Giant optical nonlinearity of graphene in a strong magnetic field
GGiant optical nonlinearity of graphene in a strong magnetic field
Xianghan Yao and Alexey Belyanin ∗ Department of Physics and Astronomy, Texas A&M University, College Station, TX, 77843 USA (Dated: 20 October 2011)We present quantum-mechanical density-matrix formalism for calculating the nonlinear opticalresponse of magnetized graphene, valid for arbitrarily strong magnetic and optical fields. We showthat magnetized graphene possesses by far the highest third-order optical nonlinearity among allknown materials. The giant nonlinearity originates from unique electronic properties and selectionrules near the Dirac point. As a result, even one monolayer of graphene gives rise to appreciablenonlinear frequency conversion efficiency for incident infrared radiation.
PACS numbers: 81.05.ue, 42.65.-k
Graphene, a two-dimensional monolayer of carbonatoms arranged in a hexagonal lattice, holds manyrecords as far as its mechanical, thermal, electrical, andoptical properties are concerned; see. e.g. [1] for thereview. With this Letter we would like to add yet an-other distinction to this list of superlatives: we show thatgraphene in a strong magnetic field has the highest in-frared optical nonlinearity of all materials we know.Strong optical nonlinearity of graphene, like most of itsunique electrical and optical properties, stems from lineardispersion of carriers near the K,K’ points of the Brillouinzone. As a result, the electron velocity induced by an in-cident electromagnetic wave is a nonlinear function ofinduced electron momentum. Nonlinear electromagneticresponse of classical charges with linear energy disper-sion has been studied theoretically in [2]. Recently, four-wave mixing in mechanically exfoliated graphene flakeswithout magnetic field has been observed at near-infraredwavelengths [3]. Effective bulk third-order susceptibilitywas estimated to have a very large value, χ (3) ∼ − esu. This is comparable in magnitude to the resonant in-tersubband χ (3) nonlinearity observed in the mid-infraredrange for low-doped quantum cascade laser structures [4],which are essentially asymmetric coupled quantum wellheterostructures.Nonlinear cyclotron resonance in graphene was consid-ered theoretically in [5], again in the classical limit, bysolving the equation of motion F = d p /dt for a mass-less charge. Classical approximation can be applied toelectrons in low magnetic field that are occupying highlyexcited Landau levels n (cid:29)
1, when energy and momen-tum quantization are neglected. Here we present rigorousquantum mechanical description of the nonlinear opti-cal response of magnetized graphene, which is valid forarbitrary magnetic fields and electron distributions overLandau levels (LLs). After finding matrix elements of theoptical transitions between LLs, we calculate the third-order nonlinear susceptibility using the density-matrixformalism and then evaluate the efficiency of the four-wave mixing process. The magnitude of χ (3) turns out ∗ Electronic address: [email protected] to be astonishingly large, of the order of 10 − esu atmid/far-infrared wavelengths in the field of several Tesla.This leads to a surprisingly high four-wave mixing effi-ciency of the order of 10 − W/W per monolayer.Linear (one-photon) absorption in monolayer and bi-layer graphene in arbitrary magnetic fields has been cal-culated in [6] using Keldysh Green’s function formalism.This approach is inconvenient when it comes to calculat-ing the nonlinear optical response. The density matrixformalism adopted in this paper provides a rigorous, intu-itive, and straightforward framework for calculating thehierarchy of nonlinear optical susceptibilities and inter-action of strong multi-frequency EM fields or ultrashortpulses with graphene. Expressions for one-photon ab-sorbance obtained in [6] can be retrieved by calculatingthe linear susceptibility in the limit of a weak monochro-matic field.In the absence of the optical field, the effective-massHamiltonian [7–9] for a graphene monolayer (in the xyplane) in the magnetic field B ˆ z , in the vicinity of Kand K’ points [10] in the nearest-neighbor