Nonlinear light mixing by graphene plasmons

Transcription

1 Supporting Information for Nonlinear light mixing by graphene plasmons Dmytro Kundys, Ben Van Duppen, Owen P. Marshall, Francisco Rodriguez, Iacopo Torre, Andrea Tomadin, Marco Polini*, and Alexander N. Grigorenko* * Corresponding author. Marco.Polini@iit.it (MP); sasha@manchester.ac.uk (ANG). Phone for correspondence: S1

2 Materials and Methods Fabrication of graphene nanoribbons The nanoribbons were fabricated using graphene grown by CVD on copper foil and mechanically exfoliated graphene. The graphene was transferred to a CaF 2 substrate by a standard procedure that includes coating graphene with PMMA, etching the copper in an ammonium persulfate solution, rinsing with deionized water, transferring the resulting membrane to the CaF2 substrate and heating in a hot plate at 150 C for 10 minutes to improve the adhesion. The PMMA was later removed with acetone and isopropanol. The nanoribbons were patterned by electron beam lithography of a thin layer of PMMA with a conductive polymer that was spin-coated on the sample. After exposure and development, the graphene in the exposed areas was etched with Ar:O 2 plasma. The rest of PMMA was removed with acetone and isopropanol. Figure S1 provides a SEM micrograph of a typical fabricated nanoribbon array. Temperature of graphene array during light mixing experiments To check the temperature of the graphene nanoribbon array during LMGP we have used FLIR T640 camera. The measurements were taken at the experimental conditions. The sample was mounted as for LMGP experiments and the camera was sat on a tripod at about cm away from the sample at small angle ~15-20 degrees from normal. The CO 2 laser spot size was the same as in LMGP experiments (of ~50 m in diameter); both lasers beams were aligned prior to the temperature reading. Figure S2 shows the typical temperature reading for a suspended graphene paper and a graphene flake on quartz substrate (we did not present the thermal image from the array due to difficulty to interpret the thermal data). One can see that the temperature reached ~300 C in the middle of the beam for graphene paper and ~200 C for a graphene flake. These present the lower limit for temperatures of graphene nanoribbons in our experiments. The actual temperatures were much higher due to larger absorption of graphene nanoribbons at the plasmon resonance and the reduced fractional areal coverage. Second order light mixing As theory describes, if inversion symmetry of the nanoribbon array is broken (for example by fabrication, disorder or light beam alignment) it is possible to observe second order light mixing and sidebands at 2 1 could be expected. In a sample made from an exfoliated graphene flake we indeed observed the second order sidebands. The graphene in this sample was more doped (E F ~ 0.4 ev). To observe LMGP in this sample, we have used a 1060 nm probe laser and the same CO 2 pump laser (of wavelength 10.6 m). In addition to the third order light mixing demonstrated in Figure S3a, we also detected the second order sidebands 2 1 shown in Figure S3b-3c. The observation of the second order sidebands is due to the absence of the inversion symmetry for graphene nanoribbon edges produced during S2

3 fabrication of nanoribbon array using exfoliated graphene. The conversion coefficients for the harmonics are given in the Table S1 below. Table S1. Sidebands Wavelengths nm nm 1060 nm nm Conversion coefficient S3

