Abstract
Pristine graphene and graphenebased heterostructures can exhibit exceptionally high electron mobility if their surface contains few electronscattering impurities. Mobility directly influences electrical conductivity and its dependence on the carrier density. But linking these key transport parameters remains a challenging task for both theorists and experimentalists. Here, we report numerical and analytical models of carrier transport in graphene, which reveal a universal connection between graphene’s carrier mobility and the variation of its electrical conductivity with carrier density. Our model of graphene conductivity is based on a convolution of carrier density and its uncertainty, which is verified by numerical solution of the Boltzmann transport equation including the effects of charged impurity scattering and optical phonons on the carrier mobility. This model reproduces, explains, and unifies experimental mobility and conductivity data from a wide range of samples and provides a way to predict a priori all key transport parameters of graphene devices. Our results open a route for controlling the transport properties of graphene by doping and for engineering the properties of 2D materials and heterostructures.
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Introduction
The unique electrical properties of graphene, such as high carrier mobility, µ > 10^{4} cm^{2}/Vs, at room temperature^{1,2,3}, offer significant advantages for applications ranging from fast electronics to touch screens and ultrasensitive photon detection^{4,5,6}. However, the emergence of graphene electronics on the market is limited by the absence of costeffective largescale production of highquality graphene with reproducible electronic properties. The best results have been achieved in exfoliated suspended singlelayer graphene (SLG) samples a few micrometres across, in which µ is limited only by the scattering of charge carriers (electrons and/or holes) by intrinsic phonons^{7,8}. Epitaxial growth of graphene by chemical vapour deposition (CVD)^{9,10} or SiCsurface growth^{11,12} methods provides costeffective growth of large (>10 mm) SLG layers. However, the mobility of suspended sheets of graphene is markedly different to that of graphene deposited on a substrate^{13,14}. The presence of charged impurities near the graphene significantly reduces µ due to the associated longrange Coulombscattering centres^{7,15}. If impurities are present, they often ionise and form chargescattering centres, which deflect the trajectories of electrons and holes in the twodimensional (2D) layer, thereby degrading the mobility. Often scattering by charged impurities has the dominant effect on the transport properties of graphene and effects due to ballistic transport are negligible^{16}. Charged impurities are introduced in the substrate and/or in the SLGcapping layer during the device processing, e.g., created by the diffusion of metallic ions present in the solvents or etching solutions used^{17}. Recent studies indicate that resonant impurities^{18} and neutral impurities and defects limit graphene mobility^{15,19,20} and can dominate at carrier densities away from neutrality point, resulting in a sublinear dependence of the conductivity on carrier concentration.
Various theoretical models have been proposed to explain the effect of impurities on the carrier mobility in SLG. It is commonly accepted that µ is inversely proportional to the charged impurity density, n_{imp}, but is independent of carrier density n^{21,22}. Scattering by chargeneutral point defects can also affect µ, which in this case is inversely proportional to the carrier density^{23,24}, making it the dominant scattering mechanism at large carrier densities. To date, qualitative and semiquantitative models have been developed to describe the conductivity minimum; however, the physical mechanism for the minimum conductivity is not yet fully understood^{21,25}. Owing to the 2D nature of SLG, the mobility is sensitive to the surrounding environment, in particular to the presence and position of the charged impurities. However, there is still limited understanding of the effect of the standoff distance, d, of the impurities from the graphene plane on the carrier mobility and other transport parameters. Additional complications arise in graphenebased heterostructures where SLG is sandwiched between two other materials with the same or different dimensionality (threedimensional (3D) bulk, 2D layers and/or 0D quantum dots (QDs)^{2,26,27,28}).
