Valley polarization induced second harmonic generation in graphene
VValley polarization induced second harmonic generation in graphene
L. E. Golub and S. A. Tarasenko , Ioffe Physical-Technical Institute, 194021 St. Petersburg, Russia and St. Petersburg State Polytechnic University, 195251, St. Petersburg, Russia
The valley degeneracy of electron states in graphene stimulates intensive research of valley-relatedoptical and transport phenomena. While many proposals on how to manipulate valley states havebeen put forward, experimental access to the valley polarization in graphene is still a challenge.Here, we develop a theory of the second optical harmonic generation in graphene and show thatthis effect can be used to measure the degree and sign of the valley polarization. We show that, atthe normal incidence of radiation, the second harmonic generation stems from imbalance of carrierpopulations in the valleys. The effect has a specific polarization dependence reflecting the trigonalsymmetry of electron valley and is resonantly enhanced if the energy of incident photons is close tothe Fermi energy.
PACS numbers: 78.67.Wj, 42.65.Ky, 73.50.Pz
The valley degree of freedom of charge carriers inmulti-valley semiconductor systems such as silicon, di-amond, graphene, carbon nanotubes, transition metaldichalcogenides, etc. attracts growing attention dueto great and yet unexplored potential of semiconduc-tor valley properties to practical applications [1, 2]. Apromising candidate for the study of valley physics intwo dimensions is graphene, a one-atom-thick layer ofcarbon [3]. Graphene technology is now well devel-oped, which enables the synthesis of large-scale defect-free monolayers as well as the production of graphenenanostructures with controllable shapes and edges [4–6].A number of proposals on how to generate the valleypolarization of carriers and valley currents in graphenehas been put forward. It was shown that the electriccurrent gets valley polarized in a graphene point contactwith zigzag edges [7], graphene layer with broken inver-sion symmetry [8], strained graphene with mass Diracfermions [9], at the boundary between monolayer and bi-layer graphene [10, 11], at a line defect [12], or if mono-layer or bilayer graphene is additionally illuminated bycircularly polarized radiation [13, 14]. Valley currentscan be induced in graphene rings by asymmetric mono-cycle electromagnetic pulses [15]. It was also proposedthat bulk valley currents in graphene and carbon nan-otubes can be excited by polarized light [16–18] or acmechanical vibrations [19]. While the above methodscan be used to create imbalance in valley populations,experimental study of valley phenomena and verificationof the theoretical proposals is still a challenge because oflack of efficient and reliable methods to probe the valleypolarization.The valleys in graphene are situated at the K and K (cid:48) points of the two-dimensional Brillouin zone which areconnected with each other by the space inversion C i [20].Each of the valleys is described by the D h small groupand lacks the center of space inversion while the overallsymmetry of free standing graphene D h = D h × C i iscentrosymmetric. It follows that the valley polarizationof carriers reduces the spatial symmetry of the structureto the symmetry of an individual valley. Such a sym- metry reduction gives rise to optical effects such as sec-ond harmonic generation (SHG) which require the spatialsymmetry breaking [16]. Here, we develop a microscopictheory of SHG in graphene and show that the effect canbe used to measure the degree and the sign of valleypolarization. We demonstrate that valley polarizationinduced