Electrically controllable cyclotron resonance
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Electrically controllable cyclotron resonance
A. A. Zabolotnykh and V. A. Volkov ∗ Kotelnikov Institute of Radio-engineering and Electronics of the RAS, Mokhovaya 11-7, Moscow 125009, Russia (Dated: January 12, 2021)Cyclotron resonance (CR) is considered one of the fundamental phenomena in conducting sys-tems. However, Kohn’s theorem excludes the possibility of electrically controlling the CR, whichsignificantly limits the scope of research and does not allow to exploit the full application potentialof this phenomenon. We demonstrate the breakdown of this theorem in the case of back-gatedtwo-dimensional (2D) electron systems, where the 2D electron sheet is separated from the metallicsteering electrode (”back-gate”) by a standard dielectric substrate. The resultant effect is attributedto the retardation of the interaction of 2D electrons and currents with their images in the back-gate.Consequently, we predict that a tremendous shift and narrowing of the CR line can be achievedby varying the electron density or the gate voltage, given the realistic parameters. The effectivecontrollability of CR opens the door for exploring new physics and CR application possibilities.
In a classical plasma, an electron exposed to the mag-netic field B rotates in a closed orbit with the angularfrequency ω c = eB/mc , where − e and m are the electroncharge and effective mass, and c is the speed of light invacuum (we note that in this paper, we use the Gaussunits). The absorption of an electromagnetic wave atthis frequency, known as cyclotron resonance (CR), haslong been used in fundamental studies and applications ofboth non-degenerate gaseous and degenerate condensed-matter plasmas. Initially, CR in two-dimensional elec-tron systems (2DESs) was observed in silicon inversionlayers [1, 2]. Nowadays, it is used extensively to charac-terize various kinds of 2DESs.Depending on the presence of a steering electrode (thegate), all 2DESs can be divided into gated and un-gated systems. Theoretically, the CR in infinite un-gated 2DES was studied in Ref. [3], see also [4, 5]. Itwas found that for the 2DES situated between two half-spaces with dielectric permittivities κ and κ , the half-width of the CR line is defined by the sum of collisionalbroadening γ = 1 /τ , with τ being the electron relax-ation time in 2DES, and collective radiative broadeningΓ κ = 2Γ / ( √ κ + √ κ ), whereΓ = 2 πe nmc , (1)and n is the 2D electron concentration. In ungated GaAsquantum wells, the collective nature of the CR decay rateΓ κ has been observed in time-domain experiments [6, 7],and interpreted as the effect of superradiance [8, 9].However, to the best of our knowledge, collective con-tributions to the CR frequency and decay rate in gated2DESs have not been previously studied in detail. Thegated 2DES has a unique feature — a technical possi-bility to control the electron density over a wide rangethrough the gate voltage. Thus, it enables, for exam-ple, fine-tuning of the electron-electron interaction pa-rameter. However, according to the famous Kohn’s theo-rem [10], manifestations of many-body effects in the CRregime are strongly suppressed: the CR frequency in a standard homogeneous 2DES is independent of electron-electron interaction and, therefore, electron concentra-tion as well. For this reason, Kohn’s theorem imposesa serious constraint on the fundamental and practicalpotential of CR physics. Nevertheless, there are spe-cial cases when the theorem can be invalidated. For in-stance, in Ref. [11] this theorem was violated by ultra-sound to observe an analog of CR on composite fermions.Other examples include non-parabolic electron dispersion[12], polarons [13], non-equilibrium and dynamic effects[14, 15], etc. Although, as a rule, these effects have weakinfluence on the CR frequency.In his work on CR, Kohn did not take into consid-eration electromagnetic retardation of electron-electroninteraction. Consequently, CR in the regime of stronglight-matter coupling is of particular interest, for reviewsee Ref. [16]. Over the past decade, the interaction be-tween the cyclotron motion of electrons and discrete pho-ton modes in specially designed microwave or THz res-onators has been investigated extensively in various inde-pendent studies [17–24]. It was found that the violationof Kohn’s theorem and the renormalization of the CRfrequency occur indeed, though within a finite range ofmagnetic fields corresponding to the anti-crossing of thebare CR frequency and the photon frequency of the res-onator.In the present Letter, we report on significant andunexpected CR renormalization in a gated 2DES —namely, the CR frequency shift and the narrowing of itslinewidth. We emphasize that, surprisingly, it takes placein