Nonlinear terahertz transmission spectroscopy on Ga-doped germanium in high magnetic fields

We report the observation of cyclotron resonance (CR) transitions of holes in the magnetotransmission spectra of gallium-doped germanium at low temperatures, using intense, pulsed THz free-electron laser radiation with a photon energy lower than the ionization energy of the Ga dopants (11 meV). The THz radiation, in the range of 12–89 ${\mathrm{cm}}^{-1}$, both creates free holes through photoionization of Ga and induces the CR of these holes. For photon energies above the lowest energy internal Ga transition (55 ${\mathrm{cm}}^{-1}$), intradopant transitions are simultaneously observed with narrow CR peaks. For energies below 55 ${\mathrm{cm}}^{-1}$, with increasing THz radiation intensity first the lowest Landau level transitions of all heavy-hole and light-hole subbands appear. This marks the onset of photoionization, which is found to be more efficient for lower laser frequencies, consistent with field-ionization (Keldysh parameter $<<$ 1). For the highest laser intensities, the CR peaks of the heavy (light) holes shift to higher (lower) magnetic field, as a result of the increasing population of the higher-energy nonequidistant Landau levels, consistent with the effective-mass theory of the hole subbands in Ge.

Figure 1

Schematic representation of the valence-band energy levels in Ge:Ga. Zero energy is defined at the bottom of the valence band. The red solid lines indicate the Ga levels; the vertical colored arrows show the lowest three intracenter transitions, labeled by C, D, and G. The vertical dashed arrow shows the spacing between the Ga ground state and the bottom of the valence band at zero magnetic field (11 meV, corresponding to the ionization energy). The solid black lines correspond to the heavy-hole $\left({\epsilon }^{-}\right)$ and light-hole $\left({\epsilon }^{+}\right)$ subbands, and the dashed black lines indicate their LLs when a magnetic field is applied. The vertical black solid arrows indicate the cyclotron resonance transitions in between the LLs.

Figure 2

Inverted FTIR transmission spectra of Ge:Ga at 1.3 K in applied magnetic fields up to 28 T. G, E, and D label the lowest energy Ga intracenter transitions (see Fig. 1). The dashed colored lines are guides to the eye, illustrating how the transitions split in magnetic fields. Each spectrum has been vertically offset corresponding to the applied magnetic field (1 T steps). The three vertical gray dashed lines label the frequencies used in the FEL low-intensity experiments reported in Fig. 5.

Figure 3

Inverted FTIR transmission spectra of Ge:Ga at 20 K in applied magnetic fields up to 25 T. The dashed lines indicate the field dependence of the CR transitions for the ${\epsilon }_2^{+}, {\epsilon }_1^{+}, {\epsilon }_1^{-}, {\epsilon }_2^{-}$ subbands. The pink dashed line indicates a hole CR transition characterized by an effective mass of $0.36m_e$ hole, which was reported earlier but the origin of which is still unknown [33]. The small sharp peaks correspond to water lines. Each spectrum has been vertically offset corresponding to the applied magnetic field (2 T steps).

Figure 4

Peak positions of all transitions detected in the low-intensity FTIR measurements. The purple (black) dots report the results of the 1.3 K (20 K) measurements. The solid lines display the calculated positions of the lowest-energy LL transitions at zero momentum for the ${\epsilon }_2^{+}, {\epsilon }_1^{+}, {\epsilon }_1^{-}, {\epsilon }_2^{-}$ subbands, using the theory of Evtuhov [34]. The pink line indicates a CR transition involving a hole with an effective mass of $0.36m_e$, which was reported earlier but the origin of which is still unknown [33].

