Optical realization of magneto-intersubband oscillations

We report on the optical realization of the magneto-intersubband oscillations that have been measured in the sub-terahertz transmittance of a GaAs quantum well with two subbands occupied. Following their dc analogue, the oscillations are periodic in the inverse magnetic field with the period governed by the subband gap. Their magnitude and polarization dependence accurately follow the presented simplified version of the dynamic magneto-intersubband oscillations equation that naturally combines dc magneto-intersabband oscillations with microwave-induced resistance oscillations (MIRO). Simultaneously measured photoresistance also reveals its strong sensitivity to the sign of the circular polarization, proving the used theoretical modeling.

We report on the optical realization of the magneto-intersubband oscillations that have been measured in the subterahertz transmittance of a GaAs quantum well with two subbands occupied.Following their dc analogue, the oscillations are periodic in the inverse magnetic field with the period governed by the subband gap.Their magnitude and polarization dependence accurately follow the presented simplified version of the dynamic magneto-intersubband oscillations equation that naturally combines dc magneto-intersabband oscillations with microwave-induced resistance oscillations (MIRO).Simultaneously measured photoresistance also reveals its strong sensitivity to the sign of the circular polarization, proving the used theoretical modeling.
Transport and optical measurements are complementary probes to study solid state structures.Despite being quite different in realization and in observed effects, they can be expected to result in one-to-one correspondence regarding the physical properties of studied systems.In contrast to a variety of transport experiments in heterostructures, several their dynamic counterparts still have to be detected and studied.
One of such examples is given by the magnetointersubband oscillations (MISO), Fig. 1 (c).They are periodic in inverse magnetic field B oscillations of the resistance R of a high mobility quantum well with two populated subbands.The frequency of these oscillations is equal to the subband separation over the cyclotron energy ratio, ∆ 12 /hω c .When this ratio is equal to an integer number, the Landau level sets of each groups intersect that results in the strong scattering between the subbands, Fig. 1 (b).The prominent feature of MISO consists in the absence of their temperature damping.These resistance oscillations have been intensively studied both in theory [1][2][3] and experimentally [4][5][6][7][8] .
The presence of the external electromagnetic field modifies the magneto-intersubband oscillations, mixing them with the microwave-induced resistance oscillations (MIRO) [9][10][11][12][13] .In its turn, the microwave-induced resistance oscillations represent the results of the optical transitions between distant Landau levels.They inevitably modify the absorption and have been thoroughly studied in optics [14][15][16] .However, the optical equivalent of MISO, when magneto-intersubband oscillations interfere with MIRO, was only presented in Ref. 17, where due to technical limitations just a qualitative comparison with the theory 11 was made.The optical response, as a more direct and simple method to analyze the radiation absorption, has helped to solve the long-standing problem of the polarization immunity of MIRO [18][19][20] .However, at some conditions 21,22 , the polarization immunity has been observed in a more general phenomenon, the cyclotron resonance absorption.This particular finding contradicts the basics of the light-matter interaction and still does not have even a qualitative explanation.In this way, the optical realization of transport effects serves as a test for existing theories in the dynamic regime, and also estab-lishes the static-dynamic correspondence in two-dimensional quantum systems.This is especially important in connection with the growing interest and development of multilayer quantum materials and their applications in optical domain 23 .
Here we present an optical analog of the magnetointersubband oscillations measured in direct transmission signal in a high-quality GaAs quantum well with two subbands occupied.The oscillations have been detected simultaneously in optical and dc transport responses.The optical oscillations and their polarization dependence provide an experimental proof for the theoretical predictions given in Ref. 11.
