Main

Several theoretical studies7,8,13,14,15,17,18 have predicted charged-impurity scattering in graphene to produce σ(n) of the form

where C is a constant, e is the electronic charge and σres is the residual conductivity at n=0 (this last term was predicted only in refs 17, 18; see the Supplementary Information for a more detailed discussion of the theory). Hwang et al. 14 first calculated the screened Coulomb potential within the random phase approximation, and used the results to determine C=5×1015 V−1 s−1. Novikov 16 noted that, beyond the Born approximation used in ref. 14, an asymmetry in C for attractive versus repulsive scattering (electron versus hole carriers) is expected for Dirac fermions. Experimentally, the behaviour described by equation (1) is ubiquitously observed1,2,3,4,5,6 in graphene, strongly suggesting that charged-impurity scattering is the dominant scattering mechanism in present samples. Here, we provide the first direct verification of equation (1) for charged-impurity scattering in graphene, and determine the constant C. We also observe the expected asymmetry for attractive versus repulsive scattering for Dirac fermions 16.

At low carrier density, the conductivity does not vanish linearly, but rather saturates to a constant value near 4e2/h (ref. 2). Early theoretical work 20,21 on massless Dirac fermions predicted σmin=4e2/πh for vanishing disorder. However, in the presence of charged impurities, a finite conductivity ∼4e2/h is predicted over a plateau of width ΔVg (refs 8, 14, 15). Here, we measure experimentally the dependence on nimp of σmin, ΔVg and the gate voltage Vg,min at which the minimum conductivity occurs, and find agreement with theoretical predictions8,14,15, indicating that disorder due to charged impurities is the relevant physics at the minimum conductivity point in present samples.

Figure 1a shows the graphene device used in this study, and Fig. 1b shows its micro-Raman spectrum; the single lorentzian D′ peak confirms that the device is single-layer graphene22 (see the Methods section). To vary the density of charged impurities, the device was dosed with a controlled potassium flux in sequential 2 s intervals at a sample temperature T=20 K in ultrahigh vacuum (UHV). The gate-voltage-dependent conductivity σ(Vg) was measured in situ for the pristine device, and again after each doping interval. After several doping intervals, the device was annealed in UHV to 490 K to remove weakly adsorbed potassium23, then cooled to 20 K and the doping experiment repeated; four such runs (runs 1–4) were carried out in total.

Figure 1: Graphene device.
figure 1

a, Optical micrograph of the device. b, 633 nm micro-Raman shift spectrum acquired over the device area, with lorentzian fit to the D′ peak, confirming that the device is made from single-layer graphene (vertical scale is the same throughout b).

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Figure 2 shows the conductivity versus gate voltage for the pristine19 device and at three different doping concentrations at 20 K in UHV for run 3 (see also the Supplementary Information for measurements on a second device). On K-doping, (1) the mobility decreases, (2) σ(Vg) becomes more linear, (3) the mobility asymmetry for holes versus electrons increases, (4) the gate voltage of minimum conductivity Vg,min shifts to more negative gate voltage, (5) the width of the minimum conductivity region in Vg broadens and (6) the minimum conductivity σmin decreases, at least initially (see also Fig. 5 below and Supplementary Information). In addition, (7) the linear σ(Vg) curves extrapolate to a finite σres at Vg,min. All of these features have been predicted7,8,13,14,15,17,18 for charged-impurity scattering in graphene; we will discuss each in detail below.

Figure 2: Potassium doping of graphene.
figure 2

The conductivity (σ) versus gate voltage (Vg) curves for the pristine sample and three different doping concentrations taken at 20 K in UHV. Data are from run 3. Lines are fits to equation (1), and the crossing of the lines defines the points of the residual conductivity and the gate voltage at minimum conductivity (σres, Vg,min) for each data set. The variation of σmin with impurity concentration is shown in Fig. 5.

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Effects (4) and (5) were observed in a previous study in which graphene was exposed to molecular species24. However, the authors reported no changes in mobility, concluding that charged-impurity scattering contributes negligibly to the mobility of graphene. As discussed further in the Supplementary Information, the previous experiments did not control the environment and had low initial sample mobility. The failure to observe effects (1)–(3) therefore is most likely due to the presence of significant concentrations of both positively and negatively charged impurities24,25, although the presence of water and resist residue19 may also be contributing factors24.

