Beatings of ratchet current magneto-oscillations in GaN-based grating gate structures: manifestation of spin-orbit band splitting
P. Sai, S. O. Potashin, M. Szola, D. Yavorskiy, G. Cywinski, P. Prystawko, J. Lusakowski, S. D. Ganichev, S. Rumyantsev, W. Knap, V. Yu. Kachorovskii
BBeatings of ratchet current magneto-oscillations in GaN-based grating gatestructures: manifestation of spin-orbit band splitting.
P. Sai , , , S. O. Potashin , M. Szo(cid:32)la , D. Yavorskiy , , G. Cywi´nski , , P. Prystawko , J. (cid:32)Lusakowski , S. D. Ganichev , , S. Rumyantsev , W. Knap , , and V. Yu. Kachorovskii , CENTERA Laboratories, Institute of High Pressure Physics,Polish Academy of Sciences, 01-142 Warsaw, Poland Centre for Advanced Materials and Technologies,Warsaw University of Technology, 02-822 Warsaw, Poland V. Ye. Lashkaryov Institute of Semiconductor Physics, 03680 Kyiv, Ukraine Ioffe Institute, 194021 St. Petersburg, Russia Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland Institute of High Pressure Physics, Polish Academy of Sciences, 01-142 Warsaw, Poland Terahertz Center, University of Regensburg, 93040 Regensburg, Germany and Laboratoire Charles Coulomb, University of Montpellier and Centrenational de la recherche scientifique, 34950 Montpellier, France (Dated: February 26, 2021, v.2)We report on the study of the magnetic ratchet effect in AlGaN/GaN heterostructures super-imposed with lateral superlattice formed by dual-grating gate structure. We demonstrate thatirradiation of the superlattice with terahertz beam results in the dc ratchet current, which showsgiant magneto-oscillations in the regime of Shubnikov de Haas oscillations. The oscillations havethe same period and are in phase with the resistivity oscillations. Remarkably, their amplitude isgreatly enhanced as compared to the ratchet current at zero magnetic field, and the envelope of theseoscillations exhibits large beatings as a function of the magnetic field. We demonstrate that thebeatings are caused by the spin-orbit splitting of the conduction band. We develop a theory whichgives a good qualitative explanation of all experimental observations and allows us to extract thespin-orbit splitting constant α SO = 7 . ± . I. INTRODUCTION
One of the most important tasks of modern optoelec-tronics is to provide efficient conversion of high-frequencyterahertz (THz) signals into a dc electrical response,for reviews see, e.g., Refs. [1–9]. In the last decades,the focus of research in this direction was on the peri-odic structures like field effect transistor (FET) arrays,grating gate, and multi-gate structures. Such structuresalso attract growing interest as simple examples of tun-able plasmonic crystals [10–14]. Plasmonic crystals al-ready demonstrated excellent performance as THz de-tectors [15–20], in close agreement with the numericalsimulations [21–25]. They are also actively studied aspossible emitters or amplifiers of THz radiation [26–28].Importantly, a non-zero dc photoresponse requiressome asymmetry of the system, which would determinethe direction of the produced dc current. Generation of dc electric current in response to ac electric field in sys-tems with broken inversion symmetry is usually calledthe ratchet effect which was studied both theoreticallyand experimentally in a great number of systems, for re-views see, e.g., Refs. [29–34]. For effective radiation con-version to dc signal in periodic structures, there shouldbe strong built-in asymmetry inside the unit cell of theplasmonic crystals. In particular, the ratchet dc cur-rent can be induced by the electromagnetic wave inci-dent on the spatially modulated system provided that the wave amplitude is also modulated but is phase-shiftedin space, for review see Ref. [30]. On the theoreticallevel, the ratchet current arises already in non-interactingapproximation (so called electronic ratchet). Electronicratchets were discussed in the two dimensional systemswith lateral gratings [30, 35–41] or arrays of asymmetricdots/antidots [42–45]. Electron-electron (ee) interactioncan dramatically increase ratchet current due to plas-monic effects [46–54].Although the ratchet effect was treated theoreticallyand observed experimentally in diverse low dimensionalspatially-modulated structures, some basic issues of thiseffect still remain puzzling. One of the interesting ques-tions that has not yet been discussed in the literature isthe manifestation of the effects of spin-orbit (SO) inter-action in the ratchet effect. In this paper, we addressthis question. We study ratchet effect in magnetic field(in what follows, we call it magnetic ratchet effect) inthe regime of Shubnikov de Haas (SdH) oscillations anddemonstrate that it is dramatically modified by SO in-teraction. Specifically, we report on the observation ofthe magnetic ratchet effect in the lateral GaN-based su-perlattice formed by dual-grating gate (DGG) structure.The specific property of the GaN systems as compared toother 2D structures including graphene based structuresis a very high value of Rashba spin-orbit coupling (at leastten times larger than in GaAs-based structures) [55–63],which is caused by high build-in electric field existing in a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b uch polar materials. This is, therefore, very favorablefor observations of SO-induced effects. We demonstrate,both experimentally and theoretically, that in quantizedmagnetic fields THz excitation results in the giant mag-netooscillations of the ratchet current coming from Lan-dau quantization, which, due to large SO band splitting,are strongly modulated as a function of the magneticfield. We reach a good qualitative agreement betweenour experimental results and theory (compare, respec-tively, Fig. 6 and Fig. 7 below).Our results provide a novel method to study the bandspin splitting. Currently the most widely used techniquesare direct measurements of magnetoresistance in SdHoscillations regime [64], the weak anti-localization ex-periments [65], optical methods [66] and photogalvanicstudies [67]. Since SdH oscillations in magnetoresistanceregime and ratchet current correlate, these two measur-ing methods are complementary, which gives the oppor-tunity to double check the results.The possibility to increase the ratchet effect in themagnetic field deserves special attention. Therefore, westart the paper with the discussion of the key points ofthe magnetic ratchet effect, see Sec. II. The rest of thepaper is organized as follows. In Sec. III we present theexperimental results on magnetic ratchet effects in GaN-based structures. In the following Sec. IV we presentthe theory and compare its results with the experimen-tal data. Sec. IV C is devoted to the discussion of theplasmonic effects. Finally, in Sec. V we summarize theresults. II. RATCHET EFFECT IN MAGNETIC FIELDS:STATE OF THE ART
Physics of the ratchet effect becomes much richer if oneapplies the magnetic field. Magnetic ratchet effect, whichis in some publications called magneto-photogalvanic ef-fect, was widely studied in different semiconductor sys-tems. The magnetic ratchet effect can be induced even inthe case of homogeneous graphene with structure inver-sion asymmetry, see e.g. [68–72]. The effect is sensitiveto disorder and can be tuned by the gate voltage [72].Furthermore, the theoretical consideration of the mag-netic ratchet effect in graphene and bi-layer grapheneshowed that it can be substantially enhanced under thecyclotron resonance (CR) condition [73]. Most recentlyit has been shown theoretically and observed experimen-tally that magnetic ratchet effect also can be drasticallyenhanced by deposition of asymmetric lateral potentialintroduced by asymmetric periodic metallic structure ontop of a structure with two dimensional electron gas [74–77]. Remarkably, magnetic ratchet strongly increases (bymore than two orders of magnitude) in the SdH oscil-lations regime. Physically, this happens due to a veryfast oscillations of the resistivity with the Fermi energy,and consequently, with the electron concentration. Asa result, inhomogeneous (dynamical and static) density modulations induced by the electromagnetic wave leadsto a very strong response.Importantly, the responsivity in the regime of SdH os-cillations increases not only in grating gate structuresbut also in single FETs [78–81]. Although