Electronic transport in two-dimensional graphene

A broad review of fundamental electronic properties of two-dimensional graphene with the emphasis on density and temperature-dependent carrier transport in doped or gated graphene structures is provided. A salient feature of this review is a critical comparison between carrier transport in graphene and in two-dimensional semiconductor systems (e.g., heterostructures, quantum wells, inversion layers) so that the unique features of graphene electronic properties arising from its gapless, massless, chiral Dirac spectrum are highlighted. Experiment and theory, as well as quantum and semiclassical transport, are discussed in a synergistic manner in order to provide a unified and comprehensive perspective. Although the emphasis of the review is on those aspects of graphene transport where reasonable consensus exists in the literature, open questions are discussed as well. Various physical mechanisms controlling transport are described in depth including long-range charged impurity scattering, screening, short-range defect scattering, phonon scattering, many-body effects, Klein tunneling, minimum conductivity at the Dirac point, electron-hole puddle formation, $p\mathrm{\text{−}}n$ junctions, localization, percolation, quantum-classical crossover, midgap states, quantum Hall effects, and other phenomena.

Figure 1

(a) Graphene honeycomb lattice showing in different colors the two triangular sublattices. Also shown is the graphene Brillouin zone in momentum space. Adapted from 74. (b) Carbon nanotube as a rolled up graphene layer. Adapted from 271. (c) Lattice structure of graphite, graphene multilayer. Adapted from 73. (d) Lattice structure of bilayer graphene. ${\gamma }_0$ and ${\gamma }_1$ are, respectively, the intralayer and interlayer hopping parameters $t$, $t_{\perp }$ used in the text. The interlayer hopping parameters ${\gamma }_3$ and ${\gamma }_4$ are much smaller than ${\gamma }_1\equiv t_{\perp }$ and are normally neglected. Adapted from 324. (e) Typical configuration for gated graphene.

Figure 2

(a) Graphene band structure. Adpated from 448. (b) Enlargment of the band structure close to the $K$ and $K^{\prime }$ points showing the Dirac cones. Adpated from 448. (c) Model energy dispersion $E=\hslash v_F|\mathbf{k}|$. (d) Density of states of graphene close to the Dirac point. The inset shows the density of states over the full electron bandwidth. Adapted from 74.

Figure 3

(a) Energy band of bilayer graphene for $V=0$. (b) Enlargment of the energy band close to the neutrality point $K$ for different values of $V$. Adapted from 313.

Figure 4

(a) Diagram showing the bands at the interfaces of a metal-oxide-silicon structure. (b) Band diagram showing the bending of the bands at the interface of the semiconductors and the two-dimensional subband.

Figure 5

(a) Graphene nanoribbon energy gaps as a function of width. Adapted from 186. Four devices (P1–P4) were orientated parallel to each other with varying width, while two devices (D1–D2) were oriented along different crystallographic directions with uniform width. The dashed line is a fit to a phenomenological model with $E_g=A/(W-W^{*})$ where $A$ and $W^{*}$ are fit parameters. The inset shows that contrary to predictions, the energy gaps have no dependence on crystallographic direction. The dashed lines are the same fits as in the main panel. (b) Evidence for a percolation metal-insulator transition in graphene nanoribbons. Adapted from 10. Main panel shows graphene ribbon conductance as a function of gate voltage. Solid lines are a fit to percolation theory, where electrons and holes have different percolation thresholds (seen as separate critical gate voltages $V_c$). The inset shows the same data in a linear scale, where even by eye the transition from high-density Boltzmann behavior to the low-density percolation transport is visible.

Figure 6

(a) Disorder averaged resistance as a function of top gate voltage for a fixed back gate density $n_{\mathrm{bg}}=5\times {10}^{11}{\mathrm{cm}}^{-2}$ and several values of the impurity density (from bottom to top $n_{\mathrm{imp}}=0$, 1, 2.5, 5, 10, and $15\times {10}^{11}{\mathrm{cm}}^{-2}$). Results were obtained using ${10}^3$ disorder realizations for a square samples of size $W=L=160\mathrm{nm}$ in the presence of a top gate placed in the middle of the sample 10 nm above the graphene layer, 30 nm long, and of width $W$. The charge impurities were assumed at a distance $d=1\mathrm{nm}$ and the uniform dielectric constant $\kappa$ was taken equal to 2.5. Adapted from 375. (b) Solid line is Eq. (2.4) for armchair boundary conditions showing the aspect ratio dependence of the Dirac point ballistic conductivity (437). For $W\gg L$, the theory approaches the universal value $4e^2/\pi h$. Circles show experimental data taken from 306, and squares show the data from 99. Inset: Illustration of the configuration used to calculate graphene’s universal minimum conductivity. For $V_g>0$, one has a $p\mathrm{\text{−}}p\mathrm{\text{−}}p$ junction, while for $V_g<0$, one has a $p\mathrm{\text{−}}n\mathrm{\text{−}}p$ junction. This illustrates the ballistic universal conductivity that occurs at the transition between the $p\mathrm{\text{−}}p\mathrm{\text{−}}p$ and $p\mathrm{\text{−}}n\mathrm{\text{−}}p$ junctions when $V_g=0$.

