Effective mass in bilayer graphene at low carrier densities: The role of potential disorder and electron-electron interaction

Rapid Communication

Effective mass in bilayer graphene at low carrier densities: The role of potential disorder and electron-electron interaction

J. Li, L. Z. Tan, K. Zou, A. A. Stabile, D. J. Seiwell, K. Watanabe, T. Taniguchi, Steven G. Louie, and J. Zhu

Phys. Rev. B 94, 161406(R) – Published 25 October 2016

Abstract

In a two-dimensional electron gas, the electron-electron interaction generally becomes stronger at lower carrier densities and renormalizes the Fermi-liquid parameters, such as the effective mass of carriers. We combine experiment and theory to study the effective masses of electrons and holes $m_{e}^{*}$ and $m_{h}^{*}$ in bilayer graphene in the low carrier density regime on the order of $1 \times 10^{11} c m^{- 2}$ . Measurements use temperature-dependent low-field Shubnikov–de Haas oscillations observed in high-mobility hexagonal boron nitride supported samples. We find that while $m_{e}^{*}$ follows a tight-binding description in the whole density range, $m_{h}^{*}$ starts to drop rapidly below the tight-binding description at a carrier density of $n = 6 \times 10^{11} c m^{- 2}$ and exhibits a strong suppression of 30% when $n$ reaches $2 \times 10^{11} c m^{- 2}$ . Contributions from the electron-electron interaction alone, evaluated using several different approximations, cannot explain the experimental trend. Instead, the effect of the potential fluctuation and the resulting electron-hole puddles play a crucial role. Calculations including both the electron-electron interaction and disorder effects explain the experimental data qualitatively and quantitatively. This Rapid Communication reveals an unusual disorder effect unique to two-dimensional semimetallic systems.

Received 6 July 2016
Revised 2 September 2016

DOI:https://doi.org/10.1103/PhysRevB.94.161406

Physics Subject Headings (PhySH)

2-dimensional systems Graphene

Random phase approximation Shubnikov-de Haas effect Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

J. Li¹, L. Z. Tan^2,3,*, K. Zou^1,†, A. A. Stabile¹, D. J. Seiwell¹, K. Watanabe⁴, T. Taniguchi⁴, Steven G. Louie^2,3,‡, and J. Zhu^1,5,§

¹Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
²Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
³Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
⁴National Institute for Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan
⁵Center for 2-Dimensional and Layered Materials, The Pennsylvania State University, University Park, Pennsylvania 16802, USA

^*Present address: Department of Chemistry, The Makineni Theoretical Laboratories, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323, USA; Corresponding author: liangtan@sas.upenn.edu
^†Present address: Department of Applied Physics, Center for Research on Interface Structures and Phenomena (CRISP), Yale University, New Haven, Connecticut 06520, USA.
^‡Corresponding author: sglouie@berkeley.edu
^§Corresponding author: jzhu@phys.psu.edu

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Vol. 94, Iss. 16 — 15 October 2016

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Images

Figure 1
Sheet resistance vs carrier density $R_{sheet} (n)$ for samples A (solid red), B (solid blue), and C (dashed blue). Samples A and B are supported on h-BN; sample C is supported on $Si O_{2}$ . The field effect mobility $μ_{FE}$ is 30 000, 20 000, and $4000 c m^{2} V^{- 1} s^{- 1}$ , respectively, for samples $A - C$ . $T = 1.6 K$ . The large resistance sample A exhibits at the CNP results from a finite band gap caused by unintentional doping. We discuss the effect of a band gap on the band mass in Fig. S4 of the Supplemental Material [39]. The inset: An optical micrograph for sample A.
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Figure 2
(a) $T$ -dependent magnetoresistance $R_{x x} (B)$ for $n_{h} = 4.7 \times 10^{11} c m^{- 2}$ at selected temperatures as indicated in the plot. (b) Oscillation amplitude $δ R_{x x} (B)$ of the data in (a) after background subtraction. The solid red curve plots Eq. (2) with fitting parameters $m_{h}^{*} = 0.0347 m_{e}$ and $τ_{q} = 140 fs$ . $T = 2.3 K$ . $δ R_{x x} (B)$ starts deviating from the fit above $B = 3 T$ . The conventional method used to extract $δ R_{x x}$ is illustrated by the blue dashed lines and produces $m^{*} = 0.0311 (2) m_{e}$ . This is 10% smaller than $m_{h}^{*} = 0.0347 m_{e}$ obtained from the global fitting. (c) $δ R_{x x} (B)$ for $n_{h} = 3.0 \times 10^{11} c m^{- 2}$ at $T = 2.3$ and $T = 15 K$ . The dashed curves are fits to Eq. (2) with $m_{h}^{*} = 0.0285 m_{e}$ and $τ_{q} = 107 fs$ . Data in (a)–(c) are from sample B. (d) The quantum scattering time $τ_{q}$ as a function of carrier density in sample A (red symbols) and sample B (blue symbols). Electrons are shown by the filled symbols, and holes are shown by the open symbols. $τ_{q}$ is about 40 fs (dashed gray line) in sample C (Ref. [10]).
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Figure 3
The effective carrier masses $m_{h}^{*}$ and $m_{e}^{*}$ as a function of the carrier density (red for electrons, blue for holes) in samples A (squares), B (stars), and C (triangles). The data on sample C are from Ref. [10]. Together, the measurement covers the density range of approximately $1.4 − 4.1 \times 10^{11} c m^{- 2}$ . The dashed curves plot $m^{*}$ calculated using a $4 \times 4$ tight-binding Hamiltonian with hopping parameters $γ_{0} = 3.43 eV, γ_{1} = 0.40 eV, γ_{3} = 0$ , and $v_{4} = 0.063$ . These values are obtained in Ref. [10] by fitting the data in sample C at high densities.
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Figure 4
Comparison of calculations and experiment at low carrier density $(0.2 - 1.3 \times 10^{12} c m^{- 2})$ . Experimental data follow the symbols used in Fig. 3. The olive dashed lines plot the calculated $m^{*}$ including the $e - e$ interaction in a random-phase approximation. The black and gray lines are calculations that further include the effect of potential disorder using $δ E = 5.4 meV$ obtained from $τ_{q}$ and the temperature dependence of the conductance. In both calculations, $γ_{0} = 3.08$ eV and $γ_{1} = 0.36 eV$ are chosen to fit the experimental data in the high-density regime. Their values differ from those obtained in Ref. [10] since the $e - e$ interaction is explicitly calculated here whereas in Ref. [10] its effect is represented by renormalizing the hopping parameters. $γ_{3} = 0$ and $v_{4} = 0.063$ are taken from Ref. [10]. The inset: a schematic of the electron-hole coexistence at low carrier densities due to disorder and its effect on the cyclotron motion.
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