Abstract
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 longlived population inversion in Landauquantized graphene (i.e. graphene in an external magnetic field) suggesting the design of tunable THz Landau level lasers. The unconventional nonequidistant quantization in graphene offers optimal conditions to overcome the counteracting Coulomb and phononassisted 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 manyparticle mechanisms, we propose two different experimentally realizable schemes to design tunable graphenebased THz Landau level lasers.
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Introduction
In 1986 H. Aoki proposed the first Landau level laser for twodimensional electron systems^{1} exploiting the discreteness of the Landau levels (LLs) to tune the laser frequency through the magnetic field. The key challenge for the realization of such a LL laser is to obtain and to sustain a longlived population inversion (PI) between LLs. This is difficult to achieve in conventional semiconductors where strong Coulomb scattering between equidistant LLs acts in favor of an equilibrium FermiDirac distribution. Similarly, phononinduced scattering can counteract the population inversion, if the phonon energy is in resonance with the interLandau level transitions involved in the lasing process.
Graphene as a twodimensional zerogap semiconductor with remarkable properties^{2} offers optimal conditions for LL lasing. Its linear electronic dispersion leads to an unconventional nonequidistant LL spacing including the appearance of a zero Landau level in an external magnetic field^{3,4}. The observation of a number of interesting effects such as the fractional quantum Hall effect^{5,6}, a giant Faraday rotation^{7}, the quantum ratchet effect^{8}, the Hofstadter butterfly^{9} and the demonstration of a tunable THz detector^{10} has already attracted enormous interest to Landauquantized graphene^{11}. The nonequidistant LLs and the specific optical selection rules allowing transitions between LLs with n → n ± 1 have been predicted to make graphene an optimal material for the realization an efficient twodimensional LL laser^{12}. A transient population inversion in graphene without a magnetic field has already been theoretically predicted^{13,14} and experimentally demonstrated^{15,16,17,18}. It emerges as a result of a relaxation bottleneck close to the Dirac point and decays mainly due to Coulombinduced recombination processes^{14}.
In this article, we predict the occurrence of a longlived population inversion in Landauquantized graphene. We present two different experimentally feasible mechanisms to achieve the population inversion induced by optical pumping, cf. Fig. 1. The first mechanism (A) is based on the specific optical selection rules in Landauquantized graphene yielding the possibility to selectively pump a single LL transition constituting an effectively threelevel laser system, cf. Fig. 1A. Using a linearly polarized optical excitation field with an energy matching the interLandau level transition LL_{−3} → LL_{+2} (and LL_{−2} → LL_{+3}), a population inversion between LL_{+1} and LL_{+2} (σ^{+}PI) (and between and between LL_{−2} and LL_{−1} (σ^{−}PI)) is generated. The second mechanism (B) exploits the scattering among electrons to achieve PI and thereby adds an additional level to the system, which can be beneficial for the efficiency of the laser, cf. Fig. 1B. Here, a linearly polarized pump pulse resonantly exciting the transitions LL_{−7} → LL_{+8} and LL_{−8} → LL_{+7} is used to induce Auger scattering that generates a population inversion between the same levels as in scheme A by populating LL_{+2} and depopulating LL_{−2}. Interestingly, scheme B provides a Coulombinduced mechanism to create PI, which is quite remarkable, because Coulombinduced Auger scattering was shown to rather reduce PI in graphene^{14} and has been believed to be the main obstacle for the realization of a graphenebased twodimensional LL laser^{19,20}. Since we preserve the electronhole symmetry, PI is obtained in the conduction band and in the valence band at the same time. It occurs between the LLs with the indices n = 1 and n = 2. However, we want to stress that also other schemes are possible to obtain PI for different LL transitions. Note that the PI transitions in the conduction and in the valence band are optically coupled by inversely circularly polarized photons, i.e. photons created in a stimulated emission process inducing the electronic transition LL_{+2} → LL_{+1} (LL_{−1} → LL_{−2}) are σ^{+} polarized (σ^{−} polarized). Hence, we label the corresponding population inversion as σ^{+} PI and σ^{−} PI, respectively.