SHG is caused by the trigonal warping of theelectron dispersion in valleys and calculate the second-order susceptibility tensor for interband optical transi-tions. The efficiency of SHG is resonantly enhanced ifthe energy of incident photons is close to the Fermi en-ergy. The second optical harmonic due to the valley po-larization is generated at the normal incidence of radia-tion and, therefore, can be discriminated from the SHGsignals stemming from structure inversion asymmetry ofgraphene flakes on substrate [21] or in-plane photon mo-mentum [22, 23] which both require the oblique incidenceof radiation. The effect is also different by symmetryfrom SHG caused by the flow of a direct electric cur-rent in the sample [24–26]. Since non-linear optical spec-troscopy is a sensitive and powerful tool to study carrierkinetics and structure symmetry with high spatial res-olution, SHG applied to graphene will enable the localprobe of valley polarization as well as the study of valleypolarization thermal fluctuations (valley noise) [27].The valley polarization induced SHG is illustrated inFig. 1. We assume that graphene is excited by a planeelectromagnetic wave with the frequency ω at the normalincidence. Both K and K (cid:48) valleys have trigonal symme-try and contribute to SHG. However, since the valleys arerelated to each other by the space inversion, SHG signalsstemming from the valleys are counter phased, Fig. 1a.Therefore, the total SHG signal vanishes at equal distri-bution of carriers in the valleys and arises in the case ofimbalance of the valley populations, see Fig. 1b. SHGhas a resonant behavior and drastically enhanced if theenergy of incident photon (cid:126) ω is close to the Fermi energyof carriers, Fig. 1b.Phenomenologically, SHG is described by the second-order susceptibility tensor χ which couples the polariza-tion amplitude at the double frequency P (2 ω ) with the a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov K K E E (a) x E ħω K K ħω ħω E ħω ħω ħω (b) Δ E F FIG. 1. Microscopic mechanism of valley polarization inducedsecond harmonic generation. (a) SHG is caused by trigonalasymmetry of the valleys. Signals stemming from the K and K (cid:48) valleys have opposite sign. (b) Imbalance of valley pop-ulations gives rise to a net SHG signal. SHG is resonantlyenhanced if the energy of incident photons is close to theFermi energy. incident radiation electric field amplitude E ( ω ), P α (2 ω ) = χ αβγ E β ( ω ) E γ ( ω ) , (1)where α , β , and γ are the Cartesian coordinates. Here,we assume that the incident radiation is linearly po-larized and its electric field has the form E ( t ) = E ( ω ) exp ( − i ωt ) + E ( ω ) exp (i ωt ) = 2 E ( ω ) cos ωt . Sym-metry analysis shows that non-zero components of thetensor χ caused by valley polarization are χ xxx = − χ xyy = − χ yxy ≡ χ , (2)where x and y are the in-plane axes perpendicular toeach other with x parallel to the K (cid:48) K direction, Fig. 1a.Equation (2) implies the polarization dependence P x (2 ω ) = χ | E ( ω ) | cos 2 α, P y (2 ω ) = − χ | E ( ω ) | sin 2 α, (3)where α is an angle between the polarization plane ofincident radiation and the x axis, Fig. 1a. Such a polar-ization behavior of SHG follows from the trigonal sym-metry of an individual valley. The generation of secondharmonic at the normal incidence of radiation and itsspecific polarization dependence given by Eq. (3) enableone to discriminate the valley-related SHG from othersources of SHG signal stemming, e.g., from structure in-version asymmetry of graphene flakes.Microscopic calculation of the second-order susceptibil-ity can be carried out in the density-matrix-theory for-malism [28]. In