Figure presents typical CR spectra in sample 1 with inverted band structure obtained with a FT spectrometer. The positions of all observed absorption peaks versus the magnetic field are plotted in Figure . The symbol size characterizes the line intensity: bigger points correspond to more intense lines. The calculated energies of allowed transitions between LLs (Δn
=1) are also plotted in Figure . There are two stronger lines in the spectra: line β
and line Π
. In high magnetic fields, line β
definitely arises from the transition between n
= − 2 and n
= − 1 LLs (cf.
]). In this case, LL n
= − 2 is fully occupied, level n
= − 1 is empty (see Figure ), and a
transition is allowed, so we have a strong line in the spectra. In moderate magnetic fields, line β
can also be attributed to a
transition in the conduction band: at B
<4 T, energies of
transitions are closed to each other; as the magnetic field decreases, the occupancy of LL n
= − 1 in the conduction band in the semimetallic sample 1 increases, so the intensity of the
transition goes down while that of the
one increases. Weak line βi
, observed in high magnetic fields below line β
, in our opinion, can be attributed to electron transitions between LL n
= − 2 and residual donor states pertained to LL n
= − 1.
Typical CR spectra for sample 1. The numbers against the CR lines are the magnetic field values in Tesla. Gray stripes are Reststrahlen bands.
Figure 3 Energies of cyclotron transitions versus the magnetic field for sample 1. Solid lines correspond to the calculated transitions with adjusted parameters; thin dotted lines, with traditional parameters. Symbols are experimental data. The size of symbols (more ...)
The second strong line Π
is a hole CR apparently. It crosses X
-axes in a nonzero magnetic field (≈5 T), which means that the transition takes place between LLs crossing approximately in this field. The only allowed transition satisfying this condition is the
one in the valence band. Some discrepancy between measured and calculated energies (see Figure ) is due to violation of axial approximation. Thus, line Π
is the first observed hole CR in HgTe QWs in quantizing magnetic fields. A weaker line Πi
can be, by analogy, attributed to the transitions between the filled LL n
= 1 in the valence band and impurity state pertained to empty LL n
In the magnetic field range 3.5 to 5 T in the CR spectra in sample 1, we have observed a weaker line α
that is known to result from the interband transition
]. In B
< 3.5 T, LL n
= 1 seems to be occupied and the absorption decreases, while in B
>5 T, the ‘initial’ level n
= 0 seems to rise over the Fermi level.
Weak high-frequency lines I1 to I3 probably resulted from some interband transitions (cyclotron or impurity). At the moment, it is difficult to identify them only because of the great number of allowed transitions between valence and conduction band LLs in this frequency range. At last, the line U whose spectral position does not depend on the magnetic field most probably resulted from transitions between impurity states pertained to LLs n = − 2 in the valence and conduction bands (since direct transitions between these two LLs are forbidden in the Faraday configuration).
Investigations of CR absorptions in sample 2 also revealed a lot of spectral features. In this sample, in addition to the magnetoabsorption study with a FT spectrometer, we also measured CR with QCLs at different 2D electron concentrations varied using the positive PPC effect. As easy to see from Figure , the rise of the electron concentration results in the increase of the CR line intensity only while its position is unchanged. This is an indication that the observed CR line resulted from transitions from one and the same LL (namely n
= 1 in the conduction band; see Figure ) because in classical magnetic fields, a gradual shift of the CR line to higher magnetic fields with the concentration increase is observed [19
Figure 4 Typical CR spectra for sample 2 measured using 3.2-THz QCL. In the absence of visible light illumination (1) and at various levels of illumination (2 to 5). The carrier density in units of 1010 cm-2 is 3.5 (1), 5.4 (2), 7.2 (3), 9.3 (4), and 10.3 (5). (more ...)
Figure 5 Energies of cyclotron transitions versus the magnetic field for sample 2. Solid lines correspond to the calculated transitions with adjusted parameters; thin dotted lines, with traditional parameters. Symbols are experimental data. The size of symbols (more ...)
Experimental data obtained in sample 2 with both the FT spectrometer and the QCLs, as well as calculated energies of allowed transitions between conduction band LLs versus magnetic field are presented in Figure . It is clearly seen that the data obtained with different techniques correspond fairly well (see lower left corner in Figure ). Besides, using QCL operating at 4.3 THz made it possible to measure CR in the phonon absorption band around 150 cm-1 (see Figure ) due to a high stability of QCL radiation intensity.
