Introduction

For disordered graphene, theoretical predictions suggest that as the disorder strength increases, the average density of states (ADOS) increases accordingly, in comparison with pristine graphene1,2. The changes of the ADOS near the charge neutrality point (NP) still remain ambiguous and under debate3,4,5,6,7,8,9. One of the theoretical predictions on disordered graphene suggested that the ADOS followed a power law, i.e. , 0 < α < 1, where EF is the Fermi energy3,4 and the exponent α decreases with increasing disorder . Thus, the ADOS is expected to increase when disorder becomes stronger in graphene.

In two-dimensional (2D) materials, quantum capacitance10,11 Cq can directly reflect the ADOS because it is proportional to the ADOS at EF due to the large contribution of electron compressibility12, i.e. , where e is the electron charge and is the ADOS at EF. The measurement of Cq is more immune to the scatterings arising from disorder compared to the DC transport measurement. The theoretical expressions of the quantum capacitance and the ADOS in single layer pristine graphene11,13 have been verified experimentally by capacitance measurements for devices with various top-gate geometries14,15,16,17.

The measurement of quantum capacitance is expected to help probe the non-conducting states, particularly in strong localization systems such as the chemically decorated graphene structures, which contain strong disorder as evidenced by the DC transport measurement18,19,20,21,22,23. So far, no experimental work seems to have been reported on the quantum capacitance of disorder graphene. Since the ADOS has been predicted to increase in disordered graphene, it is natural to expect that quantum capacitance measurements should reflect the increased ADOS directly.

On the other hand, the local density of states (LDOS) in disordered graphene nanostructures has been investigated theoretically and experimentally (mostly by scanning tunneling microscopy) and it is believed to fluctuate due to the presence of the disorder24,25,26,27,28,29. For substrate-supported pristine graphene, experimental works demonstrated that the spatial fluctuations of LDOS in the vicinity of the NP follow Gaussian distributions30,31. Gaussian distributions have been predicted for weakly disordered systems with extended states32. However, for strongly disordered systems, the fluctuations of LDOS have been predicted to be lognormal33. Recent numerical simulations showed the lognormal distribution in disordered graphene34. The lognormal distribution has also been identified in some experimental studies of 2D materials35.

To introduce strong disorder into single layer graphene, we adopted our previously developed method of Ag-decoration of graphene surfaces18. Through transport measurements of Ag-decorated graphene, we found strong Anderson localization induced by the disorder of Ag clusters. The resistance of Ag-decorated graphene largely increased right after the decoration and the variable-range-hopping (VRH) model interpreted the experimental data of graphene conductivity fairly well even at the temperature as high as 300 K18. Obviously, strong localization had been induced in such disordered graphene even at room temperature. It should be pointed out that the Ag decoration method was proved by the comparison experiments to be non-invasive and the lattice of graphene was preserved. Therefore, it makes strongly disordered Ag-decorated graphene an interesting candidate for the study of both the ADOS and its local fluctuations through capacitance measurements.

In this letter, contrary to our expectation, we found that the measured quantum capacitance Cq decreases dramatically as disorder strength (controlled by the plasma-assisted Ag-cluster decoration18) increases. To understand the reduction in a quantitative way, here we proposed a two-parameter model which incorporates both the ADOS and its local fluctuations. Based on this model, we determined simultaneously the ADOS changes of the Ag-decorated graphene and the local fluctuations. It is found that ADOS indeed increase with disorder and follow the power law . The dramatic reduction of the measured Cq is caused by large fluctuations of LDOS, which follows a lognormal distribution and is independent of the Fermi energy.

Results

The fabrication of top-gate graphene devices and the measurement of capacitance are described in the method section. Fig. 1(a) schematically shows the geometry of the Ag-decorated graphene device with a top-gate. Fig. 1(b) illustrates the three-terminal configuration of the capacitance measurement Cq(Vch) and oxide layer capacitance Cox. In the serial connection, the quantum capacitance can be obtained from the top-gate capacitance Ctg through the relation . Here Vch is the potential across the graphene sheet and can be obtained by integrating the charge conservation relation , i.e., . For pristine graphene devices, the quantum capacitance has been given by 11,13, where k denotes the Boltzmann constant and νF the Fermi velocity. We fitted the measured quantum capacitance of pristine graphene Cq0_exp to the theory Cq0_theory to obtain . The result is shown in Fig. 1(c). The consistency in the vicinity of the NP has been improved compared to previous works15,17,30. One of the possible reasons for this improvement is the use of Cu mask before Yttrium deposition. By this way, we effectively reduced contamination from e-beam lithography, which may result in electron-hole puddles30,31.

