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M. Hild, I. Yahniuk, L. E. Golub, J. Amann, J. Eroms, D. Weiss, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), S. D. Ganichev

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[Circular terahertz ratchets in a two-dimensionally modulated Dirac system](https://mdr.nims.go.jp/datasets/e3202683-2978-4a38-aa69-ad3fd7e2ee06)

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Circular terahertz ratchets in a two-dimensionally modulated Dirac systemPHYSICAL REVIEW RESEARCH 6, 023308 (2024)Circular terahertz ratchets in a two-dimensionally modulated Dirac systemM. Hild,1 I. Yahniuk,1,2 L. E. Golub ,1 J. Amann,1 J. Eroms ,1 D. Weiss ,1 K. Watanabe ,3T. Taniguchi,4 and S. D. Ganichev 1,21Terahertz Center, University of Regensburg, 93040 Regensburg, Germany2CENTERA Labs, Institute of High Pressure Physics, PAS, 01-142 Warsaw, Poland3Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan4Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan(Received 27 February 2024; accepted 16 May 2024; published 21 June 2024)We report on the observation of the circular ratchet effect excited by terahertz laser radiation in a speciallydesigned two-dimensional metamaterial consisting of a graphene monolayer deposited on a few-layer graphenegate patterned with an array of triangular antidots. We show that a periodically driven Dirac fermion system withspatial asymmetry converts the a.c. power into a d.c. current, whose direction reverses when the radiation helicityis switched. The circular ratchet effect is demonstrated for room temperature and a radiation frequency of 2.54THz. It is shown that the ratchet current magnitude can be controllably tuned by the patterned and uniform backgate voltages. The results are analyzed in the light of the developed microscopic theory considering electronicand plasmonic mechanisms of the ratchet current formation.DOI: 10.1103/PhysRevResearch.6.023308I. INTRODUCTIONSingle field-effect transistors (FETs) have proven to bepromising devices for sensitive and fast room-temperaturedetection of terahertz (THz) radiation, see, e.g., Refs. [1–9].In the past decade, it has been suggested that the perfor-mance of FET structures can be significantly improved byfabricating an asymmetric comblike dual grating-gate (DGG)FETs based on semiconductor quantum wells or graphene[10–31]. Furthermore, it has been shown that electronic andplasmonic ratchet effects in DGG-based metamaterials leadto a helicity-driven photoresponse that reverses the sign byswitching from right-handed to left-handed circular polariza-tion [10,13,18,19,25,28,32,33]. This is in contrast to the singleFET in which a helicity-driven photoresponse can only beobtained due to an interference of plasma oscillations in thechannel of the FETs connected to specially designed anten-nas [34–36], e.g., a tilted bow-tie antenna [37,38]. Therefore,metamaterials can not only improve the performance of FETdetectors, but also lead to new functionalities, in particular,the all-electric detection of radiation helicity. Thus, the searchfor novel concepts and designs of metamaterials that providethe helicity-sensitive dc current in response to terahertz radi-ation is an important and challenging task. Here, we reporton the observation and study of the helicity-driven dc currentexcited by THz radiation in a two-dimensional (2D) metama-terial consisting of a few-layer graphene gate patterned withan array of triangular antidots and placed under a graphenePublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.monolayer. An advantage of the novel design with triangularantidot metamaterials fabricated under the graphene is thatthe radiation enters directly into the graphene layer withouthaving to pass through the metal grid. This is in contrast to thewell known asymmetric dual-grating top gate ratchet struc-tures, see, e.g., Refs. [10–31]. We show that the photoresponseis due to the ratchet current caused by the combined action of aspatially periodic in-plane electrostatic potential and a period-ically modulated radiation electric field caused by near-fielddiffraction. We investigate an all-electrically tunable magni-tude of the rectified voltage that