tight-bindingmodel is given byˆ H = υ F π x − i ˆ π y π x + i ˆ π y π x + i ˆ π y π x − i ˆ π y (1)where υ F is a band parameter (10 cm/s) [11, 12], ˆ (cid:126)π =ˆ (cid:126)p + e (cid:126)A/c , ˆ (cid:126)p is the electron momentum operator, and (cid:126)A is the vector potential, which is equal to (0 , Bx ) here.To simplify notations, we write down the solutions tothe Schr¨odinger equation ˆ H Ψ = ε Ψ separately near theK and K’ point. For example, near the K point theHamiltonian is ˆ H = υ F (cid:126)σ · (cid:126)π , where (cid:126)σ = ( σ x , σ y ) isa vector of Pauli matrices. The eigenfunction is spec-ified by two quantum numbers, n and k y , where n =0 , ± , ± , · · · , and k y is the electron wave vector along ydirection [8]:Ψ n,k y ( r ) = C n √ L exp( − ik y y ) (cid:18) sgn( n ) i | n |− φ | n |− i | n | φ | n | (cid:19) (2) a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec with C n = 1 when n = 0, C n = 1 / √ n (cid:54) = 0, and φ | n | = H | n | (cid:0) ( x − l c k y ) /l c (cid:1)(cid:112) | n | | n | ! √ πl c exp (cid:34) − (cid:18) x − l c k y l c (cid:19) (cid:35) , where l c = (cid:112) c (cid:126) /eB and H n ( x ) is the Hermite Polyno-mial. The eigen energy is ε n = sgn( n ) (cid:126) ω c (cid:112) | n | , where ω c = √ υ F /l c . In the presence of the incident classical optical field (cid:126)E = (1 / eE ω e − iωt polarized along the vector ˆ e in thex-y plane (ˆ e LHS = [ˆ x − i ˆ y ] / √ e RHS = [ˆ x + i ˆ y ] / √ (cid:126)A opt = icω (cid:126)E , to the vector potential of the mag-netic field in the generalized momentum operator ˆ (cid:126)π inthe Hamiltonian. This results in adding the interactionHamiltonian ˆ H int to ˆ H , whereˆ H int = υ F (cid:126)σ · ec (cid:126)A opt (3)This linear in (cid:126)A opt expression for the interactionHamiltonian is exact, unlike the case of the kinetic en-ergy operator quadratic in momentum, where the termproportional to A is usually neglected. Note also thatEq. (3) does not contain the momentum operator andits matrix elements are simply determined by the matrixelements of (cid:126)σ . This immediately gives the selection rules[6] for the transitions between the LLs: ˆ e LHS photons areabsorbed when | n f | = | n i | +1, whereas ˆ e RHS photons areabsorbed when | n f | = | n i | −
1. Here n i and n f indicateinitial and final energy quantum numbers of LLs.Now we can write a standard time-evolution equationfor the density matrix of Dirac electrons in graphene cou-pled to an arbitrary optical field: ∂ ˆ ρ∂t = − i (cid:126) [ ˆ H + ˆ H int , ˆ ρ ] + ˆ R (ˆ ρ ) . (4)Here ˆ R (ˆ ρ ) describes incoherent relaxation due to disor-der, interaction with phonons, and many-body carrier-carrier interactions. Equations Eq. (4) have to be solvedtogether with Maxwell’s equations that contain the opti-cal polarization (cid:126)P ( (cid:126)r, t ) = (1 /V )Tr(ˆ ρ · (cid:126)µ ) (average dipolemoment (cid:104) (cid:126)µ (cid:105) per unit volume) as a source term. In theperturbative regime, they give rise to the hierarchy of theoptical susceptibilities χ ( n ) [13], but they are also validfor describing non-perturbative coupling to strong fields,interaction with ultrashort pulses etc.Since graphene is essentially a 2D system, it makessense to introduce a surface (2D) polarization P s deter-mined as an average dipole moment per unit area ratherthan unit volume. Below we will use 2D susceptibilitiesunless specified otherwise.For a weak monochromatic field one can retain onlythe term ρ (1) mn = ( ρ (0) nn − ρ (0) mm ) (cid:104) m | ˆ H int | n (cid:105) / ( ε m − ε n − (cid:126) ω − i (cid:126) γ mn ) linear with respect to the field and take the sum (cid:80) m,n ρ nm (cid:126)µ mn to obtain an expression for the linear sus-ceptibility: χ (1) ( ω ) = (cid:88) n ≥ α,α (cid:48) C n − e υ F πl c (cid:126) ωω c ( α √ n − α (cid:48) √ n − × ( ν n,α − ν n − ,α (cid:48) )( α (cid:48) √ n − ω c − α √ nω c − ω − iγ ) . (5)Here we used (cid:104) m | ˆ H int | n (cid:105) = − ( i/ω ) eυ F (cid:104) m | (cid:126)σ | n (cid:105) (cid:126)E ( ω )and (cid:104) m | (cid:126)µ | n (cid:105) = ( i (cid:126) / ( ε n − ε m )) eυ F (cid:104) m | (cid:126)σ | n (cid:105) . Note thatthe matrix element of