optoelectronics by Yvan Ngassa

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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

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.

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

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ii then the Joule • A description of power flux and absorption. Since we have defined current as proportional to ∇ρ, heating losses should take the form: x)|2 p( x) ∝ |∇ρ( which we take as the absorbed power. For simplicity, we take uniform kp (ignoring the possible changes in kp that occur near/past the junction) and solve with Fourier decomposition: ρ( k) =

1 f ( k) 1 − k 2 /kp2

where f is a modified sourcing distribution taking into account a mirror source to represent reflections: f (x, y) = f (x, y) + rf (x, −y) for reflection coefficient r. The result of this wave propagation and reflection is depicted in main text Fig. 2b, where x)|2 . we have plotted |∇ρ( In the manuscript, we provide a simulation with kp = (56 + 1.8i) µm−1 and with a Gaussian sourcing distribution f (x, y) of 10 nm width. As discussed in the manuscript, Im kp controls the ultimate size of the absorbed p( x). The width of f (x, y) is semi-crucial: it must be smaller than the plasmon wavelength in order to actually launch plasmons, but should not be unreasonably small. If f (x, y) is extremely small then there is a divergent cusp in p( x) due to the nonpropagating “near fields” around the source. With experimentally reasonable size of f (x, y) (comparable to the 25 nm tip size), the plasmon launching effect dominates over these near field effects, and the results are insensitive to the exact value, up to an overall prefactor which is normalized away in the final result. ∞ Mathematically, this occurs since our absorbed power in the absence of interference would be proportional to 0 k 3 |f (k)/(1 − k 2 /kp2 )|2 dk: the resonance in the denominator selects a narrow k range from the forcing distribution. (This tip size dependence is also apparent in the more quantitative calculations in the second part of this supplement, see Fig. S3.)

Thermal spreading

x)|2 is effectively directly converted to heat in the electron gas, raising the The absorbed power p( x) ∝ |∇ρ( electronic temperature T (x, y). Note that here we neglect the intermediate state after plasmon absorption, which is a random single-particle excitation, however the fast electron-electron interactions in graphene quickly lead to a thermalized electron gas with a minimal spreading of under 50 nm.S1 The heat diffuses laterally, however, and so the temperature-rise is spread out compared to p( x) according to a Fourier heat law, i.e. · (−κ∇T ) + g(T − Tsub ) = p, ∇ for in-plane thermal conductance κ and out-of-plane heat sinking conductance g (conduction of heat out of the electron gas, to substrate of temperature Tsub ). For simplicity we assume uniformity in κ and g (again ignoring possible changes nearby the junction) allowing us to write: δT − lT2 ∇2 δT = p/g,

where lT = κ/g is the thermal spreading lengthscale. At the graphene edge, there is a Neumann boundary condition (no heat current exiting past edge) along y = 0: ∂y T |y=0 = 0. We incorporate this by mirroring the heat source: p (x, y) = p(x, y) + p(x, −y), resulting in a simple unbounded Laplace problem that can be solved in Fourier space: T ( k) =

1 p ( k)/g. 1 + lT2 k 2

The result of this thermal spreading is depicted in main text Fig. 2c. 2

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iii C.

Photocurrent generation

Finally, the photocurrent is generated according to the thermoelectric electromotive forces produced by ∇T . For a simple 1D junction (thermocouple), the photocurrent would be given by I = (∆S)(T −Tsub )/R, for Seebeck coefficient difference ∆S, junction temperature T and total resistance R. (This assumes that all other junctions in the path remain at temperature Tsub .) For a 2D device, the correpondence is not in general so simpleS2 . Since however we have simple rectangular device geometry, then by symmetry we can convert to an effective 1D problem, and thereby take T to be the average junction temperature. We remark that since only the junction temperature matters, and since the form of p(x, y) is translationally symmetric with shifts in xtip , then the photocurrent x-dependence can be also be calculated from p directly, by convolving p with a two-tailed exponential responsivity function: I(xtip , ytip ) = dx dy p(xtip − x, y, ytip )r(x − xjunc ) (s1) r(x − xjunc ) =

∆S 1 √ e−|x−xjunc |/lT . RW 2 κg

(s2)

for device width W Again, this convolution performed with the aid of Fourier techniques. This provides a significant speed improvement in calculating the model and we have taken this faster approach to obtain Fig. 2c. Note that we can use (s2) to provide a magnitude estimate of the junction responsivity to the absorbed power: √ A typicalS3 value of ∆S ≈ 100 µV/K, with R ≈ 1 kΩ, W = 13 µm, together with 2 κg ≈ 0.5 W/K/m (estimated from Wiedemann-Franz law and the measured lT at high carrier density) yields a junction responsivity of order 20 mA/W. Thus for the observed currents of order 10 nA, we should have absorbed around 500 nW of power. In comparison our electrodynamic model (next section) estimates powers of order 30 nW absorbed in the graphene. This large discrepancy is likely due to an underestimation of the strength of the tip electric fields, or of the tip sharpness. Likewise, given these numbers we can estimate that the 10 nA demodulated current arises from a junction temperature rise of only 0.1 K, only a perturbation compared to room temperature. II.

