Hyperfine-induced spin relaxation of a diffusively moving carrier in low dimensions: Implications for spin transport in organic semiconductors

The hyperfine coupling between the spin of a charge carrier and the nuclear spin bath is a predominant channel for the carrier spin relaxation in many organic semiconductors. We theoretically investigate the hyperfine-induced spin relaxation of a carrier performing a random walk on a $d$-dimensional regular lattice, in a transport regime typical for organic semiconductors. We show that in $d=1$ and 2, the time dependence of the space-integrated spin polarization $P\left(t\right)$ is dominated by a superexponential decay, crossing over to a stretched-exponential tail at long times. The faster decay is attributed to multiple self-intersections (returns) of the random-walk trajectories, which occur more often in lower dimensions. We also show, analytically and numerically, that the returns lead to sensitivity of $P\left(t\right)$ to external electric and magnetic fields, and this sensitivity strongly depends on dimensionality of the system $(d=1$ versus $d=3)$. Furthermore, we investigate in detail the coordinate dependence of the time-integrated spin polarization $\sigma \left(\mathbf{r}\right)$, which can be probed in the spin-transport experiments with spin-polarized electrodes. We demonstrate that, while $\sigma \left(\mathbf{r}\right)$ is essentially exponential, the effect of multiple self-intersections can be identified in transport measurements from the strong dependence of the spin-decay length on the external magnetic and electric fields.

Figure 1

Lateral (a) and vertical (b) spin valves with organic active layer (gray), in which the charge carriers are restricted to move in one-dimensional channels along the $x$ direction. The separation between the injecting and detecting electrodes is $L$.

Figure 2

Diffusive random walk of a polaron (red circle) over $d=1$ and 2 dimensional regular lattices of molecular sites (gray circles). (a) In the absence of external electric field, polaron hops between the nearest-neighbor sites of $d=1$ linear chain occur with the equal rates $\nu /2$ indicated by the blue arrows. (b) In the presence of external electric field along the linear chain, hopping rates ${\nu }_f$ and ${\nu }_b$, respectively in the directions forward and backward to the electric field, are different; ${\nu }_f>{\nu }_b$. (c) With an external electric field along the $x$ axis of a regular lattice in $d=2$, hopping rates in the forward and backward directions to $\widehat{\mathbf{x}},{\nu }_f$ and ${\nu }_b$, can depend on the electric field, while the rates in the perpendicular direction are the same as without the field, and equal to $\nu /4$ each.

Figure 3

Logarithmic plot of $P\left(t\right)$ in $d=1$, for values of $\eta \equiv b_{\text{hf}}/\nu$ ranging from $0.002$ to $0.05$. When plotted against the renormalized time $\tau =\left(\nu t\right){\eta }^{4/3}$, all data points fall on a single scaling curve. The resulting scaling curve is well fitted by $exp(-8{\tau }^{3/2}/3\sqrt{2\pi })$ for small $\tau$ and $0.827exp(-1.33{\tau }^{3/4})$ for larger $\tau$. Normal plot of the scaling curve $F\left(\tau \right)$ is shown in the inset.

Figure 4

Magnetic field dependence of spin relaxation for a diffusion in $d=1$ with $\eta =0.01$. Red, green, and blue plots correspond to the magnetic field values $B=0,0.5b_{\text{hf}}$, and $2b_{\text{hf}}$, respectively. (a) $P\left(t\right)$ is plotted against $\tau =\left(\nu t\right){\eta }^{4/3}$; the red curve is the same as the one in the inset of Fig. 3. Inset: the same plot in the log scale. (b) $\sigma \left(r\right)$ is plotted versus $r$. Black curves are our exponential fits. (c) The normalized spin-decay length $l_S\left(B\right)/l_S\left(0\right)$ is plotted vs $B/b_{\text{hf}}$. Dashed line indicates $l_S\left(B\right)/l_S\left(0\right)$ without returns, plotted from Eq. (A13).

Figure 5

Electric field dependence of spin relaxation in $d=1$ with $\eta =0.01$, at zero magnetic field. With typical $a\approx 1$ nm and $T\approx 100$ K, parameter $\varepsilon \equiv eEa/k_{B}T$ corresponds to the electric field $E\approx 8.6\varepsilon$ mV/nm. The red, dark green, violet, and magenta plots correspond to $\varepsilon =0,0.15,0.3$, and 3, respectively. (a) $P\left(t\right)$ is plotted versus $\tau =\left(\nu t\right){\eta }^{4/3}$. The red curve is the same as in Fig. 4. The dashed black curve represents the exponential saturation for $\varepsilon \gg 1$. Inset: the same in the log scale. (b) $\sigma \left(r\right)$ is plotted versus $r$. Black lines are our exponential fits. (c) Normalized spin-decay length $l_S\left(\varepsilon \right)/l_S\left(0\right)$ versus $\varepsilon$ (orange). The dashed line indicates the same dependence without multiple returns, plotted from Eq. (A16).

