A Delgard

B Chenaud

U Gennser

D Mailly

P Degiovanni

C Chaubet

Effective quantum inertia of charge carriers in a macroscopic conductor

Keywords: numbers: 72.10.-d, 73.23.-b, 73.43.-f, 73.43.Fj

We study the low frequency admittance of a quantum Hall bar of size much larger than the electronic coherence length. We find that this macroscopic conductor behaves as an ideal quantum conductor with vanishing longitudinal resistance and purely inductive behavior up to f 1 MHz. Using several measurement configurations, we study the dependence of this inductance on the length of the edge channel and on the integer quantum Hall filling fraction. The experimental data are well described by a scattering model for edge magnetoplasmons taking into account effective long range Coulomb interactions within the sample. This demonstrates that the inductance's dependence on the filling fraction arises from the effective quantum inertia of charge carriers induced by Coulomb interactions within an ungated macroscopic quantum Hall bar.

By demonstrating that macroscopic conductors could exhibit robust d.c. transport properties of quantum origin, the integer quantum Hall effect (IQHE) [1][2][3][4][5] has been a major surprise. The importance of this breakthrough for metrology has been acknowledged immediately [1] and has lead to the redefinition of the Ohm at the CPEM88 conference [START_REF] Des Poids | Représentation de l'ohm à partir de l'effet hall quantique[END_REF]. The finite frequency response of quantum Hall conductors has been a subject of intense work by metrologists: the use of an a.c. bridge at finite frequency f revealed departure of the Hall resistance R H (f ) at ν = 2 from the expected value R K /2 = h/2e 2 [7][8][9][10][11]. It was then attributed to "intrinsic inductances and capacitances" [12,13]. Later, Schurr et al proposed a double shielded sample allowing for a frequency-independent resistance standard [14], but these works left open the question of the origin of these capacitances and inductances.

On the other hand, the finite frequency transport properties of quantum coherent conductors, of size smaller than the electron coherence length, are expected to be dominated by quantum effects. For low-dimensional conductors such as carbon nanotubes [15], or graphene [16], the inductance is of purely kinetic origin. Small superconducting inductors [17,18] which are now used in space industry [19] are based on the inertia of Cooper pairs. For a quantum coherent conductor, the quantum transport theory developped by Bűttiker and his collaborators [20][21][22] relates the associated R/L or RC times to the Wigner-Smith time delay for charge carriers scattering across the conductor. These remarkable predictions have been confirmed by the measurement of the finite frequency admittance of quantum Hall R-C [23] and R-L [24,25] circuits of µm-size in the GHz range at cryogenic temperatures.

In this letter, we demonstrate that, in the a.c. regime, a mm long ungated macroscopic quantum Hall bar, of size much larger than the electronic coherence length, exhibits a finite inductance as well as a vanishing longitudinal resistance. Such a purely inductive longitudinal response is expected for quantum conductors with zero backscattering: a kinetic energy cost proportional to the square of the current arises from both the Pauli principle and the linear dispersion relation for electrons close to the Fermi level. This effective inertia of carriers causes the current response to lag the applied electric field. Here, we identify an inductance of the order of tens of µH mm -1 and connect it to an effective velocity v 0 of charges along the quantum Hall bar's edges. Contrary to gated samples, in which this velocity is independant of the filling fraction [24], we show that, because of Coulomb interactions between opposite edges of the sample, it does depend on ν in our samples. Using the edge-magnetoplasmon scattering approach combined with a discrete element approach à la Bűttiker, we show that, for a sample of width W

v 0 v D = 1 + να mat π ln W/ξ H ν (1)
where v D = E y /B z is the drift velocity of non-interacting electrons at the edge, which plays the role of a Fermi velocity in their 1D linear dispersion relation along the edge [START_REF] Girvin | Modern Condensed Matter Physics[END_REF] and α mat denotes the fine structure constant (α qed in the vacuum) within the material: α mat = (α qed /ε r ) × (c/v D ). The length ξ H is an effective renormalized width of a single edge channel of the order of the width of incompressible edge channels λ H = /m * v D [START_REF] Chklovskii | [END_REF] where m * = 0.067 m e is the effective mass of electrons.

Our experimental data leads to estimates for v D and ξ H consistent with previous works in similar samples. Our work demonstrates that the purely inductive response of the macroscopic ungated quantum Hall bar FIG. 1.

a) The Device Under Test (DUT) is measured using four coaxial cables and the impedance-meter Agilent HP4294A, which measures the current I at Lcur, the potential V at Hpot, and gives G = I/V , for details see Ref. [START_REF] Kibble | Coaxial AC bridges[END_REF][START_REF]Agilent impedance measurement handbook[END_REF]. Note that the potentials of Hpot, Lpot, Hcur and Lcur are the potentials of the four connectors of the impedance-meter. b) Scheme of the multi-terminal Hall bar with only three ohmic contacts wire-bonded onto the sample holder. In this geometry, the impedance is Z (exp) 23 (ω) = -(∂V3/∂I2)(ω) for V2 = 0.

reflects the effective quantum inertia of charge carriers renormalized by Coulomb interactions within the sample. Therefore, although electron transport across such a conductor is not coherent, its d.c. and a.c. transport properties are of quantum origin, a fact that ultimately relies on the coherence of edge magnetoplasmon (EMP) modes propagating along chiral edge channels. EMP coherence has enabled the demonstration of single and double EMP Fabry-Pérot interferometers [28] as well as of a Mach-Zehnder plasmonic interferometer [29].

Our samples are processed on an AlGaAs/GaAs heterojunction with the two dimensional electron gas (2DEG) located at the hetero-interface with carrier density N s = 5.1 × 10 11 cm -2 and mobility µ = 30 m 2 /Vs. We have processed a 400 µm wide ungated Hall bar whose large size was necessary in order to exhibit a sufficiently large kinetic inductance. The sample is placed at the center of a high magnetic field at the temperature of pumped He 4 (1.5 K).

A measurement setup of impedance is depicted on Fig. 1-a.: the current is injected using H cur , and measured using L cur . The potential of H pot is measured while V = 0 and I = 0 are imposed at L pot . All measurements were performed using an impedance-meter Agilent 4294A [START_REF] Kibble | Coaxial AC bridges[END_REF][START_REF]Agilent impedance measurement handbook[END_REF] in the nominal conditions for accurate measurements. The amplitude of the voltage applied across the sample was 5 mV. In the IQHE regime, this induces a current intensity below 0.5 µA at ν = 2, a value far below the breakdown current and even far below current densities used in metrology [4,7]. For each values of magnetic field, the resistance and the reactance have been measured for 300 values of the frequency f in the range 40 Hz-100 kHz.

Due to chirality of the quantum Hall transport, an ohmic contact wire-bonded to the sample holder and so to a coaxial cable, generates a leakage current through the cable capacitance if the potential does not vanish [START_REF] Grayson | [END_REF]33]. This results in a faulty measurement [34,35]. For this reason, all results presented here have been carried out at integer filling factors, where the longitudinal resistance R xx (ω) vanishes [START_REF] Rxx | ω) denotes the frequency dependant longitudinal dependence of the quantum Hall bar[END_REF]. Furthermore, only a few of the ohmic contacts processed on the sample were wirebonded onto the sample holder as shown on Fig. 1.b. In order to measure the zero resistance state, a third contact is inserted along the edge connected to the reference potential (contact 2 on Fig. 1.b). In d.c., one would measure a potential V Hpot = 0. In a.c., V Hpot = 0 and we measure the frequency dependent impedance Z (exp) 23

(ω) = -∂V 3 /∂I 2 | V2=0 (ω). Different configurations can be obtained, by changing the location of the third contact and the sample side. In this case, the magnetic field orientation also has to be changed. In order to access the dependence on the lengths of the edge channels, we have investigated several configurations, which entailed shifting the ohmic contacts on the holder. Sometimes, we have also wire-bonded a fourth ohmic contact on the same side of the sample to connect L pot to access another edge channel length.

Figure 2 presents unfiltered and non-averaged raw data for the frequency dependent reactance X(f ) = Im(Z (exp) 23 (2πf )) in a given sample configuration for integer filling factors ν = 2, 4, 6, 8. The positive linear dependence of X(f ) is the signature of an inductive behavior. The corresponding inductance decreases with filling factor. These data are completely reproducible in the regions of magnetic fields as long as R xx = 0. This is a key point of our work: for all filling factors, the real part R(f ) = Re(Z (exp) 23 (2πf )) of the measured impedance is close to zero with values between ±0.5 Ω at low frequency as shown in the inset of Fig. 2. These results extend the work of Gabelli et al [24] in which the sample resistance was R H = R K /ν, to the case of a zero resistance macroscopic device. At higher frequencies, a small real part of the measured response function R(f ) appears. This effect is discussed in the Supplementary Material (Sec. 2) and is correlated to the deviation of the reactance X(f ) from linearity seen on Fig. 2.