Results and Discussion
While the carrier dynamics without a magnetic field has been already thoroughly studied in experiment^{17,21,22,23,24} and theory^{14,24,25,26,27,28,29}, its investigation in Landauquantized graphene has just started to pick up pace very recently^{19,30,31,32}. We have developed a theory based on the density matrix approach^{29} providing access to time and energydependent relaxation dynamics in Landauquantized graphene and revealing microscopic insights into the underlying manyparticle scattering pathways^{30,31}. The temporal evolution of LL carrier occupations and microscopic polarizations (with the fermionic creation and annihilation operators and a_{i}) determining the strength of optical LL transitions is obtained using the graphene Bloch equations in the presence of a magnetic field
The set of coupled differential equations has been obtained by exploiting a correlation expansion within the secondorder BornMarkov approximation^{29}. Here, we explicitly take into account the occupations and polarizations of the energetically lowest LLs up to n = 10, including the optical excitation as well as all energyconserving carriercarrier and carrierphonon scattering processes. The Rabi frequency appearing in Eqs (1) and (2) depends on the electron’s charge e_{0}, its free mass m_{0}, the optical matrix element M_{ij} and on the vector potential A(t). The scattering rates incorporate all energyconserving electronelectron and electronphonon scattering processes including timedependent Pauli blocking terms. The Coulomb interaction is dynamically screened taking into account the momentum dependence of the dielectric function in the random phase approximation^{11,31}. Phononinduced scattering via the dominant optical phonon modes ΓTO, ΓLO and KTO is included, where a coupling to a bath is considered. The microscopic polarization decays due to a dephasing γ(t) caused by manyparticle scattering as well as impurityinduced LL broadening. The energy difference Δω_{ij} = (ε_{i} − ε_{j})/ħ between LL_{i} and LL_{j} describes the oscillation of the corresponding polarization. Excitons are not considered, since no signatures of excitonic effects were observed in the lowenergy regime of Landauquantized graphene^{3,4}. Furthermore, we assume an impurityinduced broadening of the LLs calculated in a selfconsistent Born approximation^{31,33}, where a reasonable strength of the impurity scattering is chosen^{32,33} yielding a broadening of approximately 4 meV. Details of the calculations can be found in the supplementary material, including the tightbinding description of graphene in a magnetic field, the matrix elements and explicit expressions for the scattering rates .
We investigate the carrier dynamics in graphene in the presence of an external magnetic field of B = 4T. We consider the system to be at room temperature with initial FermiDirac distributed occupations that are optically excited by a pump pulse with a width of 1 ps, a pump fluence of ε_{pf} = 1μJcm^{−2} and an energy matching the pumped LL transitions of the respective PI scheme, cf. Fig. 1. In order to keep the excitation energydependent pulse area constant, the pump fluence in scheme B is increased to ε_{pf} = 2.27μJcm^{−2} (for details see Ref. 34). Solving the graphene Bloch equations (Eqs (1) and (2)) yields the temporal evolution of p_{ij} and ρ_{i} allowing us to investigate the interplay between optical transitions on the one side and carriercarrier and carrierphonon scattering processes on the other side. Since neutral Landauquantized graphene is symmetric for electrons and holes, we focus the discussion on the σ^{+}PI in the conduction band. Figure 2 illustrates the timedependent occupations of LL_{+1} and LL_{+2} (upper panel) as well as the resulting population inversion (lower panel). In both PI schemes, ρ_{+1}(t) changes only slightly, while ρ_{+2}(t) shows a fast increase on a subpicosecond time scale during and shortly after the optical excitation (illustrated by the yellow area in the background) followed by a slow decay on a picosecond time scale. We find a longlived population inversion that is defined by
and represented by the respective areas between ρ_{+1} and ρ_{+2} in Fig. 2. Exponential fits to the temporal evolutions of PI for schemes A and B (dashed lines in lower panel of Fig. 2) reveal their decay times τ_{A} = 39 ps and τ_{B} = 27 ps. Since scheme A provides a straightforward approach to induce PI as a direct consequence of the optical excitation, the maximal value is reached already during pumping. In scheme B, on the other hand, Coulombscattering is needed to induce PI by redistributing the optically excited charge carriers, consequently, the buildup time is longer and its maximum is reached with a delay of a few picoseconds. While both PI schemes are suited to create a significant PI with a rather long decay time, the advantage of scheme B is its additional fourth level making it a potentially better laser system which is reflected by its higher maximal PI value (cf. Fig. 1). As a consequence, optically excited charge carriers in LL_{+8} scatter down to LL_{+2} already during pumping which reduces the pumping saturation and allows the excitation of more charge carriers. The cost of the enhanced maximal PI is a faster decay resulting from the greater complexity of PI scheme B involving more LLs and therefore opening up more possible phononassisted decay channels: In scheme A the PI vanishes after ~82 ps, while in scheme B the PI lasts for ~62 ps.