the generalphysics of enhancement in both cases is connected withfast oscillations of resistivity, there is an essential differ-ence. In grating gate structures the shape of typical dc photoresponse roughly reproduces resistance oscillations,while in single FETs the typical response is π/ γ ( n ) in the SdH regime sharply depends on the dimen-sionless electron concentration n = ( N − N ) /N (here N is background concentration and N is the concentra-tion in the channel). Expanding γ ( n ) ≈ γ (0) + γ (cid:48) (0) n with respect to small n, one finds that a nonlinear term, γ (cid:48) (0) n v , appears in the Navier Stockes equation, where v is the drift velocity. This is sufficient to give a nonzeroresponse, which in a single FET arises in the second or-der with respect to external THz field (both n and v are linear with respect to this field). Therefore, in thiscase, the response is proportional to first derivative of γ (cid:48) (0) with respect to concentration (i.e. with respect toFermi energy, E F ), hence, it is π/ dc response appears only in thethird order with respect to perturbation [30]. As a conse-quence, the ratchet current is proportional to the secondderivative γ (cid:48)(cid:48) (0) (see discussion in Sec. IV) and thereforeroughly (up to a smooth envelope) reproduces resistanceoscillations.We will show that similar to other structures [74, 75,77], the amplitude of the magnetooscillations is greatlyenhanced as compared to the ratchet effect at zero mag-netic field. We experimentally demonstrate that the pho-tocurrent oscillates in phase with the longitudinal resis-tance and, therefore, almost follows the SdH oscillationsmultiplied by a smooth envelope. This envelope encodesinformation about cyclotron and plasmonic resonances.The most important experimental result is the demon-stration of the beatings of the ratchet current oscilla-tions. We interpret these beatings assuming that theycome from SO splitting of the conduction band. Thevalue of SO splitting extracted from the comparison ofthe experiment and theory is in good agreement withindependent measurements of SO band splitting [55–63].An important comment should be made about the roleof the ee interaction. Actually, the effect of the interac-tion is twofold. First of all, sufficiently fast ee collisionsdrive the system into the hydrodynamic regime. We as-sume that this is the case for our system and use the hy-drodynamic approach. Secondly, ee-interaction leads toplasmonic oscillations, so that a new frequency scale, theplasma frequency, ω p ( q ) appears in the problem, where q is the inverse characteristic of the spatial scale in the sys-2em. For a device with a short length, for example, for asingle FET, q is proportional to the inverse length of thedevice. For periodic grating gate structures, q = 2 π/L, where L is the period of the structure. At zero mag-netic field, the dc response is dramatically enhanced inthe vicinity of plasmonic resonance, ω = ω ( q ) both fora single FET with asymmetric boundary conditions [82]and for periodic asymmetric grating gate structures [54].Also, the response essentially depends on the polarizationof the radiation.Here, we calculate analytically the dc response in thequantizing magnetic field within the hydrodynamic ap-proximation for arbitrary polarization of the radiationand analyzed plasmonic effects. One of our main pre-diction is that for linearly polarized radiation, the de-pendence of the ratchet current on the direction of thepolarization appears only due to the plasmonic effects.We use the derived expression to prove that for specificparameters of our structures the plasmonic effects arenegligible, and as a consequence, the dc response doesnot depend on the polarization direction. The latter is-sue is very important for us since the direction of linearpolarization used in our experiment was not well con-trolled. We also argue how to modify the structures inorder to observe plasmonic resonances. III. EXPERIMENTA. Experimental details
We choose AlGaN/GaN heterostructure system for theexperimental study of the effect of SO splitting on mag-netic ratchet effect. Important unique properties of GaNsystem are the ability to form high density, high mo-bility two dimensional electron gas (2DEG) on the Al-GaN/GaN interface, and large Rashba spin splitting ofthe conduction band [55–63]. Density of 2DEG and theband spin splitting in this system are about an orderof magnitude higher than that in AlGaAs/GaAs system.High carrier density is an important factor because aswill be shown later the amplitude of the photoresponsein the regime of the SdH oscillations is proportional tothe square of electron density.AlGaN/GaN heterostructures were grown by Met-alorganic Vapour Phase Epitaxy (MOVPE) method inthe closed coupled showerhead 3 × . Ga . N barrier layer,1.5 nm Al . Ga . N spacer, 0.9 µ m unintentionallydoped (UID) GaN layers, and 2 µ m high resistive GaN:Cbuffer, see Fig. 1(a). Growth of all mentioned epilayerswas done on the bulk semi-insulating GaN substrates,grown by the ammonothermal method [83]. In thismethod high resistivity of substrates (typically no lessthan 10 Ω · cm) was obtained by compensation of resid-ual oxygen, incorporated during ammonothermal growth,by Mg shallow acceptors. B V PH V T G1 V T G2 GaN bulk substrateGaN:C buffer Regrown Ti/Al/Ni/Au SD GaN UIDNi/Au AlGaNbarrierAlGaNspacer2DEGregionlayer L L S S (a)(b) B V T G1 V T G2 B V XX ( с ) I DS THz
S SD D
FIG. 1. (a) Cross sectional view of the sample heterostruc-ture with DGG. (b) Scheme of the photoresponse measure-ments. (c) Scheme of the magnetoresistance measurements, R xx = V xx /I DS . μ m (a) μ m (b) TG1TG2S D
FIG. 2. Nomarski contrast microscope photos of investigatedasymmetric DGG (a), where TG1 and TG2 – two multi-fingertop gate electrodes, S and D – source and drain electrodesrespectively; magnified active region of asymmetric DGG (b). ◦ in a nitrogen atmosphere for 60 s.This procedure yielded reproducible ohmic contacts withresistances in the range of 0 . . · mm. Finishingfabrication step was the deposition of Ni/Au (100/300˚A) in order to form the DGG superlattice on the topof the AlGaN/GaN mesas. A schematic view and No-marski contrast microscope photos of fabricated devicesare shown in Figs. 1(a) and 2, respectively. The unit cellof the DGG superlattice consisted of two gates of differentlengths ( L = 1 . µ m and L = 3 . µ m) with differentspacings between them ( S = 2 . µ m and S = 5 . µ m).The cell was repeated 35 times resulting in a superlatticewith a total length of 500 µ m.All wide gates were connected forming the multi-fingertop gate electrode TG1, see Fig. 1(b,c) and Fig. 2(a).Similarly connected narrow gates formed the gate elec-trode TG2. Independent bias voltages ( V TG1 , V
TG2 )could be applied to wide and narrow gates. The width ofthe whole structure was 0.5 mm yielding the total activearea A = 0 .
25 mm . The total gate area was ≈ . .This is a large area, which is ∼ µ m and 100 µ m, respectively.This made very challenging the fabrication of the de-scribed DGG transistor with a reasonably small gateleakage current. Figure 3 shows two examples of thetransfer current voltage characteristics of the studied de-vices. Current in the subthreshold region is determinedby the gate leakage current (shown as a red dashed linefor one of the devices). As seen, the gate leakage cur-rent is rather small, significantly smaller than the draincurrent even at a very low drain voltage of V DS = 1 mV.Even for those devices with relatively high gate leakagecurrent ( f =630 GHzwas used to study the ratchet effect. The radiation wasguided onto the sample through a steel waveguide andwas modulated at a frequency of about 173 Hz. Externalmagnetic field up to 12 T was applied normally to 2DEGplane, as shown in Fig. 1(b). The photoresponse, V P H ,was measured in a cryostat at the temperature of 4.2 K inthe open circuit configuration using the standard lock-intechnique. Magnetoresistance was measured by applying - 4 - 3 - 2 - 1 0 101 x 1 0 - 6 - 6 - 6 - 6 - 6 - 6
Current (A)
G a t e v o l t a g e ( V )
T G 1 T G 2 V D S = 1 m V T = 4 . 2 K FIG. 3. Transfer current voltage characteristics for two rep-resentative devices. Red dashed line shows the gate leakagecurrent for one of the devices. a small < µ A current to the drain (see Fig. 1c).
