Plasmon Generation through Electron Tunneling in Graphene SUPPORTING INFORMATION

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1 Plasmon Generation through Electron Tunneling in Graphene SUPPORTING INFORMATION Sandra de Vega 1 and F. Javier García de Abajo 1, 2 1 ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), Spain 2 ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys 23, Barcelona, Spain (Dated: August 23, 2017) I. EFFECTIVE QUANTUM-WELL DESCRIPTION OF THE OUT-OF-PLANE GRAPHENE ELECTRON WAVE FUNCTION We describe the out-of-plane electron wave function of conduction electrons in graphene as a quantum well state. For simplicity, we assume a square well potential of depth V 0 and width a. We focus on the first excited state, which is antisymmetric along the normal direction z, just like the p z orbital in graphene, as shown in the following scheme (potential in black, wave function in blue): We denote the wave function (blue curve) as ϕ (z). In this model, the parameters V 0 and a are used to fit (i) the state binding energy to the graphene work function 1 Φ = 4.7 ev; and (ii) the centroid of the electron density away from each side of the z = 0 plane to the value obtained for the carbon 2p orbital. We find the quantum well state ϕ (z) by solving the Schrödinger equation ( h 2 /2m)d 2 ϕ (z)/dz 2 + [V (z) E]ϕ (z) = 0, where V (z) is the potential shown in the scheme above (black line). The wave function of the first excited state can be written A sin(k in z) inside the well, with k in = 2m(E + V 0 )/ h, while it decays evanescently in the outer region as sign(z)be kout z, with k out = 2mE/ h. The electron energy E < 0 is referred to the potential outside the well. The continuity of the wave function and its derivate at the well boundaries z = ±a/2 lead to the condition k in /k out = tan(k in a/2), which determines the discrete energies E of asymmetric states. Combining these conditions with normalization ( dz ϕ (z) 2 = 1), we find A 2 = (a/2 + 1/k out ) 1 and B = A e kouta/2 sin(k in a/2). The centroid of the wave function away from the z = 0 plane is calculated as dz z ϕ (z) 2 and compared with the centroid of the p z orbital of graphene d 3 r z ϕ pz (r) 2. We approximate the latter by using a tabulated 2p atomic carbon wave function, 2 ϕ 2p (r) = z j β je αjr, where the parameters α j and β j are expressed in the following table in atomic units: j α j β j In the following plot we show the 2p electron probability density integrated over parallel (x, y) directions ( dx dy ϕ 2p (r) 2, solid curve), compared with the fitted well state ( ϕ (z) 2, dashed curve):

2 2 Probability density well 2p orbital 0 0 The agreement between the two probability densities is excellent using fitted values V 0 = 45 ev and a = 2 nm. II. DERIVATION OF EQUATION 2 OF THE MAIN TEXT In this section, we start from eq 1 of the main text for the inelastic electron transition probability, Γ(ω) = 2e2 ˆ ˆ d 3 r d 3 r ψ i h (r) ψ f (r) ψ f (r ) ψ i (r ) Im{ W (r, r, ω)} i,f δ(ε f ε i + ω) f 1 ( hε i ) [1 f 2 ( hε f )] (S1) (see definitions of different elements in the Methods section), and specify it for the sandwich structure depicted in the following scheme, consisting of two graphene layers separated by a film of thickness d and permittivity ɛ: We factorize the electron wave functions as the product of in-plane ϕ and out-of-plane ϕ components. The latter is described in Sec. I and is common to all conduction electrons, so we can write ψ i (r) = ϕ i (R)ϕ (z) for initial states in layer 1, centered at z = 0, and ψ f (r) = ϕ f (R)ϕ (z d) for final states in layer 2, centered at z = d. Here, we use the notation r = (R, z), with R = (x, y). The in-plane wave functions are Dirac fermions characterized by their parallel wave vector Q = (Q x, Q y ), spin, and valley (K or K points). The transition probability must be independent of spin and valley, so we perform the calculation for only one combination of these degrees of freedom and multiply the result by a factor of 4. We further consider negative doping and a bias such that the initial and final wave functions lie within the upper (conduction) band of their respective layers. The Dirac fermions admit the expression 3 ϕ (R) = 1 ( e e iq R iφ Q ) /2 2A e iφ, (S2) Q/2 where A is the normalization area, φ Q = tan 1 (Q y /Q x ) is the azimuthal angle of Q, and the upper and lower components refer to the value of the wave function in each of the two graphene carbon sublattices. The corresponding electron energy relative to the Dirac point is hε Q = hv F Q. We have adapted eq S1 from a previous derivation 4 in order to incorporate the sum over both carbon sublattices, which is accounted for through the indicated spinor products. Before inserting eq S2 into eq S1, we recast the sum over i as i (A/4π2 ) dq i, and similarly for the sum over f. Additionally, as a consequence of translational invariance, the integrand inside d 3 r should be independent of R, so we can replace d 2 R A. Now, we express the screened interaction as W (r, r, ω) = (2π) 2 d 2 k exp[ik (R R )] W (k, z, z, ω) (see Sec. III), which allows us to carry out the integral over R analytically to yield a δ function for conservation of

