Temperature Dependent Valley Relaxation Dynamics in Single Layer WS2 Measured Using Ultrafast Spectroscopy
Cong Mai, Yuriy G. Semenov, Andrew Barrette, Yifei Yu, Zhenghe Jin, Linyou Cao, Ki Wook Kim, Kenan Gundogdu
TTemperature Dependent Valley Relaxation Dynamics in SingleLayer WS Measured Using Ultrafast Spectroscopy
Cong Mai, Yuriy G. Semenov, Andrew Barrette, Yifei Yu, ZhengheJin, Linyou Cao, ∗ Ki Wook Kim, † and Kenan Gundogdu ‡ Department of Physics, North Carolina State University, Raleigh, NC 27695, USA Department of Electrical and Computer Engineering,North Carolina State University, Raleigh, NC 27695, USA Department of Material Science and Engineering,North Carolina State University, Raleigh, NC 27695, USA
Abstract
We measured the lifetime of optically created valley polarization in single layer WS usingtransient absorption spectroscopy. The electron valley relaxation is very short ( < . One process involves direct scattering of excitons from K to K (cid:48) valleys with a spin flip-flop interaction. The other mechanism involves scattering throughspin degenerate Γ valley. This second process is thermally activated with an Arrhenius behaviordue to the energy barrier between Γ and K valleys. PACS numbers: 73.21.-b, 78.47.j-,71.35.-y a r X i v : . [ c ond - m a t . m t r l - s c i ] M a y NTRODUCTION
The discovery of graphene, a monolayer of carbon atoms, has inspired considerable in-terest in other 2D material systems in search of exotic electronic, optical and mechanicalproperties and novel practical applications[1–4]. In particular, two-dimensional transitionmetal dichalcogenides (TMDC) recently emerged as a promising material system with manypotential electronic applications. With their tunable direct gap in visible range of the opticalspectrum, absence of dangling bonds and high surface-to-volume ratio, these 2D semicon-ducting systems are ideal for field-effect transistors (FET), photovoltaics, light emittingdiodes (LEDs), molecule sensing, and electrocatalytic water splitting applications[5–14].Moreover, due to strong spin-orbit splitting they have been subject to specific spin andvalley applications. The optical band gap in these structures are located at the K and K (cid:48) = − K points at the edge of the Brillouin zone. A strong spin-orbital coupling resultsin distinct states with different spin and valley indices so that K to K (cid:48) elastic transitionrequires a spin flip for the electrons and holes. One immediate consequence of this propertyis the ability to control valley polarization, hence crystal quasi-momentum of electrons, byusing circularly polarized light, which is impossible in conventional semiconductors due tonegligible momentum of photons. This could open up opportunities for developing optoelec-tronic and valleytronic applications based on manipulation of spin and valley polarizationof charge carriers[11–13, 15–17].Detailed understanding of inter-valley relaxation dynamics is critical for the implemen-tation of valleytronic applications. Very recently, numerous experimental and theoreticalefforts have been devoted to characterization of electronic structure and optical propertiesof TMDCs[18–26]. However, there are only limited experimental studies that directly ad-dress the valley and spin relaxation process. Recently we measured valley lifetime in singlelayer MoS to be 10 ps at 74K, using time-resolved absorption spectroscopy[27]. Surpris-ingly this relaxation time is very short, as large spin splitting in the valence band and spinvalley coupling in K and K (cid:48) valleys was expected to impede hole valley scattering[11–13, 15–17, 26, 27]. As of now an accurate picture of valley relaxation mechanisms in atomicallythin TDMCs is missing. In the current work, we studied thermal dependence of valley re-laxation of excitons in monolayer WS using broadband transient absorption spectroscopy.In WS the spin-orbit splitting is significantly larger compared to MoS and other TMDCs213, 18–21]. Even so we observe that hole valley lifetime is more than an order of magnitudelarger in WS compared to MoS . Similar to MoS , electron valley relaxation is dominatedby many body interactions and faster compared to hole valley relaxation. Based on tem-perature dependence of valley lifetimes, we propose a mechanism for an exciton-mediatedvalley relaxation process for holes in single layer WS .In order to measure the valley relaxation time, we performed experiments on single layerWS samples grown on quartz substrates using a chemical vapor deposition technique (Sup.Inf.)