4 Theory A Linear response theory for plasmon modes in a graphene nanoribbon A graphene nanoribbon supports plasmons that can be excited by an external electric field E ext (r,ω) that has a non-zero component oriented perpendicularly to the ribbon. The plasmon can be described in terms of a local perturbation δn(r,ω) of the carrier density and an associated current density j(r, ω) that describes the flow of charge causing this perturbation. In this Section, we calculate these quantities in the framework of linear response theory (LRT). 1 The starting point is the combination of the continuity equation and the response of the charge carriers to an electric field applied to the system. These equations are, respectively, given by iωδn(r,ω) + j(r,ω) = 0, (1) j(r,ω) = σ D (r,ω)e tot (r,ω). (2) In Eq. (2), E tot is the total electric field, i.e. the external field plus the self-induced field induced by the movement of the charge carriers, and we have defined the local Drude conductivity as σ D (r,ω) = id(r) π(ω + iγ), (3) where D(r) = e 2 v F π n(r)/ h is the Drude weight for a liquid of massless Dirac Fermions 2 with charge e, Fermi velocity v F and n(r) is the equilibrium electron density equal to n inside the ribbon and zero outside. With the condition that the current perpendicular to the boundary of the nanoribbon needs to vanish at the edges of the nanoribbon to prevent loss of charge, one can solve Eqs. (1)-(2) numerically. The solution can be written in the LRT framework as δn(r,ω) = dr χnn(r,r R,ω)Φ ext (r,ω), (4) j i (r,ω) = dr σ R ii (r,r,ω)e i,ext (r,ω). (5) In Eq. (4) we have introduced the electrical potential Φ ext (r,ω) of the external electric field E ext (r,ω). These are related by E ext (r,ω) = Φ ext (r,ω). In Eqs. (4)-(5) the linear response functions describe the response of the electron liquid in the nanoribbon to the external electric field. The response of the carrier density, δn(r,ω), is described by the density-density response function χ R nn(r,r,ω) that can be expressed in terms of the numerically calculated density eigenmodes δn mk (r) as χnn(r,r R,ω) = C Ω2 mk δn mk(r)δn mk (r ) m,k ω(ω + iγ) Ω 2. (6) mk S4

5 The eigenmodes are determined by two quantum numbers: m and k. The latter refers to the momentum in the longitudinal direction while the former describes the profile of the modes in the transverse direction of the nanoribbon. The energy dispersion of the eigenmodes is shown in Fig. S4 as a function of longitudinal momentum k for different transverse m modes. Notice that the m = 0 mode corresponds to the standard Dirac plasmon propagating the longitudinal direction and having the characteristic k dispersion. 3 The dispersion relation for the eigenmodes is given by Ω mk = Ω pl ξmk, where ξ mk follows from the numerical solution and the frequency scale is provided by the characteristic plasmon frequency Ω pl, which is given by Ω pl = 2D εw. (7) In Eq. (7) w denotes the width of the nanoribbon, D = e 2 v F π n/ h is the massless Dirac Fermion Drude weight and ε is the average dielectric constant of the environment. In Eq. (6) the normalization constant C is given by C = ε 2πe 2 w 2 L, (8) where L is a unit of length of the nanoribbon in the x-direction. In a similar way, also the current density generated by an external electric field can be calculated in terms of current density eigenmodes j mk (r). The linear response function is now the longitudinal conductivity σii R(r,r,ω) that follows in terms of the numerically calculated current density eigenmodes as σ R xx(r,r,ω) = idδ(r r ) π(ω + iγ), (9) σyy(r,r R,ω) = C iω2 j y,mk (r) jy,mk (r ) m,k ω(ω + iγ) Ω 2. (10) mk Eq. (9) shows that the conductivity in the longitudinal direction of the nanoribbons reduces to the local Drude form. However, the conductivity in the transverse direction of the nanoribbons, Eq. (10), is clearly non-local. From Eqs. (9)-(10) one can find the absorption spectrum of radiation that is polarized respectively longitudinally or transverse to the nanoribbons. In Fig. 3(A) of the main text we show the absorption spectrum compared to the measured data for an array of nanoribbons with width w = 50 nm and damping strength γ = 0.48 mev. From this comparison, one can see a quantitative and qualitative match with experiment and it is clear that the m = 1 mode is excited when the pump laser has a wavelength λ 1 = 10 µm. Notice that because the damping γ is reasonably large, we can neglect plasmon leaking between ribbons and treat each ribbon individually. 4 Finally, notice that the relative transmission increases for large energy. This is due to Pauli blocking and has been accounted for by assuming that the inter-band processes in the nanoribbons are the same as those of a continuous graphene sheet. As shown in Fig. 3(A) of the main text, plasmons are excited when the polarization of the external electric field is perpendicular to the nanoribbons. From now on, we S5

