Photo-oscillations in MgZnO/ZnO heterostructures

We theoretically examine the characteristics of microwave-induced magnetoresistance (MIRO) and photovoltage oscillations in MgZno/ZnO heterostructures. We demonstrate that both kind of oscillations, although described with different physical properties, are intimately related sharing the same physical origin. We use the radiation driven electron orbit model showing that the interplay of radiation driven swinging Landau orbits and the scattering processes are at the heart of the oscillations in both scenarios. Thus, our simulations show that all photo-oscillations present the main features of MIRO: they are periodic with the inverse of the magnetic field and the oscillations minima are 1/4 cycle shifted.


Theoretical model
The radiation-driven electron orbits model is a previously developed theoretical approach to explain both MIRO and ZRS that were observed in irradiated GaAsAl/GaAs heterostructures. One of the main results of this theory is that the Landau orbits are driven harmonically by radiation and the corresponding guiding center describes harmonic and classical trajectories on the 2D system. Accordingly, the interplay of this driven-harmonic motion OPEN 1 Escuela Politécnica Superior, Universidad Carlos III, Leganes, 28911 Madrid, Spain. 2  www.nature.com/scientificreports/ and scattering with sample disorder are at the core of photo-oscillations. Previous to irradiating the sample, electrons interact via scattering with the system disorder giving rise to resistance. In principle, the scattering is performed randomly in any direction leading to no net effect. Nevertheless, if there exists a DC electric field on the 2D sample, that can either be externally applied or built-in 48 , a definite direction is determined that, on average, will be followed by the electron when interacting with scatterers. In each scattering jump the scattered electron advances an average distance, X 0 along the DC field direction 49 . Thus, a certain current shows up that can be measured in terms of magnetoresistance R xx 1,2 or photovoltage 48 . The scattering scenario in the dark is deeply altered when the sample is illuminated because the Landau orbits oscillate 30,31,43 . This novel situation can be experimentally observed via R xx 1,2 or, more recently, via photovoltage 48,54 . Now, under radiation, the advanced distance or spatial shift, due to scattering, turns into a harmonic function according to the radiation-driven electron orbit model 30,31,43 , where w and w c are the radiation and cyclotron frequencies respectively and A is the oscillation amplitude, where E 0 is the radiation electric field amplitude. According to Eq. (1), the radiation-driven Landau states, perform a swinging motion where the electrons interact with the lattice ions resulting in a damping process. The latter is phenomenologically introduced through the γ damping term 30,31,43 .
If we now focus on the irradiated R xx , we have to calculate first the corresponding conductivity σ xx following a semiclassical Boltzmann approach 61-63 , being E the energy, ρ i (E) the density of initial Landau states and W I is the scattering rate of electrons with sample disorder. According to the Fermi's golden rule: where N i is the number of impurities, φ i and φ f are the wave functions corresponding to the initial and final Landau states respectively and V r is the disorder scattering potential. E i and E f stand for the initial and final energies. The V r matrix element is given by 61,63 : and the term I i,f 61,63 , where X 0 and X ′ 0 are the guiding centers of the initial and final Landau states respectively and q x the x-component of − → q , the electron momentum change after the scattering event.
In the MgZno/ZnO system the main source of disorder and scattering is no longer long-range Coulomb potential centers such as remote charged impurities in AlGaAs/GaAs heterostructures. Now, short-range potential disorder is the main source of scattering. The heterointerface between MgZnO and ZnO takes in most of the disorder due to the Mg atoms that diffuse into the ZnO inversion layer. Thus, to calculate W I we have considered a simple model of a 2D neutral impurity 64 (Mg atoms) based on a circular barrier of radius a. This constant potential is given by Thus, V 0 plays the role of the scattering potential. In the calculation of W I the Fourier transform V (| − → q |) of the potential V r , needs to be obtained and accordingly is given by 64 where J 1 is the first order Bessel function and S is the sample surface. Now we consider that | − → q | is small and thus V(q) no longer depends on q and the scattering becomes isotropic. Then V (| − → q |) takes the form, V (| − → q |) ≃ πa 2 V 0 /S . In our simulations we have used for a the effective Bhor radius 64 that in the case of ZnO is of the order of 2 nm. For V 0 we have used for a neutral impurity an estimate of 61,65 V 0 ∼ 50 meV. Finally, R xx is calculated according to the usual tensorial relations, R xx = σ xx σ 2 xx +σ 2 xy , where σ xy ≃ n i e B , n i being the electrons density, and e the electron charge. Then, and according to Eq. (3), the distance X is direct responsible of the rise of MIRO when measuring R xx .
The joint effect of radiation-driven Landau states and impurity scattering as origin of photo-oscillations can be revealed by measuring photovoltage instead of irradiated R xx 59 . As we have indicated above, in order to obtain a net scattering we need a predominant direction along which the scattering jump takes place. This direction is determined by a DC electric field acting on the sample. In the case of R xx this DC field is externally applied 1,2 . Nevertheless, in the case of photovoltage it can be either built-in 48 due, for instance to the presence in the sample  59 . In it, one of the squared sample edges is connected to an external positive DC voltage, +V 0 and then a definite scattering direction is determined in the sample (see Fig. 1). As a result two lines of opposite charge rise at facing sides. The lines width is of the order of the scattering spatial shift between Landau states 30,31,43 X 0 . Then, a voltage drop, V dark , is created along the sample and can be experimentally measured. An expression for V dark can be obtained, from basic electrostatics,  and instead of V dark there is a photovoltage, V ph given by an expression similar to V dark but with X instead of X 0 .