[29–32]. In time resolved experiments, circularly polarized 60 fs pump pulses, tuned tolower energy tail of A excitonic transition (1.977 eV) to create electron-hole pairs in the Kvalley. The differential transmission of a same circularly polarized (SCP) broad band white-light continuum probe pulse measures the relaxation within the same valley and oppositecircularly polarized probe pulse (OCP) measures the population in the other valley. Thedecay of the polarization anisotropy between the SCP and the OCP spectra reveals theintervalley relaxation dynamics.Figure 1 (a) schematically displays the electronic band structure for WS at the K and K (cid:48) points, adapted from Ref. [33]. The conduction band is composed by d [1 − z ] orbitals | L, m (cid:105) of W with zero magnetic quantum number m = 0 under orbital moment L = 2 andrelatively small admixture of p [ x − iνy ] orbitals of S [17] with m = − ν , which constitutesrelatively small, 27 meV, spin splitting ( ν = ± K and K (cid:48) valleys). Thevalence band is formed by d [( x + iνy ) ] orbitals of W with m = 2 ν and some admixture of p -orbitals of S that conditions much larger, 435 meV, valence band spin splitting [17, 18].This spin splitting leads to energetically well separated A and B excitonic transitions shownby red and blue arrows in Fig. 1a. Their electro-dipole interband optical transitions arecoupled with circularly polarized light.Figure 1(b) shows transient SCP and OCP spectra for A exciton at 110 K. Initially theSCP spectrum shows a dispersive line shape in which the lower energy tail of the spectrumexhibits ground state bleaching (GSB) and stimulated emission (SE) and the higher energytail exhibits photoinduced absorption (PIA). This dispersive feature evolves into purelyabsorptive line shape in about 7.6 ps. The OCP spectrum also exhibits a dispersive lineshape, though less prominent than that of SCP. In the later time delays the anisotropybetween the SCP and OCP spectra vanishes and a sharp dip shows up at 2.04 eV.We attribute this dispersive distortion to phase-space filling effects similar to those ob-3erved in conventional semiconductors[34, 35]. Briefly the photoexcited population blue-shifts the A exciton absorption, resulting in a differential line-shape in the transient spec-trum. Monolayer materials exhibit ultra-tight quantum confinement[20, 21, 35]. Thus phase-space filling of carriers, which is related to Pauli-blocking, is very significant[26, 34, 35]. Asimilar blue shift has been observed in MoS transient absorption studies as well[26]. Theexact source of the spectral dip at 2.04 eV is unclear. It could be due to an overlappingabsorptive transition from the excited state. A better understanding of the band structureand the excited states of the WS is needed for analyzing this feature.The evolution of the polarization anisotropy, measured by the difference between theSCP and OCP spectra in Figure 1, reveals the valley relaxation time of electrons and holes.The immediate presence of A exciton feature in the OCP spectra clearly suggests that asignificant fraction of electron population, initially photo-excited in the K valley, quicklydelocalize between K and K (cid:48) valleys within 200 fs, which is within the time resolution ofthe experiment. This is because without any valley relaxation OCP should not exhibit any A exciton feature. The electron relaxation into the K (cid:48) valley (i.e. transition of the directexciton [ eK, hK ], with both carriers in K-valley, to indirect one (dark exciton) [ eK (cid:48) , hK ])leads to bleaching of the A exciton absorption in the OCP spectra. This immediate electronvalley delocalization has been observed in single layer MoS and attributed to coherentcoupling of excitonic states[26]. While the circularly polarized optical transitions are valleyspecific, the resulting optically created exciton associated with a specific valley, K or K (cid:48) ,is not an eigenstate of the full exciton problem. Strong electron-hole confinement in excitonwith energy 710 meV [37] and radius around 1 - 2 nm couples the K and K (cid:48) valleys. As aresult the electron in K valley quickly delocalizes over both valleys. This is further confirmedwith the analysis of the evolution of the B exciton transition. As observed in Figure 2 (a,b), the B transition bleached immediately after the pump pulse in both SCP and OCPspectra with equal intensity. Because the B valence levels are very high in energy andare not populated with the pump pulse, it is only sensitive to electron dynamics. Henceelectrons occupy B -levels in both valleys immediately upon photoexcitation. Because B -levels have opposite spin orientation; these spectra suggest not only that valley relaxationof the electrons is very quick but also that electron spin relaxation within the same valleyis very fast (Fig. 2 c, d).In contrast to fast conduction band spin and valley relaxation, hole delocalization is4ot favorable energetically due to the large, ∆ SO (cid:39)