Figure 5

Comparison of the FTIR and FEL measurements at low intensities (bottom curves) and FEL experiments at higher intensities (top curves). Black lines correspond to FEL spectra obtained using the 3 GHz mode, whereas magenta curves $+$ symbols show the discrete FTIR traces. The high-power FEL spectra are chosen to express the highest contrast from the low-power measurements. The frequencies and the 0 dB intensities are (a) 52.75 ${\mathrm{cm}}^{-1}$ and $320.74\mathrm{W}/{\mathrm{cm}}^2$, (b) 59 ${\mathrm{cm}}^{-1}$ and $100.81\mathrm{W}/{\mathrm{cm}}^2$, and (c) 65.3 ${\mathrm{cm}}^{-1}$ and 78.51 $\mathrm{W}/{\mathrm{cm}}^2$.

Figure 6

Intensity-dependent transmission curves as a function of magnetic field, obtained using the FEL frequencies 14.5 (a), 23 (b), and 31.5 (c) ${\mathrm{cm}}^{-1}$ at 4.8 K with the 60 MHz mode. The corresponding peak micropulse intensities near the sample at 0 dB are 7.2, 3.2, and $2.1\times {10}^5 \mathrm{W}/{\mathrm{cm}}^2$. The different curves are vertically offset for visibility.

Figure 7

(a) Intensity-dependent transmission traces as a function of magnetic field at 1.4 K, using a frequency of 21.7 ${\mathrm{cm}}^{-1}$(3 GHz mode). The micropulse peak intensity was $4.8\times {10}^4\mathrm{W}/{\mathrm{cm}}^2$ at 0 dB. A vertical offset was added to each curve for better visibility. (b)–(d) Intensity dependence of the resonance curves for each individual hole subband, where the $x$-axis is converted to the calculated effective masses using the CR condition. The amplitude values shown correspond to the resonance intensity normalized to the 0 T value. Solid lines indicate the effective masses of the lowest energy transition for each subband (the numbers label the LL transition indices) using the theory of Ref. [34]. The dotted, dashed, and dash-dotted lines mark the higher-order LL transitions. For ${\epsilon }_{1,2}^{+}$ these are $1\to 2, 2\to 3, 3\to 4$ and for ${\epsilon }_{1,2}^{-}$ they are $3\to 4, 4\to 5, 5\to 6$, respectively. The same notation is used in Figs. 8 and 9.

Figure 8

(a) Intensity-dependent transmission traces as a function of magnetic field at 1.4 K using a FEL frequency of 36 ${\mathrm{cm}}^{-1}$ (3 GHz mode). The micropulse peak intensity was $9.4\times {10}^4\mathrm{W}/{\mathrm{cm}}^2$ at 0 dB. A vertical offset was added to each curve for better visibility. (b)–(d) Intensity dependence of the resonance curves for each individual hole subband, which are labeled in the same way as in Fig. 7.

Figure 9

The frequency-field dependence of the transmission peak energies in Ge:Ga. Black circles correspond to the 20 K FTIR data, and red (blue) circles correspond to the peak positions of all experiments with the 60 MHz (3 GHz) mode of FLARE (all attenuation levels). The 3-GHz mode data points show both CR transitions and internal Ga dopant transitions. The solid lines indicate the calculated behavior of the lowest-energy CR transitions of the different valence hole subbands, and they are the same as the solid lines in Fig. 4 [34]. The dotted, dashed, and dash-dotted lines show the calculated high-order LL transitions [34]. The level indexes are indicated on the plot and are the same as those in Figs. 7 and 8.

Figure 10

Calculated Landau level energies for the different valence subbands. Transitions are shown with vertical solid lines for a radiation frequency of 36 ${\mathrm{cm}}^{-1}$ from the lowest LL (solid line) to one level higher (faded line). The vertical lines belong to the higher (more faded) LL transitions.

Figure 11

The power dependence of the relative (normalized to ${\alpha }_{\text{MAX}}$) absorption coefficients of all the ${\epsilon }_{1,2}^{+}$ and the merged HH peaks. (a),(b),(c) are from the 60 MHz mode, while (d),(e),(f) are from the 3 GHz mode. At 36 ${\mathrm{cm}}^{-1}$, we display all the HH-related peaks without distinction. They are quite close to each other, therefore there are some overlaps among them. The lines between the points are guides for the eye.