Introduction

Graphene has revealed a series of amazing properties, such as ultra-high electrical conductivity1,2, ultra-low shear viscosity to entropy ratio3, as well as exceptional structural strength, as combined with mechanical flexibility4 and optical transparency5. Many of these fascinating properties are due to the fact that, consisting of literally one single carbon monolayer, graphene represents the first instance ever of a truly two-dimensional material (the “ultimate flatland”6). Moreover, due to the special symmetries of the honeycomb lattice, electrons in graphene are shown to behave like an effective Dirac fluid of massless chiral quasi-particles, propagating at a Fermi speed of about c ~ 106 m/s3,7. This configures graphene as an unique, slow-relativistic electronic fluid, where many unexpected quantum-electrodynamic phenomena can take place8. For instance, since electrons are about 300 times slower than photons, their mutual interaction is proportionately enhanced, leading to an effective fine-structure constant . As a result of such strong interactions, it has been recently proposed that this peculiar 2D graphene electron gas should be characterized by an exceptionally low viscosity/entropy ratio (near-perfect fluid), coming close to the famous AdS-CFT lower bound conjectured for quantum-chromodynamic fluids, such as quark-gluon plasmas8. This spawns the exciting prospect of observing electronic pre-turbulence in graphene samples, as first pointed out in Ref.3 and confirmed by recent numerical simulations9.

Some of the electrical properties of graphene are still not fully understood, such as the linear increase of the electrical conductivity with the number of charge carriers, the existence of a minimum conductivity (see Ref. 10 and reference therein) and even the nature of the main scattering mechanism limiting the carrier mobility11. In fact, classical transport theories, based on short-range scattering of electrons by impurities, predict that the electric conductivity in graphene should be independent of the carrier density12. Recent works in the field13,14 have shown that such linear dependence might be potentially explained by treating the impurities as screened Coulomb scatterers. Nevertheless, some measurements of the change in the electrical conductivity upon immersion of graphene samples in high-κ dielectric media differ from this conclusion15,16. Here, we construct a model for describing the electrical conductivity in graphene by using a completely different approach, which is based on the recently developed picture of an electronic ideal relativistic fluid at the Dirac point. We demonstrate that, although this model is based on a semiclassical theory (it cannot take into account all quantum effects, e.g. Landau quantization, quantum hall effects and quantum interference), it captures the main factors that contribute to such linear behavior and the appearance of a minimum conductivity.

Since the most likely relevant limiting factor for the graphene conductivity is still subject of controversy, e.g. it can be due to random charged impurity centers13 or strong neutral defects that induce resonant scattering15,16, we will treat the impurities as hard-spheres, hindering the electron flow (scattering electrons), similarly to the way a disordered medium does in the context of fluid dynamics. The choice of hard-spheres is based on the experimental results by Monteverde et al.11, which suggested that electrons seem to collide mostly with short range scatterers of the size of a few carbon-carbon interatomic distances, like voids, adatoms, etc. Since the relativistic fluid approach is derived from the quantum Boltzmann equation (QBE) for graphene17, a hydrodynamic description of the conductivity can be expected to apply as long as the QBE collision operator takes proper account of the Coulomb interactions between electrons. Therefore, once Coulomb interactions are included in the viscosity of the fluid, the conductivity (which in our case, unlike viscosity, is a property of the flow rather than of the fluid) becomes a function of the dissipation introduced in the system by the impurities, i.e. the electron-impurity interaction.

Here, we treat graphene as a disordered medium and develop a model for its conductivity, as a function of the impurity density describing the anomalous dependence of the conductivity on the carrier density and the minimum conductivity due to the carrier density induced by the impurities. The results are compared with experimental data yielding very satisfactory agreement.

Results

Electronic fluid in graphene

Our treatment is based on the hydrodynamic equations derived by Müller et al.3,7, based on the quantum Boltzmann equation for electrons in graphene. This analysis delivers the value of the transport coefficients, namely the fluid shear viscosity, which is an input parameter in our model. The hydrodynamic approach in graphene is valid when the inelastic-scattering rate due to electron-electron interactions dominates. This is the case at low doping, at high temperatures and in moderate fields18. However, to neglect electron-phonon interactions, we will have to stay at a moderately high temperature of around 100 K19. In absence of magnetic fields, the quasiparticle distribution function, fs, evolves according to the quantum Boltzmann equation,

