugli, P. Modeling of Short-Channel Effectsin Organic Thin-Film Transistors. IEEE Transactions on Electron Devices , , 2561–2567.doi:10.1109/TED.2008.2003022.8. Marinov, O.; Deen, M.J.; Zschieschang, U.; Klauk, H. Organic Thin-Film Transistors: Part I—Compact DCModeling. IEEE Transactions on Electron Devices , , 2952–2961. doi:10.1109/TED.2009.2033308.9. Di Pietro, R.; Venkateshvaran, D.; Klug, A.; List-Kratochvil, E.J.W.; Facchetti, A.; Sirringhaus, H.; Neher, D.Simultaneous extraction of charge density dependent mobility and variable contact resistance from thinfilm transistors. Applied Physics Letters , , 193501. doi:10.1063/1.4876057.10. Natali, D.; Caironi, M. Charge injection in solution-processed organic field-effect transistors: physics,models and characterization methods. Advanced Materials (Deerfield Beach, Fla.) , , 1357–1387.doi:10.1002/adma.201104206.11. Wang, S.D.; Yan, Y.; Tsukagoshi, K. Transition-Voltage Method for Estimating Contact Resistance in OrganicThin-Film Transistors. IEEE Electron Device Letters , , 509–511. doi:10.1109/LED.2010.2044137.
12. Takagaki, S.; Yamada, H.; Noda, K. Extraction of contact resistance and channel parameters from theelectrical characteristics of a single bottom-gate/top-contact organic transistor.
Japanese Journal of AppliedPhysics , , 03DC07. doi:10.7567/JJAP.55.03DC07.13. Torricelli, F.; Ghittorelli, M.; Colalongo, L.; Kovacs-Vajna, Z.M. Single-transistor method for the extractionof the contact and channel resistances in organic field-effect transistors. Applied Physics Letters , , 093303. doi:10.1063/1.4868042.14. Kanicki, J.; Libsch, F.R.; Griffith, J.; Polastre, R. Performance of thin hydrogenated amorphous siliconthin-film transistors. Journal of Applied Physics , , 2339–2345. doi:10.1063/1.348716.15. Luan, S.; Neudeck, G.W. An experimental study of the source/drain parasitic resistance effects inamorphous silicon thin film transistors. Journal of Applied Physics , , 766–772. doi:10.1063/1.351809.16. Natali, D.; Fumagalli, L.; Sampietro, M. Modeling of organic thin film transistors: Effect of contactresistances. Journal of Applied Physics , , 014501. doi:10.1063/1.2402349.17. Deen, M.J.; Marinov, O.; Zschieschang, U.; Klauk, H. Organic Thin-Film Transistors: Part II—ParameterExtraction. IEEE Transactions on Electron Devices , , 2962–2968. doi:10.1109/TED.2009.2033309.18. Fischer, A.; Zündorf, H.; Kaschura, F.; Widmer, J.; Leo, K.; Kraft, U.; Klauk, H. NonlinearContact Effects in Staggered Thin-Film Transistors. Physical Review Applied , , 054012.doi:10.1103/PhysRevApplied.8.054012.19. Mayer, J.; Khairy, K.; Howard, J. Drawing an elephant with four complex parameters. American Journal ofPhysics , , 648–649. doi:10.1119/1.3254017.20. Gundlach, D.J.; Royer, J.E.; Park, S.K.; Subramanian, S.; Jurchescu, O.D.; Hamadani, B.H.; Moad, A.J.;Kline, R.J.; Teague, L.C.; Kirillov, O.; Richter, C.A.; Kushmerick, J.G.; Richter, L.J.; Parkin, S.R.; Jackson,T.N.; Anthony, J.E. Contact-induced