Magneto-exciton limit of quantum Hall breakdown in graphene

TL;DR

This paper investigates the magneto-exciton instability as a fundamental limit to the quantum Hall effect breakdown in graphene, revealing relativistic effects and universal behavior in monolayer and bilayer graphene.

Contribution

It demonstrates the magneto-exciton instability in monolayer graphene and links it to relativistic signatures and universal ME conductivity, extending previous bilayer graphene studies.

Findings

Magneto-exciton instability observed in monolayer graphene.

Relativistic signatures influence the breakdown velocity.

Universal ME conductivity determines the instability threshold.

Abstract

One of the intrinsic drift velocity limit of the quantum Hall effect is the collective magneto-exciton (ME) instability. It has been demonstrated in bilayer graphene (BLG) using noise measurements. We reproduce this experiment in monolayer graphene (MLG), and show that the same mechanism carries a direct relativistic signature on the breakdown velocity. Based on theoretical calculations of MLG- and BLG-ME spectra, we show that Doppler-induced instabilities manifest for a ME phase velocity determined by a universal value of the ME conductivity, set by the Hall conductance.

Equations

$$\frac{I_N}{W}=\gamma n^2\left[\frac{E}{B{v}_{bd}}-1\right],$$

Full text

Low-bias quantum Hall (QH) transport is notoriously described in terms of single-electron physics, as exemplified by the edge-channel conductance quantization used in metrology [Klitzing1980prl, ; Tzalenchuk2010nnano, ; Ribeiro2015nnano, ]. The situation differs at large bias as electrons may couple to the collective particle-hole excitation spectrum (PHES) [Goerbig2011rmp, ], described by a dispersion relation $\omega(q)$ : it includes in the integer QH case both magneto-plasmon (MP) and magneto-exciton (ME) branches [Roldan2009prb, ], and, in the fractional QH case, a magneto-roton (MR) branch [Girvin1986prl, ; Jolicoeur2017prb, ]. High-bias transport also differs in the electric field and current distributions. In a transistor or a Hall bar geometry (length $L$, width $W$), the non-dissipative Hall current penetrates the Landau insulating bulk, so that source and drain get connected via open ballistic orbits (drift velocity $v_{x}=E_{y}/B$) [Streda1984jpc, ; Panos2014njp, ]. The high-bias conductance $G_{H}$, and Hall conductivity $\sigma_{xy}=G_{H}=I_{x}/V_{y}=\nu G_{K}$, are still set by the conductance quantum $G_{K}=e^{2}/h$ and the filling factor $\nu=nh/eB$ at a carrier density $n$. This ballistic transport is ultimately limited by the quantum Hall effect breakdown (QHEBD), a bulk effect occurring at a critical voltage $V_{bd}$ (or field $E_{bd}=V_{bd}/W$, or velocity $v_{bd}=E_{bd}/B$), which is signaled by the onset of a longitudinal voltage $V_{x}=LE_{x}$ associated with a bulk backscattering current and its associated shot noise $S_{I}$. The most frequently considered QHEBD mechanism is inter-Landau-level tunneling (ILLT), a single-particle effect that sets in when the wavefunctions of neighboring Landau-levels (LL) overlap in the tilted potential under applied bias.[Eaves1986sst, ]. In the case of a massive 2d electron gas called 2DEG, ILLT has a critical Zener field $E_{Z}\sim\hbar\omega_{c}/eR_{c}$, where $\omega_{c}=eB/m^{*}$ and $R_{c}\sim\sqrt{N}l_{B}$ are the cyclotron angular frequency and radius, $m^{*}$ is the effective mass, $N$ the number of occupied LLs, and $l_{B}=\sqrt{\hbar/eB}$ the magnetic length [Eaves1986sst, ]. ILLT gives rise to quite large velocities $v_{Z}\sim\hbar/m^{*}R_{c}$ ($\sim 2.10^{5}\;\mathrm{m.s^{-1}}$ for $N=1$ at $10~{}T$ with $m^{*}\simeq 0.06~{}m_{0}$ for GaAs-based 2DEGs). Hall bar experiments indicate premature breakdowns with $v_{bd}\lesssim v_{Z}/10$ in both 2DEGs and graphene (see [Yang2018prl, ] and references therein). Several mechanisms have been considered to explain this discrepancy, such as the phonon- or impurity-assisted ILLT [Eaves1986sst, ; Chaubet1998prb, ]. Such extrinsic mechanisms are actually needed to overcome the momentum-conservation protection of ILLT, which stems from the $2k_{F}$ momentum mismatch between neighboring LL wave functions [Dmitriev2012rmp, ], where $k_{F}$ is the Fermi momentum. However, larger velocities $v_{bd}\sim v_{Z}/2$ have been reported in quantum-Hall constrictions [Bliek1986sst, ; Chida2014prb, ], thanks to a more uniform electrostatic landscape in the absence of invasive voltage probes. These experiments challenge the single-particle ILLT interpretation, and motivate alternative explanations in terms of collective excitations, such as the ME-instability scenario proposed in Ref. [Yang2018prl, ].

