Observation of Phase-Filling Singularities in the Optical Dielectric Function of Highly Doped $n$-Type Ge

Phase-filling singularities in the optical response function of highly doped ($>{10}^{19}{\mathrm{cm}}^{-3}$) germanium are theoretically predicted and experimentally confirmed using spectroscopic ellipsometry. Contrary to direct-gap semiconductors, which display the well-known Burstein-Moss phenomenology upon doping, the critical point in the joint density of electronic states associated with the partially filled conduction band in $n$-Ge corresponds to the so-called $E_1$ and $E_1+{\mathrm{\Delta }}_1$ transitions, which are two-dimensional in character. As a result of this reduced dimensionality, there is no edge shift induced by Pauli blocking. Instead, one observes the “original” critical point (shifted only by band gap renormalization) and an additional feature associated with the level occupation discontinuity at the Fermi level. The experimental observation of this feature is made possible by the recent development of low-temperature, in situ doping techniques that allow the fabrication of highly doped films with exceptionally flat doping profiles.

Figure 1

Band structure of Ge calculated with the $\mathit{k}·\mathit{p}$ method of Ref. [31], highlighting the region corresponding to the $E_1$ and $E_1+{\mathrm{\Delta }}_1$ transitions. For the $E_1$ case, we show schematically the effect of Pauli blocking following $n$-type doping. Red indicates forbidden transitions at $T=0$. Green shows allowed transitions at $T=0$. Darker greens correspond to higher energies.

Figure 2

Calculated real (blue curves) and imaginary (red curves) parts of the dielectric function and their second derivatives near the $E_1/E_1+{\mathrm{\Delta }}_1$ gaps of intrinsic and doped Ge. The top three panels (a)–(c) and (e)–(g) on each side show only the $E_1$ contribution to better illustrate the effect of doping. The bottom panels (d) and (h) incorporate the full $E_1/E_1+{\mathrm{\Delta }}_1$ for a comparison with the experiment. The arrows mark the position of the extra features due to doping. In all cases, the gray curves correspond to the undoped case.

Figure 3

(a) Imaginary part of the 77 K dielectric function for several $n$-type Ge films. The lower panels show second derivatives of the complex dielectric function of the above films. The solid lines represent simultaneous fits of the real and imaginary parts with an expression derived from Eq. (1). Traces are vertically offset by $700{\mathrm{eV}}^{-2}$ each for clarity. (b) Real parts. The doping levels that produce the best fit are indicated below the corresponding traces. (c) Imaginary parts.