Universal diffusion-limited injection and the hook effect in organic thin-film transistors

The general form of interfacial contact resistance was derived for organic thin-film transistors (OTFTs) covering various injection mechanisms. Devices with a broad range of materials for contacts, semiconductors, and dielectrics were investigated and the charge injections in staggered OTFTs was found to universally follow the proposed form in the diffusion-limited case, which is signified by the mobility-dependent injection at the metal-semiconductor interfaces. Hence, real ohmic contact can hardly ever be achieved in OTFTs with low carrier concentrations and mobility, and the injection mechanisms include thermionic emission, diffusion, and surface recombination. The non-ohmic injection in OTFTs is manifested by the generally observed hook shape of the output conductance as a function of the drain field. The combined theoretical and experimental results show that interfacial contact resistance generally decreases with carrier mobility, and the injection current is probably determined by the surface recombination rate, which can be promoted by bulk-doping, contact modifications with charge injection layers and dopant layers, and dielectric engineering with high-k dielectric materials.

2 Part 1. Derivation of Table 1 (functions for extracting contact resistance from   output characteristics) Basic assumption: mobility is weakly dependent on Vd in the studied region (the mobility extracted here is an average of all Vd region). In the following a parameter is used, L = ox .
I. R-function (for simple examination of RC) When Vd is small and c ≫ sh , c~tot (s1-2) However, measured data can be noisy and this fitting does not often show clear information.

II. G-function (for general cases to extract RC)
II-1. The total output conductance is defined by, intercept( , Vd) = ox ( g − th ) = L ( g − th ) (s1-7) Using this equation can directly derive the mobility at certain Vg from one output curve, and it is free from contact resistance.
II-4. Then, use this mobility to extract contact resistance: III. Gdif-function (for the cases that G is not sensitive to Vd) III-1. The differential output conductance is defined by, III-3. Therefore, when Vd is large, Gdif-Vd is linear, III-4. So the contact resistance can be extracted as, IV. G*-function (for the cases that RC is almost constant with Vd) IV-1. The differential output conductance is expressed by s12-s14.

IV-2. As
IV-3. Therefore, when Vd is large, G*-Vd is linear, IV-4. So Rc can be extracted as Part 2. Simulation methods and parameters ( Figure 2 in the main texts) The Id-Vd characteristics in the linear regime were simulated by the following set of equations: The expressions of Rc,int have the following models, These forms are to consider the typical cases given by Eq. 9 in the main text. The parameters for simulations are list in Table S1 (p-type device). Table S1. Parameters for simulation shown in Figure 2.
In the simulation, Vd ranges from 0 V to -60 V with a step of -1 V (

Part 4 Rc extracted by Table 1 is the interfacial resistance
Usually Rc is a combination of the interfacial injection resistance (Rc,int) and the bulk injection resistance (Rc,bulk) modulated by the gate field, [1] in which case we can still use the G-function method. We express Rc,bulk in the sum of an extended channel term (RshΔL/W) and a remainder term ΔR(Vd, Vg), as found in many inorganic and organic TFTs [2] [3] . This is shown as:

Part 5. Derivation of general surface recombination rate
The surface recombination rate rrec can be calculated as follows.
(1) Using Langevin's recombination model, carriers recombine with the mirror charges when thermal energy kT reaches the carrier-image binding energy at a distance of xC=rC/4, where rC is the Coulomb radius rC= q 2 /4πεε0kT. So by using l= xC and D=(kT/q)μint (μint is the carrier mobility near the interface), we can calculate the surface recombination velocity (unit in cm s -1 ) as, This form is exactly the same with the one derived by Scott et al [4] . In addition, the mean free time τ during which the carriers travel a distance of l before recombination Here P is the probable recombination events that one carrier can encounter in a unit time (i.e. in one second).
(2) Then the surface recombination rate rrec(Vd) at the contact interface of OTFT, defined as the number of recombination between carriers and the image-charges per unit time (in one second) and per unit area (in one cm 2 ), is the product of the surface charge density n and the surface recombination velocity S . The voltage across the interface a and local electric field at the interface and are affected by Vd. Assume the former is a = 1 ( d ) and the latter is = 2 ( d ) , and charge mobility near the interface follows the field-effect mobility int = 3 . It is expressed by (unit in cm -2 s -1 ), [4] Here is a weak function of [4]. So at zero field (Vd=0 and = 1), rrec,0 is, And so Apparently rrec,0 is proportional to carrier mobility near the interface μint and the number of chargeable sites N0, but decays fast as function of the Schottky barrier height φB. The parameter rrec,0 is independent of Vd and well characterize the interfacial injection conditions with surface recombination process. For calculations in Figure 8, the used parameters are in Table S2. The Figure 8e is drawn by plotting rrec against Rc,int (by Eq. S6-2) for different Vd and φeff. (3) Note that rrec here is an interface property and is different from the original "Langevin's rate of recombination" in the bi-polar bulk OSC, which is defined by the total number of electron-hole recombination events per unit time (in one second) and per unit volume (in one cm 3 ) and is also proportional to mobility.

Part 6. Derivation of Rc,int and Equation 16 in the main text
The voltage across the interface a and local electric field at the interface and are affected by Vd. Assume the former is a = 1 ( d ) and the latter is = 2 ( d ) , and charge mobility near the interface follows the field-effect mobility int = 3 . Yet