To obtain more insights into the underlying elementary processes, we investigate the maximal PI as a function of the magnetic field, cf. Fig. 3. For this investigation, we determine the maximal PI at different magnetic fields B, while the pulse area is held constant by scaling the pump fluence linearly with B (for details see Ref. 34). The energy of the PI transition ε_{+2} − ε_{+1} is tunable via the magnetic field and is given on the upper axis. The dynamics without phonons (thin lines) shows a weak dependence on the magnetic field: At higher B, the LLs are shifted to higher energies, thereby reducing the initial occupations in the conduction band, in particular ρ_{+1} decreases resulting in a stronger PI. However, at the same time the scattering generally becomes more efficient with increasing magnetic fields, since the degeneracy^{34} of LLs scales with B. This enhances the dephasing γ(t) of p_{ij} (cf. Eq. (2)) and reduces the pumping efficiency and consequently also the maximal PI. These counteracting effects nearly balance each other out resulting in a very weak dependence on the magnetic field. Switching on phononinduced scattering, the dependence qualitatively changes and pronounced peaks and dips emerge in the Bdependence of the maximal PI (thick lines). They indicate magnetic fields that fulfill the resonance condition between the energy of an optical phonon and interLandau level transitions involved in the respective PI scheme. This is further evidenced by showing the dynamics that only includes ΓTOphonons (dashed lines), where the number of peaks and dips is clearly reduced. In scheme A, two main resonances are present at the magnetic fields B = 2.67T and B = 3.30T. The first corresponds to the transition LL_{+2} → LL_{−3}, while the second coincides with the transition LL_{+2} → LL_{−2}, both reducing the PI by providing direct decay channels of the PI. The Bdependence in scheme B is more complex, since more LLs are involved (cf. Fig. 1) and hence more resonances occur. The three distinct dips at B = 1.45T, B = 1.60T and B = 3.30T correspond to the transitions LL_{+2} → LL_{−8}, LL_{+2} → LL_{−7} and LL_{+2} → LL_{−2} (and at the same time LL_{+8} → LL_{0}), respectively. For reasons of clarity, symmetric hole transitions are omitted for the discussion. Interestingly, phononinduced scattering can also increase the PI, as can be seen at B = 1.22T and B = 1.99T. At these magnetic field strengths, the energy of the ΓTOphonon is in resonance with the transitions LL_{+4} → LL_{−7} and LL_{+1} → LL_{−7}, respectively. The former case positively affects the pumping, while in the latter case the PI is not only increased through the depletion of LL_{+1}. It also couples the lower laser level LL_{+1} with the ground state LL_{−7} from which electrons are excited and thus allows an electron to perform cycles in the fourlevel system opening up the possibility of continuous laser action (cf. Fig. 1): First it is excited from LL_{−7} to LL_{+8}, from where it scatters down and accumulates in LL_{+2}, before it participates in a stimulated emission event that transfers it to LL_{+1}. Now, a phonon with the appropriate energy can bring the electron back to LL_{−7}. The resonance condition is fulfilled for the ΓTO, ΓLO and KTOphonon modes at the magnetic fields B = 1.99T, B = 2.11T and B = 1.41T, respectively. A magnetic field of B = 2T seems to be optimal, since no interfering resonances with other interLL transitions occur (cf. Fig. 3).