where is an external electric field, e the electric charge of the electron, Ω[fs] a collision operator that takes into account the electron-electron interactions, , the wave vector, c the Fermi speed (~ 106 m/s) and the sign s, not to be confused with the entropy density, distinguishes between electrons (+) and holes (−)18,20. At equilibrium, the probability distribution function becomes the Fermi-Dirac distribution,

where μ is the chemical potential and T denotes the temperature. Thus, in the hydrodynamic limit, from Eqs. (1) and (2) one can derive the equations for the Dirac electron fluid in graphene: , for charge conservation; , for energy density conservation and

for momentum conservation. Here, is the energy density, p the pressure, ρ the charge density, the drift velocity, and η the shear viscosity.

For the case of undoped graphene (μ = 0), the presence of charge density is due to the thermal energy and can be described by,

However, when there are impurities, they can induce electric potentials on the graphene sample and a correction due to the chemical potential must be considered,

where Φρ is a dimensionless increasing function defined in Ref. 7. Note that, in our analytical model, we will use this concept in order to introduce a minimum conductivity in the graphene sample, where the function Φρ will be modeled by a free parameter to fit the experimental data and will take into account not only the carriers generated by the impurities but also other kind of phenomena that could contribute to induce carrier density.

The shear viscosity η, in Eq. (3), can be calculated using

where Cη ~ O(1) is a numerical coefficient, is the effective fine structure constant, the relative dielectric constant of the substrate and M the number of species of free massless Dirac particles3,7. Additionally, the entropy densities can be calculated according to the Gibbs-Duhem relation . These equations have been derived under the assumption and therefore the relativistic correction term, ∝ ∂p/∂t, can be neglected, so that the classical Navier-Stokes equations are recovered. Note that, despite the high speed of the electrons, , the Reynolds number remains moderate, due to nano-metric size of the samples and the high kinematic viscosity of the electronic fluid in graphene.

Kinematic viscosity

Based on Ref. 3, the dynamic viscosity of graphene in a sample of linear size L0, is given by Eq. (6). This equation can be written in the following form:

where we have introduced the characteristic frequency ωf = c/L0 and by solving the appropriate quantum Boltzmann equation, it is concluded that . Eq. (7) can also be rewritten as , where . Note that, in order for a classical (non quantum) picture of electron fluid to apply, the energy of excitations must be much lower than the thermal energy, i.e. qf ≪ 1, the so-called collision-dominated regime. Taking a typical set of parameters (in MKS units), c = 106, L0 = 10−6, T = 100 K and , we obtain η~10−20. Since the Reynolds number is dictated by the kinematic viscosity of the fluid, ν, rather than by the dynamic one, η = ρν, with no need of involving the mass density, it is of interest to estimate the kinematic viscosity of the electron fluid in graphene.

To this purpose, we appeal to the definition of the Reynolds number as given in Ref. 3, namely:

where ν0 = cL0. By writing Re = u0L0/ν and equating with the above, we obtain

Using and , we obtain . To be noted that, in spite of its extremely low dynamic viscosity, the kinematic viscosity of graphene is about four orders of magnitude larger than that of water. These four orders of magnitude are more than compensated by the large speed of the electrons, which is ultimately responsible for the sizeable values of the Reynolds numbers which can be achieved in graphene samples at micron scales. For instance, by taking u0 = 0.1c~ 105 m/s, for a sample of 1 micron in length, we obtain Re~ 20 for the global sample and about Re~ 0.04 on the scale of the impurities.

Analytical model description

In this work, we will treat impurities as circular rigid obstacles of diameter d. This choice is not arbitrary, but it is based on the fact that some experiments11,15,16 suggest the the main scattering mechanism in graphene is due to strong neutral defects, with a range shorter than the Fermi wavelength, inducing resonant scattering. Thus, the diameter d can be interpreted as the characteristic length for the range of the interaction electron-impurity.