this approach, electron system in each valley is described by the density matrix ρ which satis-fies the quantum kinetic equation ∂ρ∂t = − i (cid:126) [ H + V, ρ ] + St ρ . (4)Here, H is the Hamiltonian in the absence of radiation, H = (cid:18) νv p − − µp νv p + − µp − (cid:19) , (5) ν is the valley index ( ν = ± K and K (cid:48) valleys,respectively), v is the electron velocity, p ± = p x ± i p y , p is the electron momentum, µ is the constant of trigonalwarping, V is the operator of electron-photon interaction, V = − ec v · A + e c (cid:88) αβ ∂v α ∂p β A α A β , (6) e is the electron charge, c is the speed of light, v = ∂H/∂ p is the velocity operator, A = − i ( c/ω ) E ( ω ) is the vectorpotential amplitude, and St ρ is the collision integral de-scribing relaxation processes.Solution of the kinetic Eq. (4) can be expanded in theseries of the electric field amplitude, ρ = ρ (0) + [ ρ (1) e − i ωt + c . c . ] + [ ρ (2) e − ωt + c . c . ] + . . . , (7)where ρ (0) is the equilibrium density matrix, ρ (1) ∝ E ,and ρ (2) ∝ E . Second harmonic is determined bythe term ρ (2) . We consider optical transitions betweenthe valence ( v ) and conduction ( c ) bands in n -dopedgraphene. Straightforward calculations show that the in-terband and intraband components of the density matrix ρ (2) for a given valley and momentum have the form ρ (2) cv = ( e/c ) ( A · v cv )[ A · ( v cc − v vv )](2 (cid:126) ω − E cv + iγ )( (cid:126) ω − E cv + iγ ) ( f v − f c ) (8)+ ( e/c ) (cid:80) αβ ( ∂v α /∂p β ) cv A α A β (cid:126) ω − E cv + iγ ) ( f v − f c ) ,ρ (2) cc = − ( e/c ) ( A · v cv )( A · v vc )( (cid:126) ω − E cv + iγ )( (cid:126) ω + E cv + iγ ) ( f v − f c ) , where v cv = v ∗ vc , v cc , and v vv are the interband and in-traband matrix elements of the velocity operator in thevalley, E cv is the energy gap between the valence andconduction bands, γ/ (cid:126) is the decay rate of the interbandcomponent of the density matrix, f v and f c are the equi-librium electron distribution functions in the valence andconduction bands. Below we assume for simplicity that γ is independent of energy. The component ρ (2) vc can beobtained from ρ (2) cv by the complex conjugation and thereplacement ω → − ω ; component ρ (2) vv is equal to − ρ (2) cc .Polarization at the double frequency can be expressedin terms of the current density at the double frequencyand is given by P (2 ω ) = ( i/ ω ) j ω = ( ie/ω ) (cid:88) p ,ν Tr (cid:16) ρ (2) v (cid:17) , (9)where the spin degeneracy is taken into account and sum-mation is performed over the momentum and the valleyindex. Calculation of Eq. (9) shows that the second-ordersusceptibility χ is the sum of intravalley contributions, χ = χ + + χ − , (10)where χ ν = − i (cid:16) eω (cid:17) (cid:88) p (cid:104) f ( ν ) v ( − ε ( ν ) p ) − f ( ν ) c ( ε ( ν ) p ) (cid:105) Φ ν ( p ) , (11)Φ ν ( p ) = (cid:40)(cid:34) | v ( ν ) x,vc | v ( ν ) x,cc (cid:126) ω + i γ − ε ( ν ) p + v ( ν ) x,vc (cid:32) ∂v ( ν ) x ∂p x (cid:33) cv (cid:35) (12) × (cid:126) ω − ε ( ν ) p + i γ − | v ( ν ) x,vc | v ( ν ) x,cc ( (cid:126) ω + i γ ) − (2 ε ( ν ) p ) (cid:41) + c.c. ( − ω ) , and ε ( ν ) p is the electron energy in the ν th valley. In deriv-ing Eqs. (11) and (12) we took into account the electron-hole symmetry: v ( ν ) vv = − v ( ν ) cc and E ( ν ) cv = 2 ε ( ν ) p . Notethat Φ + ( p ) = − Φ − ( − p ), which indicates that χ + (cid:54) = − χ − and the second harmonic is generated only for nonequaldistributions of electrons in the valleys.We consider a valley polarized degenerate electron gaswith the Fermi quasi-energies E ( ± )F = E F ± ∆ E F / K and K (cid:48) valleys, respectively, see Fig. 1b. Inthis case, the