Next, we explore the doping dependence of the PI, which opens up the possibility to control it by the application of a gate voltage resulting in a shift of the Fermi energy away from E_{F} = 0. This breaks the electronhole symmetry and allows to tune the relative population inversion between the two inversely polarized LL transitions LL_{±2} → LL_{±1}, cf. Fig. 4a. While a small positive Fermi energy gives rise to an increase of the σ^{+} PI, the impact on the σ^{−} PI is opposite. This behavior can be attributed to new scattering channels that are forbidden under electronhole symmetry, but arise as soon as this symmetry is broken. For simplicity, we focus on the simple PI scheme A in the following: An upshift of the Fermi energy results in a less efficient pumping of the transition LL_{−3} → LL_{+2} due to an enhanced Pauli blocking in comparison to the transition LL_{−2} → LL_{+3}, since LL_{+2} becomes thermally occupied. According to this, σ^{+} PI should be suppressed, while σ^{−} PI is expected to be enhanced. Interestingly, we observe the opposite behavior, as shown in Fig. 4a. To understand this, we consider the energyconserving Coulomb process involving the transitions LL_{0} → LL_{+2} and LL_{0} → LL_{−2} (outward scattering), which cancels out in an electronhole symmetric system that also exhibits the inverse process (inward scattering: LL_{+2} → LL_{0} and LL_{−2} → LL_{0}) occurring with the same probability. However, due to the asymmetric pumping and a more than halffilled LL_{0} in a ndoped sample, the Coulombinduced outward scattering prevails over the inward scattering, cf. Fig. 4b. As a result, σ^{+} PI is enhanced, while σ^{−} PI is suppressed, as observed in Fig. 4a. Shifting the Fermi energy further away from the neutral position, the PI of both transitions decreases, which can be readily understood considering the initial occupations. When the Fermi energy reaches the vicinity of LL_{+1}, its initial occupation ρ_{+1}(t_{0}) is considerably increased counteracting the buildup of a population inversion. The PI in both schemes shows a similar dependence of the doping.
Finally, we propose a pumpprobe experiment to test our predictions at sufficiently low temperatures, so that the initial occupations ρ_{+1}(t_{0}) and ρ_{+2}(t_{0}) are nearly zero: Exciting Landauquantized graphene according to one of the PI schemes A or B, cf. Fig. 1, with a probe pulse measuring the σ^{+} PI absorption, a positive differential transmission signal (DTS) indicates a faster increase of ρ_{+2} in comparison to ρ_{+1} and consequently would provide strong evidence for the occurrence of gain. More generally, in the presence of gain, the real part of the optical conductivity (which is proportional to the absorption) should become negative at the energy matching the transition LL_{+1} → LL_{+2}. Based on our calculations, the optimal experimental conditions to measure gain in Landauquantized graphene are expected at a magnetic field of B = 2T and a linearly polarized optical excitation with an energy of about ε_{+8} − ε_{−7} = 281 meV (depending on the exact value of the Fermi velocity, cf. Eq. 3 of the Supplementary material) corresponding to excitation scheme B (cf. Fig. 1).
In conclusion, based on microscopic calculations we predict the occurrence of a pronounced population inversion in Landauquantized graphene. We demonstrate that controlling the magnetic field and the doping allows to tune the energy as well as the polarization of the emitted radiation. We show that carrierphonon scattering can be exploited to boost the effect and to even open the way to continuous wave laser operation. Our microscopic insights into the carrier dynamics in Landauquantized graphene will guide future experiments towards the design of graphenebased Landau level lasers and THz emitters.
Methods
The microscopic modeling of the carrier dynamics has been performed within the formalism of density matrix theory^{29,35,36}. The magnetic field has been implemented into the equations by applying the Peierls substitution^{11,31}. A detailed description of the calculations including the electronic dispersion in the presence of a magnetic field as well as the coupling elements determining the carrierlight, carriercarrier and carrierphonon interaction can be found in the supplementary material.
Additional Information
How to cite this article: Wendler, F. and Malic, E. Towards a tunable graphenebased Landau level laser in the terahertz regime. Sci. Rep. 5, 12646; doi: 10.1038/srep12646 (2015).
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Acknowledgements
We acknowledge financial support from the EU Graphene Flagship (contract no. CNECTICT604391), the Swedish Research Council (VR) and the Deutsche Forschungsgemeinschaft (DFG) through SPP 1459. Furthermore, we thank A. Knorr (TU Berlin) and S. Winnerl (HelmholtzZentrum DresdenRossendorf) for inspiring discussions on Landau level lasers.
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Both authors developed the theoretical model, analysed and discussed the obtained results. F.W. performed the calculations and wrote the paper with major input from E.M.
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Wendler, F., Malic, E. Towards a tunable graphenebased Landau level laser in the terahertz regime. Sci Rep 5, 12646 (2015). https://doi.org/10.1038/srep12646
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DOI: https://doi.org/10.1038/srep12646
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