is different for clockwise andcounterclockwise circularly polarized radiation. The data arediscussed in the light of the developed theory, which is basedon the solution of the Boltzmann kinetic equation and welldescribes all experimental findings. The results are analyzedin terms of electronic and plasmonic mechanisms of photocur-rent generation in periodic structures. We show that the ratchetphotocurrent arises due to the noncentrosymmetric unit cell ofthe periodic structure.II. SAMPLE AND METHODSFigures 1(a) and 1(b) show the design of the investigatedmetamaterial. The monolayer of graphene encapsulated be-tween hexagonal boron nitride (hBN) is deposited on thepatterned bottom gate made of five layers of graphene. Thepattern consists of equilateral triangular antidots arranged ina square lattice; see Fig. 1(b). The antidots array has a periodof 1000 nm and the triangles have a sidelength of 600 nm.In addition, a Si wafer with a 285 nm SiO2 layer on topwas used as a uniform back gate. Two Hall bar samples withohmic chromium-gold contacts were fabricated. The lengthand width of the sample A (B) were 16 µm (7.5 µm) and2.5 µm (3.5 µm), respectively. Consequently, the area of the2643-1564/2024/6(2)/023308(9) 023308-1 Published by the American Physical Societyhttps://orcid.org/0000-0003-3818-1014https://orcid.org/0000-0003-2212-9537https://orcid.org/0000-0002-9630-9787https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0001-6423-4509https://ror.org/01eezs655https://ror.org/00fb7yx07https://ror.org/026v1ze26https://ror.org/026v1ze26https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.6.023308&domain=pdf&date_stamp=2024-06-21https://doi.org/10.1103/PhysRevResearch.6.023308https://creativecommons.org/licenses/by/4.0/M. HILD et al. PHYSICAL REVIEW RESEARCH 6, 023308 (2024)FIG. 1. Panel (a): Cross section of the two unit cells, see dashedrectangular in the panel (b). Panel b: (b) AFM image of the patternedgate formed by the periodic array of triangular antidots arranged ina square lattice. The x and y directions used for the photovoltagedetection are chosen due to the symmetry of the system, which helpsin the phenomenological treatment of the data. The x direction isalong the height of the triangles, and the y direction is along theirbase. Panel (c): A photograph of sample A (the orange area is theetch mask overlay), in which triangles are highlighted by a whiteline and the numbers enumerate the contacts. Panel (d): Experimentalconfiguration.Hall bar structures As was 40 µm2 for sample A and 26 µm2for sample B. The maximum charge densities for the studiedpatterned gate voltage Vp, ranging from −0.75 V to 0.75 Vwere 4×1011 cm−2 for electrons and 5.5×1011 cm−2 forholes. The maximum carrier mobility for electrons and holesare 104 cm2/Vs and 0.7×104 cm2/Vs, respectively. Furtherdetails on sample preparation and transport characteristics canbe found in Ref. [31], where the same samples were investi-gated.To excite ratchet currents, we used polarized THz radia-tion from a cw molecular laser optically pumped by a CO2laser with CH3OH as active medium operating at frequencyf = 2.54 THz (wavelength λ = 118 µm) and power at thesample position of about 30 mW. The sample was excited bynormally incident elliptically (circularly) polarized radiation,see Fig. 1. A Gaussian-like shape of the beam measured by apyroelectric camera [39,40] had a full width at half maximumof 1.8 mm and a beam area Abeam ≈ 0.025 cm2 at the sampleposition. Considering the area of the Hall bar, the samplewas exposed to the power Ps = P×As/Abeam. The radiationpolarization of the initially linearly polarized laser radiationwith E ‖ y was obtained using crystal quartz lambda quarter-wave plates. The plate was rotated counterclockwise by anangle ϕ between the x axis and the c axis of the plate. In thisgeometry, the degree of circular polarization varies accordingto Pcirc = (Iσ+ − Iσ−)/(Iσ+ + Iσ−) = sin 2ϕ. Here Iσ+(Iσ−)is the intensity of the right-handed (left-handed) circularlypolarized radiation. Consequently, for ϕ = 45◦ and 135◦ theradiation was right-handed (σ+) and left-handed (σ−) cir-cularly polarized, respectively. The polarization ellipses forsome angles ϕ are sketched on top of Fig. 