Abstract
The most recognizable feature of graphene’s electronic spectrum is its Dirac point, around which interesting phenomena tend to cluster. At low temperatures, the intrinsic behaviour in this regime is often obscured by charge inhomogeneity^{1,2} but thermal excitations can overcome the disorder at elevated temperatures and create an electron–hole plasma of Dirac fermions. The Dirac plasma has been found to exhibit unusual properties, including quantumcritical scattering^{3,4,5} and hydrodynamic flow^{6,7,8}. However, little is known about the plasma’s behaviour in magnetic fields. Here we report magnetotransport in this quantumcritical regime. In low fields, the plasma exhibits giant parabolic magnetoresistivity reaching more than 100 per cent in a magnetic field of 0.1 tesla at room temperature. This is ordersofmagnitude higher than magnetoresistivity found in any other system at such temperatures. We show that this behaviour is unique to monolayer graphene, being underpinned by its massless spectrum and ultrahigh mobility, despite frequent (Planckian limit) scattering^{3,4,5,9,10,11,12,13,14}. With the onset of Landau quantization in a magnetic field of a few tesla, where the electron–hole plasma resides entirely on the zeroth Landau level, giant linear magnetoresistivity emerges. It is nearly independent of temperature and can be suppressed by proximity screening^{15}, indicating a manybody origin. Clear parallels with magnetotransport in strange metals^{12,13,14} and socalled quantum linear magnetoresistance predicted for Weyl metals^{16} offer an interesting opportunity to further explore relevant physics using this well defined quantumcritical twodimensional system.
Similar content being viewed by others
Main
A variety of mechanisms—both intrinsic and extrinsic—can lead to large magnetoresistance (MR) in metallic systems. The quest to understand these mechanisms has continued for longer than a century but many gaps still remain, which is especially obvious for the MR reported in newcomer materials such as various Dirac and Weyl systems^{17,18,19,20,21,22,23,24,25}, strange metals^{12,13,14} and so on. The history and current status of the research field are briefly reviewed in Methods. Whichever mechanism is behind a particular MR behaviour, it always relies on bending of electron trajectories by a magnetic field B and, accordingly, high carrier mobility μ is an essential prerequisite for the observation of large MR. Colossal MR (reaching about 10^{6}% in a magnetic field of 10 T) was observed in a number of highμ systems at liquidhelium temperatures^{17,18,19,20,21,22,23,24,25}. However, because mobility decreases with increasing temperature T, this usually results in only a tiny MR above liquidnitrogen temperatures. Those few materials in which carriers remain highly mobile at room temperature (such as doped graphene and indium antimonide)^{26,27,28} are all noncompensated systems and, in agreement with the classical theory of normal metals^{29}, their longitudinal resistivity ρ_{xx} saturates in high B, leading again to little MR. Only the presence of extended defects^{30,31,32} or a special design of fourprobe devices^{26,33} that creates strongly nonuniform current flows can lead to large—but extrinsic—MR (Methods).
As shown below, thermally excited charge carriers in monolayer graphene (MLG) at the neutrality point (NP) exhibit an anomalously high μ exceeding 100,000 cm^{2} V^{−1} s^{−1} at room temperature, despite the fact that the system is strongly interacting^{3,4,5,6,7,8} and the electron–hole (e–h) scattering time τ_{P} is ultimately short, being limited by the uncertainty principle τ_{P}^{−1} ≈ Ck_{B}T/h where k_{B} and h are the Boltzmann and Planck constants, respectively, and C ≈ 1 is the interaction constant^{3,4,5,9,10,11,12}. Importantly, unlike any known system with high μ at room temperature, the Dirac plasma is compensated (charge neutral) so that its zero Hall response allows nonsaturating MR^{29} whereas the high μ makes it colossal. To emphasize how unique magnetoresistivity ρ_{xx}(B) of the Dirac plasma is, we provide its comparison with graphite (multilayer graphene^{34}) and chargeneutral bilayer graphene (BLG), another quantumcritical system exhibiting Planckian scattering but having massive charge carriers with modest mobilities^{9,10}.
Giant MR in nonquantizing fields
Our primary devices were multiterminal Hall bars made from MLG encapsulated in hexagonal boron nitride (hBN; Fig. 1a). We have studied several such devices and focus here on two of them (devices D1 and D2) showing representative behaviour. At low T, their mobilities exceed 10^{6} cm^{2} V^{−1} s^{−1} at characteristic carrier densities of about 10^{11} cm^{−2}, being limited by edge scattering despite the devices’ size being more than 10 μm. The typical behaviour of ρ_{xx} as a function of the gateinduced density n is shown in Fig. 1b. If the same curves are replotted on a log scale (Fig. 1c), it becomes clear that ρ_{xx} responds to gate voltage only above a certain n dependent on T. This behaviour is commonly quantified as shown in Fig. 1c,d where δn marks the gateinduced density that leads to notable changes in ρ_{xx}. At high T, the peak in ρ_{xx}(n) broadens because of thermally excited electrons and holes in concentrations n_{th} = (2π^{3}/3)(k_{B}T/hv_{F})^{2}, where v_{F} is the Fermi velocity (Methods). The extracted δn evolves proportionally to T^{2} as expected (Fig. 1d) and its absolute value is about 0.5n_{th}, which means that to make changes in ρ_{xx} visible on such log plots, gateinduced carriers are required in concentrations of about 50% of the thermal concentration. At low T, δn saturates typically at about 10^{10} cm^{−2} because of residual charge inhomogeneity (e–h puddles of submicrometre scale)^{1,2}. Below we focus on T > 100 K where thermal excitations totally overwhelm the residual δn.
The Dirac plasma’s response to small fields is shown in Fig. 1e. One can see that the longitudinal resistivity at the NP ρ_{NP} ≡ ρ_{xx}(n = 0) increases proportionally to B^{2}, as expected from the classical Drude model^{29}. However, the changes in ρ_{NP} are unexpectedly large for this T range. Indeed, if we consider 0.1 T as a characteristic field relevant for magneticsensor applications, then the relative magnetoresistivity Δ = [ρ_{xx}(B) – ρ_{xx}(0)]/ρ_{xx}(0) reaches about 110% at 300 K near the NP (Fig. 1e) and increases by a factor of 3–4 at 200 K. For comparison, Δ in normal metals rarely exceeds a small fraction of 1% above liquidnitrogen temperatures. Even highquality encapsulated bilayer, fewlayer and multilayer graphene exhibit Δ(0.1 T) reaching only about 1% at room temperature (Methods). Also, the renowned giant MR based on spin flipping in ferromagnetic multilayers yields changes in resistance that are one to two orders of magnitude smaller^{35,36} than those observed here for the Dirac plasma.
Further characterization of the e–h plasma is provided in Fig. 2. It shows that Δ rapidly diminishes away from the NP at characteristic densities n ≈ n_{th} (Fig. 2a). This is expected^{29} because, for noncompensated systems, changes in ρ_{xx}(B) should be small and saturate, if Hall resistivity ρ_{xy} > ρ_{xx} (Methods). In contrast, for chargeneutral systems (zero ρ_{xy}), the Drude model predicts nonsaturating magnetoresistivity such that Δ = μ_{B}^{2}B^{2} where μ_{B} is the mobility in nonquantizing magnetic fields (Methods). The latter expression describes well the behaviour observed in small B (Fig. 2b). Figure 2c shows the extracted μ_{B} as a function of T. The mobility exceeds 100,000 cm^{2} V^{−1} s^{−1} at room temperature and grows above 300,000 cm^{2} V^{−1} s^{−1} below 150 K. Although high μ values are well known for the Fermiliquid regime in doped graphene, it is unexpected that the mobility remains high in the presence of Planckian scattering, characteristic of the quantumcritical regime in neutral graphene^{5,6}. For comparison, bilayer and multilayer graphene also exhibit very high mobilities at liquidhelium temperature, but their ρ_{NP}(B) are practically flat at elevated T (Fig. 1e), yielding μ_{B} of only about 10,000 cm^{2} V^{−1} s^{−1} at 300 K (Extended Data Figs. 2 and 3). The marked difference in electronic quality between the e–h plasmas of relativistic and nonrelativistic fermions (in MLG and BLG, respectively) stems from the small effective mass m characteristic of the Dirac spectrum (μ ∝ m^{−1}) and its low density of states, which reduces the efficiency of electron scattering (Methods). It is noted, however, that the Dirac spectrum on its own is insufficient for achieving giant values of Δ, and the high quality of MLG devices is paramount. This is emphasized by Extended Data Fig. 8, which shows magnetotransport for nonencapsulated graphene on a silicon oxide substrate. Such lowquality MLG exhibits three orders of magnitude smaller MR.