Let us now explain how this inductive behavior arises from the quantum inertia of carriers renormalized by Coulomb interactions within the quantum Hall bar. In the presence of an a.c. drive, the propagation of electric charges is described by the dynamics of charge density waves (plasmons) whose velocity in 1D or 2D systems is modified by Coulomb interactions. The idea is that the ac drive applied to the reservoir injects a classical plasmon wave into the conductor. Knowing how this incoming wave is scattered by the conductor enables us to compute the corresponding outgoing currents and consequently, the electrical current flowing across all leads and gates connected to the conductor. This general idea was first applied to 1D gated quantum wires by I. Safi who has related its finite frequency admittance matrix to the scattering matrix of plamons propagating within the wire [START_REF] Safi | [END_REF]38]. It has lead to remarkable predictions on fractionalization of charge pulses [START_REF] Safi | Quantum Transport in Semiconductor Submicron Structures[END_REF] observed only a decade ago [START_REF] Steinberg | [END_REF]. Fractionalization has also been studied in an artificial Tomonaga-Luttinger liquid obtained from two capacitively coupled counter-propagating edge channels in AlGaAs/GaAs heterostructures [41] as well as in graphene [42]. More recently, time of flight measurements have probed the dynamics of plasmon modes in quasi-1D wires made from a tailored 2DEG at zero magnetic field and confirmed the role of Coulomb interactions in this system [43].

In the context of the quantum Hall effect, the relevant degrees of freedom are EMP modes which have been studied since the 80s [44][45][START_REF] Mikhaelov | Edge excitations of low dimensional charged systems[END_REF]. The edge-magnetoplasmon scattering approach has been used to describe the high frequency linear response of circuits built from quantum Hall edge channels [START_REF] Hashisaka | [END_REF]. It has also been used for discussing energy relaxation in a ν = 2 edge channel system [48,49] and it is the key ingredient for predicting single electron decoherence in quantum Hall edge systems [50].

We have developped an analytical model for edgemagnetoplasmon scattering in the spirit of Ref. [28] for a quantum Hall bar taking into acccount long range Coulomb interactions between edge channels of the same chirality as well as between counter-propagating edge channels. This effective edge-magnetoplasmon scattering model, analytically solved in the Supplementary Material (Sec. 3), describes the finite frequency transport in the linear regime beyond the GHz, the limit being the cyclotron frequency which is 4 THz at 1 T. Above it, inter-Landau level transitions may be excited and the bulk contribution cannot be neglected. To account for the long-range interactions in our ungated quantum Hall bars, this model assumes that all the edge channels of the same chirality have the same velocity v D and are so strongly coupled that they see the same time independant potentiel as in Ref. [25]. The two counterpropagating groups of edge channels are capacitively coupled by a geometric capacitance C H . Finally, dissipation of EMP modes have been neglected, an hypothesis a posteriori satisfied in our samples.

In the geometry of the sample depicted on Fig. 1, the low frequency expansion of the measured reactance is of the form (see Supplementary Material, Sec. 2):

Im Z (exp) 23 (ω) = iLω + O(ω 2 ) ( 2
)
where L denotes the total inductance for the quantum Hall bar delimited by a dashed red box on Fig. 1-b. Because the magnetic inductance L m is much smaller than the kinetic inductance L kin in our sample (see Supplementary Material, Sec. 1), L = L m + L kin L kin can be obtained from the edge magnetoplasmon scattering model as

L = (R K /ν) × (l/2 v 0 ) (3a) v 0 = v D × 1 + C q 2C H . ( 3b
)
where l is the length of the Hall bar (see Fig. 1-b), C q = νe 2 l/hv D is the quantum capacitance of ν edge channels and the geometric capacitance C H describes the effect of Coulomb interactions between counter propagating edge channels. This is different from the quantum RL-circuit where, because of the gating, the capacitance C H has to be replaced by the capacitance with the nearby gates leading to a renormalization of v D by 1 + C q /C H for right and left moving charge density waves [24]. Here, by contrast, the renormalization corresponds to the ratio of a C H capacitance with the series addition C q /2 of the quantum capacitances of counterpropagating edges. Eq. ( 3) suggests that the inductance can be interpreted as a kinetic inductance associated with an effective time of flight l/v 0 . As discussed in the Supplementary Material (Sec. 3), v 0 is neither the drift velocity for noninteracting electrons nor even a renormalized electron's velocity within chiral edge channels but an effective velocity arising from the combination of their quantum inertia and Coulomb interactions within the quantum Hall bar. This inertia is of quantum origin, reflecting the minimal energy associated with an electrical current due to Fermi statistics of electrons, augmented here by the effects of Coulomb interactions. At fixed v D , the main ν-dependence of v 0 arises from the linear increase of C q with ν in (3b) but we expect a sublinear ν-dependence coming from the dependence of the geometric capacitance C H on the structure and geometry of the quantum Hall edge channels. In the Supplementary Material (Sec. 4), we show that the main subdominant contribution induced by the geometric capacitance dependence on ν, which arises from the linear dependence on ν of the width of the edge channels, is logarithmic:

C H (ν, l) = πε r ε 0 l ln W/ξ H ν (4)
where W denotes the sample width and ξ H is a length proportional to the width W H of a single quantum Hall edge channel. Following Ref. [START_REF] Chklovskii | [END_REF],

W H = (1 + π 2 α mat ) /m * v D ,
which is of the order of a few tens of nm for AlGaAs/GaAs quantum Hall systems. The dimensionless factor ξ H /W H depends on the specific microscopic model used to describe the edge channel geometry [START_REF] Chklovskii | [END_REF]51]. Eq. ( 4) can therefore be considered as a universal scaling formula leading to the ν-dependence of v 0 for ungated samples given by Eq. ( 1). Let us now compare this prediction to the experimental data. Fig. 3 contains the first main quantitative result of this work, i.e. the quantum inductance as function of 1/ν for configurations B > 0. Values have been obtained from the reactance data depicted on Fig. 2 and on the inset of Fig. 3 using the slope at low frequency of f → X(f ) datasets. Three configurations in which L pot and H pot are plugged to different contacts (see Fig. 3) have been studied. The main result is the dependence of the inductance on 1/ν which is increasing but with a slight non-linearity according to Eq. (3b) and (4).

To verify this sub-dominant behavior, we have extracted the velocity v 0 using Eq. (3a) from each value of the inductance. These values are reported in Fig. 4 as a function of ν. This is the second main quantitative ex- perimental result of this work. Each family of points corresponds to a specific sample configuration for which the sample has been heated up, re-bonded and cooled down again. These manipulations affect the electrostatic arrangement of charges at the edge, thereby modifying v D from one experiment to the other. Fig. 4 thus presents predictions for v 0 (ν) from Eq. ( 1) for different values of v D . Using five different values of v D as a single fit parameter in Eq. ( 1) for five different sets of experimental data gives the dashed lines on Fig. 4. This demonstrates a good agreement between the experimental results for v 0 as a function of ν deduced from admittance measurements using Eqs. (3) and predictions from Eq. ( 1). The highest values of v D obtained here are comparable to the ones obtained on ungated samples (as ours) in Ref. [52] but five times higher than those in Refs. [24,53] obtained on samples with side gates. This is expected since gating leads to a higher geometric capacitance which brings down the velocity v 0 much closer to v D . Last but not least, because of the variabilty of the electrostatic environment in ungated samples, the spread of values for v D found here is not surprising because the datasets depicted on Fig. 4 correspond to different experiments performed on both sides of the sample. Additional results obtained on samples from different wafers are presented in Sec. 5 of the Supplementary Information provide additional evidence of the robustness of our analysis.

To summarize, we have shown that, at low frequencies, a macroscopic quantum Hall bar is a perfect 1D conductor exhibiting a vanishing longitudinal resistance and a finite inductance. By fitting its dependence on the filling fraction and on the sample geometry using a simple long range effective Coulomb interaction model, we have shown that it reflects the effective quantum inertia of charge carriers within the edges of the quantum Hall bar. Contrary to the case of superconductors where carrier in-ertia arises from the effective mass of the Cooper pairs, here it reflects how Coulomb interactions alter the propagation of low-frequency massless edge-magnetoplasmon modes. A quantum Hall bar of size much larger than the electron's coherence length thus appears as a quantum conductor not only in the d.c. regime but also at low frequencies in the a.c. regime. Although the low frequency response is not universal, its dependence on the filling fraction and sample geometry can be understood using a simple model which is formally similar to the one used for gated nano-fabricated samples [START_REF] Hashisaka | [END_REF]. Therefore macroscopic samples can provide a rescaled test-bed for studying the scattering properties of edge-magnetoplasmons in µm-sized quantum coherent samples. Studying a.c. transport properties of macroscopic samples up to radiofrequencies could thus open the way to realizing controlled quantum linear components for quantum nanoelectronics in 1D edge channels, with possible applications to electron [54] and micro-wave quantum optics in ballistic quantum conductors [55].

Let us first consider non-interacting chiral charge carriers with propagation velocity v D (see Fig. 5). A net current flow corresponds to a chemical potential between chiral edge channels of opposite chirality. The corresponding kinetic energy is then obtained by adding together the energy of all quantum states which have been occupied when biasing chiral edge states with different chemical potential ∆µ = -eV :

E chir = k ε(k) = ν × g(ε)l × (eV ) 2 /2 , (A1)
where g(ε) = 1/hv D is the density of states (DOS) per unit length for a 1D-channel, ν the number of channels, l the edge-state length. The voltage drop across the sample is

V = R H I ν , where R H = R K /ν = h/e 2 ν
denotes the quantum Hall resistance at filling fraction ν and I ν the current. The resulting kinetic energy is then quadratic in the current, as expected for a linear inductor:

E chir (I ν ) = νe 2 hv D × l 2 × R K I ν ν 2 = h e 2 × l νv D × I 2 ν 2 .
(A2) For a quantum Hall bar, we have two counterpropagating groups of ν chiral edge channels. Assuming that the two chiralities carry equal charge of the total current I ν = ±I/2, this leads to a total kinetic energy

E K (I) = 2E chir (I/2) = h e 2 × l νv D × 2(I/2) 2 2 . (A3)
Using E K (I) = LI 2 /2, we obtain the following formula for the inductance L of the quantum Hall bar:

L K = R K ν × l 2v D (A4)
Note the appearance of the half of the time of flight l/v D of the electrons across the quantum Hall bar which is ultimately due to the existence of two counter-propagating groups of copropagating edge channels.