the low-frequency regime, when the bare cyclotron fre-quency is much less than the characteristic frequency ofthe Fabry-Perot resonator formed by the 2DES and themetal gate, with a dielectric substrate in between, as il-lustrated in Fig. 1. Furthermore, we assert that the givenlow-frequency CR renormalization can be large, even inconventional 2D electron structures with the back-gate.Therefore, it can be rather easily observed experimen-tally.We establish the CR renormalization to be governed by (cid:1) d z FIG. 1. Diagram of the system under consideration: 2Delectron sheet ( z = 0), a dielectric substrate ( − d < z < z ≤ − d ) form an analog of theFabry-Perot resonator. the retardation parameter A defined as the ratio of thecharacteristic velocity in a gated 2DES ( V p ) to the speedof light in the dielectric substrate separating the 2DESand the metal gate ( c/ √ κ ). Formally, the characteristicvelocity equals to that of 2D plasma waves in the gated2DES [26]: V p = r πne d κ m , (2)where d is the distance between the gate and 2DES. Theretardation parameter can be written in three equivalentforms, as follows: A = V p κ /c = 2 d Γ /c = 4 πne d/ ( mc ) . (3)We find that the renormalized CR frequency can be ex-pressed in terms of A through a simple relation: ω ren = ω c / (1 + A ) . (4)It should be stressed that the given retardation parame-ter can be easily modified by changing electron concen-tration by the chemical doping or with the gate voltage,which enables the electrical control of the renormalizedcyclotron resonance frequency.To determine the CR frequency and linewidth, we con-sider the absorption of the electromagnetic plane waveincident normally onto the gated 2DES, as depicted inFig. 1. Given the 2DES sheet and the top surface ofan ideal metal gate situated, respectively, at z = 0and z = − d , the incident and reflected electromag-netic waves take the forms of E i exp( − iωz/c − iωt ) and E r exp( iωz/c − iωt ). To calculate the absorption, we fol-low the classic approach based on Maxwell’s equationsand Ohm’s law j = b σ E , with b σ denoting the 2DES con-ductivity tensor.As a matter of convenience, we introduce the ”circular”variables: E i ± = E ix ± iE iy , E r ± = E rx ± iE ry , and σ ± = σ xx ∓ iσ xy . Then, following a standard procedure Ref. [3],the amplitude of the reflection coefficient r ± = E r ± /E i ± can be expressed as: r ± = 1 − i √ κ cot( ω √ κ d/c ) − πσ ± /c i √ κ cot( ω √ κ d/c ) + 4 πσ ± /c . (5)From the equation above, we can obtain an estimateof the resonance frequency and line width, consideringthat these properties are directly related to the poles ofreflection (as well as absorption) coefficient. Given theFabry-Perot frequency ω F P = c/ ( √ κ d ), we introduce di-mensionless frequencies Ω c = ω c /ω F P and Ω = ω/ω
F P .Then, using the Drude model for the conductivity ten-sor [27], we arrive at the following equation for thecomplex-valued frequency Ω corresponding to the polesof Eq. (5):(Ω ± Ω c + iγ/ω F P )(cot Ω − i/ √ κ ) = − A . (6)As for the physical meaning of this relation, we note thatparameter A describes not only the electromagnetic re-tardation but also a collective contribution of the 2DESto the net response, since A ∝ n . In the zeroth ap-proximation A = 0, and in the ”clean” limit ( γ → c . Increasing A results in the interaction of bare cyclotron motion withthe photonic modes of the resonator, leading to the col-lective renormalization of CR.Considering a high-quality resonance with Im Ω ≪ Re Ω and γ ≪ ω c in the low-frequency limit | Ω | ≪ ± Ω c A − i A Ω c √ κ (1 + A ) − i γ/ω F P A . (7)The real and imaginary parts of the resultant frequencydescribe, respectively, the position and broadening of therenormalized CR line. It should be noted that in thiscase, the line-broadening term includes both radiativeand collisional contributions. We also note that the signnotation in the numerator of the first term in Eq. (7)indicates the different sign of circular polarization.Now, let us find the shape of the renormalized cy-clotron line from the direct calculation of the energyabsorption coefficient P ± = 1 − | r ± | . Thus, using theDrude model for the conductivity tensor, we derive theexact expression for P ± , as shown in the SupplementalMaterial [27]. Due to its cumbersome form, we do notinclude it in the main body of the paper. Importantly,of particular interest to us is the low-frequency regimeΩ ≪
1, i.e. ω ≪ c/ ( d √ κ ), where we can neglect allbut the dominant terms in the full expression for P ± toobtain the simplified relation: P ± = 4 γA d/c ( ω ± ω c ω + A ) + γ ω + γA dc + d ( ω ± ω c ) c , (8) (cid:0) (cid:2) c A b s o r p ti on , a r b . un it s / A = FIG. 2. Absorption in the gated 2DES, calculated fromthe exact equation [27] as a function of the radiation fre-quency ω , given γ/ω c = 0 . κ = 12 .