B. Experimental results
First, we describe the results of the magnetotransportmeasurements, which are summarized in Figs. 4, and 5.The overall shape of the DGG structures was close tothe square. Therefore, in the magnetic field perpendic-ular to the drain to source plane investigated structuresexhibited geometrical magnetoresistance [85]. The fullgeometrical magnetoresistance is observed either in thedisk Corbino geometry or in the samples with
W >> L .For the arbitrary shaped rectangular samples, the geo- (cid:1)
Rxx / R B ( T ) m = 4700 cm / V s
FIG. 4. Magneto resistance as a function of B in weak mag-netic fields for a representative sample. Dashed line showsthe linear fit. . 1 0 0 . 1 5 0 . 2 0 0 . 2 5- 1 5- 1 0- 505 B ( T - 1 ) (cid:1) Rxx ( W ) V T G 1 = V T G 2 = 0 (cid:1)
Rxx (arb.units) n / 1 0 ( c m - 2 ) FIG. 5. Resistance as a function of the inverse magnetic field1 /B at V TG1 = V TG2 = 0. The inset shows shows the re-sult of the Fourier transform of the oscillations in Fig. 4 withfrequency taken in the units of the electron concentration, n = 2 eν/h . metrical magneto resistance can be approximated as [86]: (cid:52) RR ∼ = ( µB ) (cid:18) − . LW (cid:19) . (1)This allowed us to extract the electron mobility. Fig-ure 4 shows the experimental dependence of the magne-toresistance obtained for a weak magnetic field for oneof the samples as a function of B . The estimate yields µ = 4700 cm /Vs.The concentration in the channel can be extracted fromthe magneto resistance in the higher magnetic fields. Fig-ure 5 shows the resistance SdH oscillations as a functionof the inverse magnetic field 1 /B measured for both topgate voltages equal to zero. The concentration is givenby N = 2 eh (cid:52) (1 /B ) = 2 eνh , (2)where (cid:52) (1 /B ) and ν are the period and frequencyof SdH oscillations, respectively. The inset in Fig. 5shows the result of the Fourier transform of the resis-tance magnetooscillations with the frequency taken inthe units of the electron concentration, N = 2 eν/h .Two peaks in the inset correspond to the concentrations N = 9 . × cm − and N = 8 . × cm − , whichwe interpret as electron densities in the gate free andunder the gate areas, respectively.Irradiating the unbiased structures we detected a pho-tosignal caused by the generation of the ratchet pho-tocurrent. Figure 6(a) shows the photoresponse mea-sured for asymmetric gate voltages applied: V TG1 = − V, V
TG2 = 0. The photoresponse current was cal-culated as I xx = V PH /R xx . In low and zero magneticfields the response is positive and weakly depends onthe magnetic field. When gate voltages were changedto V TG1 = 0 , V
TG2 = − V the magnitude of the responsewas approximately the same, but of the negative sign.The change of the signal sign upon inversion of the lat- ( c )( b ) Photocurrent
Ixx (nA)Photocurrent
Ixx (nA) B ( T ) B ( T ) Photocurrent
Ixx (nA) B ( T ) ( a ) - 1 5- 1 0- 5051 01 5 V T G 1 (cid:1) (cid:3) (cid:1) (cid:5) (cid:2) (cid:4) V T G 2 = 0 V V T G 1 (cid:1) (cid:3) (cid:1) (cid:5) (cid:2) (cid:4) V T G 2 = 0 V (cid:1)
Rxx ( W ) (cid:1) Rxx ( W ) (cid:1) Rxx ( W ) V T G 1 (cid:1) (cid:3) (cid:1) (cid:5) (cid:2) (cid:4) V T G 2 = 0 V - 1 5- 1 0- 5051 01 5 - 2 . 0- 1 . 5- 1 . 0- 0 . 50 . 00 . 51 . 0
FIG. 6. Photocurrent and resistance SdH oscillations as afunction of magnetic field from B = 0 to B = 12 T (a). (b)and (c) show close look to intermediate and low magnetic fieldregions, respectively. | E ( x ) | dV ( x ) dx , (3)where V ( x ) and E ( x ) are spatially modulated by gratinggates static potential and electric field amplitude (over-line shows average over the modulation period). Ex-change of the gate voltages applied to the TG1 and TG2results in the change of sign of dV /dx and, consequentlyin the sign inversion of the ratchet current.The increase of magnetic field results in the sign-alternating oscillations with the amplitude by orders ofamplitude larger than the signal obtained for zero mag-netic field. Moreover, the envelope of the oscillationsexhibits large beatings as a function of magnetic field.Comparison of the observed oscillations with the SdHmagnetoresistance oscillations demonstrates that at highmagnetic fields both, photocurrent and resistivity oscilla-tions, have the same period and phase. Importantly, SdHeffect also show similar beatings of the envelope function.To facilitate the comparison of the phase of the oscilla-tions, we zoom the data of the panel (a) for the range offields B = 6 ÷
10 T in the panel (b) Fig. 6.The overall behavior of the observed current, besidesbeatings, corresponds to that of the magneto-ratchetcurrent most recently detected in CdTe-based quantumwells [74, 75] and graphene [77]. Importantly, the oscilla-tions of the magneto-ratchet current are in phase with theSdH oscillations. As we discussed in Sec. II, this differsfrom the photocurrent magnetooscillations detected insingle transistors [81] described by the theoretical modelof Lifshits and Dyakonov [79]. We will show in Sec. IVthat the theory of the magneto-ratchet effect predicts thephotoresponse to be proportional to the second deriva-tive of the magnetoresistance, and, hence describes wellthe experimental findings. Note, that in the low mag-netic field range, see Fig. 6(c), the oscillations of thephotoresponse and magnetoresistance are phase shifted.This is an indication that at low magnetic fields we haveat least two different competing mechanisms of the de-tection. Below we focus on the case of sufficiently highmagnetic fields and postpone discussion of possible com-peting mechanisms at low fields for future research. Im-portantly, the theory demonstrates that, in agreementwith the experiment (see Fig. 6) the photoresponse inthe regime of SdH oscillations is significantly enhancedas compared to that at zero magnetic field. This point isvery important in view of possible applications and de-serves a special comment. The theoretical limit for theresponse of the detectors based on the direct rectificationis defined by the device built-in nonlinearity. In Schot-tky diodes and FETs the maximum current responsivityis approximately e/ ηkT , where e is the elemental chargeand η is the ideality factor of the Schottky barrier or sub-threshold slope of FET transfer characteristic [87, 88]. The responsivity of the real device is usually orders ofmagnitude smaller due to the parasitic elements and notperfect coupling. However, increasing of the theoreticallimit still should be beneficial for the increasing of the re-sponsivity in real devices. Reducing temperature indeedleads to the responsivity increase but only to a certainlimit. As shown in [89] the temperature decrease below30K does not lead to the increase of responsivity. Thislow-temperature saturation is caused by the increase offactor η with temperature decrease which is known inSchottky diodes and FETs [90]. In the magnetic field un-der the regime of SdH oscillations resistance of FET verysharply depends on the gate voltage and provides the op-portunity to go beyond e/ ηkT limit. We can speculatethat about an order of magnitude increase of the respon-sivity in an external magnetic field in Fig. 6 demonstratesthe increase of the physical responsivity beyond the fun-damental limit.Importantly, the theory presented below also describeswell the observed oscillations of the envelope amplitudeof the photoresponse, which are clearly seen in Fig. 6. Itshows that the oscillations are due to the spin-splittingof the conduction band and can be used to extract thisimportant parameter. IV. THEORY