2.The radiation was modulated by an optical chopper at afrequency 130 Hz and the photovoltage U was measured usingthe standard lock-in technique. The samples were placed in aFIG. 2. Photovoltage normalized to the radiation power Ps as afunction of the angle ϕ, which determines the radiation helicity. Thedata are obtained for sample A. Panels (a) and (b) show the Ux/Ps andUy/Ps measured across the long (contacts 1-5) and short (contacts8-2) sides of the Hall bar for different back gate voltages and forVp = 0. The curves are fits according to Eq. (1) and (2) with thefitting parameters given in Table I. The ellipses on top illustrate thepolarization states for different angles ϕ.vacuum chamber with a z-cut crystal quartz optical windowcovered with a thin black polyethylene foil to prevent illu-mination by visible and near infrared light. The voltage Uxwas measured at the contacts 5-1 and Uy was measured at thecontacts 8-2, with contacts 1 and 2 grounded. The generatedratchet current density was calculated as j = U/(Rsw), whereTABLE I. Parameters used for fitting of experimental curves inFig. 2.Parameter Vbg (V)(V/W) −5 0 5U circ 17 −17 −25U0 14 12 −31UxUL1 −41 53 59UL2 5 −28 −17Ũ circ −19 20 −1Ũ0 25 4 −22UyŨL1 16 0 −14ŨL2 21 21 −3023308-2CIRCULAR TERAHERTZ RATCHETS IN A … PHYSICAL REVIEW RESEARCH 6, 023308 (2024)Rs � Rin is the two-point sample resistance, Rin is the inputimpedance of the amplifier, and w is the Hall bar width. Allexperiments were conducted at room temperature.III. RESULTSWhen the structures were illuminated with elliptically po-larized radiation, we measured a photosignal Ux from thesource-drain contact pair, i.e., via the long side of the Hall barand the signal Uy perpendicular to it from the contact pair 8-2,Fig. 1(c). Figure 2 shows the photosignal as a function of theangle ϕ, which controls the radiation helicity and the degreeof linear polarization. The data are presented for sample Aand show that the overall polarization dependence can be wellfitted byjx ∝ Ux =U circ sin 2ϕ + U0− UL1 (1 + cos 4ϕ)/2 − UL2sin 4ϕ2, (1)jy ∝ Uy = Ũ circ sin 2ϕ + Ũ0− ŨL1 (1 + cos 4ϕ)/2 − ŨL2sin 4ϕ2. (2)The ϕ-dependent terms in Eqs. (1) and (2) are the Stokesparameters describing the degrees of linear and circular po-larization of the elliptically polarized radiation:PL1 = −1 + cos 4ϕ2, PL2 = − sin 4ϕ2, Pcirc = sin 2ϕ.(3)As a central result, the figure shows that in all experimentaltraces obtained for different back gate voltages, the signalsexcited by left and right circularly polarized radiation aredifferent in magnitude and, in some traces, of opposite sign.This is due to the significant contribution of the circular pho-toresponse given by the first term on the right-hand side ofboth equations, which describes the photocurrent proportionalto the degree of the circular polarization Pcirc. Although weobtained similar results for samples A and B, in the following,we focus on the data obtained in sample A and present the datafor sample B in the Appendix.Figure 2 and Eqs. (1) and (2) show that the helicity-drivenphotocurrent is superimposed by the polarization-independentratchet current given by the terms U0 and Ũ0 and the linearratchet current given by the terms proportional to the co-efficients UL1, ŨL1,UL2, ŨL2. These contributions have beenpreviously detected and studied using linearly polarized radi-ation on the same structure; see Ref. [31]. For pure circularlypolarized radiation obtained at ϕ = 45◦ and 135◦, PL1 andPL2 vanish. Consequently, the photosignal in response to thecircularly polarized radiation is given byU circx = ±U circ + U0, U circy = ±Ũ circ + Ũ0, (4)where the + and − signs correspond to the right- and left-handed circularly polarized radiation, respectively.Figure 2 and the insets in Figs. 3 and 4 show that the signalsin response to right- and left- handed circularly polarizedradiation are sensitive to the gate voltage: they are consis-tently different in magnitude, and for certain gate voltagesopposite in sign. To analyze the gate voltage dependenciesFIG. 