Organic Semiconductor Growth and Transistor Performance as a Function of the Density of the Octadecylsilane Dielectric Modification Layer

Abstract

The central research focus in organic electronics has been improvement of charge transport in organic thin film transistors, the building blocks of organic circuits.

Appendix 2.A: Growth of a Kossel Crystal

Due to its significance on the mobility and conductivity of pentacene thin films, this appendix will introduce general concepts related to crystallization from the vapor phase, and the heterogeneous nucleation of 2D versus 3D crystals. A Kossel crystal is one where all the atoms/molecules are assumed to be cubic in geometry [18]. This is the simplest kind of crystal; more complex geometries often lead to equations which are analytically impossible to solve. Comparing with nucleation of a liquid droplet from a supersaturated vapor, the crystallization of solid crystals is more complex due to the various surfaces with their (often) distinct surface energies [18]. Consider a homogeneous (i.e., not on a substrate or surface) 3D Kossel crystal in equilibrium with the vapor phase (constant temperature and constant volume) then the change in Helmholtz free energy (dF) is zero and can be expressed:

$$ dF = - P_{v} dV_{v} - P_{c} dV_{c} + \sum_{n} \sigma_{n} dA_{n} = 0 $$

(2A.1)

$$ - (P_{c} - P_{v} )dV_{c} + \sum_{n} \sigma_{n} dA_{n} = 0 $$

(2A.2)

where P _v is the pressure in the vapor phase, P _c is the vapor pressure of the crystal, V _v and V _c are the vapor and crystal volumes, σ _n is the surface energy of surface n with corresponding area A _n . Equation 2A.2 is the simplified form of Eq. 2A.1 since at equilibrium the total volume is constant (i.e. (dV _v  = −dV _c )). The volume of a crystal can also be expressed as a sum of volumes of pyramids with heights h _n and areas A _n , as suggested by Wulff in 1901 [18, 38]. V _c and dV _c can then be expressed:

$$ V_{c} = \frac{1}{3}\sum_{n} h_{n} A_{n} $$

(2A.3)

$$ dV_{c} = \frac{1}{2}\sum_{n} A_{n} dh_{n} $$

(2A.4)

To second order, the very small changes to the total volume dV _c can be accounted for by assuming constant area with infinitesimal changes in pyramid height dh _n (see Markov, Ref. [18] for more details). Reinserting into Eq. 2A.2

$$ \sum_{n} \left[ {\sigma_{n} - \frac{1}{2}(P_{c} - P_{v} )h_{n} } \right]dA_{n} = 0 $$

(2A.5)

Since each of the changes in area (dA _n ) are not related, the first term in the bracket must equal zero

$$ P_{c} - P_{v} = 2\frac{{\sigma_{n} }}{{h_{n} }} = {\text{constant}} $$

(2A.6)

This is a restatement of Wulff’s rule which states: “at equilibrium, the distances of the crystal faces from a point within a crystal (called Wulff’s point which can arbitrarily be chosen as the center of the crystal) are proportional to their corresponding specific surface energies of these faces” [18, 38]. This concept is extremely important in determining whether 2D or 3D crystal growth dominates. Since the chemical potential difference is directly related to the difference in pressures of the two phases by the molar volume of the crystal phase (V _c ), Eq. 2A.6 can also be written in a more convenient form:

$$ \Updelta u = u_{v} - u_{v} = 2\frac{{\sigma_{n} v_{c} }}{{h_{n} }} $$

(2A.7)

This is an important result which mathematically expresses the physical concept that that the supersaturation is the same over the crystal surface, and the growth mode (values of h _n ) is directly related to the supersaturation. Again, the discussion above was given for a homogenous crystal. For heterogeneous nucleation, which is relevant for organic semiconductor nucleation in OTFTs, Eq. 2A.7 must be slightly modified to include the interaction or adhesion energy (σ _i ) between the crystal and the substrate upon which it is nucleating:

$$ \frac{\Updelta u}{{2v_{c} }} = \frac{{\sigma_{0} - \sigma_{1} }}{{h_{n} }} = {\text{constant}} $$

(2A.8)

where σ _o refers to the homogenous case (surface energy); when σ _i is zero then the homogenous Eq. 2A.7 is retained. For values where h _n  > 1, 3D crystals will form, whereas for h _n  = 1, desirable 2D nucleation prevails. Thus, the term σ _i, which relates the strength of interaction between the semiconductor and the substrate is a key parameter in determining whether 2D or 3D growth will prevail [18]. The chemical potential driving force and the interfacial energies will determine the growth mode. Define the total change in surface energy upon nucleation on a foreign substrate by ∆σ where:

$$ \Updelta \sigma = \sigma + \sigma_{i} - \sigma $$

(2A.9)

σ is the surface energy of the crystal, σ _i is the interfacial surface energy (whose magnitude can be either positive or negative) and σ _s is the surface energy of the substrate [18]. There are three basic cases (Fig. 2.9).

Fig. 2.9

The relevant surface/interfacial energies used to determine the equilibrium shape of a crystal

Case 1: ∆σ < 0, this case results when the interaction with the surface is greater than the interlayer interaction energies. Of course in this case, 3D nucleation is prohibited and 2D nucleation can occur at ∆μ = 0, and even at undersaturation ∆μ < 0 (provided that |∆μ| < |A_m∆σ| where A _m is molecular area).

Case 2: ∆σ = 0 indicates a balanced force between interlayer interaction energy and molecule substrate interactions. This is the general case for nucleation for a material on it crystal of itself (homogenous nucleation). Again in this case, 3D nucleation is thermodynamically impossible, and 2D wetting occurs for supersaturated systems ∆μ > 0.

Case 3: ∆σ > 0, or when the system’s surface energy increases can give rise to both 2D and 3D growth depending on ∆μ. This is the general case which was discussed in Chap. 2. The barrier for 3D nucleation, $\Updelta G_{3D}^{ * }$, is inversely related to (∆μ)² (Eq. 2.3) and is possible for all values of ∆μ > 0. Again 2D nucleation becomes possible only at supersaturations greater than ∆μ ₂ , where the change in surface free energy upon nucleation is ∆μ ₂  = A _m ∆σ. This is a logical conclusion, since there must be a driving force greater than the gain in surface energy for nucleation, for the total free energy of the system to decrease. As ∆μ increases beyond ∆μ₂, there exists a critical supersaturation ∆μ _cr (where ∆μ _cr  = 2∆μ ₂ at which the $\Updelta G_{3D}^{ * } = \Updelta G_{2D}^{ * }$) or consequently the height the 3D island is one monolayer high (i.e. a 2D crystal). The extension of Wulff’s rule shows that under equilibrium a Kossel crystal will try to maintain its height/length ratio [18, 38] (Fig. 2.10).

Fig. 2.10

A schematic showing how past the critical supersaturation a 3D Kossel crystal must become 2D in order to maintain equilibrium shape from Ref. [18]

In the analysis presented in this chapter on pentacene growth, the chemical potential driving force was fixed, and thus the energetics which determined growth mode are related to the interfacial energies. This allowed for estimation of the interaction energy between pentacene and the different OTS layers. In the following chapter it was determined that on crystalline OTS the pentacene molecule substrate interaction energy is greater than the interlayer interaction energy and in fact this would fall under case 1 (∆σ < 0) presented above.

The important caveat which must be mentioned is that for systems far from equilibrium (high supersaturations) cannot be addressed using methodology discussed in this chapter, which use thermodynamic models for treating nucleation and crystal shape.