8, and c/ ( dω c ) =10. Solid lines correspond to indicated values of A =0 . , . , . , . , . ,
1. Dashed line designates the asymp-totic relation of the absorption in (8) calculated in the limit ωd √ κ /c ≪ which can be used to find the resonant frequency ω ren and linewidth ∆ ω analytically. We obtain that in clas-sically strong magnetic field ( γ ≪ ω c ), the resonant fre-quency takes the form described in (4) — identical to thereal part of Eq. (7), as expected.Provided that γ ≪ ω c , the linewidth of the resonancebecomes: ∆ ω = 2 γ + A ω ren d/c A . (9)Here, the total linewidth ∆ ω is the sum of renormal-ized collisional broadening proportional to γ and radia-tive broadening. The half-width ∆ ω/ A . For comparison, the dashedcurve indicates the line shape obtained from the asymp-totic relation in (8) for A = 1. Clearly, the asymptoticresult is in excellent agreement with the exact calcula-tion. In addition, Fig. 3 shows the resonance positionand linewidth as a function of the retardation parame-ter. Comparing the given numerical and analytical datalikewise makes it evident that calculations based on theexact expression for P ± [27] (green solid lines) perfectlyagree with the asymptotic results from (4) and (9) (bluedashed lines). We also note that when expressed in termsof A , the linewidth ∆ ω reaches its maximum value A m A (cid:3) r e n (cid:4) c A Δ ωω c // ( b ) ( a ) FIG. 3. Dependence of the position ω ren (a) and linewidth∆ ω (b) of the absorption maximum on the parameter A (3),computed for γ/ω c = 0 . κ = 12 .
8, and c/ ( ω c d ) = 10.Green (solid) lines indicate the exact numerical calculation of ω ren and ∆ ω [27]; blue (dashed) lines correspond to the ana-lytical expressions (4) and (9) for ω ren and ∆ ω , accordingly.The maximum linewidth is reached at the value of A m definedby Eq. (10). defined by: A m = s ω c dγc + ω c d γ c − − ω c dγc . (10)Next, let us analyze the absorption at the resonancefrequency, P ± ( ω ren ), as a function of the retardation pa-rameter, A ∝ √ n . Considering the extreme cases, werecognize that in the limit of A →
0, the absorption ap-proaches zero, as 2DES becomes virtually absent. Simi-larly, in the limit of A → ∞ , 2DES attains infinite con-ductivity, which also leads to zero absorption due to fullreflection of the incident radiation. Therefore, at finitevalues of A , the absorption is expected to have one ormore maxima. Furthermore, we find that under the con-dition of ”weak” magnetic field, i.e. ω c d/ ( γc ) <
4, theabsorption resonance P ± ( ω ren ) exhibits a single maxi-mum at A = 1. On the other hand, given ”strong”magnetic field, i.e. ω c d/ ( γc ) >
4, there occur two localmaxima of P ± ( ω ren ) at A , defined as follows: A , = − ω c d γc ± s ω c d γ c − ω c dγc , (11)with the local minimum of P ± ( ω ren ) situated betweenthe maxima, at A = 1.Now, let us compare the absorption of a circular polar-ized electromagnetic wave in gated and ungated 2DESsexposed to the magnetic field. In the case of the un-gated 2DES in vacuum [3], the absorption maximum ap-pears exactly at the cyclotron frequency ω c , while thehalf-linewidth of the resonance equals γ + Γ. By con-trast, in the gated 2DES, the factor (1 + A ) − leadsto the reduction in the linewidth, as well as the shiftin the absorption peak away from ω c , as defined inEq. (4). In addition, the radiative contribution to thelinewidth becomes significantly suppressed due to the fac-tor d ω ren /c ≪