3. Normalized photovoltages U circ/Ps (a) and Ũ circ/Ps (b) asa function of the back gate voltage Vbg. The data are obtained bymeasuring the photoresponse to right- and left-handed radiation andcalculating U circ = (U σ+x − U σ−x )/2 and Ũ circ = (U σ+y − U σ−y )/2.The data are obtained for three values of the patterned gate voltagesVp = 0, ±0.75 V. Curves 1©, 2©, and 3© show back gate voltage depen-dencies of the two-point sample resistance (right y axis) obtained forthe patterned gate voltages Vp = 0, −0.75, and 0.75 V, respectively.Left insets in panels (a) and (b) show the back gate dependencies ofthe signals Ux and Uy excited by right (σ+) and left (σ−) circularlypolarized radiation obtained for Vp = 0 and −0.75 V, respectively.Right insets show the Hall bar with triangular antidots and contactsused for signal detection.of the helicity-driven photoresponses U circ and Ũ circ, weused the fact that these contributions change sign by revers-ing the radiation helicity and calculated them according to(U σ+x,y − U σ−x,y )/2.The gate dependencies of the circular photosignal areshown in Figs. 3 and 4. The curves are obtained by varyingone gate voltage Vbg or Vp and holding the other constant. Notethat due to the patterned gate placed between the graphenelayer and the uniform back gate, a voltage applied to the lattergate is in fact not uniform and introduces an asymmetry of theelectrostatic potential acting on the graphene layer. Figures 3and 4 show that varying the asymmetric potential obtainedby changing either Vbg or Vp changes the signal in a similarway: It increases or decreases significantly near the resis-tance maximum and varies only slightly at high gate voltages.At the same time, depending on the value and sign of the023308-3M. HILD et al. PHYSICAL REVIEW RESEARCH 6, 023308 (2024)FIG. 4. Normalized photovoltages U circ/Ps, panel (a), andŨ circ/Ps, panel (b), as a function of the patterned gate voltage Vp.The data are obtained for three values of the back gate voltagesVbg = 0,±5 V. Curves 1©, 2©, and 3© show the patterned gate voltagedependencies of the two-point sample resistance (right y axis) ob-tained for the back gate voltages Vbg = 0, −5, and 5 V, respectively.Insets show the back gate dependencies of the signal excited by right(σ+) and left (σ+) circularly polarized radiation. Insets show the Hallbar with triangular antidots and contacts used for signal detection.constant gate voltage, it may (i) change sign near the resis-tance maximum, see, e.g., the traces in Fig. 3(a) for Vp =−0.75 V or Fig. 3(b); or (ii) not change sign, see, e.g., Fig. 4.A comparison of the results obtained for the source-draincontacts with those obtained for the contact pair across theHall bar shows that the circular effect for the direction alongthe heights of the triangles and their bases consistently haveopposite signs.IV. THEORYObservation of the photocurrent excited in an unbiasedgraphene sample using normally incident homogeneous radia-tion shows that the photoresponse is caused by the asymmetricpatterned gate placed under the graphene layer. The asym-metric periodic array reduces the symmetry of the systemallowing the generation of such a photocurrent. A previousstudy of the ratchet effect excited by THz radiation in thesame samples demonstrated that the system under study hasC1 point symmetry without any nontrivial symmetry elements[31]. In the theory, the ratchet current is caused by the si-multaneous action of the asymmetric static potential V (r) andthe THz near-field with amplitude E0(r), formed by radiationdiffraction at the edges of the triangular antidots. Both thepotential and the near-field are 2D-periodic with the periodof the structure. The low symmetry of the studied system iscaptured by the following vector parameter:�2D = E20 (r)∇V (r), (5)where the overline indicates the averaging over the 2D period.Although both the near-field intensity and the periodic poten-tial are zero on average, �2D is finite due to the low symmetryof the structure.For structures with C1 point symmetry, both Cartesiancomponents of the vector �2D, �x and �y, are nonzero. Thisimplies that the ratchet current allowed under normal lightincidence in both x and y directions in the structure plane isjx = −�yγ Pcirc + �x(χ0 + χ1PL1) + �yχ2PL2, (6)jy = �xγ Pcirc + �y(χ0 − χ1PL1) + �xχ2PL2. (7)Here jx,y are the components of the electric current density,and the values