Here we report a theoretical and experimental study of the effects of charged impurities and optical phonons on the charge transport properties of graphene. We develop a conductivity model for graphene based on the convolution of carrier density, which accounts for temporal and spatial fluctuations of the carrier density. This model accurately reproduces the experimental data and provides a robust, simple way to model the graphene conductivity as a function of the impurity density and carrier concentration. We show that the experimentally measured parameter δn, which is the full width at half maxima of the ρ(V_{g}), enables us to fit the whole conductivity curve σ(V_{g}) and to determine the exact shape of the conductivity minimum plateau for a wide range of graphene devices. Hence, several properties of electron transport in graphene can be determined using δn including the mobility and concentration dependence on V_{g}. This model is verified by numerical kspace simulations of carrier transport. We use the Discontinuous Galerkin (DG) technique to numerically solve the Boltzmann transport equation and investigate the effect of charged impurity scattering on the electron/hole mobility. We demonstrate that such processes give rise to universal mobility characteristics over a wide range of carrier densities and in the presence of multiple sources of scattering. The calculations are supported by experimental results obtained on both pristine and surfacedecorated graphene devices, which have the following structures Si/SiO_{2}/Graphene and Si/SiO_{2}/Graphene/2D(0D), respectively. Our investigations show how these scattering processes give rise to mobility characteristics, which are universal over a wide range of graphene devices, and thus potential sources of scattering. Our results enable new understanding, based on firstprinciples calculations, of the link between different transport parameters, which are of fundamental and applied interest.
Results and discussion
Modelling the transport properties of graphene
In our work, we consider graphene sheets with charged impurities at a distance, d_{imp}, from the plane of the graphene and optical phonons with energy ħω (Fig. 1a). We model the effect of impurities and optical phonon scattering on the following transport properties of graphene: the carrier concentration (n), mobility (µ), conductivity (σ) and resistivity (ρ) at the Dirac point (σ_{min} and ρ_{max}, respectively). The graphene conductivity in the vicinity of the Dirac point can be strongly affected by a number of different phenomena besides impurity scattering, including ballistic transport effects^{29}, quantum capacitance^{7,30} and temperature^{31,32}. As a result, the device conductivity and carrier density are nonzero even when the Fermi energy, ε_{F}, is at the Dirac (charge neutrality) point. Therefore, a simple model for the Drude conductivity:
with a constant mobility, µ, is not applicable for small gate voltages, V_{g}, assuming the classical capacitance model for the graphene’s carrier number density
where n_{0} = n_{c}(V_{g} = 0) is the sheet density of the graphene doping (Supplementary Note 1). Spatial fluctuations of the local electrostatic potential in the graphene layer and the presence of electron and hole ‘puddles’ (Fig. 1b) are thought to explain the nonzero conductivity and resistance (σ_{min} and ρ_{max} in Fig. 1c, d) observed at the Dirac point^{21,33}. Electrons and holes play equal roles in determining the graphene conductivity with no scattering at the borders between the n and ptype graphene areas due to the Klein paradox^{34}.
We now consider spatial fluctuations in the carrier number density and their dependence on the gate voltage. Combined with the Drude conductivity, this model accurately describes the shape of the measured ρ(V_{g}) curve (Fig. 1d).
