Disorder-induced temperature-dependent transport in graphene: Puddles, impurities, activation, and diffusion

We theoretically study the transport properties of both monolayer and bilayer graphene in the presence of electron-hole puddles induced by charged impurities that are invariably present in the graphene environment. We calculate the graphene conductivity by taking into account the non-mean-field two-component nature of transport in the highly inhomogeneous density and potential landscape, where activated transport across the potential fluctuations in the puddle regimes coexists with regular metallic diffusive transport. The existence of puddles allows the local activation at low carrier densities, giving rise to an insulating temperature dependence in the conductivity of both monolayer and bilayer graphene systems. We also critically study the qualitative similarity and the quantitative difference between monolayer and bilayer graphene transport in the presence of puddles. Our theoretical calculation explains the nonmonotonic feature of the temperature-dependent transport, which is experimentally generically observed in low mobility graphene samples. We establish the two-component nature (i.e., both activated and diffusive) of graphene transport arising from the existence of potential fluctuation induced inhomogeneous density puddles. The temperature dependence of the graphene conductivity arises from many competing mechanisms, even without considering any phonon effects, such as thermal excitation of carriers from the valence band to the conduction band, temperature-dependent screening, thermal activation across the potential fluctuations associated with the electron-hole puddles induced by the random charged impurities in the environment, leading to very complex temperature dependence, which depends both on the carrier density and the temperature range of interest.

Figure 1

Normalized density of states for both electron and hole in MLG. The solid and dashed lines are for the DOS in inhomogeneous and homogeneous systems, respectively. The electron (hole) band tail locates at $E<0$ ($E>0$), which gives rise to electron (hole) puddles at $E<0$ and $E>0$.

Figure 2

(a) Electron density of MLG at CNP as a function of temperature for different $s$. At $T=0$ the density is given by $n_0=D_1{s}^2/4$. (b) Temperature-dependent electron density of MLG at finite $E_F$ for different $s$. For $s/E_F\ne 0$ the leading-order behavior is quadratic. (c) Total electron densities (solid lines) and hole densities (dashed lines) of MLG as a function of $E_F$ for two different $s=30$ and 70 meV. The densities at the band tails are given by $n_e(E_F=0)=n_h(E_F=0)=D_1{s}^2/4$.

Figure 3

${\sigma }_t\left(T\right)$ of MLG at charge neutral point for different $s$ [Eq. (22a) and $n_e$ as given in Eq. (6)]. Inset shows the thermally activated conductivity of MLG as a function of temperature, where ${\sigma }_a\left(T\right)/{\sigma }_a\left(0\right)=1+e^{{\beta }^2{s}^2/2}\mathrm{erfc}(\beta s/\sqrt{2})$.

Figure 4

Calculated total conductivity ${\sigma }_t\left(T\right)/{\sigma }_t\left(0\right)$ of MLG with the following parameters: $n_i={10}^{12}$ cm${}^{-2}$ and $n_d{V}_0^2=2$ (eV Å)${}^2$. (a) ${\sigma }_t\left(T\right)$ for $E_F=120$ meV and for different $s$. (b) ${\sigma }_t\left(T\right)$ of MLG for $s=80$ meV and for several $E_F=80$, 100, 120, 160 meV, which correspond to the net carrier densities $n=n_e-n_h\simeq 0.9\times {10}^{12}$, $1.2\times {10}^{12}$, $1.5\times {10}^{12}$, and $2.3\times {10}^{12}$ cm${}^{-2}$.

Figure 5

Normalized density of states for both electron and hole in BLG. The solid and dashed lines are for the DOS in inhomogeneous and homogeneous systems, respectively. The electron (hole) band tail locates at $E<0$ ($E>0$), which gives rise to electron (hole) puddles at $E<0$ and $E>0$.

Figure 6

(a) Electron density of BLG at CNP as a function of temperature for different $s$. At $T=0$ the density is given by $n_0=D_{0}s/\sqrt{2\pi }$. (b) Temperature-dependent electron density of BLG at finite $E_F$ for different $s$. For $s/E_F\ne 0$ the leading-order behavior is quadratic while at $s=0$ the density is exponentially suppressed. (c) Total electron densities (solid lines) and hole densities (dashed lines) of BLG as a function of $E_F$ for two different $s=30$ and 70 meV. The linear line represents the density difference $n=n_e-n_h=D_0{E}_F$, which linearly depends on the Fermi energy. The densities at the band tails are given by $n_e(E_F=0)=n_h(E_F=0)=D_{0}s/\sqrt{2\pi }$.

Figure 7

${\sigma }_t\left(T\right)$ of BLG at charge neutral point for different $s$ [Eq. (22a) with $n_e$ for BLG]. Inset shows the thermally activated conductivity in BLG as a function of temperature, where ${\sigma }_a\left(T\right)/{\sigma }_a\left(0\right)=1+e^{{\beta }^2{s}^2/2}\mathrm{erfc}(\beta s/\sqrt{2})$, the same as for the MLG case.

Figure 8

Calculated total conductivity ${\sigma }_t\left(T\right)/{\sigma }_t\left(0\right)$ of BLG with the following parameters: $n_i={10}^{12}$ cm${}^{-2}$ and $n_d{V}_0^2=2$ (eV Å)${}^2$. (a) ${\sigma }_t\left(T\right)$ for $E_F=60$ meV and for different $s$. (b) ${\sigma }_t\left(T\right)$ for $s=40$ meV and for several $E_F=20$, 40, 60, 80 meV, which correspond to the net carrier densities $n=n_e-n_h=0.55\times {10}^{12}$, $1.1\times {10}^{12}$, $1.6\times {10}^{12}$, and $2.2\times {10}^{12}$ cm${}^{-2}$.

