Hall Effect Gyrators and Circulators
HHall Effect Gyrators and Circulators
Giovanni Viola and David P. DiVincenzo
1, 2, 3 Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany Peter Gr¨unberg Institute: Theoretical Nanoelectronics, Research Center J¨ulich J¨ulich-Aachen Research Alliance (JARA), Fundamentals of Future Information Technologies (Dated: June 19, 2018)The electronic circulator, and its close relative the gyrator, are invaluable tools for noise man-agement and signal routing in the current generation of low-temperature microwave systems forthe implementation of new quantum technologies. The current implementation of these devicesusing the Faraday effect is satisfactory, but requires a bulky structure whose physical dimensionis close to the microwave wavelength employed. The Hall effect is an alternative non-reciprocaleffect that can also be used to produce desired device functionality. We review earlier efforts touse an ohmically-contacted four-terminal Hall bar, explaining why this approach leads to unaccept-ably high device loss. We find that capacitive coupling to such a Hall conductor has much greaterpromise for achieving good circulator and gyrator functionality. We formulate a classical Ohm-Hallanalysis for calculating the properties of such a device, and show how this classical theory simplifiesremarkably in the limiting case of the Hall angle approaching 90 degrees. In this limit we find thateither a four-terminal or a three-terminal capacitive device can give excellent circulator behavior,with device dimensions far smaller than the a.c. wavelength. An experiment is proposed to achieveGHz-band gyration in millimetre (and smaller) scale structures employing either semiconductorheterostructure or graphene Hall conductors. An inductively coupled scheme for realising a Hallgyrator is also analysed.
PACS numbers: 71.10.Pm
I. INTRODUCTION
The Faraday-effect circulator is an unsung workhorseof the contemporary surge of low temperature microwavedevice physics, playing a key role in permitting low noisecontrol and measurement of superconducting qubits andresonators. The essence of the three-port circulator is itsnon-reciprocal routing of signals: electromagnetic radia-tion is passed cyclically from one port to its neighbor –radiation in at port one goes out at port two, in at twogoes out at three, and in at three goes out at one, seeFig. 1. The S matrix describing the circulator is sim-ply [1] S = . (1)Here and later, the S matrix relates the incoming am-plitudes of electromagnetic waves to the outgoing ampli-tudes. We are not referring to the quantum S matrix forelectronic wave function amplitudes.Fig. 2 shows a circulator in place in a contemporaryqubit experiment [4]. A typical present-day experimentinvolving just a single superconducting qubit requires nofewer than four circulators [5] for the proper managementof signals used to do high-fidelity, rapid measurementson the qubit. (During the writing of this manuscript,a three-qubit experiment was reported [6] with no fewerthan eleven circulators!) While highly reliable and rea-sonably close to ideal in their designed frequency bandof operation, they are quite bulky. The few-centimeterlinear dimension of a circulator operating in the few- gigahertz frequency range is explained very simply: theFaraday effect causes circulation by a wave-interferencephenomenon [1], requiring a physical scale on the orderof the wavelength. Naive scaling of experiments to, say,hundreds of qubits would require an impractically largevolume of low-temperature space devoted to circulators.A primary objective of the present work has been to iden-tify a new physical basis for the circulator which permitsvery significant miniaturization. FIG. 1. The conventional symbol for the three-port circula-tor [1], indicating counterclockwise circulation (1 → → a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r FIG. 2. A standard Faraday circulator mounted in a low-temperature experiment. The circulator, with its enclosingmagnetic shield, is contained in the rectangular steel-colorerdbox (dimensions 6cm × × It is well known how to achieve miniaturatization ofthe circulator function to far below wavelength scale us-ing operational amplifier circuits [7]. But even if suchelectronic circuits could work at cryogenic temperatures,their power dissipation and noise performance would beunacceptable for current applications. Very compact gy-rators may be achievable with SQUID structures, ac-cording to a preliminary study [8]. Another form of ac-tive gyrator that is workable at low temperature is cur-rently considered in the area of parametric Josephson de-vices [9]; while these are integrated-circuit devices [10],their employment of resonator structures [11–14] meansthat their compactness is not guaranteed. These schemesseem to be related to theoretical ideas from long ago forrealising non-reciprocal devices with parametrically mod-ulated linear components [15–17]; these theories were ap-parently never put into practice (see also p. 30 of [2]).Several devices are straightforward derivatives of thecirculator, which deserve consideration in the own right.If one of the ports of the circulator is terminated in amatched load, that is, with a resistance Z equal to thewave or source impedance chosen for signal propagationin the system (often 50 or 100Ω, this is just the ratio ofthe guided wave’s voltage to its current), one obtains the isolator , in which signals are perfectly transmitted in onedirection between the remaining ports, and perfectly ab-sorbed in the other direction. In a recent experiment [5] FIG. 3. The gyrator, a two-port non-reciprocal device. a) The“quasi-optical” conventional symbol [1], emphasising thatthat the gyrator imparts a phase inversion to signals prop-agating in one direction only. b) The lumped-element 4-terminal symbol of the gyrator. According to Tellegen’s def-inition [18], the port currents and voltages satisfy the equa-tions V = ZI , V = − ZI . Z is the gyration resistance. three of the four circulators are configured as isolators,serving the function of blocking noisy (high temperature)radiation from entering the low-temperature part of theexperiment. If one port is unterminated, the resultingdevice is uninteresting: the reverse-direction signal un-dergoes reflection at the open port and is transmittedidentically to transmission in the other direction – bothforward- and reverse-propagating signals are transmit-ted without modification. If one circulator port is termi-nated in a short circuit, the resulting highly non-trivialtwo-port device is known as a gyrator . It is a maximallynon-reciprocal device in its effect on the phase of sig-nals. Forward-directed signals acquire no phase shift,while back-propagating signals are inverted, being phaseshifted by π , coming from the phase-inverting reflectionat the short-circuit termination. This is indicated in theconventional symbol for the gyrator (Fig. 3 (a)).The gyrator is arguably more fundamental than thecirculator, and it will be the main focus of study in thispaper. Historically the gyrator predates the circulatorand was responsible for its discovery. There are two dis-tinct methods, to be reviewed below, for obtaining circu-lator action using a gyrator. Before introducing these, itis best that we first summarize a few of the basic pointsabout the mathematical description of the gyrator [1, 2].The scattering matrix of the ideal gyrator is [1] S = (cid:18) −
11 0 (cid:19) . (2)The impedance and admittance matrices of the“matched” gyrator, for which the internal impedance andthe source impedance both equal Z , are given by thestandard matrix formulas [1] Z = Z ( I + S )( I − S ) − = Z (cid:18) −
11 0 (cid:19) . (3) Y = Z − = − Z (cid:18) −
11 0 (cid:19) . (4)Port impedances or admittances become especially usefulquantities in the “near field” or “lumped device” limit,when the device dimensions are much smaller than thewavelength of interest [1]. This is not particularly trueof the present-day Faraday devices, but will be true forthe devices we analyse here. In this near-field setting it isalways possible to identify four nodes (a pair for each portof the device) at which to define the port currents andvoltages that are related by the impedance or admittancematrices.This lumped-device point of view is embodied in theother standard symbol for the gyrator shown in Fig. 3 (b).This symbol, and the current-voltage relations shown,were introduced in the seminal paper of Tellegen [18] re-porting his invention of the gyrator concept. Tellegenhad realised [19] that the lumped-element model of elec-tric circuits was incapable of describing non-reciprocalbehavior, which could readily arise in general electromag-netic theory. He surmised that the gyrator is a minimaladdition to linear network theory to make it complete,i.e., to describe any arbitrary electromagnetic linear re-sponse. Subsequent work proved this surmise to be cor-rect [2]. Tellegen noted that the gyrator is a legitimatepassive circuit element, neither storing nor dissipatingenergy. He was hopeful [18–20] that a physical imple-mentation of this device would be possible. A partialrealisation of his gyrator was achieved in subsequent in-vestigations of the magneto electric effect [21]. However,considering that the ideal gyrator, as he defined it, hasthe response Eq. (3) for all frequencies , this realisationmust necessarily involve some approximation. FIG. 4. Hogan’s interferometric implementation [22, 23] of thethree-port circulator using one gyrator, using quasi-opticalnomenclature. One of the beams emerging from the Mach-Zehnder interferometer is reflected back into the structurewith a mirror. In Hogan’s original discovery, magic-tees areemployed rather than half-silvered mirrors (in microwave par-lance, these are directional couplers).