To first approximation, the carrier number density at a given gate voltage, n(V_{g}), can be modelled as the moving average (convolution) of n_{c} (Eq. (2)) over a window of width δn, which is the characteristic amplitude of the carrier density fluctuations in the graphene layer. This is equivalent to the convolution of the n_{c}(V_{g}) function with a box function f(n_{c}) of width δn (see Supplementary Video (convolution2.avi) and detailed description in Supplementary Note 1), which gives \(n = n_{\mathrm{c}} \ast\, f(n_{\mathrm{c}})\), where \(f(n_{\mathrm{c}}) = \left\{ \begin{array}{*{20}{c}} \frac{1}{{\delta n}} & {\mathrm{for}}  \frac{{\delta n}}{2}\,<\,n_{\mathrm{c}} < + \frac{{\delta n}}{2} \hfill\\ 0 & {\mathrm{for}}  \frac{{\delta n}}{2}\,> \, n_{\mathrm{c}}\;{\mathrm{or}}\;n_{\mathrm{c}} > + \frac{{\delta n}}{2} \end{array} \right.\). Using the linear n_{c}(V_{g}) dependence in Eq. (2) gives
This expression for n(V_{g}) is equal to the constantcapacitance model for gate voltages where \(n_{\mathrm{c}}(V_{\mathrm{g}})\,> \,\frac{{\delta n}}{2}\) and has a parabolic form for gate voltages close to the Dirac point, where \(n_{\mathrm{c}}(V_{\mathrm{g}})\,<\,\frac{{\delta n}}{2}\). Using Eq. (3), we can determine the conductivity as
From this expression, it can be shown that δn equals the full width half maximum of the peak resistivity around the Dirac point. In addition, when n_{c} = 0, δn = 4n_{NP}, where n_{NP} = σ_{min}/eμ, is the residual carrier density at the Dirac (neutrality) point. Our model thus provides a simple expression which enables us to extend the linear conductivity model^{8} to values of n close to the Dirac point, which was not possible with previous models^{21}. Thereby, the full observed σ(V_{g}) and ρ(V_{g}) dependences can be accurately reproduced (Fig. 1c, d).
To model δn(V_{g}) and μ(V_{g}) curves in more detail, we consider semiclassical kspace simulations of the electron and hole dynamics. Graphene has a linear energywavevector dispersion relation, with electron energy in graphene ε = ±ħvk, where v = 10^{6} ms^{−1} is the speed of electrons in graphene, for the electrons and holes at the two inequivalent valleys, K and K′, in reciprocal space. Charge carriers undergo diffusive scattering transport, which we describe using a semiclassical Boltzmann transport approach. The influence of perturbations, such as impurities and phonons on the scattering of electrons is calculated using the Fermi golden rule for transition rates between states. The electrons are initially assumed to obey a Fermi–Dirac distribution, f_{0}(k). Interband transitions are neglected such that the valence band is assumed to be full throughout the time evolution when the gate voltage is positive, i.e., when the chemical potential lies within the conduction band. We assume full ionisation of all the impurities and their distribution to be independent on gate voltage. In the highgate voltage regime, this assumption is confirmed by the linear dependence of n(V_{g}), where the value of n is determined using the equation for the field effect capacitance and verified using the Hall effect measurements.
The spatially homogeneous Boltzmann transport equation,
describes the evolution of the occupancy, f(t, k), of state k at time t. The first term on the righthand side of Eq. (5) describes the acceleration of electrons under an applied electric field, E, and the collision term is given by the detailed balance equation,
where S_{k→k′} is the transition rate of carriers from a state of crystal momentum ħk to a new state with momentum ħk′. Equation (6) represents the collision integral. In the particular case of elastic scattering, S_{k→k′} = S_{k′→k} and the products of the two distributions cancel. We solve Eq. (5) using the DG approach^{35} (for detailed solution, see Supplementary Note 2) for the steadystate distribution function, f(k). We then determine the mobility, μ, for an applied electric field, E, from the drift velocity:
Alternatively, to find approximate analytical solutions to Eq. (5), we can assume a small shift in the initial distribution function, f_{0}(k), proportional to the ensemble momentum relaxation time, τ_{m}. This results in the linearised Boltzmann (LB) approximation^{36} for the mobility, which, at zerotemperature, is related to the relaxation time at the chemical potential, ε_{F}, via
We calculate the momentum relaxation time, τ_{m}(k), as the sum over all possible transition rates, S_{k→k′}, modified by the deflection angle, θ_{k,k′}, between the incoming and outgoing vectors:
where A is the area of graphene unit cell.