ELECTRODYNAMIC MODEL OF ABSORBED POWER

For calculating the light energy absorbed in the graphene, we used an electrodynamic tip model based the tip charge as calculated from an electrostatic “lightning rod” model.S4 This tip charge is then fed into an electrodynamic model of the sample response, from which absorption is obtained. Our goal here is not to completely model the tip response, which is a very complex taskS4 , but only to correctly model the local shape and intensity of near fields around the tip apex, which allows to quantitatively compare the coupling efficiency to graphene plasmons of differing wavelengths. For this purpose the lightning-rod model is appropriate, since it supplies the detailed shape of fields around the tip apex. The tip model is not completely accurate, as it neglects retardation effects from the oscillating field, the influence of light scattered off the sample, and the back-action of the sample on the tip (re-screening). Nevertheless this is a quantitative tip model in both field intensity and field sharpness, both of which strongly influence the plasmon launching process. The most significant uncertainty in our model is in estimating the overall strength of the tip fields, which depends on optical factors such as the proper focussing of the infrared light on the tip, back-reflections from the sample, etc. A.

Electrostatic calculation of tip charge distribution

We use a boundary element model of a conical tip, adapted from Ref. S4. The tip is represented by a series of finely spaced charge “rings”, distributed along a hyperboloid of revolution. For regularization purposes, we radially smear the ring charges. By varying the charge on these rings, we equalize the potential at each rings, which represents the shielding effect of the conductive metal tip. This is done while under the influence of a linear incident potential, representing a uniform electric field. We work in cylindrical coordinates. In vacuum, a ring of charge ρ0 at radius r and height z produces an electrostatic potential of the form: 2 ρ0 4rr φ(r, z) = K , (s3) 4πε0 π (r + r )2 + (|z − z | + d)2 (r + r )2 + (|z − z | + d)2 NATURE MATERIALS | www.nature.com/naturematerials

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FIG. S1. Charge ring positions for representing the tip. Left: all ring positions (large scale). Right: Zoom in on tip apex region. The dashed circle shows the nominal 25 nm radius of curvature.

where K is the complete elliptic integral of the first kind. The parameter d here represents a radial smearing, which can be set to zero in order to obtain a perfect ring of charge. In what follows, we take d = 0 to regularize the solution, i.e., remove divergences. The above expression has a convenient 2D Fourier transform (i.e., the Hankel transform in the radial coordinate),

φ(k, z) =

ρ0 1 J0 (r k)e−(|z−z |+d)k , 4πε0 k

(s4)

which will be useful to transfer the potential into the electrodynamic simulation. We distribute i = 1 . . . 2755 rings equidistantly along a hyperbola in r, z, which has a 25 nm radius of curvature, a half-angle of 53◦ (based on SEM images from the tip manufacturer), and extends to a height of 5000 nm (Fig. S1). The radius and angle are critical for determining the k-distribution, i.e., the sharpness, of the nearfields, whereas the hyperbola’s height only affects the nearfield intensity. (Note that the appropriate effective height may be reduced relative to the true tip height, due to retardation effects in the actual dynamic electromagnetic fields.S4 ) We then take (s3) to calculate the full matrix of potentials φij at each ring site, induced by charges at all other ring sites, with a radial spreading parameter of d = 1 nm. We then solve for the charge distribution that correctly compensates for a linear vertical applied potential, under the constraint of a zero total charge and with an freely adjustable value of the uniform potential on the tip. We finally simplify this charge distribution by projecting down to the plane ztip of the tip apex: we calculate the vertical electric field Ez (r) in the plane located at the height of the tip apex. In Fig. S2 we plot this field. This field can be reproduced by a simpler planar charge distribution, infinitesimally above ztip : ρ0 (k) = 2ε0 Ez (k) This “flat tip” distribution produces the same electric fields below the tip, but is more easily used in the subsequent calculations. To produce the power levels we have taken an incident field of E0 = 300 × 103 V/m. This is an estimate based the 0.5 numerical aperture of focussing in the nearfield microscope and 10 mW incident laser power. It may underestimate the actual field level at the tip due to sample reflection effects, especially due to the presence of metal gates. We stress that the tip sharpness can also strongly influence the nearfield strength at the relevant k-vectors. 4

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FIG. S2. Vertical electric field Ez at tip apex plane, calculated in electrostatic boundary element method. This is normalized to the incident field E0 .