1, according to Eq. (9). Thus, in thegated 2DES, one can obtain that when the radiative con-tribution to the linewidth dominates that of collisionalorigin, the linewidth narrows with increasing electronconcentration: ∆ ω ∝ A − ∝ n − , which is very muchunlike the case of the ungated 2DES.Consequently, we demonstrate that Kohn’s theoremcan be violated strongly, depending on the value of A .In contrast to the well-known effects of strong light-matter coupling [16], the theorem breaks down in thelow-frequency regime ω ≪ c/ ( √ κ d ). We attribute thisto the retarded interaction of electrons and currents inthe 2DES with their images in the metal gate.For practical purposes, we estimate the retarda-tion parameter A in 2DES based on a back-gatedGaAs/AlGaAs quantum well, given the following char-acteristic parameters: d = 0 . n = 5 · cm − ,and m = 0 . m , where m is the free-electron mass.As a result, we find A ≈
1, which in clean samples corre-sponds to the resonant frequency equal half the bare ω c .Therefore, CR renormalization in standard back-gatedsemiconductor structures can be far from negligible.In summary, we have conducted the analytical andnumerical investigation of the absorption of electromag-netic wave incident normally onto the gated or back-gated 2DES in the presence of a perpendicular magneticfield. Importantly, the study takes into account the effectof electromagnetic confinement in the natural resonatorformed by the 2DES sheet and the metallic back-gate,with a dielectric substrate in between. Unexpectedly, wefind the CR renormalization, i.e., the shift in the reso-nance frequency (4) and narrowing of the linewidth (9),to occur in the low-frequency regime when the radia-tion frequency is much smaller than the Fabry-Perot fre- quency of the resonator. We establish that given renor-malization is controlled by the retardation parameter(3), which depends on the electron concentration in the2DES. We prove that this parameter can be large enough,even in standard back-gated samples. As a result, it canlead to a tremendous shift and narrowing of the CR line.Therefore, gated and especially back-gated 2DESs provevery promising for exploring new physical effects, for in-stance, the experimental studies of the extreme regimesof light-matter coupling.We would like to thank I. V. Kukushkin and V. M.Muravev for numerous stimulating discussions. The workof A.A.Z. was supported by the Foundation for the Ad-vancement of Theoretical Physics and Mathematics ”BA-SIS” (project No. 19-1-4-41-1). The work was donewithin the framework of the state task and supportedby the Russian Foundation for Basic Research (projectNo. 20-02-00817). Supplementary Material I. Conductivity tensorbased on the Drude model
Consider a 2DES in a constant magnetic field B ap-plied perpendicular to the 2DES plane. In the frame-work of the Drude model, the dynamical longitudinal σ xx and transverse σ xy conductivities of the 2DES can be ex-pressed as follows: σ xx = e nm /τ − iω (1 /τ − iω ) + ω c , (12) σ xy = e nm − ω c (1 /τ − iω ) + ω c , where n is the electron concentration in the 2DES, − e and m are the electron charge and effective mass, ω c = | e | B/ ( mc ) is the electron cyclotron frequency inthe 2DES, and τ is the electron relaxation time. Supplementary Material II. The exact equation forthe absorption coefficient