Magnetic inductance

A quantum Hall bar can be viewed as a conductor built from two linear wires of length l, and of diameter or width W H separated by a distance W which is the width of the quantum Hall bar. The magnetic inductance of the resulting dipole can be computed from Biot-Savart's law [56]. Note that the wire's diameter W H (ν) usually scales as ν (see Ref. [START_REF] Chklovskii | [END_REF] and Sec. D 1). In the quantum Hall bar geometry, the total current I flowing across the quantum Hall bar corresponds to ±I/2 along each of the two long counter-propagating edges. Consequently, in the limit l, W W H (ν), the geometric inductance of the quantum Hall bar is approximately equal to

L m = lµ π ln 2W W H (ν) -1 (A5)
where µ = µ r µ 0 is the magnetic permeability of the material expressed as the product of its relative permeability µ r by the vacuum permittivity µ 0 = 1/ε 0 c 2 . For a nonmagnetic material such as AlGaAs/GaAs, µ r = 1 and therefore µ = µ 0 . Using this expression and Eq. (A4), the ratio of the magnetic to the kinetic inductance is given by:

L m L K = 4α qed v D c × ν × ln 2W W H (ν) -1 (A6)
where α qed is the fine structure constant. The magnetic character of this ratio appears through the multiplication factor v D /c 1. Together with the smallness of α qed and the logarithmic dependance in the aspect ratio W/W H (ν) ∼ W/νW H (1), this explains why the magnetic inductance is always much smaller than the kinetic one for ν 100.

In his pioneering work on finite frequency charge transport in mesoscopic conductors [20], Büttiker stressed the importance of interactions: time dependent drives applied to reservoirs lead to charge pumping in the conductor which, in return, alter the electrical potential within the conductor. As a result, its transport properties can no longer be derived by using the d.c. response at zero bias. Büttiker then developed a self consistent mean-field approach to this problem, taking into account how the time dependent charge density within the conductor generates a time-dependent potential and thereby alters the transport [22]. In the linear regime, the a.c. response of the conductor is described by the admittance matrix

G αβ (ω) = ∂I α (ω) ∂V β V =0 (B1)
giving the average current entering the conductor from lead α when a voltage drive at the same frequency ω/2π is applied to the reservoir β. Charge conservation and gauge invariance imply that the finite frequency admittance matrix satisfy the general sum rules

α G αβ (ω) = 0 (B2a) β G αβ (ω) = 0 (B2b)
which is ensured by total screening of effective Coulomb interactions.

At low frequency, this admittance matrix is expanded in ω, the zero-th order term being its d.c. conductance

G (dc)
αβ . The first order term is related to the emittances introduced by Büttiker et al. :

G αβ (ω) = G (dc) αβ -iωE αβ + O(ω 2 ) . ( B3
)
For quantum Hall edge channels, Christen and Bűttiker [57] fully exploited the chirality of charge transport to compute the dc conductances G (dc)

αβ as well as the emittances E αβ using a discrete element description of the circuit.

Here we will show how to describe the finite frequency transport using the building blocks of a discrete element description of quantum Hall bars and the finite frequency admittance properties obtained from the edgemagnetoplasmon (EMP) scattering approach. The latter has been first pioneered in the context of quantum wires [START_REF] Safi | [END_REF]38] before being used for quantum Hall edge channels [48] where it is instrumental for computing single electron decoherence [50,58]. The details of how the admittances are obtained from the EMP scattering approach will be described later in Sec. C.

Depending on the sign of the diagonal emittances, the quantum conductor under consideration exhibits a capacitive (E αα > 0) or inductive (E αα < 0) behavior. Basic examples of these behaviors, involving two different two-contact devices, have been given by Christen and Bűttiker [57]. A two-contact Hall bar is predicted to act as an inductance while a ring shaped sample (the socalled Corbino geometry) acts as a capacitance. Hence both geometries exhibit opposite diagonal emittances, which can be understood in terms of reaction of the circuit to an injected charge. Indeed, as shown on Fig. 1 of [57] a charge injected by contact 1 into a Hall bar is transmitted to contact 2, while the charge induced by interedge Coulomb interactions created at contact 2 is transmitted to contact 1. On the contrary, in a Corbino sample, the injected charge returns to the contact it comes from, as does the image charge. It was experimentally shown by Delgard et al. [59] that Corbino emittances will exhibit this predicted capacitive behavior. Here, we will highlight the inductive behavior of quantum Hall bars in a multi-contact geometry that ensures vanishing longitudinal resistance.

We shall now discuss how, for two different sample configurations, the measured response function relates to the finite frequency admittance of the quantum Hall bar depicted as a red dashed rectangle in the two forthcoming figures, taking into account capacitive leaks at the contacts.

Three contact geometry

Let us first consider the three contact geometry depicted on Fig. 6. Contacts 1 and 2 are directly connected to reservoirs whereas contact 3 is not but we take into account the cable capacitance C 0 of the cable. In this section, we shall compute the experimentally measured finite frequency impedance

Z (exp) 23 (ω) = - ∂V 3 (ω) ∂I 2 (ω) V2=0 (B4)
up to second order in ω and show how it relates to the properties of the quantum Hall bar B and to the cable capacitance C 0 (see Fig. 6). In this approach, the quantum Hall bars A and B delimited on Fig. 6 are characterized by their frequencydependent EMP scattering matrix. For the quantum Hall bar A, the EMP scattering matrix, which we denote here by A(ω) relates the ingoing and outgoing currents through [60]

j (out) (ω) = A(ω) j (in) (ω) (B5)
in which the edges of the Hall bar are labeled by their chirality. In the same way, the EMP scattering matrix of the quantum Hall bar B will be denoted by B(ω). These matrices are computed in Sec. C but for the moment, we won't need their explicit form. At contact 3, the incoming and outgoing currents propagating within the edge channels are also related by an input/output relation that reflects the time delay associated with the R H C 0 relaxation time of the contact:

i 3,out (ω) = i 3,in (ω) 1 -iR H C 0 ω (B6)
Eliminating the edge channel currents internal to the sample directly leads to the finite frequency admittances connecting the currents I α entering from the reservoirs to the drive voltage V 1 :

R H I 2 = - A RR B RR 1 -A RL T B LR V 1 (B7)
in which the matrix elements of the Hall bar's current scattering matrices A and B are involved as well as T (ω) = 1/(1 -iR H C 0 ω), the edge current transmission of the Ohmic contact with cable capacitance C 0 . For simplicity, the ω dependence of V 1 , I 2 and of the scattering matrix elements of the A and B quantum Hall bars as well as of T have been omitted. The potential of the contact 3 can also be computed as

-iωC 0 R H V 3 = (T -1)B LR A RR 1 -A RL T B LR V 1 (B8)
which finally leads to It is then useful to relate

Z (exp) 23 (ω) R H = 1 1 -iR H C 0 ω B LR (ω) B RR (ω) . ( B9
Z (exp) 2,3
to the impedance of the Hall bar B, which we know has a real part ∼ R H , with a small deviation for ω = 0, as observed by metrologists studying the quantum Hall resistance a low frequencies [15][16][17]. This is achieved using the relation [38,48] between the quantum Hall bar finite admittance matrix and its edgemagnetoplasmon scattering matrix to obtain the following expression, valid when the left and right moving edges of the quantum Hall bar are in total mutual influence: (ω) = 0. However, we will now show that a non vanishing r(ω) = Re(z H (ω)) -1 is responsible for the non-linearity of X(f ) seen on Fig. 2 of the paper. Such a feature has been observed for long time by metrologists studying quantum Hall resistance at low frequencies [7,10,11]. In the metrology community, this finite frequency deviation r(ω) from the dc quantum Hall value is commonly attributed to parasitic effects such as self and resistance of ohmic contacts and bond wires and thus depend on configurations and samples. A quadratic dependence on the frequency is usually found in the form r(ω) = a 2 (ω/2π) 2 . The a 2 parameter has been measured in many configurations and samples and has been found to be positive or negative but is always around (or below) 10 -7 /kHz 2 [7,10,11].

