Effective mass of electrons and holes in bilayer graphene: Electron-hole asymmetry and electron-electron interaction

Precision measurements of the effective mass $m^{*}$ in high-quality bilayer graphene using the temperature dependence of the Shubnikov–de Haas oscillations are reported. In the density range $0.7\times {10}^{12}$ $<$ $n$ $<$ $4.1\times {10}^{12}$ cm${}^{-2}$, both the hole mass $m_{\mathrm{h}}^{*}$ and the electron mass $m_{\mathrm{e}}^{*}$ increase with increasing density, demonstrating the hyperbolic nature of the bands. The hole mass $m_{\mathrm{h}}^{*}$ is approximately 20–30% larger than the electron mass $m_{\mathrm{e}}^{*}$. Tight-binding calculations provide a good description of the electron-hole asymmetry and yield an accurate measure of the interlayer hopping parameter $v_4=0.063$. Both $m_{\mathrm{h}}^{*}$ and $m_{\mathrm{e}}^{*}$ are suppressed compared with single-particle values, suggesting renormalization of the band structure of bilayer graphene induced by electron-electron interaction.

Figure 1

Sheet conductance $\sigma \left(V_{\mathrm{bg}}\right)$ of sample A. From top to bottom: (a) $T=15,100,150,\text{and}250$ K and (b) in reverse order. The charge neutrality point is at $V_{\mathrm{bg}}=7$ V.

Figure 2

(a) SdH oscillations ${\rho }_{\mathrm{xx}}\left(B\right)$ at $T$ $=$ 1.5–50 K for $n_{\mathrm{e}}$ $=$ $3.26\times 10$${}^{12}$cm${}^{-2}$. (b) ${\rho }_{\mathrm{xx}}\left(B\right)$ at $T$ $=10$ and 40 K for $n_{\mathrm{h}}$ $=0.89\times 10$${}^{12}$ cm${}^{-2}$. Dashed lines are fittings with ${\tau }_{\mathrm{q}}$ $=42$ fs and $m_{\mathrm{h}}^{*}$ $=$ 0.036$m_{\mathrm{e}}$. A smooth background has been subtracted. (c) $\delta {\rho }_{\mathrm{xx}}/T$ versus $T$ in a semilog plot for $n_{\mathrm{e}}$ $=3.26\times 10$${}^{12}$ cm${}^{-2}$ at $B$ $=$ 7.53 T [down triangle in (a)] and for $n_{\mathrm{h}}$ $=0.89\times 10$${}^{12}$ cm${}^{-2}$ at $B=4.70$ T [up triangle in (b)]. The symbols correlate. Dashed lines are fittings with $m_{\mathrm{e}}^{*}$ $=$ 0.041$m_{\mathrm{e}}$ (down triangles) and $m_{\mathrm{h}}^{*}$ $=$ 0.036$m_{\mathrm{e}}$ (up triangles). (d) ${\tau }_{\mathrm{q}}$ versus $n$ for electrons (red triangles) and holes (black squares). All data in (a)–(d) are from sample A.

Figure 3

(a) Measured $m_{\mathrm{h}}^{*}$ and $m_{\mathrm{e}}^{*}$ vs $n$ for samples A and B. The symbols are indicated in (a) and used in (a)–(c). Solid blue lines are fittings to sample A with ${\gamma }_0$ $=$ 3.419 eV, ${\gamma }_1$ $=$ 0.40 eV, and $v_4=0.063$. The magenta dash-dotted line is a fitting to sample B with ${\gamma }_0$ $=$ 3.447 eV, ${\gamma }_1$ $=$ 0.40 eV, and $v_4=0.063$. The yellow dashed lines correspond to ${\gamma }_0$ $=$ 3.167 eV, ${\gamma }_1$ $=$ 0.30 eV, and $v_4=0.063$. (b) Comparison of measured $m^{*}$ and calculated $m_0^{*}$ for sample A. From top to bottom: ${\gamma }_0$ $=$ 2.72 (olive), 3.09 (wine), and 3.42 eV (blue). ${\gamma }_1$ $=$ 0.40 eV, $v_4=0.063,$ and $\Delta$ $=$ 0.018 eV for all traces. (c) Ratio $m^{*}$/$m_0^{*}$ vs $n$ for sample A. From top to bottom: ${\gamma }_0$ $=$ 3.42, 3.09, and 2.72 eV. Dashed lines are visual guides.

Figure 4

Warped Fermi surfaces in momentum space for $E_{\mathrm{F}}=100$ meV ($n\sim 4\times {10}^{12}$ cm${}^{-2}$) (outer contour) and $E_{\mathrm{F}}=25$ meV ($n\sim 0.7\times {10}^{12}$ cm${}^{-2}$) (inner contour). The parameters used are ${\gamma }_0=3.1$ eV, ${\gamma }_1=0.4$ eV, ${\gamma }_3=0.31$ eV, $\Delta =0$, and $V\left(n\right)=0$.

Figure 5

Measured (symbols) and calculated (lines) $m^{*}$ of sample A. Blue solid lines correspond to an initial doping of $V_{\mathrm{bg}}$ $=$ $+$7 V, and orange dashed lines correspond to zero doping. The other parameters used are ${\gamma }_0$ $=$ 3.419 eV, ${\gamma }_1$ $=$ 0.4 eV, ${\gamma }_3$ $=$ 0.0 eV, $v_4$ $=$ 0.063, and $\Delta$ $=$ 0.018 eV.

Figure 6

Fitting results of ${\gamma }_0$ as a function of ${\gamma }_1$. The linear fit corresponds to ${\gamma }_0$ $=$ 2.411 $+$ 2.52${\gamma }_1$.