Results
In Fig. 2  In Fig. 3 we plot the radiation frequency dependence of irradiated magnetoresistance in MgZnO/ZnO. In the upper panel we present irradiated R xx vs B for nine different frequencies from 80 to 128 GHz. All the curves are 1/4 cycle shifted irrespective of the frequency. We observe that MIRO displace to higher B as frequency increases, increasing as well the number of oscillations. The extrema change their positions in the B axis according to a www.nature.com/scientificreports/ definite dependence; for the minima according to w/w c = 1/4 + j , j and for the maxima, w/w c = 3/4 + j , j. This dependence is revealed in the lower panel where we represent B versus radiation frequency for five cases corresponding to the extrema labelled in Fig. 2 lower panel. We have carried out fits on every group of points obtaining very clear straight lines, as expected according to the two previous formula that relate w and w c . The variation of extrema positions with B according to a straight line is another genuine characteristic of MIRO and has been experimentally observed in previous semiconductor platforms. Again the calculated results are in qualitatively good agreement with the experimental ones 54 . Figure 4 shows the power dependence of irradiated magnetoresistance in MgZnO/ZnO. In the upper panel we present irradiated R xx vs B for ten different power values ranging from 0 to 1350 µ W. As the power increases the MIRO amplitude rises too following a square root law: R xx ∝ P 0.5 . The P increase does not affect the oscillations position that keep constant. The power Law dependence is exhibited in the lower panel where we present R xx amplitude vs P. We fit the calculated points obtaining a square root dependence (dashed red curve) of R xx with P: R xx = 0.89P 0.48 . Despite the controversy between linear and square root dependence, we have to admit that a good number of experiments show a mixed behaviour between linear and sublinear. The most recent experiments with encapsulated monolayer graphene 56 , show a linear dependence at low power and as the latter increases, the dependence becomes sublinear.
We now focus on the photovoltage results. Thus, in the upper panel of Fig. 5 we exhibit photovoltage vs B for a frequency of 95 GHz and T = 1 K. The curve is qualitatively very similar to the one of irradiated R xx in Fig. 2a.
In the middle panel we exhibit photovoltage amplitude vs B. Both curves in (a) and (b) present the main characteristics of MIRO that were already obtained when calculating magnetoresistance: photovoltage turns out to be periodic with 1/B (not shown in the figure) and minima positions are 1/4 cycle shifted. The extrema positions coincide with the ones obtained in irradiated R xx . Finally in the lower one we observe that photovoltage follows a square root law when it comes to radiation power dependence.

Conclusions
Summing up, we have theoretically studied the microwave-induced resistance oscillations and photo oscillations experimentally found in MgZnO/ZnO heterostructures. We have used the radiation-driven electron orbit model to depict a common microscopic model for both kind of oscillations. We have come to the conclusion that the interplay between the radiation-driven Landau orbits that perform harmonic trajectories and the interaction with

Data availability
All data generated or analysed during this study are included in this published article.