crystallinity for high-performance soluble acene-based transistors andcircuits. Nature Materials , , 216–221. doi:10.1038/nmat2122.21. Petritz, A.; Krammer, M.; Sauter, E.; Gärtner, M.; Nascimbeni, G.; Schrode, B.; Fian, A.; Gold, H.;Cojocaru, A.; Karner-Petritz, E.; Resel, R.; Terfort, A.; Zojer, E.; Zharnikov, M.; Zojer, K.; Stadlober, B.Embedded Dipole Self-Assembled Monolayers for Contact Resistance Tuning in p-Type and n-Type OrganicThin Film Transistors and Flexible Electronic Circuits. Advanced Functional Materials , , 1804462.doi:10.1002/adfm.201804462.22. Shockley, W. A Unipolar "Field-Effect" Transistor. Proceedings of the IRE , , 1365–1376.doi:10.1109/JRPROC.1952.273964.23. Hall, R.B. The Poole-Frenkel effect. Thin Solid Films , , 263–271. doi:10.1016/0040-6090(71)90018-6.24. Vissenberg, M.C.J.M.; Matters, M. Theory of the field-effect mobility in amorphous organic transistors. Physical Review B , , 12964–12967. doi:10.1103/PhysRevB.57.12964.25. Horowitz, G.; Hajlaoui, M.E.; Hajlaoui, R. Temperature and gate voltage dependence of hole mobilityin polycrystalline oligothiophene thin film transistors. Journal of Applied Physics , , 4456–4463.doi:10.1063/1.373091.26. Marquardt, D. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. Journal of the Societyfor Industrial and Applied Mathematics , , 431–441. doi:10.1137/0111030.27. Hong, J.P.; Park, A.Y.; Lee, S.; Kang, J.; Shin, N.; Yoon, D.Y. Tuning of Ag work functions by self-assembledmonolayers of aromatic thiols for an efficient hole injection for solution processed triisopropylsilylethynylpentacene organic thin film transistors. Applied Physics Letters , , 143311. doi:10.1063/1.2907691.28. Borchert, J.W.; Peng, B.; Letzkus, F.; Burghartz, J.N.; Chan, P.K.L.; Zojer, K.; Ludwigs, S.; Klauk, H.Small contact resistance and high-frequency operation of flexible, low-voltage, inverted coplanar organictransistors. submitted .29. Ante, F.; Kälblein, D.; Zaki, T.; Zschieschang, U.; Takimiya, K.; Ikeda, M.; Sekitani, T.; Someya, T.; Burghartz,J.N.; Kern, K.; Klauk, H. Contact Resistance and Megahertz Operation of Aggressively Scaled OrganicTransistors. Small , , 73–79. doi:10.1002/smll.201101677.30. Rödel, R.; Letzkus, F.; Zaki, T.; Burghartz, J.N.; Kraft, U.; Zschieschang, U.; Kern, K.; Klauk, H. Contactproperties of high-mobility, air-stable, low-voltage organic n-channel thin-film transistors based on anaphthalene tetracarboxylic diimide. Applied Physics Letters , , 233303. doi:10.1063/1.4811127.31. Bittle, E.G.; Basham, J.I.; Jackson, T.N.; Jurchescu, O.D.; Gundlach, D.J. Mobility overestimationdue to gated contacts in organic field-effect transistors. Nature Communications , , 10908.doi:10.1038/ncomms10908.
32. Uemura, T.; Rolin, C.; Ke, T.H.; Fesenko, P.; Genoe, J.; Heremans, P.; Takeya, J. On the Extraction of ChargeCarrier Mobility in High-Mobility Organic Transistors.