Let us now assume that the electronic fluid moves in the x direction as a consequence of an applied electric field E and (incompressible limit). Therefore, we can calculate the force acting on a single impurity due to the electronic flow, as , where is the stress tensor defined by Πij = pδij + ρruiuj + πij, with the viscous tensor πij = η(∂ui/∂xj + ∂uj/∂xi) and is a unit vector perpendicular to the impurity circle. Here the indices i, j = 1, 2 denote the coordinates x and y. Thus, solving the equations for the Dirac electron fluid, in the steady state (all time derivatives are neglected) and following an analogous procedure as in classical hydrodynamics21, we obtain for the drag force, FD = Fx,

where λ is a dimensionless parameter that depends on the Reynolds number. Here, v is the velocity of the fluid very far from the impurity. We first simulated single impurities with different sizes and different fluid velocities (see Fig. 1), obtaining that a linear approximation is justified in the range of relevance to this work. Note that, for a perfect fluid (ν = 0), there is no drag force. However, from the point of view of the kinematic viscosity, electrons in graphene are far from being a perfect fluid. Therefore, we expect them to play a crucial effect on the drag force controlling the total conductivity of the sample.

Figure 1
figure 1

Drag force acting on a single impurity.

Drag force FD acting on a single impurity as a function of the graphene flow drift velocity for different impurity diameters. The solid lines represent the linear dependency of the drag force with the velocity of the fluid. In the inset, the dependence of the dimensionless parameter λ on the impurity diameter is shown.

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Let us denote by φ the impurity fraction, namely the ratio between the area not occupied by the impurities and the total area of the system, φ = 1 − Nπd2/(4A), with N the number of impurities in a sample of area A. We can thus propose the relation between v and the total current density as J = φρv. Note that φρ is the existent electronic charge density in the graphene sample, since the volume fraction 1 − φ is occupied by the impurities.

Let us consider a representative elementary area of the sample of length dx in the direction of the flow and dy across it. With circular impurities in this area, we can write . The total force acting on the electronic fluid due to impurities (equal to the total force acting on the impurities due to the fluid), is given by . Here, the distance between impurities is taken sufficiently large enough to prevent the flow close to an impurity from affecting the flow nearby another impurity.

In order to describe correctly the physics of graphene, we need to include in our model an extra feature. Due to the linear Dirac-Weyl spectrum of graphene and the non-existence of a gap between the conduction and valence bands, the slightest amount of impurities or an external potential will induce charge carriers in the graphene sample22,23, see Eq. (5). Thus, in our model, the total amount of carriers induced by the impurities will be proportional to the impurity concentration, (1 − φ)A, leading to an extra carrier density in the fraction of the sample occupied by the electronic fluid, φA.The extra carriers are then given by γ(1 − φ)/γ, where γ is the proportionality constant that characterizes the impurity-electron interaction.

Summing the forces, namely the Lorentz and drag forces, acting on the elementary area leads to: φρEdxdy + γ(1 − φ)Edxdy/φ − Ftot = 0 and by inserting the value of Ftot, we obtain

Replacing Eq. (10), taking into account that J = φρv and Ohm's law, we can identify the conductivity as:

where we have introduced the coefficients σ0 = ρ2πd2/(4ηλ) and σmin = σ0γ/ρ. This equation represents the key result of our paper. An analogous derivation, for fluid dynamics in disordered media, can be found in Ref. 24, 25. Note that σ0 also can be written as σ0 = n(e2πd2c2/12kBTνλ), where n = ρ/e is the electronic number density and ν the kinematic viscosity. From this expression, we can see that the conductivity in graphene depends linearly on the carrier density, thus implying a constant mobility μ = σ/ne, in agreement with experimental observations1,26. In addition, our model can also explain why the mobility remains almost constant in the range of temperatures where ν ∝ T−1 (see Ref. 9) and the presence of a minimum conductivity in graphene, second term on the rhs of Eq. (12), σmin = eπd2c2γ/12kBTνλ, which is independent of the carrier and impurity densities. Indeed, this model cannot explain, as other theoretical models, the sublinear behavior of the conductivity for the zero range impurity because, in that range, point defects and boundary conditions start to be dominant. In addition, it cannot describe the electron-phonon interaction either, since these have been excluded at the outset. All features above will make the object of future extensions of this work.