χ0, χ1,2, and γ describe the polarization-independent, linear, and circular ratchet currents, respectively.The Stokes parameters of the radiation are defined by the com-plex polarization vector e: PL1 = |ex|2 − |ey|2, PL2 = exe∗y +e∗xey, Pcirc = −i[e×e∗]z, where we take into account that theradiation propagates along the −z direction. The circularratchet current reverses its direction when switching fromσ+ to σ− polarization because it is proportional to the de-gree of circular polarization Pcirc. For circularly polarizedradiation the circular ratchet current is superimposed on thepolarization-independent current, whereas for elliptically po-larized light the linear ratchet effect may also contribute,see Eqs. (6) and (7). As mentioned above, the theory ofthe polarization-independent and linear ratchet currents ispresented and discussed in detail in Ref. [31], so in the fol-lowing, we focus on the theory of the helicity-driven ratchetcurrent.To develop a microscopic description of the circular ratchetcurrent, we use the drift-diffusion modeling based on thesolution of the Boltzmann kinetic equation for the distributionfunction f (p, r)∂ f∂t+ vp · ∇ f + F(r, t ) · ∂ f∂ p= St[ f ]. (8)Here vp = v0 p/p, where v0 is the Dirac fermion velocity ingraphene, St is the collision integral, and the force F is asum of the static contribution due to the potential V (r) anda dynamic force caused by the near field:F(r, t ) = −∇V (r) + qE0(r)[e exp(−iωt ) + c.c.], (9)where q is the elementary charge positive for holes and neg-ative for electrons. By treating the periodic potential gradientand the radiation electric field as small perturbations, we iter-ate the kinetic Eq. (8) and find the static and homogeneouscorrection to the distribution function δ f (p) ∝ �x,y. This023308-4CIRCULAR TERAHERTZ RATCHETS IN A … PHYSICAL REVIEW RESEARCH 6, 023308 (2024)allows us to calculate the ratchet current density as follows:j = 4q∑pvpδ f (p). (10)Remarkably, since we take into account either ∇xV or ∇yV ,the final expression for j coincides with the result for the1D modulation with a potential that depends only x or y.Therefore, the 2D character of the modulation is accountedfor by the lateral asymmetry parameter �2D, while the factorγ in Eqs. (6) and (7) is determined only by the properties ofthe 2D carriers in graphene above the periodic gate. This prob-lem for �2D ‖ x has already been considered for graphene inRef. [13]. Generalizing to the case of 2D modulation with�x  = 0 and �y  = 0 yields, in agreement with the phenomeno-logical Eqs. (6) and (7), the following expression for thecircular ratchet current density:jcircx = −�yγ Pcirc, jcircy = �xγ Pcirc, (11)whereγ = q3v40τ3trω2F (ωτtr )2s2π h̄2εF[1 + (ωτtr )2][ω2 + (ω2 − ω2pl)2τ 2tr] . (12)Here, εF is the Fermi energy, ω is the radiation frequency,and τtr is the transport relaxation time. We take here intoaccount a resonant enhancement of the near-field at the plas-mon frequency ω = ωpl [25,33]. In the studied 2D squarelattice with the period d , we have ωpl = s√q2x + q2y , whereqx = qy = 2π/d , and s is the plasmon velocity.The frequency dependence of the ratchet current variesstrongly with the type of the elastic scattering potential ofthe carriers in graphene [13]. This is captured in Eq. (12)by the dimensionless factor F (�) where � = ωτtr . For theshort-range (SR) and long-range Coulomb (Coul) scatteringpotentials it is given byFSR(�) = −�(2�4 + �2 + 8)(�2 + 4)2, FCoul(�) = 1�. (13)The divergence at � → 0 that appears in FCoul is smeared byplasmon or energy relaxation processes. The energy relaxationis thought to be caused by the electron-phonon scatteringwhich was discussed in detail in Ref. [13]. Therefore, incalculations at ωpl = 0 we take FCoul = �/[�2 + (τtr/τε )2],where τε � τtr is the energy relaxation time. The frequencydependence of the amplitude of the circular ratchet currentgenerated, e.g., in the y direction jcircy = �xγ normalized toj0 = �xq3v40τ3tr/(4s2π h̄2εF) is shown in Fig. 5 for two con-sidered types of elastic scattering potentials. For Coulombimpurity scattering at ωpl = 0 we take (τtr/τε )2 = 0.2 [41].V. DISCUSSIONNow, we discuss the obtained experimental results in thelight of the developed theory. We begin with the polarizationdependence of the observed photoresponse. As discussed inSec. III, all curves obtained for different combinations ofthe gate voltages are well described by Eqs. (1) and (2);see Fig. 2. Comparing these equations with the theoreticallyobtained Eqs. (6) and (7) for the ratchet current and notingFIG. 