Tuning Many-Body Interactions in Graphene: The Effects of Doping on Excitons and Carrier Lifetimes

Kin Fai Mak, Felipe H. da Jornada, Keliang He, Jack Deslippe, Nicholas Petrone, James Hone, Jie Shan, Steven G. Louie, and Tony F. Heinz

Phys. Rev. Lett. 112, 207401 – Published 20 May 2014

Abstract

The optical properties of graphene are strongly affected by electron-electron ( $e − e$ ) and electron-hole ( $e − h$ ) interactions. Here we tune these many-body interactions through varying the density of free charge carriers. Measurements from the infrared to the ultraviolet reveal significant changes in the optical conductivity of graphene for both electron and hole doping. The shift, broadening, and modification in shape of the saddle-point exciton resonance reflect strong screening of the many-body interactions by the carriers, as well as changes in quasiparticle lifetimes. Ab initio calculations by the $G W$ Bethe-Salpeter equation method, which take into account the modification of both the repulsive $e − e$ and the attractive $e − h$ interactions, provide excellent agreement with experiment. Understanding the optical properties and high-energy carrier dynamics of graphene over a wide range of doping is crucial for both fundamental graphene physics and for emerging applications of graphene in photonics.

Received 26 December 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.207401

Authors & Affiliations

Kin Fai Mak¹, Felipe H. da Jornada^2,3, Keliang He⁴, Jack Deslippe^2,3,5, Nicholas Petrone⁶, James Hone⁶, Jie Shan⁴, Steven G. Louie^2,3, and Tony F. Heinz^1,*

¹Departments of Physics and Electrical Engineering, Columbia University, New York, New York 10027, USA
²Department of Physics, University of California, Berkeley, California 94720, USA
³Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
⁴Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106, USA
⁵National Energy Research Scientific Computing Center, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
⁶Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA

^*Corresponding author. tony.heinz@columbia.edu

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Vol. 112, Iss. 20 — 23 May 2014

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Images

Figure 1
Optical response of graphene as a function of doping, showing results of experiment (black line or symbols), of $G W$ calculations (green line), and of $G W$ -BSE calculations (red line). (a) The optical conductivity $σ (E)$ of graphene as a function of photon energy $E$ for different doping densities $n$ . A redshift of the experimentally observed peak in the UV is observed for both electron and hole doping. The position of the peak and its variation with doping are captured by $G W$ -BSE calculations. The $G W$ calculations, which omit $e − h$ interactions, overestimate the resonance energy for all doping densities. (b) The resonance energy $ω_{0}$ as a function of doping density, with combined data for electron and hole doping. (c) The shift $Δ ω_{0}$ in resonance energy from that at charge neutrality as a function of doping, with combined data for electron and hole doping. The net shift (red line) calculated within the $G W$ -BSE method is decomposed into two opposing contributions from screening of the electron self-energy (green line) and of the excitonic response (blue line). Effects due to finite lifetime of the excited electron and hole have not been included in the $G W$ and $G W$ -BSE calculations for the theoretical curves in this figure.
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Figure 2
Ab initio calculations of the optical conductivity spectra $σ (E)$ including lifetime effects. The calculated $G W$ -BSE conductivity spectra with quasiparticle lifetime effects included (red dashed lines) are compared to the experimental spectra (black lines) for four different doping densities. By evaluating the influence of both $e − e$ and $e -ph$ decay channels for the optically excited states, we obtain agreement with the experimental spectra for all doping levels.
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Figure 3
Decay rates of highly excited quasiparticles in graphene. The linewidth of the saddle-point exciton (as discussed in the text) is shown as a function of absolute doping level (for doping densities shown in Fig. 2), given in terms of the Fermi energy, for measurements with both electron and hole doping. The dots are the experimental values, while the columns show the calculated contributions from $e -ph$ (blue) and $e − e$ (orange) interactions. The horizontal dotted line is a guide to the eye.
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Figure 4
Calculated redistribution of oscillator strength. The $k$ -space distribution of the oscillator strength $I (k)$ is plotted for (a) a low carrier doping density ( $n = 0.4 \times 10^{13} {cm}^{- 2}$ ) and (b) a high doping density ( $n = 13 \times 10^{13} {cm}^{- 2}$ ). The corresponding energy distribution of oscillator strength $I (Δ ω)$ is also shown for (c) the low and (d) the high doping levels.
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