Hexagonal boron nitride is an indirect bandgap semiconductor

Transcription

1 Hexagonal boron nitride is an indirect bandgap semiconductor G. Cassabois, P. Valvin, B. Gil Laboratoire Charles Coulomb, UMR 5221 CNRS-Université de Montpellier, F-34095, Montpellier, France A. EFFECTIVE TEMPERATURE OF THE THERMALIZED EXCITON GAS We consider here the problem of phonon-assisted recombination of an indirect exciton. In the case of hbn, the fundamental indirect exciton of lowest energy is located at the point of the reciprocal space. Neglecting the wavevector of light in the recombination process, momentum conservation imposes for the phonon wavevector, later labelled since is in the middle of the Brillouin zone, around the so-called conservation then writes : points. Energy where is the phonon energy at, and is the emitted photon energy. In the case of a gas of excitons thermalized along the exciton dispersion curve, we have to consider exciton states at higher energy than so that energy conservation now reads : where and are the in-plane and out-of-plane exciton masses, is the in-plane phonon group velocity around points, and the phonon effective mass characterizing the decrease of the phonon energy along the z-axis. As far as the in-plane motion is concerned ( ), the quadratic energy term becomes dominant over the linear one for wavevectors such that the phonon energy is much larger than, which is in the 5-10 mev range, i.e. more than one order of magnitude lower than the extension of the high-energy tails where exciton thermalization is monitored. In that case, the relation between the excess kinetic energy of the exciton and the emitted photon energy is given by : NATURE PHOTONICS 1

2 so that a Boltzmann distribution of excitons at the lattice temperature transforms in a signal intensity following a Boltzamnn law of effective temperature such that. In the opposite case of the out-of-plane motion ( ), one obtains : where is the reduced mass given by. Consequently, the relation between the excess kinetic energy of the exciton and the emitted photon energy now becomes : so that the PL signal intensity follows a Boltzamnn law of effective temperature such that : The usual situation in semiconductor materials corresponds to a negligible k- dependence of the phonon energy compared to the excitonic one, and leads again to. In hbn, the strong anistropy of the band structure with flat bands along the z-axis ( ) results in a huge increase of the density of states due to the large effective mass so that exciton thermalization is mostly monitored along the z-axis. The 1.85 value of the ratio shows that the exciton and phonon dispersions have approximately the same flatness with. B. SYMMETRY SELECTION RULES I. Hexagonal boron nitride symmetries - Group theory analysis Ab initio band structure calculations ([1-5] in the Supplementary information, corresponding to References [4-8] of the paper) predict that the conduction band minimum sits at the M point of the Brillouin zone, while the valence band maximum occurs near the K point. More precisely, it is located at T 1, lying in the ( K) direction, with a T 1 K distance of the order of few percents of the K one [5]. Because of such a small deviation, we did not mention it in the core of the paper where we preferred to write «around the K point» for the sake of clarity. However, in order to perform a proper group theory analysis, we have considered this particularity in the discussion below. 2 NATURE PHOTONICS

3 SUPPLEMENTARY INFORMATION Consequently, we present the analysis of the symmetry group at the following different points of the Brillouin zone : (1) the zone center, (2) M point, (3) T 1 point, and (4) K point. 1. The D 6h group at zone center is a 24-fold non abelian group with 10 irreducible representations. A cartesian basis that can be defined at is the following one : (x,y,z) with z colinear to the 6-fold symmetry axis. 2. At the M edge of the Brillouin zone (corresponding to the conduction band minimum), the group of vector k is D 2h, an Abelian group with 8 unidimensional irreducible representations, four of them being even with respect to the space inversion symmetry. The salient feature of this point group is the existence of three orthogonal two-fold rotation axes passing though the inversion point ; orthogonal to each other. Each of these two-fold axes is parallel to two symmetry planes and orthogonal to the remaing one. The cartesian basis that can be defined at M is the following one : ( ~ x, ~ y, ~ z ) where and ~ z and ~ y are colinear to ~ the -M and -A directions, respectively, and are chosen as basis vectors of the B1u and ~ B2u symmetries respectively, ~ x is orthogonal to the ( ~ z, ~ y ) plane and chosen as basis vector of the B 3u symmetry. The products ~ x ~ x, ~ y ~ y, and ~ zz are transforming like ~ A g ; products ~ x ~ y, ~ x ~ z, and ~ y ~ z are transforming like ~ B1g, respectively. ~ B2g, ~ B3g, Fig.1 (Supplementary information) : Symmetry elements of the D 2h group : planes (blue and black), two-fold axes (blue arrows) ; the inversion symmetry is located at the intersection of the symmetry planes. NATURE PHOTONICS 3