One clear strategy for approximate realisation is to ob-tain the gyrator characteristic over a limited band of fre-quencies. This is what was accomplished, a few years af-ter Tellegen’s initial theoretical proposal, by Hogan [23].His invention uses the non-reciprocal rotation of the po-larisation of quasi-free space propagating microwaves,arising from the Faraday effect occurring in magnetisedferrite materials (see also [22]). A functioning gyratorwas constructed that accurately approximated the idealgyrator response in a band around a few GHz. Hogan noted that by placing the gyrator in a interferometerstructure – in optics language it is a Mach-Zehnder typeinterferometer – “circulation” could be achieved (Fig. 4).It was immediately recognised that this circulator wouldhave many direct applications. Subsequent refinement ofthe structure, with a simplification and symmetrizationof the interferometer structure, was rapidly achieved, andthe Faraday circulator had achieved essentially its mod-ern form by around 1960 [24].In his original work Tellegen [18, 19] envisioned reali-sations involving non-reciprocal [25] electric or magneticpolarisations (see [21]). He did not envision a realisationbased on non-reciprocal electrical conduction , but an ef-fort was quickly made by other workers to employ thisnon-reciprocity – the Hall effect – to realise a gyrator.We will describe this effort in Sec. II which, unlike therealisation based on the Faraday effect, ended with anapparently definitive failure. In the present work we re-examine this failure, showing that there is an alternativeapproach to Hall effect gyration that is in fact successful.It should actually be superior to the Faraday gyratorin several respects, most notably that it should permitmuch, much greater miniaturisation of the gyrator, andtherefore of the corresponding circulator.The remainder of this paper will proceed as follows:Sec. II reviews the previous construction and analysis ofthe resistive gyrator. We make the case in Sec. III forwhy a reactive-coupling approach has the prospect formaking a fundamentally better Hall gyrator. The specificcase of the capacitively coupled gyrator is taken up inSec. IV. The extremal case of 90-degree Hall angle leadsto tremendous simplifications as discussed in Sec. IV A.Our analysis is applied to a two-terminal device in Sec.IV B, and to the four-terminal device in IV C, with thelatter giving good gyrator characteristics. The responseof this device is significantly different if the capacitivecontacts are smoothed, as analyzed in Sec. IV D. Threeterminal devices can also lead directly to a circulator,as analysed in Secs. IV E and IV F. The dual approachof inductive coupling is analysed in Sec. V, where scal-ing arguments are given indicating why the capacitiveapproach is to be favoured. A discussion of how thecurrent ideas might be put into practice in current ex-perimental graphene-sandwich structures is given in Sec.VI. Sec. VII gives conclusions, with some observationson the new problems posed for the quantum theory bythe present device concepts.
II. THE “GERMANIUM GYRATOR”
At the same time as Hogan’s work on the Faradaygyrator, another group of researchers (also at the BellTelephone research laboratories) took up experiments torealise gyration using the Hall effect, and a set of resultswere reported employing doped germanium, motivatedby the basic strategy that a low carrier density metalwill exhibit a large Hall effect [26].
FIG. 5. A Hall bar geometry with four ohmic contacts, witha uniform magnetic field H pointing in the z direction pro-ducing a Hall effect in the electric conduction equation (6)for the material. Experiments on a thin, three-dimensionaldoped germanium crystal were reported in 1953 [27] whichattempted to realise the gyrator (four-terminal labelling cor-responding to Fig. 3), with a signal field E x exciting a Hallcurrent J y . Intrinsic inhomogeneities in the field distribu-tion cause this device to have high losses, preventing it fromsuccessfully approximating the ideal gyrator. For large Hallangle, the losses become concentrated at “hot spots” at thepoints R and R (cid:48) indicated. Let us summarise the basic approach, following [27].A crystal is connected ohmically to four contacts, seeFig. 5. The material is three dimensional but thin (thetwo-dimensional electron gas had not been discovered in1953), thin enough that the conduction can be describedtwo-dimensionally; it is assumed that there is a uniformmagnetic field H perpendicular to the thin conductor.Contacts 1 and 1’ (see Fig. 5), the “current leads” inmodern parlance, will define one port of the gyrator, andcontacts 2 and 2’ – the “voltage leads” – will define thesecond port.We consider the action of this device within the classi-cal, Ohm-Hall framework. Here we follow the contempo-rary theoretical analysis of Wick [28] which gave an ex-cellent accounting of the experiments performed at thattime on the germanium gyrator [27]. The four contactsare equipotentials with (possibly time-dependent) poten-tials V , V (cid:48) , V , and V (cid:48) . It is assumed that there are noaccumulations of charge inside the conductor, and thatthe time dynamics is quasi-static, so that the potentialsatisfies the two-dimensional Laplace equation ∇ V ( x, y ) = 0 . (5)The contacts then define “Dirichlet” boundary condi-tions. The boundary conditions away from the contactsmust be established by a consideration of the conductionprocess. The Ohm-Hall formulation of linear electric con-duction in a magnetic field is the spatially local law − (cid:126) ∇ V = (cid:126)E = ρ(cid:126)j − R H (cid:126)j × (cid:126)H. (6)The standard approximate formula for the Hall coeffi-cient R H = 1 / ( en ) shows why a large Hall effect is ex-pected in a material, such as doped germanium, with asmall value of the carrier density n .According to Eq. (6) the electric field (cid:126)E and the cur-rent density (cid:126)j are not collinear, but have a fixed angle between them, the Hall angle θ H : θ H = tan − HR H ρ . (7)Inverting Eq. (6) and writing in componentwise formgives the matrix equation (cid:18) j x j y (cid:19) = σ (cid:18) cos θ H sin θ H − sin θ H cos θ H (cid:19) (cid:18) E x E y (cid:19) (8)= (cid:18) σ xx σ xy σ yx σ yy (cid:19) (cid:18) E x E y (cid:19) , σ ≡ (cid:112) ρ + ( HR H ) . This equation defines the components of the conductivitytensor σ ij .With this in hand we can state the remaining boundarycondition. Away from the contacts on the boundary nocurrents should flow in and out of the conductor, that is,ˆ n · (cid:126)j ( (cid:126)r ) = 0 , (cid:126)r ∈ S. (9)Here ˆ n is the normal vector to the boundary surface S .For an ohmic conductor without a Hall effect, Eq. (9)would lead to “Neumann” (normal derivative) boundaryconditions. However, due to the non-collinear relation-ship between ∇ V and j , Eq. (8,9) imply the rotatedderivative boundary conditionˆ n H · ∇ V = 0 , (10)or more explicitlycos θ H ∂V∂n + sin θ H ∂V∂s = 0 . (11)In words, this condition relates the normal derivative of V to its tangential derivative (i.e., along the boundarycoordinate s , see also Fig 7) in proportions determinedby the Hall angle. ˆ n H is the normal unit vector rotatedby (minus) the Hall angle. FIG. 6. A circuit representation of the response that is achiev-able using the