Signature of electronic correlations in the optical conductivity of the doped semiconductor Si:P

Electronic transport in highly doped but still insulating silicon at low temperatures is dominated by hopping between localized states; it serves as a model system of a disordered solid for which the electronic interaction can be investigated. We have studied the frequency-dependent conductivity of phosphorus-doped silicon in the terahertz frequency range $(30\mathrm{GHz}–3\mathrm{THz})$ at low temperatures $T⩾1.8\mathrm{K}$. The crossover in the optical conductivity from a linear to a quadratic frequency dependence as predicted by Efros and Shklovskii [Sov. Phys. JETP 54, 218 (1982)] is observed qualitatively; however, the simple model does not lead to a quantitative agreement. Covering a large range of donor concentration, our temperature- and frequency-dependent investigations reveal that electronic correlation effects between the localized states play an important and complex role at low temperatures. In particular, we find a superlinear frequency dependence of the conductivity that highlights the influence of the density of states, i.e., the Coulomb gap, on the optical conductivity. When approaching the metal-to-insulator transition by increasing doping concentration, the dielectric constant and the localization length exhibit critical behavior.

Figure 1

Temperature dependence of the real part of the conductivity ${\sigma }_1$ for various frequencies and the temperature-dependent dc results for Si:P crystals with (a) $n=1.6\times {10}^{18}{\mathrm{cm}}^{-3}$ and (b) $n=1.97\times {10}^{18}{\mathrm{cm}}^{-3}$.

Figure 2

Frequency dependence of a Si:P sample with $n=1.6\times {10}^{18}{\mathrm{cm}}^{-3}$ at $T=1.8\mathrm{K}$. The straight lines are fits with linear and quadratic behaviors, respectively. The dashed line is a fit with the prediction of Shklovskii and Efros (Ref. 9) given in Eq. (7). The error bars are of the size of the symbols.

Figure 3

Frequency-dependent conductivity in the Coulomb-glass regime for the silicon samples with phosphorus concentration $n=2.29\times {10}^{18}{\mathrm{cm}}^{-3}$ and $n=2.57\times {10}^{18}{\mathrm{cm}}^{-3}$. The fits correspond to a superlinear behavior as indicated in the graph with $I_0=45\mathrm{meV}$. At higher frequencies, the transition to quadratic behavior is apparent. Linear behavior is plotted as gray lines for comparison.

Figure 4

Frequency-dependent conductivity of the four experimentally accessible Si:P samples. The respective phosphorus concentration is indicated in the graph in units of ${10}^{18}{\mathrm{cm}}^{-3}$. The fits correspond to the superlinear and approximately quadratic behaviors.

Figure 5

Temperature dependence of the dielectric constant ${ϵ}_1$ for Si:P samples with different phosphorus concentrations $n$ as indicated. The curves correspond to the left and right axes as indicated; also note the different temperature axes for the two frames. The data points are mean values averaged over the frequency range investigated.

Figure 6

Dielectric constant ${ϵ}_1$ of Si:P as a function of phosphorus concentration measured at $T=1.8\mathrm{K}$. The solid line represents a fit by ${ϵ}_1-{ϵ}_{\text{host}}\propto {(n_c∕n-1)}^{-{\zeta }^{″}}$ with ${\zeta }^{″}=1.2$.

Figure 7

Dielectric susceptibility $4\pi \chi$ of Si:P crystals versus the reduced doping concentration $1-n∕n_c$ with $n_c=3.5\times {10}^{18}{\mathrm{cm}}^{-3}$ at $T=1.8\mathrm{K}$.

Figure 8

Scaling behavior of the localization length in Si:P. The critical concentration is $n_c=3.5\times {10}^{18}{\mathrm{cm}}^{-3}$.