Bound-state third-order optical nonlinearities of embedded germanium nanocrystals
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug Bound-state third-order optical nonlinearities of embedded germanium nanocrystals
Hasan Yıldırım ∗ Department of Physics, Bilkent University, Ankara 06800, Turkey andNanoscience Laboratory Department of Physics, University of Trento Via Sommarive, 14 38100 Trento, Italy
Ceyhun Bulutay † Department of Physics, Bilkent University, Ankara 06800, Turkey (Dated: November 19, 2018)Embedded germanium nanocrystals (NCs) in a silica host matrix are theoretically analyzed toidentify their third-order bound-state nonlinearities. A rigorous atomistic pseudopotential approachis used for determining the electronic structure and the nonlinear optical susceptibilities. Thisstudy characterizing the two-photon absorption, nonlinear refractive index, and optical switchingparameters reveals the full wavelength dependence from static up to the ultraviolet spectrum andthe size dependence up to a diameter of 3.5 nm. Similar to Si NCs, the intensity-dependent refractiveindex increases with decreasing NC diameter. On the other hand, Ge NCs possess about an orderof magnitude smaller nonlinear susceptibility compared to Si NCs of the same size. It is observedthat the two-photon absorption threshold extends beyond the half band-gap value. This enablesnonlinear refractive index tunability over a much wider wavelength range free from two-photonabsorption.
PACS numbers: 42.65.-k, 42.65.Ky, 78.67.Bf
I. INTRODUCTION
Semiconductor nanocrystals (NCs) benefit from the accumulated knowledge in semiconductor physics and thematurity of the semiconductor industry as well as the new opportunities provided by the nanoscience, hence theyoffer unique optical properties. In particular, Si and Ge NCs attract an increasing attention because of their low-cost and microelectronic-compatible photonic applications ranging from light emitting diodes and lasers to solar cellsand other photonic devices.
Even though Si and Ge are both Group IV elements, there are a number of notabledifferences between them such as the band edge effective mass of the carriers are smaller for Ge, whereas the dielectricconstant is larger which results in bulk exciton radius about five times larger for Ge compared to Si. As a result, theconfinement effects will be felt starting from larger sizes. Moreover, Ge is a weakly indirect band-gap semiconductorwith the direct to indirect band-gap ratio being 1.2, in contrast to 2.9 in Si. Furthermore, the narrower band-gap of thebulk as well as the NC Ge can be preferred in certain applications to harvest the near infrared part of the spectrum. Finally, the proximity of direct-gap optical transitions in bulk Ge to the fiber optic communication wavelength of1.5 µ m range is particularly important. This has recently stimulated extensive interest; notably tensile-strained Gephotodetectors on Si platform has been demonstrated and a tensile-strained Ge-based laser is proposed. If the lattercan also be experimentally demonstrated, this will mark the dawn of the germanium photonics era.For all-optical switching and sensor protection applications as well as in the absorption of the subband-gap light forthe possible solar cell applications, the nonlinear refractive index coefficient also known as the optical Kerr index, n ,and two-photon absorption coefficient, β , are the two crucial third-order optical nonlinearities which play an importantrole. Recent experiments show that Ge NCs have enhanced third-order optical nonlinearities. However,differences in the sample preparation methods, the choice of the matrix, the excitation laser wavelength and the sizedistribution of the NCs contribute to the wide variance within these results as shown in Table I. A number of theseinvestigations have observed two characteristic temporal nonlinear response contributions, distinguished as fast andslow, but there is no quantitative agreement among themselves.
Regarding the origin of the nonlinear response,some of these reports have stressed the role of the excited-state contribution produced by the linear absorption, also the involvement of the trap/defect states was addressed.