Above, we experimentally demonstrated that the pho-tocurrent oscillates and the shape of these oscillationsalmost follow the SdH resistivity oscillations. In this sec-tion, we consider gated 2DEG and demonstrate that theresults can be well explained within the hydrodynamicapproach.The effect which we discuss here is present for the sys-tem with an arbitrary energy spectrum. However, calcu-lations are dramatically simplified for the parabolic spec-trum, so that we limit calculations to this case only. Weassume that electron density in the structure is periodi-cally modulated by built-in static potential and study op-tical dc response to linearly-polarized electromagnetic ra-diation which is also spatially modulated with the phaseshift ϕ with respect to modulation of the static potential.A variation of individual gate voltages of the DGG lat-eral structure allows one to change controllably the signof V ( x ) and, consequently, the direction of the ratchetcurrent. Furthermore, the phase of the oscillations of themagnetic ratchet current is sensitive to the orientationof the radiation electric field vector with respect to theDGG structure as well as to the radiation helicity. In thelatter case, switching from right- to left-circularly polar-ization results in the phase shift by π , i.e., at constantmagnetic field the helicity-dependent contribution to thecurrent changes the sign. We consider magnetic field in-duced modification of the zero B -field electronic ratcheteffect. We also analyze our results theoretically for dif-ferent relation between ω and ω p ( q ) having in mind tofind signature of the plasmonic effects.6 . Model We model electric field of the radiation, E ( x, t ) = E ( x ) e − iωt + c.c., and the static potential, V, as follows E x ( x, t ) = [1 + h cos( qx + ϕ )] E cos α cos ωt, (4) E y ( x, t ) = [1 + h cos( qx + ϕ )] E sin α cos( ωt + θ ) , (5) V ( x ) = V cos qx, (6)where h (cid:28) ϕ is the phase,which determines the asymmetry of the modulation, α and θ are constant phases describing the polarization ofthe radiation. These phases are connected with the stan-dard Stockes parameters (normalized by E ) as follows: P = 1 , P L1 = sin(2 α ) cos θ,P L2 = cos(2 α ) , P C = sin(2 α ) sin θ. (7)Within this model, asymmetry parameter [see Eq. (3)]becomes Ξ = E hV q sin ϕ . (8)As seen, Ξ is proportional to the sine of the spatial phaseshift ϕ. Hydrodynamic equations for concentration and veloc-ity look ∂n∂t + div [(1 + n ) v ] = 0 , (9) ∂ v ∂t + ( v ∇ ) v + γ ( n ) v + ω c × v + s ∇ n = a . (10)Here n = N − N N , (11) N = N ( x, t ) is the concentration in the channel and N its equilibrium value, a = − e E m + em ∇ V, E = (cid:20) E x E y (cid:21) , (12) ω c = e B /m eff c is the cyclotron frequency in the exter-nal magnetic field B , s is the plasma waves velocity, γ ( n ) = 1 /τ tr ( n ) is momentum relaxation rate. The non-linearity is encoded in hydrodynamic terms ∂ ( n v ) /∂x, ( v ∇ ) v as well as independence of transport scatteringrate on the concentration. Specifically, we use the ap-proach suggested in Ref. [79]. We assume that γ ( n )is controlled by the local value of the electron concen-tration, n, which, in turn, is determined by the localvalue of the Fermi energy, n ( r ) = ( E F ( r ) − E ) /E (herewe took into account that the 2D density of states isenergy-independent). Due to the SdH oscillations, scat-tering rate is an oscillating function of E F , and, conse-quently, oscillates with n. In the absence of SO coupling, γ ( x, t ) = γ [ n ( x, t )] is given by [79] γ ( x, t ) = γ (cid:26) − δ cos (cid:20) πE F ( x, t ) (cid:126) ω c (cid:21)(cid:27) , (13) where δ = 4 χ sinh χ exp (cid:18) − πω c τ q (cid:19) (14)is the amplitude of the SdH oscillations, χ = χ ( ω c ) = 2 π T (cid:126) ω c ,T is the temperature in the energy units, E F ( x, t ) = E F [1 + n ( x, t )] is the local Fermi energy, which is relatedto concentration in the channel as N ( x, t ) = νE F ( x, t )(here ν is the density of states), and τ q is quantumscattering time, which can be strongly renormalized byelectron-electron collisions in the hydrodynamic regime.We assume that 2 π T + π (cid:126) /τ q (cid:29) (cid:126) ω c . Then, δ (cid:28) , (15)and the second term in the curly brackets in Eq. (13)is very small. Hence γ ( n ) is very close to the value oftransport scattering rate γ at zero magnetic field.Eq. (13) can be generalized for the case of non-zero SOcoupling by using results of Refs. [75, 91, 92] γ ( n ) = γ (cid:20) − χ sinh χ (16) × exp (cid:18) − πω c τ q (cid:19) cos (cid:18) πE F (1 + n ) (cid:126) ω c (cid:19) cos (cid:18) π ∆ (cid:126) ω c (cid:19)(cid:21) . Here we assumed that there is linear-in-momentum spin-orbit splitting of the spectrum, E ( k ) = (cid:126) k / m ± ∆ , where ∆ = α SO k F , (17)is characterized by coupling constant α SO . Experimen-tally measured values of α SO lays between 4-10 meV˚A[55–63]. For such values of α SO and typical values ofthe concentration, one can assume ∆ (cid:28) E F and ne-glect dependence of k F on n . Equation (16) was de-rived under the assumption that the quantum scatter-ing time τ q is the same in two spin-orbit split subbands.Actually, this assumption is correct only for the modelof short range scattering potential where both transportand quantum scattering rates are momentum indepen-dent. For any finite-range potential, the quantum scat-tering times in two subbands differ, because of small dif-ference of the Fermi wavevectors k = (cid:112) m ( E F + ∆) / (cid:126) and k = (cid:112) m ( E F − ∆) / (cid:126) . Denoting these times as τ and τ , we get instead of Eq. (16): γ ( n ) γ = 1 − χ sinh χ (cid:88) i =1 , exp (cid:18) − πω c τ i (cid:19) cos (cid:20) πE i ( n ) (cid:126) ω c (cid:21) , (18)where E ( n ) = E F (1 + n ) + ∆ , E ( n ) = E F (1 + n ) − ∆ . (19)7etailed microscopical calculation of τ , for specificmodel of the scattering potential is out of the scope ofthe current work. Here, we use τ , as fitting parameters.Let us now expand γ ( n ) near the Fermi level: γ ( n ) = γ (0) + γ (cid:48) (0) n + γ (cid:48)(cid:48) (0) n , (20)where γ (cid:48) and γ (cid:48)(cid:48) are, respectively, first and second deriva-tives with respect to n taken at the Fermi level. Sinceoscillations are very fast, we assume γ (cid:48) γ ∝ γ (cid:48)(cid:48) γ (cid:48) ∝ E F (cid:126) ω c (cid:29) . (21)Due to these inequalities oscillating contribution to theratchet current can be very large and substantially exceedzero-field value [77].Here, we focus on SdH oscillations of the ratchet cur-rent, so that we only keep oscillations related to depen-dence of γ on n and, moreover, skip in Eq. (20) the termproportional to γ (cid:48) . We use the same method of calculation as one de-veloped in Ref. [54]. Specifically, similar to impurity-dominated regime [30] we use the perturbative expansionof n and v and dc current, J dc = − eN (cid:104) (1 + n ) v (cid:105) t,x (22)over E and V. Non-zero contribution, ∝ E V , arises inthe order (2 ,
1) [see Eq. (3)].