the interaction Hamiltonian canbe written as − ˜ (cid:126)µ mn (cid:126)E , where ˜ (cid:126)µ mn = ( i/ω ) eυ F (cid:104) m | (cid:126)σ | n (cid:105) ,and ˜ (cid:126)µ mn = (cid:126)µ mn when ε n − ε m = (cid:126) ω .We assumed for simplicity that the relaxation termfor the off-diagonal density matrix elements R mn = − γ mn ρ mn and all γ ’s are the same. For easy comparison,we used the same notations for LLs as in [6]: α, α (cid:48) = ± denote whether the corresponding state belongs to theconduction (+) or valence (-) band and ν n,α are the fill-ing factors of LLs; a complete occupation corresponds to ν = 2. The degeneracy of a given LL is 2 / ( πl c ) includingboth spin and valley degeneracy. After calculating the di-mensionless linear absorbance as (2 πω/c )Im[ χ (1) ( ω )] weobtain the same result as in [6].Now we consider a specific example of the nonlinear op-tical interaction, namely the four-wave mixing. Considera strong bichromatic field (cid:126)E = (1 / (cid:126)E exp( − iω t ) + (cid:126)E exp( − iω t ) + c . c . ) normally incident on the graphenelayer. Here ω is nearly resonant with the transition from n = − n = 2 and (cid:126)E has left circular polarization.The frequency ω is nearly resonant with the transitionfrom n = 0 to n = ± (cid:126)E has linear polarization,so that it couples both to transition − → → (cid:126)E at fre-quency ω = ω − ω nearly resonant with the transitionfrom n = 2 to n = 1 is generated.Efficient nonlinear mixing becomes possible due tostrong non-equidistancy of the LLs and unique selectionrules ∆ | n | = ± n greater than 1, for example the transition from state n = − n = 2. This transition would be for-bidden in conventional LL systems with ∆ n = ± υ F /ω , i.e. they are similar toeach other within a factor of 2 and are very large: ofthe order of 10-100 ˚ A in the mid/far-IR range. This, incombination with sharp peaks in the density of states atLLs enables a strong nonlinear response.For simplicity we assume that the incident field isnot strong enough to significantly modify populationsand all states below n = 0 are fully occupied. Thenthe optical fields interact resonantly only with states n = − , , ,
2, which we renamed to n = 1 , , , H ) mn is diagonal, with diagonal elements beingthe energies of corresponding LLs, and the interaction ( a ) ( b ) - | p | E (cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)JJJJJJJ JJJJJJJ(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10) υ F | p | ~ ω c √ ~ ω c − ~ ω c −√ ~ ω c n = 0 n = 1 n = 2 n = − n = − | i | i| i| i ˆ e LHS (cid:10)(cid:10)(cid:10)(cid:10)(cid:29) ˆ e LHS (cid:10)(cid:10)(cid:10)(cid:10)(cid:29) ˆ e RHS
QQQs ˆ e RHS
FIG. 1: (a): Landau levels near the K point superimposed onthe electron dispersion without magnetic field E = ± υ F | p | .(b): A scheme of resonant four-wave mixing process in thefour-level system of LLs with energy quantum numbers n = − , , +1 , +2. The case of exact resonance is shown. Polar-ization of light corresponds to the allowed transitions. Hamiltonian is given by the matrix − ˜ (cid:126)µ mn (cid:126)E as specifiedabove. This approximation is similar to the one adoptedin [4, 14–16] for analyzing resonant nonlinear processesin coupled quantum-well heterostructures. The resultingthird-order nonlinear optical susceptibility at frequency ω = ω − ω is χ (3) ( ω ) = − (2 /πl c ) µ ˜ µ ˜ µ ˜ µ ( i (cid:126) ) Γ × (cid:18) − ρ − ρ Γ ∗ Γ ∗ + ρ − ρ Γ ∗ Γ ∗ + ρ − ρ Γ Γ + ρ − ρ Γ Γ ∗ (cid:19) (6)Here the complex detuning factors are Γ = γ + i (( ε − ε ) / (cid:126) − ω ), Γ = γ + i (( ε − ε ) / (cid:126) − ω ), Γ = γ + i (( ε − ε ) / (cid:126) − ω ), and Γ = γ + i (( ε − ε ) / (cid:126) − ω ).In deriving Eq. (6) from Eq. (4) we assumed that pop-ulations ρ mm are constant and solved for the off-diagonaldensity matrix elements. To get an order-of-magnitudeestimate, we assume that all fields are in exact resonanceand all dephasing rates are the same, so that Γ ij = γ .We also assume for definiteness that state 1 is fully oc-cupied while states 2,3, and 4 are empty. Coming backto original notations of LLs in Fig. 1a, this means thatthe n = 0 LL is empty, i.e. the Fermi level is betweenstates n = 0 and n = −