5. Frequency dependence of the circular ratchet current fortwo types of elastic scattering. The curves are calculated for ωpl = 0(red) and for ωplτtr � 1; see vertical dashed lines.that the Stokes parameters in our experiments vary accord-ing to Eq. (3), we see that both sets of equations representidentical dependencies on the angle ϕ. The three last terms ineach equation with the parameters U0 ∝ �xχ0, UL1 ∝ �xχ1,UL2 ∝ �yχ2, Ũ0 ∝ �xχ0, ŨL1 ∝ �yχ1, and ŨL2 ∝ �xχ2 rep-resent the polarization-independent and linear ratchet effects,which are discussed in a separate publication [31]. Here, wefocus on the helicity-sensitive contribution defined by theparameters U circ ∝ �yγ and Ũ circ ∝ −�xγ . Equations (11),obtained from both the symmetry arguments and the micro-scopic theory, show that the circular ratchet currents excitedin x- and y direction have opposite directions provided that�x and �y have the same sign. This is in agreement withthe experimental data showing that the corresponding signalshave opposite signs; see, e.g., Fig. 2 and insets in Figs. 3and 4.A fingerprint of the ratchet effects is that it is proportionalto the lateral asymmetry parameters �x and �y; see Eqs. (5)and (11). These parameters are defined, on the one hand,by the near field, and, on the other hand, by the asymmetryof the electrostatic potential, which can be varied by thegate voltages applied to the back and patterned gates. Thesimulation performed in Ref. [31] shows that both, patternedand back gate voltages, introduce an asymmetric electrostaticpotential acting on the electron gas in graphene. Varying thegate voltage from negative to positive values changes themagnitude and the sign of the asymmetry. Note that evenfor zero potential at both gates, an asymmetry is createdby the built-in potential caused by the conducting patterned023308-5M. HILD et al. PHYSICAL REVIEW RESEARCH 6, 023308 (2024)FIG. 6. Gate voltage dependencies of the normalized circularphotocurrent density jcirc/Ps = U circ/(RswPs ) measured in the di-rection along the height of triangles (contacts 1–5). Curves 1©, 2©,and 3© in panel (a) show the back gate voltage dependencies of thetwo-point sample resistance (right y axis) obtained for the patternedgate voltages Vp = 0, −0.75, and 0.75 V, respectively. Curves 1©, 2©,and 3© in panel (b) show the patterned gate voltage dependencies ofthe two-point sample resistance (right y axis) obtained for the backgate gate voltages Vbg = 0, −5, and 5 V, respectively.gate deposited under the graphene. The calculations showthat the asymmetry along the x direction is indeed presentand can be reversed by reversing the polarity of one of thegates. In particular, in the discussed metamaterial the uni-form bottom gate potential leads to a change of ∇V andthus of the lateral asymmetry parameter. This is because theapplied back gate voltage is periodically screened by the pat-terned gate. It can even lead to the formation of lateral p-njunctions.Figures 3 and 4 show that in the range of Vbg from −2to 2 V and Vp from 0.1 to 0.3 V the signal measured acrossthe long (short) side of the Hall bar increases (decreases)linearly with the gate voltages. At higher |Vbg| and |Vp| thesignal saturates or even slightly decreases; see Figs. 3 and 4.The behavior of the ratchet current calculated according toj = U/(Rsw) is slightly different. This is due to the reductionof the sample resistance with respect to its value at zero gatevoltages, see curves 1, 2, and 3 in Figs. 3(a) and 4(a), whichaffects the gate voltage dependence of the current, especiallyat high gate voltages. The gate dependence of the photocurrentis shown in Fig. 6, which demonstrates that its behavior onlyslightly differs from that of the photovoltage: In most cases itsaturates at high gate voltages.Now, we comment the sign of the photocurrent. At firstglance, one could expect, since the asymmetry of the elec-trostatic