ohmically-contacted Hall bar of Fig. 5. Thelossy (diagonal) part of the impedance matrix R fundamen-tally cannot be smaller than the antisymmetric lossless gyra-tion resistance Z . Wick [28] gave a very general solution to this prob-lem, for a 2D conductor of arbitrary polygonal shape, us-ing conformal mapping techniques; his analysis was usedrepeatedly in subsequent studies of such problems [29],up to the present [30, 31]. His key observation, for thepresent purposes, is that gyration can only be poorlyapproximated by this device. He establishes his “nogo” result with a simple argument which shows that if I = I (cid:48) = 0 (i.e., no total current flowing through leads2 and 2 (cid:48) ), the potentials V and V (cid:48) cannot lie outsidethe range between V and V (cid:48) . This permits him to provethat there must be an input resistance R at both inputports, at least as large as the gyration resistance. Thus,the best approximation to a gyrator that can be achievedwith the “resistive gyrator” is as shown in Fig. 6. Wicktakes pains to point out that this result is independent ofthe sample shape, and is also true “in the limit of infinitemagnetic field.” By this he means the limit θ H → π/ σ being finite, or, in other words, σ xy = − σ yx , σ xx = σ yy = 0 . (12) θ H = π/ θ H > π/ θ H → π/ etal. [46] found that this boundary charge smearing in factleads only to small quantitative errors compared withthe line-charge model. Despite all potential difficulties, the classical theory has indeed been extremely successfulin giving detailed, quantitative predictions of transportbehavior in suspended graphene Hall bars [47].The quantum and classical models are even in agree-ment on the question of where the two-terminal dissipa-tion occurs in the quantum Hall state. Although Wick’sargument is clearly correct, it nevertheless may be viewedas paradoxical that the classical model is capable of de-scribing any dissipation in the θ H → π/ P diss. = (cid:90) A (cid:126)E · (cid:126)j dxdy = σ cos θ H (cid:90) A | (cid:126) ∇ V | dxdy. (13)Since cos θ H = 0, there is “obviously” no dissipation pos-sible. This argument is wrong because the fields do nothave finite limiting behaviour as θ H → π/
2. As reviewedclearly by Rendell and Girvin [29], the fields become di-vergent at the ends of the ohmic contacts, on either theleft or right sides according to the direction of the mag-netic field (R and R’ points in Fig. 5 for the orienta-tion of H in the figure) depending on the sign of θ H ,as | θ H | → π/
2. The fields are well behaved elsewhere;thus the argument of Eq. (13) is almost right: Jouleheating goes to zero everywhere, except for “hot spots”(becoming Dirac delta functions, in fact) at the R and R (cid:48) points. This hot spot behaviour is observed experimen-tally [41, 48], and also has a simple interpretation in aquantum treatment[44], where the dissipation is ascribedto a sudden change of the local chemical potential as thequantum edge states enter the lead reservoirs. III. REACTIVE COUPLING APPROACH
This last observation has directed the approach thatwe report in this paper, which analyses alternative deviceschemes that will achieve gyration in the “quantum” Halllimit θ H → π/ P diss. in Eq.(13) from being zero is a singularity arising from the in-compatibility of the ohmic and insulating boundary con-ditions, we can investigate contactless, or reactive, meansof contacting the Hall conductor. We find both an induc-tive and a capacitive scheme in which the new bound-ary conditions avoid dangerous boundary singularities as θ H → π/
2. The fields have finite limits everywhere, andthe argument given above applies: as cos θ H → P diss. goes to zero – the “quantum” Hall state indeed givesa dissipationless device. A pure gyrator is not directlyachieved, but with proper choice of design excellent ap-proximations to gyration should be achievable in con-venient frequency regions, and with physical device di-mensions far smaller than for the corresponding Faradaygyrator.While both the inductive and capacitive schemes haveappealing features, we believe that the capacitive cou-pling scheme has the greatest potential for being realisedexperimentally, and has the greatest potential miniatur-isability; thus we will explore this scheme in the greatestdetail in the following. IV. CAPACTIVELY COUPLED HALL EFFECTGYRATOR
We will now state a new boundary condition that isappropriate for the case of a segment of boundary of aHall conductor forming one side of a capacitive couplingas shown in Fig. 7. Such a capacitor will be charac-terised by having a capacitance per unit perimeter length c ( s ). While at this point in our discussion c ( s ) should beviewed purely as a phenomenological capacitance func-tion, it will be important for the physical discussion givenin Sec. VI that this function incorporate the full electro-chemical capacitance to the Hall material, including thequantum capacitance [49]. Writing c ( s ) this as a functionof the perimeter coordinate s allows the possibility thatthe capacitor has smoothly variable strength around theperimeter. We will see that piecewise constant capaci-tances are completely reasonable, in the sense that stepchanges in capacitance do not lead to any singular be-haviour of the fields, unlike for the case of abrupt endingof ohmic contacts. FIG. 7. Arrangement for four-terminal capacitive couplingto a 2D Hall conductor. The coordinate measured along theperimeter is labeled s , the origin of this coordinate is labelled O , ending at the same point at perimeter length P . Terminalsegments T are in capacitive contact with external electrodesat a.c. potentials ¯ V . The left and right L/R endpoints ofthe T segments are labeled. T segments are separated byuncontacted insulating segments U . For Hall angle θ H = π/ We consider the external capacitor electrode to be a good conductor, and thus all at a single potential ¯ V . Ifat point s on the perimeter the potential at the edge ofthe Hall conductor is V ( s ), then the displacement currentdensity j D ( s ) at that point of the capacitor, equal to thecurrent density inside the Hall material directed normalto the edge ˆ n · (cid:126)j ( s ), is given by the ordinary capacitanceequationˆ n · (cid:126)j ( s, t ) = j D ( s, t ) = c ( s ) ddt ( ¯ V ( t ) − V ( s, t )) . (14)The static case is uninteresting, and we have made allquantities explicit functions of time t . Following Eqs.