4 The compatibility relations between D 6h and D 2h write here : A 1g ~ A g A 2u ~ B 2u E 1u ~ B 1u + ~ B 3u E 1g ~ B 1g + ~ B 3g E 2g ~ A g + ~ B 2g Obtained from ( x, y, z ) ( ~ y, ~ x, ~ y ) ( xz, yz ) ( ~ z ~ y, ~ x ~ y ) (x 2 y 2, xy ) (~ z 2 ~ x 2, ~ z ~ x ) The symmetries of the acoustic phonons transform as follows : A 2u +E 1u ~ B 2u + ~ B 1u + ~ B 3u For the Raman active phonons we obtain : E 2g ~ A g + ~ B 2g and we get : A 2u ~ B 2u and E 1u ~ B 1u + ~ B 3u for the Infra-Red active modes. The ZA(M) and ZO(M) phonons transform like ~ B 2u. The LA(M) and LO(M) modes which produce a polarization parallel to the propagation direction transform like ~ B 1u. The TA(M) and TO(M) modes transform like ~ B 3u. 3. At the T 1 ( or ) point of the Brillouin zone (close to the K point and corresponding to the valence band maximum), the symmetry is reduced to C 2v, a four-fold Abelian group. Again we define a three-dimensional cartesian basis that is now noted as ( ^x, ^y, ^z ). The two-fold rotation axis is colinear to ^z which is chosen to transform like ^A 1, while vectors ^x and ^y generate representations labeled ^B 1 and ^B 2 respectively. The products ^x ^x, ^y ^y, and ^z ^z also transform like ^A 1 ; products ^x ^y, ^x ^z, and ^y ^z are transforming like ^A 2, ^B 1 and ^B 2 respectively. 4 NATURE PHOTONICS

5 SUPPLEMENTARY INFORMATION Fig.2 (Supplementary information) : Symmetry elements of the C 2v group : the two-fold rotation axis is colinear to ^z which is chosen to transform like ^A 1, the symmetry planes (blue) are parallel to it and orthogonal to each other. The compatibility relations between D 6h and C 2v write here : A 1g ^A 1 A 2u ^B 2 E 1u ^B 1 +^A 1 E 1g ^A 2 +^B 2 E 2g ^A 1 +^B 1 Obtained from ( x, y, z ) ( ^z, ^x,^y ) ( xz, yz ) ( ^z ^y, ^x ^y ) (x 2 y 2, xy ) ( ~ z 2 ~ x 2, ~ z ~ x) The symmetries of the acoustic phonons transform as follows : A 2u +E 1u ^B 2 +^B 1 +^A 1 For the Raman active phonons we obtain : E 2g ^A 1 +^B 1 and we get : A 2u ^B 2 and E 1u ^B 1 +^A 1 NATURE PHOTONICS 5

6 for the Infra-Red active modes. We remark that Raman modes E 2g and Infra-Red modes E 1u have similar symmetry. Then their dispersion relations are allowed to anticross in the -K direction away from zone center. This is effectively reported in Kern, Kresse and Hafner PRB 59,8551, (1999) and also in Michel and Verneck PRB 83,115328, (2011) as: «Considering the high-frequency optical branches which correspond to in-plane displacements we notice in Fig. 3(b) that in addition to the splitting of the E 1u mode at, the E 2g2 mode, which is degenerate at, evolves into two branches away from to M (K); the lower branch coincides almost with the lowest E 1u branch while the higher E 2g2 branch crosses the lower E 1u branch and shows a pronounced over- bending.» The ZA(T 1) and ZO(T 1) phonons transform like ^B 2. The LA(T 1) and LO(T 1) modes transform like ^A 1. The TA(T 1) and TO(T 1) modes transform like ^B 1. Fig.3 (Supplementary information) : Summary of the symmetries with respect to the phonon dispersion relations (from [6], corresponding to Ref.[30] of the paper). 4. At the K point of the Brillouin zone the symmetry is D 3h with a three fold axis in the -A direction. This corresponds to the A 2'' symmetry while vectors that generate the 6 NATURE PHOTONICS