6 will assume that the electric field is oriented in the lateral direction in order to excite plasmons. Further, because the spot size of the laser is much larger than the width of the nanoribbon, we assume that the associated electric field is homogeneous, i.e. E ext (r,ω) = π(e 1 δ(ω ω 1 )+E 1 δ(ω +ω 1))ŷ, corresponding to the electric field of a laser with angular frequency ω 1 polarized perpendicular to the nanoribbons. This allows to calculate the time dependent carrier density profiles δn (t) (r) and current density profiles δ j (t) y (r) as δn (t) (r) = 1 2 δn(r,ω 1)e iω 1t + c.c., (11) j (t) y (r) = 1 2 j y(r,ω 1 )e iω 1t + c.c.. (12) Figs. 1(A) and (B) of the manuscript show the time evolution of these profiles over one oscillation cycle. In the experiments, we have adjusted the pump laser to be in resonance with the m = 1 mode of the sample. Notice that as the electric field is homogeneous and oriented perpendicular to the nanoribbon, Eqs. (6) and (10) show that the eigenmodes can only be excited if m is odd. This is also shown in Fig. 1(A) as the carrier density profile is odd with respect to the center of the nanoribbon. Assuming that only the m = 1 mode is excited, the plasmon energy can be calculated as 2ξ αee hv F hω pl = E F. (13) εw In Eq. (13) the factor stemming from the numerical calculation is ξ = ξ 10 = Eq. (13) coincides with Eq. (1) of the manuscript. From the measured absorption spectrum in Fig. 3(A) of the manuscript, we can infer an environmental dielectric constant ε = 1.86 B Deriving the third-order conductivity In this Section we derive the third-order conductivity from an expansion of the finite temperature longitudinal conductivity. B.1 Finite temperature longitudinal conductivity of a drifting Dirac liquid The longitudinal optical conductivity at angular frequency ω of a graphene sheet with a finite drift velocity v was calculated in Ref. 5 It was shown that the conductivity depends on the angle φ between the direction of the electric field probing the system and the direction of the drift velocity. This results in a birefringent character of the response of the electron liquid. In this study, we fix the current density of the plasmons in the y direction, i.e. perpendicular to the nanoribbons, and the probe field in the x direction, i.e. longitudinal to the nanoribbons. Neglecting the small imaginary part, the zero temperature longitudinal conductivity σ xx then becomes 5 σ xx (u) = σ 0 Θ(u 1) + σ 0 π 1 Θ(1 u)θ(1 + u) ( ) arccos( u) u 1 u 2, (14) S6

7 where σ 0 = e 2 /(4 h) is the universal conductivity and the quantity u is defined as u = ω 2 µ βω. (15) In Eq. (15) µ is the chemical potential and β = v /v F is the reduced drift velocity. In order to find a finite temperature form of Eq. (14), we can use an identity due to Maldague 6 σ xx (ω,t ) = dy σ xx (ω) µ=y 4k B T cosh 2 ( y µ 2k B T ) = ωβ β 1 8k B T du σ xx (u) cosh 2 ( ω(βu 1) 2µ 4k B T ) +(µ µ). (16) In the second part of Eq. (16) we have made use of the expression for u in Eq. (15) to simplify the integration and show the symmetry of the conductivity under a change in sign of µ. The integration presented in Eq. (16) cannot be performed analytically. However, for small β, one can expand the integrand up to second order in β as cosh 2 ( ω(βu 1) ± 2µ 4k B T ) = cosh 2 ( u ±)[ 1 + tanh ( u ±) uωβ 2k B T + 1 ( 3tanh 2 ( u ±) 1 )( uωβ 4 2k B T In Eq. (17) we have introduced the variable u ± = (ω ±2µ)/(4k B T ) for notational ease. Using the expansion, one can find the finite temperature conductivity up to second order in β as σ xx (ω,t ; µ,β) = σ 0 2 ( tanh(u + ) + tanh(u ) ) σ 0 2 ) 2 ] (17) [ ]( 5 tanh(u+ ) cosh 2 (u + ) + 3 tanh(u ) ωβ cosh 2 (u ) 8k B T (18) Notice that because of the symmetry of the conductivity with respect to the sign of β, no odd powers will contribute in the expansion. The mixing of two photons discussed in the Article is the result of oscillations in the local drift velocity and chemical potential. Therefore, we are interested in the change of conductivity due to an oscillation δ µ around a background chemical potential µ. Up to second order in β and δ µ/ µ Eq. (18) reduces to σ xx (ω, µ,t ;δ µ,β) = σxx(ω, 0 µ,t ) + σ xx µ,1 (ω, µ,t ) δ µ µ + σ µ,2 xx (ω, µ,t ) ( δ µ µ ) 2 + σ β,2 xx (ω, µ,t )β 2. (19) The components in Eq. (19) are functions of the angular frequency ω, the background ) 2. S7