We first examine the behaviour of σ(Vg) at high carrier density. For Vg not too near Vg,min, the conductivity can be fitted (Fig. 2) by

where μe and μh are the electron and hole field-effect mobilities, cg is the gate capacitance per unit area, 1.15×10−4 F m−2, and σres is the residual conductivity that is determined by the fit. The mobilities are reduced by an order of magnitude during each run, and recover on annealing. The electron mobilities ranged from 0.081 to 1.32 m2 V−1 s−1 over the four runs, nearly covering the range of mobilities reported so far in the literature (∼0.1–2 m2 V−1 s−1) (refs 2, 3, 6).

For uncorrelated scatterers, the mobility depends inversely on the density of charged impurities, 1/μ∝nimp, and equations (1) and (2) are identical. We assume nimp varies linearly with dosing time t as potassium is added to the device. Figure 3 shows 1/μeand 1/μh versus t, which are linear, in agreement with 1/μ∝nimp, hence verifying that equation (1) describes charged-impurity scattering in graphene. We estimate the dosing rate dnimp/dt=(2.6–3.2)×1015 m−2 s−1 and the maximum concentration of (1.4–1.8)×10−3 potassium per carbon (see the Supplementary Information). From this point, we parameterize the data by 1/μe, proportional to the impurity concentration (the data set for μe is more extensive than for μh because of the limited Vg range accessible experimentally). 

Figure 3: Inverse of electron mobility 1/μe and hole mobility 1/μh versus doping time.
figure 3

Experimental error determined from standard error propagation is less than 4% (see the Methods section). Lines are linear fits to all data points. Inset: The ratio of μe to μh versus doping time. Error bars represent experimental error in determining the mobility ratio from the fitting procedure (see the Methods section). Data are from run 3 (same as Fig. 2).

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Figure 3, inset shows that, although the μe and μh are not identical, their ratio is fairly constant at μe/μh=0.83±0.01 (see the Methods section). Novikov16 predicted μe/μh=0.37 for an impurity charge Z=1; however, the asymmetry is expected to be reduced when screening by conduction electrons is included.

As K-dosing increases and mobility decreases, the linear behaviour of σ(Vg) (Fig. 2) associated with charged-impurity scattering dominates, as predicted theoretically14. At the lowest K-dosing level, sublinear behaviour is observed for large |Vg−Vg,min| as anticipated. The dependence of the conductivity on carrier density n∝|Vg–Vg,min| is expected to be σ∝na with a=1 for charged impurities and a<1 for short-range and ripple scattering (see the Supplementary Information). Adding conductivities in inverse according to Matthiessen’s rule indicates that scattering other than by charged impurities will dominate at large n, with the crossover occurring at larger n as nimp is increased14. A previous study3 also found more linear σ(Vg) for devices with lower mobility. Thus, our data indicate that the variation in observed field-effect mobilities of graphene devices is determined by the level of unintentional charged impurities.

We now examine the shift of the curves in Vg. Figure 4 shows Vg,min as a function of 1/μe. Run 1 differs from runs 2–4, presumably owing to irreversible changes as potassium reacts with charge traps on silicon oxide and/or edges and defects of the graphene sheet. After run 1, subsequent runs are very repeatable, other than an increasing rigid shift to more negative voltage in the initial gate voltage of minimum conductivity. (The same distinction between first and subsequent experiments is seen in Fig. 5 as well.) It might be expected that the minimum conductivity would occur at the induced carrier density that precisely neutralizes the charged-impurity density: n=−Z nimp or ΔVg,min=−nimpZ e/cg (ref. 24), where e is the elementary charge and Z e is the charge of the potassium ion. This prediction is shown as the long-dashed line in Fig. 4; the experimental data show a distinctly different effective power-law dependence. Adam et al. 15 proposed that the minimum conductivity in fact occurs at the added carrier density at which the average impurity potential is zero, , where is a function of nimp, the impurity spacing d from the graphene plane and the dielectric constant of the SiO2 substrate. The theory also assumes that Z=1; experimentally, a reasonable evaluation26 of Z for dilute potassium on graphite is ∼0.7 (see the Supplementary Information for further discussion of Z≠1). The theoretical lines in Fig. 4 are given by the exact result of Adam et al. 15, and follow an approximate power-law behaviour of ΔVg,min∝nimpb with b=1.2–1.3, which agrees well with experiment. The only adjustable parameter is the impurity–graphene distance d; we show the results for d=0.3 nm (a reasonable value for the distance of potassium on graphene26,27,28) and d=1.0 nm (the value used by Adam et al.). As ΔVg,mingives an independent estimate of nimp, the quantitative agreement in Fig. 4 verifies that C=5×1015 V−1 s−1 in equation (1), as expected theoretically.

