optoelectronics

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SUPPLEMENTARY INFORMATION i DOI: 10.1038/NMAT4755

Supplementary material for Thermoelectric detection and imaging of propagating graphene plasmons CONTENTS

I. Interference model for fringes in photocurrent A. Wave propagation B. Thermal spreading C. Photocurrent generation

i i ii iii

II. Electrodynamic model of absorbed power A. Electrostatic calculation of tip charge distribution B. Electrodynamic absorption References

iii iii v vi

I.

INTERFERENCE MODEL FOR FRINGES IN PHOTOCURRENT

As a simple model of our scanning photocurrent patterns I(xtip , ytip ), we describe them as a two-dimensional problem: the tip-launched plasmons are represented as two-dimensional waves, which decay into a two-dimensional heat distribution within a two-dimensional thermoelectric system. In making such a model, we inevitably must discard the overall prefactor which sets the strength of our plasmon launching, since this is quite difficult to map into two dimensions. We therefore also discard all other constant proportionality factors and focus on the xtip , ytip dependence of the result. This spatial form depends on remarkably few variables: kp (complex plasmon wavevector), r (complex edge-reflection coefficent), and lT (cooling length), as described in the main text.

A.

Wave propagation

The graphene plasmon is an oscillating 2D charge plasma wave ρ(x, y)e−iωt at fixed frequency ω. This follows a conservation law: ρ(x, y) +

i ∇ · j(x, y) = f (x, y) ω

where f (x, y) is the forcing distribution from the tip, centered at xtip , ytip . In the above, j(x, y) is the current, which is a conductive reaction to the electric field generated from the charge wave. To obtain a wavelike response, the current can be approximated as a plasmonic (i.e., inductive, or delayed) response to a respulsive force from local charge gradients: j(x, y) = −iωkp−2 ∇ρ(x, y). The complex variable kp determines the propagation wavevector including attenuation. Taken together, these equations are a Helmholtz equation in ρ: · (kp−2 ∇ρ(x, y)) + ρ(x, y) = f (x, y) ∇ In the following we assume the plasmon wavevector kp has positive real and imaginary parts, i.e., positive wavelength and damping. To understand power absorption in our devices we need two ingredients: • A description of the reflected wave. The edge boundary condition to first approximation is that current falls to zero, however this is not quite correct. In fact the edges serve as partial reflectors (absorbing). To simply model reflected waves we can take a mirror source, multiplied by a complex reflection factor r with |r| ≤ 1. NATURE MATERIALS | www.nature.com/naturematerials

© 2016 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

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optoelectronics by Yvan Ngassa - Issuu