Z (exp) 23 (ω) R H = z H (ω) -1 1 -iR H C 0 ω (
To discuss the experimental results, which include both the real and imaginary parts of Z (ω)/R H is then

Z (exp) 23 (ω) R H r(ω) -ω 2 L eff (ω)C 0 (B12a) + iω L eff (ω) R H + R H C 0 r(ω) + O(ω 3 ) (B12b)
In our experiments, the reactance Im Z (exp) 23

(ω) is obviously dominated by the kinetic inductance L:

Im Z (exp) 23 (ω) = ωL , (B13)
but r(ω) is seen through the deviations of the imaginary part from linearity (see Fig. 2 of the paper). It is nevertheless small and therefore, the dominant contribution to the real part Re Z (exp) 23

(ω) is a negative quadratic one directly proportional to the cable capacitance: -ω 2 LC 0 , as observed in the inset of Fig. 2 of the paper. Both real and imaginary part of admittance seen in Fig. 2 suggest a quadratic behavior for r(ω). To second order in ωR K C 0 :

Re Z (exp) 23 (ω) R H = (R H C 0 ω) 2 r 2 - L/R H R H C 0 (B14)
where r(ω) r 2 (ωR

H C 0 ) 2 .
In our experiments, the cable capacitance is C 0 = 218 pF. We deduce from Fig. 2 of the paper that, at ν = 2 and ω/2π = 100 kHz, ωr(ω)R 2 H C 0 ≈ 2.2 Ω. This gives r(ω) ≈ 10 -4 at 100 kHz thereby corresponding to a 2 = 4π 2 r 2 (R H C 0 ) 2 ≈ 10 -8 /kHz 2 , a value in total agreement with the deviations reported in the litterature. Consequently, the observed deviations from pure inductive behavior can be explained by the combined effect of the cable capacitance and of the deviations of Re(Z H (ω)) from its dc value R H . Let us now turn to the case of a four contact sample depicted on Fig. 7. This geometry represent a four contact measurement of the central part B of the device. Here, the parts A and C of the device play the role of the leads that connect B to the current injection reservoirs which are the Ohmic contacts 1 and 2. The Ohmic contacts 3 and 4 are used for measuring the voltage difference between the two port of the would-be dipole B.

In this section, the capacitances C 0 of the cables will be omitted for simplicity. This does not change the physics of the problem but greatly simplifies the computations. Nonetheless, we know from Eq. (B6) that the same prefactor (1 -iR H C 0 ω) -1 will appear in the results when capacitances are taken into account.

We assume that A, B and C can be modeled as quantum Hall bars characterized by their edgemagnetoplasmon scattering matrices. Exactly as before, this amounts to neglecting Coulomb interactions outside the Hall bars themselves, an hypothesis consistent with the total screening hypothesis stated in Sec. C 1. Let us denote by M and N the points on edge channel 1, which respectively face contacts 4 and 3. Introducing the EMP scattering matrices A, B and C respectively associated with the three quantum Hall bars A, B and C represented on Fig. 7, the dynamics of the three quantum Hall bars is described by:

i M i 4,out = A i 1,in i 4,in (B15a)
i N i 3,out = B i M i 3,in (B15b) i 1,out i 2,out = C i N i 2,in (B15c)
in which, for compactness, the ω dependence is not recalled. In the experimental configuration depicted in Fig. 7, V 3 = 0 and we measure V 4 . We calculate I 2 and V 4 , in order to find Z (exp) 42

= V 4 /I 2 . The lead potentials fix the incoming currents injected by the reservoirs. Moreover, because of the high impedance of the voltmeter connected to H pot and L pot , we neglect the currents leaking into these Ohmic contacts. We thus have:

i 1,in = G H V 1 , i 2,in = G H V 2 (B16a) i 3,in = 0, i 4,in = G H V 4 (B16b) i 3,in = i 2,out , i 4,in = i 3,out (B16c) I 2 = i 2,in -i 1,out . (B16d)
From Eqs. (B15a) and (B15b), we obtain :

G H V 4 = i 4,in = i 3,out i 3,out = B LR (A RR G H V 1 + A RL G H V 4 ) (B17)
or, equivalently:

V 4 = B LR A RR 1 -B LR A RL V 1 . (B18)
On the other hand, calculating I 2 :

I 2 = G H V 2 -i 2,out = G H V 2 (1 -C RL ) -C RR i N , (B19)
using now Eq. (B15c), this becomes a current division relation

I 2 = C i N where C = -C RR C LL -C LR + C LR C RL C LL . (B20)
Eqs. (B15a) to (B15b) then shows that i N = B RR i M . We now use Eq. (B15a) to express i M in terms of G H V 1 and G H V 4 . Using Eq. (B18), we obtain :

i N = B RR A RR 1 -B LR A RL G H V 1 . (B21)
This gives us the total current entering lead 2 in terms of the applied voltage V 1 at the right Ohmic contact:

I 2 = C B RR A RR 1 -B LR A RL G H V 1 (B22)
Using one last time the voltage division relation (B18), the dimensionless impedance z 24 = Z (exp) 24

/R H = G H V 4 /I 2 is thus finally given by:

z 24 = B LR B RR × C . (B23)
As in the three contact case, the fraction B LR /B RR appears in the results showing that the four point measurement does gives us information on the B quantum Hall bar. However, the matrix C is involved via the quantity C. Nevertheless. it is possible to show that, to the lowest order frequency expansion, z 24 does not depend on the physical relevant information contained within the C matrix. For this we once again use the low frequency expansion of the C scattering matrix (see Eq. (C12)):

C LL = C RR = 1 + C RL (B24) C LR = C RL = -iωl C v D (2 + να ν ) = -iωτ C (B25)
where l C is the length of Hall bar C, and

τ C = l C 2v D (1+ναν /2)
is the transit time across the Hall bar C. These formulas allow the calculation of quantity C from its definition by Eq. (B20) at the first order in ωτ C :

C = -1 + 2iωτ C + o (ωτ C ) 2 . (B26)
Finally, using Eq. (B23), In this Section, we present the computation of the EMP scattering matrix for an ungated quantum Hall bar and derive the corresponding description in terms of discrete elements valid at low frequency.

z 24 = -iωτ B (1 -iωτ B )(-1 + 2iωτ C ) iωτ B + o (ωτ C ) 2 (

The model

We consider a quantum Hall bar of length l in the integer quantum Hall regime with filling fraction ν ≥ 1 as depicted on Fig. 8. We denote by R (resp. L) the upper (resp. lower) edge channels which are right (resp. left) movers as depicted on Fig. 8. We denote by j (in) α the incoming current entering the Hall bar on side α = R, L and j (out) α the outcoming one. Our goal is to relate the outcoming currents to the incoming ones in the presence of Coulomb interactions.

Here v D denotes the bare drift velocity, which for simplicity we take identical for all edge channels. Coulomb interactions are described by assuming a discrete element model à la Christen-Büttiker [57] in which all the electrons within channels of the same chirality see the same time dependent potential U R (t) (for right movers) and U L (t) (for left movers). These potentials are related to the total charge stored within the L and R edge channels by a capacitance matrix

Q = C U (C1) in which Q = Q R Q L , U = U R U L (C2)
and C denotes the capacitance matrix

C = C H -η C H -η C H C H , (C3)
where 0 ≤ η ≤ 1 depending on the screening of Coulomb interaction between channels of opposite chiralities by nearby gates and other on-chip conductors. Eq. (C1) plays the role of a solution of the Poisson equation relating the electrical potential to the charge density. In this work, unless otherwise stated, we assume that the left and right chiral edge channels are in total influence and therefore that η = 1. The capacitance C H can then be computed as the geometric capacitance of a stripline capacitor (see Sec. D).

The charges within the edge channels are directly related to the incoming and outcoming currents through charge conservation which, in the Fourier domain and in vector notation, is written as

j (in) (ω) -j (out) (ω) = -iωQ(ω) . ( C4
)
The last equation we need is the equation of motion for the charge or current density within the Hall bar [50]:

Q(ω) + C q (l, ν; ω)U(ω) = l v D f ωl v D j (in) (ω) , (C5)
in which f (x) = (e ix -1)/ix, and

C q (l, ν; ω) = νe 2 l hv D f ωl v D (C6)
represents the effective quantum capacitance at frequency ω/2π for the edge channels of length l and filling fraction ν. Eq. (C5) expresses that the charge stored within each edge channels comes from what is injected by the reservoirs and from the shift of the bottom of the electronic bands by the electric potential seen by all electrons.

Edge magnetoplasmon scattering

Using Eqs. (C1) and (C4) to eliminate Q(ω) and U(ω) from Eq. (C5) leads to

(C q (l, ν; ω) + C) j (out) (ω) = C q (l, ν; ω) + e iωl/v D C j (in) (ω) . ( C7
)
Solving this equation leads to the scattering matrix S(ω) relating incoming to outgoing electrical currents at ω.

From the bosonization point of view, this is the scattering matrix for the bosonic counter-propagating edgemagnetoplasmon modes carrying the total charges of the ν right and ν left moving edge channels. This scattering matrix depends on a dimensionless coupling constant α ν characterizing the strength of Coulomb interactions within each edge channel by the way of the ratio of the bare quantum capacitance of a single edge channel with drift velocity v D to the geometric capacitance [START_REF]The l dependance of CH is restored here to stress that we are in the limit of a long quantum Hall bar ([END_REF]:

α ν = C q (l, ν; ω = 0) C H (l)ν = e 2 l/hv D C H (l) . (C8)
The higher this number, the higher is Coulomb's energy e 2 /C H (l) with respect to the kinetic energy scale hv D /l. The geometric capacitance can be evaluated using standard electrostatics as explained in Sec. D. The result comes under the form

C H (l) = ε 0 ε r l f bar (W, W H (ν)) (C9)
in which ε r is the relative permittivity of the material and f bar (W, W H (ν)) is a geometrical factor that depends on the width W of the quantum Hall bar and on W H (ν), the width of the system of ν copropagating edge channels at the edge of the quantum Hall bar. The latter is proportional to the filling fraction ν and, in the W H (ν) W limit, the geometric factor f bar depends logarithmically on the aspect ratio W/W H (ν) as shown in Sec. D. The dimensionless coupling constant

α ν = 2 α mat f bar (W, W H (ν)) (C10)
is therefore proportional to the effective fine structure constant within the material

α mat = e 2 4πε r ε 0 v D = α qed ε r c v D . (C11)
up to a subdominant logarithmic dependance in ν arising from classical electrostatics (see Sec. D). The ω dependence is an X = ωl/v D dependence involving the free electron time of flight l/v D along the edges of the quantum Hall bar. Solving Eq. (C7) leads to the edge magnetoplasmon scattering amplitudes

S RL (ω) = S LR (ω) = -iXf (X) 2 + να ν f (X) (C12a) S RR (ω) = S LL (ω) = 1 -S RL (ω) . (C12b)
Note that the relation between diagonal and off diagonal S matrix elements given by Eq. ( C12b) is only valid for total screening (η = 1). One should also note that because of the symmetric setup considered here, the edge magnetoplasmon scattering matrix is determined by its diagonal coefficient S d (ω) = S RR (ω) = S LL (ω) and its off-diagonal coefficient S od (ω) = S LR (ω) = S RL (ω). They satisfy the unitarity conditions

|S d (ω)| 2 + |S od (ω)| 2 = 1 (C13a) Re(S d (ω)S od (ω) * ) = 0 . ( C13b
)
which ensure energy conservation within the quantum Hall bar.