It is instructive to compare the found μ_{B} with the zerofield mobility μ_{0}. The latter can be evaluated using the standard Drude formula ρ_{NP}^{−1} = 2n_{th}eμ_{0}, where e is the electron charge and the factor 2 accounts for equal concentrations of electrons and holes at the NP. Figure 2d shows that ρ_{NP} quickly decreases with increasing T from liquidhelium temperatures to about 100 K but, as the Dirac plasma gets established (n_{th} >> residual δn), ρ_{NP} becomes almost T independent with a constant value of about 1 kΩ above 150 K (also, inset of Fig. 3b and Extended Data Fig. 1). The saturating behaviour of ρ_{NP} is attributed to the onset of the quantumcritical regime in which the scattering is dominated by the Planckian frequency, τ_{P}^{−1}. Indeed, ρ_{NP} ≈ 1 kΩ yields C ≈ 0.7 close to unity, as expected^{3,4,5,9,10,11,12}. This analysis also agrees with that of the quantumcritical behaviour reported for BLG^{9,10} (Methods) and conclusions about MLG from other measurements^{5}.
Figure 2c shows that μ_{0} evolves proportionally to 1/T ^{2}, as expected for Planckian systems with a Dirac spectrum (Methods). Surprisingly, μ_{0} is two to three times smaller than μ_{B}. As shown in Methods, this happens because μ_{B} is less sensitive than μ_{0} to the dominating e–h scattering. Qualitatively, the difference can be understood as arising from different relative motions of electrons and holes in zero and finite B. In zero B, the electric field forces electrons and holes to move in opposite directions so that e–h collisions are efficient in impeding a current flow. In contrast, cyclotron motion causes a drift of both electrons and holes in the same direction. Therefore, e–h collisions do not affect the Hall currents responsible for magnetoresistivity. This explanation is further substantiated by our measurements using screened graphene devices (encapsulated MLG with metallic gates placed at a distance of about 1–3 nm)^{15}. The screening is found to suppress Coulomb scattering, which results in smaller C and, therefore, higher μ_{0} (Extended Data Fig. 4a). However, the same screening has little effect on ρ_{NP}(B) and hence μ_{B} (Extended Data Fig. 4b), in agreement with theory. This consideration is equally applicable for e–h plasma of massive fermions and, indeed, a similar difference between μ_{0} and μ_{B} is observed for neutral BLG (Extended Data Fig. 2). The above analysis allows us to conclude that the anomalously large MR in low B arises owing to ultrahigh mobility of Dirac fermions, combined with ineffectiveness of e–h scattering in suppressing Hall currents.
Strange linear MR in the extreme quantum limit
In high B, magnetotransport in the Dirac plasma exhibits profound changes such that, above a few tesla, ρ_{NP}(B) evolves from being parabolic to linear (Fig. 3 and Extended Data Fig. 5). Slopes of this linear MR are found to be similar for all the studied devices (Fig. 3b, inset) and almost independent of T. The crossover between parabolic and linear dependences is marked by a flattened section on the curves which appears at T below 200 K. We attribute the flattening to the onset of Landau quantization (Extended Data Fig. 6). This attribution also agrees with the fact that at B ≈ 3–5 T, the main cyclotron gap between the zeroth and first Landau levels (LLs) reaches about 800 K, notably exceeding the thermal smearing k_{B}T. As for the MR magnitude, Δ reaches about 10^{4}% at 10 T and, despite the linear (slower than parabolic) dependence in quantizing fields, this is again among the highest for roomtemperature experiments^{32}. Comparison with multilayer and lowquality graphene (Extended Data Figs. 3 and 8) shows the importance of both the Dirac spectrum and the electronic quality for such a giant MR response. Another notable feature of magnetotransport in the plasma of the zeroth LL is that ρ_{NP} at a given B increases with increasing T (Fig. 3a and Extended Data Fig. 5). This contradicts the orthodox MR behaviour observed in all other systems, which results in lower Δ at higher T because of increased scattering^{29} (Methods). To shed light on strange magnetotransport, we have also tested how ρ_{NP}(B) is affected by proximity screening. Although the parabolic dependence of ρ_{NP} in low B was practically unaffected (as discussed above), the screening greatly suppressed MR in quantizing B (Fig. 3b). The linear slopes of ρ_{NP}(B) decrease from 5–8 kΩ T^{−1} in our primary devices to 1–3 kΩ T^{−1} in those with screening (inset of Fig. 3b), implying that magnetotransport in the zeroth LL depends on Coulomb interactions.
In discussing the highB behaviour, we first note that the previously reported linear MR can in most cases be attributed to complex current flows that become increasingly nonuniform as ρ_{xy} ∝ B increases (Methods). The involved mechanisms are based on either the spatial inhomogeneity or the presence of edges. To check for possible edge effects in our case, we have studied Corbino disks fabricated from encapsulated MLG and found very similar ρ_{NP}(B) (Extended Data Fig. 7). Thus, for our zerothLL plasma with zero ρ_{xy}, those extrinsic mechanisms can be ruled out (Methods). It may also be tempting to evoke Abrikosov’s linear MR^{16} predicted to occur in threedimensional (3D) semimetals with Diraclike spectra in the extreme quantum limit. However, the Born approximation used in the 3D model cannot be justified for twodimensional (2D) transport in a smooth background potential^{37} because charge carriers remain localized within electron and hole puddles.
For the lack of a theory suitable to describe the observed linear MR, we employ the simple Drude model by considering cyclotronorbit centres as quasiparticles that circle along equipotential contours and, also, diffuse between them owing to electron scattering. The density of such quasiparticles is determined by the capacity of the LL, n_{LL} = 2B/ϕ_{0} >> n_{th}, where ϕ_{0} = h/e. For charge neutrality, the Drude model yields ρ_{NP}(B) ≈ ρμ^{2}B^{2} (Methods), where n and μ in the standard expression ρ = 1/neμ should be substituted with n_{LL} and μ_{Q}, respectively, to reflect the density and mobility for the plasma of the zeroth LL. This leads to
The linearity in B arises from the fact that the B^{2} dependence inherent for compensated semimetals is moderated by the linear increase in the carrier density in the zeroth LL. Next, to estimate μ_{Q}, we assume that Planckian scattering moves quasiparticles by a typical distance \({\ell }\) between equipotentials, resulting in the diffusion coefficient \(D\approx {\ell }^{2}/{\tau }_{{\rm{p}}}={{v}_{{\rm{T}}}}^{2}{\tau }_{{\rm{p}}}\) with the corresponding thermal velocity \({v}_{{\rm{T}}}\equiv {\ell }\,/{\tau }_{{\rm{p}}}\). Diffusion within individual puddles leaves carriers localized inside. Only if a quasiparticle covers a distance of approximately ξ between neighbouring puddles, those processes contribute to macroscopic currents along the electric field and, hence, global conductivity. Accordingly, the timescale relevant for electron transport in the zeroth LL is given by τ_{tr} ≈ ξ^{2}/D >> τ_{p} and the corresponding diffusion coefficient can be written as D_{tr} = v_{T}^{2}τ_{tr} ≈ v_{T}^{2}ξ^{2}/D = ξ^{2}/τ_{p}. Then, using the Einstein–Smoluchowski equation, we find the transport mobility μ_{Q} = eD_{tr}/k_{B}T ≈ eξ^{2}/k_{B}Tτ_{p} = Ceξ^{2}/h, where both v_{T} and \({\ell }\) fell out from the final expression. This result suggests that the MR of the zero LL is linear in B and independent of T, as observed experimentally, and may also explain the suppression by proximity screening as smaller C result in lower μ_{Q}. Furthermore, equation (1) can be rewritten as
where ℓ_{B} is the magnetic length. Equation (2) closely resembles the result of a formal extension of Abrikosov’s model into the 2D case^{37}. Although the above consideration catches the main physics and qualitatively agrees with our observations, further work is required to develop a microscopic theory of magnetotransport in the 2D Boltzmann plasma at the zeroth LL.