450 meV, spin orbital splitting, whichimposes the energy gap between the parallel spin states in the two valleys (Fig. 1a). Hencehole valley relaxation takes a much longer time. The total loss of the polarization anisotropyin the SCP and OCP spectra show that it is around 100 ps. We note that this is much longercompared to MoS where the valley relaxation takes place in about 10 ps.In order to resolve hole valley relaxation mechanism we performed transient absorptionexperiments in a range of temperatures from 74 K to 298 K. For each temperature, the pumppulses were tuned to the lower energy side of the A excitonic transition and resulting dynam-ics were probed with a broadband white light continuum pulse. At each temperature theearly spectra show similar dynamics to 110 K data suggesting fast spin and valley relaxationdynamics for the electrons. However relaxation of the circularly polarized anisotropy clearlyevolves at a different rate, revealing the thermal dependence of the hole valley relaxationdynamics.In order to quantitatively extract the hole valley relaxation time, we analyzed the decay ofpolarization anisotropy at the A exciton transition by taking the difference of SCP and OCPspectra[38]. Neither single nor double exponential decay functions fit the anisotropy decay.Therefore we used a triple exponential fit. For all temperatures the first time constant is in100 fs range, within the resolution of the experiment. The second time constant is in 1.5-3ps range (Sup. Inf.). The slow component is the only one that exhibits clear temperaturedependence varying from 88 ps to 8 ps at 74 K and 298 K respectively (Fig. 3). In thefollowing this slow component is used for theoretical analysis of the valley relaxation process.We considered several previously proposed inter-valley relaxation mechanisms to explainthe experimental results. Our initial analysis suggests single carrier relaxation mechanismscannot be responsible for the temperature dependence that we observe. For instance scatter-ing with nonmagnetic impurities has been suggested as a potential valley relaxation process.Reference [39] shows that this mechanism requires a ∼ k dependence on hole momentum,which leads to a stronger temperature dependence ( T ) than our observation. Relaxationthrough flexural phonon modes was also considered as a potential mechanism[44]. It pre-dicts an inverse relation between mobility and relaxation rate, and order of magnitude longervalley lifetimes compared to our results, hence it is unlikely to be responsible for the valleyrelaxation in WS .We considered spin relaxation mechanisms such as Elliot-Yafet (EY) and Dyakonov-Perel5DP) processes as well. These processes require scattering of carriers under inhomogeneousmagnetic field. For instance in conventional semiconductors, Dressalhause effect leads tosuch an effective field. Unlike many conventional semiconducting materials in TDMCs thespin quantization axis retains normal direction over all of Brillouin zone except the specificpoints where spin splitting reduces to zero. Such a property is clearly demonstrated in Ref.[17] based on calculations involving 80 energy bands of MoS . Our first principle calculationsfor WS also reproduce the absence of transversal spin components over all of Brillouin zone.Thus the hole/electron diffusive motion cannot mix the states with opposite spin directionsalong the normal to layer plane (i.e. z axis). Therefore Elliot-Yafet and Dyakonov-Perelmechanisms are irrelevant.Note however that in a realistic structure the spin quantization axis deviates from thenormal direction due to inversion asymmetry induced by the substrate. Indeed, modeling thesubstrate induced asymmetry by an effective external electric field, we found the spin quan-tization axis deviates from z-direction. Proportionality of this deviation to quasi-momentumshift from extremum points represents the Rashba effect with Hamiltonian H R = α ϕ | ∆ k | ,(∆ k = ± K − k , ϕ is an angle between ∆ k and in-plane axis x ), which in addition revealstrigonal anisotropy of α ϕ in the vicinities of each K and K (cid:48) = − K points. The correspondentspin-relaxation rate is, τ − s = 2 (cid:126) (cid:28) α ϕ ∆ k τ k τ k ω SO + 1 (cid:29) (1)where brackets mean thermal averaging, ω SO = ∆ SO / (cid:126) , and τ k is momentum relaxationtime. Our first principle calculations estimate the Rashba