B. Calculations and results
Let us formulate the key steps of calculations. Due tothe large parameter, E F / (cid:126) ω c (cid:29) , the main contributionto the rectified ratchet current comes from the non-linearterm γ (cid:48)(cid:48) v n / n and v in linear(with respect to E and V ) approximation, substitutingthe result into non-linear term and averaging over timeand coordinate, we get γ (cid:48)(cid:48) (0) (cid:104) v n (cid:105) x,t / (cid:54) = 0 . Next, onecan find rectified current J dc by averaging of Eq. (10)over t and x . This procedure is quite standard, so thatwe delegate it to the Supplemental Material (similar cal-culations were performed in Ref. [54] for zero magneticfield). The result reads J dc J = γ (cid:48)(cid:48) (0) γ R . (23)Here J = − (cid:18) eE m (cid:19) (cid:18) eV q ms (cid:19) eN h sin ϕγ (24)is the frequency and magnetic field independent parame-ter with dimension of the current (physically, J gives the typical value of current for the case,when all frequenciesare of the same order, ω ∼ ω c ∼ qs ∼ γ ∼ /τ , ), thedimensionless factor γ (cid:48)(cid:48) /γ accounts for SdH oscillationsand dimensionless vector R = γ ( P a + P L1 a L1 + P L2 a L2 + P C a C ) | ω − ( ω − iγ ) | ( γ + ω )( γ + ω ) | D ωq | . (25)depends on the radiation polarization encoded in the vec-tors a i = (cid:20) a ix a iy (cid:21) ( i = 0 , L1 , L2 , C) and also contains information about cy-clotron and magnetoplasmon resonances which occur for ω = ω c and ω = (cid:112) ω + s q , respectively. The latterresonance appears due to the factor D ωq in the denomi-nator of Eq. (25). Analytical expressions for a i and D ωq are quite cumbersome and presented in the Supplemen-tary material [see Eqs. (47), (48), (49), (50) and (51)].The second derivative of the scattering rate with re-spect to n is calculated by using Eq. (18): g ( ω c ) = γ (cid:48)(cid:48) (0) γ = 2 χ ( ω c )sinh[ χ ( ω c )] (cid:18) πE F (cid:126) ω c (cid:19) (26) × (cid:88) i =1 , e − π/ω c τ i cos [2 πE i (0) / (cid:126) ω c ] . Here, E (0) = E F + ∆ , E (0) = E F − ∆ [see Eq. (19)].The function g ( ω c ) rapidly oscillates due to the factorscos [2 πE i (0) / (cid:126) ω c ] . For ω c → − π/ω c τ i ) , so that discussedmechanism has nothing to do with the zero field ratcheteffect.The smooth envelope of the function g ( ω c ) reads˜ g ( ω c ) = 2 χ ( ω c )sinh[ χ ( ω c )] (cid:18) πE F (cid:126) ω c (cid:19) (27) × (cid:12)(cid:12)(cid:12) e − π/ω c τ + 2 πi ∆ / (cid:126) ω c + e − π/ω c τ − πi ∆ / (cid:126) ω c (cid:12)(cid:12)(cid:12) . Function g ( ω c ) shows rapid SdH oscillations with thebeats due to the spin orbit coupling. As seen from thebehavior of the envelope function ˜ g ( ω c ) , the beats aremost pronounced for τ = τ , when ˜ g ( ω c ) is proportionalto cos(2 π ∆ / (cid:126) ω c ) and therefore vanishes at the values of ω n c , obeying 2 π ∆ /ω n c = π/ πn. For τ (cid:54) = τ , envelopefunction is nonzero at these points, ˜ g ( ω n c ) (cid:54) = 0 , and beatsare less pronounced.Now, we are ready to explain why the response in theSdH oscillation regime is much larger than at zero mag-netic field. The enhancement of the response as com-pared to the case B = 0 , is due to the factor δ (cid:18) πE F (cid:126) ω c (cid:19) (cid:29) . (28)One can check that for experimental values of parametersinequalities Eq. (15) and (28) are satisfied simultaneously8 IG. 7. Theoretically calculated ratchet magnetooscillationsfor the following parameters: (cid:15) = 9 , α SO = 7 . T =4K, m eff = 0 . m e , d = 2 . · − cm, L = 15 · − cm, N =8 · cm − , α = 0 , τ tr = 1 . × − s , τ = 1 . × − s , τ =10 − s. in a wide interval of magnetic fields, 1 < B < E in the g ( ω c )the response increases with the concentration in contrastto conventional transistor operating at B = 0 , where re-sponse is inversely proportional to the concentration athigh concentration [82] and saturates at low concentra-tion when a transistor is driven below the threshold [93].This means that the use of AlGaN/GaN system for de-tectors operating in the SdH oscillation regime is veryadvantageous because of the extremely high concentra-tion of 2DEG.Let us discuss the polarization dependence of the re-sponse. Importantly, vectors a i , responsible for polariza-tion dependence, contain q -independent terms and termsproportional to ω q = s q . The latter describe plasmoniceffects. As seen from Eqs. (47),(48), (49), for small q (or/and small s ), vectors a L1 and a L are small, ∝ q . In other words, for our case of linearly-polarized radia-tion with polarization directed by angle α, the depen-dence of the rectified current on α appears only due tothe plasmonic effects. For experimental values of the pa-rameters, the value of plasmonic frequency, sq was suffi-ciently small, about 0 . · s − , which is much smallerthan the radiation frequency (for f = 0 . ω = 2 πf ≈ . · s − ). As follows from this esti-mate, the plasmonic effects are actually small and canbe neglected.Then, the response does not actually depend on polar-ization angle α. This justifies our experimental approach,where α is not well controlled. Within this approxima-tion, one can put q → P C = 0), we get (cid:20) J x dc J y dc (cid:21) = 2 J g ( ω c ) γ ω c ( γ + ω ) | ( ω + iγ ) − ω | (cid:20) − ω c γ (cid:21) . (29) FIG. 8. Envelope of the ratchet current magnetooscillationsfor different ratio of quantum times τ and τ (other param-eters are the same as in Fig. 7). This expression simplifies even further in the resonantregime, ω ≈ ω c (cid:29) γ : (cid:20) J x dc J y dc (cid:21) = J g ( ω c ) γ ω [( ω − ω c ) + γ ] (cid:20) − ω c γ (cid:21) . (30)This expression shows rapid oscillations, described byfunction g ( ω c ) , which envelope represent a sharp CR withthe width γ. Let us now compare theoretical results with experi-mental observations. In Fig. 7 we plot the x − componentof the rectified dc current (this component was actuallymeasured in the experiment), calculated with the use ofEq. (23), as a function of the magnetic field. We as-sumed that the radiation is linearly-polarized along x axis[ α = θ = 0, P C = P L1 = 0 , P L2 = 1 , see Eqs. (4),(5), and FIG. 9. Envelope of the ratchet current magnetooscillationsfor different values of the quantum times τ and τ with thefixed ratio τ /τ (other parameters are the same as in Fig. 7). IG. 10. Envelope of the ratchet current magnetooscilla-tions for different values of the electron concentration (otherparameters are the same as in Fig. 7). (7)] and used experimental values of parameters: m eff =0 . m e , d = 2 . · − cm, L = 15 · − cm, (cid:15) = 9 , T = 4K, n = 8 · cm − , τ tr = γ − = 10 − s, ω = 3 . · s − .The best fit was obtained for α SO = 7 . ± . τ , as the fitting parameterschoosing τ tr = 1 . τ = 1 . τ . We see that exactly thisbehavior is observed in the experiment (see Fig. 6). Mostimportantly, we reproduce experimentally observed beatsof SdH oscillations using the value of α SO consistent withprevious experiments. In Figs. 8, 10, and 9 we show de-pendence of the smooth envelope of the current, ˜ J x , onmagnetic field for different values of τ , and differentconcentrations. As we explained above, the most pro-nounced modulation is obtained for τ = τ (see Fig. 8).Dependence on concentration appears both due to thefactor ( E F / (cid:126) ω c ) in g ( ω c ) and due to the dependence of∆ on k F . Evidently, Eq. (29) can be presented as a product ofa smooth function describing cyclotron resonance (CR)and rapidly oscillating function, which encodes informa-tion about SO splitting. This is illustrated in Fig. 11.
C. Role of the plasmonic effects
Above, we demonstrated that plasmonic effects can beneglected for our experimental parameters and, as a con-sequence, the response is insensitive to the direction ofthe linear polarization. However, the role of plasmoniceffects is not fully understood. The point is that theexisting ratchet theory assumes a weak coupling with adiffraction grating. In such a situation, the plasmon wavevector, which determines plasma oscillations frequency,is set by the total lattice period: q = 2 π/L . In the ex-periment, L = 13.95 µ m, i.e. is very large and, as a FIG. 11. DC response for small q [shown in bottom panel,described by Eq. (29)] is given by the product of smoothfunction R x [blue curve in upper panel], which shows CRand rapidly oscillating function g ( ω c ) [orange curve in upperpanel, described by Eq. (26)], which contains beats of SdHoscillations. consequence, the plasma frequency corresponding to thefull period is small. This frequency does not appear inthe experiment, as follows from the theoretical picturespresented above (see Fig. 12a). If we assume, that thecoupling is not so weak, then the plasmons determinedonly by the gate region, should show up. Then q is de-termined only by the gate length L g < L and it shouldmanifest itself, as it is shown in Fig. 12. Namely, plasmawave effects should lead to the plasmonic splitting of theCR.The role of the plasmonic effects can be enhanced ei-ther by decreasing the period of the structure, which im-plies increasing of q or by decreasing transport scatteringrate γ as illustrated in Fig. 12, where cyclotron and mag-netoplasmon resonances in the smooth function R x [seeEq. (25)] are shown for different values of L and γ. Al-though, for large values of γ (Fig. 12a) plasmonic effectsare fully negligible for small q (large L ) and function R x shows only CR at ω c = ω, with decreasing L there ap-pears a week plasmonic resonance at ω c = (cid:112) ω − s q . For smaller γ (Fig. 12 b, c), both cyclotron and magne-toplasmonic resonances become sharper. For very small γ (Fig. 12c), plasmonic resonance appears even for verysmall q. IG. 12. Cyclotron and magnetoplasmon resonances inthe smooth function R x [see Eq. (25)] for different values ofwavevector q = 2 π/L (determined by size of the unit cell L ) and transport scattering rate γ. For large values of γ (a)plasmonic effects are fully negligible for small q (large L ) andfunction R x shows only CR at ω c = ω (this case correspondsto our experimental situation). With decreasing L there ap-pears a week plasmonic resonance at the value of ω c givenby (cid:112) ω − s q . For smaller γ (b,c), both cyclotron and mag-netoplasmonic resonances become sharper. For very small γ (d), plasmonic resonance appears even for very small q . V. CONCLUSION
To conclude, we presented observation of the magneticratchet effect in GaN-based structure superimposed withlateral superlattice formed by dual-grating gate (DGG)structure. We showed that THz excitation results in thegiant magnetooscillation of the ratchet current in theregime of SdH oscillations. The amplitude of the oscil-lations is greatly enhanced as compared to the ratcheteffect at zero magnetic field. We demonstrate that thephotocurrent oscillates as the second derivative of thelongitudinal resistance and, therefore, almost follow theSdH resistivity oscillations multiplied by a smooth enve-lope. This envelope encodes information about cyclotronresonance. One of the most important experimental re-sults is the demonstration of beats of the ratchet cur-rent oscillations. We interpret these beats theoreticallyassuming that they come from SO splitting of the con-duction band. The value of SO splitting extracted fromthe comparison of the experiment and theory is in goodagreement with independent measurements of SO bandsplitting. We also discuss conditions required for the ob-servation of magnetoplasmon resonances.
