Negative correlation between charge carrier density and mobility fluctuations in graphene

By carrying out simultaneous longitudinal and Hall measurements in graphene, we find that the $1/f$ noise for the charge carrier density is negatively correlated to that of mobility, with a governing behavior that differs significantly from the relation between their mean values. The correlation in the noise data can be quantitatively explained by a single-parameter theory whose underlying physics is the trapping and detrapping of the fluctuating charge carriers by the oppositely charged Coulomb scattering centers. This can alter the effective density of long-range scattering centers in a transient manner, with the consequent fluctuating effect on the mobility.

Figure 1

(a) Graphene flake of sample A is measured in perpendicular magnetic field under various temperatures. Longitudinal voltage $\left(V_{xx}\right)$ and Hall voltage $\left(V_H\right)$ are collected simultaneously and subsequently analyzed. Sample A is etched into the standard Hall-bar geometry, shown in the inset by the atomic force microscope (AFM) image. The geometric parameters are $L=1.6\mu \mathrm{m},W=1.2\mu \mathrm{m},$ and $d=0.4\mu \mathrm{m}.$ (b) Sample B in square (cross-bar) geometry is imaged by AFM. The scale bar on the lower right is 1 μm. Measurement probes are labeled as shown. Sample B's geometric parameters are $L=W=0.6\mu \mathrm{m}.$

Figure 2

Longitudinal resistance, measured as a function of back gate voltage at 150 K, is shown on the left axis for $B=0$ (black circles), 0.5 T (red line). The coincidence of the resistances measured under two different magnetic fields, in the region away from the charge neutral point, indicates the absence of quantum correction of longitudinal resistance at low magnetic field. Conductance at 0.5 T is plotted with black squares (scale is shown on the right) together with the green fitting curve that accounts for the long- and short-range scatterings. The deviation at $|V_G-V_{\mathrm{CNP}}|>5\mathrm{V}$ from the blue reference curve (only long-range scatterings) indicates the importance of the short-range scattering mechanisms. Data are from sample A.

Figure 3

(a) Power spectrum density of longitudinal noise with current levels varying from zero to 0.05, 0.1, 0.2, and 0.4 μA are shown for $V_G-V_{\mathrm{CNP}}=-36\mathrm{V}$, where a low-pass filter with a cutoff frequency of 2.6 Hz has been applied. Black curve represents the noise background. Hall noise, measured simultaneously, is plotted in (b). (c) Noise magnitudes and their corresponding current dependence at different back gate voltages are plotted in log-log scale for the longitudinal and Hall voltage PSDs. All straight lines have a slope of 2, indicating a quadratic current dependence. Data in this figure are from sample B, measured at $T=180\mathrm{K}.$

Figure 4

(a) Longitudinal (black squares) and Hall voltages (blue circles) plotted as a function of gate voltage. (b) Conductance (black squares) and carrier density (blue circles) values are derived from (a). Separate fittings for electron and hole lead to the same slope but different $V_{\mathrm{CNP}}$ (0.3 V for hole and 1.33 V for electron, which is ascribed to the hysteresis effect). (c) Normalized longitudinal noise PSD (solid black squares) and Hall noise PSD (open blue symbols) plotted as functions of gate voltage for $T=150$ and 200 K. Away from the CNP, the Hall noise is shown to be larger than that for the longitudinal noise. (d) Mobility noise (solid black squares) displays no strong relation with gate voltage. Derived values of time-averaged mobility are plotted in open symbols, with the scale shown on the right. Data in this figure are from sample A at $T=150\mathrm{K}.$

Figure 5

Instantaneous Hall mobility μ plotted as a function of the instantaneous carrier density $n$, for data taken at 200 K. The time-averaged mean values of $(\mu ,n)$ are shown in the inset, where the solid square dots are the experimental data and the black curve represents the best fit. At each gate voltage (in intervals of 1 V, varying from 5 to 19 V for the electrons), the fluctuating pairs of $(\mu ,n)$ are plotted in different colors. The thin red curves are fitting results from our trapping and detrapping model.

Figure 6

Cartoon picture for the trapping and detrapping model. For different gate voltages, the trapping and detrapping of carriers not only can vary the net charge carrier density at any moment, but also would “neutralize” the charged impurities, thereby leading to altered (instantaneous) mobility.

Figure 7

A log-log plot of fluctuating pair $\left(\mu ,\tilde{n}\right)$ with data collected at gate voltages of 5 V (red), 7 V (crimson), 10 V (dark blue), 15 V (green), and 20 V (black). An excellent linear relation is obtained, with all the data collapsing on a straight line as predicted by Eq. (7c). Data in this figure are from sample A measured at $T=150\mathrm{K}.$

Figure 8

(a) Correlation coefficient (from sample A) as a function of gate voltages obtained from ten sets of voltage signals with each set lasting 125 s, with a sampling rate of 512 Hz. Data were taken at $T=150\mathrm{K}$. (b) Correlation coefficient (from sample B) plotted as a function of gate voltages. Blue shaded region represents the range of correlation coefficient measured in semiconductors on sapphire (SOS) systems Ref. [28]. The inset image is the fluctuating longitudinal voltage (black) and Hall voltage (red) from sample B are shown in time domain. The fluctuations for longitudinal and Hall voltages are defined as $\Delta V_{xx\left(H\right)}\left(t\right)\equiv \left[V_{xx\left(H\right)}\left(t\right)-V_{xx\left(H\right)}^{\mathrm{average}}\right]/V_{xx\left(H\right)}^{\mathrm{average}}$. Data were taken at $T=180\mathrm{K}$.