8 chemical potential µ, the electron temperature T. They are given by σxx(ω, 0 µ,t ) = σ 0 ( tanh(ū + ) + tanh(ū ) ), (20) 2 σ xx µ,1 (ω, µ,t ) = σ 0 ( cosh 2 (ū + ) cosh 2 (ū ) ) µ 2 2k B T, (21) σ xx µ,2 (ω, µ,t ) = σ ( 0 tanh(ū + )( ) ) 2 cosh 2 (ū + ) + tanh(ū ) µ 2 cosh 2 (ū, (22) ) 2k B T σxx β,2 (ω, µ,t ) = σ [ ]( ) 0 5 tanh(ū+ ) 2 cosh 2 (ū + ) + 3 tanh(ū ) ω 2 cosh 2 (ū. (23) ) 8k B T B.2 Relation between (µ,β) and (n, j) The optical conductivity presented in Eq. (19) depends on the chemical potential µ, the magnitude of the drift velocity β and the temperature T. These parameters of the system are determined by the magnitude of the current density j = j and carrier density n calculated in Section A. At finite temperature T they are related as: n(t ; µ,β) = N f(k B T ) 2 2π( hv F ) 2 Li 2 ( e µ/(kbt ) ) + Li 2 ( e µ/(kbt ) ) (1 β 2 ) 3/2, (24) j(t ; µ,β) = N f(k B T ) 2 2π( hv F ) 2 Li 2 ( e µ/(kbt ) ) Li 2 ( e µ/(kbt ) ) (1 β 2 ) 3/2 ev F β. (25) In Eqs. (24)-(25) the functions Li 2 are the dilogarithm functions, 7 N f = 4 is the number of Fermion flavors in the system and e is the elementary charge. Solving Eqs. (24)- (25) for the chemical potential and drift velocity constitutes the link between Sec. A and Sec. B.1. Even though there is no analytical solution for these equations, we can determine the relation up to second order in β and δ µ. The density perturbation δn is defined with respect to the equilibrium carrier density n n(t ; µ,0) as δn n(t ; µ + δ µ,β) n. (26) Up to second order in δ µ = δ µ/e F and β, where E F = hv F π n is the Fermi level, the relations are as follows: δn n j j ( ( = 2T log 1 + e µ /T ) ( + log 1 + e µ /T )) δ µ + e µ /T 1 e µ /T + 1 (δ µ ) β 2, ( ) = 2T Li 2 ( e µ /T ) + Li 2 ( e µ /T ) + 2T ( log ( 1 + e µ /T ) log (27) β ( 1 + e µ /T )) βδ µ. (28) In these expressions, we have defined T = k B T /E F and µ = µ/e F. In the last expression, we have defined the critical current density j = e nv F. Solving Eqs. (27)-(28) for S8

