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Carrier Multiplication in Graphene Under Landau Quantization ARTICLE Received 17 Sep 2013 | Accepted 21 Mar 2014 | Published 16 Apr 2014 DOI: 10.1038/ncomms4703 Carrier multiplication in graphene under Landau quantization Florian Wendler1, Andreas Knorr1 & Ermin Malic1 Carrier multiplication is a many-particle process giving rise to the generation of multiple electron-hole pairs. This process holds the potential to increase the power conversion efficiency of photovoltaic devices. In graphene, carrier multiplication has been theoretically predicted and recently experimentally observed. However, due to the absence of a bandgap and competing phonon-induced electron-hole recombination, the extraction of charge carriers remains a substantial challenge. Here we present a new strategy to benefit from the gained charge carriers by introducing a Landau quantization that offers a tunable bandgap. Based on microscopic calculations within the framework of the density matrix formalism, we report a significant carrier multiplication in graphene under Landau quantization. Our calculations reveal a high tunability of the effect via externally accessible pump fluence, temperature and the strength of the magnetic field. 1 Institute of Theoretical Physics, Nonlinear Optics and Quantum Electronics, Technical University Berlin, Hardenbergstrasse 36, Berlin 10623, Germany. Correspondence and requests for materials should be addressed to F.W. (email: fl[email protected]). NATURE COMMUNICATIONS | 5:3703 | DOI: 10.1038/ncomms4703 | www.nature.com/naturecommunications 1 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4703 n 1961 Schockley and Queisser1 predicted a fundamental limit n of B30% for the power conversion efficiency of single- +4 Ijunction solar cells. Their calculation is based on a simple +3 model, in which the excess energy of absorbed photons is +2 assumed to dissipate as heat. This is a good assumption for most IE AR conventional semiconductors, whereas for low-dimensional +1 nanostructures with strong Coulomb interaction, the excess energy can be exploited to generate additional electron-hole pairs. In such structures with highly efficient Auger processes, the Schockley–Queisser limit can be exceeded up to 60%2. 0 Furthermore, the gained multiple electron-hole pairs can also increase the sensitivity of photodetectors and other optoelectronic devices. Carrier multiplication (CM) was first theoretically predicted3 –1 and measured4 for semiconductor quantum dots exhibiting a strong spatial confinement in all three directions. Multiplication –2 of photo-excited carriers has also been found in one-dimensional –3 5,6 nanostructures, such as carbon nanotubes . In graphene, the –4 carrier confinement combined with the linear bandstructure was predicted to account for highly efficient Auger processes7 giving Figure 1 | Auger scattering in Landau-quantized graphene. Sketch of the rise to a considerable CM8,9. Just recently, these predictions were nine energetically lowest Landau levels of graphene with the Dirac confirmed by Brida et al. within a joint experiment–theory study cone in the background. The optical excitation via a linearly polarized laser also including new insights into collinear scattering in pulse induces the transitions À 4- þ 3 and À 3- þ 4, cf. the yellow graphene10, which has been controversially discussed in the arrows. The energy-conserving process of IE involves the inter-Landau level literature. However, to realize graphene-based photovoltaic transition þ 4- þ 1, which provides the energy to excite an electron devices, the key problem of charge carrier extraction in gapless from n ¼ 0ton ¼þ1, resulting in an increase of the number of charge nanostructures needs to be solved. Here we suggest a strategy carriers, cf. the red arrows. The inverse process of AR is shown by the green based on Landau quantization of graphene to address arrows. this substantial challenge, where a magnetic field is used to induce gaps between discrete Landau levels. Graphene in high magnetic fields has already attracted an immense interest in recent years11–14. Unlike an ordinary two-dimensional electron relaxation dynamics within the energetically lowest Landau levels. gas, it exhibits an unique Landau level structure, where the energy We exploit a correlation expansion in second order Born–Markov is proportional to theÀÁ squarep rootffiffiffiffiffiffi of the Landau level index approximation21,22 to derive Bloch equations for graphene under n ¼ 0, ±1, ±2,y En /Æ jjn , cf. Fig. 1. This is a direct Landau quantization: X consequence of the linear density of states in the low-energy in out r_ ðtÞ¼ À2 Re½O 0 ðtÞp 0 ðtÞ þ S ðtÞ½À1 À r ðtÞ S ðtÞr ðtÞ; region and results in an anomalous quantum Hall effect11,12. n