Motional narrowing under Markovian and non-Markovian hopping transitions in inhomogeneous broadened absorption line shape

Inspired by recent experiments showing a minimum of electron spin or paramagnetic resonance (ESR and EPR) line width as a function of inverse temperature, we studied the motional narrowing effect by considering a combined model of carrier transitions and static dispersion of the angular frequency giving rise to an inhomogeneous broadening in the spectrum. The dispersion of the angular frequency results from the distribution of the local field. The transition between the sites under inhomogeneous static local field induces adiabatic relaxation of the spin. We also considered the on-site inherent (nonadiabatic) relaxation of the spin. We obtained the exact solution of the spin correlation function by explicitly considering transitions between two sites for both Markovian and non-Markovian transition processes. The absorption line shape is expressed in terms of the Voigt function, which is a convolution of a Gaussian function and a Lorentzian function. Using the known properties of the Voigt function, we discuss the correlation between the change in the full-width at half-maximum and the change in line shape, both of which are induced by motional narrowing. By assuming thermal activation processes for both the hopping transition and the on-site inherent relaxation, we show that the minimum of the width appears as a function of inverse temperature as observed experimentally in organic materials. Contrary to the general belief, we also show that the narrowing of the Gaussian line shape under a local random field did not necessarily lead to a Lorentzian line shape in particular under the presence of heavy tail property in the waiting time distribution of hopping transitions.

Figure 1

Schematic picture of the Kubo-Anderson-Kitahara model is shown. The model comprises a spin executing transitions between a pair of statistically equivalent sites labeled by a and b. We consider a large number of the pairs and study the ensemble average of spins. The local field at each site obeys a Gaussian distribution. As a result of local field dispersion, the angular frequency of the spin at each site obeys a Gaussian distribution centered on the Larmor angular frequency. The carrier transition rate between the sites is given by $\gamma /2$. We also include the spin relaxation at each site with the rate ${\gamma }_s$.

Figure 2

(a) Spin correlation function is shown as a function of time. (b) The power spectrum is shown as a function of angular frequency. In both plots, we set ${\mathrm{\Gamma }}_s=0.01$ and $\mathrm{\Gamma }=1$; the solid lines represent the numerically exact results. In panel (a) the red circles represent the results obtained from Eq. (10); the red dashed line represents the approximate solution obtained from Eq. (8). In panel (b) the red circles represent approximate results obtained from Eq. (19) by summing the first four terms.

Figure 3

Frequency dependence of the power spectrum illustrating (a) FWHM and (b) the equivalent width (integral breadth). FWHM is defined by the width when the height of the spectrum is half of its maximum value. The equivalent width is defined using the maximum height, and the area under the spectrum as a function of the angular frequency. The kurtosis parameter is defined by the equivalent width divided by FWHM. The spectral lines are drawn for ${\mathrm{\Gamma }}_s=1$ and $\mathrm{\Gamma }=1$.

Figure 4

Normalized FWHM using ${\mathrm{\Delta }}_0$ is shown against $\mathrm{\Gamma }=\gamma /{\mathrm{\Delta }}_0$. The thick solid lines represent the numerical results of Eq. (4) using Eq. (10). The long dashed lines and dots represent the results of Eq. (20) and those of Eq. (21), respectively. The (red) thin lines indicate $\mathrm{\Gamma }$ dependence given by Eq. (22). The (red) circles indicate FWHM calculated from the approximate expression given by Eq. (19).

Figure 5

The kurtosis parameter is shown as a function of $\mathrm{\Gamma }$ for various values of ${\mathrm{\Gamma }}_s$. The kurtosis parameter values are 1.06 and 1.57 for the Gaussian spectrum and the Lorentzian spectrum, respectively.

Figure 6

(a) FWHM normalized by ${\mathrm{\Delta }}_0$ and (b) kurtosis parameter plotted against inverse temperature. The kurtosis parameter values are 1.06 and 1.57 for the Gaussian spectrum and the Lorentzian spectrum, respectively. The dashed lines are obtained for $E_a=0.075$ eV, and the solid lines indicate the results of the Marcus equation for $\lambda =0.3$ eV. The parameter values are $c_s=1025.86$, $c=1.79$, $E_{as}=0.3$ eV, and $T_m=300$ K for the Marcus equation.

Figure 7

First derivative of the absorption spectrum plotted against angular frequency for ${\mathrm{\Gamma }}_s=0.01$ and $\mathrm{\Gamma }=1$. The solid line represents the convergent values of Eq. (26). The summation of the first four terms yields the convergent results. The (red) dashed and (blue) dots indicate the fitting of the solid line by a Gaussian function and a Lorentzian function, respectively. (We find $\sigma =0.891$ for the Gaussian function [Eq. (15)] and $r=0.882$ for the Lorentzian function [Eq. (16)].) The kurtosis parameter value is 1.11.