Journal of the American Chemical Society pyrimidine dipoles. This is best seen when averaged over the length scale of interatomic distances (red curve).94,95 The collectively generated potential-energy gradient exists only with- in the SAM and can be associated with an eﬀective internal ﬁeld. The magnitude of that “collectively-induced” ﬁeld can be esti- mated to be 1.6 eV/nm from the overall potential energy increase and the thickness of the monolayer. Such simple model con- siderations have been found to be of great explanatory power in previous studies of inorganic systems,96 98 and in the following sections we describe how they are applicable also to oligomeric SAMs made of molecules that consist of polar repeat units. In passing, we mention that one should be able to observe this potential gradient, for example, by high-resolution X-ray photo- electron spectroscopy experiments,99,100 as it results in the core electrons of carbon and nitrogen atoms in the backbone being located at diﬀerent relative electrostatic energies. Also in oligopyrimidine SAMs, the overall magnitude of ΔEvac is reduced by the above-mentioned depolarization eﬀects, as can be inferred from its sublinear increase with packing density shown by the green diamonds in Figure 3b. A comparison between the trend for 3N (green diamonds) and 3Nup (black squares) shows that the thiol docking group does not qualitatively inﬂuence the overall electrostatic situation. In 3Ndown, the pyrimidines are oriented diﬀerently (see Figure 1a), which reverses the sign of ΔEvac, keeping its magnitude, however, largely unchanged (orange disks; note the reversed scale). The overall situation of the free-standing SAM in Figure 3a can be viewed in analogy to the well-understood case of an (inorganic) semiconductor quantum-well under the inﬂuence of an electric ﬁeld. In semiconductor quantum-wells the ﬁeld is typically externally applied, although the role of internal ﬁelds is well acknowledged (for instance, see ref 101). Here we are dealing exclusively with the latter: a packing-density dependent collectively induced ﬁeld that originates from the distributed molecular dipoles within the system. In both, semiconductor quantum-wells and oligopyrimidine SAMs, one has to deal with inﬁnitely extended periodic systems in the x- and y-direction, while there are a ﬁnite number of repeat units in z-direction. As a consequence, the wave functions of the quantum-well retain their Bloch-type character in x- and y-direction (i.e., they can be described as lattice-periodically modulated plane waves), while in z-direction the so-called envelope-function approximation comes into play.102 There, the plane-wave parts of the Bloch- type states, which require inﬁnite periodicity, are replaced by the eigenfunctions of the quantum-well (reﬂecting the well proper- ties); the lattice-periodic parts of the wave functions still deter- mine which band the states are associated with. Describing the orbitals of oligomers (like terpyrimidine) in such a quasiband structure picture is a well established concept103 which is consistent with H€uckel theory104 and has been conﬁrmed experimentally by inelastic electron scattering105 107 and photo- electron spectroscopy.108 111 The envelope functions play a deﬁning role for the properties of the orbitals and are key to understanding the electronic structure of the SAMs discussed here. When, for the sake of simplicity, inﬁnitely high barriers are assumed, the well-known eigenfunctions of such a quantum-well in the presence of a constant potential gradient (i.e., a homogeneous electric ﬁeld) are linear combinations of Airy functions.112 The few lowest are depicted in the left panel of Figure 3c. In contrast to the ﬁeld-free situation (middle panel of Figure 3c), they are polarized by the electric ﬁeld. The lowest state shows pronounced localization on ARTICLE Figure 4. Density of states (DOS) for the 3N (upper panels) and 3P (lower panels) molecules (left panels) and high-coverage SAMs (θ = 1/2, right panels), aligned at the average electrostatic energy across the SAM, at E = (Elveafct + Erviagcht)/2 (cf. Figure 3a). The thick black curves are Gaussian-broadened (σ = 0.1 eV) convolutions of the results of the calculation. The insets show orbital/band charge densities of the frontier states, and the band gap is indicated. It is determined from the onsets of the respective nonbroadened DOS peaks. the low-energy (left) side of the well and for moderate ﬁelds the probability density for higher states increases in the high-energy (right) side of the potential well.112 114 This phenomenon of qualitatively diﬀerent localization for diﬀerent states has been denoted as anomalous polarization in the literature,113,114 and is relevant for the discussion below. For describing both the valence and the conduction band, one has to keep in mind that holes carry a positive charge, which reverses the corresponding potential well leading to the situation depicted in the right panel of Figure 3c. The latter is characteristic of the quantum-conﬁned Stark eﬀect (QCSE). It is exploited, for example, in absorptive modulators, where the conﬁnement imposed by the well-structure is necessary to prevent dissociation of the excitons.115,102 The simple picture in Figure 3c implies that (i) the gap between the occupied and unoccupied states should decrease upon increasing the ﬁeld strength (in the oligopyrimi- dines this means increasing the packing density); (ii) electrons and holes are localized at opposite edges of the quantum structure; (iii) the degree of this localization should increase with the ﬁeld. Collectively Induced QCSE: Band Gap and Localization. We start our discussion with terpyrimidine, where no thiol group is attached (3N in Figure 1a). Comparing the DOS of this molecule with that of the corresponding SAM (θ = 1/2) in the upper panel of Figure 4, it is found that the HOMO LUMO gap is reduced by no less than 0.8 eV. In contrast, the gap of the nonpolar terphenyl (3P in Figure 1a) remains essentially un- changed upon monolayer formation (Figure 4, lower panel). Consequently, it is the continuous electric field induced within the ensemble of 3N molecules which is responsible for the observed reduction of the band gap in the spirit of the above- described QCSE. This is consistent with the results of others for molecules45 50,52 and SAMs45,46,53 in electric fields. It also 18638 dx.doi.org/10.1021/ja203579c |J. Am. Chem. Soc. 2011, 133, 18634–18645