Electrodynamic absorption

The final tip model is obtained by oscillating the charge distribution ρ0 (k) at a frequency represented by an oscillating longitudinal in-plane current j( k) at height ztip .

ω 2π

= 28 THz, which is

jtip ( k) = ω kρ0 (k) k2 The sample response to the oscillating charge distribution is calculated by way of the ω, k-dependent transfer matrix method, which we previously used to calculate plasmon dispersion (see Supplement of Ref. S5). In this method, we approximate the graphene, BN, and metal layers as infinite uniform sheets, i.e., we neglect that there is a split in the metal and that the graphene/BN have edges. We use the same hBN dielectric function as in our previous workS5 , taken from Caldwell et al.,S6 ; the hBN relative permittivity at this frequency was taken as 8.27 + 0.16i in-plane and 1.88 + 0.04i out-of-plane. The graphene is represented by a local conductivity function calculated including thermal smearing effects,S7 which (in this case) are more crucial for accuracy than the inclusion of nonlocality effects seen in the full, zero temperature RPA, since they allow a smooth calculation at lower carrier densities.. A new feature in this case is the underlying metal gate layer (rather than SiO2 ), though this has negligible hybridization with the plasmon due to the 27 nm thickness of the bottom hBN layer: as it is much thicker than the out-of-plane decay length of the plasmon (≈ 3–7 nm) for these frequencies, the metal does not affect the plasmon properties in this case. k), as well as the graphene current This method yields a corresponding electric field at the graphene layer, E( jgr (k) = σ E(k). The resulting power absorbed in the graphene is given by the integral ∞ P (ztip ) = dr 2πr 12 Re[ jgr (r)∗ · E(r)]. 0

We have included that the power depends functionally on ztip , important for the demodulation process. In Fig. S3, we plot the respective power function in k-space, which (by Parseval’s theorem) gives an equivalent power as the real-space one. Finally, to compare to our data (demodulated photocurrent) we observe the dependence of all quantities on ztip , and apply to them the same demodulation procedure as in experiment. In particular if P (ztip ) is the absorbed power as defined above, for static tip height, then the n-th order demodulated power for a tapping tip is given by π dθ cos(nθ)P (A − A cos θ). Pn = 0

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FIG. S3. Plot of k-dependent absorbed power distribution p(k) = πk Re[σ]|E(k)|2 , such that the total power is the area under ∞ the curve, i.e., P = 0 p(k) dk. Calculation was done for n = 7.6 × 1012 cm−2 . A peak can be clearly seen at the plasmon resonance; the dashed line indicates the value (56 µm−1 ) extracted from experimental fringe spacings in the main text figure 1.

FIG. S4. ztip -dependence of absorbed power at 28 THz. The upper curve is for n = 0 (neutral graphene with only thermal carriers, with σ = (49.8 − 1.2i) µS). The lower curve is for n = 7.6 × 1012 cm−2 (highly doped, σ = (2.4 + 206.5i) µS for a scattering time of 500 fs).

where A is the tip tapping amplitude.

[S1] Koppens, F. H. L. et al. Photodetectors based on graphene, other two-dimensional materials and hybrid systems. Nature Nanotech. 9, 780–793 (2014). [S2] Song, J. C. W. & Levitov, L. S. Shockley-ramo theorem and long-range photocurrent response in gapless materials. Physical Review B 90, 075415 (2014). [S3] Gabor, N. M. et al. Hot carrier-assisted intrinsic photoresponse in graphene. Science 334, 648–652 (2011). 6

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vii [S4] McLeod, A. S. et al. Model for quantitative tip-enhanced spectroscopy and the extraction of nanoscale-resolved optical constants. Phys. Rev. B 90, 085136 (2014). [S5] Woessner, A. et al. Highly confined low-loss plasmons in graphene–boron nitride heterostructures. Nature Mater. 14, 421–425 (2015). [S6] Caldwell, J. D. et al. Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride. Nat. Commun. 5, 5221 (2014). [S7] Falkovsky, L. A. & Varlamov, A. A. Space-time dispersion of graphene conductivity. Eur. Phys. J. B 56, 281–284 (2007).

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