constant mediated by potentialdrop ∆ U between two sides of a layer. Averaging over angle ϕ , we find α = β ∆ U , where β (cid:39) . · − cm, which results in an insignificant contribution ( τ − s ∼ − (cid:10) τ − k (cid:11) ) of thismechanism to intervalley spin relaxation even at ∆ U =100 meV.We conclude that the previously discussed single particle relaxation mechanisms are notsufficient to explain exciton intervalley relaxation. On the other hand, the excitonic electron-hole exchange interaction lifts orthogonality of spin states attributed to different valleys.This effect describes a minimal exciton spin Hamiltonian H = τ SO σ z + τ δ SO s z + ∆ eh σ s , (2)where σ and s are the Pauli matrixes for electron and hole spins, τ = ± SO and δ SO are the hole and electron spin splitting induced by spin-orbital interaction and∆ eh is the strength of electron-hole exchange interaction. It is important to note that thisinteraction mixes the exciton spin states located in different valleys. This is because electro-dipole interband transitions occur between electronic states with same spin that generateselectron-hole pair with opposite spin directions, i.e. with zero projections s z + σ z = 0. Thusthe transitions between K and − K excitons would be proportional to the spin flip-flop factor T = |(cid:104)↑ , ↓ |↓ , ↑ (cid:105)| , where |↓ , ↑(cid:105) corresponds to s z = − /
2. If the spin-independent intervalleyrelaxation rate is τ − KK (cid:48) the net exciton relaxation rate will be τ − ex = τ − KK (cid:48) eh (∆ SO − δ SO ) + 4∆ eh . (3)Exciton scattering on any (not only spin-associated) local defects or phonons actualizesuch transitions. In contrast to free carriers, excitons are composed of band states involvingthose far away from the extramums K or K (cid:48) , due to their short radius. These statesadmix the conduction band state | , (cid:105) to valence bands of both valley states which liftstheir orthogonality. For instance, small deviation ∼ . | K − K (cid:48) | from extremum leads tonon-negligible admixture of conduction band by a factor of about ξ ∼ . K − exciton scatters to anindirect one [ K e ( K (cid:48) e )Γ h ] and then to direct K (cid:48) − exciton. This mechanism is not subjected tospin restrictions with factor in Eq. (3) due to zero spin-orbital splitting (∆ SO = δ SO = 0) atthe Γ point. Moreover, since Γ state is composed from | , (cid:105) and | , (cid:105) orbitals, the K − Γoverlap is substantially stronger than direct inter-valley overlap. On the other hand thedifference ∆ exK Γ in direct exciton energy E ex [ K e K h ] = E e ( K ) − E v ( K ) − E X ( K ) and indirectone E ex [ K e Γ h ] = E e ( K ) − E v (Γ) − E X ( K e Γ h ) imposes a thermal activation process withArrhenius-like temperature dependence τ − K Γ = r K Γ exp( − ∆ exK Γ /kT ) , (4)where ∆ exK Γ = E v ( K ) − E v (Γ) + E X ( K ) − E X ( K e Γ h ), the E e ( v ) ( B ) is the energy at theconduction (valence) B − band edge [ B = K, Γ] and E X ( K ) and E X ( K e Γ h ) are the bindingenergies for direct and indierct excitons. Our first principal calculations (Sup. Inf.) give E v ( K ) − E v (Γ) = 310meV, which is slighly differerent from previous calculations[21]. In7he 2D Wannier-Mott exciton model the difference E X ( K ) − E X ( K e Γ h ) = ∆ E X basicallystems from different reduced effective masses µ ( K ) and µ (Γ) so that ∆ E X = E X ( K )[1 − µ (Γ) /µ ( K )][38].We estimated the reduced effective masses µ ( K ) and µ (Γ) based on our first principle en-ergy band structure calculations (Sup. Inf.)[40–42]. The resulting binding energy differencebetween the direct K − exciton and indirect [ K e ( K (cid:48) e )Γ h ] exciton, ∆ E X (cid:39) −
170 meV, givesan energy barrier of ∆ exK Γ (cid:39)
140 meV.Quantitative estimation of r K Γ , ∆ eh and τ − KK (cid:48) would be grounded on theory of excitonformation in TMDCs, which is not completed yet. Therefore we will treat these parametersas phenomenological ones. We observe that low-temperature thermal dependence of τ − KK (cid:48) (Fig. 3b) is similar to that of phonon-assistant momentum relaxation rate τ − p = τ − p ( T )calculated from first principles (Sup. Inf.)[43]. For free holes τ − h can be approximatedwith τ − h ( T ) = r h [1 + ( T /T )] at r h = 7 . · ps − and T (cid:39)
200 K in wide range (70 K to200 K) of temperatures. Adapting similar dependence for exciton momentum relaxation andcombining τ − e,h ( T ) with Eq. (4) the net result of both mechanisms describes the temperaturedependence in the form τ − ex = r KK (cid:48) (1 + T /T ) + r K Γ exp( − ∆ exK Γ /kT ) . (5)Fig. 3b shows that only two free parameters, r KK (cid:48) = 0 .