The effect of screening by the electron and hole gases is included by introducing a random phase approximation for the dielectric screening function^{37,38}
where ṽ_{2D} = e^{2}/2ϵ_{0}ϵ_{r}q is the unscreened Coulomb potential in Fourier space, Π(q) is the static polarisation function and the reciprocal space variable q = k′ − k. As the conduction band distribution function changes throughout the simulation, the screening function should be carefully considered. However, the valence band distribution is constant throughout as the band is assumed to be full. The polarisation function for screening by a full valence band is^{38}
For the conduction band, at T = 0, maximum screening occurs in the Thomas–Fermi limit, q → 0, and is given by
where D(ε) = 2ε/π(ħv)^{2} is the density of states. Throughout the simulation, the integral in Eq. (12) does not change, due to conservation of charge. As both Eqs. (11) and (12) are independent of the evolution of the distribution function, we define a timeindependent tworegime screening function, where Thomas–Fermi screening is assumed for low energy scattering and the valence electron screening is assumed for highenergy electrons^{21}, i.e., we set
where r_{s} = e^{2}/(4πϵ_{0}ϵ_{r}ħv) and q_{s} = 4k_{F}r_{s}. For graphene on SiO_{2}, we take ϵ_{r} ≈ 2.45^{22}. A Coulombic scattering potential is assumed for charged impurities near the graphene plane:
where d_{imp} is the distance of the impurities from the graphene plane. It is noteworthy that in this model we consider randomly distributed impurities and disregard any possible spatial correlation of charges below and above the graphene plane^{39,40}. Then, the transition rate is
where q = 2k sin(θ_{k,k′}/2) for elastic scattering. Defects that perturb the band structure over a small spatial area are characterised by the shortrange scattering potential U(r) = U_{0}H(R − r), where H is the Heaviside step function and R gives the spatial extent of the perturbation. This potential represents any chargeneutral point defects within the lattice. The rate of carrier scattering transitions due to such defects is
where A_{sr} = πR^{2} is the effective crosssection of defects with an areal density n_{sr}.
In our calculations, we assume a low temperature and a phonon occupation of N ≈ 0 (k_{B}T ≪ ħω) and perform our numerical calculations using an initial Fermidistribution of low finite temperature, T = 20 K, to avoid discontinuities over the discretised kspace. Therefore, one might not expect phonons to have a significant effect on the transport properties compared to that of scattering by impurities^{7}. However, for low carrier densities, we find carriers can be accelerated to high energies (~100 meV) resulting in a ‘hot electron’ distribution (Joule heating), as was observed previously, e.g., in metals^{41}. In this case, inelastic optical phonon scattering becomes important in relaxing the energy of the carriers. Hot electron phenomenon is a particularly important consideration for transport in graphene due to weak electronacoustic phonon scattering and relatively high optical phonon energies. Our calculations show that the hot electron effect is significant even for electric fields as low as ~100 V/m, comparable to commonly used experimental values (for steadystate characteristics of the Boltzmann equation, see Supplementary Note 3). We use the optical phonon scattering rates calculated using density functional theory in refs. ^{42,43}. Near the Γ− points of the reciprocal lattice, the energy and coupling strength of both transverse and longitudinal optical phonons are reported to be ħω_{O} ≈ 165 meV and β_{O} ≈ 10 eV/Å, respectively^{44}. Therefore, we can combine the transition rates of the two modes to obtain a single overall optical scattering rate
Phonons at the Kpoints cause intervalley scattering at a rate
where ħω_{K} ≈ 124 meV and β_{K} ≈ 3.5 eV/Å^{44}, and ρ_{m} = 7.6 × 10^{−7} kg/m^{2} is the mass density of graphene.
We consider a residual charge density of electronhole puddles at the Dirac point, due to inhomogeneity in the impurityinduced potential (Fig. 1b), which limits the minimum conductivity. To calculate the residual charge, and thus the minimum chemical potential in our calculations, we use Eq. (8) in ref. ^{21}, derived assuming a random distribution of impurities. Here we assume that the transition from the residual chargedominated minimum carrier concentration to the linearly V_{g}dependent concentration occurs when the gateinduced charged density, n(V_{g} − V_{0}), is equal to the residual charge density, n_{NP}, where V_{0} is the position of the Dirac point.