Figure 1
figure 1

(a) The geometry of a top-gate Ag-decorated graphene device.The small white islands represent the Ag clusters. (b) The equivalent circuit of the three-terminal capacitance measurement. The p-Si substrate is grounded to avoid its parasitic capacitance. (c) The quantum capacitance of one pristine graphene device with . The inset shows the optical image of the device. The dashed line indicates the outline of the graphene flake and the scale bar is 5 μm. (d) The measured quantum capacitance of three Ag-decorated graphene devices sputtered for 1 s, 5 s and 10 s, respectively.

Fig. 1(d) shows the measured quantum capacitances Cq(Vch) of three typical Ag-decorated graphene devices. Samples Ag1, Ag2 and Ag3 were decorated by Ag clusters via sputtering (5 W) for 1 s, 5 s and 10 s, respectively. Compared to the results recorded from the pristine graphene Cq0_exp in Fig. 1 (d), the quantum capacitance obtained from Ag-decorated graphene devices is much smaller. The Cq(Vch) curves of Ag-decorated graphene devices are all asymmetrical between the electron and hole bands. This asymmetry was also observed in transport measurements18. As the Ag decoration is increased, the quantum capacitance of the devices decreases dramatically, which seems to be counterintuitive. Direct interpretation of these results via leads to disagreement with the theoretical predictions of the ADOS in disordered graphene.

In order to quantitatively understand the dramatic reduction of quantum capacitance in disordered graphene samples, we propose here a two-parameter model. In this model we incorporate both the ADOS of the disordered graphene and its LDOS fluctuations. To simulate the quantum capacitance, we first divide the total area A of graphene evenly into N small regions Ai, each has a size A/N as shown in Fig. 2(a). For simplicity, we set A = 1. In each region Ai, we define a normalized random variable with , where ρi is the LDOS in Ai and is the ADOS of the system. For a given EF, the local quantum capacitance in the region Ai can be written as

As indicated by the equivalent circuit shown in Fig. 2(b), the total capacitance of the graphene device Ct is the summation of all parallel capacitances of N serial capacitances of Co and Cqi, where Co = Cox/N is the oxide layer capacitance in each small region. Thus, the quantum capacitance can be simulated using the following equation:

which depends only on the ADOS and the normalized distribution of δi. In order to simulate the quantum capacitance, we need to choose a correct distribution for . In strongly disordered graphene, the distribution of LDOS has been studied and found to be lognormal34. The lognormal distribution has also been evidenced experimentally in other 2D disordered materials35. Thus, to simulate Eq. (2), we propose a lognormal distribution for δ, i.e.,

or, with . Because the variance of δ is , ξ denotes the degree of the fluctuations. Theoretically, the ADOS in disordered graphene have been predicted to increase with disorder strength. One of the proposed analytical expressions for ADOS is a non-universal power law referred to quasi-particles in d-wave superconductor, which was supported by numerical exact diagonalizations (ED) for different types of disorders3,4. In our simulation, we assume that the ADOS follow the non-universal power law. Thus, the ADOS at the Fermi energy (EF = eVch) is given by

where is the residual ADOS at NP, λ is the cut-off energy and λ = 3t, a is the lattice constant and t is the nearest-neighbor hopping energy3,4 (t = 2.8 eV). The exponent in Eq. (4) represents the disorder strength and its value decreases with increasing disorder strength.

Figure 2
figure 2

(a) The top-gate Ag-decorated graphene device is divided into N small areas in the FLDOS model.Co and Cqi are the oxide layer capacitance and the quantum capacitance in Ai, respectively. (b) Equivalent circuit of the top-gate graphene device in the FLDOS model. The total capacitance is the summation of parallel capacitances of all serial capacitances of Co and Cqi. (c) The experimental quantum capacitance (scatters) and the simulation results by the FLDOS model (solid lines).

The simulated quantum capacitance Cq_mod of Eq. (2) now depends only on two parameters: the disorder strength α and the degree of LDOS fluctuations ξ. By using Eq. (2), We performed Monte Carlo simulations and found that Cq_mod always converges to a well-defined value if N is sufficiently large for a chosen set of (α,ξ). In Fig. 2(c), we plot the simulated results for all the three disordered samples shown in Fig. 1(d). It shows good agreements between the simulated and measured quantum capacitance in a wide range of Fermi energies. To avoid ambiguity in the vicinity of the NP30,31, the simulation started from Vch = 0.15 V. Good agreements are also obtained for the hole branch. (see Supplementary Information). However, for Sample Ag3, some deviations between the measured and simulated results are noticed. This indicates that for a strongly disordered sample, the ADOS might deviate from the power law behavior of Eq. (4).