7 SUPPLEMENTARY INFORMATION horizontal plane orthogonal to it transform like E'. The cascading relationship between the symmetries of D 6h with those of D 3h and at the end those of C 2v of interest for us here is : A 2u A 2 ' ' ^B 2, E 1u E ' ^A 1 +^B 1 and E 2g E ' ^A 1 +^B 1 The Raman and Infra-Red modes have similar symmetry. There is an LA-TA splitting at K which splits the E' degeneracy. II. Selection rules for phonon-assisted transitions 1. One-photon transitions We consider the phonon-assisted transition from the top-most valence band at T 1 (symmetry ) towards the bottom of the conduction band at M (symmetry ). The transition probability for a one-photon phonon-assisted process is proportional to: where is the momentum-conservative electric dipole operator, represents the interaction operator with the phonon field that transfers some momentum via the exciton-phonon interaction, and represents all intermediary virtual or real state that satisfies k- conservative selection rule and is even in real space as the valence band and dipole operator are both odd in real space. In our experimental configuration, the Poynting vector of the photon that impacts the valence electron is colinear to the z axis of the crystal ( symmetry), the symmetry of the electric field is in the most general case. According to the rule of symmetry, and as given by the multiplication tables, matrix element does not vanish when the symmetry of is even in real space and can be written. We now consider the different possible symmetries for the exciton-phonon interaction : namely (ZO and ZA), (TA and TO) and (LA and LO) and again apply group theory following the method of Bassani and Hassan [7]. The symmetry of the exciton-phonon interaction operator is the product of the symmetry of the phonon under consideration with the symmetry of the excitonic envelope function which has to be even in real space so that does not vanish for parity reasons, which NATURE PHOTONICS 7

8 restricts exciton envelope functions to those with symmetry. Then one finds that matrix element involving operator vanishes when the phonon symmetry is, that is to say emission (or absorption) of one phonon ZA or ZO is forbidden in the context of the dipole interaction. Note : These selection rules can be changed if performing the experiment in the context of another experimental configuration. Let us choose the Poynting vector of the photon perpendicular to the z axis of the crystal and let us choose it to transform like for carrying on the demonstration of selection rules for the phonon assisted process within the context of the group theory analysis. There are two possible symmetries for the electric field of the photon : (then the electric field of the photon is colinear with the z axis of the crystal) or (the electric field of the photon is orthogonal to the ( ) plane). In the first case, non vanishing matrix elements occur for a transition from a valence band state of symmetry (respectively ) to an intermediate state of symmetry (respectively ). Regarding selection rules for, non-vanishing elements are and, which indicates that emission or absorption of one ZA or ZO phonon is allowed in the context of the dipole interaction, with this polarisation of the photon. Therefore, the optical feature recorded in that case is obviously different from the one recorded in the previous configuration [8,9]. Repeating this experiment, but after rotating the polarisation of the photon so that it becomes perpendicular to the z axis of the crystal and so that it transforms like, the selection rules of the first experimentally examined configuration are recovered, that is to say the emission or absorption of one ZA or ZO phonon is forbidden in the context of the dipole interaction, with this polarisation of the photon. 2. Two-photon transitions The transition probability is in that case proportional to: 8 NATURE PHOTONICS

9 SUPPLEMENTARY INFORMATION where and represent here two sets of intermediary virtual or real state that satisfy k-conservative selection rule and are respectively even and odd in real space (as the valence band and dipole operator are both odd in real space). It is interesting to outline the impact of the odd symmetry of states in real space which, compared to the case of the one-photon process, that requires to switch the parity of operator and thus the parity of the excitonic envelope function so that does not vanish. This is not really a surprise as one-photon transition probe s-states whilst two-photon transitions probe p-states. Excitonic envelope function are a priori possible with ( state, labelled in the core of the paper), or and ( states, labelled in the core of the paper ) symmetries. We still consider here our experimental configuration where the Poynting vector of the photon that impacts the valence electron is colinear to the z axis of the crystal ( symmetry). After carefully browsing the different situations we find that the symmetry of is to be or. After carefully browsing all the possible situations, we find that two-photon assisted transition towards the state of the indirect exciton ( symmetry of the excitonic envelope function) is possible via assistance of either a ZA or a ZO phonon whilst two-photon assisted transitions towards the state of the indirect exciton are possible via assistance of either a TA or a TO phonon ( symmetry of the excitonic envelope function) and via assistance of either a LA or a LO phonon ( symmetry of the excitonic envelope function). C. Photoluminescence excitation spectroscopy (PLE) in hbn : detection energy & line profile I. Detection energy In Fig.2(a) in the core of the manuscript, we compare the luminescence band at 5.89 ev with the PLE spectrum from Ref.[32] recorded at 5.78 ev, whereas, strictly speaking, one would need the PLE spectrum recorded at 5.89 ev. In our two-photon excitation spectroscopy, we have observed that the emission bands centered at 5.78 ev and 5.89 ev always present the same intensity ratio when changing the laser energy. Consequently, the two-photon PLE spectra recorded at 5.78 ev and 5.89 ev are identical. NATURE PHOTONICS 9