For all numerical simulations, we apply an electric field, E = 10^{4} V/m (0.1 V drop across a 10 µmlong SLG), corresponding to a regime of lowfield mobility, where μ is independent of the applied electric field strength (for details of numerical simulations, see Supplementary Note 3). For comparison, we also calculate the mobility using the LB formalism, Eq. (8), with the scattering time calculated using Eq. (9), in which the integrals are evaluated numerically. The LB method gives exact solutions when the electric field is sufficiently small and the hot electron effects are negligible. However, for small densities, carriers can be accelerated to high energies resulting in a hot carrier distribution which is far from thermal equilibrium. In this case, the LB approximation diverges from the accurate numerical solution provided by the DG approach (for details of DG approach, see Supplementary Note 3).
Figure 2a shows the calculated dependence of μ on n for n > n_{NP}. The low carrier mobility μ(n ≈ n_{NP}) corresponds to the regime where the chemical potential is near the Dirac point. With increasing n, we observe an initial increase of μ. This is followed by a peak and a monotonic decrease of μ at large n. This dependence arises from the competition between scattering by longrange Coulombic impurities and shortrange defects. Shortrange defect scattering is found to be dominant at large n, as expected from comparison of the momentum relaxation time for shortrange defects, τ_{sr} ~ n^{−1/2}, calculated using the Born approximation, and longrange impurities, τ_{imp} ~ n^{1/2}. Beyond the Born approximation, for sufficiently strong defect scattering, the exponent of n in the momentum relaxation time, τ_{sr}, can increase towards that of longrange impurity scattering^{33}. The dependence of mobility on carrier concentration, μ(n), is affected by the density of impurities, n_{0}, and by their distance from the graphene, d_{imp}. Hence, both δn and the low carrier mobility, μ(n ≈ n_{NP}), depend on n_{0} and d_{imp}. As shown in Fig. 2b, the mobility increases as d_{imp} is increased. Furthermore, for low impurity densities, and thus small residual charge densities, the mobility given by the DG simulations differs from that obtained from the LB calculations, whereas the two methods give μ values that converge at higher impurity densities.
Both the LB model and the DG simulations assume that initially, at t = 0, the electron gas is in thermal equilibrium and obeys the Fermi–Dirac distribution. Equation (8) assumes a linear shift in the momentum of the ensemble, proportional to the ensemble relaxation rate, τ(ε_{F}), whereas the DG simulations include the time evolution of the momentum distribution, described by the full Boltzmann transport equation, Eq. (5). Therefore, the discrepancy between the two methods seen at low impurity densities can be understood by consideration of the steadystate distribution functions. Figure 2c, d shows the final distribution of electrons obtained for a small impurity density, using the DG simulation. We obtain similar results by Monte Carlo simulations^{35,45} (Monte Carlo simulations are described in part (A) of Supplementary Note 2) as shown in Fig. 2d. In both the DG and Monte Carlo simulations, with increasing t we observe continuous spreading of the electron distribution in kspace, until the occupied kvalues become limited by inelastic phonon scattering (see also Supplementary Note 3). Hence, a hot electron regime is realised, which is not captured within the LB approximation.
The effect of d_{imp} on the electrical properties of graphene is summarised in Fig. 3. Our calculations demonstrate that the linewidth δn of the ρ(V_{g}) curve broadens with decreasing d_{imp} (Fig. 3a). Combining the results of Figs. 2b and 3a, the mobility decreases with increasing δn (Fig. 3b), with the broadening of δn being larger for smaller d_{imp} at a given value of mobility. At small values of δn, and hence small impurity densities n_{0}, we observe discrepancy between the DG and LB calculations of μ (as shown in Figs. 2b and 3b). Despite the discrepancy in μ(n_{0}) and μ(δn), we find that both methods yield a similar δn(n_{0}) profile. We now compare our calculations to experimental measurements.