01 ps − and r K Γ = 24 ps − , describeall data with experimental accuracy.To conclude, we present ultrafast valley relaxation dynamics measurements in monolayerWS . The intervalley scattering lifetime in monolayer WS is much longer than that ofmonolayer MoS . The thermal dependence of the valley relaxation rate indicates that, dueto strong exciton binding energy and exchange interaction, electron-hole spin flip-flop mech-anism becomes an efficient spin relaxation channel. Because such excitonic relaxation mech-anisms will not affect free carrier valley relaxation, we predict much longer valley lifetimefor free carriers, which is important for future efforts in developing spintronic/valleytronicdevices based on TMDCs.KWK acknowledges the support from SRC/NRI SWAN.8 [email protected] † [email protected] ‡ [email protected][1] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V.Khotkevich, S. V. Morozov, and A.K. Geim, Proc. Natl. Acad. Sci. U.S.A. , 10451 (2005).[2] A. K. Geim, Science , 1530 (2009).[3] S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui , J. A. Gupta, H. R. Guti´errez, T. F. Heinz, S.S. Hong , J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa , R.D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi , M. G. Spencer, M. Terrones, W.Windl, and J. E. Goldberger, ACS Nano , 2898 (2013).[4] D. Jariwala, V. K. Sangwan, L. J. Lauhon, T. J. Marks, and M. C. Hersam, ACS Nano ,1102 (2014).[5] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, Nat. Nanotechnol. , 699 (2012).[6] B. Radisavljevic, A. Radenovic, J. Brivio, J. V. Giacometti, and A. Kis, Nat. Nanotechnol. ,147 (2011).[7] L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y.-J. Kim, R.V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A.H. Castro Neto, and K. S. Novoselov, Science , 1311 (2013).[8] R. S. Sundaram , M. Engel, A. Lombardo, R. Krupke, A. C. Ferrari, Ph. Avouris, and M.Steiner, Nano Lett. , 1416 (2013).[9] F. K. Perkins, A. L. Friedman, E. Cobas, P. M. Campbell, G. G. Jernigan, and B. T. Jonker,Nano Lett. , 668 (2013).[10] K. He, C. Poole, K. F. Mak, and J. Shan, Nano Lett. , 2931 (2013).[11] A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli, and F. Wang, NanoLett. , 1271 (2010).[12] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev. Lett. , 136805 (2010).[13] D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, Phys. Rev. Lett. , 196802 (2012).[14] Y. Yu, S. Huang, Y. Li , S. N. Steinmann, W. Yang , and L. Cao, Nano Lett., , 553 (2014).
15] K. F. Mak, K. He, J. Shan, and T. F. Heinz, Nat. Nanotechnol. , 494 (2012).[16] H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Nat. Nanotechnol. , 490 (2012).[17] T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu and J.Feng, Nat. Commun. , 887 (2012).[18] K. Ko´smider and J. Fern´andez-Rossier, Phys. Rev. B , 075451 (2013).[19] H. R. Guti´errez, N. Perea-L´opez, A. L. El´ıas, A. Berkdemir, B. Wang, R. Lv, F. L´opez-Ur´ıas,V. H. Crespi, H. Terrones, and M. Terrones, Nano Lett. , 3447 (2013).[20] H. Zeng, G.-B. Liu, J. Dai, Y. Yan, B. Zhu, R. He, L. Xie, S. Xu, X. Chen, W. Yao, and X.Cui, Sci. Rep. , 1608 (2013).