Figure 4: Shift of minimum conductivity point with doping.
figure 4

The gate voltage of minimum conductivity Vg,min as a function of inverse mobility, which is proportional to the impurity concentration. All four experimental runs are shown. Each data set has been shifted by a constant offset in Vg,min to make Vg,min(1/μe→0)=0, to account for any rigid threshold shift. The offset (in volts) is −10,3.1, 5.6 and 8.2 for the four runs, respectively, with the variation likely to be due to accumulation of K in the SiO2 on successive experiments. The open circles are Vg,min obtained directly from the σ(Vg) curves rather than fits to equation (1) because the linear regime of the hole side of these curves is not accessible owing to heavy doping. The solid and short-dashed lines are from the theory of Adam et al. 15 for an impurity–graphene distance d=0.3 nm (solid line) and d=1 nm (short-dashed line), and approximately follow power laws with slopes 1.2 and 1.3, respectively. The long-dashed line shows the linear relationship ΔVg,min=nimpZ e/cg, where nimp=(5×1015 V−1 s−1)/μ and Z=1.

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Figure 5: Change in behaviour near the minimum conductivity point with doping.
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a, The minimum conductivity and the residual conductivity (defined in text) as a function of 1/μe (proportional to the impurity density). b, The plateau width ΔVg as a function of 1/μe. In a and b, data from all four experimental runs are shown, as well as the theoretical predictions of the minimum conductivity and plateau width from Adam et al. 15 for d=0.3 nm (solid line) and d=1 nm (short-dashed line). Error bars represent experimental error in determining σres and ΔVg from the fitting procedure (see the Methods section); σmin is measured directly.

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We now turn to the behaviour near the point of minimum conductivity. Figure 5a shows the minimum conductivity σmin and residual conductivity σres as a function of 1/μe, and Fig. 5b shows the plateau width ΔVg as a function of 1/μe; ΔVg is the difference between the two values of Vg for which σmin=σ(Vg) in equation (2). The predictions from the theory of Adam et al. 15 for σmin and ΔVg are also shown. The minimum conductivity drops on initial potassium dosing, and shows a broad minimum near 4e2/hbefore gradually increasing with further exposure. Notably, the cleanest samples show σmin significantly greater than 4e2/h, and strongly dependent on charged-impurity density, indicating that the universal behaviour20,21 of σminassociated with the Dirac point is not observed even in the cleanest samples. The irreversible change in the value of σmin between run 1 and runs 2–4 is larger than the entire variation within runs 2–4. This difference between initial and subsequent runs indicates that the initial K-dosing and anneal cycle introduces other types of disorder (possibly short-range scatterers induced by irreversible chemisorption of potassium on defects or reaction of potassium with adsorbates) that have a comparable or greater impact on σmin than charged impurities. That, for some disorder conditions (run 1), σmin varies significantly with nimp, but for other conditions (runs 2–4), the decrease in σmin saturates rapidly with increasing nimp, and is nearly constant for a very broad range of doping, suggests that the substantial variations reported in the literature (some groups report that σmin is a universal value2, whereas other groups observe variation in σminfrom sample to sample3) are probably due to poor control of the chemical environment of the devices measured. The observed residual conductivity σres is finite and surprisingly constant (Fig. 5a); it is only weakly dependent on doping, and shows little variation between the first run and subsequent runs. Finite σres has been predicted theoretically17,18 for graphene with charged impurities; however, the magnitude has not been calculated. The change of ΔVg with doping (Fig. 5b) agrees only qualitatively with the theory, which predicts larger values and a sublinear dependence on doping. However, the quantitative disagreements between experiment and theory in Fig. 5a,b are connected: mobility, minimum conductivity and residual conductivity determine ΔVg.