VI. ACKNOWLEDGEMENTS
A financial support of the IRAP Programme of theFoundation for Polish Science (grant MAB/2018/9,project CENTERA) is greatfully acknowledge.The partial support by the National Science Cen-tre, Poland allocated on the basis of Grant Nos.2016/22/E/ST7/00526, 2019/35/N/ST7/00203 andNCBIR WPC/20/DefeGaN/2018 is acknowledged.S.D.G. thanks DFG-RFBR project (numbers Ga501/18-1 and 21-52-12015) and Volkswagen Stiftung Coopera-tion Program (97738). The work of S.P. and V.K. wasfunded by RFBR, project numbers 20-02-00490 and21-52-12015, and by Foundation for the Advancement ofTheoretical Physics and Mathematics.11
1] S. A. Maier,
Plasmonics: Fundamentals and Applications (Springer, 2007).[2] D. K. Gramotnev, S. I. Bozhevolnyi, Nature Photonics , 83 (2010).[3] F. H. L. Koppens, D. E. Chang, F. Javier Garcia deAbajo, Nano Lett. , 3370 (2011).[4] Joerg Heber, Nature Materials , 745 (2012).[5] A. N. Grigorenko, M. Polini, K. S. Novoselov, NaturePhotonics , 749 (2012).[6] Peter Nordlander, Nature Nanotechnology , 76 (2013).[7] M. Glazov and S. Ganichev, Phys. Rep. , 101 (2014).[8] Koppens, F. H. L.; Mueller, T.; Avouris, P.; Ferrari, A.C.; Vitiello, M. S.; Polini, M. Nature Nanotechnology ,780 (2014).[9] Jacob B. Khurgin, Nature Nanotechnology , 2 (2015).[10] G. C. Dyer, G. R. Aizin, S. Preu, N. Q. Vinh, S. J. Allen,J. L. Reno, and E. A. Shaner, Phys. Rev. Lett. ,126803 (2012).[11] G. R. Aizin, G. C. Dyer, Phys. Rev. B , 232108 (2012).[13] Gregory C. Dyer, Gregory R. Aizin, S. James Allen, Al-bert D. Grine, Don Bethke, John L. Reno, and Eric A.Shaner, Nature Photonics , 925 (2013).[14] Lin Wang, Xiaoshuang Chen, Weida Hu1, Anqi Yu, andWei Lu, Appl. Phys. Lett. , 243507 (2013).[15] X. G. Peralta, S. J. Allen, M. C. Wanke, N. E. Harff, J.A. Simmons, M. P. Lilly, J. L. Reno, P. J. Burke, and J.P. Eisenstein, Appl. Phys. Lett. , 1627 (2002).[16] E. A. Shaner, Mark Lee, M. C. Wanke, A. D. Grine, J.L. Reno, and S. J. Allen, Appl. Phys. Lett. , 193507(2005).[17] E. A. Shaner, M. C. Wanke, A. D. Grine, S. K. Lyo, J.L. Reno, and S. J. Allen, Appl. Phys. Lett. , 181127(2007).[18] A. V. Muravjov, D. B. Veksler, V. V. Popov, O. V. Polis-chuk, N. Pala, X. Hu, R. Gaska, H. Saxena, R. E. Peale,and M. S. Shur, Appl. Phys. Lett. , 042105 (2010).[19] G. C. Dyer, S. Preu, G. R. Aizin, J. Mikalopas, A. D.Grine, J. L. Reno, J. M. Hensley, N. Q. Vinh, A. C.Gossard, M. S. Sherwin, S. J. Allen, and E. A. Shaner ,Appl. Phys. Lett., , 083506 (2012).[20] X. Cai, A. B. Sushkov, R. J. Suess, M. M. Jadidi, G.S. Jenkins, L. O. Nyakiti, R. L. Myers-Ward, S. Li, J.Yan, D. K. Gaskill, T. E. Murphy, H. D. Drew, and M.S. Fuhrer, Nat. Nanotechnol. , 814 (2014).[21] G. R. Aizin, V. V. Popov, and O. V. Polischuk Appl.Phys. Lett. , 143512 (2006).[22] G. R. Aizin, D. V. Fateev, G. M. Tsymbalov, and V. V.Popov Appl. Phys. Lett. , 163507 (2007).[23] T. V. Teperik, F. J. Garci’a de Abajo, V. V. Popov, andM. S. Shur Appl. Phys. Lett. , 251910 (2007).[24] V. V. Popov, D. V. Fateev, T. Otsuji, Y. M. Meziani, D.Coquillat, and W. Knap, Appl. Phys. Lett. , 243504(2011).[25] V. V. Popov, D. V. Fateev, E. L. Ivchenko, and S. D.Ganichev. Phys. Rev. B, , 235436 (2015).[26] Y. M. Meziani, H. Handa, W. Knap, T. Otsuji, E. Sano,V. V. Popov, G. M. Tsymbalov, D. Coquillat, and F.Teppe Appl. Phys. Lett. , 201108 (2008). [27] T. Otsuji, Y. M. Meziani, T. Nishimura, T. Suemitsu,W. Knap, E. Sano, T. Asano, and V. V. Popov, J. Phys.:Condens. Matter , 384206 (2008).[28] S. Boubanga-Tombet, W. Knap, D. Yadav, A. Satou, D.B. But, V. V. Popov, I. V. Gorbenko, V. Kachorovskii,and T. Otsuji, Phys. Rev. X , 031004 (2020).[29] P. Reimann, Phys. Rep. , 57 (2002).[30] E.L. Ivchenko and S. D. Ganichev, Pisma v ZheTF ,752 (2011) [JETP Lett. , 673 (2011)].[31] P. Hanggi and F. Marchesoni, Rev. Mod. Phys. , 387(2009).[32] S. Denisov, S. Flach, and P. H¨anggi, Phys. Rep. , 77(2014).[33] D. Bercioux and P. Lucignano, Rep. Prog. Phys. ,106001 (2015).[34] C. O. Reichhardt and C. Reichhardt, Annu. Rev. Con-dens. Matter Phys. , 51 (2017).[35] P. Olbrich, E. L. Ivchenko, R. Ravash, T. Feil, S. D.Danilov, J. Allerdings, D. Weiss, D. Schuh, W. Wegschei-der, and S. D. Ganichev, Phys. Rev. Lett. , 090603(2009).[36] Yu.Yu. Kiselev and L.E. Golub, Phys. Rev. B , 235440(2011).[37] P. Olbrich, J. Karch, E. L. Ivchenko, J. Kamann, B.M¨arz, M. Fehrenbacher, D. Weiss, and S. D. Ganichev,Phys. Rev. B , 165320 (2011).[38] A. V. Nalitov, L. E. Golub, E. L. Ivchenko, Phys. Rev.B , 115301 (2012).[39] G. V. Budkin and L. E. Golub, Phys. Rev. B , 125316(2014).[40] P. Olbrich, J. Kamann, M. K¨onig, J. Munzert, L. Tutsch,J. Eroms, D. Weiss, Ming-Hao Liu, L. E. Golub, E. L.Ivchenko, V. V. Popov, D. V. Fateev, K. V. Mashinsky,F. Fromm, Th. Seyller, and S. D. Ganichev, Phys. Rev.B, , 075422 (2016).[41] S. D. Ganichev, D. Weiss, and J. Eroms, Ann. Phys. ,1600406 (2018).[42] A. D. Chepelianskii, M. V. Entin, L. I. Magarill, and D.L. Shepelyansky, Phys. Rev. E , 041127 (2008).[43] S. Sassine, Yu. Krupko, J.-C. Portal, Z.D. Kvon, R. Mu-rali, K.P. Martin, G. Hill, and A.D. Wieck, Phys. Rev. B , 045431 (2008).[44] L. Ermann and D. L. Shepelyansky, Eur. Phys. J. B ,357 (2011).[45] E.S. Kannan, I. Bisotto, J.-C. Portal, T. J. Beck, and L.Jalabert, Appl. Phys. Lett. , 143504 (2012).[46] V. V. Popov, D. V. Fateev, T. Otsuji, Y. M. Meziani, D.Coquillat, and W. Knap, Appl. Phys. Lett. , 243504(2011).[47] V. V. Popov, Appl. Phys. Lett. , 253504 (2013).[48] T. Otsuji, T. Watanabe, S. A. B. Tombet, A. Satou, W.M. Knap, V. V. Popov, M. Ryzhii, and V. Ryzhii, IEEETrans. Terahertz Sci. Technol. , 63 (2013).[49] T. Watanabe, S. A. Boubanga-Tombet, Y. Tanimoto, D.Fateev, V. Popov, D. Coquillat, W. Knap, Y.M. Meziani,Yuye Wang, H. Minamide, H. Ito, and T. Otsuji, IEEESensors J. , 89 (2013).[50] Y. Kurita, G. Ducournau, D. Coquillat, A. Satou1, K.Kobayashi, S. Boubanga Tombet, Y. M. Meziani, V. V.Popov, W. Knap, T. Suemitsu, and T. Otsuji, Appl.Phys. Lett. , 251114 (2014).