[21] Z. Y. Zhu, Y, C. Cheng and U. Schwingenschlogl, Phys. Rev. B , 153402 (2011).[22] W. Zhao, Z. Ghorannevis, L. Chu, M. Toh, C. Kloc, P.-H. Tan, and G. Eda, ACS Nano ,791 (2013).[23] A. M. Jones, H. Yu, N. J. Ghimire, S. Wu, G. Aivazian, J. S. Ross, B. Zhao, J. Yan, D. G.Mandrus, D. Xiao, W. Yao, and X. Xu, Nat. Nanotechnol. , 634 (2013).[24] L. Sun, J. Yan, D. Zhan, L. Liu, H. Hu, H. Li, B. K. Tay, J.-L. Kuo, C.-C. Huang, D. W.Hewak, P. S. Lee, and Z. X. Shen, Phys. Rev. Lett. , 126801 (2013).[25] H. Shi, R. Yan, S. Bertolazzi, G. Gao, A. Kis, D. Jena, H. Xing and L. Huang, ACS Nano ,1072 (2013).[26] S. Sim, J. Park, J. Song, C. In, Y. Lee, H. Kim, and H. Choi, Phys. Rev. B , 075434 (2013).[27] C. Mai, A. Barrette, Y. Yu, Y. G. Semenov, K. W. Kim, L. Cao, and K. Gundogdu. NanoLett., , 202 (2014).[28] Q. Wang, S. Ge, X. Li, J. Qiu, Y. Ji, J. Feng and D. Sun. ACS Nano , 11087 (2013).[29] Y. Yu, S. Hu, L. Su, L. Huang, Y. Liu, Z. Jin, A. A. Purezky, D. B. Geohegan, K. W. Kim,Y. Zhang, L. Cao, arXiv:1403.6181 (2014)[30] Y. -H. Lee, L. L. Yu. H. Wang, W. J. Fang, X. Ling, Y. M. Shi, C. T. Lin, J. K. Huang, M.T. Chang, C. S. Chang, M. Dresselhaus, T. Palacios, L. J. Li, and J. Kong, Nano Lett. ,1852 (2013).[31] S. Najmaei, Z. Liu, W. Zhou, X. Zou, G. Shi, S. Lei, B. I. Yakobson, J.-C. Idrobo, and P. M.Ajayan and J. Lou, Nat. Mater. , 754 (2013).[32] A. M. van der Zande, P. Y. Huang, D. A. Chenet, T. C. Berkelbach, Y.M. You, G.-H. Lee, T.F. Heinz, D. R. Reichman, D. A. Muller and J. C. Hone, Nat. Mater. , 554 (2013).
33] G. Sallen, L. Bouet, X. Marie, G. Wang, C. R. Zhu, W. P. Han, Y. Lu, H. Tan, T. Amand,B. L. Liu, and B. Urbaszek, Phys. Rev. B , 081301, (2012).[34] Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery,and N. Peyghambarian, Phys. Rev. Lett. , 2446 (1986).[35] D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. V. Stryland,Adv. Opt. Photonics , 60 (2010).[36] A. Kuc, N. Zibouche, and T. Heine. Phys. Rev. B , 245213 (2011).[37] N. Peyghambarian, H. M. Gibbs, J. L. Jewell, A. Antonetti, A. Migus, D. Hulin, and A.Mysyrowicz Phys. Rev. Lett. , 2433 (1984).[38] B. Zhu, X. Chen, X. Cui, arXiv:1403.5108v1 (2014).[39] H. Lu, W. Yao, D. Xiao, S. Shen, Phys. Rev. Let. , 016806, (2013).[40] P. Giannozzi et al., J. Phys.: Condens. Matter , 395502 (2009).[41] A. Klein, S. Tiefenbacher, V. Eyert, C. Pettenkofer, and W. Jaegermann, Phys. Rev. B ,205416 (2001).[42] H.-P. Komsa, and A.V. Krasheninnikov, Phys. Rev. B , 085318 (2013).[43] X. Li, J., T. Mullen, Z. Jin, K.M. Borysenko, M. Buongiorno Nardelli, and K.W. Kim, Phys.Rev. B , 115418 (2013)[44] Y. Song, and H. Dery. Phys. Rev. Let. ,026601, (2013).[45] K. C. Hall, K. G¨undogdu, E. Altunkaya, W. H. Lau, M. E. Flatt´e, T. F. Boggess, J. J. Zinck,W. B. Barvosa-Carter, and S. L. Skeith, Phys. Rev. B , 115311 (2003).[46] K. Zerrouati, F. Fabre, G. Bacquet, J. Bandet, J. Frandon, G. Lampel, and D. Paget, Phys.Rev. B , 1334 (1988).[47] M. I. D’yakonov and V. I. Perel, Zh. Eksp. Teor. Fiz. 60, 1954 (1971) [Sov. Phys. JETP ,1053].[48] Y. Yafet, Phys. Rev. , 478 (1952).[49] R. J. Elliott, Phys. Rev. , 266 (1954). K’a b cd e f
Energy (eV)
Ground state bleach
Stimulated emissionPhotoinduced abs.