Finite frequency impedances

In the spirit of Büttiker, let us derive a discrete element circuit description of the quantum Hall bar. In full generality, this description involves a three terminal circuit in order to take into account the electrostatic coupling to external gates which are assumed to be at the ground here (see Fig. 9). In our experimental, ungated samples, this coupling is expected to be small as we shall see. Note that the symmetry of the Hall bar under magnetic field reversal justifies considering two identical impedances Z(ω) on both sides of the central node A on Fig. 9. For this tri-terminal circuit, one can combine the relation between the potentials V G , V D , V A and V 0 = 0 (by gauge invariance) and the current sum rule I G + I D + I 0 = 0 to relate I G and I D to V G and V D :

Z + Z 0 Z 0 Z 0 Z + Z 0 • I G I D = V G V D . (C14)
On the other hand, the finite frequency admittance of the quantum Hall bar can be obtained from the edge magnetoplasmon scattering matrix which, in the present case is symmetric. We can then infer expressions for the impedances Z(ω) and Z 0 (ω) in terms of the EMP scattering matrix of the quantum Hall bar:

Z 0 (ω) = R H S d (ω) (1 -S od (ω)) 2 -S d (ω) 2 , (C15a)
Z(ω) = R H 1 + S d (ω) -S od (ω) . (C15b)
Note that Eqs. (C15a) and (C15b) have been obtained without any explicit assumption of total screening. In the case of η = 1, Z 0 is expected to be infinite and is indeed found to be infinite since, in this case, S d (ω) + S d (ω) = 1 (see Eq. ( C12b)).

To obtain the simplest effective circuit description, we just derive the dominant terms of the low frequency expansion of Z(ω) and 1/Z 0 (ω) by expanding the edgemagnetoplasmon scattering matrix in powers of ω. This leads to 1/Z 0 (ω) = 0 (total screening) and

Z(ω) = R H 2 1 - iωl/v D 2 + να ν (C16)
which shows that the impedance Z(ω) can be viewed as the series addition of a resistance R H /2 (ν parallel single channel contact resistances R K /2) and an inductance

L Z such that 2L Z R H = l/2 v D 1 + να ν /2 . ( C17
)
The quantum Hall bar is then a dipole with impedance 2Z(ω) which then behaves as an inductance in series with the quantum Hall resistance R H at low frequency. Since 2L Z is the total inductance L of the quantum Hall bar, it follows that the electronic time of flight l/v D appearing in Eq. ( A4) is renormalized by Coulomb interactions. The corresponding renormalized velocity

v 0 = v D 1 + να ν 2 (C18)
can, in a sense, be interpreted as an effective charge velocity.

To understand this more precisely, let us imagine that the quantum Hall bar could be viewed as an ideal coaxial cable with ν channels with plasmonic velocity v P . This would lead to a diagonal plasmon scattering matrix: S od (ω) = 0 and S d (ω) = e iωl/v P . The corresponding admittance Z(ω) would then be given by

Z tl (ω) = R H 2 1 1 + e iωl/v P (C19)
where the index tl emphasizes the transmission line description considered here. Its low frequency expansion would correspond to the series addition of the contact resistance R H /2 with an inductance L

Z given by 2L

(tl) Z R H = l 2 v P . (C20)
Consequently, we also recover the analogue of Eq. (A4) with v P playing the role of v D . However, in this transmission line model, the admittance 1/Z 0 would not vanish, thereby showing that such a transmission line model cannot describe the totally screened situation. The edge magnetoplasmon scattering matrix derived in Sec. C 2 shows that, in the totally screened case, R H /Z 0 vanishes.

Appendix D: Geometric capacitance

In this section, we compute the geometric capacitance of the quantum Hall bar using a model a capacitor FIG. 9.

The lumped element circuit equivalent to simple quantum Hall bar depicted on Fig. 8. The 0 lead corresponds to metallic gates and conductors to which the edge channels are capacitively coupled in the case of partial mutual capacitive coupling between the R and L channels (η < 1). We expect Re(Z(ω)) = RH /2 and Im(Z(ω)) to be inductive at low frequency whereas Re(Z0(ω)) = 0 and Z0(ω) is expected to be capacitive in general, and to be infinite in the totally screened case (η = 1).

built from two coplanar strips corresponding to the two counter-propagating edge channel systems.

Width of edge channels

The model of Ref. [START_REF] Chklovskii | [END_REF] by Schklovskii et al. describes the structure of quantum Hall edge channels when taking into account the effect of Coulomb interactions. It predicts the width λ H (denoted by a ν in Ref. [START_REF] Chklovskii | [END_REF]) of the incompressible channels in term of the density gradient at the edge as well as the width b ν of compressible channels as a function of λ H .

The width of incompressible stripes is obtained by identifying the Landau gap with the energy variation of an electron in the confining potential U (see Figure 10):

a ν × e∇U = eB m * , (D1)
where ∇U denotes the gradient of U at the edge of the sample. This leads to:

a ν = m * v D , (D2)
where we have defined v D = E y /B z as the drift velocity along the edge. Note that this expression is formally the Compton wavelength associated with a relativistic particle of mass m * and effective speed of light v D . This explains why we denote it by λ H in the following. This result is not surprising, the width of the incompressible stripe is the minimal one for closing the cyclotron gap for an electronic excitation in the confinement potential, as shown in Ref. [START_REF] Chklovskii | [END_REF]. Once the width of an incompressible stripe is known, the width of a compressible stripe is

b ν = π 4 λ 2 H a B ( D3
)
where a B = 2 ε/m * e 2 denotes the effective Bohr radius in the material.

Capacitance computation

The quantum Hall edge channel system has a width

W H (ν) = ν m * v D 1 + π 2 α mat , (D4)
which is of the order of 90 nm per edge channel in Al-GaAs/GaAs systems. In the transverse direction, the electrons are confined within a triangular potential well. The typical extension of the electron's wavefunction in the transverse direction is of the order of 5 nm [START_REF] Fang | [END_REF], therefore suggesting a rather flat shape of the edge channel. A first estimate of the geometric capacitance of the edge channel system can thus be obtained by computing the geometric capacitance per unit of length of an infinite pair of coplanar strips of width w separated by a distance d [START_REF] Iossel | The calculation of electrical capacitance[END_REF]: .

C H (l) l = ε 0 ε r K( √ 1 -x 2 ) K(x) , (
(D6)

On the other hand, using a more precise self-consistant solution of the electrical potential within quantum Hall edge channel arising from the repartition of quantum electrons, Hirai and Komiyama [51] have shown that the capacitance can be obtained as the geometric capacitance of two parallel wires: Fig. 3 of the paper shows, for a positive orientation of magnetic field, the dependence of the quantum inertia on the inverse of the filling factor. In Fig. 11 of this Supplementary Material, these results have been obtained for a negative orientation of magnetic field. Dashed curves are obtained using Eq. (1) of the paper, using a proper velocity v D for each configuration (see caption).

Influence of filling factor for other samples

For sample A we have shown that reactance increases linearly with frequency and that the slope depends strongly on the filling factor. Here, we observe the same behavior on other samples (see Figs. 12 to 15): the slope decreases while the filling factor increases. For all samples and configurations the inductance decreases with density of states. Quantum inertia appears then to be a robust effect against variability of experimental conditions, even in macroscopic samples. Still, the dependence of the inductance on the filling factor is not fully linear. These results corroborates the conclusions obtained from sample A : the velocity v 0 depends on the filling factor.

Influence of configurations for other samples

Figs. 16 to 19 depict the reactance as a function of frequency for three samples at different filling factors. For each sample at a given filling factor, we observe how the configuration (shown as an inset for each curve) modifies the slope of the reactance. Each configuration has its proper length and the larger the edge states the higher the quantum inertia. The length concerned here is the distance between potentials L p and H p . Meanwhile, differences in velocity v D for distinct configurations imply that the ratio between quantum inertia and length of edge states not a constant. Configurations are obtained after wire-bonding the samples. This entails heating the samples and cool them down again afterwards, and this causes the velocity v 0 to change a bit from one configuration to another.

FIG. 2 .

 2 FIG. 2. The reactance X(f ) as a function of the frequency f for different filling factors and B<0, in the measurement configuration shown here. Inset: the longitudinal resistance R(f ) vanishes quadratically for integer filling factors at low frequency.

FIG. 3 .