Figure 7

Demonstration of one-parameter scaling at the Dirac point. From 31. Main panel shows the conductivity as a function of $L^{*}/\xi$, where $L^{*}=f(K_0)L$ is the scaled length and $\xi$ is the correlation length of the disorder potential. Note that $\beta =d\ln \sigma /d\ln L>0$ for any disorder strength. The inset shows explicit comparison of the $\beta$ function for the Dirac fermion model and for the symplectic (AII) symmetry class. From 334.

Figure 8

Picture proposed by 334 to understand the difference in topological structure between the massless Dirac model and the random spin-orbit symmetry class.

Figure 9

Diagrammatic representation for (a) diffuson, (b) Cooperon, and dressed Hikami boxes [(c) and (d)]. Adapted from 251.

Figure 10

Left panel: Schematic of different magnetoresistance regimes. (i) For strong intervalley scattering (i.e. ${\tau }_i\ll {\tau }_{\varphi }$ and ${\tau }_z\ll {\tau }_{\varphi }$), Eq. (2.18) gives weak localization or $\delta \sigma \sim \rho (B)-\rho (0)<0$. This is similar to quantum transport in the usual 2DEG. (ii) For weak intervalley scattering (i.e., ${\tau }_i\gg {\tau }_{\varphi }$ and ${\tau }_z\gg {\tau }_{\varphi }$), one has weak antilocalization, characteristic of the symplectic symmetry class. (iii) For ${\tau }_i\gg {\tau }_{\varphi }$, but ${\tau }_z\ll {\tau }_{\varphi }$, Eq. (2.18) gives a regime of suppressed weak localization. Right panel: Experimental realization of these three regimes. Graphene magnetoconductance is shown for carrier density (from bottom to top) $n=2.2\times {10}^{10}$, $1.1\times {10}^{12}$, and $2.3\times {10}^{12}{\mathrm{cm}}^{-2}$, at $T=14\mathrm{K}$. The lowest carrier density (bottom curve) has a small contribution from short-range disorder and shows weak antilocalization [i.e., the zero-field conductivity is larger than at finite field. $\sigma (B=0)={\sigma }_0+\delta \sigma$ and $\sigma (B>B^{*})={\sigma }_0$, with $\delta \sigma >0$. $B^{*}$ is the phase-breaking field]. In contrast, the highest density data (top curve) has a larger contribution of intervalley scattering and shows weak localization, i.e., $\delta \sigma <0$. From 425.

Figure 11

(a) The measured conductivity $\sigma$ of graphene as a function of gate voltage $V_g$ (or carrier density). The conductivity increases linearly with the density. Adapted from 74. (b) $\sigma$ as a function of $V_g$ for five different samples For clarity, curves are vertically displaced. The inset shows the detailed view of the density-dependent conductivity near the Dirac point for the data in the main panel. Adapted from 420. (c) $\sigma$ vs $V_g$ for the pristine sample and three different doping concentrations. Adapted from 86. (d) $\sigma$ as a function $V_g$ for pristine graphene (circles) and after deposition of 6 monolayers of ice (triangles). Inset: Optical microscope image of the device. Adapted from 229.

Figure 12

The ratio of transport scattering time (${\tau }_t$) to quantum scattering time (${\tau }_q$) as a function of density for different samples. Dashed (solid) lines indicate the theoretical calculations (218) with Coulomb disorder ($\delta$-range disorder). Adapted from 201.

Figure 13

Polarizability $\Pi (q,T)$ in units of the density of states at the Fermi level $D_0$. $\Pi (q,T)$ of monolayer graphene (a) as a function of wave vector for different temperatures and (b) as a function of temperature for different wave vectors. (c), (d) BLG polarizability. (e), (f) The 2DEG polarizability.

