Cyclotron resonance of carbon-doped two-dimensional hole systems: From the magnetic quantum limit to low magnetic fields

We report on cyclotron resonance (CR) experiments of high-quality C-doped two-dimensional hole systems in GaAs/AlGaAs quantum wells covering the microwave and far-infrared regimes. At high magnetic field $B$ we observe a strong $B$ dependence of the cyclotron mass $m_c$ which can be reproduced by calculations based on a $4\times 4$ Luttinger Hamiltonian in the axial approximation. At small $B$ one main and up to three secondary CRs are found, which continuously evolve with smooth magnetic-field dependences. This has not been observed and predicted so far. In the regime of the magnetic quantum limit a further CR is observed, which can be attributed to a transition between Landau levels originating from the two different spin-split heavy- and light-hole subbands.

Figure 1

(Color online) Experimental spectra in the (a) FIR and (b) MW regimes at 1.6 K for a 15-nm-wide quantum well at $p_s=1.3\times {10}^{11}{\text{cm}}^{-2}$. (a) Experimental spectra for different fixed fields $B$. The spectra are normalized to other magnetic fields with resonances far away from the resonance under investigation and are offset for clarity. Arrows indicate the CR. Between $B=5.5$ and 8 T the CR is not on display: due to the reduced beam-splitter efficiency the signal-to-noise ratio is quite low in the regime of 2.1–2.5 meV. The positions of the cyclotron resonance shifts stronger for higher magnetic fields than for lower fields. The resonance clearly broadens at lower magnetic fields. (b) Experimental spectra for different fixed MW frequencies. Note that in contrast to (a) the measurements in the MW regime were performed at fixed frequencies while sweeping the magnetic field. The spectra are normalized to the transmission signal at $B=0\text{T}$ and are offset for clarity. Arrows indicate the cyclotron resonance. The additional resonances which appear next to the indicated resonances are discussed later in the text.

Figure 2

(Color online) Magnetic-field dependence of the cyclotron resonance energy $E$ and corresponding cyclotron mass $m_c$ measured at $T=1.6\text{K}$ for three different hole densities $p_s$ ranging from $0.7\times {10}^{11}$ to $1.6\times {10}^{11}{\text{cm}}^{-2}$. The low-frequency regime will be shown on an expanded scale later. Crosses indicate the CR measured with FIR and the open circles indicate those measured with MW. The small gap in the MW part has experimental reasons. The positions of integer filling factors $\nu$ are indicated by perpendicular lines. The dispersion of the cyclotron resonance energy shows for all three hole densities a strongly nonproportional dependence on $B$ which indicates a strong dependence of the cyclotron mass on $B$. Differences in $m_c$ dispersions for different hole densities are particularly obvious for small magnetic fields $B$.

Figure 3

(Color online) Experimental transmission in the MW regime at $T=1.6\text{K}$ for $p_s=1.3\times {10}^{11}{\text{cm}}^{-2}$. Multiple CRs are observed. Solid line arrows indicate the main CR, while dashed line arrows indicate secondary CRs. Up to three secondary CRs can be resolved simultaneously and followed over certain $B$ regimes.

Figure 4

(Color online) Magnetic-field dependence of the main and secondary cyclotron resonances’ energy $E$ and the resulting cyclotron mass $m_c$ for low $B<4\text{T}$ measured at $T=1.6\text{K}$ for three different hole densities $p_s$ ranging from $0.7\times {10}^{11}$ to $1.6\times {10}^{11}{\text{cm}}^{-2}$. Shaded areas indicate the frequency gaps of the experimental setup. (Red) crosses denote the main CR measured in the FIR regime and (red) closed circles denote those measured in the MW regime. (Blue) open squares indicate the energy of the secondary CRs measured in the MW regime. The positions of integer filling factors $\nu$ are indicated by perpendicular lines.

Figure 5

Calculated (a) potential profile and (b) subband dispersion for $B=0\text{T}$, $p_s=1.3\times {10}^{11}{\text{cm}}^{-2}$, and $T=0\text{K}$. The subbands are labeled corresponding to the $z$ component of angular momentum $m$ at $k_{\parallel }=0$. $E_F$ denotes the Fermi energy.

Figure 6

Calculated density-of-states effective mass $m_{\text{DOS}}$ of the topmost subband. $m_{\text{DOS}+}$ and $m_{\text{DOS}-}$ correspond to the upper and lower spin subbands of Fig. 5.

Figure 7

(Color online) (a) Calculated Landau levels evolving from the spin-split highest heavy-hole subband together with the Fermi energy $E_F$ (bold blue line). One possible transition of the CR is indicated by the dashed arrow. (b) Corresponding calculated cyclotron masses $m_c$ (solid line) and measured $m_c$ (markers) for $p_s=1.3\times {10}^{11}{\text{cm}}^{-2}$. (Red) crosses denote the $m_c$ calculated from the main cyclotron resonance CR measured in the FIR regime and (red) closed circles denote those measured in the MW regime. (Blue) open squares indicate the $m_c$ deduced from the secondary CRs measured in the MW regime. The details of the regime $B<3\text{T}$ will be shown later.

Figure 8

(Color online) Measured (red crosses) and calculated $m_c$ for different sets of Luttinger parameters.

Figure 9

(Color online) Measured and calculated $m_c$ for two different hole densities $p_s$. Line styles are given in the inset.

Figure 10

(Color online) Measured and calculated $m_c$ in the low magnetic-field regime for the hole densities (a) $p_s=1.3\times {10}^{11}{\text{cm}}^{-2}$, (b) $0.7\times {10}^{11}{\text{cm}}^{-2}$, and (c) $1.6\times {10}^{11}{\text{cm}}^{-2}$. (Red) crosses denote $m_c$ extracted from the main cyclotron resonance CR measured in the FIR regime and (red) closed circles denote those measured in the MW regime. (Blue) open squares indicate $m_c$ deduced from the secondary CRs measured in the MW regime. (Black) solid lines indicate the calculated $m_c$ according to the allowed transitions. Dashed lines indicate the calculated $m_c$, which correspond to forbidden transitions at $T=0\text{K}$ due to an empty initial or a completely filled final state. Shaded areas indicate the nonavailable gaps in the MW spectrum of our experimental setup.

Figure 11

(Color online) (a) Experimental transmission at 1.21 meV (293 GHz) for different temperatures for $p_s=1.3\times {10}^{11}{\text{cm}}^{-2}$. Solid line arrows indicate the main CR, while dashed line arrows indicate secondary CRs. (b) Corresponding resonance positions versus temperature up to $T=4.2\text{K}$. Closed circles indicate the main CR, while open squares (triangles) indicate the higher- (lower-) lying secondary CR.

Figure 12

(Color online) High-energy CR in the FIR regime, measured for $p_s=1.3\times {10}^{11}{\text{cm}}^{-2}$ at $T=1.6\text{K}$.

Figure 13

(Color online) Magnetic-field dependence of the $m_c$ of the measured second CR in the FIR regime and the calculated $m_c$, which are possible according to the denoted transitions. The results for $p_s=1.3\times {10}^{11}$ and $1.6\times {10}^{11}{\text{cm}}^{-2}$ are on display. The linewidth is implicitly indicated by the thickness of the symbols. Line styles are given in the inset.

Figure 14

(Color online) (a) Experimental spectra in the FIR regime at $B=12\text{T}$ for different temperatures $T$. (b) Temperature dependence of the zero-field transport mobility and of the cyclotron mobility. Styles are given in the inset.