Outlook
The Dirac plasma in graphene exhibits the one of the highest MRs observed above liquidnitrogen temperatures in both low and high fields. In low B, only ferromagnetic devices employing spin tunnelling^{38} or the use of fourprobe geometry^{26,33} allow stronger electronic response to magnetic fields. In contrast to the latter phenomena, the giant MR of graphene stems from its magnetoresistivity ρ_{xx}(B). In quantizing fields, graphene experiences a system transformation, becoming an e–h plasma residing in the zeroth LL. Our observations are also relevant to the physics of strange metals that exhibit Planckian scattering. Strange metals display the renowned linear T dependence of their resistivity, in obvious contrast to our case. However, this difference arises only because strange metals have a fixed carrier density whereas the carrier density and effective mass in the Dirac plasma increase with T, leading to the constant ρ_{NP}. Moreover, strange metals also exhibit linear MR that is weakly T dependent. This MR remains unexplained, although a recent ansatz^{13,14} suggests that in Planckian systems, τ^{−1} should be defined by the largest relevant energy scale, either k_{B}T or some magneticfieldinduced energy proportional to B. The ansatz does not seem work for the Dirac plasma because the only relevant and sufficiently large magnetic energy is the cyclotron gap. It evolves as B^{1/2} rather than linearly in B. Notwithstanding any differences, Planckian systems in high fields remain poorly studied, and graphene’s Dirac plasma offers a model system to understand the relevant physics. The possibility to modify magnetotransport by tuning electron–electron interactions using proximity screening is especially appealing in this context.
Methods
Brief history of linear MR
Studies of the electrical response of metals to magnetic fields go back to experiments by Lord Kelvin and Edwin Hall over oneandahalf centuries ago^{39,40}. Although the subject continued to attract sporadic attention during the following decades (see, for example, ref. ^{41}), the first systematic study of MR phenomena is usually credited to Pyotr Kapitsa. In 1928–1929, he reported highfield studies of MR in 37 different materials^{42,43}. This research brought up two major findings. First, some materials (for example, bismuth, arsenic, antimony and graphite) were found to exhibit MR exceeding 100% in a magnetic field of 30 T at room temperature, much higher than the others in that study. So large MR could not be explained by contemporary theories. Second, despite different absolute values of MR, all the studied materials followed a universal B dependence. In small fields, it was always parabolic, in agreement with the already accepted understanding that cyclotron motion of currentcarrying electrons should bend their trajectories and, hence, increase resistivity. However, in fields above several tesla, MR was found to increase linearly, which was unexpected.
The first puzzle of large MR values was solved relatively quickly, owing to the development of the band theory. Most of the materials exhibiting large roomtemperature MR in Kapitsa’s experiments appeared to be semimetals so that the electric current was carried by both electrons and holes. It is now well understood that the reduced Hall effect in this case leads to ρ_{xx} evolving in high fields approximately as 1/σ_{xx}, where σ_{xx} is the longitudinal conductivity. This is in contrast to the case of one type of charge carriers where ρ_{xx} ≈ σ_{xx }ρ_{xy}^{2} (‘Drude model for chargeneutral graphene’ below). The second puzzle of linear MR has attracted numerous theories and explanations. In general, there are several mechanisms that can cause linear MR and, even today, its observation often leads to controversies because it is difficult to pinpoint the exact origin.
One of the first mechanisms causing linear MR was proposed by Lifshitz and Peschanskii^{44}. In 1959, they considered magnetotransport in polycrystalline metals with open Fermi surfaces. For certain orientations of the magnetic field with respect to crystallographic axes, such metals flaunt open cyclotron orbits that result in nonsaturating MR proportional to B^{2} (refs. ^{45,46}). However, this quadratic behaviour occurs within only a narrow interval of angles, which decreases proportionally to B^{−1}. For the other angles, cyclotron orbits remain closed, and MR attributable to them saturates in high B. Averaging over all angles for polycrystalline samples resulted in linear MR, and this result helped to explain many—but not all—observations in the literature. Those ideas were further developed by Dreizin and Dykhne^{47} who obtained MR proportional to B^{4/3} and MR proportional to B^{2/3}, depending on whether a metal with an open Fermi surface was compensated or not, respectively. Moreover, the authors presented a magnetotransport theory for not only polycrystalline but also inhomogeneous conducting media. Depending on the Fermi surface and compensation between charge carriers, various powers of B could be obtained including, for example, linear MR in compensated semimetals with 2D disorder^{47}.
The MR theory relying on materials’ inhomogeneity was expanded both theoretically and experimentally in the 1970s and 1980s. It was shown that macroscopic strain^{48}, voids^{49,50,51,52} and thickness variations^{53,54} could lead to linear MR in high B (μB >> 1). The next step was taken in 2003 by Parish and Littlewood who considered the case of very strong inhomogeneity that could not be described by the earlier theories^{55}. Using a random 2D resistance network, they obtained linear MR that starts from small magnetic fields (μB < 1) and could explain the behaviour observed in some disordered semiconductors^{55}. The fundamental reason for MR in all the cases involving inhomogeneous media is the following. In the presence of regions with different magnetotransport coefficients, the arising Hall voltages (large for μB >> 1) necessitate substantial changes in the electric current distribution to satisfy boundary conditions at interfaces between different regions. As a result, the electric current becomes increasingly inhomogeneous, being squeezed into narrow streams near the interfaces. This current inhomogeneity increases the effective resistance of the medium^{53,54}.
A different mechanism was suggested by Abrikosov^{16,56,57}. He pointed out that some materials exhibiting linear MR were neither polycrystalline nor inhomogeneous but single crystals with closed Fermi surfaces including graphite, bismuth and other materials^{58,59}. To explain these observations, Abrikosov considered a Weyl (3D Diraclike) spectrum so that, in quantizing B, all charge carriers collapsed onto the lowest (zeroth) LL. Assuming a scattering potential caused by screened charged impurities, linear MR was predicted in this case. Because of the essential role played by Landau quantization, the effect was called quantum linear MR^{16,56,57}. The Abrikosov mechanism attracted considerable interest and was invoked as an explanation for many experiments^{60,61}, even though the concerned materials often poorly matched the assumptions required by the theory (including being 2D rather than 3D systems). Unfortunately, Abrikosov provided no explanation for the physics behind his theory and only recently^{37} it has been shown that his analysis is equivalent to calculations of diffusion of cyclotronorbit centres in an electrostatic potential. This conceptual overlap requires mentioning of the earlier theories by Kubo and Ando for diffusion of cyclotronorbit centres^{62,63}. Furthermore, within the selfconsistent Bohr approximation, linear MR was shown to appear in the 2D case for strongly screened charged impurities whereas, for shortrange scattering, MR becomes sublinear^{64}. The formal extension of Abrikosov’s theory into two dimensions also leads to linear MR^{37}.
Finally, two other mechanisms that result in linear MR have to be mentioned. First, Alekseev with colleagues showed that e–h annihilation at the edges of 2D semimetals could lead to linear MR^{65,66}. This mechanism can be ruled in or out by comparing magnetotransport in Hallbar and Corbinodisk devices, as done in our work. Second, socalled strange metals often exhibit resistivity that increases linearly not only with temperature but also with magnetic field^{13,14}. Although such linear MR does not follow from the socalled holographic approach^{11}, it was suggested^{13,14,67} that the quantumcritical scattering rate τ^{−1} ≈ E/h could be controlled by the maximum relevant energy E in the uncertainty equation, that is, by either k_{B}T or μ_{Bohr }B where μ_{Bohr} is the Bohr magneton. This could then explain both (T and B) linear dependences in strange metals. It is also worth mentioning that linear MR was recently reported in two other 2D strongly interacting systems, namely, twisted tungsten diselenide^{68} and magicangle graphene^{69}. It was suggested that the MR had the same origin as in strange metals.
Earlier studies of MR in graphene and Diractype materials
Over the past decade, there have been numerous studies of magnetotransport using newly available materials such as graphene (see, for example, refs. ^{30,31,32,33,34,61,70,71,72,73,74}), topological insulators (see, for example, refs. ^{75,76,77}) and highmobility Dirac and Weyl semimetals (see, for example, refs. ^{17,18,19,20,21,22,23,24,25}). Graphene has attracted particular attention as a promising material for magneticfield sensors owing to its high μ at room temperature. The first generation of graphene devices (graphene placed on oxidized silicon and socalled epitaxial graphene) exhibited relatively low μ, and their MR was also relatively low, reaching only about 100% in fields above 10 T (refs. ^{30,31,32,61,70,71,72}; ‘MR of lowmobility graphene’ below). The MR typically originated from charge inhomogeneity and other disorder, although some reports suggested^{61} the observation of Abrikosov’s linear MR in doped multilayer graphene. Later research ruled out this explanation, arguing that the observed linear MR originated from a polycrystalline disorder^{31,55}.