The fitting parameters for the three samples are listed in Table 1. It is clearly seen that α decreases with increasing disorder. The corresponding ADOS are plotted in Fig. 3(a), which shows explicitly that ADOS increases with disorder strength. This is consistent with the previous theoretical predictions of ADOS in disordered graphene3,4 and our transport measurements18. Table 1 also indicates that the residual DOS ρ0 at the NP cannot be ignored in such highly disordered graphene samples. Thus, the dramatic reduction in measured quantum capacitance found in Fig. 1(d) is caused by the large fluctuations of LDOS. Indeed, Table 1 shows that the value of ξ increases with disorder strength.

Table 1 Fitting parameters and DOS at EF = 0, 0.25 eV
Figure 3
figure 3

(a) The simulated ADOS of three samples which follows Eq.(4) with α = 0.74,0.69,0.49 for Ag1, Ag2 and Ag3, respectively and the ADOS of pristine graphene for reference.(b) shows the reduction factor β of experimental (scatters) and simulation results by the FLDOS model (solid lines).

To quantify the reduction of quantum capacitance, here we define a dimensionless reduction factor , where is the quantum capacitance if the DOS is assumed to be uniform. Actually, Cq* is also the real quantum capacitance of the disordered graphene. As we will show later that Cq* can only be achieved when the oxide layer capacitance Cox is much greater than Cq*. In Fig. 3(b), we plot the reduction factor for both the measured quantum capacitance, i.e., and the simulated quantum capacitance, i.e., . Here, for all three samples. It is clearly seen that more reduction is found for more disordered samples.

From the above simulation results, we would like to point out another interesting property of the LDOS fluctuations in disordered graphene. The fact that, for each disordered sample in Fig. 2(c) or 3(b), the quantum capacitance of disordered graphene can be excellently described by only two parameters in a wide range of Fermi energies indicates that the distribution of the normalized LDOS δ given in Eq. (3) is independent of EF for a given disorder sample. To confirm this, we have performed the following simulations based on the Anderson tight-binding model for disordered systems36. We calculated the LDOS ρi in graphene with a given site disorder W by using the Haydock-Heine-Kelly recursion method. (see Supplementary Information for details). For two different values of W, the results of the distributions for three different values of EF are shown in Figs. 4(a) and 4(b), where the hopping energy t is taken as the units of energy. These results show that for each given value of W, the distributions of δ at three different Fermi energies can all be described nicely by the same lognormal distribution (solid curve). It is also seen that the variance of the distribution is larger for larger W. Thus, these simulation results support our two-parameter model that the distribution of δ depends only on the disorder strength and is independent of the Fermi energy EF.

Figure 4
figure 4

(a) shows the distributions of normalized LDOS at EF = 0.05 t,0.15 t,0.20 t (t = 2.8 eV), when W = 0.50 t, which all follow one lognormal distribution.(b) also indicates the single lognormal distribution of LDOS at different EF when W = 1.50 t.

Finally we would like to study the role played by the Yttrium oxide layer in the observed reductions of quantum capacitance of disordered graphene. The presence of a finite capacitance of the Yttrium oxide layer Cox allows local fluctuations of channel potential across the graphene sheet in each small region. It is such fluctuations that lead to the reduction of the observed quantum capacitance of the graphene. For this purpose, we rewrite Eq. (2) in the following form:

where . In this form, Eq. (5) also exhibits an interesting scaling behavior, i.e., the reduction factor depends only on the ratio of Cox to Cq*, not on Cox and Cq* individually. Here we focus on one particular disorder strength and study the reduction factor as a function of r by using different thicknesses of the Yttrium oxide layer. We also test the r-scaling law by using data obtained at different Fermi energies, where Cq* varies.