(9) and (10), the normal current is proportional to therotated projection of the field gradient:ˆ n · (cid:126)j ( s, t ) = − σ ˆ n H · ∇ V ( s, t ) = c ( s ) ddt ( ¯ V ( t ) − V ( s, t )) . (15)We may write this equation in the frequency domain,giving our final boundary-condition equation: − σ ˆ n H · ∇ V ( s, ω ) = iωc ( s )( ¯ V ( ω ) − V ( s, ω )) . (16)While this is a perfectly well-posed mixed, inhomoge-neous boundary condition for the Lapace equation, weare not aware that it has been previously examined. Itis applicable around the entire boundary, as the regu-lar insulating boundary condition for the Hall conductor(Eqs. (10,11) above) corresponds to a region of bound-ary with c ( s ) = 0. Ohmic boundary conditions are insome sense treated by taking c ( s ) → ∞ , but this casewill not come up in the following, and by keeping c ( s )finite we avoid the singular behaviour of the fields dis-cussed above. Note that our boundary condition (16) iscomplex-valued; this has the normal interpretation fora.c. electrical problems, that the real part of the field isthe in-phase response and the imaginary part of the fieldis the out-of-phase or quadrature response; that is, thewhen driven with a field at frequency ω , the temporal re-sponse is Re( V ( r, ω )) cos ωt + Im( V ( r, ω )) sin ωt . Whilewe will apply these boundary conditions for ω up to mi-crowave frequencies, we will consider only cases wherethe device dimensions are much smaller than the wave-length of radiation at these frequencies; in this near fieldlimit the quasi-static analysis of the bulk conduction asdetermined by the Laplace equation Eq. (5) still applies.The boundary conditions for the ohmic and capacitivecontacts (Eqs. (10,11) and (16)) have different behav-ior under conformal transformation. The ohmic case isconformally invariant since it only fixes the direction of (cid:126)E with respect to the boundaries. Instead, Eq. (16) isa condition on the values of (cid:126)E along a direction, henceit is not conformally invariant. Therefore the conformalmapping methods [28, 30, 50] cannot easily be appliedto the electrostatic problem in our case; however, as wewill show shortly, we can calculate all device quantitiesof interest (analytically for θ H = 90 ◦ ) without resort toconformal mapping techniques.The problem of finding the two-port response, e.g., theadmittance matrix Y (or Z or S ), is now straightfor-wardly posed: given a geometry as in Fig. 7, we identifyfour terminal segments T , (cid:48) and T , (cid:48) and four uncon-tacted, insulating segments U , (cid:48) and U , (cid:48) . There is nocapacitance in the U regions. c ( s ) will be nonzero alongthe T segments; we will analyse both the case of con-stant capacitance per unit length, and the case wherethe capacitance goes smoothly to zero at the ends ofthese segments. Smoothing will cause significant differ-ences in the response, but this difference does not mod-ify the main features in the relevant frequency range.The a.c. terminal potentials ¯ V i =1 , (cid:48) , , (cid:48) will, as statedabove, be taken as constants (possibly complex) in eachof the terminal segments T i . Solving the field prob-lem as a function of ω gives normal boundary currentsˆ n · (cid:126)j ( s, ω ) = − σ ˆ n H · ∇ V ( s, ω ); integrating gives the ter-minal currents I i ( ω ) = (cid:90) T i ˆ n · (cid:126)j ( s, ω ) ds = − σ (cid:90) T i ˆ n H · ∇ V ( s, t ) ds. (17)These terminal currents are linear functions of the ter-minal potentials: I i ( ω ) = (cid:88) j =1 , (cid:48) , , (cid:48) y ij ¯ V j . (18)The coefficients in this equation are admittances, but onefurther calculation is needed to obtain the two-port ad-mittance matrix Y from them. We must enforce the con-dition that the terminal pairs T − T (cid:48) and T − T (cid:48) actas ports. The pair T i − T j is a port if I i = − I j . Onemust determine the relative potential between our twoports, as measured by, e.g., ¯ V − ¯ V , which will cause theport condition I = − I (cid:48) to be satisfied; then the othercondition I = − I (cid:48) is automatically satisfied, since thetotal current entering the Hall conductor is zero. Thenthe port currents are functions of the port voltages, viz., I = Y ( ¯ V − ¯ V (cid:48) ) + Y ( ¯ V − ¯ V (cid:48) ) ,I = Y ( ¯ V − ¯ V (cid:48) ) + Y ( ¯ V − ¯ V (cid:48) ) , (19)thus defining the 2 × Y . Wewill now investigate under what conditions the gyratormatrix Eq. (4) is obtained.One further comment about going from terminal toport response: in the electrical literature, it is often as-sumed without discussion [2] that the ports are electri-cally isolated, meaning that there is identically vanishingdependence on the potential difference between the twoports ( ¯ V − ¯ V in the analysis above). Under these cir-cumstances the port current condition is also automati-cally satisfied. This isolation is not present in our device(e.g., current can, in principle, flow from terminal 1 to 2).It is understood that in many circumstances this input-output isolation is not necessary for proper functioningof the device; if it is needed, it can be achieved by sep-arate isolation (e.g., transformer coupling). This issuewill arise one further time in the present paper, in theanalysis of the three-terminal gyrator in Sec. IV E. A. Response requires only boundary calculationfor Hall angle π/ We now proceed to explicit calculations of several Hallgyrator structures. Since, as we confirm shortly, losslessoperation is achieved in the case of Hall angle equal toits extremal value of π/
2, and so θ H = π/ θ H = π/
2, permittingclosed-form solutions for a wide class of device structures.This simplification arises from examining our boundarycondition equation (16) for this case; recalling Eq. (11),we obtain − σ ∂V ( s, ω ) ∂s = iωc ( s )( ¯ V ( ω ) − V ( s, ω )) . (20)This boundary condition equation now contains only thetangential derivative of the potential, which involves onlypotential values at the boundary. Thus, this equation isnow a closed one-dimensional condition in the boundarycoordinate s , fully determining the field on the boundarywithout reference to the interior of the conductor. Thefield in the interior of the conductor is still well defined,but is entirely a slave of the boundary potential; thefull field can be calculated by considering the perimeterfield as a Dirichlet boundary condition. But all device-response coefficients are purely functions of the perimeterfield, so the interior field need never be calculated. Twodimensional plots of the in-phase and out-phase fields V ( s, ω ), for the Hall bar with four capacitive contacts(the same setup as in Fig. 5) is shown in the Fig. 8 for afrequency for which perfect gyration occurs.Note that the calculation of the terminal currents alsotakes a much simpler form in this case; Eq. (17) becomes I i ( ω ) = − σ (cid:90) T i ∂V ( s, ω ) ∂s ds = σ ( V ( s = L i , ω ) − V ( s = R i , ω )) . (21)Thus, the current is simply given by the difference of thepotential from the left point of the capacitor L i to theright point R i (see also Fig. 7) [51]. Furthermore, thefield solution is completely independent of the shape ofthe boundary; it can be deformed at will (as suggestedby Fig. 7), and the boundary potential and all deviceresponse coefficients will be unchanged as long as theperimeter length P and the capacitance function c ( s ) areunchanged. Note that for general boundary conditions,the solution on the perimeter can be written as an inte-gral over the perimeter (cf. Eq. (7.2.12) of Ref. [50]);but the kernel of this integral is a Green function which,in the general case, is globally sensitive to the detailedstructure of the entire conductor. Thus, our situation isquite special.It is valuable to note that the homogeneous part of Eq.(20) is a one-dimensional Dirac eigenvalue equation, withperiodic boundary conditions from 0 to P , with c ( s ) play-ing the role of the position-dependent mass of that Diracequation. The two-component Dirac spinor consists ofthe real and imaginary part of V . The eigenfrequencies ω n of this equation are equally spaced: ω n = 2 nπσ (cid:82) P c ( s ) ds . (22)We will see that these eigenmodes have the physicalmeaning of chiral edge magnetoplasmons of the Hall con-ductor; they will set the scale of frequency at which in-teresting gyrator behavior occurs.Magnetoplasmons have been investigated thoroughlyin 2D Hall conductors [52, 53], including in the quantumHall regime [54, 55]. In Sec. VI we will discuss details ofhow this work has developed up to the present, and whatsuggestions it makes for the physical implementation ofthe devices analysed here. B. Two-terminal device