Figure 6

Spin relaxation in a finite chain of length $L$. When plotted versus $\tau =\left(\nu t\right){\eta }^{4/3}$ and $\lambda =L{\eta }^{2/3}$, including (a) $\lambda =0.75$, (b) $\lambda =1.25$, and (c) $\lambda =2$, all $P(t,L)$ points fall on universal curves. Cyan lines are stretched-exponential fits $P(t,L)\sim exp(-\alpha t^{\beta })$, with (a) $\beta =0.59$, (b) $\beta =0.61$, and (c) $\beta =0.64$. Magenta lines are our theoretical fits with $exp(-8{\tau }^{3/2}/3\sqrt{\pi })$. (d) The resulting universal scaling function $F(\tau ,\lambda )$ is plotted for the above values of $\lambda$. The dashed line illustrates saturation of $F(\tau ,\lambda )$ for $\lambda \to \infty$.

Figure 7

Spin polarization at the detection electrode ${\sigma }_{*}\left(L\right)$ for a system with $\eta =0.01$. (a) Log plot of ${\sigma }_{*}\left(L\right)$ in zero electric field and different magnetic fields: $B=0$ (black), $0.5b_{\text{hf}}$ (red), $2b_{\text{hf}}$ (green). (c) Log plot of ${\sigma }_{*}\left(L\right)$ in three different electric fields, $\varepsilon =0$ [black; the same as in (a)], $\varepsilon =0.03$ (blue), $\varepsilon =0.06$ (magenta), in zero magnetic field. The black and cyan lines in (a) and (c) are our exponential fits. (b), (d) Dependence of the (normalized) diffusion length $l_S/l_S\left(0\right)$ on external magnetic and electric fields, respectively.

Figure 8

Spin relaxation of a carrier diffusing over $d=2$ regular lattice, with $\eta \equiv b_{\text{hf}}/\nu =0.025$. (a) Log plot of $P\left(t\right)$ against $\nu t$ (black). The cyan dashed curve is plotted from Eq. (14). The magenta dashed line is our exponential fit. (b) Magnetic field dependence of the (normalized) spin decay length; $l_S/l_S\left(0\right)$ is plotted vs $B$ for a diffusion in $d=1$ with ${\eta }_1=0.021$ (orange), $d=2$ with $\eta =0.025$ (black), and strong-collision approximation to $d=3$ with ${\eta }_3=0.032$ (dashed line). (c) Electric field dependence of the spin decay length. $l_S/l_S\left(0\right)$ is plotted vs $\varepsilon =eEa/k_{B}T$, for a diffusion in $d=1$ with ${\eta }_1=0.021$ (orange), $d=2$ with $\eta =0.025$ (black), and $d=3$ with ${\eta }_3=0.032$ in strong-collision approximation (dashed line).

Figure 9

Spin relaxation in a 1D chain with Gaussian energetic disorder. Simulations are performed for a system with $\eta \equiv b_{\text{hf}}/\nu =0.01$, at zero magnetic and electric fields. The red, green, blue, and magenta plots correspond to ${\delta }_d=0$, 1, 2, and 3, respectively. (a) $P\left(t\right)$ is plotted versus $\tau =\left(\nu t\right){\eta }^{4/3}$. The red curve is the same as in Figs. 4 and 5. Inset: the same in the log scale. (b) $\sigma \left(r\right)$ is plotted versus $r$. The black lines are exponential fits to the simulation results. (c) Dependence of the spin-decay length $l_S$ on the disorder strength ${\delta }_d$.

Figure 10

Dependence of the spin-decay length on external magnetic and electric fields, in the presence of energy disorder. Plotted are the results for a system with energy disorder ${\delta }_d=1,\eta =0.01$ (orange), and a system without energy disorder ${\delta }_d=0,\eta =0.035$ (dark green). The systems are chosen to have the same zero-field spin-decay length $l_S\left(0\right)=6.5$. (a) $l_S\left(B\right)$ is plotted versus $B/b_{\text{hf}}$, at zero electric field. (b) $l_S\left(E\right)$ is plotted versus $\varepsilon \equiv eEa/k_{B}T$, at zero magnetic field. Disorder enhances the sensitivity of $l_S$ to the external fields.

Figure 11

Fit of the magnetoresistance traces calculated from Eq. (17) (red and blue curves), to the experimental data from Fig. 4(a) of Ref. [6] (red and blue squares). $l_S\left(B\right)$ is that shown in Fig. 10 in orange. Electrode coercive fields defining $m_{1,2}\left(B\right)$ are equal to those in Fig. 4(a) of Ref. [6]: $B_{c1}=3.3$ mT, $B_{c2}=11$ mT. The curves are calculated at $b_{\text{hf}}=2$ mT and $d/l_S\left(0\right)=1.54$. Insets show the magnetization orientations of the ferromagnetic electrodes.

Figure 12

$exp\left[K_2^E\left(t\right)\right]$ is plotted against $\left(\nu t\right)$ from Eq. (B13) with $\eta =0.05$, for $\varepsilon =0$ (red), $\varepsilon =0.5$ (green), $\varepsilon =1$ (blue), and $\varepsilon =4$ (orange). Open black circles are simulated points for $P\left(t\right)$ with corresponding values of parameters. The dashed cyan line indicates the large-$\varepsilon$ asymptote $exp[-4{\eta }^2(\nu t-2)]$.