QHEBD was recently investigated in bilayer graphene (BLG) transistors using shot noise as a probe of ballistic transport breakdown [Yang2018prl, ]. Interestingly, doped BLG emulates a massive 2DEG with $m^{*}\simeq 0.03~{}m_{0}$. In the two-terminal transistor geometry, the breakdown was monitored by the sharp onset of the microwave shot-noise current $I_{N}=S_{I}/2e$ above the noiseless ballistic Hall background. Breakdown noise is characterized by a large differential noise conductance $G_{N}=\partial I_{N}/\partial V$ exceeding the DC Hall conductance $G_{H}$. These large values signal a strongly superpoissonian backscattering shot noise which has been interpreted in Ref.[Yang2018prl, ] as a signature of a collective magneto-exciton (ME) instability, calling for a kinematic origin of breakdown. The $\omega(q\sim k_{F})$ sector of the 2DEG-PHES, which is relevant for breakdown in 2DEGs, being essentially interaction independent (see [Roldan2009prb, ] and discussion below), the ME-instability velocity $v_{ME}^{BLG}\sim\hbar/m^{*}R_{c}$ turns out to be similar to the interaction-free Zener limit $v_{Z}$, providing a clue to the apparent single-particle ILLT puzzle [Yang2018prl, ]. Even though the ME scenario can hardly be distinguished from ILLT according to the breakdown threshold in 2DEGs, it does explain the superpoissonian noise as a mere consequence of its collective nature. Note that the ME-instability has also been considered to interpret quantum Hall fluid flows across an ionized impurity in Ref.[Martin2003prl, ], and DC magnetoresistance resonances in monolayer graphene (MLG) in Ref.[Greenaway2021ncomm, ].

The present work extends the noise investigation to MLG, that sustains a qualitatively different PHES due to its relativistic Landau level ladder, and a more pronounced effect of interactions on the ME branches of the PHES, as explained in Ref.[Roldan2009prb, ]. This peculiarity of MLG is revisited below, and in Supplementary Information Section III, with new RPA-calculations of the spectral function and magneto-optical conductivity $\sigma_{MO}$, accounting for screening by both hBN-encapsulation and local back-gating. Noise measurements, performed in high-mobility hBN-encapsulated graphene transistors, present a magnetic field and doping independent breakdown velocity $v_{bd}^{MLG}\simeq 1.4\;10^{5}\;\mathrm{m/s}$. Calculations of the PHES for our transistor geometry indicate that this constant breakdown velocity is actually determined by an empirical but universal impedance matching criterion: $\sigma_{MO}\sim 10^{-2}~{}NG_{K}$, where $NG_{K}$ is the Hall conductance. We conclude the paper by a comparison between MLG and BLG breakdown velocities at large doping, illustrating this qualitative difference between massive and massless ME-instability supported by RPA theory.