The studied GaAs quantum wells have been grown by molecular beam epitaxy on a GaAs substrate [24][25][26][27] .The sample size was about 10×10 mm 2 , the Ohmic contacts at the corners were used in the van der Pauw geometry and they were fabricated by burning indium droplets.After exposure to the room light the total electron density and average mobility were n = 7.7 × 10 11 cm −2 and µ = 1.9 × 10 6 cm 2 /Vs, respectively.The optical measurements were performed in the Faraday geometry, see Fig. 1, the sample was irradiated from the substrate side through a 8-mm metal aperture.A backwardwave oscillator was used as a source of the normally incident continuous monochromatic radiation.The transmittance through the sample was measured using a He-cooled bolometer.In parallel with the transmittance, the photoresistance δ R (the difference of the resistance signals in the presence and absence of irradiation) was measured using the doublemodulation technique 16,20,28 .All presented results were obtained at radiation frequency f = ω/2π = 213 GHz and temperature 1.8 K.
In Fig. 1 (c) we show the magnetic field dependences of the sample resistance, measured when the radiation source is switched off (black) and on (red) at 213 GHz right-hand (σ + ) circular polarization (note the shift of the red curve).Let us first discuss the magnetoresistance without illumination with sub-terahertz light.Several types of oscillations can be seen in these data.First, high-frequency magneto-intersubband oscillations that survive up to the lowest magnetic fields.At higher magnetic fields they are supplemented by the Shubnikov -  de Haas oscillations from both subbands.These types of oscillations can be observed without sample irradiation.Lowpower sub-terahertz radiation adds a kind of cos(2πω/ω c ) modulation to the R(B) dependence (red curve in Fig. 1 (c)).This term originates from the radiation-induced transitions between distant Landau levels, which is the mechanism responsible for the microwave-induced resistance oscillations in onesubband systems 13,29 .Since the intersubband energy difference ∆ ∼ 10 meV (see later for an exact number) is higher than the radiation energy hω ≈ 0.9 meV, the MIRO-like modulation has a smaller frequency compared to the intersubband oscillations.

B (T)
In Fig. 1 (d) we show the magnetic field dependence of the transmittance |t| 2 measured using the right-hand (σ + ) and left-hand (σ − ) circular polarizations.Nearly zero transmittance at |B CR | ≈ 0.54 T corresponding to the cyclotron resonance allows us to determine the cyclotron effective mass m CR ≈ 0.071 m 0 .In Fig. 1 (e) we show the expanded view near zero magnetic field in Fig. 1 (d).Two types of oscillations can be seen in these data as well.Their amplitude is higher at positive (negative) magnetic fields for the σ + (σ − ) polarization, having the maximum radiation absorption at the cyclotron resonance field.The lower-frequency oscillations are governed by the ratio ω/ω c and they essentially represent the dynamic analogy of MIRO that was studied in Ref. 16.The higher-frequency oscillations are the optical realization of the magnetic-intersubband oscillations that are the main result of our work.
Let us have a closer look at the observed oscillations in transport and optics.The photoresistance δ R and the oscillating parts of the transmittance |t| 2 − ⟨|t| 2 ⟩ are shown in Fig. 2 (a) and (b), respectively.Both curves are sensitive to the degree of the circular polarization (see the discussion below), and the corresponding FFT curves are presented in Fig. 2 (c).From these data, it is evident that phototransport and transmittance reveal the same sets of oscillations, namely, the magneto-intersubband oscillations.Thereby, we can use the knowledge on the photoresistance oscillations to analyse the FFT frequencies.