Advanced Materials (Deerfield Beach, Fla.) , , 151–155. doi:10.1002/adma.201503133.33. Fishchuk, I.I.; Arkhipov, V.I.; Kadashchuk, A.; Heremans, P.; Bässler, H. Analytic model of hoppingmobility at large charge carrier concentrations in disordered organic semiconductors: Polarons versus barecharge carriers. Physical Review B , , 045210. doi:10.1103/PhysRevB.76.045210.34. Zschieschang, U.; Ante, F.; Kälblein, D.; Yamamoto, T.; Takimiya, K.; Kuwabara, H.; Ikeda, M.;Sekitani, T.; Someya, T.; Nimoth, J.B.; Klauk, H. Dinaphtho[2,3-b:2’,3’-f]thieno[3,2-b]thiophene (DNTT)thin-film transistors with improved performance and stability. Organic Electronics , , 1370–1375.doi:10.1016/j.orgel.2011.04.018. upplementary Materials: Critical Evaluation ofOrganic Thin-Film Transistor Models Markus Krammer , James W. Borchert , Andreas Petritz , Esther Karner-Petritz , GerburgSchider , Barbara Stadlober , Hagen Klauk and Karin Zojer * V GS = - . V - . V - . V - . V - . V (a) channel length L / µ m O N - r e s i s t a n c e r ON / k Ω c m r S,0 = 0.14 k Ω cm, r D,0 = 0.00 k Ω cm − − − V GS = - . V - . V - . V - . V - . V (b) channel length L / µ m O N - r e s i s t a n c e r ON / k Ω c m r S,0 = 0.07 k Ω cm, r D,0 = 0.07 k Ω cm − − − V GS = - . V - . V - . V - . V - . V (c) channel length L / µ m O N - r e s i s t a n c e r ON / k Ω c m r S,0 = 0.00 k Ω cm, r D,0 = 0.14 k Ω cm − − − Figure S1.
Transmission line method (TLM)[1,2] performed on a simulated set of devices with theparameters extracted for the real set of devices with TLM (intrinsic channel mobility µ = /Vs,threshold voltage V T = − L T = µ m and combined contact resistance r C ,0 = r S ,0 + r D ,0 = Ω cm) apportioning r C ,0 in three different ways to the source resistance r S ,0 and drain resistance r D ,0 . The TLM is evaluated at a non-vanishing drain-source voltage V DS = − V DS → I D is calculated at V DS = − V GS = -1.67 V, -2.00 V, -2.33 V, -2.67 Vand -3.00 V. Those drain currents are used to get an estimate for the ON-resistance r ON = WV DS / I D .The such calculated values for r ON were used to perform a TLM. The inserts show a magnificationof the intercept of all fit lines in the region of negative channel lengths L . In ( a ), the entire contactresistance r C ,0 was assigned to the source side, resulting in an intercept that is smeared out towardsmore negative channel lengths. The extracted parameters of µ = /Vs, V T = − L T = µ m and r C ,0 = Ω cm reflect this behaviour with a too high transfer length L T and atoo low r C ,0 . The threshold voltage is shifted by V DS /2 and the mobility is nearly not affected. In ( b ), r C ,0 was equally distributed over r S ,0 and r D ,0 , leading to a precise intercept. The extracted parameters µ = /Vs, V T = − L T = µ m and r C ,0 = Ω cm perfectly match the inputparameters except the threshold voltage which is shifted by V DS /2. In ( c ), r C ,0 was entirely attributedto r D ,0 , giving rise to an intercept smeared out towards more positive channel lengths. The extractedparameters are changed in the opposite direction compared to ( a ): µ = /Vs, V T = − L T = µ m and r C ,0 = Ω cm. Only the threshold voltage is shifted in the same way as in ( a )and ( b ) by V DS /2. a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec − − − − (a) L = 2 µ m d r a i n c u rr e n t I D / µ A − − − (b) L = 8 µ m − − − (c) L = 40 µ m − − − − (d) L = 80 µ m T L M p a r a m e t e r s − − − − V DS = − V V DS = − V (e) d r a i n c u rr e n t I D / µ A − − − (f) − − − (g) − − − − (h) T S F A c o n s t . m o b ili t y − − − − − − − (i) gate source voltage V GS / V d r a i n c u rr e n t I D / µ A − − − − − − (j) gate source voltage V GS / V − − − − − − (k) gate source voltage V GS / V − − − − − − − (l) T S F A v a r . m o b ili t y gate source voltage V GS / V Figure S2.