Dimensionless numbers

For the numerical validation and in order to obtain general results, we will use dimensionless numbers. For this purpose, we can rewrite Eq. (3) alternatively as

and therefore we obtain,

where ν represents the kinematic viscosity. Let us define the following relations: , , (x, y, z) = L0(x′, y′, z′), , ρ = ρ0ρ′ and , where the prime quantities are dimensionless and u0, t0, L0, ρr0, ρ0 and E0 are characteristic values for the respective physical quantities. Thus, using the state equation , we can deduce and . Replacing these relations in Eq. (14), multiplying by and using the relation , we obtain,

To simplify this equation, we can equal the characteristic velocity to the Fermi speed, u0 = c. Therefore, we obtain

We can identify two characteristic dimensionless numbers. The first one is the Reynolds number, which is, and the second one, which we call “ number” is defined by , where V0 = E0L0 is the characteristic electrical potential. Using these definitions, we get

Note that this equation is dimensionless and therefore the universal features of the dynamics of the system are controlled only by the numbers Re and : the latter measures the strength of the electric drive, while the former scales inversely with the dissipation opposing this drive. For notational simplicity, we will remove primes, leading to

Numerical results

Fig. 2 illustrates the speed of the fluid for two different impurity densities, dark and yellow colors denoting low and high speeds respectively. An electric field of 1.77 V/m was applied in x direction (from left to right). Here we can see that for high impurity fraction (see Fig. 2, top), the speed of the fluid presents fluctuations on larger scales affecting the total conductivity of the sample. From the calculation of the electric current density and the electric field, we obtain the Ohm's law, giving a slope σ.

Figure 2
figure 2

Speed of the electronic flow.

Absolute value of the velocity in graphene with multiple impurities, for two different impurity fractions, 0.952 (bottom) and 0.999 (top). The electric field is applied in the x direction (from left to right) and set up to 1.77 V/m.

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The conductivity σ is calculated from the numerical slopes and plotted as a function of the impurity fraction. The inset of Fig. 3 reports the comparison between the analytical solution, using Eqs. (12), showing an excellent agreement with the numerical data. For the fitting parameters, we obtain σ0 = (9.9 ± 0.1) × 10−2 e2/h and σmin = 3.4 ± 0.6. Note that there is a difference between our analytical model and the numerical simulations for the minimum conductivity. This difference is due to the fact that, for high impurity densities, the flow around one impurity starts to affect the flow around the others and therefore, Eq. (12) needs some additional terms. In particular, the approximation does not hold anymore and non-linear correction terms should be included. Thus, while the minimum conductivity given by the analytical model is ~ 3.4 e2/h, the simulations give ~ 4 e2/h. We have verified that the conductivity of graphene, as computed in our model, does not show any appreciable dependence on the size of the system.

Figure 3
figure 3

Comparison between our analytical model and experimental data.

Comparison between our results (stars) and experimental data for the conductivity σ, as a function of n/ni. Data from Ref. 1 are represented by up and down triangles, from Ref. 23 by circles and squares and from Ref. 31 by diamonds and crosses, for electrons and holes respectively. Solid lines from bottom to top, theory for separations δ = 0 according to Ref. 13, our results and theory for δ = 0.2 nm according to the previous reference. In the inset (top), we show the conductivity as a function of φ2/(1−φ), with an inset to observe the minimum conductivity. In the inset (bottom) we amplify the region close to the Dirac point.

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In order to compare with experiments, we express the conductivity in terms of the ratio n/ni, where ni = (2.91 × 1016 m−2)(1−φ), in our case. According to this expression and setting n = n0, we obtain that ξ ≡ (1/α)n/ni = (1−φ)−1, with α = 4.85 × 10−3. Inserting this result into Eq. (12), we obtain σ/σ0 = ξ(1−1/ξ)2. Note that for values ξ ≫ 1, i.e. n/ni ≫ α = 4.85 × 10−3, this equation tends to

This corresponds to the linear dependence obtained by different theoretical models for graphene13,14,22,23,27,28,29,30.

In Fig. 3, we see the dependence between the conductivity and the ratio n/ni and we clearly observe the prediction for the minimum conductivity of our model. The experimental data have been taken from Refs. 1, 23, 31 and compared with the results of the present work, showing good agreement. In Fig. 3, we also compare with the model proposed by Hwang et al.13, where the impurities are located in a plane (substrate) parallel to the layer of graphene, with a separation δ between the layers. In Fig. 4, we compare our results with Coulomb impurity charges in random phase approximation (RPA)13. In the RPA model, the Boltzmann transport equation is used with impurities that are located randomly in the graphene sample. Our model shows good agreement in the slope with the RPA model, however, we achieve higher values due to the shift made by the minimum conductivity.

Figure 4
figure 4

Comparison between analytical approaches.

Conductivity σ as a function of n/ni for different types of scattering models13. RPA is the conductivity calculated by using a random phase approximation with Coulomb scatterers. The unscreened Coulomb interaction would yield a conductivity smaller than the minimum value in graphene, over the entire range of gate voltages.