10 The one-photon PLE spectrum from Ref.[32] was probably recorded at 5.78 ev rather than at 5.89 ev because the emission intensity is roughly ten times higher at 5.78 ev. Even in that case, the authors had to use some time-gate techniques to get a reasonable signal-to-noise ratio. Although the one-photon PLE spectrum recorded at 5.89 ev is not available in Ref.[32], we can guess that it is identical to the published spectrum recorded at 5.78 ev, just as for twophoton excitation. As a matter of fact, in our one-photon experiments, we have systematically observed that the intensity ratio between the 5.78 ev and 5.89 ev bands was constant on changing the excitation power, or temperature. Indeed, in our interpretation, such an intensity ratio is an intrinsic property related to the different coupling efficiencies to optical phonons and acoustic phonons. It is thus valid to use the PLE spectrum from Ref.[32] in Fig.2(a). II. Line profile We provide here the interpretation for the existence of peaks in the absorption profile of hbn. In the framework of the pioneering study of Elliott [Phys. Rev. 108, 1384 (1957)] on the absorption profile in an indirect bandgap semiconductor, the indirect transitions to exciton states should give rise to a continuous absorption, instead of the well-defined zero-phonon line in a direct bandgap semiconductor. The striking observation of peaks in the excitonic optical response in hbn arises from the q-dependence of the exciton-phonon matrix elements, which is usually neglected, and which becomes essential in the phonon-assisted optical processes in hbn because the involved phonons are in the middle of the Brillouin zone with a finite group velocity. Taking this q-dependence into account leads to the interpretation of many features of the optical response in hbn: asymmetric profile with a steep increase, order of magnitude of the linewidth, and dependence with the phonon group velocity. The reference paper by Elliott addresses the Intensity of optical absorption by excitons in both direct and indirect bandgap materials in a very general framework. However, in order to analytically derive some expressions, several approximations are performed: 10 NATURE PHOTONICS

11 SUPPLEMENTARY INFORMATION -band edge vicinity: p 1388, left column,...for consideration of the band edge we are concerned only with states of K close to this value [=conduction band minimum K 0 ]. -exciton binding: p 1388, left column, next sentence For weak electron-hole interaction such states are built predominantly from hole and electron states close to the extrema. -matrix elements: p 1388, left column, the electron-phonon and hole-phonon matrix elements are taken constant (Eq.(4.4)&(4.5)). Under these assumptions, the absorption profile essentially reflects the joint density of states, which, for a single exciton band in 3D, is [Eq.(4.8) of Elliott]: i.e. the usual square root dependence with energy. As a matter of fact, the description of the absorption profile far from the band edge is beyond the approximation of constant matrix elements. The exciton-phonon interaction in semiconductor compounds has been the subject of many studies, for instance: -Segall and Mahan, Phys. Rev. 171, 935 (1968), -Rudin, Reinecke and Segall, Phys. Rev. B 42, (1990). The exciton-phonon matrix elements depend on (i) the phonon nature and the subsequent coupling (Fröhlich interaction for optical phonons, deformation-potential and piezo-electric coupling for acoustic phonons), and on (ii) a form factor accounting for the overlap of the phonon wavefunction with the initial and final exciton states. This form factor reads: where is the phonon wave-vector, and and are the initial and final exciton states. Its analytical expression is given in terms of hypergeometric functions in [Trallero-Giner, Cantarero, and Cardona, PRB 40, 4030 (1989)], and when the initial and final states are 1s-excitons, we have : NATURE PHOTONICS 11

12 where is the exciton Bohr radius. We thus see that the form factor has a high-q cut-off given by the inverse of the Bohr radius, meaning that for high-energy phonons, their wave-function is varying very rapidly compared to the electron-hole relative motion, on the Bohr radius scale. Adopting the notation of Section A of our Supplementary Information, at the band edge, i.e.. As far as the exciton Bohr radius in hbn is concerned, we take a Bohr radius of the order of 3. This value is obtained by taking our 130 mev-estimation of the exciton binding energy, the dielectric constants of Geick et al., Phys. Rev. 146, 543 (1966) [ and ] and the effective masses of Xu and Ching, PRB 44, 7787 (1991) [,,, and ]. As a consequence, the product is of the order of 6.8 for, i.e. already larger than 4 in the denominator of the form factor. This means that the wave-numbers of phonons assisting the excitonic recombination in hbn are sitting above the Bohr radius cutoff so that the high-q decrease of the form factor will strongly modify the absorption profile calculated by Elliott, leading to the appearance of a peak in the absorption spectrum. Fig.4 (Supplementary information) : Spectrum of absorption assisted by a given phonon type in hbn calculated (dashed line) within the framework of the Elliott model, and (solid line) by taking into account the q-dependence of the exciton-phonon form factor in hbn. 12 NATURE PHOTONICS