51] S. A. Boubanga-Tombet, Y. Tanimoto, A. Satou, T.Suemitsu, Y. Wang, H. Minamide, H. Ito, D. V. Fa-teev, V. V. Popov, and T. Otsuji, Appl. Phys. Lett. ,262104 (2014).[52] L. Wang, X. Chen, and W. Lu, Nanotechnology ,035205 (2015).[53] P. Faltermeier, P. Olbrich, W. Probst, L. Schell, T.Watanabe, S. A. Boubanga-Tombet, T. Otsuji, and S.D. Ganichev, J. Appl. Phys. , 084301 (2015).[54] I.V. Rozhansky, V.Yu. Kachorovskii, and M.S. Shur,Phys. Rev. Lett. , 246601 (2015).[55] W. Weber, S.D. Ganichev, Z.D. Kvon, V.V. Bel’kov, L.E.Golub, S.N. Danilov, D. Weiss, W. Prettl, Hyun-Ick Cho,and Jung-Hee Lee, Appl. Phys. Lett. , 262106 (2005).[56] C¸ Kurdak, N. Biyikli, ¨U ¨Ozg¨ur, H. Morko¸c, and V. I.Litvinov, Phys. Rev. B, , 022111(2006).[58] S. Schmult, M. J. Manfra, A. Punnoose, A. M. Sergent,K. W. Baldwin, and R. J. Molnar, Phys. Rev. B ,033302 (2006).[59] A. E. Belyaev, V. G. Raicheva, A. M. Kurakin, N. Klein,and S. A. Vitusevich, Phys. Rev. B , 035311 (2008).[60] W. Weber, L. E. Golub, S. N. Danilov, J. Karch, C. Re-itmaier, B. Wittmann, V. V. Bel’kov, E. L. Ivchenko, Z.D. Kvon, N. Q. Vinh, A. F. G. van der Meer, B. Mur-din, and S. D. Ganichev. Phys. Rev. B, (24), 245304(2008).[61] J. Y. Fu and M. W. Wua, J. Appl. Phys. , 093712(2008).[62] W. Stefanowicz, R. Adhikari, T. Andrearczyk, B. Faina,M. Sawicki, J. A. Majewski, T. Dietl, and A. Bonanni,Phys. Rev. B , 205201 (2014).[63] K.H.Gao X.R.Ma.H.Zhang.Z.Zhou, T. Lin, Physica B:Condensed Matter Volume , 412370 (2020).[64] Y.A. Bychkov and E.I. Rashba, JETP Lett., , 78(1984).[65] W. Knap, C. Skierbiszewski, A. Zduniak, E. Litwin-Staszewska, D. Bertho, F. Kobbi, J. L. Robert, G. E.Pikus, F. G. Pikus, S. V. Iordanskii, V. Mosser, K.Zekentes, and Yu. B. Lyanda-Geller, Phys. Rev. B ,3912 (1996).[66] Mikhail I. Dyakonov, Spin Physics in Semiconductors, ed.M.I. Dyakonov (Springer-Verlag Berlin Heidelberg 2018).[67] S.D. Ganichev and L.E. Golub, Review, Phys. Stat. SolidiB , 1801, (2014).[68] V. Falko, Fizika Tverdogo Tela , 29 (1989).[69] V. V. Belkov, S. D. Ganichev, E. L. Ivchenko, S. A.Tarasenko, W. Weber, S. Giglberger, M. Olteanu, H. P.Tranitz, S. N. Danilov, P. Schneider, W. Wegscheider,D. Weiss, and W. Prettl, J. Phys. Cond. Matt. , 3405(2005).[70] S. A. Tarasenko, Phys. Rev. B , 085328 (2008).[71] S. A. Tarasenko, Phys. Rev. B , 035313 (2011).[72] C. Drexler, S. A. Tarasenko, P. Olbrich, J. Karch, M.Hirmer, F. M¨uller, M. Gmitra, J. Fabian, R. Yakimova,S. Lara-Avila, S.Kubatkin, M. Wang, R. Vajtai, P. M.Ajayan, J. Kono, and S. D. Ganichev, Nat. Nanotechnol. , 104 (2013). [73] N. Kheirabadi, E. McCann, V. I. Fal’ko. Phys. Rev. B , 075415 (2018).[74] G. V. Budkin, L. E. Golub, E. L. Ivchenko, and S. D.Ganichev, JETP Lett. , 649 (2016).[75] P. Faltermeier, G. V. Budkin, J. Unverzagt, S. Hubmann,A. Pfaller, V. V. Bel’kov, L. E. Golub, E. L. Ivchenko, Z.Adamus, G. Karczewski, T. Wojtowicz, V. V. Popov, D.V. Fateev, D. A. Kozlov, D. Weiss, and S. D. Ganichev,Phys. Rev. B , 155442 (2017).[76] P.Faltermeier, G.V. Budkin, S.Hubmann, V.V.Bel’kov,L.E. Golub, E.L. Ivchenko, Z. Adamus, G. Karczewski,T. Wojtowicz, D.A. Kozlov, D.Weiss, and S.D.Ganichev,Physica E , 178 (2018).[77] S. Hubmann, V. V. Bel’kov, L. E. Golub, V. Yu. Ka-chorovskii, M. Drienovsky, J. Eroms, D. Weiss, and S. D.Ganichev Phys. Rev. Research , 033186 (2020).[78] M. Sakowicz, J. A. Lusakowski, K. Karpierz, W. Knap,M. Grynberg, K. Kohler, G. Valusis, K. Golaszewska, E.Kaminska, A. Piotrowska, P. Caban, and W. Strupinski,Acta Physica Polonica A , 5 (2008).[79] M. B. Lifshits and M. I. Dyakonov, Phys. Rev. B ,121304(R) (2009).[80] S. Boubanga-Tombet, M. Sakowicz, D. Coquillat, F.Teppe, W. Knap, M. I. Dyakonov, K. Karpierz, J.(cid:32)Lusakowski, and M. Grynberg, Appl. Phys. Lett. ,072106 (2009).[81] Klimenko, O. A., Mityagin, Y. A., Videlier, H., Teppe,F., Dyakonova, N. V., Consejo, C., Bollaert, S., Murzin,V. N., Knap, W., Appl. Phys. Lett. , 2 (2010).[82] M. Dyakonov and M. Shur, Phys. Rev. Lett. , 2465(1993).[83] R. Dwili´nski, R. Doradzi´nski, J. Garczy´nski, L. P.Sierzputowski, A. Puchalski, Y. Kanbara, Yagi, K., Mi-nakuchi, H., Hayashi, H., Journal of Crystal Growth, (2008).[84] W. Wojtasiak, M. G´oralczyk, D. Gryglewski, M. Zajac,R. Kucharski, P. Prystawko, A. Piotrowska, M. Ekielski,E. Kaminska, A. Taube, and M. Wzorek, Micromachines, (11), 546 (2018).[85] H. J. Lippman, F. Kuhrt, Z. Naturforsch, Teil. A (462), 474 (1958).[86] Schroder D. K., Semiconductor Material and DeviceCharacterization (John Wiley & Sons, Hoboken, NewJersey 2006).[87] A. M. Cowley, and H. O. Sorensen, IEEE Transactionson microwave theory and techniques, (1966).[88] V. Yu. Kachorovskii, S. L. Rumyantsev, W. Knap andM. Shu, Appl. Phys. Lett. , 223505 (2013).[89] O. A. Klimenko, W. Knap, B. Iniguez, D. Coquillat, Y.A. Mityagin, F. Teppe, N. Dyakonova, H. Videlier, D.But, F. Lime, J. Marczewski, and K. Kucharski, J. Appl.Phys. , 014506 (2012).[90] Simon M. Sze, Kwok K. Ng. Physics of SemiconductorDevices (John Wiley & Sons, Hobojen, New Jersey,2007),3rd Edition.[91] S.A. Tarasenko, N.S. Averkiev, JETP Lett. , 552-555(2002).[92] N.S. Averkiev, M.M. Glazov, S.A. Tarasenko, Solid StateCommunications , 543–547 (2005).[93] W. Knap, V. Kachorovskii, Y. Deng, S. Rumyantsev, J.-Q. L¨u, R. Gaska, and M. S. Shur, J. Appl. Phys. ,9346 (2002). II. SUPPLEMENTAL MATERIAL