Transfer characteristics corresponding to the output characteristics shown in Figure 3 andFigure 5 in the main manuscript. The measured values are drawn as gray symbols and the fittedcharacteristics are shown as black lines. The four different channel lengths L refer to the shortest( L = µ m), the second shortest ( L = µ m), an intermediate ( L = µ m) and the longest ( L = µ m)channel (from left to right). The topmost plots show the calculated characteristics corresponding to theparameters extracted by TLM, the middle ones to the TSFA with constant mobility and the lowermostones to the TSFA with field- and charge-carrier-density-dependent mobility. Compared to the outputcharacteristics (cf. Figure 3 and Figure 5 in the main manuscript), the deviations of the calculated curvesfrom the measured ones are more difficult to see. Only for the top left characteristics (TLM parametersfor L = µ m), a disagreement is clearly obvious. To see the differences between the different fittingparameters (corresponding to different rows), a close look at the branching point at a gate-sourcevoltage of about V GS = − − − − (a) V GS = − V − V − V − V − V − V − V L = 2 µ m d r a i n c u rr e n t I D / µ A − − − − − (b) L = 8 µ m − − − (c) L = 40 µ m − − − − − (d) L = 80 µ m − − − − − − (e) drain source voltage V DS / V d r a i n c u rr e n t I D / µ A − − − − − (f) drain source voltage V DS / V − − − − − − (g) drain source voltage V DS / V − − − − − − − (h) drain source voltage V DS / V Figure S3.
Measured output characteristics of DNTT-based bottom-gate, bottom-contact thin-filmtransistors (TFTs) with the shortest ( L = µ m), the second shortest ( L = µ m), an intermediate( L = µ m) and the longest ( L = µ m) channel length plotted as gray line and calculated outputcharacteristics as black lines. The parameters for the calculated characteristics are the ones fitted withthe field- and charge-carrier-density-dependent mobility model except the source and drain resistance r S ,0 and r D ,0 . In the first row, the contact resistances of the TFT with the smallest channel length( L = µ m) are used and in the second row, the contact resistances of the longest channel ( L = µ m)are used for all calculated characteristics. In ( a ) and ( h ), the contact resistances are the optimized ones,respectively, resulting in a match of measurement and calculation. From ( b ) to ( d ), the too low contactresistances cause an increasing overestimation of the calculated drain current I D and from ( e ) to ( g ),the drain current is underestimated due to too high contact resistances. Summarizing, the choice of thecontact resistance has an important influence on the output characteristics underlining the fact that thechannel-length dependence of the contact resistance is indeed significant. − − − − − − V GS = -1.00 V-1.33 V-1.67 V-2.00 V-2.33 V-2.67 V-3.00 V drain source voltage V DS / V d r a i n c u rr e n t I D / µ A r S,0 = 0.07 k Ω cm µ = 3.2 cm Vs r S,0 = 0.7 k Ω cm µ = 6.1 cm Vs Figure S4.
Calculated output characteristics (solid line) for a TFT with channel length L = µ massuming the parameter set extracted by TLM (intrinsic channel mobility µ = /Vs, thresholdvoltage V T = − L T = µ m and source and drain resistance r S ,0 = r D ,0 = Ω cm). To show the correlation between contact resistance and mobility, a second outputcharacteristics (dashed line) is plotted, for which only the mobility µ = /Vs and the sourceresistance r S ,0 = Ω cm are changed. Simultaneously increasing µ and r S ,0 results in a nearly perfectconformity of the output characteristics belonging to the highest gate-source voltage V GS = − V GS . For a low contact resistance, thesaturation current increases quadratically with V GS whereas for a high contact resistance, the saturationcurrent increases linearly with V GS . As a consequence, increasing µ and r S ,0 makes the spacing between I D in the saturation regime more uniform, which is able to partly compensate an incorrect mobilitymodel.1. Kanicki, J.; Libsch, F.R.; Griffith, J.; Polastre, R. Performance of thin hydrogenated amorphous siliconthin-film transistors. Journal of Applied Physics , , 2339–2345. doi:10.1063/1.348716.2. Luan, S.; Neudeck, G.W. An experimental study of the source/drain parasitic resistance effects inamorphous silicon thin film transistors. Journal of Applied Physics ,72