Quantum dynamics of nuclear spins and spin relaxation in organic semiconductors

We investigate the role of the nuclear-spin quantum dynamics in hyperfine-induced spin relaxation of hopping carriers in organic semiconductors. The fast-hopping regime, when the carrier spin does not rotate much between subsequent hops, is typical for organic semiconductors possessing long spin coherence times. We consider this regime and focus on a carrier random-walk diffusion in one dimension, where the effect of the nuclear-spin dynamics is expected to be the strongest. Exact numerical simulations of spin systems with up to 25 nuclear spins are performed using the Suzuki-Trotter decomposition of the evolution operator. Larger nuclear-spin systems are modeled utilizing the spin-coherent state $P$-representation approach developed earlier. We find that the nuclear-spin dynamics strongly influences the carrier spin relaxation at long times. If the random walk is restricted to a small area, it leads to the quenching of carrier spin polarization at a nonzero value at long times. If the random walk is unrestricted, the carrier spin polarization acquires a long-time tail, decaying as $1/\sqrt{t}$. Based on the numerical results, we devise a simple formula describing the effect quantitatively.

Figure 1

Spin relaxation in various applied magnetic fields, obtained with the numerically exact simulation of quantum spin dynamics. The numerical results for $P\left(t\right)$ are plotted with open symbols, while the black lines are the results of the semiclassical approximation. (a) System of $L=5$ sites, $N=4$ nuclear spins per site, with $\eta \equiv \left(b_{\text{hf}}{\tau }_0\right)=0.1$ and the hyperfine couplings $a_1=0.83b_{\text{hf}}, a_2=0.9b_{\text{hf}}, a_3=1.05b_{\text{hf}}$, and $a_4=1.18b_{\text{hf}}$. Plotted are the results for relaxation in the applied fields $B_0=0$ (red), $B_0=0.8b_{\text{hf}}$ (green), and $B_0=3b_{\text{hf}}$ (blue). (b) System of $L=7$ sites and $N=3$ nuclear spins per site, with $\eta =0.167$ and $a_1=1.3b_{\text{hf}}, a_2=1.15b_{\text{hf}}$, and $a_3=0.99b_{\text{hf}}$. Red, dark green, and violet symbols represent the applied fields $B_0=0, B_0=0.9b_{\text{hf}}$, and $B_0=3.6b_{\text{hf}}$, respectively.

Figure 2

Numerically exact calculation results from Fig. 1 (open symbols) are plotted together with the results obtained from the $P$-representation simulations (black lines). The comparison confirms the high accuracy of the $P$-representation approach to this problem.

Figure 3

Long-time behavior of $P\left(t\right)$ for $L=5, N=4$, and $\eta =0.25$ (orange), $\eta =0.167$ (green), $\eta =0.1$ (red), $\eta =0.07$ (blue), and $\eta =0.05$ (magenta).

Figure 4

Dependence of the long-time plateau value of $P\left(t\right)$ on the number of molecular sites $L$ and number of spins per site $N$ for $\eta =0.1$. (a) $P\left(t\right)$ is plotted for $L=5$ and five different $N$ ranging from 3 to 20. For $N\to \infty$, the curves saturate at the dashed line, which is the semiclassical approximation result for the same $L$. (b) $P\left(t\right)$ is plotted for five different $L$ values ranging from 4 to 20 and $N=4$ fixed. The dashed line is the saturation curve, $N=4, L\to \infty$.

Figure 5

Large-$L$ saturation curves for $\eta =0.1$ and $N=1$ (magenta), $N=2$ (blue), $N=3$ (green), and $N=4$ (red). Black lines are plotted from Eq. (6). The statistical error bars are of the order of the symbols size.