The next generation of graphene devices using encapsulation with hBN exhibited exceptional electronic quality^{2,78}. So far, magnetotransport in grapheneonhBN devices has been studied at elevated T only for fewlayer graphene^{34} and MLG away from the NP^{33}. Few and multilayer graphene (graphite) exhibit a relatively low μ at elevated temperatures. This results in small quadratic MR in low B and also limits MR in high fields^{32,34}, in agreement with our results in ‘MR in multilayer graphene’ below. Doped highμ graphene exhibits saturating magnetoresistivity and its magnitude is small, as expected. It is noted, however, that if one uses a geometry that instigates a nonuniform current flow, it is possible to enhance the apparent MR in fourprobe measurements, for example, using the socalled extraordinary MR configuration^{26,33}. In such a geometry, the central part of a MLG device is replaced with a highly conducting metal (for example, gold). In zero B, the current mainly flows through the metal, despite being injected into graphene. The magnetic field curves the current trajectories and forces charge carriers to move through graphene, which is much more resistive than gold films. Accordingly, the apparent fourprobe MR could reach extraordinary values of about 10^{7}% at 9 T and room temperature. This is comparable to typical changes in Hall voltage that also require a fourprobe geometry. It is noted that this extraordinary MR is not an intrinsic property of a material and, accordingly, translates into only modest changes for any twoprobe measurements. Until now, no studies of magnetotransport at elevated T have been reported for chargeneutral MLG with high μ.
For completeness, let us mention extensive magnetotransport studies of 3D counterparts of graphene, which are different topological insulators, Dirac and Weyl semimetals, and other clean semimetals such as tungsten telluride (also suggested to be a Weyl semimetal^{79}). Many of them showed huge MR, which in some cases exceeded 10^{6}% at liquidhelium temperatures^{17,18,19,20,21,22,25}. Such colossal values were attributed to the high mobility of charge carriers in these materials (μ reaching above 10^{6} cm^{2} V^{−1} s^{−1} at 4 K, similar to encapsulated graphene). However, the mobility rapidly decayed with increasing T, which resulted only in a tiny lowB MR at elevated T. This is not the case for MLG that exhibits high μ at room temperature even at the NP, which results in the colossal quadratic MR in low B, as reported in this work.
Drude model for chargeneutral graphene
To evaluate the magnetotransport properties of our devices, we have used the standard twocarrier model for electrons and holes, which allows the longitudinal and Hall conductivities to be written as^{80}
where n_{e(h)} is the carrier density of electrons (holes) and μ_{e(h)} is the corresponding mobility. The relative magnetoresistivity is defined as
where \({\rho }_{xx}(B)=\frac{{\sigma }_{xx}(B)}{{\sigma }_{xy}^{2}(B)+{\sigma }_{xx}^{2}(B)}\). For the case of a compensated semimetal with n_{e} = n_{h} and equal mobilities for electrons and holes (μ_{e} = μ_{h} = μ), the above equations yield
This expression was used in this work to extract the magnetotransport mobility μ_{B} from parabolic dependences of ρ_{NP}(B) in small B.
Our analysis of the ρ_{xx}(n)peak broadening and the zerofield mobility μ_{0} (see main text) have relied on theoretical expressions for the density of thermally excited electrons at the NP, n_{th}. For MLG, this electron density is given by
where \(\hbar \) = h/2π is the reduced Planck constant, f(E) is the FermiDirac distribution function, DOS is the density of states of MLG and x = E/k_{B}T. Holes are excited with the same density. Thermally excited Dirac fermions with a typical energy k_{B}T can be assigned with the effective mass m* that is also T dependent
This expression can be obtained from the Boltzmann equations calculating the response of chargeneutral graphene to electric field and enforcing the resulting conductivity into a Drudelike form. It is noted that the above mass is proportional to the typical energy (thermal energy k_{B}T of electrons and holes in the Dirac plasma) divided by their velocity squared, as expected for ultrarelativistic particles.
Using the same approach for BLG, we obtain its density of thermally exited electrons
The above expressions for n_{th} and m* have been used to evaluate conductivities of both chargeneutral MLG and BLG based on the standard Drudelike expression
where τ is the scattering time, and the factor of 2 accounts for equal densities of thermally excited electrons and holes.
Additional examples of magnetotransport measurements for MLG
Several (more than ten) monolayer devices (Hall bars and Corbino disks) were studied during the course of this work. To indicate variations in their magnetotransport behaviour, below we present measurements for another Hall bar (device D2) exhibiting notably higher remnant δn at low T. Its resistivity ρ_{NP}(T ) at the NP is plotted in Extended Data Fig. 1a. Similar to device D1 (Fig. 2d), ρ_{NP} of device D2 decreases with T and saturates above 200 K. In this device, the saturation occurs at higher T than in device D1 because of stronger inhomogeneity (compare Fig. 1d and Extended Data Fig. 1b). Despite an orderofmagnitude different inhomogeneities, both devices exhibit practically the same saturation value, ρ_{NP} ≈ 1 kΩ. The same was valid for the other MLG devices.
As discussed in the main text, we attribute the Tindependent ρ_{NP} in MLG to the entry of the Dirac plasma into the quantumcritical regime^{3,4,5,9,10,11,12}. In this regime, the electron scattering time is determined by Heisenberg’s uncertainty principle, \({\tau }_{{\rm{p}}}^{1}=C\frac{{k}_{{\rm{B}}}T}{h}\) where C is the interaction constant of about unity and depends on screening^{3,4,11,12}. By plugging this scattering rate into equation (10) and using the effective mass from equation (8) and the carrier density given by equation (7), we obtain the quantumcritical resistivity
which is independent of T. The observed ρ_{NP} ≈ 1 kΩ yields the interaction constant C ≈ 0.7, close to unity, as expected for Planckianlimit scattering^{3,4,5,9,10,11,12}.
As for the MR behaviour of device D2, Extended Data Fig. 1c shows that ρ_{NP} is parabolic in low B, similar to the case of device D1 in Fig. 1. The absolute value of Δ for device D2 is also similar, albeit slightly smaller, reaching 90% at 0.1 T and room temperature. The 20% reduction can be attributed to the lower electronic quality and homogeneity of device D2. Extended Data Fig. 1d shows zerofield and magnetotransport mobilities for device D2, which were extracted using the same approach as described in the main text. Both mobilities are slightly lower than those in Fig. 2. Nonetheless, at room temperature, μ_{B} in device D2 still exceeds 100,000 cm^{2} V^{−1} s^{−1}. Overall, the results presented in Extended Data Fig. 1 corroborate our conclusions that the Dirac plasma flaunts exceptionally high carrier mobility at elevated T, with no analogues among compensated metallic systems. The figure also reiterates the considerable differences between μ_{B} and μ_{0}, which were discussed in the main text and explained in ‘Difference between zerofield and magnetotransport mobilities’ below.
Electron–hole plasma in BLG
To emphasize how unique the Dirac plasma in MLG is, let us compare its magnetotransport properties with those of the closest electronic analogue, an e–h plasma at the NP in BLG. To this end, we fabricated and studied BLG devices that were also encapsulated in hBN to achieve high μ. They were doublegated and shaped into the standard Hall bars. At liquidhelium temperatures and away from the NP, the devices exhibited ballistic transport across their entire widths of about 10 μm. This was observed directly using bend resistance measurements^{81}. The doublegating was required to tune the carrier density to the NP while maintaining zero bias between the two graphene layers. The latter ensured that no gap opened at the NP^{82}, which otherwise would complicate the comparison^{10}.
The typical behaviour of BLG’s resistivity in zero B is shown in Extended Data Fig. 2a. Similar to MLG (Fig. 2d and Extended Data Fig. 1a), ρ_{NP}(B = 0) of chargeneutral BLG reaches a few kiloohms at liquidhelium temperatures, but rapidly decreases to about 1 kΩ at higher T and becomes T independent above 50 K (Extended Data Fig. 2b). Such behaviour of highquality BLG has already been reported recently, and constant ρ_{NP} was attributed to the e–h plasma entering the quantumcritical regime^{9,10}. Indeed, plugging the quantumcritical scattering rate \({\tau }_{{\rm{p}}}^{1}=C\frac{{k}_{{\rm{B}}}T}{h}\) into equation (10) and using the thermally excited density from equation (9), we obtain the resistivity for the e–h plasma in BLG
The T independent value of ρ_{NP} stems from the fact that both n_{th} and scattering frequency \({\tau }_{{\rm{p}}}^{1}\) evolve linearly with T. Equation (12) differs from equation (11) for MLG by only a factor of 2. From the data in Extended Data Fig. 2, we obtain C ≈ 1.4, close to unity as expected and in agreement with the previous reports^{9,10}. This value is two times larger than C for the Dirac plasma in MLG. We are unaware of any theory that would allow quantitative comparison between C in the two graphene systems. Nonetheless, the smaller value of the interaction constant in MLG compared with BLG could probably be understood as owing to the lower density of states in the Dirac spectrum.