The samples analyzed in this experiment have been previously used in the investigations of the Schwinger effect in Ref.[Schmitt2023nphys, ] and/or flicker noise in Ref.[Schmitt2023arXiv, ]; they are described in Supplementary Information (Table.SI-1). The transistors are embedded in coplanar wave-guides for DC and microwave noise characterization at $4\;\mathrm{Kelvin}$ (see measurement setup in Fig.1-a). The experiment is performed in the microwave frequency range to overcome flicker noise, which dominates up to the low-GHz range at large currents [Schmitt2023arXiv, ], and to access the QHEBD shot noise of interest. Data presented below concentrate on the hBN-encapsulated, bottom-gated, graphene sample AuS2 ($L\times W\times t_{hBN}=16\times 10.6\times 0.032~{}\mu\mathrm{m}$) which is described in Fig.1 and in Ref.[Schmitt2023nphys, ]. Graphene conductance is calculated after correcting for the (small) contact resistance effect. Low-bias magneto-conductance $G(V_{g},B)=\partial I/\partial V$ (Fig.1-b), and the $\partial G/\partial V_{g}(V_{g},B)$ fan-chart (Fig.1-c), show clear MLG-quantization down to low fields, i.e. for $B\gtrsim 0.5\;\mathrm{T}$ in accordance with the large $\mu\simeq 32\;\mathrm{m^{2}/Vs}$ mobility. The specific MLG quantization, with plateaus at $\nu=2(2N+1)$, is clearly observed; the tiny width of the plateaus in Fig.1-b signals the absence of disorder-induced localized bulk states, which warrants the absence of electrostatic-disorder. Plateaus gate voltages allow for the calibration of the gate capacitance at $C_{g}=1\;\mathrm{mF/m^{2}}$ for a thickness of the bottom-hBN $t_{hBN}=32\;\mathrm{nm}$ with $\epsilon_{hBN}=3.4$ [Pierret2022MatRes, ]. The large biases entail prominent drain-gating effects, eventually leading to a pinch-off, as reported in Ref.[Schmitt2023nphys, ] including for AuS2. This effect is compensated here by following the gating procedure described in Ref.[Yang2018nnano, ] and routinely used in Refs.[Yang2018prl, ; Baudin2020adfm, ; Schmitt2023nphys, ; Schmitt2023arXiv, ]; it consists in applying a bias-dependent gate voltage $V_{g}(V)=V_{g}(0)+\beta V$, $\beta\sim 0.4$ being adjusted to keep the resistance maximum at charge neutrality independent of bias at zero magnetic field. Fig.1-d shows typical microwave $S_{I}(f)$ shot-noise spectra in increasing bias. Noise is expressed below in terms of the noise current $I_{N}=S_{I}/2e$ for an easy comparison with DC transport current.

The high-bias magneto-transport and noise characteristics of sample AuS2 are described in Fig.2. The current voltage relation $I(V)$, measured at $B=0.5\;\mathrm{T}$ in Fig.2-a, shows a smooth crossover between the quantum Hall regime where $I\simeq I_{H}=\nu G_{K}V$ (inset) and the extremely high-bias metallic-like regime where the differential conductance recovers its zero-field value, which is set by the Zener-Klein conductivity [Yang2018nnano, ; Yang2018prl, ]. The breakdown voltage $V_{bd}\approx 0.6\;\mathrm{V}$ (black line) appears as a gradual deviation from the $I_{H}(V)$ Hall regime. By contrast, the current-noise characteristics $I_{N}(V)$, measured in the same conditions in Fig.2-b, clearly distinguish two regimes : a quasi-noiseless quantum Hall regime for $V\leq V_{bd}=0.6\mathrm{V}$, characterized by a residual contact noise conductance $I_{N}/V\sim 0.1\;\mathrm{mS}$, and a large differential noise conductance $G_{N}(n)=\partial I_{N}/\partial V\gtrsim 1\;\mathrm{mS}$ for $V\geq V_{bd}$. The intersection between the two lines provides an unambiguous determination of the breakdown voltage $V_{bd}$ which agrees with the transport determination in Fig.2-a. In both $I(V)$ and $I_{N}(V)$, the breakdown voltage is found to be nearly doping-independent, as opposed to the high-bias noise conductance $G_{N}\propto n^{2}$ in Fig.2-b. Fig.2-c shows the current noise $I_{N}(V)$ for different magnetic fields at a fixed large doping $n=2.10^{12}\;\mathrm{cm^{-2}}$. It highlights the strong dependence of both $V_{bd}\propto B$ (inset) and $G_{N}\propto G_{H}\propto 1/B$, leading to a field-independent zero-bias extrapolate (not shown in the figure). These doping and field dependencies can be cast into the scaling displayed in Fig.2-d, where noise data, collected over a broad $[n,B]$ range, are found to collapse on the universal master line