The lowest oscillation frequency comes from the microwave oscillations and is equal to f MIRO = Bω/ω c = mω/e, giving the quasiparticle effective mass value m = 0.068m 0 .This effective mass is about 4% lower than the cyclotron effective mass m CR ≈ 0.071 m 0 that we attribute to the electronelectron interaction in the studied structure 16,[30][31][32][33][34] .Two other FFT peaks are given by the combined frequencies, f MISO ±  f MIRO , since just transport intersubband oscillations have nearly no sensitivity to temperature variations 11 and, hence, are not seen in photoresistance.However, since the scattering rate that governs the system conductivity has an intersubband contribution, the MIRO-induced photoresistance has this type of contribution as well.The formation of the combined frequencies f MISO ± f MIRO represents the interference of MIRO and MISO effects.The periodicity of the MISO governed by cos(2π∆ 12 /hω c ) = cos(πh|n 1 − n 2 |/eB) allows us to find the difference in subband densities Considering the total density from the Hall measurements equal to 7.7 × 10 11 cm −2 , we get n 1 = 5.5 × 10 11 cm −2 and n 2 = 2.2 × 10 11 cm −2 .These values coincide with the SdH analysis of the transport data, proving our model.Now, let us analyze the transmittance oscillations quantitatively.The magnetic field dependence of the transmittance |t(B)| 2 of the circularly polarized light through a dielectric slab containing an isotropic 2DES can be fitted using a standard equation 14,20 where σ ± is the dynamic conductivity for the two-subband system (which is specified later), plus and minus signs correspond to the right and left-handed circular polarization, respectively, Z 0 ≈ 377 Ω is the impedance of vacuum, and two complex parameters s 1 = 1/2(cos(kd) − iε −1/2 sin(kd)), s 2 = 1/2(cos(kd) − i √ ε sin(kd)) describe the Fabry-Pérot interference in the substrate, and are controlled by the wave number k, the thickness d and the dielectric constant ε of the substrate, respectively.
To reproduce the oscillations in the measured |t| 2 (B) dependence, we calculate σ ± taking into account the quantum correction to the conductivity due to the Landau quantization.Following Ref. 11, away from the cyclotron resonance, when µ|B CR ∓ B| ≫ 1, we obtain , where e = |e| is the elementary charge and In these expressions, ν tr j j ′ are the transport scattering rates for the transitions between subbands j and j ′ , ω c = e|B|/m, δ j = exp(−πν j /ω c ) are the Dingle factors expressed through the quantum scattering rates ν j = ν j j + ν 12 .
In order to compare the theory to experimental data, we simplify the equation by assuming that the transport and quantum relaxation rates are the same for both subbands: ν tr 11 = ν tr 22 and ν 11 = ν 22 .This assumption is justified since there is no strong classical magnetoresistance (Fig. 1 (c)), otherwise the simplified theory would not work, and the magnetoresistance itself can be used to extract the parameters 8 .These assumptions also mean that the Dingle factors are equal as well, δ 1 = δ 2 ≡ δ = exp(−π/µ q |B|), where µ q is the quantum mobility.In these approximations, the rate ν 0 = ν tr 11 + ν tr 12 coincides with the transport relaxation rate ν tr entering the static mobility, µ = e/mν tr ; otherwice the connection between ν 0 and ν tr is more complicated 8 .Finally, neglecting the imaginary part of δ σ ± in view of its relative smallness, Imδ σ ± /Imσ ± ≪ Reδ σ ± /Reσ ± , we obtain the final expression for the conductivity correction related to the magnetointersubband oscillations where γ = ν tr 12 /ν tr j j denotes the ratio of intresubband and intrasubband scattering rates, which is considered below as an adjustable parameter.In the case of only one subband being occupied (or there is no scattering between subbands) there are no intersubband oscillations, the second term in the brackates is equal to zero, and the remaining cos(2πω/ω c ) oscillations represent the quantum, MIRO-like, oscillations of transmittance that were studies in Ref. 16.If there are two subband occupied but there is no radiation applied, the first cosine is equal to unity and we obtain standard transport magneto-intersubband oscillations.If both factors are relevant, one can detect the dynamic magneto-intersubband oscillations (Fig. 2).