13 SUPPLEMENTARY INFORMATION In Fig.4 (Supplementary information), we compare the absorption profile calculated by Elliott (dashed line) with the absorption profile taking into account the q-dependence of the form factor (solid line). More precisely, we multiply the absorption profile calculated by Elliott by the square of the form factor discussed above, that is: where is the absorbed photon energy, the band edge energy, and the phonon group velocity, with the value of LA phonons in hbn in Fig.4 [6]. Such an estimation does not take into account the detailed expression of the matrix element, in particular the coupling term of either Fröhlich, deformation-potential or piezoelectric type. Therefore, it is only a first approximation, nevertheless we see that we fairly reproduce our data. In particular we do not only account for the asymmetric profile with a steep increase but also for the order of magnitude of the linewidth (~15 mev), as well as for its dependence with the group velocity (pointed out in the core of the text). As far as other indirect semiconductors are concerned, the fundamental originality of hbn arises from the fact that, in hbn, the conduction band AND the valence one are not at the zone center. In diamond, silicon, germanium, GaP etc..., the valence band is at the zone center and the conduction band has its minimum around high symmetry points of the Brillouin zone. This means that the relevant phonons for indirect recombination are located around the same high symmetry points. Consequently, their group velocity is zero and the phonon dispersion relation is quadratic in a first order approximation. In that case, the square of the form factor would approximately scale as: where is the exciton-phonon reduced mass around arising from the quadratic dispersions of the phonons and excitons [Elliott, Phys. Rev. 108, 1384 (1957)]. We illustrate the impact of the form factor in a standard indirect bandgap semiconductor, by taking the example of diamond, for which we have the following NATURE PHOTONICS 13

14 parameters:,, [in Physics and Applications of CVD Diamond, edited by Koizumi, Nebel, and Nesladek, John Wiley & Sons (2008)], and (under the usual assumption that the phonon dispersion is negligible compared to the excitonic one) with the effective masses given in the Ioffe online-handbook at We obtain the absorption spectrum displayed below in Fig.5 (Supplementary information), where no sharp peak is observable. We instead observe a maximum around 5.6 ev and a smooth decrease at high energy, in excellent agreement with the PLE reported by Dean and Male [J. Phys. Chem. Solids 25, 1369 (1964)]. Fig.5 (Supplementary information) : Spectrum of absorption assisted by a given phonon type in diamond calculated (dashed line) within the framework of the Elliott model, and (solid line) by taking into account the q-dependence of the exciton-phonon form factor in diamond. References [1] Xu, Y.-N. & Ching, W. Y. Calculation of ground-state and optical properties of boron nitrides in the hexagonal, cubic, and wurtzite structures. Phys. Rev. B 44, (1991). [2] Furthmüller, J., Hafner, J. & Kresse, G. Ab initio calculation of the structural and electronic properties of carbon and boron nitride using ultrasoft pseudopotentials. Phys. Rev. B 50, (1994). [3] Blase, X., Rubio, A., Louie, S. G. & Cohen, M. L. Quasiparticle band structure of bulk hexagonal boron nitride and related systems. Phys. Rev. B 51, (1995). 14 NATURE PHOTONICS

15 SUPPLEMENTARY INFORMATION [4] Arnaud, B., Lebègue, S., Rabiller, P. & Alouani, M. Huge Excitonic Effects in Layered Hexagonal Boron Nitride. Phys. Rev. Lett. 96, (2006). [5] Gao, S.-P. Crystal structures and band gap characters of h-bn polytypes predicted by the dispersion corrected DFT and GW method. Solid State Communications 152, (2012). [6] Serrano, J. et al. Vibrational Properties of Hexagonal Boron Nitride: Inelastic X-Ray Scattering and Ab Initio Calculations. Phys. Rev. Lett. 98, (2007). [7] Bassani, F. & Hassan, A.R. Analysis of indirect two-photon interband transitions and of direct three photon transitions in semiconductors, Il nuovo cimento 7, 313 (1972). [8] Watanabe, K. & Taniguchi, T. Jahn-Teller effect on exciton states in hexagonal boron nitride single crystal. Phys. Rev. B 79, (2009). [9] Majety, S. et al. Band-edge transitions in hexagonal boron nitride epilayers. Applied Physics Letters 101, (2012). NATURE PHOTONICS 15