In addition, we analysed δn(T ) for our BLG devices using the same approach as described for MLG in the main text. Above 50 K, δn in Extended Data Fig. 2c exceeds the remnant charge inhomogeneity (in the limit of low T ) by a few times, which ensures that the smearing of the peak in ρ_{xx} at T > 100 K was dominated by e–h excitations. Extended Data Fig. 2c also shows that δn in BLG increased linearly with T, in agreement with equation (9) and qualitatively different from the quadratic behaviour of δn(T ) in MLG (equation (7) and Fig. 1d). Using the usually assumed value m* ≈ 0.03 m_{e} for BLG (where m_{e} is the free electron mass), we find δn ≈ 0.5 n_{th}, similar to the case of MLG as discussed in the main text.
The response of BLG’s e–h plasma to small B is shown in Extended Data Fig. 2d. Similar to the case of MLG, Δ evolves proportionally to B^{2} but its absolute value is two orders of magnitude smaller than that in MLG, reaching only 1.5% at 0.1 T at room temperature. For completeness, we have evaluated the mobilities for the compensated e–h plasma in BLG, using the same approach as in the main text. Both magnetotransport and zerofield mobilities (μ_{B} and μ_{0}, respectively) are plotted in Extended Data Fig. 2e. They are found to be an order of magnitude lower than those for the Dirac plasma, which is the underlying reason behind the twoordersofmagnitude smaller lowB MR in BLG compared with MLG (Δ ∝ μ^{2}). It is noted that μ_{0} for BLG is approximately two times lower than μ_{B} (Extended Data Fig. 2e), similar to the case of MLG in Fig. 2c. The difference between μ_{0} and μ_{B} is again attributed to electrons and holes moving against and along each other for longitudinal and Hall flows, respectively, as discussed in the main text and detailed in ‘Difference between zerofield and magnetotransport mobilities’ below.
Our experiments show that charge carriers in the Dirac plasma are several times more mobile than electrons and holes at the NP in BLG. The reason for the exceptionally high mobility in the Dirac plasma is twofold. First, the scattering rate \({\tau }_{{\rm{p}}}^{1}\propto C\) is approximately two times lower in MLG compared with BLG, as discussed above. Second, the effective mass for Dirac fermions at room temperature can be estimated from equation (8) as m* ≈ 0.01 m_{e}, which is three times lower than the effective mass of charge carriers in BLG. Taken together, this suggests that the zerofield mobility \({\mu }_{0}=e\tau /{m}^{* }\) for the Dirac plasma should be a factor of 6 higher than that for BLG’s e–h plasma, in qualitative agreement with the experiment (compare Extended Data Figs. 1d and 2e).
Magnetotransport in multilayer graphene
Another electronic system with highmobility charge carriers at room temperature is multilayer graphene (thin films of graphite). The material is an intrinsic semimetal with electrons and holes being present in approximately the same concentrations^{83}. It is instructive to compare the magnetotransport properties of this nearly compensated semimetal with those of the Dirac plasma.
Our graphite devices were several nanometres thick (10–20 graphene layers) and shaped into Hall bars. To preserve the high electronic quality, the multilayer films were again encapsulated with hBN. Measurements for one of the devices are shown in Extended Data Fig. 3. Graphite’s magnetoresistivity was found to increase quadratically in fields below 1 T. At room temperature, Δ was about 1.4% at 0.1 T, similar to the case of BLG and two orders of magnitude smaller than the MR of the Dirac plasma. Above 1 T, graphite exhibited notable deviations from the parabolic dependence bending towards a lower power and becoming practically linear in B at low T and above a few tesla. Roomtemperature Δ reaches 80% and 3,500% at 1 T and 9 T, respectively, in agreement with a previous report for fewlayer graphene^{34}. Although MLG exhibits a few times larger Δ in high B, it is possible that the linear MR in graphite (first reported a century ago^{42,43} and still not fully understood; see ‘Brief history of linear MR’) has the same origin as in MLG. This possibility requires further investigation because graphite’s electronic spectrum is complicated and, also, strongly evolves with magnetic field^{83}.
To evaluate the magnetotransport mobility μ_{B} in graphite, we used the same approach as for MLG and BLG. The results are plotted in Extended Data Fig. 3b. At room temperature, μ_{B} for the e–h system in graphite was found to be about 10,000 cm^{2} V^{−1} s^{−1}, that is, a factor of more than 10 lower than that for the Dirac plasma in MLG (Fig. 2с) but close to μ_{B} found for the e–h plasma in BLG (Extended Data Fig. 2e). This is perhaps not surprising as electronically, graphite is often considered as a stack of graphene bilayers. The provided comparison of graphene with its bilayers and multilayers highlights the unique nature of the Dirac plasma and its anomalously high mobility that results in the giant MR response, especially in low B. It is noted that μ_{B} for multilayer graphene can be extracted more accurately, using both Hall and longitudinal measurements, which does not require the used assumption of e–h symmetry at the NP. The latter analysis^{83} yields practically the same μ_{B} as our intentionally simplified approach.
Difference between zerofield and magnetotransport mobilities
Magnetotransport in graphene’s Dirac plasma was first analysed by Müller and Sachdev^{84} and later by Narozhny with colleagues^{85,86}. Below we provide analogous calculations, for completeness and to simplify our evaluation of the magnetoresistivity observed experimentally.
In the presence of electric E and magnetic B fields, the Boltzmann equations for electrons and holes at the NP can be written as
where u_{e} and u_{h} are the drift velocities of electrons and holes, respectively, τ_{eh} is the e–h scattering time and τ is the electron–impurity and/or electron–phonon scattering times. The effective mass m* for the Dirac plasma is given by equation (8).
Taking the sum and difference between the top and bottom expressions in equation (13), we obtain
where \({\tau }_{0}^{1}={\tau }_{{\rm{eh}}}^{1}+{\tau }^{1}\) is the total scattering rate. Plugging u_{e} + u_{h} obtained from the top expression of equation (14) into the lefthand side of the bottom one, we obtain
where \({{\mu }_{{\rm{B}}}}^{2}=\frac{{e}^{2}{\tau }_{0}\tau }{{m}^{* 2}}\) and \({\mu }_{0}=\frac{e{\tau }_{0}}{{m}^{* }}\). As shown below, these coefficients determine the magnetotransport and zerofield mobilities. If equation (15) is placed into the lefthand side of the first line of equation (14), this leads to
where z is the unit vector in the direction of magnetic field. Combining equations (15) and (16) allows us to find
Equation (17) yields \({\sigma }_{xx}=\left(n+p\right)e\frac{{\mu }_{0}}{1+{{\mu }_{{\rm{B}}}}^{2}{B}^{2}}={2n}_{{\rm{th}}}e\frac{{\mu }_{0}}{1+{{\mu }_{{\rm{B}}}}^{2}{B}^{2}}\) where n and p are the densities of thermally excited electrons and holes, respectively (n = p = n_{th}). To obtain ρ_{xx}(B) at the NP, we take into account that for a compensated e–h plasma the Hall conductivity σ_{xy} = 0 and ρ_{xx} = 1/σ_{xx}, which leads to
The first term defines the zeroB resistivity of the Dirac plasma and, as expected, depends on the total scattering rate 1/τ_{0}. However, the second term is proportional to \({{\mu }_{B}}^{2}/{\mu }_{0}=e\tau \,/{m}^{* }\), that is, the absolute value of MR ρ_{xx}(B) − ρ_{xx}(0) is independent of e–h collisions and depends on only impurity and/or phonon scattering.
As for relative MR, we obtain
The above analysis suggests different zerofield and magnetotransport mobilities, and their ratio is given by
Our experiments found typical μ_{B}/μ_{0} of about 3, in agreement with the expectation that e–h scattering in the Dirac plasma should be the dominant scattering mechanism at room temperature.
Effect of proximity screening on mobility and MR
The observed difference between mobilities extracted from zerofield and magnetotransport measurements implies that μ_{0} and μ_{B} should be affected differently by screening. The latter mobility should be less sensitive to screening because e–h scattering does not contribute to Hall currents, as discussed above.
We have verified these expectations using MLG devices with proximity screening^{15}. Such devices have previously been studied in the doped regime where electron scattering was found to be notably reduced by the screening^{15}. Electron–hole interactions in chargeneutral graphene can also be expected to be modified by such proximity screening. We studied three MLG devices in which the graphite gate served as a metallic screening plate and was separated from graphene by a thin hBN layer (thicknesses of about 0.9 nm, 1.2 nm and 2.4 nm; inset of Extended Data Fig. 4a). In the particular case of the 2.4nm device shown in Extended Data Fig. 4, we have found the screening to reduce ρ_{NP} by a factor of about 2 below 250 K compared with similarquality MLG devices without screening. The reduction in ρ_{NP} yields a smaller interaction constant (about 0.4) and translates into higher μ_{0}. It is noted that the difference between ρ_{NP} observed for screened and unscreened devices reduces at higher T (Extended Data Fig. 4a). This can be attributed to the fact that the screening is sensitive to the average separation between charge carriers, which is proportional to n^{−1/2}. As the density of thermally excited carriers increases with T, the screening efficiency is reduced^{15}.