Universal mobility characteristics
We apply our analysis to experimental results reported previously for >20 devices fabricated using exfoliated and CVDgrown graphene deposited on Si/SiO_{2} substrate. We use both pristine graphene devices and graphene heterostructures incorporating 2D layers (InSe, hBN) or 0D nanostructures (colloidal QDs, inorganic perovskites)^{46,47,48} (Fig. 3a). In these devices, impurities at a distance, d_{imp}, from the 2D plane of graphene act as scattering centres. We fit the measured σ(V_{g}) dependencies and determine values of μ and δn (fit of σ(V_{g}) is described in Supplementary Note 1 and phenomenological fit of experimental data is in Supplementary Note 4). As shown in Fig. 3b, the mobility increases with decreasing δn. The experimental values measured in pristine graphene devices are in good agreement with the results of our DG simulations with d_{imp} = 2 nm. Interestingly, our model provides good fit for highmobility exfoliated graphene, where other scattering mechanisms could play a significant role. Since the convolution model is based on experimentally determined value of δn, it accounts for all different scattering mechanisms (for universality of analytical convolution model, see Supplementary Note 5).
We note that our fit (Fig. 3b) uses δn calculated from the full width at half maximum of the σ(V_{g}) curve rather than from the value of n_{0} extracted from the gate voltage at which σ(V_{g}) = σ_{max}. By using Eq. (4) and assuming the universal minimum conductivity for pristine graphene as σ_{min} ≈ 4e^{2}/h^{29} and constant mobility (with respect to carrier density), we obtain a simple inverse power law for the dependence of μ on δn.
Equation (19) includes one measured parameter δn, which simplifies the data analysis, as demonstrated on the experimental data from a wide range of devices (experimental results are included in Supplementary Note 4). Overall, our model, which considers the effect of impurity scattering to be dominant on mobility, describes well all examined types of graphene: highmobility exfoliated graphene and lowmobility CVDgrown graphene. We stress that, remarkably, even in devices where other transport mechanisms are important, e.g., ballistic transport in highpurity exfoliated graphene, their electronic properties can be determined using the measured value of δn.
Recently, the decoration of graphene devices with other lowdimensional materials, such as 0D (colloidal PbS quantum dots^{46} or CsPbI_{3} perovskite^{47}) and 2D (InSe flakes)^{48} materials has been used to functionalise these devices, e.g., for photon sensing^{5,47,48}. The properties of the graphene heterostructures are greatly affected by both the unintentional presence of charge impurities in the vicinity of graphene (as described above by d_{imp}) and those deliberately introduced by the top layer (d_{top}) in graphene heterostructures (Fig. 4a), which we model as a distribution of impurities at an effective distance, d_{eff}. We note that in surfacedecorated graphene devices, the distance between the graphene plane and the top layer can be controlled, e.g., by introducing a dielectric layer such as hBN, thus providing a tool for tailoring the electrical properties. The relationship between mobility and the gatevoltage offset is μ ∝ 1/n_{0} for most pristine devices^{21}. However, for devices with high densities of correlated unipolar charges^{39,40} or uncorrelated bipolar charges^{49}, spatial correlation between charges must be considered. This is particularly important when the dopants are mobile and able to adopt low energy, correlated configurations. Such effects were recently demonstrated for quantum dotdecorated graphene and validated using Monte Carlo simulations^{40,49}.