[TABLE]

where $E=V/W$, $\gamma=40\times 10^{-32}~{}\mathrm{Am^{3}}$, and $v_{bd}=1.4\times 10^{5}~{}\mathrm{m/s}$. While the scaling differs from that of BLG [Yang2018prl, ], the noise amplitudes are comparable, with $I_{N}/W=40~{}\mathrm{A/m}$ for $V=2~{}V_{bd}$ at $n=10^{12}\;\mathrm{cm^{-2}}$. This noise scaling, with a doping-independent $v_{bd}$, contrasts with the doping-dependent ILLT breakdown threshold (dashed line in Fig. 2c-Inset). As quasi-particle interactions are controlled by the doping-independent fine-structure constant $\alpha_{g}=e^{2}/4\pi\hbar\epsilon_{0}\epsilon_{r}v_{F}$, the observation of a doping-independent $v_{bd}$ suggests a breakdown mechanism controlled by interactions. Besides, the current noise intensity $S_{I}\propto n^{2}$ corresponds to a doping-independent velocity noise $S_{v}\propto S_{I}/n^{2}$, suggesting a kinematic interpretation of breakdown such as that provided by the ME-instability.

To base this qualitative interpretation on a more quantitative analysis, we recalculate below the MLG-PHES of Ref.[Roldan2009prb, ], adapting it for geometry and material parameters that are suitable for our experimental conditions. Figures 3-(a,b) show the PHESs calculated in the RPA approximation of Ref.[Roldan2009prb, ]. It is adapted for AuS2-sample geometry by including the screening by the local bottom-gate and the hBN-encapsulation, as explained in Supplementary Section-III.A. In the context of velocity-induced instability, we have plotted the magneto-optical conductivity spectrum $\Re[\sigma_{MO}(q,\omega)]$ (denoted $\sigma_{MO}$ below), which is deduced from the usual spectral function $\Im[\Pi^{RPA}(q,\omega)]$ using the $\Re[\sigma_{MO}(q,\omega)]=-\frac{\omega e^{2}}{q^{2}}\Im[\Pi^{RPA}(q,\omega)]$ relation. Note that $\sigma_{MO}$ suffers from non-physical divergence in the low-$q$ PHES-limit, blurring the magneto-plasmon branch and hiding the existence of a spectral gap that appears more clearly in the spectral function (see $\Pi^{RPA}$ spectra in Supplementary Fig. SI-5). This $N$-dependent bandgap is equal to the MLG cyclotron gap $\omega_{c}^{MLG}=(\sqrt{N+1}-\sqrt{N})v_{F}/l_{B}\simeq v_{F}/R_{c}$ (with $R_{c}$ the cyclotron radius), similarly to BLG in Figs.3-(d,e), where $\omega_{c}^{BLG}=1/m^{*}l_{B}^{2}$ is $N$-independent (see the spectral function in Fig. SI-7). Conductivity spectra are plotted for $N=3$ (panel a) and $N=12$ (panel b), displayed in logarithmic scale to map their steep $\omega$ and $q$ dependencies, and normalized to the Hall conductivity $NG_{K}$ (per spin and valley) for a direct comparison of electronic and collective electron-hole excitation’s conductivity. The momentum and energy scales are expressed in MLG-relevant dimensionless units $ql_{B}$ and $\omega l_{B}/v_{F}$, which imply a magnetic-field independence of the ME branches phase velocity $v_{ME}=\omega_{ME}/q$. Remarkably the ME optical conductivity $\sigma_{MO}$ is steeply increasing with $v_{ME}$, with $\sigma_{MO}\sim(3.10^{-4}$-$3.10^{-2})\;NG_{K}$ for $v_{ME}=(0.06$-$0.35)\;v_{F}$.