In this Supplemental Material to the paper “Beatings of ratchet current magneto-oscillations in GaN-based gratinggate structures: manifestation of spin-orbit band splitting.” we present technical details of the calculations
We start with continuity equation and Navie-Stokes equation written in components (notations are the same as inthe main text) ∂n∂t + ∂v x ∂x = 0 ,∂v x ∂t + γv x + ω c v y + s ∂n∂x = em (cid:26) V ∂ cos qx∂x − E x (1 + h x cos ( qx + φ )) cos ωt (cid:27) − γ (cid:48)(cid:48) n v x ,∂v y ∂t + γv y − ω c v x = − eE y m (1 + h y cos ( qx + φ )) cos ( ωt + θ ) − γ (cid:48)(cid:48) n v y . (31)Here E x = E cos α, E y = E sin α (32)describe linearly-polarized wave. Following procedure described in the main text, we take into account in theseequations only non-linearity related to dependence of γ on n (assuming that term with second derivation of γ (cid:48)(cid:48) (0)dominates) linearizing all other terms.We search the solution as perturbative expansion over E and V . The nonzero rectified electric current J dc = − eN j dc , j dc = (cid:104) (1 + n ) v (cid:105) t,x (33)appears in the third order ( ∝ E V ): j dc ≈ j , dc , . In order to find j , dc , we need to calculate − γ (cid:48)(cid:48) (cid:10) n v (cid:11) x,t = − γ (cid:48)(cid:48) (cid:10) n , ( x ) n , ( x, t ) v , ( x, t ) (cid:11) x,t ∝ E V sin φ (cid:54) = 0 . (34)Hence, we need to calculate n , ( x ) , n , ( x, t ) , v , ( x, t ). A. Calculation in the order (0,1)
In this order, one puts E = 0 , and there is only correction to the electron concentration induced by the inhomo-geneous static potential v , = 0 , n , ( x ) = (cid:18) eV ms (cid:19) cos qx. (35) B. Calculation in the order (1,0)
We search for solution in the form v , ( x, t ) = (cid:18) eE m (cid:19) (cid:2) V , ( t ) + h V , ( t ) sin ( qx + φ ) (cid:3) ,n , ( x ) = (cid:18) eE hm (cid:19) N , ( t ) sin ( qx + φ ) . (36)The current is expressed in terms of these notations as follows j , dc = − γ (cid:48)(cid:48) eV N sin φms γ + ω c (cid:18) γ − ω c ω c γ (cid:19) (cid:68) N , ( t ) · V , ( t ) (cid:69) t . (37)The amplitudes entering Eq. (36) can be found by substituting Eq. (36) in the starting equations and separatingterms of the order (1 ,
0) 14 γ − iω ) V ,ωx + ω c V ,ωy = − eE x m e − iωt , ( γ − iω ) V ,ωy − ω c V ,ωx = − eE y m e − iωt e − iθ , (38) (cid:18) γ − iω + i s q ω (cid:19) V ,ωx + ω c V ,ωy = − eE x h x m e − iωt , ( γ − iω ) V ,ωy − ω c V ,ωx = − eE y h y m e − iωt e − iθ , − iωN − qV ,ωx = 0 . (39)Solution of these linear equations reads (cid:18) V , x V , y (cid:19) = eE m e − iωt ( ω + iγ ) − ω c (cid:18) − i ( ω + iγ ) cos ( α ) − ω c sin ( α ) e − iθ ω c cos ( α ) − i ( ω + iγ ) sin ( α ) e − iθ (cid:19) + c.c, (40) (cid:32) V , x V , y (cid:33) = − eE h m e − iωt ( ω + iγ ) D qω (cid:18) iω ( ω + iγ ) cos ( α ) + ωω c sin ( α ) e − iθ − ωω c cos ( α ) + i ( ω ( ω + iγ ) − s q ) sin ( α ) e − iθ (cid:19) + c.c, (41) N , = q eE h m (cid:2) ( ω + iγ ) cos ( α ) − iω c sin ( α ) e − iθ (cid:3) e − iωt ( ω + iγ ) D qω + c.c. (42)Here D qω = ω ( ω + iγ ) − ω c ωω + iγ − q s . (43)Then, we find (cid:68) N , ( t ) · V , ( t ) (cid:69) t = (cid:18) eE m (cid:19) qh ( ω + iγ ) D qω [( ω − iγ ) − ω c ] (cid:18) C x C y (cid:19) + c.c, (44)where C x = i [cos α ( ω + γ ) + ω c sin α − α sin α ( ω c ω sin θ + γω c cos θ )] ,C y = i [cos α ω c ( ω c + iγ ) + ω c sin α ( ω − iγ ) + cos α sin α ( i cos θ ( ω + γ − ω c ) − sin θ ( ω + γ + ω c ))] , (45) J dc = − qγ (cid:48)(cid:48) sin φ ( eE ) heV N m s ω + iγ ) D qω [( ω − iγ ) − ω c ] ( γ + ω c ) (cid:18) γ − ω c ω c γ (cid:19) (cid:18) C x C y (cid:19) + c.c. (46)Introducing now Stokes parameters, after some algebra, we reproduce Eqs. (23) and (25) of the main text with a = 2 ω ω c (cid:12)(cid:12) ω − ( ω − iγ ) (cid:12)(cid:12) (cid:20) − ω c γ (cid:21) (47)+ s q (cid:20) ( γ + 2 ω )( γ + ω ) + ( γ − ω ) ω γω c (cid:2) γ − ω + ω + 2 ω ( γ + 2 ω ) (cid:3) (cid:21) , a L1 = s q γ ( γ + ω + ω c ) (cid:20) − ω c (3 γ + ω − ω ) γ ( γ + ω − ω ) (cid:21) , (48) a L2 = s q γ ( γ + ω + ω c ) (cid:20) γ ( γ + ω − ω ) ω c (3 γ + ω − ω ) (cid:21) , (49) a C = ω ( ω + γ + ω ) (cid:12)(cid:12) ω − ( ω − iγ ) (cid:12)(cid:12) (cid:20) ω c − γ (cid:21) (50)+ s q ω ( γ + ω + ω c ) (cid:20) − ω c (3 γ + ω − ω ) γ ( γ + ω − ω ) (cid:21) ,D ωq = ω ( ω + iγ ) − q s − ω ωω + iγ .iγ .