The influence of proximity screening on magnetotransport in the Dirac plasma is found to be notably different from the case of zero B. Extended Data Fig. 4b shows that changes in ρ_{NP} as a function of B remained practically the same for devices with and without screening. This agrees with the results in ‘Difference between zerofield and magnetotransport mobilities’, which predict that changes in ρ_{NP}(B) should be insensitive to e–h scattering and, therefore, unaffected by proximity screening, in contrast to ρ_{NP}(B = 0) that is dominated by this scattering mechanism.
For quantitative analysis of the observed screening effects, we have extracted e–h and electron–impurity (inelastic) scattering times (τ_{eh} and τ, respectively) for the devices with and without proximity screening. To this end, we used the fact that the first (zero B) term in equation (18) depends on both τ_{eh} and τ whereas the second term is determined only by τ. The results are plotted in Extended Data Fig. 4c. Both screened and unscreened devices exhibit similar τ that is, several times longer than τ_{eh}. As expected, the proximity screening significantly suppresses electron interactions so that at about 150 K, τ_{eh} is twice longer in the devices with proximity screening than for the standard encapsulated graphene. The difference is reduced at higher T, with possible reasons for this being mentioned earlier in this section.
Linear magnetoresistivity in high fields
As discussed in the main text, the parabolic MR is observed only in small magnetic fields up to about 0.1 T. In higher B, a linear MR behaviour emerges. We observed the linear dependence over a wide range of magnetic fields up to 18 T, the highest B available in our experiments. This is shown in Extended Data Fig. 5 for device D2. Again, the slope of ρ_{NP}(B) depends weakly on T, and its absolute value is close to that exhibited by device D1 (within 20%), as shown in Fig. 3a. Overall, the described highB behaviour was very similar for all five such MLG devices that we studied (inset of Fig. 3b). It is noted that the absence of T dependence for highB MR indicates that manybody gaps caused by lifting of spin and valley degeneracies play little role within the discussed range of T and B. Otherwise, the gaps’ smearing should have led to a strong T dependence.
Landau quantization at room temperature
We have attributed the observed linear MR in high B to the transition of the Dirac plasma into the quantized regime where the linear spectrum of MLG splits into dispersionless LLs. This condition is an essential prerequisite for discussing magnetotransport for the compensated Boltzmann gas in the zeroth LL.
In MLG, the main cyclotron gap at the filling factor ν = 2 in units of the kelvin (K) is given by^{87} \(E[{\rm{K}}]={{\rm{v}}}_{{\rm{F}}}{(2{\rm{e}}\hslash {\rm{B}})}^{1/2}\approx 400\times \sqrt{{\rm{B}}[{\rm{T}}]}\). The gap’s size notably exceeds the thermal energy at room temperature already in fields of a few tesla. Previously, the Landau quantization has been reported for ultrahigh magnetic fields of 30–40 T where even the quantum Hall effect was observed at room temperature^{87}. To demonstrate that Landau quantization in our devices becomes important at room temperature already in moderate B, Extended Data Fig. 6a shows the fan diagram measured for one of our Corbino devices at room temperature. The found peaks in inverse conductivity follow the main gaps at ν = ±2, as expected, and become clearly visible at B above 6 T. The Landau quantization is also visible in ρ_{xx} measured in the standard Hallbar geometry (Extended Data Fig. 6b). These observations support the description of highB transport in neutral MLG in terms of the zeroth LL for the discussed temperature range up to 300 K.
Linear MR in Corbino devices
We have also used our Corbino devices to rule out edge effects in the appearance of strange linear MR. Extended Data Fig. 7 shows that the linear dependence ρ_{NP}(B) was also observed in this geometry, exhibiting little difference with respect to the behaviour reported for the fourprobe Hallbar devices. Indeed, MR of Corbino disks is found to be weakly dependent on T and exhibit slopes with values close to those observed in the Hallbar geometry (compare Fig. 3a and Extended Data Fig. 5). This proves that the linear MR is an intrinsic (bulk) effect and, for example, it is not related to e–h annihilation at graphene edges^{66} or to spin and valley Hall currents reported for neutral graphene^{88}.
MR of lowmobility graphene
To illustrate the importance of high quality for the reported MR behaviour of MLG in both low and high B, we have measured lowmobility devices obtained by exfoliation of graphene onto an oxidized silicon wafer (inset of Extended Data Fig. 8a). At liquidhelium temperatures, such devices exhibited strong charge inhomogeneity with δn ≈ 10^{11} cm^{−2} (Extended Data Fig. 8a), which was nearly two orders of magnitude higher than that for hBNencapsulated graphene (Fig. 1c). Even at 300 K, thermally excited density n_{th} remained smaller than the residual δn, which means that electron transport near the NP in such devices was dominated by charge inhomogeneity (e–h puddles) at all T in the experiment. Accordingly, although ρ_{NP} decreased with increasing T (Extended Data Fig. 8), similar to the case of our highmobility devices, it only reached about 4 kΩ at room temperature, significantly away from the intrinsic value of about 1 kΩ for the Dirac plasma in the quantumcritical regime.
In small magnetic fields, ρ_{NP} for MLG on silicon dioxide evolved quadratically with B (Extended Data Fig. 8b). The measured Δ was found to be more than two orders of magnitude smaller than in highquality MLG (<1% at 0.1 T), which corresponds to about 8,500 cm^{2} V^{−1} s^{−1} at the NP. With increasing B above 1 T, the MR of graphene on silicon dioxide deviated from the parabolic dependence and became sublinear at high T (Extended Data Fig. 8c), in agreement with the previous reports^{30,70}. Such sublinear behaviour may be attributed to shortrange scattering^{73}, which is present in graphene on silicon dioxide^{89}, but further research is required to unambiguously identify the origins of highB MR in lowmobility MLG. Nonetheless, our observations clearly show the importance of electronic quality for the observation of the linear magnetoresistivity.
Data availability
All relevant data are available from the corresponding authors. Source data are provided with this paper.
References
Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).
Yankowitz, M., Ma, Q., JarilloHerrero, P. & LeRoy, B. J. van der Waals heterostructures combining graphene and hexagonal boron nitride. Nat. Rev. Phys. 1, 112–125 (2019).
Kashuba, A. B. Conductivity of defectless graphene. Phys. Rev. B 78, 085415 (2008).
Fritz, L., Schmalian, J., Müller, M. & Sachdev, S. Quantum critical transport in clean graphene. Phys. Rev. B 78, 085416 (2008).
Gallagher, P. et al. Quantumcritical conductivity of the Dirac fluid in graphene. Science 364, 158–162 (2019).
Crossno, J. et al. Observation of the Dirac fluid and the breakdown of the Wiedemann–Franz law in graphene. Science 351, 1058–1061 (2016).
Ku, M. J. H. et al. Imaging viscous flow of the Dirac fluid in graphene. Nature 583, 537–541 (2020).
Block, A. et al. Observation of giant and tunable thermal diffusivity of a Dirac fluid at room temperature. Nat. Nanotechnol. 16, 1195–1200 (2021).
Nam, Y., Ki, D.K., SolerDelgado, D. & Morpurgo, A. F. Electron–hole collision limited transport in chargeneutral bilayer graphene. Nat. Phys. 13, 1207–1214 (2017).
Tan, C. et al. Dissipationenabled hydrodynamic conductivity in a tunable bandgap semiconductor. Sci. Adv. 8, eabi8481 (2022).
Zaanen, J. Planckian dissipation, minimal viscosity and the transport in cuprate strange metals. SciPost Phys. 6, 061 (2019).
Phillips, P. W., Hussey, N. E. & Abbamonte, P. Stranger than metals. Science 377, eabh4273 (2022).
Hayes, I. M. et al. Scaling between magnetic field and temperature in the hightemperature superconductor BaFe_{2}(As_{1−x}P_{x})_{2}. Nat. Phys. 12, 916–919 (2016).
GiraldoGallo, P. et al. Scaleinvariant magnetoresistance in a cuprate superconductor. Science 361, 479–481 (2018).
Kim, M. et al. Control of electron–electron interaction in graphene by proximity screening. Nat. Commun. 11, 2339 (2020).
Abrikosov, A. A. Quantum linear magnetoresistance. Europhys. Lett. 49, 789–793 (2000).
Ghimire, N. J. et al. Magnetotransport of single crystalline NbAs. J. Phys. Condens. Matter 27, 152201 (2015).