3 parallel momentum, δ(q f Q i + k ), and this in turn can be used to perform the integral over Q f. Putting these elements together, eq S1 becomes ˆ ˆ ( Γ(ω) = e2 A 8π 4 h d 2 k d 2 Q i (e iφ Q f /2 e iφ Q /2 f e iφ Qi ) /2 2 ) e iφ Q i /2 f 1 ( hv F Q i ) [1 f 2 ( hv F Q i k )] ˆ ˆ dz dz ϕ (z)ϕ (z d)ϕ (z )ϕ (z d) Im { W (k, z, z, ω) } δ(v F ( Q i k Q i ) + ω ev b / h). (S3) Notice that the Fermi-Dirac distributions are referred to the Dirac points of their respective graphene layers. However, the electron energy in layer 2 is shifted by the bias energy ev b relative to layer 1 (last term inside the δ function). We also note that the spinor product yields 1 + exp[i(φqi φ Qf )] 2 = 2 [ 1 + Q i (Q i k )Q 1 i Q i k 1], where we have expressed the angle between Q i and Q f = Q i k in terms of the inner product of these two vectors. Finally, inserting this expression into eq S3, and noticing that the result is independent of the direction of k once the Q i integral has been carried out, we can make the substitution dφ k 2π for the azimuthal integral and divide the result by the graphene area A to readily obtain eq 2 of the main text. Equation 2 is the expression that we use in our numerical simulations of the tunneling current, in which the azimuthal φ Qi integral is carried out analytically by using the relation δ[f (φ Qi )] = j δ(φ Q i q j )/ F (q j ) for the δ function (notice that the poles of F (φ Qi ) are of first order). 3 III. DERIVATION OF EQUATION 3 OF THE MAIN TEXT The screened interaction W (r, r, ω) is defined as the scalar potential produced at r by a charge placed at r and oscillating with frequency ω. Translational invariance allows us to write ˆ W (r, r, ω) = d2 k (2π) 2 eik (R R ) W (k, z, z, ω), so it is natural to work in k space (i.e., we assume an overall e ik (R R ) dependence). A point charge placed at z produces a direct scalar potential (2π/k )e k z z in vacuum (i.e., this is the direct Coulomb interaction term in eq 3). Additionally, inside the bulk of an anisotropic dielectric (permittivity ɛ z along z, and ɛ x along x and y), the Poisson equation ɛ φ = 0 has solutions φ = e ±iqz, where q = k ɛx /ɛ z and we take the square root to yield Re{q} > 0; this allows us to write the point-charge potential as [2π/(ɛ z q)]e q z z inside that medium. Now, the induced potential has the form Ae k z below the sandwich (z < 0), De k (d z) above it (z > d), and Be qz +Ce q(d z) inside the dielectric (0 < z < d). Here, the coefficients A, B, C, and D are used to satisfy the boundary conditions, namely: (1) the continuity of the potential at each graphene layer j = 1, 2; and (2) the jump of normal displacement is equal to 4π times the induced charge. From the continuity equation, the induced charge can be expressed as the divergence of the current, and this in turn as the product of the conductivity σ j times the in-plane electric field. The jump of normal displacement at layer j is then given by 4πik 2 σ j/ω times the potential. Solving the resulting system of four equations for each position of the external charge z, we obtain, after some tedious but straightforward algebra, the expression for the screened interaction W (k, z, z, ω) presented in eq 3 of the main text. Alternatively, a more direct Fabry-Perot-like derivation can be made in terms of the transmission and reflection coefficients of the dielectric/graphene/vacuum interface defined in the main text (search for A j, A j, B j, and B j ). IV. ADDITIONAL NUMERICAL SIMULATIONS We present additional simulations of the dispersion diagrams and tunneling currents for various combinations of the graphene Fermi energies in Figures S1-S5. We also show in Figures S6 and S7 calculations similar to those of the main text for graphene layers separated by vacuum instead of hbn. We assume a conservative graphene plasmon lifetime of 66 fs in all cases. Electronic address: javier.garciadeabajo@nanophotonics.es 1 Yu, Y.-J.; Zhao, Y.; Ryu, S.; Brus, L. E.; Kim, K. S.; Kim, P. Nano Lett. 2009, 9, Clementi, E.; Roetti, C. At. Data Nucl. Data Tables 1974, 14, Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. Rev. Mod. Phys. 2009, 81, García de Abajo, F. J. Rev. Mod. Phys. 2010, 82,