 3 FIG. 3. (Color online) For three measurement configurations with B > 0 depicted on the figure, the inductance increases with 1/ν as expected by Eq. (3a). Dashed lines correspond to our theory for (red data) l = 600 µm with vD = 8 × 10 5 ms -1 , (black data) l = 1000 µm with vD = 0.16 × 10 5 ms -1 and (blue data) l = 1600 µm with vD = 0.2 × 10 5 ms -1 . Inset: reactance X(f ) as a function of the frequency for different filling factors and B > 0.

FIG. 4 .

 4 FIG. 4. Low frequency charge carrying edge-magnetoplasmon velocity v0 as a function of filling factor ν. Dashed curves correspond to Eq. (1) for the values, from bottom to top, vD = (0.15; 0.8; 4; 12; 30) × 10 5 ms -1 .

FIG. 5 .

 5 FIG. 5.Dispersion relations for non-interacting electrons propagating along a right-moving chiral edge channel R and a left moving one L with propagation velocity ±vD. In the presence of a d.c. voltage bias V , a chemical potential imbalance -eV appears between the two chiral edge channels. At filling fraction ν, such a bias generates a net current I = (e 2 ν/h)×V .

FIG. 6 .

 6 FIG. 6. The device under test when three Ohmic contacts are wire-bonded onto the sample holder. The quantum Hall bar B, which is the dipole under test, is shown as a dashed red box. The quantum Hall bar A is shown as a dotted blue box. The capacitance of the coaxial cable connected to contact 3 is denoted by C0. In this experimental setup, V2 = 0 whereas the drive V1 is non zero and the experimentally measured impedance is Z (exp) 23 = -∂V3/∂I2.

)

  Note that, at low frequency (ωR H C O 1), the result only depends on the finite frequency properties of the B quantum Hall bar i.e. the dipole under test in this setting. b. Effect of the finite frequency deviation of Re(Zxy(ω)) from RH on Z (exp) 23

  , let us therefore write z H (ω) = 1 + r(ω) + iωτ H (ω) (B11) in which r(ω) denotes the frequency-dependent deviation to R H (when non zero), and τ H (ω) = L eff (ω)/R H is the dimensionless frequency-dependent effective RLtime of the quantum Hall bar (L = L eff (ω = 0)). The low frequency expansion of the experimentally measured impedance Z (exp) 23

3 .

 3 FIG. 7. The device under test has now four contacts connected to sample holder, two of them being on same side. The quantum Hall bar is divided into three parts, shown as colored box, A (blue), B(red) and C(green). Hpot and Lpot are connected respectively to contacts 4 and 3. In this configuration, the low frequency impedance depends only on the central Hall bar B.

  B27)showing that the measured inductance corresponds to the transit time in Hall bar B. Thus, the length involved in the determination of velocities is l = l B .Appendix C: Discrete element description from edge-magnetoplasmon scattering

FIG. 8 .

 8 FIG. 8. A simple quantum Hall bar at filling fraction ν = 3 connecting the G lead to the D lead. Along the R edge channels, electrons are right movers flowing from lead G to lead D whereas for edge channels L, they are left movers.

FIG. 10 .

 10 FIG. 10. Edge channel structure arising from Coulomb interaction effects in a mean field theory approach [27]: incompressible stripes alternate with compressible stripes in which electrons are located right at the Fermi energy.

  D5) in which K(x) denotes the elliptic integral of the first kind [64] evaluated at x = d/(d + 2w) where, in our case w = W H (ν) and d + 2w = W is the total width of the quantum Hall bar. Using known asymptotics for the elliptic integral, we find the geometrical capacitance for the quantum Hall bar of width W and length l at filling fraction ν C H (ν, l) πε 0 ε r l ln W 8W H (ν)

FIG. 11 .

 11 FIG. 11. Kinetic Inductance as a function of the inverse of filling factor for a negative orientation of magnetic field . Dashed line represents the theoretical model (Eq.(1) of the paper) with following parameters : vD = 8 × 10 5 ms -1 for the larger configuration, vD = 11 × 10 5 ms -1 for the intermediate one, vD = 22 × 10 5 ms -1 for the short one. Length of edge states are still the same (see Fig. 3 of the paper).

FIG. 12 .

 12 FIG. 12. Reactance as a function of frequency, for sample B and ν = 4 and 4. Configuration is shown in inset.

FIG. 14 .

 14 FIG. 14. Reactance as a function of frequency, for sample D and ν = 2 and 4. Configuration is shown in inset.

FIG. 16 .

 16 FIG.16. Reactance as a function of frequency, for sample A at ν = 6 and for the three configurations.

FIG. 19 .

 19 FIG. 19. Reactance as a function of frequency for sample D at ν = 4 for two configurations. Note the smaller values for X(Ω) compared to the other curves which may explain the more noisy dataset for the four-terminal geometry.

ACKNOWLEDGMENTS

We warmly thank G. Fève, Ch. Flindt and B. Plaçais for useful discussions and suggestions concerning this manuscript. This work has been partly supported by ANR grant "1shot reloaded" (ANR-14-CE32-0017).

Supplementary Information

In this Supplementary Information, we discuss a simple evaluation of the kinetic inductance of a quantum Hall bar at filling fraction ν, compare it to its geometric inductance. We then move directly towards a more realistic evaluation of experimental relevant quantities. First, we connect these quantities to the way a quantum Hall bar scatters ac currents. To compute this scattering properties, we use the edge magnetoplasmon scattering method which takes into account Coulomb interactions between the opposite channels of the quantum Hall bar. Moreover, we connect these scattering properties to the finite frequency admittance of the quantum Hall bar.

Appendix A: Kinetic and magnetic inductance

In this section, we compare the kinetic inductance of a quantum Hall bar at filling fraction ν to its geometric inductance.

Appendix B: Experimental signals

In this section, we discuss how the quantities that are measured in the experiment are related to the a.c. response of a quantum Hall bar even in the presence of experimental imperfections such as cable capacitances.

Appendix E: Supplementary results

In this Section, we show further experimental results obtained on sample A (the sample discussed in the paper) and on three other samples manufactured from three different AlGaAs/GaAs heterojunctions. Their wafer characteristics (electron density and mobility of the twodimensional electron gas) are reported in table I. Samples have been processed on these heterojunctions using the same design and dimensions as for sample A. Exeperimental results obtained on these samples have been obtained using exactly the same protocols and they are completely similar to the results presented in the paper.

We first present complementary results on sample A, then we present results obtained on the other samples, which concern influence of filling factor ν and influence of edge channel length.


Corentin Lefebvre

Lucas Fortier

Norbert Hoffmann
email: norbert.hoffmann@univ-reims.fr

Photochemical Rearrangements in Heterocyclic Chemistry

Keywords: Heterocycles, Rearrangements, Photochemistry, Hydrogen Atom Transfer, Isomerization Graphical Heterocyclic chemistry

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Photochemical reactions are characterized by electronic excitation caused by light absorption. The change of electronic configuration causes a considerable change in chemical reactivity when compared to reactions at the ground state. [START_REF] Klán | Photochemistry of Organic Compounds[END_REF] For example, electron transfer is facilitated. A photochemical reaction can also be induced by energy transfer from the excited species. [START_REF] Dantas | [END_REF] , 3 These processes are now intensively investigated in connection with many applications to organic synthesis. 4 These studies essentially contributed to the recent rebirth of synthetic organic photochemistry. The traditionally strong interaction between organic photochemistry and physical chemistry 5 enables a high level of characterization and understanding of these reactions and particularly the mechanisms which also facilitate the optimization and application in various fields.

In connection with the electronic configuration change, very often typical characteristics such as regio or stereochemistry are inversed in photochemical reactions when compared to corresponding ground state reactions. Complex structures in a broader scope become more easily available. 6 Typical examples are the di-π-methane rearrangement and corresponding heteroatom oxa-and aza-di-π-methane rearrangements. [START_REF] Riguet | Comprehensive Organic Synthesis[END_REF] Often photochemical products possess an increased reactivity in particular the ground state reactions and can serve as interesting synthesis intermediates for natural product synthesis or for the preparation of new compounds in many fields such as pharmaceutical and agro-chemistry or for material sciences. In these domains, heterocyclic compounds play a key role. Access to new heteropolycyclic compounds is a daily challenge to fill the blanks in the chemical space. [START_REF] Virshup | [END_REF] Often heterocyclic substituents are considered as bioisosteres [START_REF]Bioisosteres in Medicinal Chemistry[END_REF] in the context of a rational drug development. It should further be pointed out that the photon as a traceless reagent accounts for the green chemistry principles. 10 , 11 In this article, we would like to discuss photochemical rearrangements of heterocyclic compounds [START_REF] Hoffmann | Handbook of Photochemistry and Photobiology[END_REF] , [START_REF] Arai | [END_REF] involving heteroatom isomerization, hydrogen atom transfer (HAT) or electrocyclization.

Photochemical heteroatom isomerization of heterocyclic compounds

Photochemical reactions provide interesting tools to change the position of the heteroatoms or the substituents in heterocyclic compounds. In this way, the structural diversity is increased which is an important topic when new biologically active compounds are looked for. In material science such an approach is also interesting when heterocycles play a key role as building blocks. Typical examples are pyrazole and imidazole. The inhibition of the phosphodiesterase (PDE) 5 is increased when the pyrazole moiety in 1 is replaced by an imidazole ring in 2 (Figure 1). 14 Similar effects have been observed for purin and corresponding pyrazol derivatives as inhibitors of RNA or DNA glycosilase activity of shiga toxin 1. 15 Figure 1.