In summary, the dependence of conductivity of graphene on the density of charged impurities has been demonstrated by controlled potassium doping of clean-graphene devices in UHV at low temperature. The minimum conductivity depends systematically on charged-impurity density, decreasing on initial doping, and reaching a minimum near 4e2/h only for non-zero charged-impurity density, indicating that the universal conductivity at the Dirac point2,20,21 has not yet been probed experimentally. The high-carrier-density conductivity is quantitatively consistent with theoretical predictions for charged-impurity scattering in graphene7,8,13,14,15,17,18. The addition of charged impurities produces a more linear σ(Vg), and reduces the mobility, with the constant C=μ nimp=5×1015 V−1 s−1, in excellent agreement with theory. The asymmetry for repulsive versus attractive scattering predicted for massless Dirac quasiparticles16 is observed for the first time. Finally, the minimum conductivity point15 occurs at the applied gate voltage at which the average impurity potential is zero and not at the voltage at which the gate-induced carrier density neutralizes the impurity charge.

Other observations indicate the need for fuller experimental and theoretical understanding. The irreversible changes in the behaviour around Vg,min between the first and subsequent doping runs indicate that the precise value of the minimum conductivity depends on the interplay of more than one type of disorder, and hence cannot be explained by existing theories7,8,9,10,12,14,15,17,18. An interesting new feature, the residual conductivity, may point to physics beyond the simple Boltzmann transport picture17,18. Further experiments including introducing short-range (neutral) scatterers to graphene will be useful in addressing these questions. Full understanding may require scanned-probe studies of graphene under well-controlled environmental conditions19, which can completely characterize the disorder due to defects, charged and neutral adsorbates and ripples, as well as probe the electron scattering from each29.

Methods

Graphene is obtained from Kish graphite by mechanical exfoliation30 on 300 nm SiO2 over doped Si (back gate), with Au/Cr electrodes defined by electron-beam lithography (Fig. 1a). Raman spectroscopy confirms that the samples are single-layer graphene22 (Fig. 1b). After fabrication, the devices are annealed in H2/Ar at 300 ∘C for 1 h to remove resist residues19. Gas flows are 1,700 ml min−1 (H2) and 1,900 ml min−1 (Ar) at 1 atm; gases are flowing throughout heating and cooling. The devices are mounted on a liquid-helium-cooled cold finger in a UHV chamber, so that the temperature of the device can be controlled from 20 to 490 K.

Following a vacuum bakeout, each device is annealed in UHV at 490 K overnight to remove residual adsorbed gases. Experiments are carried out at pressures lower than 5×10−10 torr and device temperature T=20 K. Potassium doping is accomplished by passing a current of 6.5 A through a getter (SAES Getters) for 40 s before the shutter is opened for 2 s. The getter temperature during each potassium dosage was 763±5 K as measured by optical pyrometry. The stability of the potassium flux was monitored by a residual gas analyser positioned off-axis and behind the sample (see the Supplementary Information). All measurements were carried out on one four-probe device shown in Fig. 1a, although several two-probe devices showed similar behaviour (see Supplementary Information, Figs S1,S3–S5).

Conductivity σ is determined from the measured four-probe sample resistance R using σ=(L/W)(1/R). Because the sample is not an ideal Hall bar, there is some uncertainty in the (constant) geometrical factor L/W. We estimate L/W=0.80±0.09, where the error bars represent ± one standard deviation. This 11% uncertainty in L/W translates into an 11% uncertainty in the vertical axes of Figs 2 and 3, the horizontal axes of Figs 4 and 5b and both axes of Fig. 5a. Such scale changes are comparable to the spread among different experimental runs, and do not alter our conclusions. Notably, the uncertainty represents a systematic error, so relative changes in, for example, the minimum conductivity with charged-impurity density are still correct.

Best fits to equation (1) were determined using a least-squares linear fit to the steepest regime in the σ(Vg) curves. The steepest regime of the σ(Vg) curves was determined by examining dσ/dVg; the fit was carried out over a 2 V interval in Vg around the maximum of dσ/dVg. Other criteria for determining the maximum field-effect mobility give similar results. The experimental errors in μe and μh are determined by the fitting procedure described above; the errors in Vg,min, σres, ΔVg (plateau width) and μe/μh are then calculated using equation (1) and standard error propagation. The errors (standard deviation) in μe, μh and Vg,min were typically less than 4%. σmin is measured directly, and has less than 1% error. Error bars (± one standard deviation) are shown in Fig. 3, inset for the errors in μe/μh, and in Fig. 5 for the errors in σres and ΔVg. The weighted mean of μe/μh at non-zero dosing time is 0.83 and the weighted standard deviation of the mean is 0.01.