The main lines in absorption spectra in sample 2 are α
, and δ
. This sample has a normal band structure; therefore, all the transitions take place within the conduction band. The LL structure is analogous to that of sample 100708 studied earlier (see Figure one in [14
]). Line α
corresponds to the transition
from the lowest LL in the conduction band. In high magnetic fields over 4 T, the LL filling factor is less than unity and all the electrons in the QW occupy LL n
=0; therefore, only CR line α
is observed. However, in lower magnetic fields, the electrons populate the next LL n
= − 1 (see Figure one in [14
]) and the transitions
) are observed. At still smaller magnetic fields, the third LL in the conduction band is occupied that leads to a decrease in the intensity of transition
) and in the appearance of line δ
The observed intensive absorption line γ−
is to be considered separately. Its position corresponds fairly well to the transition between two lowest LLs
. In magnetic fields over 5.5 T, where this line is observed, LL n
= 0 is filled while that of n
= − 1 is empty. However, according to our calculations within the axial model, the square of the electrodipole matrix element for this transition is by 4 orders of magnitude less than that for transition
). Actually, the
transition corresponds to electron spin resonance that should not be observed in the Faraday configuration. Nevertheless, line γ−
is clearly seen in the absorption spectra. Probably, this line resulted from transitions between shallow-donor impurity states pertained to LLs 0 and − 1. It is also possible that because of the absence of the axial symmetry in reality, the square of the matrix element for this transition will be significantly higher. In any case, the origin of line γ−
(which has been observed in a number of samples with normal band structure) requires further investigations.
Weak line αi seems to result from the transition between the 1s-like state of residual shallow donors pertained to LL n = 0 and the excited 2p + -like state pertained to LL n = 1. In contrast to impurity lines βi and Πi observed in sample 1 with inverted band structure (Figure ), the energy of the transitions corresponding to line αi exceeds that of line α since the binding energy of the 1s-like ground state is greater than that of the excited 2p+-like state. The origin of other weak lines observed in the absorption spectra in sample 2 requires further studies.
The last sample 3 under study contains narrow HgTe QW with nominal width of 6.3 nm that should correspond to zero bandgap [1
]. In contrast to the previous one, this sample demonstrated negative persistent photoconductivity at illumination by visible light down to electron freezing out. The latter enables us to measure the spectrum of interband photoconductivity (Figure ; cf.
]). One can see a distinct low-frequency edge of the conductivity at 380 cm-1
(47 meV). According to the theoretical model used, the gap value of 47 meV corresponds to a significantly narrower QW with normal band structure. Therefore, general features of the LL fan chart in this sample are the same as those in sample 2 (see Figure one in [14
Figure 6 Typical CR spectra and photoconductivity spectrum for sample 3. The numbers against the CR lines are the magnetic field values in Tesla. Arrows indicate the observed cyclotron peaks. Gray stripes are Reststrahlen bands. The ‘bandgap’ mark (more ...)
Typical CR spectra in sample 3 are plotted in Figure , and the overall data are presented in Figure together with calculated energies of CR transitions versus the magnetic field. There are four main lines in the spectra to be considered: α
, and U2
. The nature of lines α
is well known. As discussed above, line α
corresponds to the transition from the lowest LL in the conduction band
. Line β
corresponds to the interband transition from the top LL in the valence band to the conduction band
. The intensity of this line decreases in magnetic fields below 3 T (see Figure ) because of partial filling of the ‘final’ LL for this transition n
= − 1. To the best of our knowledge, this is the first observation on the interband transition in the HgTe QW with normal band structure. At present, such transitions have been observed in HgTe QWs with inverted band structure only (see, e.g., [6
]). It should be mentioned that the extrapolation of the spectral position of line β
= 0 gives slightly less bandgap (340 cm-1
) than that obtained from the photoconductivity spectrum (measured on another sample cut from the same wafer). Therefore, in our calculations, we used a compromised (between CR and photoconductivity data) QW width of 4.8 nm. Let us note that in this sample 3 with the narrowest QW, the linewidth of the interband transition (β
) exceeds significantly that of the intraband one (α
), while in broad QWs, they are approximately the same (see, e.g., Figure ; [7
]). To our opinion, a spreading of the interband line β
in sample 3 with narrow QW resulted from the enhanced role of one-monolayer fluctuations (about 0.5 nm) of this narrow QW width which, in turn, leads to bandgap fluctuations.
Figure 7 Energies of cyclotron transitions versus the magnetic field for sample 3. Solid lines correspond to the calculated transitions with adjusted parameters; thin dotted lines, with traditional parameters. Symbols are experimental data. The size of symbols (more ...)
The nature of the most intense low-frequency line in CR spectra U1 is not quite clear. It persists up to the maximal magnetic field used (11 T) when the LL filling factor is much less than unity, so it cannot be attributed to transitions between higher LLs in the conduction band. On the other hand, the transition energies are much less than the bandgap. Therefore, the only reasonable explanation is to attribute this absorption line to intracenter excitation of residual donors. In wide QWs (such as in sample 2), the shallow-donor binding energies are small compared to those of the CR ones because of small electron effective masses (of the order 10-2m0, where m0 is the free electron mass). However, in narrow QWs, the donor binding energies increase significantly since the QW potential pushes the donor wavefunction to the impurity ion. A weaker absorption line U2 seems to result from some impurity interband transition since the line is as broad as line β. As a whole, the accordance between measured and calculated data in sample 3 with the narrowest QW (Figure ) is worst than those in samples 1 and 2 with wider QWs. The latter means that the theoretical model for the description of such narrow QWs is to be elaborated.