potential reverses its sign near zero gate voltages,the current should also reverse its direction. However, thereal situation is more complex. Equation (12) shows that thecurrent is proportional to the third power of the carriers’charge; thus, varying the gate voltage may also cause the signof the current to change. Consequently, the reversing of thesign of the electrostatic potential does not necessarily reversethe ratchet current direction. This is indeed observed in theexperiment, which shows that for some values of one fixedgate voltage, the variation of the other changes the currentsign, while for some values of the fixed gate the current cankeep its direction; see Figs. 3(a) and 4. Such a behavior isspecific to the used design of our 2D metamaterial, wherethe patterned gate is placed between the graphene layer andthe uniform back gate. Because of the triangular antidots,both gates change the carrier density and the carrier signinhomogeneously, either in the area outside the antidots (n, pcontrolled by Vp) or above the antidots (n∗, p∗ controlled byVbg), the 2D map of the sample resistance showing the gatevoltage regions with dominating n, p, n∗, p∗ carrier types ispresented in Ref. [31]. By the same arguments, the lateralasymmetry parameters in these two regions, �2D and �∗2D, canalso be different in magnitude and even in sign. Since the signof the ratchet current is defined by the product q3�x,y and/orq∗3�∗x,y, if both q and �2D change sign, then the directionof the ratchet current remains unchanged. Otherwise, if oneof the signs holds and the other reverses, then the currentdirection changes.Finally, we discuss the frequency dependence of the cir-cular ratchet current, which was obtained theoretically; seeEqs. (11) and (12) and Fig. 5, but has not been studied ex-perimentally so far. The figure shows that the behavior of thecircular ratchet current depends strongly on the value of ωplτtr .This is caused by different mechanisms of the ratchet currentformation, electronic and plasmonic, realized in the limitsωplτtr � 1 and ωplτtr � 1, respectively. The frequency depen-dence for the first case is shown by the red curves in Fig. 5. Itcan be seen that the circular ratchet current behaves nonmono-tonically with frequency for both types of scattering potentialsand has a maximum at ωτtr � 1. This behavior is characteris-tic for all circular photocurrents caused by the photogalvaniceffect, the dynamic Hall effect, and edge photocurrents, forreviews see Refs. [42,43], and has the same physical back-ground. At zero frequency, the circular polarization is absent,and subsequently circular photocurrents disappear. At finitefrequency, model considerations show that a retardation ofcarrier motion with respect to the radiating electric field iscrucial for the formation of helicity-driven photocurrents. Theretardation is caused by carrier scattering, and the circularphotocurrent approaches maximum at ωτtr � 1 for both typesof scattering, in particular at ωτε ≈ 1 for Coulomb impu-rity scattering. At higher frequencies, the current decreasesrapidly. Depending on the scattering potential, the asymptoteat ωτtr � 1 is ∝ 1/ω3 for short-range scattering but a muchfaster decrease ∝ 1/ω5 is realized for long-range Coulombimpurities.Increasing the plasmon frequency leads to the plasmonicratchet effect, which drastically changes the frequency depen-dence of the circular ratchet current; see blue and magentacurves in Fig. 5 obtained for different values of ωpl > 1/τtr .For both scattering potentials, in common is that the023308-6CIRCULAR TERAHERTZ RATCHETS IN A … PHYSICAL REVIEW RESEARCH 6, 023308 (2024)TABLE II. Parameters used for fitting of experimental curves inFig. 