9 δ µ and β, we find δ µ = M n,1 ( µ,t ) δn ( ) δn 2 ( ) 2 n + M n,2( µ,t ) + M j ( µ,t ), (29) n j j β = B j ( µ,t ). (30) j j In these equations, the coefficients are given by M n,1 ( µ,t ) = ( 2T µ ) ( ( 1 log 1 + e µ /T ) ( + log 1 + e µ /T )) 1, (31) 1 (e µ /T 1) M n,2 ( µ,t ) = 8 µ (T ) 3 (e µ /T + 1)(log ( 1 + e µ /T ) + log ( 1 + e µ /T ) ) 3,(32) 3 ( M j ( µ,t ) = 16 µ (T ) 5 Li 2 ( e µ /T ) + Li 2 ( e µ /T )) 2 (33) ( ( log 1 + e µ /T ) ( + log 1 + e µ /T )) 1, (34) B j ( µ,t ) = (2(T ) 2 ) 1 ( Li 2 ( e µ /T ) + Li 2 ( e µ /T )) 1. (35) Using these definitions, we can write the absorption Eq. (19) as an expansion in the quantities δn and j: σ xx (ω, µ,t ;δ µ,β) = σxx(ω, 0 µ,t ) + σxx n,1 (ω, µ,t ) δn n + σ n,2 xx (ω, µ,t ) ( δn n ) 2 + σ j,2 xx (ω, µ,t ) ( j j ) 2. (36) By substituting Eqs. (11)-(12) into Eq. (36), we obtain Eq. (4) of the manuscript. For completeness, we report the coefficients of Eq. (36): σxx(ω, 0 µ,t ) = σ 0 ( tanh(ū + ) + tanh(ū ) ), (37) 2 σxx n,1 (ω, µ,t ) = σ xx µ,1 (ω, µ,t )M n,1 ( µ,t ), (38) σxx n,2 (ω, µ,t ) = σ xx µ,1 (ω, µ,t )M n,2 ( µ,t ) + σ xx µ,2 (ω, µ,t )(M n,1 ( µ,t )) 2, (39) σxx j,2 (ω, µ,t ) = σxx β,2 (ω, µ,t )(B j ( µ,t )) 2 + σ xx µ,1 (ω, µ,t )M j ( µ,t ). (40) B.3 Third-order conductivity The non-local third-order conductivity σ xyyx(r,r (3),r ;ω 2 2ω 1 ) couples the x-component of the current density J x (3) (r,ω 2 2ω 1 ) at position r to electric fields in the system as J x (3) (r,ω 2 2ω 1 ) = dr dr σ xyyx(r,r (3),r ;ω 2 2ω 1 )E 1,y (r,ω 1 )E 1,y (r,ω 1 )E 2,x (r,ω 2 ). (41) Notice that in writing Eq. (41), we have assumed that the system responds locally to the probe field E 2. Eq. (36) shows that the local response of the system containing S9