4 4 0 1 (c) (d) (e) (f) FIG. S1: Additional dispersion diagrams. Same as Figure 2a of the main text for various combinations of the gap distance and the graphene Fermi energies (see labels). (c) (d) FIG. S2: Additional calculations of the energy- and momentum-resolved tunneling current. Same as Figure 2b of the main text for fixed gap distance d = 2 nm and various combinations of the graphene Fermi energies (see labels).

5 5 (c) FIG. S3: Additional dispersion current diagrams. Same as Figure 3 of the main text with J(k, ω) plotted in logarithmic scale.! "!(") (1/nm!) Ef1 = 0.75 ev Ef2 = 0.5 ev !(") (1/nm!) Ef1 = 0.75 ev Ef2 = 5 ev Energy (ev) Energy (ev) Ef1 = ev Ef2 = 510 ev !(") (1/nm!) (c) Ef1 = ev Ef2 = 0.5 ev Energy (ev) (d) FIG. S4: Additional calculations of the spectrally resolved tunneling current. Same as Figure 4a of the main text for fixed gap distance and various combinations of the graphene Fermi energies (see labels).

6 6 Electron # current e! / (s!nm") (10 8 s 1 nm 2 ) FIG. S5: Additional calculations of the tunneling current. Same as Figure 4b of the main text for fixed gap distance and various combinations of the graphene Fermi energies (see labels). 1 optical plasmon acoustic plasmon 1 nm 1 ev 0.5 ev 1 nm 1 ev 0.5 ev ev C B (c) (nm 2 ) FIG. S6: Energy- and momentum-resolved electron tunneling with a vacuum gap. Same as Figure 2 of the main text with the hbn film replaced by vacuum. (nm 2 ) B C (ev) 1 nm 1 ev 0.5 ev ! (ev) Electron current (10 6 s 1 nm 2 ) # e! / (s!nm") 5 3 (nm) scaling FIG. S7: Probability of plasmon-generation by electron tunneling with a vacuum gap. Same as Figure 4a,b of the main text with the hbn film replaced by vacuum.