Figure 14

(a) Calculated graphene conductivity as a function of carrier density ($n_i$ is an impurity density) limited by Coulomb scattering with experimental data. Solid lines (from bottom to top) show the minimum conductivity of $4e^2/h$, theory for $d=0$ and 0.2 nm. The inset shows the results in a linear scale assuming that the impurity shifts by $d=0.2\mathrm{nm}$ for positive voltage bias. Adapted from 211. (b) Graphene conductivity calculated using a combination of short- and long-range disorder. In the calculation, $n_d/n_i=0$, 0.01, and 0.02 (top to bottom) are used. In inset the graphene mobility as a function of dielectric constant ($\kappa$) of substrate is shown for different carrier densities $n=0.1$, 1, and $5\times {10}^{12}{\mathrm{cm}}^{-2}$ (from top to bottom) in the presence of both long-ranged charged impurity ($n_i=2\times {10}^{11}{\mathrm{cm}}^{-2}$) and short-ranged neutral impurity ($n_d=0.4\times {10}^{10}{\mathrm{cm}}^{-2}$). $V_0=10\mathrm{eV}{\mathrm{nm}}^2$ is used in the calculation.

Figure 15

Hall mobility as a function of temperature for different hole densities in (a) monolayer graphene, (b) bilayer graphene, and (c) trilayer graphene. The symbols are the measured data and the lines are fits. Adapted from 472.

Figure 16

The measured conductivity of bilayer graphene as a function of gate voltage $V_g$ (or carrier density). The measured conductivity increases linearly with the density. Adapted from 317.

Figure 17

(a) Density dependence of bilayer graphene conductivity with two scattering sources: screened long-range Coulomb disorder and short-ranged neutral disorder. (b) Density dependence of BLG conductivity for different temperatures: $T=0$, 50, 100, 150, 200, and 300 K (from bottom to top). Top inset shows $\sigma$ as a function of $T$ in presence of short-range disorder. Bottom inset shows $\sigma$ as a function of $T$ in presence of screened Coulomb disorder for different densities $n=[5,10,30]\times {10}^{11}{\mathrm{cm}}^{-2}$ (from bottom to top). Adapted from 109.

Figure 18

(a) Experimental mobility $\mu$ as a function of density for two Si-MOSFET samples at a temperature $T=0.25\mathrm{K}$. (b) Calculated mobility with two different scatterings, i.e., charge impurities and surface-roughness scatterings. (c) Measured conductivity $\sigma (n)$ for Si-MOSFET as a function of electron density $n$ for two different samples. The solid lines are fits to the data of the form $\sigma (n)\propto A(n-n_p{)}^p$. The upper and lower insets show the exponent $p$ and critical density $n_p$, respectively, as a function of temperature. Solid lines are a guides for the eyes. Adapted from 431.

Figure 19

(a), (b) Experimentally measured (symbols) and calculated (lines) conductivity of two different $n$-GaAs samples. The high density conductivity limited by the charged impurities fit well to the experimental data. Adapted from 112. (c) Mobility of $p$-GaAs 2D system vs density at fixed temperature $T=47\mathrm{mK}$. (d) The corresponding conductivity vs density (solid squares) along with the fit generated assuming a percolation transition. The dashed line in (c) indicates the $\mu \sim p^{0.7}$ behavior. Adapted from 293.

Figure 20

(a) Experimental resistivity $\rho$ of Si-MOSFET as a function of temperature at 2D electron densities (from top to bottom) $n=[1.07,1.10,1.13,1.20,1.26,1.32,1.38,1.44,1.50,1.56,1.62,\mathrm{\text{and}}1.68]\times {10}^{11}{\mathrm{cm}}^2$. Inset (b) shows $\rho$ for $n=[1.56,1.62,\mathrm{\text{and}}1.68]\times {10}^{11}{\mathrm{cm}}^2$. (c) Theoretically calculated temperature and density-dependent resistivity for sample A for densities $n=[1.26,1.32,1.38,1.44,1.50,1.56,1.62,\mathrm{\text{and}}1.68]\times {10}^{11}{\mathrm{cm}}^2$ (from top to bottom). Adapted from 431. (e) Experimental $\rho (T)$ for $n$-GaAs (where $n_c=2.3\times {10}^9{\mathrm{cm}}^{-2}$). The density ranges from $0.16\times {10}^{10}$ to $1.06\times {10}^{10}{\mathrm{cm}}^{-2}$. Adapted from 280. (f) Temperature dependence of the resistivity for $p$-GaAs systems for densities ranging from $9.0\times {10}^9$ to $2.9\times {10}^9{\mathrm{cm}}^{-2}$. Adapted from 293.

Figure 21

Temperature-dependent resistivity of graphene on ${\mathrm{SiO}}_2$. Resistivity of two graphene samples as a function of temperature for different gate voltages. Dashed lines are fits to the linear $T$ dependence with Eq. (3.32). Adapted from 87.