7.Vbg (V)Parameter (V/W) −7 0 7Ux U circ 7 −10 −16U0 4 10 −7UL1 −7 19.5 16UL2 4 2 2maximum of the circular ratchet current shifts to frequenciesclose to ωpl. The width of the plasmon resonance is con-trolled by the quality factor given by ωplτtr . For short-rangescattering, the amplitude of the current at the resonance iscomparable for different values of ωpl. In sharp contrast, forthe Coulomb impurity scattering, an increase of ωpl results ina drastic decrease of the current amplitude at resonance [44].Therefore, it is important to know which type of scatteringpotential is responsible for the ratchet current. The fact that thedirection of the current is opposite for two types of scatteringpotentials may be helpful in judging this; see Fig. 5.To conclude the discussion on the frequency dependence ofthe circular ratchet current, we emphasize that the plasmonicratchet effect for short-range scattering [45] allows for design-ing a helicity-sensitive photodetector for the frequency rangeon demand, because the plasma frequency can be tuned in awide range by the period of the periodic array.VI. SUMMARYIn summary, experimental results and developed theoryshow that excitation of graphene-based 2D metamaterialsby circularly polarized THz radiation results in the helicitysensitive ratchet current. In this proof-of-principle work, thecircular ratchet current is demonstrated for devices at roomtemperature and radiation frequency of 2.54 THz. At the sametime, the developed theory and previous work on ratchet cur-rents in 1D graphene-based metamaterials [25,29] guaranteethat such structures should also be efficient over a wide rangeof temperatures and frequencies. Consequently, the proposeddesign of the metamaterial can be considered as a perspectivefor the development of novel wide-band room-temperatureTHz detection. As a prospect, the results of the developedtheory show that metamaterials with different lattice periodand cell size, characterized by different plasmon frequencies,can be used for the development of resonant helicity-sensitiveterahertz detectors with the desired central frequency and en-hanced responsivity.ACKNOWLEDGMENTSWe acknowledge the financial support of the DeutscheForschungsgemeinschaft (DFG, German Research Founda-tion) via Project ID No. 448955585 (Ga501/18), Project IDNo. 314695032–SFB 1277 (Subproject No. A09), Project IDNo. 426094608 (ER 612/2-1), and of the Volkswagen StiftungProgram (Grant No. 97738). S.D.G. and I.Y. are also gratefulfor the support of the European Union (ERC-ADVANCEDFIG. 7. Normalized photovoltage Ux/Ps measured in sample B(contacts 1–5) as a function of the angle ϕ, which determines theradiation helicity. The data are obtained for three values of the backgate voltage Vbg holding patterned gate at zero bias. Curves are fitsafter Eq. (14) with fitting parameters given in Table II. The insetshows a photograph of sample B (the orange area is the etching maskoverlay). Note that, in contrast to the measurements performed onsample A the Hall bar is oriented vertically, i.e., at zero angle ϕ theradiation electric field is parallel to the x axis. The ellipses on topillustrate the states of polarization for various angles ϕ.TERAPLASM Grant No. 101053716). K.W. and T.T. ac-knowledge support from the JSPS KAKENHI (Grants No.20H00354 and No. 23H02052) and World Premier Interna-tional Research Center Initiative (WPI), MEXT, Japan.APPENDIX: RESULTS OBTAINED ON SAMPLE BHere we present the data obtained on sample B; for itsdesign and structure see the inset in Fig. 7 and Sec. II. Figure 7shows the variation of the photosignal magnitude with theangle ϕ′. As in sample A we obtain a substantial contributionof the helicity sensitive photoresponse.FIG. 8. Back gate voltage dependence of the circular Ucirc andpolarization-independent U0 photosignals. The inset shows the backgate dependence of the photovoltages excited by right- and left-handed circularly polarized radiation.023308-7M. HILD et al. PHYSICAL REVIEW RESEARCH 6, 023308 (2024)The overall polarization dependence can be well fitted byjx ∝ Ux =U circ sin 2ϕ′ + U0+ UL1 (1 + cos 4ϕ′)/2 + UL2sin 4ϕ′2. (14)Note that, due to technical reasons in our measurements onsample B, we had to change the definition of the angle ϕ:the linear polarization of the laser light was oriented alongthe x axis, and the angle ϕ′ was counted from the y direc-tion. This resulted in the change the Stokes parameters of thecontributions proportional to the coefficients UL1 and UL2 butdid not affect the contributions U circ and U0. Consequently,for equation for the photoresponse excited by circularly po-larized radiation, Eq. (4) used for the data treatment remainsunchanged. 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