University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications, Department of Physics and Astronomy Research Papers in Physics and Astronomy 2011 Effective mass of electrons and holes in bilayer graphene: Electron-hole asymmetry and electron-electron interaction K. Zou The Pennsylvania State University X. Hong University of Nebraska–Lincoln, [email protected] J. Zhu The Pennsylvania State University, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/physicsfacpub Part of the Physics Commons Zou, K.; Hong, X.; and Zhu, J., "Effective mass of electrons and holes in bilayer graphene: Electron-hole asymmetry and electron-electron interaction" (2011). Faculty Publications, Department of Physics and Astronomy. 96. https://digitalcommons.unl.edu/physicsfacpub/96 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications, Department of Physics and Astronomy by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. PHYSICAL REVIEW B 84, 085408 (2011) Effective mass of electrons and holes in bilayer graphene: Electron-hole asymmetry and electron-electron interaction K. Zou,1 X. Hong,1,2 and J. Zhu1,3 1Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA 2Department of Physics and Astronomy, University of Nebraska–Lincoln, Lincoln, Nebraska 68588, USA 3Materials Research Institute, The Pennsylvania State University, University Park, Pennsylvania 16802, USA (Received 20 July 2011; published 22 August 2011) 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 × 1012 <n<4.1 × 12 −2 ∗ ∗ 10 cm , both the hole mass mh and the electron mass me increase with increasing density, demonstrating ∗ the hyperbolic nature of the bands. The hole mass mh is approximately 20–30% larger than the electron mass ∗ me . Tight-binding calculations provide a good description of the electron-hole asymmetry and yield an accurate = ∗ ∗ measure of the interlayer hopping parameter v4 0.063. Both mh and me are suppressed compared with single- particle values, suggesting renormalization of the band structure of bilayer graphene induced by electron-electron interaction. DOI: 10.1103/PhysRevB.84.085408 PACS number(s): 73.43.Qt, 71.70.Gm, 73.20.At I. INTRODUCTION II. SAMPLE PREPARATION Bilayer graphene may be a technologically important Bilayer graphene flakes are exfoliated onto 290-nm SiO2/Si material in electronics and photonics due to its tunable wafers from highly ordered pyrolytic graphite and identified by band gap. The fundamental property that underpins such optical microscopy and Raman spectroscopy. They are further applications—its band structure—has been the subject of confirmed by their quantum Hall sequence. Conventional many recent theoretical1–4 and experimental studies using electron-beam lithography is used to pattern the flakes into angle-resolved photoemission spectroscopy,5 infrared and Hall bars. Raman measurements,6–8 cyclotron mass measurements,9 and compressibility measurements.10,11 On a single-particle level, the band structure of the bilayer is thought to be well described III. MEASUREMENTS by a tight-binding Hamiltonian2,3 with a few leading-order Transport measurements are carried out in a He4 system Slonczewski-Weiss-McClure parameters, i.e., γ0, γ1, γ3, and using standard low-frequency lock-in technique. The field γ4. Experimental knowledge of these hopping parameters effect mobility μFE = (1/e)(dσ/dn) of our pristine bilayer 2 in the bilayer varies, with γ1 = 0.40 eV fairly accurately graphene ranges from 3000 to 12 000 cm /Vs. Data from two known7,8 and the rest much less known. For example, ex- samples (A and B) are presented in this paper. perimental values of γ4, which controls the band asymmetry, In Fig. 1, we plot the sheet conductance σ versus the 6–8,10 range from 0.11 to 0.19 eV. back-gate voltage Vbg of sample A at selected temperatures Meanwhile, electron-electron (EE) interactions in single- between 15 and 250 K. At 15 K, the mobility μFE of sample layer and bilayer graphene are predicted to be strong and A is approximately 4800 cm2/Vs for holes and 3100 cm2/Vs peculiar. Interesting collective states emerge in a magnetic for electrons. Sample B has a higher mobility of 6300 cm2/Vs field.12,13 The many-body corrections to Fermi liquid param- for holes and 6800 cm2/Vs for electrons. The conductance eters such as the compressibility κ and the effective mass of bilayer graphene samples shows a variety of temperature ∗ m are expected to be substantial at currently accessible