Using the Drude model for the conductivity tensorwith the finite electron relaxation time τ (12), we findthe exact expression for the energy absorption coefficient P ± , given the circular polarization of the incident wave: P ± = 8Γ γ (cid:16) √ κ ( ω ± ω c ) cot (cid:16) ωd √ κ c (cid:17) + 2Γ (cid:17) + γ (cid:16) κ cot (cid:16) ωd √ κ c (cid:17) + 1 (cid:17) + 4Γ γ + ( ω ± ω c ) , (13)where γ = 1 /τ corresponds to the collisional broadening,and Γ = 2 πe nmc (14) designates the radiative broadening of the absorptionlinewidth. ∗ [email protected][1] G. Abstreiter, P. Kneschaurek, J. P. Kotthaus, and J. F.Koch, Phys. Rev. Lett. , 104 (1974)[2] S. James Allen, Jr., D. C. Tsui, and J. V. Dalton, Phys.Rev. Lett. , 107 (1974)[3] W. Chiu, T. K. Lee, and J. J. Quinn, Surf. Sci. , 182(1976)[4] S. A. Mikhailov, Phys. Rev. B , 165311 (2004)[5] D. A. Rodionov and I. V. Zagorodnev, JETP Lett. ,126—130 (2019)[6] Q. Zhang, T. Arikawa, E. Kato, J. L. Reno, W. Pan, J.D.Watson, M. J. Manfra, M. A. Zudov, M. Tokman, M.Erukhimova, A. Belyanin, and J. Kono, Phys. Rev. Lett. , 047601 (2014)[7] Q. Zhang, M. Lou, X. Li, J. L. Reno, W. Pan, J. D.Watson, M. J. Manfra, and J. Kono, Nat. Phys. , 1005(2016)[8] R. H. Dicke, Phys. Rev. , 99 (1954)[9] G. Mazza and A. Georges, Phys. Rev. Lett. , 017401(2019)[10] W. Kohn, Phys. Rev. , 1242 (1961)[11] I. V. Kukushkin, J. H. Smet, K. von Klitzing, W.Wegscheider, Nature , 409-412 (2002)[12] J. Keller, G. Scalari, F. Appugliese, S. Rajabali, M. Beck,J. Haase, C. A. Lehner, W. Wegscheider, M. Failla, M.Myronov, D. R. Leadley, J. Lloyd-Hughes, P. Nataf, andJ. Faist, Phys. Rev. B , 075301 (2020)[13] C. M. Hu, E. Batke, K. K¨ohler, and P. Ganser, Phys.Rev. Lett. , 1904 (1996)[14] T. Maag, A. Bayer, S. Baierl, M. Hohenleutner, T. Korn,C. Sch¨uller, D. Schuh, D. Bougeard, C. Lange, R. Huber,M. Mootz, J. E. Sipe, S. W. Koch, and M. Kira, Nat. Phys. , 119–123 (2016)[15] M. Mittendorff, F. Wendler, E. Malic, A. Knorr, M. Or-lita, M. Potemski, C. Berger, W. A. de Heer, H. Schnei-der, M. Helm, and S. Winner, Nat. Phys. , 75–81(2015).[16] P. Forn-Diaz, L. Lamata, E. Rico, J. Kono, and E.Solano, Rev. Mod. Phys. , 025005 (2019)[17] V. M. Muravev, I. V. Andreev, I. V. Kukushkin, S. Sch-mult, and W. Dietsche, Phys. Rev. B , 075309 (2011)[18] V. M. Muravev, P. A. Gusikhin, I. V. Andreev, and I. V.Kukushkin, Phys. Rev. B , 045307 (2013)[19] G. Scalari, C. Maissen, D. Tur˘cinkov´a, D. Hagenm¨uller,S. De Liberato, C. Ciuti, C. Reichl, D. Schuh, W.Wegscheider, M. Beck, and J. Faist, Science , 1323(2012).[20] C. Maissen, G. Scalari, F. Valmorra, M. Beck, J. Faist,S. Cibella, R. Leoni, C. Reichl, C. Charpentier, and W.Wegscheider, Phys. Rev. B , 205309 (2014).[21] G. L. Paravicini-Bagliani, F. Appugliese, E. Richter,F.Valmorra, J. Keller, M. Beck, C. Rossler, T. Ihn, K. En-sslin, G. Scalari, and J. Faist, Nat. Phys. , 186 (2019).[22] A. Bayer, M. Pozimski, S. Schambeck, D. Schuh, R. Hu-ber, D. Bougeard, and C. Lange, Nano Lett. , 6340(2017)[23] X. Li, M. Bamba, Q. Zhang, S. Fallahi, G. C. Gardner,W. Gao, M. Lou, K. Yoshioka, M. J. Manfra, and J.Kono, Nat. Photon. , 324 (2018)[24] L. V. Abdurakhimov, R. Yamashiro, A. O. Badrutdi-nov, and D. Konstantinov, Phys. Rev. Lett. , 056803(2016)[25] M. Geiser, F. Castellano, G. Scalari, M. Beck, L. Nevou,and J. Faist, Phys. Rev. Lett. , 106402 (2012)[26] A. V. Chaplik, Sov. Phys. JETP35