Despite the different μ(n_{0}) characteristics of decorated and pristine graphene, remarkably, we find that both types of devices exhibit the universal scaling behaviour shown in Fig. 4b. Different surfacedecorated devices follow a common trend observed in pristine graphene. In particular, the experimental results for the InSe, perovskite and PbS decorated SLG are best fitted by DG calculations when d_{eff} = 1 nm. Therefore, we find that the relationship between μ and δn is consistent throughout all of the devices, as can be expected from the analytical expression given in Eq. (19), with modifications to only the effective distance of the impurities. Flexibility to modify composition and/or geometry of a heterostructure offers opportunities to tune the distribution and standoff distance of ionised impurities, hence changing d_{eff} and providing a tool to control transport properties of these devices. We note, that our model is valid for all devices where the position of ionised impurities is not affected by V_{g}. In rare cases, at high V_{g} regime, the ionisation of donor impurities can be affected by applied gate voltage (e.g., see ref. ^{50}) and the corresponding change of d_{imp} would need to be accounted for.
Our model links together three key transport parameters of SLG devices: μ, n_{0} and δn, where δn can also be used to calculate ρ_{max} and σ(V_{g}) (for phenomenological equations for graphene transport parameters, see Supplementary Note 6). Remarkably, this model can be used to extrapolate the whole R(V_{g}) dependence from a single R(V_{g}) = R_{max} measurement and for a wide range of different graphene devices (see Fig. 1c, d). Our approach is based on experimental value of δn, which accounts for presence of scattering centres, but does not distinguish their nature. We envisage that majority of ionised scattering centres present in our devices originates from substrate impurities and from impurities introduced from top layer (2D or 0D). The effects of other types of ionised impurities (substitutional doping, functional groups, etc.^{51}) merits further studies.
Of particular interest is the applicability of our model to a wide variety of different graphene types and to different device structures and geometries. Consequently, the model has the potential to both predict and explain the observed behaviour of newly emerging device concepts and graphene types. Recently, the need for upscaling of graphene growth and device manufacturing has led to significant research focus on Molecular Beam Epitaxial growth^{52}, liquid exfoliation of 2D materials^{53} and additive manufacturing (3D printing) of graphene devices^{54,55}. Our preliminary results indicate that our model can be optimised and expanded to explain and predict the properties of 3D printed graphene devices, by accounting for flaketoflake hopping of charges^{56}.
Conclusions
We have developed a universal analytical convolution model of electron transport in graphene and graphene heterostructures, supported by numerical timedependent analysis of the charge carrier distributions. Our model includes the effects of impurities and optical phonons on the observed charge transport properties of graphene devices. We find that the properties of a wide range of devices, from highquality graphene with low carrier density to graphenebased heterostructures, exhibit universal behaviour that can be accurately described with this model. Our calculations combine multiple parameters that affect charge transport in graphene and facilitate the design, accurate ab initio prediction of key transport parameters and analysis of future electronic and optoelectronic devices based on 2D materials. Furthermore, our results may be generalised to predict and improve the electrical behaviour of 2D multilayers made by 3D printing or from reduced graphene oxide, which are promising candidates for the scalable highyield manufacture of largearea optoelectronic devices that harness the unique properties of 2D materials.
Data availability
All relevant data are available from the authors upon request. Contact authors are Mark Fromhold (Mark.Fromhold@nottingham.ac.uk) and Lyudmila Turyanska (Lyudmila.Turyanska@nottingham.ac.uk).
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Acknowledgements
This work was funded by the Engineering and Physical Sciences Research Council [grant number EP/P031684/1]. We acknowledge support from the DSTL and the European Union’s Horizon 2020 research and innovation programme under grant agreement Graphene Core 3.
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J.G. and O.M. performed modelling studies. N.C. analysed experimental data. J.G., O.M., F.W., N.C., M.G., A.P., R.W., C.T., L.T. and T.M.F. contributed to design of the study, analysis and interpretation of results. J.G., O.M., F.W., N.C., M.G., A.P., R.W., C.T., L.T. and T.M.F. cowrote and approved the manuscript.
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Gosling, J.H., Makarovsky, O., Wang, F. et al. Universal mobility characteristics of graphene originating from charge scattering by ionised impurities. Commun Phys 4, 30 (2021). https://doi.org/10.1038/s42005021005182
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DOI: https://doi.org/10.1038/s42005021005182
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