The effect of screening is quite substantial in MLG, as depicted in Supplementary Section Fig.SI-6, and much more prominent than in BLG (Fig.SI-8), especially at large $q$. The white lines in Figs.3-(a,b) correspond to the Doppler shifted electronic energy $\omega=v_{bd}q$ of drifting electrons, calculated at the measured breakdown velocity $v_{bd}=0.14v_{F}$ of Fig.2. In both the $N=3$ (panel a) and the $N=12$ (panel b) examples, this line separates a high ME-conductivity domain for $\omega\gtrsim v_{bd}q$, where $\sigma_{MO}\gtrsim 10^{-2}NG_{K}$, from a low conductivity domain for $\omega\lesssim v_{bd}q$. The observation of a $N$- and $B$-independent ME-instability, at a velocity $v_{BD}\simeq v_{ME}=\mathrm{cst.}$ controlled by a fixed $\sigma_{MO}\sim 10^{-2}NG_{K}$ constraint, is consistent with a collective wave interpretation, even if the value of the impedance threshold remains to be established theoretically. It is obviously consistent with our experimental observation in Fig.2 of an $n$- and $B$-independent breakdown velocity, a feature observed in all tested Au-gated samples (see Supplementary Table SI-1). Unlike in BLG, the $v_{bd}=\mathrm{cst.}$ breakdown velocity of MLG, inferred from the above conductivity criterion, exceeds the ILLT $v_{Z}\propto\sqrt{B}$, especially at low-$B$.

Let us recall that the situation is different in BLG. Figures 3-(d,e) reproduce the theoretical analysis for a similar BLG sample, such as that measured in Ref.[Yang2018prl, ]. The energy ($\omega/\omega_{c}$) and momentum ($q/k_{F}$) reduced units are adapted for a massive 2DEG-like BLG, but the reduced conductivity scale $\sigma_{MO}/NG_{K}$ is the same. The two panels correspond to the same $N=6$, but different magnetic fields $B=5\;\mathrm{T}$ (panel d) and $B=1\;\mathrm{T}$ (panel e). Contrarily to MLG, the phase velocity $\omega_{ME}/q$ is not magnetic-field independent in this representation, as $\omega_{c}\propto B$ and $k_{F}\propto 1/l_{B}\propto\sqrt{B}$ have different B-dependencies. As a consequence, positioning an identical Doppler line on the two reduced-units plots amounts taking a $v_{bd}\propto\sqrt{B}$. Figs.3-(d,e) show that this criterion also corresponds to a consistent $\sigma_{MO}\sim 10^{-2}NG_{K}$ criterion for the ME-instability, which is met in BLG at ($q,\omega$)-localized ME conductivity peaks. This impedance analysis shows that for BLG, and more generally 2DEGs, the ME-instability and Zener-ILLT, which are basically different, give consistent and similar breakdown velocities in BLG, confirming earlier statement of Ref.[Yang2018prl, ]. Finally, Fig.3-c illustrates the qualitative difference between MLG and BLG in a plot of $v_{bd}(B)$ at a large $n=2.10^{12}\;\mathrm{cm^{-2}}$ (BLG data are reproduced from Fig.2 of Ref.[Yang2018prl, ]), with $v_{ME}^{MLG}\simeq 0.14v_{F}$ (blue line) and $v_{ME}^{BLG}=\hbar/m^{*}l_{B}\sqrt{N}\propto\sqrt{B}$ (red line for $N=5$) [Yang2018prl, ]. Let us note that the magnetic field dependencies $v_{bd}^{MLG}=\mathrm{Cst.}$ and $v_{bd}^{BLG}\propto\sqrt{B}$ merely reflect the energy dependence of the Fermi velocity, $v_{F}^{MLG}(\varepsilon_{L})=\mathrm{Cst.}$ and $v_{F}^{BLG}(\varepsilon_{L})\propto\sqrt{\varepsilon_{L}}$, when taken at the Landau energy $\varepsilon_{L}=\hbar\omega_{c}\propto B$.