The pyrazol derivative 1 possesses a higher activity than the corresponding imidazole derivative 2.

Since pyrazol and imidazole compounds are obtained by different synthetic approaches, a reaction enabling the systematic interconversion of derivatives of both compound families is highly appreciated. The member number of compounds families could be significantly increased. According to Scheme 1 in a photochemical reaction, pyrazols can be transformed into imidazols. 16 Such reactions are part of the the larger family of circumabulatory rearrangements. [START_REF] Childs | [END_REF] Upon irradiation, the trimethylpyrazol derivative 3 is transformed into the imidazole derivatives 4 and 5. 18 In a disrotatory [π2+σ2] electrocyclic reactions (blue arrows), the bicylic intermediate 6 is formed. 1,3-Sigmatopic shifts (red arrows) lead to the 7 and 8.

Intermediates 7 and 8 are transformed into the final products 5 and 4 respectively via disrotatory [σ2+π2] electrocyclic reactions (blue arrows). Compound 5 can also be generated by homolytic cleavage of the N-N bond in 3 leading to intermediates 9 and 10. Furthermore, it was observed that compound 4 can be transformed into 5.

Scheme 1. Photochemical transformation of a pyrazol derivative into imidazoles.

The photochemical transformation of benzopyrazoles into the corresponding imidazole derivatives can efficiently be carried out (Scheme 2). 19 Thus compounds 11 were transformed into the imidazole derivatives 12. The transformation of similar regioisomers 13 into 14 was more efficient. In fact, these pyrazol compounds possess a quinoid electronic structure. In the latter reaction the intermediates 15 and 16 has been discussed. 20 When the irradiation was carried out at low temperature (-60°C), compound 15 could be identified. Upon warmup, 15 gave 13 (not 14). Thus the final product 14 is formed from intermediate 16.

Scheme 2. Photochemical rearrangement of benzopyrazol derivatives into corresponding benzoimidazol derivatives. The irradiation has been carried out with high pressure Hg-vapor lamps.

Isoxazoles contain a nitrogen next to an oxygen atom. When compound 17 was irradiated at λ = 254 nm, three compounds 18, 19 and 20 along with some benzoylacetone were isolated (Scheme 3). 21 An advanced mechanistic study revealed that the azirine 18 is an intermediate in the formation of the oxazoles 19 and 20. More interestingly, when 18 was irradiated at 300 nm, the starting isoxazole 17 was obtained. In the first case, the formation of a zwitterion intermediate was suggested while in the second case, a diradical intermediate at the triplet state should be generated. By the way, this is an example which infringes the general accepted principle that the a photochemical reaction always occurs form the lowest excited singlet or triplet state as it was formulated in analogy with the Kasha and Vavilov roules (see ref. 1, page 40). Especially in cases of intramolecular reactions which are faster than inter conversion processes, such infringes are observed.

Scheme 3. Photochemical rearrangement of an isoxazol compound.

Similar reactions have been carried out with benz-or naphthisoxazoles 21a,b (Scheme 4). 22 Also in these cases, the spirocyclic azirine 22a,b was detected as reaction intermediate. Photochemical or thermal rearrangement generates the final oxazole derivatives 23a,b.

Scheme 4. Mechanistic study of the rearrangement of naphthisoxazoles.

The corresponding isothiazol 24 can be transformed into thiazol 25 (Scheme 5). 23 , 24 In an acidic media and when the heterocyclic compound is protonated at the nitrogen atom, the reaction is reversible. Under similar conditions, a migration of the substituents was also observed. Thus the photochemical transformation of 26 yields mainly 27 and lower amounts of 28. 25 In the main reaction, the phenyl substituent migrates. In a computational study, it has been shown that bicylic intermediates are involved (compare 6-8 in Scheme 1). 26 More generally in such reactions, a competition between this mechanism and mechanisms involving homolytic or heterolytic fragmentation of the bond between the two heteroatoms is discussed. 27 Scheme 5. Rearrangement of heteroatoms and migration of a substituent in isothiazole and thiazol derivatives.

Five membered heterocyclic aromatic compounds with three heteroatoms such as oxadiazoles are also very important structural elements in bioactive compounds and materials. 28 In these compounds similar photochemical steps are observed. 29 In particular, the cleavage of the N-O bond is one of the first steps. 30 Not only heteroatoms and substituents but also functional groups can migrate or be transformed. In the photochemical reaction of compound 29 to 30, a urea function is transformed into a carbamate (Scheme 6). 31 Intermediates which are formed in such reactions can be trapped by an intramolecular addition which leads to rearranged heterocycles. Under mild reaction conditions, the isoxdiazole derivative 31 was transformed into triazol derivative 32. 32 An attack of amine nitrogen atom of the hydrazone on the nitrogen in 2 position (33) with formation of intermediate 34 was discussed as key step in this photochemical reaction. Scheme 6. Photochemical rearrangements with isoxadiazole derivatives.

Photoredox catalysis applied to organic synthesis considerably contributes to the renaissance of organic photochemistry. [START_REF]Visible Light Photocatalysis in Organic Chemistry[END_REF] These conditions have also been applied to rearrangement reactions of heterocycles. When the isoxazolone derivative 35 is irradiated in the presence of the photoredox catalyst fac-Ir(ppy)3, the oxazol derivative 36 is obtained in high yield (Scheme 7). [START_REF] Mei | [END_REF] In this reaction, a decarboxylation step is involved. The rearrangement with ring extension (37) is a competitive process. Under optimized conditions using 4DPAIN as photocatalyst and triethyl amine as sacrificial electron donor, the latter process become the only reaction and compound 37 was isolated in good yields. Decarboxylation is suppressed by an electron transfer step because, in corresponding intermediate, a carboxyl radical is reduced to a carboxylate anion. The oxazole reaction was performed with 26 additional substrates and the oxazinone reaction (37) was pformed with seven additional derivatives. A similar study with corresponding thiocarbonyl derivatives have been carried using direct excitation, without photocatalysis. 35 A variety of derivatives has been transformed in this way.

Scheme 7. Photoredox catalytic rearrangements of an isoxazolone derivative.

Photochemical rearrangements induced by hydrogen atom transfer (HAT).

In chemical 36 and biological reactions 37 , hydrogen as the smallest and lightest atom, can easily be transferred. This transfer leads to the formation of high energy radical species that can rearrange or react with others species to form new molecules. Using photochemical HAT, the use of toxic or expensive chemicals can be avoided.

Photochemical 1,4-hydrogen abstraction is often observed. Pyrrole is a common heterocycle found in many natural products and drugs on the market. While this latter is easily accessible via photoredox catalysis, 38 Some of them can be synthetized through direct irradiation from azetidinyl ketones such as 38 (Scheme 8). 39 The reaction happens in a two steps mechanism 40 , where electron is transferred first from the nitrogen to the excited ketone (39) and then the proton follows (40, 41). A dehydration is involved in the formation of the final product 43 from intermediate 42. It was shown that this photoreaction occurred at a triplet excited-state. A singlet excited-state could be involved when an alkyl instead of an aryl substituent is used. In this case, a Norrish type II photoelimination at the alkyl substituent takes place.

Scheme 8. Synthesis of pyrroles from azetidinyl ketones via 1,4-hydrogen abstraction.

Another pathway to the synthesis of pyrroles has been proposed by Aoyama with thioamide derivatives. 41 After direct irradiation, compound 44 undergoes 1,4-HAT to provide diradical 45 (Scheme 9). Radical combination leads to the intermediate 47. After elimination of H2S, the pyrrole derivative 49 is formed. The enamine derivative 46 is obtained by hydrogen transfer from thiol function into the 6 position. Small quantities of 46 are transformed into 48 by an electrocyclic reaction involving the aromatic substituent. The reactions of six different derivatives have been reported. This latter structure can be found in agrochemical fungicide. 42 Scheme 9. Synthesis of pyrroles from thioamides derivatives.

1,4-HAT is also employed as a key-step in the synthesis of bioactive natural products such as rocaglamides, forbaglins and aglains (Scheme 10). 43 Such molecules are isolated from the plant Aglaia and are accessible via a subsequent dipolar cycloaddition. At the excited state, in the flavon compounds 50, hydrogen transfer occurs. 44 Such reaction steps are also discussed in the context of phototautomerisation. The resulting intermediates 51 undergoes cycloaddition leading adducts such as compounds 52. A ketol rearrangement yields compounds 53 and 54.

Scheme 10. Formation of bioactive natural compounds core of forbaglin and rocaglamide via 1,4-HAT and [3 + 2] cycloaddition.

Enantioselective HAT has also been carried out recently. 45 A study of Xia et al. showed the possibility to use 3-hydroxyquinolinones instead of 3-hydroxychromenones to make this photoreaction. 46 Natural compounds can be synthetized through photoinduced 1,5-HAT. Wender et al. proposed a photochemical access to a precursor of taxol C,D ring system (Scheme 11, equation 1). 47 Subsequent photoenolization from 1,5-HAT is also possible prior to cycloaddition (equation 2). 48 This was performed to the synthesis of anti-tumor drugs Pleurotin. 49 More recently, Reddy et al. used this photochemical cascade towards the synthesis of pyrroloquinazoline core of Vasicinone and Luotonins, bioactive natural products (equation 3). 50 Scheme 11. -Synthesis of precursors of natural products Taxol, Pleurotin, Vasicinone and Luotonin B.