Figure 9

Standard deviation of potential fluctuation $s$ versus the screening constant $q_{TF}$ (log-log plot) in MLG by varying the Fermi level. The dotted blue line is for $n_{\mathrm{imp}}=1.0\times {10}^{12}$ cm${}^{-2}$, $z_0=1$ nm, and $d=100$ nm. The solid red line is for $n_{\mathrm{imp}}=0.5\times {10}^{12}$ cm${}^{-2}$, $z_0=1$ nm, and $d=200$ nm. The dashed green line is for $n_{\mathrm{imp}}=0.5\times {10}^{12}$ cm${}^{-2}$, $z_0=1$ nm and $d=100$ nm. The dot-dashed black line is for $n_{\mathrm{imp}}=0.5\times {10}^{12}$ cm${}^{-2}$, $z_0=2$ nm and $d=100$ nm.

Figure 10

Calculated the density of states of electron $D_e\left(E_F\right)$ in MLG versus electron density using the following parameters: the insulator thickness $d=100$ nm, the impurity distance from the interface $z_0=1$ nm. The solid red line is for unperturbed density of states. The dashed green, dotted blue, and dot-dashed black lines correspond to $n_{\mathrm{imp}}=0.5$, $1.0,$ and $2.0\times {10}^{12}$ cm${}^{-2}$, respectively.

Figure 11

Calculated density of states of electron $D_e\left(E\right)$ of MLG versus energy $E$ for different impurity configurations and carrier densities $n$. The solid red lines are for the noninteracting MLG system. (a) Calculated $D_e\left(E\right)$ in MLG for the insulator thickness $d=100$ nm, the impurity distance from the interface $z_0=1$ nm, and carrier density $n=2.0\times {10}^{12}$ cm${}^{-2}$. The dashed green, dotted blue, and dot-dashed black lines correspond to $n_{\mathrm{imp}}=0.5$, $1.0$, $2.0\times {10}^{12}$ cm${}^{-2}$, respectively. (b) Calculated $D_e\left(E\right)$ in MLG for $d=100$ nm, $n_{\mathrm{imp}}=0.5\times {10}^{12}$ cm${}^{-2}$, and $n=2.0\times {10}^{12}$ cm${}^{-2}$. The dashed green, dotted blue, and dot-dashed black lines correspond to $z_0=1$, 2, 3 nm, respectively. (c) Calculated $D_e\left(E\right)$ in MLG for $d=100$ nm, $z_0=1$ nm, and $n_{\mathrm{imp}}=0.5\times {10}^{12}$ cm${}^{-2}$. The dashed green, dotted blue, and dot-dashed black lines correspond to $n=0.5$, $1.0$, $2.0\times {10}^{12}$ cm${}^{-2}$, respectively.

Figure 12

Standard deviation of potential fluctuation $s$ versus the screening constant $q_{TF}$ (log-log plot) in BLG by varying the Fermi level. The dotted blue line is for $n_{\mathrm{imp}}=1.0\times {10}^{12}$ cm${}^{-2}$, $z_0=1$ nm and $d=100$ nm. The solid red line is for $n_{\mathrm{imp}}=0.5\times {10}^{12}$ cm${}^{-2}$, $z_0=1$ nm, and $d=200$ nm. The dashed green line is for $n_{\mathrm{imp}}=0.5\times {10}^{12}$ cm${}^{-2}$, $z_0=1$ nm, and $d=100$ nm. The dot-dashed black line is for $n_{\mathrm{imp}}=0.5\times {10}^{12}$ cm${}^{-2}$, $z_0=2$ nm, and $d=100$ nm.

Figure 13

Calculated density of states of electron $D_e\left(E_F\right)$ of BLG versus electron density using the following parameters: the insulator thickness $d=100$ nm, the impurity distance from the interface $z_0=1$ nm. The solid red line is for unperturbed density of states. The dashed green, dotted blue, and dot-dashed black lines correspond to $n_{\mathrm{imp}}=0.5$, $1.0,$ and $2.0\times {10}^{12}$ cm${}^{-2}$, respectively.

Figure 14

Calculated density of states of electron $D_e\left(E\right)$ of BLG versus energy $E$ for different impurity configuration and carrier densities $n$. The solid red lines are for the noninteracting BLG system. (a) Calculated $D_e\left(E\right)$ in BLG for the insulator thickness $d=100$ nm, the impurity distance from the interface $z_0=1$ nm, and carrier density $n=2.0\times {10}^{12}$ cm${}^{-2}$. The dashed green, dotted blue, and dot-dashed black lines correspond to $n_{\mathrm{imp}}=0.5$, $1.0$, $2.0\times {10}^{12}$ cm${}^{-2}$, respectively. (b) Calculated $D_e\left(E\right)$ in BLG for $d=100$ nm, $n_{\mathrm{imp}}=0.5\times {10}^{12}$ cm${}^{-2}$, and $n=2.0\times {10}^{12}$ cm${}^{-2}$. The dashed green, dotted blue, and dot-dashed black lines correspond to $z_0=1$, 2, 3 nm, respectively. (c) Calculated $D_e\left(E\right)$ in BLG for $d=100$ nm, $z_0=1$ nm, and $n_{\mathrm{imp}}=0.5\times {10}^{12}$ cm${}^{-2}$. The dashed green, dotted blue, and dot-dashed black lines correspond to $n=0.5$, $1.0$, $2.0\times {10}^{12}$ cm${}^{-2}$, respectively.