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Discussion

We have developed an analytical model which accounts for a linear behavior of the conductivity with the electron density n, as well as with the ratio n/ni, in the limit n/ni ≫ 4.5×10−3. In addition, it can also model the minimum conductivity in graphene as a consequence of the carrier density induced by the presence of impurities. Our model is based on a hydrodynamic description of electron flow in graphene, whereby Coulomb interactions are included through the viscosity of the electron fluid and is valid in the collision-dominated regime. In this model, the impurities are treated as hard-sphere obstacles submerged on the electronic fluid, based on the fact that some experiments11,15,16 suggest that strong short-range neutral scatterers are the main scattering mechanism in graphene. Although this idea and the one about the long-range Coulomb scatterers are still object of controversy, the fact that the present analytical model can account for the conductivity of graphene suggests that indeed the short-range scattering models might be appropriate for graphene.

This work is based on the hydrodynamic description of electrons in graphene proposed in Ref. 3, 7, which is a model developed for undoped graphene that neglects the electron-impurity and electron-phonon interactions. Here we have -extended- this approach by adding the electron-impurity interactions through a macroscopic porous media approach. Since this approach rests on basic conservation laws, it is supposedly very robust and independent on the validity of an underlying quantum Boltzmann equation, so long the microscopic interactions justify the build-up of a macroscopic viscosity (no superconductivity or other macroscopic quantum effects of that sort). Thus, our model is able to reproduce experimental results to a satisfactory degree of accuracy.

For the set of parameters investigated in the present work, linear Ohm's law appears to apply throughout. However, based on Ref. 9, by increasing the size of the impurities (less screening), non-Ohmic behavior could occur, due the onset of pre-turbulent phenomena within the graphene sample. It would be very interesting to verify such possibility by future experiments, as well as the inclusion of the electron-phonon interaction to model both, suspended and supported samples, at higher temperatures.

Methods

For the simulation, we use the hydrokinetic fluid solver proposed by Mendoza et al.32,33,34, adapted to two-dimensional flow in graphene9. The simulation was implemented on a grid of size 256 × 512 cells, representing a rectangular graphene sample of size 1.5 × 3 μm. We set up samples with a fixed number of impurities located randomly on the grid, each impurity covering one cell size, varying φ between 0.4 and 0.999. The Dirac-quasiparticle fluid in graphene has a kinematic viscosity ν = 8.57 × 10−3 m2/s (see Ref. 9) and by taking the Fermi speed u0 = 106 m/s as a characteristic speed, we obtain a Reynolds number Re = 350. Equating Re for both systems, in physical and numerical units, the cell size and time step are fixed to δx = 5.86 nm and δt = 5.86 fs. For a given temperature, T0 = 100 K in our case, we can calculate the carrier density induced by the thermal energy with Eq. (4), n0 = 1.411 × 1014 m−2 and therefore, using the approximate relation 3,7, the energy density and the density ρr0 = 5.84 × 10−19 kg/m2. Using the electron charge, we obtain the charge density, ρ0 = 2.26 × 10−5 C/m2. In numerical units, these values correspond to n0 = 4.845 × 10−3, and ρr0 = ρ0 = 1.0, where the charge of the electron is e = 2.064 × 102. Using the characteristic velocity u0, we can calculate the value of the characteristic current density J0 = ρ0u0 = 22.6 A/m or J0 = ρ0u0 = 1.0 in physical and numerical units, respectively. On the other hand, to obtain realistic values of , we use a characteristic electric field of E0 = 4.41 V/m, which in numerical units corresponds to E0 = 10−6. In this work, takes values from 10−5 to 2 × 10−4.

To model the extra carrier density induced by the impurities, as described in Eq. (5), we introduce an extra density charge Δρ localized on each impurity position. Therefore, each impurity contributes a quantity ρ* = Δρ dx2/A to the total charge of the sample, such that ρ = ρth + ρ* N = ρth + Δρ ni, where ni, denotes the impurity density. This linear dependence between ρ and ni is in qualitative agreement with experimental data23. We made several simulations for different values of Δρ, finding that Δρ = 60 leads to a minimum conductivity of 4 e2/h. In our analytical model, this value corresponds to γ = 60, in numerical units. The simulations ran up to 5 × 105 time steps.