While it has no application for gyration (two terminaldevices must be reciprocal), the solution to the simpletwo-terminal problem is instructive, especially for the in-sight that it gives into the edge magnetoplasmon dynam-ics in this system. We consider the special case of thetwo capacitors with constant capacitance per unit lengthattached to the Hall conductor. Suppose the length ofthe capacitor is L and the capacitance per unit length is c , so that the total lead capacitance is C L = cL . Thenthe (scalar) admittance of the device is calculated to be Y ( ω ) = iσ tan ωC L σ . (23)This solution is still correct for the case when the twoleads have different widths, but the c - L products shouldbe the same. The placement of the leads around theperimeter is arbitrary, the lengths of the insulating re-gions between the leads are irrelevant, and the conductorcan be of arbitrary shape (including sharp turns). Inte-rior holes in the conductor also play no role. Note thatthe poles of this admittance coincide with magnetoplas-mon eigenfrequencies (as defined in Eq. (22)), and thatthe low-frequency limit, iωC L /
2, is that of two capacitors C L in series.In fact, the admittance function (23) is a familiar one.It is identical to that of a segment of transmission linewith characteristic impedance 1 /σ and transit time (wavevelocity times length) of τ = C L / σ , open-circuited atthe end. An important feature of this transmission-lineresponse is that a short voltage pulse applied to it isperfectly reflecting, but with a transit-time delay of 2 τ .When this pulse is applied to the two-terminal Hall de-vice, where does the pulse live during this 2 τ transittime? The answer is that when the pulse arrives at thecapacitor electrodes, it produces a non-zero field in the FIG. 8. Two dimensional plots of the in-phase (left panel) andout-of-phase (right panel) potential fields for a capacitivelycoupled four-terminal Hall-bar device, for θ H = π/
2. Thefrequency of the applied field is ν gy = σ/ C L (Eq. (29)),the first perfect gyration frequency. Contacts span the entire(length=2) of the top and bottom edge of the bar, with ¯ V = ± . V . The position of the contacts are indicated with theblack bar. Length-2 contacts are centered on the left andright of the bar, with ¯ V = 0. The capacitance function c ( s )is a constant on all the terminal boundaries. Perfect gyrationrequires that there be current only from the left to the rightterminal, and it be in-phase. We see that the in-phase currentdensity (left panel), which follows the potential contours inaccordance with the guiding-center principle, does indeed flowsmoothly from one contact to the other. Note that thesecontacts are not equipotentials, as they would be for ohmiccontacts; at this perfect gyration frequency, however, the topand bottom terminals are at equipotentials. There is a non-zero out of phase current density flow (right panel), but it ispurely local to the contact, and results in no net current. Hall conductor only in the immediate vicinity of the rightedge (points R i ) of the two capacitors. This localisededge field propagates, in a dispersionless way, counter-clockwise around the edge of the conductor, with velocity v pl = L/ τ = σ/c. (24)After time 2 τ these two edge excitations reach the leftend (points L i ) of the capacitors, causing a re-emissionof the radiation pulse back into the leads. For normaldevice parameters, this plasmon propagation velocity isfar smaller than the speed of light; thus, this “simulatedtransmission line” is very compact compared with thecorresponding real transmission line. C. Four terminal device: the gyrator
We now return to the four terminal device, revisitingthe approach of Mason et al. [27] (Fig. 5), with ohmiccontacts replaced by capacitive contacts. For the case ofuniform capacitance, c ( s ) =const., and four equal con-tact capacitors with capacitance C L , the exact solutionfor the two-port response matrices is elementary. Theadmittance is Y P ( ω ) = σ (cid:18) i tan ωC L σ − ωC L σ − sec ωC L σ i tan ωC L σ (cid:19) , (25)which when inverted gives the two-port impedance Z P ( ω ) = 1 σ (cid:18) − i cot ωC L σ − − i cot ωC L σ (cid:19) . (26)Note that Y and Z satisfy the conditions for multiportlossless response, which are [2] that the imaginary partof the matrix be symmetric and an odd function of fre-quency, while the real part is antisymmetric and an evenfunction of frequency. (These conditions are also triv-ially satisfied for the one-port device in Eq. (23)). Thiscondition is equivalent to P diss. = 0 (see Eq. (13)). Thisconfirms that the argument given using Eq. (13) appliesto our calculation, given that there is no singularity in-volved in going to the θ H → π/ θ H = π/ θ H = π/ − (cid:15) : to lowest order in (cid:15) the field solutions canbe taken to be independent of (cid:15) , so that the total dissi-pation of the device will, using Eq. (13), be proportionalto (cid:15) , but with “nonuniversal” coefficients (i.e., dependenton the details of the device geometry).The scattering matrix is obtained using the formula S = ( Z + Z I ) − ( Z − Z I ) [1] with Z = 1 /σ : S P ( ω )= 2 d − (cid:18) cos ( ωC L σ ) 2 sin ( ωC L σ ) − ( ωC L σ ) cos ( ωC L σ ) (cid:19) (27a) d = 3 cos( ωC L σ ) − i sin( ωC L σ ) − S P is unitary (because the device is lossless)and non-symmetric (because it is non-reciprocal).The relation to gyration is especially easy to see usingthe impedance matrix Eq. (26). There is a series of fre-quencies at which perfect gyration (Eq. (3)) is achieved, given by the equation cot ωC L σ = 0; these frequencies are ω gy = πσC L (1 + 2 n ) , n ≥ ν gy = σ C L (1 + 2 n ) [Hz] n ≥ Y , cf. Fig. 10. Thus thegyration is fundamentally non-resonant, and is presentto good approximation over relatively wide ranges of fre-quency. D. Smoothed capacitances: two-port case
While many aspects of the device geometry are ir-relevant for the port response, the details of the edge-capacitance function c ( s ) do matter. We now study thedevice behavior if the capacitors are smoothed, so that c ( s ) goes continuously to zero at the edges of the capaci-tors. Closed-form expressions of the solution of Eq. (20)are obtainable for many different tapering functions; aconvenient analytic form is c ( s ) = (cid:40) c | s | < L c sech (cid:16) | s |− L λ (cid:17) | s | > L (30)Assuming that the insulating region between contacts ismany λ s long so that these is negligible overlap betweenthese capacitance functions, we find the two-port admit-tance to be Y P,λ = (31) σ sin (cid:0) Lcω σ (cid:1) cos (cid:16) (2 λ + L ) cωσ (cid:17) (cid:32) − i cos( ( L +2 λ ) cω σ ) sin( ( L +2 λ ) cω σ ) − sin( ( L +2 λ ) cω σ ) − i cos( ( L +2 λ ) cω σ ) (cid:33) . We see that for small rounding, this response exhibits aslow modulation in frequency (on the scale of ω = σcλ ),with the low-frequency behavior matching the unrounded( λ = 0) response calculated above. Fig. 9 plots this re-sponse and the relevant component of the scattering ma-trix S P,λ or simply S for slightly rounded capacitances.As Fig. 10 shows, at low frequency perfect gyration oc-curs at the regularly-spaced frequencies as indicated byEqs. (28,29). This is indicated by | S − S | attainingthe value 2; because of unitarity, this can only occur if S = S = 0 and S = − S , the conditions for a per-fect gyrator. We see in fact that at low frequency thiscondition is satisfied over wide bands. As the modulationdue to the rounded contacts begins to have an effect, thefrequency dependence of | S − S | is modified, but itstill returns to the ideal value of 2 frequently, albeit overnarrower frequency ranges.We note that this modulation causes the perfect gyra-tion points to change from having a real-valued S matrix( S , S = ±