Relying on the good mapping of the ME-scenario with experiment, we exploit the RPA calculations further in Supplementary Section III-B and III-C to model breakdown in varied graphene geometries such as graphene in vacuum or semi-infinite hBN embedding, keeping a systematic benchmark of MLG and BLG cases, and assuming the existence of a universal impedance-matching condition. For MLG, we show in Figs.SI-6-(a,b) that screening by the bottom gate in AuS2 (panel a) is equivalent to a semi-infinite, $\epsilon_{r}=100$ dielectric (panel b), meaning that both PHESs correspond to the fully screened conductivity. Effect of interactions, which is maximal for un-gated suspended graphene ($\epsilon_{r}=1$ in panel c), amounts to suppressing the conductivity amplitude below the $\sigma_{MO}\sim 10^{-2}NG_{K}$ ME-instability threshold over most of the ME-spectrum leading to an enhanced breakdown velocity $v_{bd}\simeq 0.5v_{F}$ (white line). Given the impedance matching condition $\sigma_{MO}\sim 10^{-2}NG_{K}$, we conclude that the ME-instability velocity of MLG is a constant, in the range $v_{ME}=[0.14,0.5]\;v_{F}$ that depends on screening. The same analysis is performed for BLG in Figs.SI-8, showing that the large-$q$ PHES sector is to a large extent insensitive to screening, yielding a Zener-like breakdown velocity $v_{ME}^{BLG}=\frac{\hbar}{m^{*}R_{c}}$.

In conclusion, we have shown that bulk quantum Hall breakdown is controlled by the magneto-exciton instability in both MLG and BLG with a threshold $v_{drift}\geq v_{ME}$, which is reminiscent of the Cerenkov effect [Landau1969MIR, ]. More precisely, instability is defined by a universal conductivity criterion $\sigma_{MO}\sim 10^{-2}NG_{K}$. This universal criterion explains the qualitative differences between the massless MLG, and massive BLG. Whereas the BLG-ME instability mimics single-particle ILLT, that of MLG is sensitive to screening by the embedding dielectric and local gates. Screening reduces the breakdown velocity, and gated transistors correspond to the fully screened regime. Both studies promote shot-noise as a sensitive probe of quantum Hall transport, RPA as a relevant theoretical tool to tackle interactions and screening, and high-velocity transport as a sensitive probe of the large-momentum collective excitations, as suggested by Landau [Landau1941pr, ]. Understanding the combined effects of Landau quantization and interactions in the collective modes of the integer quantum Hall effect is a prerequisite before addressing the more challenging case of the fractional regime, where elusive magneto-rotons may come into play. Finally and on a broader scope, let us mention that the magneto-exciton instability is a quantum-Hall-matter light coupling effect, which belongs to a domain of current interest [Appugliese2022science, ].

Supplementary Information

A Supplementary Information is available. It presents a similar analysis of the experimental data on the other devices of the series, as well as an extended discussion on the magneto-optical conductivity spectrum in monolayer and bilayer graphene. In this respect, it focuses on the contrasted role of interactions in these two cases.

Acknowledgements.

AS thanks Prof. C.R. Dean for hospitality and introducing him to the fabrication of high-quality graphene-hBN heterostructures. The research leading to these results has received partial funding from the European Union Horizon 2020 research and innovation program under grant agreement No.881603 ”Graphene Core 3”, and from the French ANR-21-CE24-0025-01 ”ELuSeM”.

Conflict of interest

The authors have no conflict of interest to disclose.

Authors contribution statement

AS, BP and EB conceived the experiment. AS conducted device fabrication and measurements, under the guidance of MR in the early developments. TT and KW have provided the hBN crystals. AS, MOG and BP developed the models and theoretical interpretations. AS, GF, JMB, GM, CV, BP and EB participated to the data analysis. BP wrote the manuscript with assistance of AS and EB, and contributions from the coauthors.

Data availability statement

Data are available on a public Zenodo repository.