To fit the experimental |t| 2 (B) data shown in Fig. 1 (d) and (e) we combine the Drude conductivity σ D ± , Eq. 3 with its correction δ σ ± from Eq. 5, and insert the obtained total conductivity of the system to Eq. 1. Two fitting parameters, the quantum mobility µ q = 2 × 10 5 cm 2 /Vs and the relative strength of the intersubband scattering γ = 0.4, allow us to obtain a very good fit of the transmittance, which additionally proves the validity of the simplifications.Now we will turn to the observed polarization dependence of the measured photosignal.In Fig. 2 (a) we show the magnetic field dependences of the photoresistance measured at two circular polarizations.This polarization dependence is governed by the polarization dependence of the cyclotron absorption and does not depend on the exact mechanism of the oscillations 13,20 .Thus, the ratio of the photoresistance at positive and negative magnetic fields should be equal to the corresponding ratio of the cyclotron absorption, δ R(B)/δ R(−B) = A(B)/A(−B).The absorption can be calculated as A = Z 0 |t ± | 2 Re σ D ± , therefore, the photoresistance ratio for the first oscillations harmonics should be equal in our case to about 18 (at B ≈ 0.43 T).The experimentally obtained ratio is equal to only about 4, which most probably comes from polarization distortions due to metallic aperture.As has been investigated elsewhere 20 , the boundary conditions at the aperture lead to a local admixture of a linear polarization, which finally makes the δ R(B) dependence more symmetric.The experimentally observed asymmetry can be explained by 88% to 12% admixture of right and left circular polarizations, respectively.
In this work we demonstrated the optical analogy of the magneto-intersubband oscillations in the sub-terahertz transmittance signal.The detected oscillations precisely follow the theoretical predictions extended to the dynamic domain.This work, together with Refs.16 and 35, consistently demonstrates and proves the ability to detect and study the highfrequency dissipative transport features using contactless optical methods.Moreover, the dynamic oscillations reveal another advantage of studying the optical response to simultaneously get the information about the Drude parameters and their corrections by electron-electron correlations.Taking into account that the optical approach does not require high-quality structures and any specific fabrication processing, it can be used to test a variety of 2DES, including multi-valley systems and the steadily growing family of two-dimensional materials.
The observed strong polarization dependence of the photoresistance in a sub-THz frequency range together with the observation of the corresponding oscillations in the transmittance additionally prove the validity of the used theoretical models used in Ref. 11.
While presented experiments only prove the theoretical pre-dictions, there are established in dc transport effects that do not have high-frequency extensions yet.First, the study of the transmission in the conditions of zero resistance state (ZRS) 13 , when 4-point resistance is equal to zero.The current and not yet full understanding of this phenomenon 36 is based on the domain structure, when the local conductivity is either positive or negative that can be spontaneously switched 37 .
The switching leads to the breaking of the Drude approach, making such a study especially promising.Another possible direction for investigations is related to the optical realization of the weak localization effects 38 .Here, the dc studies of the effect in the absence and presence of the radiation were performed 39,40 .However, the optical measurements as well as the dynamic extension of equations are still missing.Finally, the recently discovered polarization immunity problem of the cyclotron absorption observed at some conditions 21 is also waiting for its explanation.All these and other studies of subtle optical properties of 2DES require a solid understanding of experimentally proved theories that, among the other things, we present in this work.

FIG. 1 .
FIG. 1.(a) Experimental setup, optical and phototransport measurements were done in parallel.(b) A Landau level fan chart for a twosubband system, E 1 and E 2 represent bottoms of two subbands, the subband spacing ∆ 12 = E 2 − E 1 .Arrows indicate the conditions of the resonant scattering between subbands when their Landau levels intersect.(c) Magnetic field dependences of the measured resistance R(B) when radiation is off (black) and on (red), the red curved is vertically shifted by 1.5 Ω for clarity.Microwaves result in the additional modulation of the resistance oscillations.(d) Magnetic field dependences of the transmittance |t ± | 2 (B) measured at f = 213 GHz using right-hand (black, σ + ) and left-hand (blue, σ − ) circularly polarized radiation.(e) A zoom-in of panel (d) near zero field, where the studied oscillations are clearly seen.The theory curves are based on Eqs.1-3, and 5.

FIG. 2 .
FIG. 2. (a) and (b) Magnetic field dependences of photoresistance δ R(B) and transmittance oscillations |t| 2 − ⟨|t| 2 ⟩ for right-hand (red, σ + ) and left-hand (blue, σ − ) circularly polarized radiation.A similar asymmetry for the curves at ±B is seen.(c) A fast Fourier transform (FFT) of the data from panel (a) and (b) for the right-hand circularly polarized radiation.The curves on all panels are vertically shifted for clarity.