Liang, T. et al. Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd_{3}As_{2}. Nat. Mater. 14, 280–284 (2015).
Shekhar, C. et al. Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP. Nat. Phys. 11, 645–649 (2015).
Ali, M. N. et al. Large, nonsaturating magnetoresistance in WTe_{2}. Nature 514, 205–208 (2014).
Luo, Y. et al. Hall effect in the extremely large magnetoresistance semimetal WTe_{2}. Appl. Phys. Lett. 107, 182411 (2015).
Tafti, F. F., Gibson, Q. D., Kushwaha, S. K., Haldolaarachchige, N. & Cava, R. J. Resistivity plateau and extreme magnetoresistance in LaSb. Nat. Phys. 12, 272–277 (2016).
Gao, W. et al. Extremely large magnetoresistance in a topological semimetal candidate pyrite PtBi_{2}. Phys. Rev. Lett. 118, 256601 (2017).
Kumar, N. et al. Extremely high magnetoresistance and conductivity in the typeII Weyl semimetals WP_{2} and MoP_{2}. Nat. Commun. 8, 1642 (2017).
Singha, R., Pariari, A. K., Satpati, B. & Mandal, P. Large nonsaturating magnetoresistance and signature of nondegenerate Dirac nodes in ZrSiS. Proc. Natl Acad. Sci. USA 114, 2468–2473 (2017).
Solin, S. A., Thio, T., Hines, D. R. & Heremans, J. J. Enhanced roomtemperature geometric magnetoresistance in inhomogeneous narrowgap semiconductors. Science 289, 1530–1532 (2000).
Rode, D. L. Electron transport in InSb, InAs, and InP. Phys. Rev. B 3, 3287–3299 (1971).
Wang, L. et al. Onedimensional electrical contact to a twodimensional material. Science 342, 614–617 (2013).
Pippard, A. B. Magnetoresistance in Metals(Cambridge Univ. Press, 1989).
Gopinadhan, K., Shin, Y. J., Yudhistira, I., Niu, J. & Yang, H. Giant magnetoresistance in singlelayer graphene flakes with a gatevoltagetunable weak antilocalization. Phys. Rev. B 88, 195429 (2013).
Kisslinger, F. et al. Linear magnetoresistance in mosaiclike bilayer graphene. Nat. Phys. 11, 650–653 (2015).
Hu, J. et al. Roomtemperature colossal magnetoresistance in terraced singlelayer graphene. Adv. Mater. 32, 2002201 (2020).
Zhou, B., Watanabe, K., Taniguchi, T. & Henriksen, E. A. Extraordinary magnetoresistance in encapsulated monolayer graphene devices. Appl. Phys. Lett. 116, 053102 (2020).
Gopinadhan, K. et al. Extremely large magnetoresistance in fewlayer graphene/boron–nitride heterostructures. Nat. Commun. 6, 8337 (2015).
Baibich, M. N. et al. Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices. Phys. Rev. Lett. 61, 2472–2475 (1988).
Binasch, G., Grünberg, P., Saurenbach, F. & Zinn, W. Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange. Phys. Rev. B 39, 4828–4830 (1989).
Kazantsev, A., Berdyugin, A., Geim, A. & Principi, A. On the origin of Abrikosov’s quantum linear magnetoresistance. Preprint at http://arxiv.org/abs/2208.06273 (2022).
Ikeda, S. et al. Tunnel magnetoresistance of 604% at 300K by suppression of Ta diffusion in CoFeB/MgO/CoFeB pseudospinvalves annealed at high temperature. Appl. Phys. Lett. 93, 082508 (2008).
Thomson, W. XIX. On the electrodynamic qualities of metals:—effects of magnetization on the electric conductivity of nickel and of iron. Proc. R. Soc. Lond. 8, 546–550 (1857).
Hall, E. H. On the new action of magnetism on a permanent electric current. Lond. Edinb. Dublin Phil. Mag. J. Sci. 10, 301–328 (1880).
Becker, J. A. & Curtiss, L. F. Physical properties of thin metallic films. I. Magnetoresistance effects in thin films of bismuth. Phys. Rev. 15, 457–464 (1920).
Kapitza, P. The study of the specific resistance of bismuth crystals and its change in strong magnetic fields and some allied problems. Proc. R. Soc. Lond. A 119, 358–443 (1928).
Kapitza, P. The change of electrical conductivity in strong magnetic fields. Part I.—experimental results. Proc. R. Soc. Lond. 123, 292–341 (1929).
Lifshitz, I. M. & Peschanskii, V. G. Galvomagnetic characteristics of metals with open Fermi surface. Sov. Phys. JETP 35, 875–883 (1959).
Lifshitz, I. M., Azbel’, M. I. A. & Kaganov, M. I. The theory of galvanomagnetic effects in metals. Sov. Phys. JETP 4, 41–54 (1957).
Alekseevskii, N. E. & Gaidukov, Y. P. Measurement of the electrical resistance of metals in a magnetic field as a method of investigating the Fermi surface. Sov. Phys. JETP 36, 311–313 (1959).
Dreizin, Y. A. & Dykhne, A. M. Anomalous conductivity of inhomogeneous media in a strong magnetic field. Sov. Phys. JETP 36, 127–136 (1973).
Amundsen, T. & Jerstad, P. Linear magnetoresistance of aluminum. J. Low Temp. Phys. 15, 459–471 (1974).
Sampsell, J. B. & Garland, J. C. Current distortion effects and linear magnetoresistance of inclusions in freeelectron metals. Phys. Rev. B 13, 583–589 (1976).
Stroud, D. & Pan, F. P. Effect of isolated inhomogeneities on the galvanomagnetic properties of solids. Phys. Rev. B 13, 1434–1438 (1976).
Beers, C. J., van Dongen, J. C. M., van Kempen, H. & Wyder, P. Influence of voids on the linear magnetoresistance of indium. Phys. Rev. Lett. 40, 1194–1197 (1978).
Yoshida, K. Structural magnetoresistance of indium containing granular glass. J. Phys. F 11, L245–L248 (1981).
Bruls, G. J. C. L., Bass, J., van Gelder, A. P., van Kempen, H. & Wyder, P. Linear magnetoresistance caused by sample thickness variations. Phys. Rev. Lett. 46, 553–555 (1981).
Bruls, G. J. C. L., Bass, J., van Gelder, A. P., van Kempen, H. & Wyder, A. P. Linear magnetoresistance due to sample thickness variations: applications to aluminum. Phys. Rev. B 32, 1927–1939 (1985).
Parish, M. M. & Littlewood, P. B. Nonsaturating magnetoresistance in heavily disordered semiconductors. Nature 426, 162–165 (2003).
Abrikosov, A. A. Quantum magnetoresistance. Phys. Rev. B 58, 2788–2794 (1998).
Abrikosov, A. A. Quantum magnetoresistance of layered semimetals. Phys. Rev. B 60, 4231–4234 (1999).
Xu, R. et al. Large magnetoresistance in nonmagnetic silver chalcogenides. Nature 390, 57–60 (1997).
Yang, F. Y. et al. Large magnetoresistance of electrodeposited singlecrystal bismuth thin films. Science 284, 1335–1337 (1999).
Hu, J. & Rosenbaum, T. F. Classical and quantum routes to linear magnetoresistance. Nat. Mater. 7, 697–700 (2008).
Friedman, A. L. et al. Quantum linear magnetoresistance in multilayer epitaxial graphene. Nano Lett. 10, 3962–3965 (2010).
Kubo, R., Miyake, S. J. & Hashitsume, N. Quantum theory of galvanomagnetic effect at extremely strong magnetic fields. Solid State Phys. 17, 269–364 (1965).
Ando, T. & Uemura, Y. Theory of quantum transport in a twodimensional electorn systems under magnetic fields. J. Phys. Soc. Jpn 36, 959–967 (1974).
Klier, J., Gornyi, I. V. & Mirlin, A. D. Transversal magnetoresistance in Weyl semimetals. Phys. Rev. B 92, 205113 (2015).
Alekseev, P. S. et al. Magnetoresistance in twocomponent systems. Phys. Rev. Lett. 114, 156601 (2015).
Alekseev, P. S. et al. Magnetoresistance of compensated semimetals in confined geometries. Phys. Rev. B 95, 165410 (2017).
Varma, C. M. Quantumcritical resistivity of strange metals in a magnetic field. Phys. Rev. Lett. 128, 206601 (2022).
Ghiotto, A. et al. Quantum criticality in twisted transition metal dichalcogenides. Nature 597, 345–349 (2021).
Jaoui, A. et al. Quantum critical behaviour in magicangle twisted bilayer graphene. Nat. Phys. 18, 633–638 (2022).