Kutateladze's team developed a strategy to use a highly stereoselective photoreaction for the synthesis of complex polyheterocycles. Enol/imine intermediates 56 are generated by photoinduced 1,5-HAT (Scheme 12). 51 , 52 After hydrogen abstraction, a [4 + 2] cycloaddition leads to the formation of 57 and a [4 + 4] cycloaddition generates the oxamorpham core 58 from compound 55. The same reaction mechanism is involved in the formation of 60 and 61 from 59. It should be pointed out that in the case of the photoadducts obtained from 55, a Suzuki reaction is carried out. Such photochemical hydrogen atom or proton transfer processes between heteroatoms like oxygen or nitrogen (see also Scheme 10) are also discussed in the context of excited state intramolecular proton transfer (ESIPT). 53 Scheme 12. Formation of polyheterocycles via photoenolisation.

Nitrogen-containing heterocycles are ubiquitous in nature. [START_REF] Brown | Ring Nitrogen and Key Biomolecules : the biochemistry of N-heterocycles[END_REF] Many efforts are carried out by synthetic chemists to produce new azaheterocyclic building-blocks to feed pharmaceutical market as they have a big presence in it. [START_REF] Mcgrath | [END_REF] Photoinduced hydrogen abstraction can be carried out on N-heterocycles to access interesting compounds. After irradiation of molecule 62, compound 63 is obtained due to [2 + 2] cycloaddition followed by a photochemical rearrangement (Scheme 13). 56 Longer irradiation triggers 1,5-HAT leading to the intermediate 64. The transformation to compound 65, a bridged azocanone, resembles to the Norrish type II reaction. This kind of core stucture is found in synthetic precursors of alkaloids such as strictamine, rhazinoline, melinonine and strychnoxanthine. 57 , 58 Scheme 13. Formation of a bridged azocanone: potential precursor for the synthesis of alkaloid structures.

Hoffmann's group proposed a way of synthetizing a trifluoromethyl substituted pyrrolo-oxazolone (Scheme 14). 59 In this study, they show the regioselectivity of photoinduced HAT. In contrary of previous research with furanone derivatives, 60 they found that reaction selectivity is not induced in the HAT step, but in the cyclisation step. Two compound families resulting either from the formation of a C-C bond (67 and 68) or a C-N bond 69 can be obtained in good yield from the reaction of oxazolone derivatives 66. This latter compound is a core structure of molecules used for diabetes treating. 61 Scheme 14. Photoinduced intramolecular radical cyclisation via 1,5-HAT with imines.

1,6-Hydrogen abstraction has frequently been applied to natural product synthesis. Natural molecules are a source of potential drug leads. From mankind's origin to nowadays, therapeutic treatments is mainly based on molecular structures from fauna and flora molecular wealth. It is estimated that only around 36 % of new chemical entities for drugs are not based or inspired by natural products. 62 Among the latters, alkaloid synthesis has been highlighted as their group is wide and offers numerous biological properties (Scheme 15). [START_REF] Aniszewski | Alkaloids -Chemistry, Biology, Ecology, and Applications[END_REF] Gramain et al. succeeded in making Isoretronecanol via photoinduced 1,6-HAT. [START_REF] Gramain | [END_REF] In this photochemical key step, the pyrrolidine derivative 70 was transformed into the bicyclic lactam 71. Isoretronecanol a pyrrolizidine type alkaloid found in various plant species. Wu et al. proposed a similar pathway to the synthesis of alkaloid-like benzopyrrolizidine structure, found in Linamycins. 65 In this synthesis, 72 was transformed into 73 via a 1,6-hydrogen transfer.

Scheme 15. Access to Isoretronecanol and Linamycin core via 1,6-HAT.

Others precursors of natural products are accessible via 1,6-HAT like Combrenin, 66 Lurbinectedin, 67 Tawianiaquinol A, 68 Pauwlonin, 69 Coumestan, Coumestrol, 70 and Aflatoxins 71 (Scheme 16).

Scheme 16. Synthesis of various precursors of natural products via 1,6-HAT.

Macrocyclizations via photoinduced hydrogen transfer has been intensively studied with phthalimides systems. 72 , 73 , 74 Hu et al. proposed another system to carry out 1,n-HAT (Scheme 17). 75 After irradiation of 74 an electron is transferred to the excited ketone, followed by a proton. The mechanism of transferring hydrogen unfolds in two-steps. 40 Macrocyclization then occurs by recombination to 75. In the same year, Yamazaki et al. succeeded in making oxathiocanone core 77 from 76 with the same reaction involving two-steps mechanism for the hydrogen atom transfer. 76 Scheme 17. Macrocyclization with thioether derivatives via hydrogen atom transfer.

Other examples of macrocyclization via 1,n-HAT have been reported in connection with studies on ketoprofen-quencher conjugates 78 and benzophenone-methionine dyads 79 (Scheme 18). 77 , 78 , 79 These studies mimic and highlight stereodifferentiating interactions between photosensitizing drugs and macromolecular structures in the excited state in order to understand mechanistic overviews of reaction steps in biological systems.

Scheme 18. Examples of 1,12-HAT and 1,15-HAT for macrocyclization.

Electrocyclization reactions with pyridine derivatives

Taylor and Paudler reported that substituted pyridones are rapidly converted by UV light irradiation to a solid dimer which reverts to the original pyridone upon heating above its melting point. 80 Only a few years later, in 1964, Corey and Streith studied the photoreactivity of N-methyl-2-pyridone 80 (Scheme 19). 81 Irradiation of diluted solutions of these compounds at -20°C led to the corresponding bicyclic isomers 81.

Scheme 19. Photochemical electrocyclization N-methylpyridone 80.

Photopyridone's relative stabilities were then intensively discussed during the 70's. De Selms and Schleigh concluded that unsubstituted photoisomers, are relatively unstable. 82 However, substituted 2pyridones led to relatively high degree of stability of the corresponding photopyridones. 81 In a more exhaustive publication, Furrer obtained similar results and proved the high thermal stability of the resulting photoproducts. 83 Asymmetric electrocyclization reactions induced by irradiation of 2-pyridones using chiral auxiliarities were also reported by Sato et al. 84 The authors used enantiomerically pure 4-menthyloxy pyridone 82 (Scheme 20). A variety of enantiomerically pure β-lactam derivatives with antibacterial activities are available with this reaction.

Scheme 20. Asymmetric electrocyclization of a pyridine derivative.

The reaction is also used for the synthesis of more complex heterocyclic compounds. Thus Tsichiya et al. reported the synthesis of 1,4-oxazepinones 84 and 1,4-diazepinones 85 from N-protected 2-pyridones 83, using photochemical and thermal rearrangements (Scheme 21). 85 In this reaction sequence, a heteroatom is inserted between the 4 and 5 position.

Scheme 21. Synthesis of 7 membered heterocycles using the photochemical electrocylization of pyridines.

Although the chromophore of pyridinium salts resembles to that one of pyridones, the photoreactivity is different. In these compounds an electrocyclic reaction is observed involving postions 2 and 6. Such reactions were first studied during the 70's by Kaplan et al. 86 87 Further transformations yielded a corresponding cyclopentenylamine derivative 91. Which was transformed into the bicyclo[3.1.0]hexane scaffold 92. These compounds act as a mimic of a distorted sialic acid bound in the neuraminidase active site. Using continuous flow technology, the photochemical reaction was carried out on multi gram scale with a productivity of 3.7 g L -1 h -1 . 88 Scheme 22. Photochemical electrocyclization of a pyrdinium salts.

The photochemical pathway gives an attractive approach, leading to versatile intermediate compounds which can be used for the total synthesis of molecules such as mannostantin A, trehazolin derivatives 89 , cephalotaxine 90 or even Lactacystin 91 (Figure 2). Mechanistic and theoretical studies performed by Damiano et al. explain in details how the irradiation of pyridinium salts provides the stereocontrolled synthesis of a huge range of molecular structures such as bicyclic aziridines, fused heterocycles and functionalised aminocyclopentenes. 90 Figure 2. Natural product targets which are accessible by a photochemical electrocyclization of pyridinium precursors.

The reaction was also carried out with complex compounds. Using the stereoselective properties these photochemical reactions, Hanaoka et al. first showed that it was possible to stereoselectively synthesize fumariline (an alkaloid isolated from Fumaria officinalis), involving an electrocyclization with phenol betaine 93 which was obtained from protoberberine (Scheme 23). 92 Scheme 23. Photochemical electrocyclization applied to the synthesis of fumariline.

Conclusion

As in many cases of organic photochemical reactions, a systematic scope exploitation of photochemical reactions of heterocyclic compounds has rarely been performed in the past. This is deplorable since such reactions are of high interest in the research of biologically active compounds and many other domains. Photochemical heteroatom isomerization of the heteroatoms or substituents as well as electrocyclization and hydrogen atom transfer reactions considerably enlarge the access to different heterocyclic compounds. During recent years, new techniques have been developed such as LED lamps or continuous flow techniques adapted to photochemical reactions. Also photoredox catalysis offers numerous new possibilities for the transformation of heterocyclic compounds. This research domain should be revisited by the photochemical and organic chemists' communities.

  Scheme 24. Photochemical and thermal reactions starting with pyridine N-imides.

  Upon irradiation methylpyridinium chloride 86 in aqueous KOH led to stereoselective formation of the aziridine containing bicyclic allylic alcohol 87 (Scheme 22). Intermediate 88 results from a photochemical electrocyclization. When treated under acidic or similar conditions, the aziridine ring is opened. Bennet et al. obtained the 6azabicyclo[3.1.0]hexenol compound 90 from the photochemical transformation of 89.