distribution functions satisfy the condition f ( ν ) v ( − ε ( ν ) p ) − f ( ν ) c ( ε ( ν ) p ) = θ ( ε ( ν ) p − E ( ν )F ). For small valleypolarization, when | ∆ E F | (cid:28) E F , Eq. (10) yields χ ≈ ∂χ + ∂E F ∆ E F = i (cid:16) eω (cid:17) ∆ E F (cid:88) p δ ( ε (+) p − E F )Φ + ( p ) . (13)The trigonal warping of the electron energy spectrumin graphene responsible for SHG is small and can be con-sidered as a perturbation. To first order in the warpingparameter µ , the electron energy and the velocity matrixelements have the form ε ( ν ) p = v p − νµp cos 3 ϕ p , (14) v ( ν ) x,cc = v cos ϕ p + νµp cos 4 ϕ p − ϕ p , (15) v ( ν ) x,vc = i v sin ϕ p + i νµp sin 4 ϕ p − ϕ p , (16) (cid:32) ∂v ( ν ) x ∂p x (cid:33) cv = 2i µν sin ϕ p , (17) g = 0 . 1 E F g = 0 . 2 E F Re c (nm2/V) (cid:1) w / E F Im c (nm2/V) (cid:1) w / E F g = 0 . 1 E F g = 0 . 2 E F FIG. 2. Dependence of the real and imaginary parts of χ on the incident photon energy. The curves are plotted afterEq. (18) for ∆ E F /E F = 0 . E F = 100 meV, µ (cid:126) /v = 0 . γ . where ϕ p is the azimuthal angle of the p vector.Finally, summing up Eq. (13) over the momentum weobtain χ = µ e (cid:126) πv E ∆ E F E F [ G ( ω ) + G ∗ ( − ω )] , (18)where the complex function G ( ω ) is given by G ( ω ) = − i E ( (cid:126) ω ) ( (cid:126) ω + i γ − E F ) (19) × (cid:18) (cid:126) ω (cid:126) ω + i γ/ − E F − E F (cid:126) ω + i γ + 2 E F (cid:19) . As discussed above, the second-order susceptibility givenby Eq. (18) is proportional to the valley polarization∆ E F /E F . Therefore, optical response at the double fre-quency can be used to measure the valley polarizationin graphene. Moreover, specific polarization dependenceof SHG determined by non-zero components of the ten-sor χ , see Eq.(2), enables to discriminate the effect frompossible background noise.Figure 2 shows the dependence of the real and imagi-nary parts of χ on the energy of incident photons. Valleypolarization induced SHG demonstrates a resonant be-havior at (cid:126) ω ≈ E F , which is described by χ ≈ i µ e (cid:126) πv E F ∆ E F E F (cid:126) ω − E F + i γ/ . (20)The resonance is situated in the spectral range whereone-photon direct optical transitions are forbidden. Mi-croscopically, it originates from a strong difference in therates of two-photon absorption in the K and K (cid:48) val-leys due to different occupations of the final states, seeFig. 1b. Additional resonance at (cid:126) ω = 2 E F is situatedat the edge of fundamental absorption band and stemsfrom a difference in the one-photon absorption rates inthe valleys.The calculation yields χ ≈ . /V for the valley po-larization ∆ E F /E F = 0 .
1, Fermi energy E F = 100 meV,photon energy (cid:126) ω = E F , broadening γ = 10 meV, and µ (cid:126) /v = 0 . χ is rather highand comparable to the nonlinear susceptibility of dopedgraphene induced by in-plane electric current with thedensity 1 A/cm [26]. We also note that nonlinear suscep-tibility of the same order of magnitude has been recentlymeasured in MoS and WS monolayers, where the effectcomes from the lack of crystal lattice space inversion [29–31].To summarize, we have shown that valley polarizationof free carriers in graphene can be probed by the effectof second optical harmonic generation. The effect hasa specific light polarization dependence caused by thetrigonal symmetry of electron valleys in graphene.The work was supported by the Russian Foundationfor Basic Research and EU project POLAPHEN. [1] N. Rohling and G. Burkard, Universal quantum com-puting with spin and valley states, New J. Phys. ,083008(2012).[2] E. A. Laird, F. Pei, and L. P. 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