dependence, depending on the carrier density and mobility. densities.14–18 These renormalization effects are related to, Near the charge neutrality point, all our samples show an but also quantitatively different from, those observed in insulating-like T dependence (dσ/dT >0), as shown in Fig. 1. conventional two-dimensional electron gases (2DEGs),19,20 This behavior is due to the thermal excitation of carriers out due to the chirality of single-layer and bilayer graphene.16 of electron-hole (EH) puddles, as demonstrated in Ref. 21. For example, instead of an enhancement,20 the effective mass As the carrier density increases, dσ/dT eventually becomes of bilayer graphene is predicted to be increasingly suppressed negative (metallic) in the highest-quality samples. This trend at lower carrier densities.16 No experimental evidence of such is shown by the hole branch in Figs. 1(a) and 1(b), where 12 −2 renormalization effect has been reported so far. the crossover density is approximately nh = 2.1 × 10 cm . In this work, we report measurements of the effective For samples with lower mobilities, the insulating-like T ∗ mass m in bilayer graphene samples for a wide range dependence persists to high densities, an example of which of carrier densities using high-quality Shubnikov–de Haas is given by the electron branch in Fig. 1(b). (SdH) oscillations. The interlayer hopping parameter γ4 is This complex behavior is in contrast to that of single-layer determined to be γ4 = 0.063(1)γ0, with the highest accuracy graphene, for which a metallic-like temperature dependence ∗ reported so far. The magnitude and density dependence of m dominates over a wide range of densities due to phonon deviate from tight-bind calculations, providing evidence for scattering.22–24 The qualitative features of our data are consis- EE-interaction-induced band renormalization. tent with the model proposed in Ref. 25, where σ (T ) combines 1098-0121/2011/84(8)/085408(6)085408-1 ©2011 American Physical Society K. ZOU, X. HONG, AND J. ZHU PHYSICAL REVIEW B 84, 085408 (2011) T (K) (a) 15K (b) 100K 2.0 02040 150K 800 1.5K 3.0 10K (a) (c) 250K 100 20K ) 30K K 1.5 ) 40K Ω ) 2.8 Ω 600 50K 10 ( /T ( / xx ρ (mS xx σ 2.6 1.0 δρ 400 1 1 2.4 (b) (d) 0.5 60 ) 0 55 2.2 Ω (k 50 (fs) 0.0 q xx τ -50 -45 -40 -35 -30-20 0 20 40 ρ -1 45 V (V) 10K bg 40K 40 -2 FIG. 1. (Color online) Sheet conductance σ (Vbg) of sample A. 0246801234 12 2 From top to bottom: (a) T = 15,100,150,and 250 K and (b) in reverse B (()T) n (10 /cm ) order. The charge neutrality point is at Vbg = 7V. FIG. 2. (Color online) (a) SdH oscillations ρxx(B)atT = 1.5– = × 12 −2 = metallic and insulating trends arising from the conduction 50 K for ne 3.26 10 cm .(b)ρxx(B)atT 10and40K = × 12 −2 = of the majority and minority carriers, respectively. The true for nh 0.89 10 cm . Dashed lines are fittings with τq 42 ∗ = metallic T dependence of a bilayer graphene 2DEG emerges fs and mh 0.036me. A smooth background has been subtracted. = × 12 −2 only in high-quality samples and/or at high carrier densities. (c) δρxx/T versus T in a semilog plot for ne 3.26 10 cm at = = × 12 −2 In Fig. 1, the different T dependence of the two carriers in the B 7.53 T [down triangle in (a)] and for nh 0.89 10 cm at = same sample points to an intrinsic EH asymmetry of bilayer B 4.70 T [up triangle in (b)]. The symbols correlate. Dashed lines are fittings with m∗ = 0.041m (down triangles) and m∗ = 0.036m graphene, which we further examine below. e e h e To probe the band structure of bilayer graphene, we measure (up triangles). (d) τq versus n for electrons (red triangles) and holes ∗ (black squares). All data in (a)–(d) are from sample A. the effective mass m as a function of the carrier density using SdH oscillations. This technique is well established in 2DEGs ∗ ∗ = but requires high-quality oscillations to reliably extract m . Dashed lines are fittings generated with me 0.041me and m∗ = 0.036m , respectively, where m is the electron rest mass. Figure 2(a) shows the SdH oscillations ρxx(B) of sample A at h e e 12 −2 They both fit very well. Overall, Eq. (1) provides an excellent a high electron density ne = 3.26 × 10 cm and varying temperatures. The oscillations have an early onset, appear description of the δρxx(T,B) data in the entire density range studied, with the uncertainty of m∗ increasing from 0.0001m sinusoidal, and are free of beating. The amplitude δρxx is given e 26 to 0.0015me from high to low densities.