We measured the quantum capacitance of the same Ag-decorated graphene from devices with different Cox by tuning the thickness of the Yttrium oxide layer. The decorations of all these samples were carried out simultaneously under the same conditions as Sample Ag1 in order to guarantee the same strength of disorder. The results of reduction factor for different Yttrium oxide layer thicknesses, which correspond to and different Fermi energies (Vch = 0.15, 0.25, 0.35, 0.45 V) are shown in Fig. 5. All these data overlap nicely with the simulated solid curve obtained by using α = 0.74, ξ = 1.66 shown in Table 1 for Sample Ag1. Fig. 5 verifies the r-scaling behavior of our two-parameter model with a lognormal distribution of LDOS. We also notice that the experimental data slightly deviates from the model when Cox becomes smaller. In this case, the oxide layer becomes thicker and the induced polarization effect might enhance the fluctuations of LDOS, leading to a smaller β. In the inset of Fig. 5, we show the β function obtained from different values of ξ. It is found that β decreases with increasing fluctuations of LDOS as expected. It also shows that β increases with r and saturates to 1 at very large r (unachievable practically), indicating that the real quantum capacitance of disordered graphene could only be achieved when Cox Cq*.

Figure 5
figure 5

The inset shows the reduction factor β (calculated by the FLDOS model) reaches 1 when Cox is very large at different ξ.The reduction factor β (scatters) of graphene decorated by Ag for 1 s at 5 W with agrees with the r-scaling of the FLDOS model (solid lines).

Discussion

Recently, experimental works demonstrated that the spatial LDOS fluctuations in the vicinity of the NP follow a Gaussian distribution30,31. Compared to the lognormal distribution which can describe LDOS in strongly disordered systems, the Gaussian distribution is only capable of describing systems with weak disorder. We have also performed simulations of Eq. (5) based on the Gaussian distribution. However, we do not found any significant reduction in quantum capacitance. (see Supplementary Information). In strongly disordered systems, the envelope of the localized wave functions decays exponentially from the center of the localization sites36,37, creating strong spatial fluctuations of LDOS. This results in a dominance of LDOS at lower values than its mean and gives rise to a positive skewness. Although the variance of both Gaussian and lognormal distributions can represent the degree of fluctuations, the positive skewness of the lognormal distribution could be the underlying reason why it can correctly describe the LDOS fluctuations in disordered graphene, where significant LDOS fluctuations occur.

In conclusion, we have established a simple two-parameter(α,ξ) model for the quantum capacitance of disordered graphene. This model incorporates both the power-law dependence of the ADOS and the lognormal distribution of LDOS. By measuring the quantum capacitance of disordered graphene we simultaneously determine both the ADOS and its local fluctuations of the system. The model gives quantitative agreements with the measured quantum capacitance of three different strongly disordered Ag-decorated graphene samples. For Ag-decorated graphene samples, a given Ag-decoration corresponds to a configuration of disorder induced in the graphene sheet. The presence of disorder increases the ADOS and also induces large spatial fluctuations in LDOS compared to pristine graphene. The reduction of measured quantum capacitance is caused by large fluctuations of LDOS mediated by the presence of the insulating oxide layer. The validity of the model is also supported by our simulation results of LDOS using the Anderson model that the distribution of normalized LDOS is lognormal and independent of Fermi energy for a fixed disorder.

Methods

Device fabrication

The pristine graphene devices were prepared on p-Si substrates coated with 300 nm SiO2 insulating layers, as reported previously18. The top-gate electrodes were then added using a standard e-beam lithography technique17,38,39. The insulating layer between the top-gate electrode and graphene was fabricated by depositing a thin layer of Yttrium on the graphene devices, followed by oxidation at 180°C for 10 min in air. We used a mask (thin Cu foils) in order to minimize the influence of direct electron beam irradiation23 when patterning the insulating layer. The silver clusters were first deposited on the pristine graphene by the method reported previously18. Then, the graphene devices were immediately transferred into a high vacuum chamber for Yttrium deposition by electron-beam evaporation.

Capacitance measurement

The capacitance was measured by a Keithley CV analyzer (model 590) with a sensitivity of ~0.1fF. As illustrated in Fig. 1(b), the capacitance measurement was performed in a three-terminal configuration and the p-Si substrate was grounded to avoid its parasitic capacitance. The residual capacitance of the circuit in the same geometry of the devices without graphene is smaller than 0.3fF. The area of graphene for capacitance measurement was normally larger than 30 μm2 as shown in the inset in Fig. 1(c). Pristine graphene devices were used to calibrate the capacitance of the Yttrium oxide layer (Cox) of the Ag-decorated ones in the same batch. We extracted the quantum capacitance of a large number of devices for the same Ag decoration in different batches and the results were similar.