1) to being complex valued; the S matrixacquires an overall reciprocal phase factor. Since this is0 - - - i Y (cid:144) Σ - - Y (cid:144) Σ Ω C L Σ S - S ¤ (cid:144) FIG. 9. The behavior of the admittance (Eq. (25)) and of the scattering parameters (cf. Eqs. (3,4) with Z = 1 /σ ) for the twoport device with smoothed capacitive contacts. The capacitance function is as given in Eq. (30) with λ = L/
12. We show alarge range of frequency, including about 100 poles of the admittance, and about 50 good gyration points as given by Eq. (28),in order that the slow modulation of ω on the scale σ/cλ can be seen. The dashed lines showing this modulation sinusoid areguides to the eye. Top panel: (11) component (pure imaginary) of the admittance matrix Y P . Middle panel: (12) componentof the admittance matrix Y P . Bottom plot: | ( S P,λ ) , − ( S P,λ ) , | /
2. Due to unitarity, this quantity can only attain thevalue unity if good gyration is achieved ( S matrix proportional to Eq. (2)). We see that despite the modulation caused bysmoothing, perfect gyration occurs regularly along the frequency axis, at points close to those given by the formula Eq. (28)for the unrounded case. Ω C L Σ(cid:45) (cid:45)
FIG. 10. Close up of Fig. 9: | ( S P,λ ) , − ( S P,λ ) , | / | ( Y P,λ ) | and | ( Y P,λ ) | in green, blue and red respectively,at low frequency. Over the first period of the response (i.e.,for 0 < ωC L / σ < π ) the response shown here is almost in-distinguishable from that of the constant capacitance, Eqs.(25,32). what one obtains for a perfect gyrator with a change ofreference plane [1], it is fair to refer to this still as perfectgyration. It would, however, require some reconsidera-tion of the Hogan construction Fig. 4; the reference armof this interferometer would have to have a corresponding phase change, which may cause it to be a significant frac-tion of a wavelength in size. To make this constructioncompact in this case, the reference arm phase delay couldbe simulated by two cascaded Hall effect gyrators, chosento given the correct overall net reciprocal phase [18].Finally, we note that we have used a source impedance Z = 1 /σ for calculating S . We find that for Z > /σ the perfect gyration condition | S − S | = 2 continuesto be satisfied for a regularly spaced set of frequencies.For Z < /σ (the more likely case, see the discussion inSec. V) perfect gyration no longer occurs at low frequen-cies; but with finite rounding, at higher frequency perfectgyration again occurs. However, for large impedance mis-match Z << /σ good gyration occurs only over verynarrow ranges of frequency. E. Three-terminal device and the Carlinconstruction
Carlin [56, 57] (see also [2]) noted that there are severalalternatives to the Hogan construction (Fig. 4) for real-izing a three-port circulator using a gyrator. They arearguably more direct in that they do not require an inter-1
FIG. 11. The construction of Carlin [2, 56, 57] for realis-ing a circulator of Fig. 1 and Eq. (1), given a nonreciprocalthree-terminal device (terminals defined at solid dots) withadmittance as in Eq. (32). Anticipating Sec. VI and Fig.15, we depict the capacitive contacts as strips overlappingthe edge of a rectangular piece of Hall material; the capaci-tances should be the same, and may or may not be rounded.Note that the primed terminals of each of the three externalports are tied together, but should be kept away from the Halldevice so that they have no ohmic or capacitive contact to it. ferometer. In Carlin’s original construction he employsthe classic Tellegen gyrator (Fig. 3(b)) tied to a commonground, i.e., with terminals 2 and 2’ short circuited. Thisapproach cannot be applied directly to our four-terminalHall gyrator, because of the lack of input-output isola-tion mentioned above (Sec. IV). However, Carlin’s con-struction can be stated more directly: a three-terminaldevice with the right non-reciprocal admittance matrix(see Eq. (32)) can be converted to a circulator with eitherof the two Carlin constructions Figs. 11, 12. The “dual”construction of Fig. 12 actually gives a phase-invertingcirculator, with S being the negative of Eq. (1); we arenot aware of any current application of the circulator inwhich the phase of S is relevant.Perfect Carlin circulation is obtained with a three-terminal device with the admittance matrix Y T ( ω ) = ia b − b ∗ − b ∗ ia bb − b ∗ ia (32)when a = 0, Im( b ) = 0, and Re( b ) = 1 /Z , the sourceimpedance. This is obtained by using a three terminalHall device with equal contact capacitances C L . (Thenaive procedure of short circuiting 2 and 2’ in the four-terminal device would lead to one contact effectively hav-ing capacitance 2 C L .) For the case of constant capaci-tance functions, the response is as in Eq. (32), with a = 2 σ sin ωC L σ ωC L σ (33) b = σ − − iωC L σ ωC L σ (34) FIG. 12. A dual construction of Carlin [56, 57] for realisinga circulator with the same three-terminal device as in Fig.11. Phase-inverted circulation is achieved, i.e., the S matrixis the negative of Eq. (1). If the device is matched ( Z = 1 /σ ) perfect circulation isobtained at the frequencies ν circ = σ C L (1 + 2 n ) [Hz] , n = 1 , , ... (35) F. Rounded capacitances: three-terminal case
This response matrix can also be easily calculated inthe case of rounded capacitance, Eq. (30). The result is Y T,λ = ia λ b λ − b ∗ λ − b ∗ λ ia λ b λ b λ − b ∗ λ ia λ , (36a) a λ =2 σ sin (cid:16) cω ( λ + L ) σ (cid:17) − sin (cid:0) cλωσ (cid:1) (cid:16) cω (2 λ + L ) σ (cid:17) , (36b) b λ = σ exp( − icλωσ ) (cid:0) − − icLωσ ) (cid:1) (cid:16) cω (2 λ + L ) σ (cid:17) . (36c)This response again has the same slow modulation in fre-quency as in the four-terminal case. In Fig. 13 we charac-terise the quality of the resulting impedance-matched cir-culator by computing the quantity | ( S T,λ ) +( S T,λ ) +( S T,λ ) | which, due to the unitarity of the S matrix, canbe equal to three only for the case of an ideal circulator(independent of references phases). We see that at lowfrequency perfect functioning is obtained; the response isnot as robust as in the two-port case, in the sense thatwhen the modulation due to the rounding becomes im-portant, circulation is degraded (to recur again at higherfrequency). Another interesting functionality emerges:as the anti-clockwise circulation degrades, clockwise cir-culation as measured by | ( S T,λ ) +( S T,λ ) +( S T,λ ) | Ω C L Σ (cid:160) S (cid:164) (cid:43) (cid:160) S (cid:164) (cid:43) (cid:160) S (cid:164) Ω C L Σ (cid:160) S (cid:164) (cid:43) (cid:160) S (cid:164) (cid:43) (cid:160) S (cid:164) FIG. 13. Characterization of the scattering matrix of thefirst Carlin construction Fig. 11 with rounded capacitances.Rounding is as in Eq. (30) with λ = L/