10 plasmons is time dependent. Therefore, through Eq. (3) of the manuscript we can find the time-dependence of the current density as J x (r,t) = 1 2 σ (t) xx (r,ω 2 )E 2,x (r,ω 2 )e iω 2t + c.c.. (42) We are interested in the Fourier component of Eq. (42) at frequency ω 2 2ω 1. This will be provided by the last two terms of Eq. (36) as these are quadratic in the pump field E 1. To calculate the Fourier component, notice that from Eq. (12) it follows that ( j (t) y (r)) 2 = jy (r,ω 1 ) 2 + ( j y (r,ω 1 )) 2 e 2iω 1t + c.c.. (43) The first term in the right-hand side of Eq. (43) is constant as a function of time and generates an intensity-dependent refractive index as is common in third-order non-linear optical media. 8 This affects the intensity of the probe beam at frequency ω 2. The last term, however, contributes a sum-frequency component and generates the sought-after component at ω 2 2ω 1. Inserting Eq. (43) trough Eq. (36) in Eq. (42) and performing the time Fourier transform, one finds the contribution to the ω 2 2ω 1 component as J x (3), j (r,ω 2 2ω 1 ) = dr dr σxx j,2 (ω, µ,t ) π σyy(r,r R,ω 1 )σyy(r,r R,ω 1 ) 4 j 2 E 1,y (r,ω 1 )E 1,y (r,ω 1 )E 2,x (r,ω 2 ). (44) Similarly, also the density-induced contribution can be calculated as J x (3),n (r,ω 2 2ω 1 ) = dr dr σxx n,2 (ω, µ,t ) π y χnn(r,r R,ω 1 )y χnn(r,r R,ω 1 ) 4 n 2 E 1,y (r,ω 1 )E 1,y (r,ω 1 )E 2,x (r,ω 2 ). (45) In Eq. (45) we have assumed the pump field E 1 to be homogeneous on the scale of the nanoribbons width such that the electric potential can be approximated as Φ 1 (r,ω 1 ) = ye 1 (ω 1 ), where y is the lateral coordinate. Combining Eqs. (44)-(45) and comparing them to Eq. (41) we finally obtain an expression for the third-order conductivity as σ xyyx(r,r (3),r ;ω 2 2ω 1 ) = σxx j,2 (ω, µ,t ) π σyy(r,r R,ω 1 )σyy(r,r R,ω 1 ) 4 j 2 + σxx n,2 (ω, µ,t ) π y χnn(r,r R,ω 1 )y χnn(r,r R,ω 1 ) 4 n 2.(46) Eq. (46) clearly shows that the third-order conductivity is a non-local quantity through the non-locality of the first-order response functions σ R yy(r,r,ω) and χ R nn(r,r,ω). S10

11 Figure S1. SEM micrograph of a typical nanoribbon array used for nonlinear light mixing by graphene plasmons. S11

12 a b c Figure S2. Thermal images of graphene illuminated by the full power of CO 2 laser. (a) A low magnification image for graphene paper. (b) A zoomed FLIR image for a suspended graphene paper. (c) A zoomed FLIR image of graphene flake on quartz substrate. The hotspot size is 1 mm. S12

13 Intensity, V Intensity, V Intensity, V a Detecting at 1324 nm sideband Time Interval scan 6100 OFF ON OFF ON OFF b Time, s Detecting at 1177 nm sideband Time interval scan OFF ON OFF ON OFF c Time, s Detecting at 963 nm Time interval scan ON OFF ON OFF ON Time, s Figure S3. Second and third order light mixing by graphene plasmons. (a) The temporal scan of the third order mixing at with the pump laser on and off. (b) The second order mixing at 2 1. (c) The second order mixing at 2 1. The seed laser (operating wavelength of 1060nm) power was 34mW, the pump laser (operating wavelength of 10.6 µm) power was 200 mw. S13

14 Fig. S4: Dispersion of the eigenmodes of the nanoribbon with width w for different values of m. The Type I plasmon indicates the normal plasmon propagating in the longitudinal direction of the nanoribbons. The gapped modes are the modes considered in this manuscript that will support the nonlinear photon mixing.

15 References (1) Giuliani, G. F. and Vignale, G. Quantum Theory of the Electron Liquid; Cambridge University Press: Cambridge, (2) Torre, I.; Tomadin, A.; Krahne, R.; Pellegrini, V.; Polini, M. Phys. Rev. B 2015, 91, (3) Castro Neto, A. H.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. Rev. Mod. Phys. 2009, 81, (4) Nikitin, A. Y.; Guinea, F.; Garcia-Vidal, F. J.; Martin-Moreno, L. Phys. Rev. B 2012, 85 (8), (5) Van Duppen, B.; Tomadin, A.; Grigorenko, A. N.; Polini, M. 2D Mater. 2016, 3, (6) Maldague, P. F. Surf. Sci. 1978, 73, (7) Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Dover Publications: Washington D.C., 1984 p (8) Boyd, R. W. Nonlinear optics; Academic press: San Diego, S15