Periodic polymers with increasing repetition unit: Energy structure and carrier transfer

We study the energy structure and the transfer of an extra electron or hole along periodic polymers made of $N$ monomers, with a repetition unit made of $P$ monomers, using a tight-binding wire model, where a site is a monomer (e.g., in DNA, a base pair), for $P$ even, and deal with two categories of such polymers: made of the same monomer (GC…, GGCC…, etc.) and made of different monomers (GA…, GGAA…, etc.). We calculate the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) eigenspectra, density of states, and HOMO-LUMO gap and find some limiting properties these categories possess, as $P$ increases. We further examine the properties of the mean over time probability to find the carrier at each monomer. We introduce the weighted mean frequency of each monomer and the total weighted mean frequency of the whole polymer, as a measure of the overall transfer frequency content. We study the pure mean transfer rates. These rates can be increased by many orders of magnitude with appropriate sequence choice. Generally, homopolymers display the most efficient charge transfer. Finally, we compare the pure mean transfer rates with experimental transfer rates obtained by time-resolved spectroscopy.

Figure 1

Eigenspectra of I polymers. HOMO regime (left) and LUMO regime (right).

Figure 2

Eigenspectra of D polymers as well as of I1 (G…) and I1 (A…) polymers plotted together. HOMO regime (left) and LUMO regime (right).

Figure 3

Density of states of I polymers, for the HOMO (left) and the LUMO (right) regime.

Figure 4

Density of states of D polymers as well as of I1 (G…) and I1 (A…) polymers plotted together, for the HOMO (left) and the LUMO (right) regime.

Figure 5

Energy gaps. Upper panel: I2 (GC…), I4 (GGCC…), I6 (GGGCCC…), I8 (GGGGCCCC…), I10 (GGGGGCCCCC…), I20 (GGGGGGGGGGCCCCCCCCCC…) polymers, as well as I1 (G…) polymers. The horizontal green line at 3.5 eV shows the energy gap of the monomer (G-C base pair). Inset: HOMO and LUMO band upper and lower limits. Other variants of I polymers follow the same trend, e.g., the gap of I3 (GGC…) is $\approx 3.36$ eV. Lower panel: D2 (GA…), D4 (GGAA…), D6 (GGGAAA…), D8 (GGGGAAAA…), D10 (GGGGGAAAAA…), D20 (GGGGGGGGGGAAAAAAAAAA…) polymers, as well as the union of I1 (G…) and I1 (A…) polymers. The horizontal green line at 3.5 eV (purple line at 3.4 eV) shows the energy gap of the G-C (A-T) base pair. Inset: HOMO and LUMO band upper and lower limits.

Figure 6

Mean (over time) probabilities to find an extra hole in a GGGGCCCC polymer for all possible initial placements.

Figure 7

Mean (over time) probabilities to find an extra hole (left) and electron (right), initially placed at the first monomer, in a GGGCCC… polymer ($P=6$) made up of $N=P+\tau ,\tau =1,\cdots ,P$ monomers.

Figure 8

Mean (over time) probabilities to find an extra hole in a GGGGAAAA polymer for all possible initial placements.

Figure 9

Total weighted mean frequency (TWMF) as a function of the number of monomers $N$ in the polymer, having placed the carrier initially at the first monomer, for I1 (G…), I2 (GC…), I4 (GGCC…), I6 (GGGCCC…), I8 (GGGGCCCC…), I10 (GGGGGCCCCC…), and I20 (GGGGGGGGGGCCCCCCCCCC…) polymers, for the HOMO and the LUMO regimes.

Figure 10

Total weighted mean frequency (TWMF) as a function of the number of monomers $N$ in the polymer, having placed the carrier initially at the first monomer, for I1 (G…), I1 (A…), D2 (GA…), D4 (GGAA…), D6 (GGGAAA…), D8 (GGGGAAAA…), D10 (GGGGGAAAAA…), and D20 (GGGGGGGGGGAAAAAAAAAA…) polymers, for the HOMO and the LUMO regimes.

Figure 11

Pure mean transfer rate $k$ of type I1 (G…), I2 (GC…), I4 (GGCC…), I6 (GGGCCC…), I8 (GGGGCCCC…), and I10 (GGGGGCCCCC…) polymers, as a function of the number of monomers $N$ in the polymer, for $N$ equal to natural multiples of their $P$, for HOMO (upper panel) and LUMO (lower panel).

Figure 12

Pure mean transfer rate $k$ of type I1 (G…), I1 (A…), D2 (GA…), D4 (GGAA…), D6 (GGGAAA…), D8 (GGGGAAAA…), and D10 (GGGGGAAAAA…) polymers, as a function of the number of monomers $N$ in the polymer, for $N$ equal to natural multiples of their $P$, for HOMO (upper panel) and LUMO (lower panel).