12. Upper panel: | ( S T,λ ) + ( S T,λ ) + ( S T,λ ) | . Due to unitarity the maxi-mal value this sum can attain is 3, and at such a point perfectcounterclockwise (1 → → →
1) circulation is achieved.We see that when the slow modulation due to the round-ing begins to occur, perfect gyration is lost. Bottom panel: | ( S T,λ ) + ( S T,λ ) + ( S T,λ ) | . We see that the modula-tion causes this quantity to periodically attain the value 3,meaning that perfect clockwise circulation (1 → → → occurs, which becomes almost perfect in a range of fre-quencies. Thus, we have a set of interesting alternativesfor achieving circulation. Compared with the Hogan cir-culator, the Carlin circulators are more flexible, but aremore sensitive to capacitance rounding and do not workproperly when there is an impedance mismatch. V. INDUCTIVELY COUPLED HALL EFFECTGYRATOR
As pointed out in Tellegen’s original work [18], bothelectric and magnetic effects can be considered for gyra-tion. We therefore briefly take up a dual approach to us-ing the Hall effect for gyration, in which the lead couplingis magnetic rather than electric. This approach leads toa very elegant view of the response of a Hall structure tomagnetic induction, but we consider this approach lesspromising for application and will only give a sketch ofthe results.Inductive coupling requires loops of conductor, thus weconsider the nonplanar Hall-material geometry shown in
FIG. 14. Top: two-loop, nonplanar Hall conductor for re-alizing an inductively coupled gyrator. The magnetic fieldtexture necessary to produce the Hall effect in this conduc-tor is shown; threading fluxes Φ ext and Φ ext can apply elec-tromotive forces E i = ˙Φ exti around the two loops. Bottom:periodic representation of conductor, in which the loops areunwrapped at the wavy lines in the top figure. Solution to thefield problem for Hall angle θ H = π/ E , . Fig. 14. Topologically this surface is a torus with a holecut into it; such a geometry was actually considered, fora Faraday material, by Tellegen in his later work [20].While the topology we consider here has been standard inthought experiments for understanding the quantum Halleffect [58], and very analogous “crossover” Hall topolo-gies have been noted for the achievement of interestingeffects for quantum error correction [59, 60], it must beunderstood that there is no material system in whichthere is a known technique for actually producing a ma-terial with a large Hall effect in such a topology. It is forthis reason that we do not anticipate that experimentscan be performed to pursue this idea; but its principlesare interesting to elucidate nonetheless.Returning to Fig. 14, we suppose that one port pro-vides input by the a.c. signal applied as a magnetic fluxΦ ext ( t ). The time derivative of this flux produces ane.m.f. E around loop 1. Since there is then a nonzeroline integral of the electric field around this loop, thepotential field V ( r ) strictly speaking does not exist; but3since ∇ · E = 0, one can locally define a potential thatsatisfies the Laplace equation; but it will be multivalued,increasing by E each time a path is taken around theloop. This can be unwrapped into a periodic represen-tation as shown in Fig. 14. In the limit of θ H = π/ c ( s ) = 0. This says that thepotential, in this periodic representation, is an equipo-tential on each of the periodic images of the loop edgesas shown in Fig. 14. Furthermore, the very simple rela-tion between conductor current and boundary potentials,Eq. (21) means that the relation between the loop cur-rent and the e.m.f. in the other loop is perfect gyration,independent of frequency and dependent only on topol-ogy: I = σ E , I = − σ E . (37)The trouble with this approach, other than the extremedifficulty of producing non-planar conductors exhibitinga large Hall effect, is the need to couple externally to thevariables of Eq. (37), which requires two transformer-likestructures. The weakness of magnetic coupling makesthis problematic.We have found that the inductance L of this cou-pling structure imposes a lower cutoff on the frequency atwhich gyration becomes effective. This frequency scaleslike ω cutoff ∼ R gy /L = 1 /σL . The scale of inductance isset by L ∼ µµ d , where d is the physical scale of the de-vice. If the scale of σ is the quantum scale h/e , then thecutoff, expressed as a wavelength, is given by the scale of d ∼ wavelength /α , where α is the fine structure constant.This suggests that the physical scale of the inductive de-vice needs to be ∼
137 times larger than wavelength of thea.c. radiation on which it operates. The normal methodof combatting this size penalty in transformer structuresis to use high permeability materials (high µ ) and en-hancing the inductance by multiple turns of conductor.While this is a successful strategy for ordinary transform-ers, it is problematic here because the high permeabilitywould need to be retained at high applied magnetic field(see following section), and, even worse, that the Hallconductor would need to be formed into some multi-turncorkscrew. Given that even the one-turn structure ofFig. 14 is beyond any present capability, we would notjudge these strategies for making an inductively coupledgyrator very promising.It is worth noting that applying the same scaling ar-gument to the capacitively coupled gyrator goes muchmore optimistically: the characteristic frequency goeslike ω cutoff ∼ /RC = σ/C , the scale of C is C ∼ (cid:15)(cid:15) d [61], so that if again we take σ ∼ h/e , then we infer d ∼ α × wavelength , (38)that is, the natural scale of our capacitive device is 137times smaller than the wavelength, that is, 137 timessmaller than the natural scale of Hogan’s Faraday-effectcirculator. This comparison is perhaps unfair, since the desired admittance scale of 1 /
50Ω wipes out the factorof α from Eq. (38); on the other hand, it is very easyto make capacitors whose capacitance far exceeds the di-mensional estimate just used (viz., the parallel plate ca-pacitor with area much larger than thickness), and wehave seen that there are impedance-matching possibili-ties in the calculations given above so that, at least toachieve gyration over narrow bandwidths, matching 1 /σ to 50Ω need not be necessary. For engineering applica-tions, the natural impedance-match condition 1 /σ = 50Ωwould, of course, be ideal. Two routes are available forthis: First, σ can be some integer multiple ν of e /h ;filling factor ν in the range of 10-20 is feasible. Second,a stack of Hall conductors can be put in parallel, fur-ther increasing the total conductance. Of course, to keepthe gyration frequencies in the desired range while in-creasing σ , the total capacitances would also have to becorrespondingly increased (cf. Eq. (28)). FIG. 15. An exploded view of a sandwich structure, basedon the capabilities recently reported in [62]. A graphene flakeis encapsulated between two layers of insulating boron ni-tride (BN). Four edge electrodes grown above the structureas shown could serve as the four capacitive contacts of thetwo-port gyrator.