Figure 22

Temperature-dependent resistivity for four different MLG samples (symbols). The solid curve is the best fit by using a combination of $T$ and $T^5$ functions. The inset shows $T$ dependence of maximum resistivity at the neutrality point for MLG and BLG (circles and squares, respectively). Adapted from 317.

Figure 23

(a) Acoustic phonon-limited mobility of $n$-GaAs 2D system as a function of density for two different temperatures. (b) Calculated $n$-GaAs mobility as a function of temperature for different impurity densities. At low temperatures ($T<1\mathrm{K}$) the mobility is completely limited by impurity scattering. Adapted from 215.

Figure 24

Calculated graphene mobility limited by the acoustic phonon with the deformation-potential coupling constant $D=19\mathrm{eV}$ (a) as a function of temperature and (b) as a function of density. Adapted from 214.

Figure 25

Left panel: Raman spectra (wavelength 633 nm) for (a) pristine graphene and (b) graphene irradiated by 500 eV ${\mathrm{Ne}}^{+}$ ions that are known to cause vacancies in the graphene lattice. Right panel: Increasing the number of vacancies by ion irradiation caused a transition from the pristine graphene (where Coulomb scattering dominates) to the lower curves where scattering from vacancies dominate. Also shown is a fit to Eq. (3.36) from 412 that describes scattering off vacancies that have midgap states. From 90.

Figure 26

Resistance $R=1/G$ (left) and conductivity (right) obtained using $\sigma =[WdR/dL{]}^{-1}$ as a function of sample length $L$. The three curves shown are for $W/\xi =200$, $K_0=2$, and $\pi n{\xi }^2=0$, 0.25, and 1 [from top to bottom (bottom to top) in left (right) panel]. Dashed lines in the right panel show d$\sigma /d\ln L=4e^2/\pi h$. The inset in the left panel shows the crossover to diffusive transport ($L\gg \xi$).

Figure 27

Semiclassical conductivity ${\sigma }^{\prime }=\underset{L\to \infty }{lim}[\sigma (L)-{\pi }^{-1}\ln (L/\xi )]$ vs disorder strength at the Dirac point (left) and at carrier density $\pi n=K_0/(\pi \xi {)}^2$, corresponding to the edge of the minimum conductivity plateau of 14 (right). Data points are from the numerical calculation for $L=50\xi$ and the (solid) dashed curves represent the (self-consistent) Boltzmann theory. Adapted from 9.

Figure 28

Exchange, solid line, and RPA correlation potentials, dashed line, as functions of the density $n$ for $r_s=0.5$. The dash-dotted line shows the full exchange-correlation potential $V_{\mathrm{xc}}$. The dotted line is the quantum Monte Carlo exchange-correlation potential of a standard parabolic-band 2D electron gas (26) with effective mass $0.067m_e$ placed in background with dielectric constant 4. Adapted from 365.

Figure 29

TFD results as a function of position for a single disorder realization with $\kappa =2.5$, $d=1\mathrm{nm}$, and $n_{\mathrm{imp}}={10}^{12}\mathrm{c}{\mathrm{m}}^{-2}$. (a) Bare disorder potential $V_D$. (b) Carrier density obtained neglecting exchange-correlation terms. (c) Carrier density obtained including exchange-correlation terms. (d) Probability density distribution at the CNP with exchange (solid line) and without (dashed line). Adapted from 377.

Figure 30

TFD disorder averaged results for $\kappa =2.5$. The solid (dashed) lines show the results obtained including (neglecting) exchange and correlation terms (a) $n_{\mathrm{rms}}$ and (b) $\xi$ as a function of $n_{\mathrm{imp}}$. (c) Area $A_0$ over which $|n(\mathbf{r})-⟨n⟩|<n_{\mathrm{rms}}/10$. (d) Average excess charge $\delta Q$ vs $n_{\mathrm{imp}}$. Adapted from 377.

Figure 31

(a) Density distribution averaged over disorder for different values of the applied gate voltage assuming $\kappa =2.5$, $d=1\mathrm{nm}$ and $n_{\mathrm{imp}}={10}^{12}\mathrm{c}{\mathrm{m}}^{-2}$. (b) $n_{\mathrm{rms}}/⟨n⟩$ as a function of $⟨n⟩$ for $d=1\mathrm{nm}$ and different values of $n_{\mathrm{imp}}$: circles, $n_{\mathrm{imp}}=1.5\times {10}^{12}\mathrm{c}{\mathrm{m}}^{-2}$; squares, $n_{\mathrm{imp}}={10}^{12}\mathrm{c}{\mathrm{m}}^{-2}$; triangles, $n_{\mathrm{imp}}=5\times {10}^{11}\mathrm{c}{\mathrm{m}}^{-2}$. In (b) the solid (dashed) lines show the results with (without) exchange and correlation terms. Adapted from 377.

