Debye relaxation in high magnetic fields

Dielectric relaxation is universal in characterizing polar liquids and solids, insulators, and semiconductors, and the theoretical models are well developed. However, in high magnetic fields, previously unknown aspects of dielectric relaxation can be revealed and exploited. Here, we report low-temperature dielectric relaxation measurements in lightly doped silicon in high dc magnetic fields $B$ both parallel and perpendicular to the applied ac electric field $E$. For $B\parallel E$, we observe a temperature and magnetic-field-dependent dielectric dispersion $\epsilon \left(\omega \right)$ characteristic of conventional Debye relaxation where the free-carrier concentration is dependent on thermal dopant ionization, magnetic freeze-out, and/or magnetic localization effects. However, for $B\perp E$, anomalous dispersion emerges in $\epsilon \left(\omega \right)$ with increasing magnetic field. It is shown that the Debye formalism can be simply extended by adding the Lorentz force to describe the general response of a dielectric in crossed magnetic and electric fields. Moreover, we predict and observe a new transverse dielectric response $E_H\perp B\perp E$ not previously described in magnetodielectric measurements. The new formalism allows the determination of the mobility and the ability to discriminate between magnetic localization/freeze-out and Lorentz force effects in the magnetodielectric response.

Figure 1

Sample configuration and temperature-dependent dielectric response. (a) Lead configurations for the sample used in this work. C and D are the capacitive leads in the direction of the ac electric field $E_{\text{ac}}$, and the $V_H$ leads are the transverse contacts. The reversible dc magnetic field is along $z$ for $B\perp E$ (shown in figure for $\theta =90°$), and along $x$ for $B\parallel E$ $\left(\theta =0°\right)$. (b) Dielectric response of silicon sample 1 vs temperature for $\omega /2\pi =30.5\text{kHz}$ at zero magnetic field. Near $T=24\text{K}$, $\omega \tau \sim 1$ ($\omega \tau$ decreases for increasing $T$). Inset: Cole-Cole plot for ${\epsilon }^{″}$ vs ${\epsilon }^{\prime }$. The best fit of Eq. (2) to the data is for $\gamma =0.87$.

Figure 2

Temperature dependence of the real $\left({\epsilon }^{\prime }\right)$ and imaginary $\left({\epsilon }^{″}\right)$ dielectric signals for silicon sample 1 for $B\perp E$. [(a)–(f)] Frequencies of 1, 3, 30, and 100 kHz at constant magnetic fields of 0, 10, and 32 T. Deviations in the low-frequency response, e.g., Fig. 2d, arise from glassy behavior (Ref. 10). [(g)–(h)] Magnetic field dependence of ${\epsilon }^{\prime }$ and ${\epsilon }^{″}$ at constant frequency from 0 to 32 T. At high fields additional small resonances from minority dipolar states can be resolved.

Figure 3

Angular dependence of ${\epsilon }^{″}$ for silicon sample 1 at 100 kHz and 8 T. The anomalous dispersion appears when there is a finite component of $B$ perpendicular to $E$. ($0°$: $B\parallel E$ and $90°$: $B\perp E$).

Figure 4

Model for the dielectric response. A displacement of charge $\pm q$ due to an external field $E$ in a dielectric produces a polarization field $E_p$ which increases linearly with the displacement $x$. For a charge density $n$ $\left(=\text{Ne}\right)$, the polarization charge per unit area is $nx$, and the polarization field is therefore $E_p=nx/{\epsilon }_s$. As discussed in the text, in the absence of a driving field, an initial displacement will relax exponentially to zero with a time constant $\tau$.

Figure 5

Predictions of the model. (a) ${\epsilon }_x^{\prime }/{\epsilon }_s$ and (b) ${\epsilon }_x^{″}/{\epsilon }_s$ vs temperature from Eq. (7) at different magnetic fields for $B\perp E$. (c) ${\epsilon }_y^{\prime }/{\epsilon }_s$ and (d) ${\epsilon }_y^{″}/{\epsilon }_s$ vs temperature at different magnetic fields for $B\perp E$ from Eq. (8). Transverse signals only appear for $B\ne 0$

Figure 6

Longitudinal (C and D) and transverse $\left(V_H^{\prime },V_H^{″}\right)$ dielectric response for $B\perp E$ at 10 KHz and 1.23 T for sample 2. Left panels: capacitance and Hall signals for forward and reverse field. Right panels: difference signals showing transverse components [compare with Eq. (8) and Fig. 5]. The signs for curves $\delta C$ and $\delta D$ have been reversed to account for the $180°$ ambiguity in the phase of the lead configuration.