nn nn n n n n ð1Þ n0 In this article, we predict the occurrence of CM in Landau- quantized graphene, where the energy gap between the Landau GðtÞ _ 0 0 0 0 0 levels is of advantage for the extraction of gained charge carriers. pnn ðtÞ¼iDonn pnn ðtÞþOnn ðtÞ½ÀrnðtÞÀrn0 ðtÞ ‘ pnn ðtÞ: ð2Þ The magnetic field can be tuned so that the inter-Landau level spacings do not fit the energy of the most relevant optical It is a coupled system of differential equations for the carrier phonons resulting in a strongly suppressed carrier-phonon occupation probability rn(t) in the Landau level n and the scattering. Although at first sight the non-equidistant Landau microscopic polarization pnn0(t) being a measure for optical inter- level spectrum seems to suppress Auger processes, a number of Landau level transitions according to optical selection rules, cf. energetically degenerate inter-Landau level transitions exists Fig. 1. The equations explicitly include the carrier–light providing the basis for efficient Auger scattering, cf. Fig. 1. This interaction, all energy conserving electron–electron scattering allows to exploit the advantage of graphene having a strong processes, as well as the coupling with most relevant optical Coulomb interaction while the acoustic phonon coupling is weak phonons. The carrier–light interaction enters the equations via e in comparison to an ordinary two-dimensional electron gas in a the Rabi frequency O 0 ðtÞ¼ 0 M 0 Á AðtÞ with the elementary nn m0 nn semiconductor heterostructure. As a result, graphene under charge e0, the electron mass m0, the optical matrix element Mnn0 Landau quantization presents optimal conditions for the and the vector potential A(t). The energy difference Donn0 ¼ : appearance of a significant carrier multiplication. (En À En0)/ between the involved Landau levels defines the resonance condition. The magnetic field is incorporated into the equations by exploiting the Peierls substitution (see Results Supplementary Note 1), accounting for the change of electron Theoretical approach. While the carrier dynamics in graphene momentum induced by the confinement into cyclotron 15–19 14,23 in=out has been intensively investigated over the last few years , orbits . The appearing in- and out-scattering rates Sn there has been only a single study in the presence of a strong describe Coulomb- and phonon-induced many-particle scattering magnetic field by Plochocka et al.20 In the latter study, the processes. The rates include Pauli-blocking terms and the relaxation dynamics between higher Landau levels (nE100) is corresponding matrix elements, cf. Supplementary Notes 2–4 investigated in a differential transmission experiment revealing for more details. Owing to the energy mismatch of the optical the importance of Auger scattering in a magnetic field and an phonon modes with the inter-Landau level transitions at the overall suppression of scattering rates in comparison to the no- assumed magnetic field strength, optical phonon scattering is field case. Based on the density matrix formalism21,22, we perform strongly suppressed. Processes involving acoustic phonons and time-resolved microscopic calculations of the ultrafast carrier two-phonon relaxation processes can occur24–26; however, both 2 NATURE COMMUNICATIONS | 5:3703 | DOI: 10.1038/ncomms4703 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4703 ARTICLE are much slower compared with the Coulomb-induced dynamics occupation in Landau level n ¼þ1 that is not driven at all and are neglected in the following. The scattering processes not by the optical pulse, increases until saturation is reached. The only change the carrier occupation rn(t) but also contribute to the occupation r1 remains constant if the Coulomb interaction is dephasing G of the microscopic polarization pnn0(t). The main switched off, cf. the blue dashed line in Fig. 2. This behavior can contribution of G stems from electron-impurity scattering, which be understood as following: first, the exciting laser pulse generates also induces a broadening of the Landau levels. This broadening a non-equilibrium carrier distribution by transferring electrons is explicitly calculated in the self-consistent Born approximation from n ¼À3ton ¼þ4 (yellow arrows in Fig. 1). Then, Auger- following the approach of Ando27, cf. Supplementary Note 5, type processes including impact excitation (IE) and Auger where the electron-impurity scattering strength is chosen in recombination (AR) redistribute the carriers between the equi- agreement with previous theoretical studies28,29. The validity of distant Landau levels n ¼þ4, þ 1 and 0 (red and green arrows the applied Markov approximation has been tested by performing in Fig. 1). Our calculations clearly reveal
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