Figure 32

(a) Carrier density profile map at the CNP measured with an SET. The contour marks the zero-density contour. Adapted from 296. (b) Spatial map, on a $80\mathrm{nm}\times 80\mathrm{nm}$ region, of the energy shift of the CNP in BLG from STM $dI/dV$ map. Adapted from 119. (c) $60\mathrm{nm}\times 60\mathrm{nm}$ constant current STM topography, and (d) simultaneous $dI/dV$ map, at the CNP for MLG, $(V_{\mathrm{bias}}=-0.225\mathrm{V},I=20\mathrm{pA})$. Adapted from 468.

Figure 33

Solid (dashed) lines show the EMT-TFD results with (without) exchange. (a) $\sigma$ as function of $V_g$ for three different values of $n_{\mathrm{imp}}$. The dots at $V_g=0$ are the results presented by 14 for the same values of $n_{\mathrm{imp}}$. (b) ${\sigma }_{\min }$ as a function of $r_s$ for $d=1$ and $n_{\mathrm{imp}}={10}^{11}{\mathrm{cm}}^{-2}$. Solid squares show the experimental results of 229. (c) ${\sigma }_{\min }$ as a function of $n_{\mathrm{imp}}$, $r_s=0.8$ and $d=1\mathrm{nm}$. For comparison the results obtained by 14 are also shown by the dot-dashed line. (d) ${\sigma }_{\min }$ as a function of the inverse mobility as measured by 86. $r_s$ for $d=1$ and $n_{\mathrm{imp}}={10}^{11}{\mathrm{cm}}^{-2}$. (a)–(c) Adapted from 374. (d)  Adapted from 86.

Figure 34

Collapse of the conductivity data obtained for RRNs with various $-1/2<p<1/2$ and values of the parameter $\gamma$ onto a single curve. Points: numerical results; line: best fit obtained using the equations in the text. Adapted from 81.

Figure 35

Illustration of the integer QHE found in 2D semiconductor systems, (a), incorporated from 290 and 370. (b) Illustration for SLG. (c) Illustration from BLG. The sequences of Landau levels as a function of carrier concentrations n are shown as dark and light peaks for electrons and holes, respectively. Adapted from 341. (d) ${\sigma }_{xy}$ and ${\rho }_{xx}$ of MLG as a function of carrier density measured experimentally at $T=4\mathrm{K}$ and $B=14\mathrm{T}$. Adapted from 343.

Figure 36

(a) ${\sigma }_{xy}$ as a function of gate voltage at different magnetic fields: 9 T (circle), 25 T (square), 30 T (diamond), 37 T (up triangle), 42 T (down triangle), and 45 T (star). All data sets are taken at $T=1.4\mathrm{K}$, except for the $B=9\mathrm{T}$ curve, which is taken at $T=30\mathrm{mK}$. Left upper inset: $R_{xx}$ and $R_{xy}$ for the same device measured at $B=25\mathrm{T}$. Right inset: Detailed ${\sigma }_{xy}$ data near the Dirac point for $B=9\mathrm{T}$ (circle), 11.5 T (pentagon), and 17.5 T (hexagon) at $T=30\mathrm{mK}$. Adapted from 466. (b) Longitudinal resistance $R_{xx}$ as a function of gate voltage $V_g^{\prime }=V_g-V_0$ at 0.3 K and several values of the magnetic field: 8, 11, and 14 T. The inset shows a graphene crystal with Au leads deposited. The bar indicates $5\mu \mathrm{m}$. At $V_g^{\prime }=0$, the peak in $R_{xx}$ grows to $190\mathrm{k}\Omega$ at 14 T. Adapted from 77.

Figure 37

Graphene fractional quantum Hall data, from (a) 128 and (b) 52, observed on two probe suspended graphene samples.

Figure 38

(a) Measured Hall conductivity ${\sigma }_{xy}$ in BLG as a function of carrier density for $B=12$ and $B=20\mathrm{T}$ at $T=4\mathrm{K}$. (b) Measured longitudinal resistivity in BLG at $T=4\mathrm{K}$ and $B=12\mathrm{T}$. The inset shows the calculated BLG bands close to the CNP. Adapted from 341. (c) Two-terminal conductance, $G$, as a function of carrier density at $T=100\mathrm{mK}$ for different values of the magnetic field in suspended BLG. Adapted from 142.