Towards a tunable graphene-based Landau level laser in the terahertz regime

Terahertz (THz) technology has attracted enormous interest with conceivable applications ranging from basic science to advanced technology. One of the main challenges remains the realization of a well controlled and easily tunable THz source. Here, we predict the occurrence of a long-lived population inversion in Landau-quantized graphene (i.e. graphene in an external magnetic field) suggesting the design of tunable THz Landau level lasers. The unconventional non-equidistant quantization in graphene offers optimal conditions to overcome the counteracting Coulomb- and phonon-assisted scattering channels. In addition to the tunability of the laser frequency, we show that also the polarization of the emitted light can be controlled. Based on our microscopic insights into the underlying many-particle mechanisms, we propose two different experimentally realizable schemes to design tunable graphene-based THz Landau level lasers.


Wave function and dispersion in Landau-quantized graphene
The magnetic field B = ∇ × A, which shall point into the z-direction B = (0, 0, B), is introduced by applying the Peierls substitution p → π = p + e 0 A(r) (1) to the effective Hamiltonian of graphene in the low-energy regime with the canonical momentum p = q, the kinetic momentum π, the vector potential A(r), the Fermi velocity v F = 1 nm/fs [1], and the valley index ξ = ±1 [2]. The Schrödinger equation is readily solved and yields the Landau level spectrum and the spinor where e 0 is the elementary charge, λ = ±1 is the band index, n = 0, 1, 2, . . . is the Landau level index, and m is a quantum number that can be associated with the position of the cyclotron orbits in the graphene plane [2]. The tight-binding wave function is a linear combination of wave functions of the two sublattices A and B and reads with the coefficients c A and c B that are given by the spatial representation of the spinor in Eq. 4 through the relation and where φ(r − R l ) denotes the p z -orbital of the carbon atom at position R l . The explicit form of R|n, m is given by [3] R|n, m = 1 with the magnetic length l B = /(e 0 B).

Optical matrix element
An analytic expression of the optical matrix element M i,f =´drΨ * f (r)∇Ψ i (r) is obtained using the wave function from Eq. 5 and reads [4] M with the compound index i = (n i , m i , λ i , ξ i ), the unit vector in j-directionê j , and the [4] which removes the dependence of the free electron mass m 0 from the Rabi frequency The Kronecker deltas express the graphene-specific optical selection rules n → n ± 1 allowing transitions only between Landau levels with an index n differing by 1. These inter-Landau level transitions can be induced using circularly polarized radiation, where σ ± -polarized light corresponds to ∆n = ±1 respectively [6,7].
When a combination of both circular polarization directions is used, e.g. in a linearly polarized excitation field, both transitions (∆n = ±1) are pumped at the same time.

Coulomb matrix element
The double integration occurring in the Coulomb matrix element is simplified introducing the Fourier transformation which results in the splitting into two separate integrals Γ 13 (q) and Γ 24 (−q). Here, V q = e 2 0 /(2 0 r Aq) is the Fourier transform of the Coulomb interaction with the vacuum and relative permittivities 0 and r , and the area of graphene A. We consider graphene on a SiC substrate with a relative permittivity of r ≈ ( SiC + air )/2 ≈ 3.3 [8]. The integrals Γ if (q) =´dr Ψ * f (r)e iqr Ψ i (r) are evaluated using the tight-binding wave functions (Eq. 5) yielding with the momentum transfer in polar coordinates q = (q, ϕ), and the form factors [2] nm|e ±iqr |n m = e Note that the angular integration in Eq. 11 yields a Kronecker delta δ n 1 −m 1 +n 2 −m 2 ,n 3 −m 3 +n 4 −m 4 expressing the angular momentum conservation of Coulomb scattering. Dynamical screening is taken into account by replacing in Eq. 11, where the energy in the dielectric function (q, ω) == 1 − V q Π 0 (q, ω) is defined as ω = 1 − 3 [9], and Π 0 (q, ω) denotes the polarizability which is given by in the random phase approximation [2,10]. Here, n FD is the Fermi-Dirac distribution, and a constant broadening Γ = 4 meV is introduced. The latter is a Landau level broadening induced by electron-impurity scattering. The form factor F λn,λ n (q) reads [2] F λn,λ n (q) =

Phonon matrix elements
The carrier-phonon matrix element g i,f p,µ =´drΨ * f (r)V c-ph (p)Ψ i (r) depends on the potential V c-ph (p) that was explicitly calculated by Ando in the long-wavelength limit [11]. Plugging the wave function Eq. 5 into the above expression with the potential V c-ph (p) of the respective optical phonon mode yields with the coupling parameter β ≈ 2 [11], the phonon momentum transfer in polar coordinates p = (p, ϕ p ), the graphene mass density M = 7.6 × 10 −8 gcm −2 , and the lattice spacing a 0 = 0.2461 nm [5]. Furthermore, the + sign refers to the longitudinal modes (ΓLO, KLO) and the -sign to the transverse modes (ΓTO, KTO). While analytic expressions are obtained in the case of Γ phonons, the carrier-phonon interaction strengths g 2 KTO DFT = 0.0994 eV 2 · A uc /A, and g 2 KLO DFT = 0.00156 eV 2 · A uc /A are based on numerical calculations within density functional theory (DFT) that where performed by Piscanec et. al [12]. Here, A uc = √ 3a 2 0 /2 is the unit cell area and A is the area of graphene which cancels after performing the momentum sum in the scattering rates. Note that the interaction strengths in the case of the Γ phonons are also in agreement with the corresponding DFT calculations [12]. The energies of the phonon modes are considered to be constant, i.e. p,ΓTO = 192 meV, p,ΓLO = 198 meV, p,KTO = 162 meV, and p,KLO = 151 meV [13,14]. Finally, the form factors n f , m f |e ipr |n i , m i are given by Eq. 12. The KLO-phonon mode is omitted in the numerical calculations, because its interaction strength is negligible compared to the three other modes.
Since a phonon can absorb (or provide) any angular momentum, there are no selection rules for the carrier-phonon interaction and restrictions of the corresponding scattering channels are only imposed by the energy conservation.

Scattering rates
Many-particle scattering rates appearing in the Bloch equations, cf. Eqs. 1-2 in the main part of the paper, read in second-order Born-Markov approximation for the Coulomb inter- and for the carrier-phonon interaction with the Coulomb matrix element V 12 34 , the carrier-phonon matrix element g i,f p,µ , the phonon occupation n p,µ , the energy differences E iabc = i + a − b − c and E em/ab = f − i ± p,µ for the emission (+) and absorption (−) of a phonon of the mode µ. We assume a finite Landau level broadening induced by electron-impurity scattering that is expressed by a Lorentzian L Γ ( E) = Γ π ( E 2 + Γ 2 ) .