Cho, S. & Fuhrer, M. S. Charge transport and inhomogeneity near the minimum conductivity point in graphene. Phys. Rev. B 77, 081402 (2008).
Pisana, S., Braganca, P. M., Marinero, E. E. & Gurney, B. A. Tunable nanoscale graphene magnetometers. Nano Lett. 10, 341–346 (2010).
Liao, Z.M. et al. Large magnetoresistance in few layer graphene stacks with current perpendicular to plane geometry. Adv. Mater. 24, 1862–1866 (2012).
Alekseev, P. S., Dmitriev, A. P., Gornyi, I. V. & Kachorovskii, V. Yu. Strong magnetoresistance of disordered graphene. Phys. Rev. B 87, 165432 (2013).
Vasileva, G. Y. et al. Linear magnetoresistance in compensated graphene bilayer. Phys. Rev. B 93, 195430 (2016).
Wang, X., Du, Y., Dou, S. & Zhang, C. Room temperature giant and linear magnetoresistance in topological insulator Bi_{2}Te_{3} nanosheets. Phys. Rev. Lett. 108, 266806 (2012).
Zhang, S. X. et al. Magnetoresistance up to 60 tesla in topological insulator Bi_{2}Te_{3} thin films. Appl. Phys. Lett. 101, 202403 (2012).
Piatrusha, S. U. et al. Topological protection brought to light by the timereversal symmetry breaking. Phys. Rev. Lett. 123, 056801 (2019).
Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419–425 (2013).
Soluyanov, A. A. et al. TypeII Weyl semimetals. Nature 527, 495–498 (2015).
Ziman, J. M. Principles of the Theory of Solids (Cambridge Univ. Press, 1964).
Mayorov, A. S. et al. Micrometerscale ballistic transport in encapsulated graphene at room temperature. Nano Lett. 11, 2396–2399 (2011).
Zhang, Y. et al. Direct observation of a widely tunable bandgap in bilayer graphene. Nature 459, 820–823 (2009).
Yin, J. et al. Dimensional reduction, quantum Hall effect and layer parity in graphite films. Nat. Phys. 15, 437–442 (2019).
Müller, M. & Sachdev, S. Collective cyclotron motion of the relativistic plasma in graphene. Phys. Rev. B 78, 115419 (2008).
Narozhny, B. N., Gornyi, I. V., Titov, M., Schütt, M. & Mirlin, A. D. Hydrodynamics in graphene: linearresponse transport. Phys. Rev. B 91, 035414 (2015).
Narozhny, B. N., Gornyi, I. V., Mirlin, A. D. & Schmalian, J. Hydrodynamic approach to electronic transport in graphene: hydrodynamic approach to electronic transport in graphene. Ann. Phys. 529, 1700043 (2017).
Novoselov, K. S. et al. Roomtemperature quantum Hall effect in graphene. Science 315, 1379–1379 (2007).
Abanin, D. A. et al. Giant nonlocality near the Dirac point in graphene. Science 332, 328–330 (2011).
Ni, Z. H. et al. On resonant scatterers as a factor limiting carrier mobility in graphene. Nano Lett. 10, 3868–3872 (2010).
Acknowledgements
We acknowledge financial support from the European Research Council (grant VANDER), the Lloyd’s Register Foundation, Graphene Flagship Core3 Project, and the EPSRC (grants EP/W006502, EP/V007033 and EP/S030719); J.L. and A.P. were supported by the EU Horizon 2020 programme under the Marie SkłodowskaCurie grants 891778 and 873028, respectively; A.P. and A.E.K. acknowledge support from the Leverhulme Trust (grant RPG2019363).
Author information
Authors and Affiliations
Contributions
A.I.B., L.A.P. and A.K.G. designed and supervised the project; N.X., P.K. and Z.W. fabricated the graphene devices; A.M. provided multilayer graphene devices; J.L., A.I.B., L.A.P., J.B. and C.M. carried out the electrical measurements; A.I.B., J.L., L.A.P. and A.K.G. analysed data with help from V.I.F., A.P., A.E.K., A.A.G., N.X. and P.K.; A.K.G. and A.I.B. wrote the manuscript with contributions from I.V.G., A.P., A.E.K. and V.I.F. All authors contributed to discussions.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature thanks Jurgen Smet and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
Extended Data Fig. 1 MR behaviour for another MLG device.
a, Its zeroB resistivity at the NP as a function of T. b, δn as a function of T. The shadowed areas in a and b indicate the range in which the thermally excited density n_{th} is less than the remnant inhomogeneity. c, LowB resistivity for two characteristic T. Dashed curve: parabolic dependence. The black circles indicate Δ in the characteristic field of 0.1 T. d, μ_{B} and μ_{0} for this device as a function of T. Dashed curves: guides to the eye.
Extended Data Fig. 2 MR and mobility of e–h plasma in BLG.
a, Its zeroB resistivity near the NP as a function of gateinduced carrier density. b, Resistivity of BLG at the NP as a function of T. c, Thermal smearing δn was extracted using the same approach as described in the main text. d, Magnetoresistivity at the NP in low B. e, Carrier mobility at the NP as a function of T. μ_{B} and μ_{0} were extracted from MR and zerofield measurements, respectively. Dashed lines: 1/T dependences. All the measurements were carried out at zero displacement.
Extended Data Fig. 3 Magnetoresistivity of multilayer graphene.
a, ρ_{xx}(B) for a 3.5 nm thick graphite film (∼10 graphene layers) measured between 75 and 300 K in steps of 25 K. Inset: Loglog plot of the MR at three characteristic T. Dashed line: quadratic dependence. b, Magnetotransport mobility for graphite evaluated using parabolic fits of ρ_{xx}(B) over an interval of ±0.5 T. Dashed curve: 1/T dependence. We measured several graphite devices, and all exhibited very similar MR behaviour. It changed little if additional carriers were induced near the surface by gate voltage^{83}.
Extended Data Fig. 4 Influence of proximity screening.
a, Resistivity of chargeneutral MLG with and without screening. Inset: schematics of our devices where the proximity screening is provided by a bottom graphite gate. b, Changes in graphene’s resistivity in small B for devices with and without proximity screening. The black circle marks Δ at 50 mT (colourcoded). c, Inelastic and electron–hole scattering times (top and bottom panels, respectively) for devices with and without proximity screening.
Extended Data Fig. 5 Another example of quantum linear MR.
Resistivity of MLG at the NP over a large range of B. Temperatures are between 100 and 250 K in steps of 50 K; device D2. Dashed line: Guide to the eye with a slope of 6.5 kOhm T^{−1}.
Extended Data Fig. 6 Roomtemperature Landau quantization in moderate magnetic fields.
a, Conductivity σ_{xx} of MLG as a function of B and carrier density at 300 K. The measurements are for a Corbinodisk device. The vertical yellow line indicates the NP, and the red lines follow ν = ±2. b, RoomT resistivity for MLG measured in the Hallbar geometry at three representative B. The traces are shifted for clarity by 0.1 kΩ. The vertical arrows mark ν = ±2.
Extended Data Fig. 7 Quantum linear MR in Corbino devices.
ρ_{NP}(B) measured for one of our Corbinodisk devices at different T (colourcoded). The contact resistance was about 0.1 kOhm (measured in the limit of high n and assumed to change little near the NP). Black line: guide to the eye with the slope 4.8 kΩ T^{−1}. Insert: false colour micrograph of the device. Green, hBN on top of graphene; gold, metallic contacts; purple, polymer bridges over the outer ring contact, which are required for the metallization to reach the inner contact.
Extended Data Fig. 8 Magnetotransport in grapheneonsilicon oxide.
a, Resistivity at the NP as a function of n plotted for two characteristic T. The crossing of the two dashed lines indicates the charge inhomogeneity level. The inset shows an optical image of the studied device. Scale bar, 10 µm. b, Resistivity at the NP as a function of B. The open circles mark MR values at 0.1 T. Inset: same curves replotted on a log scale. The dashed line is the parabolic fit for the 250K curve below 0.2 T. c, Resistivity of chargeneutral grapheneonSiO_{2} over a large range of B at different T. No clear linear MR is observed at any T for such MLG devices.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Xin, N., Lourembam, J., Kumaravadivel, P. et al. Giant magnetoresistance of Dirac plasma in highmobility graphene. Nature 616, 270–274 (2023). https://doi.org/10.1038/s41586023058070
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41586023058070
This article is cited by

Magnetoresistivecoupled transistor using the Weyl semimetal NbP
Nature Communications (2024)

Extreme magnetoresistance at highmobility oxide heterointerfaces with dynamic defect tunability
Nature Communications (2024)

Infrared nanoimaging of Dirac magnetoexcitons in graphene
Nature Nanotechnology (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.