Polaron spin echo envelope modulations in an organic semiconducting polymer

We present a theoretical analysis of the electron spin echo envelope modulation (ESEEM) spectra of polarons in semiconducting $\pi$-conjugated polymers. We show that the contact hyperfine coupling and the dipolar interaction between the polaron and the proton spins give rise to different features in the ESEEM spectra. Our theory enables direct selective probe of different groups of nuclear spins, which affect the polaron spin dynamics. Namely, we demonstrate how the signal from the distant protons (coupled to the polaron spin via dipolar interactions) can be distinguished from the signal coming from the protons residing on the polaron site (coupled to the polaron spin via contact hyperfine interaction). We propose a method for directly probing the contact hyperfine interaction, that would enable detailed study of the polaron orbital state and its immediate environment. We also analyze the decay of the spin echo modulation, and its connection to the polaron transport.

Figure 1

Conjugated polymer PPV (a) and its derivative, MEH-PPV (b). Upper and middle panels show the chemical structures and the unit cells. The principal $x,y$ axes of the C–H proton hyperfine tensors at B and ${\mathrm{C}}^{\prime }$ carbon sites are different from those at ${\mathrm{B}}^{\prime }$, C, E, and F carbon sites, while the $z$ axes are the same and perpendicular to the plane of the picture (in MEH-PPV there are no C–H protons at ${\mathrm{B}}^{\prime }$ and C sites). (Bottom) Half-widths of the spatial extents of polarons (orange ovals), according to Ref. [26].

Figure 2

The ESEEM pulse sequences considered in the text. (a) Primary ESEEM. (b) Stimulated ESEEM.

Figure 3

(a) and (c) Functions $F_c\left(\tau \right)$ and $F_d\left(\tau \right)$, introduced in Eqs. (17) and (18), respectively, are plotted with blue. (b) and (d) The respective cosine Fourier transforms, ${\tilde{F}}_c\left(\omega \right)$ and ${\tilde{F}}_d\left(\omega \right)$, are plotted with magenta, in the same units.

Figure 4

The primary ESEEM spectrum $\tilde{E}$, calculated from orientation disorder averaged Eq. (10) numerically, is plotted in black. (a) The cosine Fourier transform of the sum, $\langle E_c\left(2\tau \right)\rangle +\langle E_d\left(2\tau \right)\rangle$, is plotted with yellow dashed line, from Eq. (16). (b) Zoom in of the region indicated in the left panel with a rectangle. ${\tilde{F}}_c(\omega -{\omega }_I)$ and ${\tilde{F}}_d(\omega -{\omega }_I)$ are plotted with the cyan and magenta dotted lines, respectively. It is seen that the spectral peak at ${\omega }_I=14.7$ MHz is exclusively due to the distant protons, whereas the side bands come from the contact protons.

Figure 5

The stimulated ESEEM $\langle E({\tau }_n,T)\rangle$, calculated from orientation disorder averaged Eq. (11) numerically, is plotted against $T$ at fixed ${\tau }_n=(\pi /{\omega }_I)n$ for $n=3,{\tau }_3\approx 102$ ns (a), and $n=4,{\tau }_4\approx 136$ ns (c). The corresponding spectra $\tilde{E}({\tau }_n,\omega ),n=3$ (b) and 4 (d), are plotted against $\omega$. The strong reduction of the peak at ${\omega }_I=14.7$ MHz for even $n$, allowing the observation of the contact proton hyperfine coupling, is obvious.

Figure 6

The decay of echo modulation for slow polaron hopping, $\eta \equiv \nu /{\omega }_{\text{hf}}\ll 1$. The right-hand sides of Eq. (24) are plotted for $\alpha \equiv T/T_0\gg 1$ (green) and $\alpha =2$ (magenta). The corresponding left-hand sides are plotted with black dotted lines. The hopping attempt frequencies and corresponding values of $\eta$ are $\nu =0.4$ MHz, $\eta =0.0085$ (a), $\nu =1$ MHz, $\eta =0.021$ (b), $\nu =3$ MHz, $\eta =0.063$ (c), and $\nu =10$ MHz, $\eta =0.21$ (d). The plots clearly confirm the validity of Eq. (24).

Figure 7

Comparison of the polymer chain orientation disorder averaged exact relation Eq. (10) (black points) and approximation Eq. (A4) (red line).

Figure 8

Plots of the amplitude ${\mathrm{\Lambda }}_d({\tau }_n,T)$ vs $T$ at fixed ${\tau }_n=(\pi /{\omega }_I)n$, for (a) odd $n=1,3,\cdots ,29$, and (b) even $n=2,4,\cdots ,30$. The plots demonstrate the reduction of ${\mathrm{\Lambda }}_d({\tau }_n,T)$ when going from odd to even $n$. For small $n$, the decrease of ${\mathrm{\Lambda }}_d({\tau }_n,T)$ from odd to even $n$ is more than two orders of magnitude. For large $n$, it is more that 15 times.

Figure 9

The amplitude ${\mathrm{\Lambda }}_c({\tau }_n,T)$ is plotted vs $T$ at fixed ${\tau }_n=(\pi /{\omega }_I)n$, for (a) odd $n=1,\cdots ,19$, and (b) even $n=2,\cdots ,20$. Though the individual curves are not well resolved, it is seen that there is no notable difference in the orders of magnitude of ${\mathrm{\Lambda }}_c({\tau }_n,T)$ with even and odd $n$.

Figure 10

Schematic definitions of polaron random walk trajectories, $\mathbf{R}\left(t\right)$, for free induction decay (a) and primary echo (b). The blue lines denote the pulses. The green lines show the detection points. The red bars are polaron random hops.

Figure 11

Illustration of notations for polaron random walk trajectories during the stimulated pulse sequence. The blue bars symbolize the pulses. The green bars show the detection. The red lines are polaron random hops. (a) Trajectories of type ${\mathbf{R}}_0$, Eq. (B17); no polaron hop occurs in the time interval $T$. (b) Trajectories of type ${\mathbf{R}}_1$, Eq. (B17); at least one hop occurs in the time interval $T$.