VI. EXPERIMENTAL CONCEPTS FORCAPACITIVE GYRATOR
Here we will explore the relation of our capacitive gyra-tor proposal to experimental observations in recent yearsinvolving magnetoplasmonic phenomena in Hall conduc-tors, both in III-V heterostructures and in graphene. Un-der conditions of the quantum Hall effect, θ H = π/ c does not follow a simple classicalpicture. In fact, it is quantitatively confirmed [64, 65]that the quantum capacitance picture, as analysed theo-retically by B¨uttiker and coworkers [66–68], is necessary4for explaining the observed dynamics.The quantum effect involved in the quantum capaci-tance is the Pauli exclusion principle. Unlike in an idealclassical metal, electric charge cannot be added or re-moved from the conductor without a change of the elec-trochemical potential. This manifests itself as an extraeffective capacitance, in parallel with the classical geo-metrical capacitance, given by the equation C q = e dNdE . (39)Here dN/dE is the density of levels around the Fermienergy. In the ideal quantum Hall state this is quitesmall, so that C q is small and can easily dominate overthe geometrical capacitance. In this state there are nobulk states at the Fermi energy, so that only states at theedge of the conductor contribute. Within the standardedge state picture, the edge state capacitance per unitlength [64], per edge state (corresponding to filling factor ν = 1), is [44] c q = e h v drift . (40)Here v drift , the velocity of the electron wave functionson the edge, has another simple classical meaning: it isthe drift velocity of a ballistic direction subject to crossedmagnetic and (confining) electric fields. From Eq. (24)we see that if the edge capacitance is c q , which will betrue as long as the geometric capacitance is in excess ofthis modest value, then the magnetoplasmon velocity isessentially equal to the drift velocity. This is a very spe-cial coincidence of the quantum chiral edge state situa-tion, in general plasmon velocities and Schr¨odinger wavevelocities are determined by very different parameters.There has been a very recent surge of interest in theseinvestigations in the new graphene quantum Hall system.The same chiral plasmon physics is also readily observedin this system [69]. Precise magnetoplasmon parame-ters have recently been observed in graphene flakes [70],with measured edge quantum capacitance per unit lengthfound to be c q = 100 pF/m , very consistent with theoret-ical estimates for graphene based on Eqs. (39,40). Thelatest report of this work, has, in fact, clearly indicatedthe potential for graphene chiral magnetoplasmons formicrowave circulators and other applications [71].The results of this paper indicate definite directionsand design criteria that can put this realisation into prac-tice. To properly interface the plasmonic excitations,whose physics has now been well documented, with thein- and out-propagating guided electromagnetic waves ofa real device, our results indicate that all contacts to thedevice should be capacitive, and not the combination ofcapacitive and ohmic contacts that are currently used inphysical experiments. Our results further indicate thatthe physical scale d of the device (see Fig. 15), in orderfor there to be successful gyrator and circulator actionin the GHz frequency range, should be in the millime-tre range, given the measured values of c q . (There could be some advantage in going to III-V heterostructure Hallconductors; especially with soft edge confinement, thedrift velocities can be smaller than in graphene, with acorrespondingly larger c q and smaller length scale for theGHz device.)According to our work, the optimal device would havemost of its perimeter occupied by contact capacitors, tomaximise C L and to minimise any stray capacitance ofuncontacted edges to ground. That is, all displacementcurrents should travel in and out of the conductor via thecontact capacitors; as noted earlier, displacement cur-rents to ground must be avoided. Note that a gate (topor bottom), even one with a very large geometric capac-itance, is not a concern here, since its quantum capaci-tance is virtually zero (because there is no bulk density ofstates, see Eq. (39)), so it will carry no ac displacementcurrent.We can mention one other scenario, in which the op-timisation of the device structure would be quite differ-ent. One can, with a very slight adjustment of parame-ters (e.g., magnetic field) work not in the fully developedquantum Hall regime, when θ H is precisely 90 degrees,but rather in regime of non-maximal Hall effect, e.g., θ H = 85 degrees. This would make the device lossy, but,especially in the isolator application, some small degreeof loss is not very detrimental to its operation. In thisregime, away from the quantum Hall “plateaus”, bulkdensity of states is present, meaning that dN/dE , and c q , is much larger. Under these circumstances, an en-hanced geometrical capacitance, achieved by making atop capacitor extending into the bulk of the conductorsome distance from the edge (as suggested by Fig. 15),could lead to a much more miniaturised device. Roughcalculations suggest that GHz operation could then evenbe achieved for d in the range of d = 10 µm .At this length scale, a new encapsulation technique[62] indicated in the figure, which involves sandwichingan isolated flake of graphene between two extremely thin(c. 10nm) layers of insulating boron nitride (BN), hasmade available graphene samples with very small disor-der (which could permit high Hall angle to be achievedfor larger filling fraction ν and/or at higher tempera-tures). It has been known for some time that the quan-tum Hall effect is rather robustly achievable in graphene,with σ = h/e corresponding to one filled Landau level.Larger σ , corresponding to filling multiple Landau levels,is also achievable, and would permit operation at smallermagnetic field. Magnetic fields on the Tesla scale will berequired; one might speculate that micromagnet struc-tures could permit a very compact encapsulated devicewith small fringing fields.A small modification of the ohmic contacting techniquepioneered in [62] should permit very well-controlled fab-rication of the lead capacitors indicated in the figure.A new difficulty would arise because, unlike in the fullydeveloped quantum Hall situation, the bulk density ofstates would be nonzero and a gate capacitor would con-vey undesired displacement current in and out of the sam-5ple, depending on the details of the bulk charge trans-port mobility [72]. Thus, consideration would have to begiven to making the bulk of the conductor floating, orcontrolled only by a very low C g , remote gate capacitor. VII. CONCLUSIONS AND OUTLOOK –QUANTUM EFFECTS
While the use of Hall conduction for the achievementof gyration and circulation was declared impossible in1954, the results of this paper indicates that this con-clusion was premature; with current device capabilities,such a gyrator might actually be possible in the near fu-ture. It is curious that the fundamentally different possi-bilities offered by reactive rather than galvanic couplingto the Hall conductor were not already examined a longtime ago. Capacitive coupling was always, of necessity,the method of contact for the two-dimensional electrongas (2DEG) formed by electrons floating on the surfaceof liquid helium. But throughout the large literature onthis subject [73–77] it seems that this coupling scheme iswas always viewed only as a means to learn the basic re-sponse coefficients of this electronic system, rather thanan interesting device feature in its own right. In metro-logical discussions [78, 79] careful accounting of capaci-tive effects has been made, but only in a setting wherethe basic coupling is ohmic. Finally, there is other litera-ture in which transport through semiconductor 2DEGs isachieved with capacitive coupling [80, 81], but with theorientation that the experimental data extractable fromcapacitive vs. ohmic contacts are equivalent, without anyattention given to the difference that this might produce.The present study is obviously incomplete, in thatno quantum analysis has been provided for the func-tionalities that we have studied. The classical Ohm-Hall approach has proven its worth in modelling thephenomenology of Hall-conduction devices from the1950s [27] up to the present [30, 31, 47]. While we canexpect that some new quantum or mesoscopic phenom-ena would manifest themselves in the capacitively cou-pled devices that we have analysed here, perhaps at lowtemperature or in very clean systems, we can feel com-forted that since the properties we have discussed hereare fundamentally classical, they should be robust evenin the face of considerable disorder, or at (moderately)high temperature. While the achievement of Hall anglesvery precisely equal to 90 degrees is very important inmetrological applications, it is not so important here; aHall angle of 85 degrees would still permit excellent gy-rator, isulator or circulator action. Quantum considerations are clearly very significant insetting limits on the validity of the results derived here.The classical theory has no limit on the linearity of theresponse; we should expect departures from linearity atleast when the potential drops in the device reach theLandau-level energy spacing. Likewise, operating fre-quencies are certainly limited to below the inter-Landaulevel transition frequency. In a classical theory any plas-mon velocity is possible,with a straightforward geometricdependence on edge capacitance; the quantum descrip-tion, as we have seen, intimately links the edge plasmonvelocity to the drift velocity, itself fixed by the phasevelocity of electron Schr¨odinger waves. Finally, the clas-sical theory has no lowest length scale of validity, whilethe quantum magnetic length is clearly a lower limit onthe device dimensions that can reasonably be considered.Fortunately, there is a strong basis for further work onthe quantum aspects of this problem, as established inthe theoretical work of B¨uttiker and co-workers in thetransmission theory of admittance and dynamic conduc-tance [82, 83]. Recent work of Aita et al. [84] offers signif-icant progress in defining the basic elements of a theoryincluding electron correlation effects, going beyond theHartree treatment of previous work. Time will tell whattools will be needed to model important new aspects ofthis problem.While the Hall effect was declared unsuitable for therealisation of gyrators and circulators sixty years ago,we can hope that, after a long period of quiescence,the simple idea of reactive coupling to the Hall conduc-tor will lead to a successful revival of this idea, withnovel, miniaturised devices providing useful alternativesfor constructing new low-temperature quantum technolo-gies.
ACKNOWLEGEMENTS
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