Undoubtedly, more experiments are needed to reachto a coherent understanding. On the other hand, to the best of our knowledge, there is no theoretical study identifyingthe wavelength and size dependences of n and β in Ge NCs. Therefore, a rigorous theoretical work may guide andinspire further experimental studies on the foregoing investigations. Moreover, it would help in assessing the potentialrole of Ge NCs, if any, in nonlinear device applications mentioned above.In this paper, our aim is to present such a theoretical account concerning n and β in Ge NCs revealing theirsize scaling and wavelength dependence from static up to ultraviolet region together with a comparison with Si NCs.Furthermore, we deal with NCs embedded in a wide band-gap matrix representing silica which is the most commonchoice in the actual structures as can be observed in Table I. Since we do not consider any interface defects, strainand thermal effects or the compounding contribution of the excited carriers through linear absorption to the nonlinearprocesses, our results may serve as a benchmark of the ideal Ge NC bound-state ultrafast third-order nonlinearities.In Sec. II we describe the theoretical approach for the electronic structure and the expressions for nonlinear opticalquantities. The results and discussions are provided in Sec. III followed by a brief conclusion. TABLE I: The summary of existing experimental studies on the third-order nonlinear optical parameters, n and β of Ge NCs.The sample diameter, D , laser excitation wavelength, λ exc , host matrix and sample preparation information are provided.Unspecified data is left as blank.Reference D (nm) λ exc (nm) n (cm /GW) β (cm/GW) Matrix PreparationRef.10 3 800 2.7-6.9 × − silica ion implantationRef.11 6 ± × − ± × − × − ± II. THEORY
The electronic structure of nanoclusters are accurately obtained routinely by means of density functional theory-based ab initio techniques. However, a several nanometer-diameter NC system including the embedding host matrixcontains thousands of atoms. This large number currently precludes the use of such ab initio pseudopotential planewave techniques. An alternative route is based on the use of semi-empirical pseudopotential description of the atomicenvironment in conjunction with the linear combination of Bloch bands as the expansion basis. In the case ofembedded Si and Ge NCs, this yields results in well agreement with experimental data for the interband, intrabandoptical absorption and the Auger recombination and carrier multiplication. All of these are governed by quantumprocesses taking place over several electronvolt energy range. As a matter of fact, this feature forms an essentialsupport for applying the approach to the characterization of the third-order nonlinear susceptibilities up to a photonenergy of 4 eV. We refer to our previous work for further details on the electronic structure. The correspondingelectronic structure for embedded Ge NCs of different sizes are shown in Fig. 1. The evolution of the effective gap, E G towards the bulk value (as marked by the gray band) can be observed as the NC diameter increases which is thewell-known quantum size effect.In this work, the electromagnetic interaction Hamiltonian is taken as − e r · E , in other words, the length gaugeis used. The third-order optical nonlinearity expressions based on the length gauge have proved to be successful inatomic-like systems but not in bulk systems because the position operator introduces certain difficulties which canactually be overcome. Nevertheless, for bulk systems the velocity gauge has been preferred which on the other handpossesses unphysical divergent terms at zero frequency (not present in the length gauge) that poses severe obstaclesin evaluating the nonlinear optical expressions. Hence, we have preferred the length gauge for the evaluation of thethird-order optical expressions due to resemblance of the band structure of NCs to atomic-like systems (cf. Fig. 1).The susceptibility expression is obtained through perturbation solution of the density matrix equation of motion. Throughout this work, we distinguish the quantities which refer to unity volume filling factor by an overbar, where f v = V NC /V SC is the volume filling factor of the NC in the matrix, V NC and V SC are the volumes of the NC andsupercell, respectively. The final expression is given by χ (3) dcba ( − ω ; ω γ , ω β , ω α ) ≡ χ (3) dcba ( − ω ; ω γ , ω β , ω α ) f v , = e V NC ~ S X lmnp r dmn ω nm − ω " r cnl ω lm − ω r blp r apm f mp ω pm − ω − r alp r bpm f pl ω lp − ω ! − r cpm ω np − ω r bnl r alp f pl ω lp − ω − r anl r blp f ln ω nl − ω ! , (1)where the subscripts { a, b, c, d } refer to Cartesian indices, ω ≡ ω γ + ω β + ω α , ω ≡ ω β + ω α , ω ≡ ω α are the inputfrequencies, r nm is the matrix element of the position operator between the states n and m , ~ ω nm is the differencebetween energies of these states, S is the symmetrization operator, indicating that the following expression should beaveraged over the all possible permutations of the pairs ( c, ω γ ), ( b, ω β ), and ( a, ω α ), and finally f nm ≡ f n − f m where f n is the occupancy of the state n . The r nm is calculated for m = n through r nm = p nm im ω nm where m is the freeelectron mass, and p nm is the momentum matrix element. Hence, after the solution of the electronic structure, the FIG. 1: The energy levels of Ge NCs for different NC sizes. All plots use the same energy reference where the bulk Ge bandgap is marked by the gray band. computational machinery is based on the matrix elements of the standard momentum operator, P , the calculation ofwhich trivially reduces to simple summations.The above susceptibility expression is evaluated without any approximation taking into account all transitionswithin the 7 eV range. This enables a converged spectrum up to the ultraviolet spectrum. In the case of relativelylarge NCs the number of states falling in this range becomes excessive making the computation quite demanding.For instance, for the 3 nm NC the number of valence and conduction states (without the spin degeneracy) become3054 and 3314, respectively. As another technical detail, the perfect C v symmetry of the spherical NCs resultsin an energy spectrum with a large number of degenerate states. However, this causes numerical problems in thecomputation of the susceptibility expression given in Eq. (1). This high symmetry problem can be practically removedby introducing two widely separated vacancy sites deep inside the matrix. Their sole effect is to introduce a splittingof the degenerate states by less than 1 meV.When solids are excited with light having a frequency below the band-gap at intensities high enough, third-orderchanges in the refractive index and the absorption are observed due to the virtual excitations of the bound charges.Accounting for these effects, the refractive index and the absorption become, respectively, n = n + n I , α = α + βI , where n is the linear refractive index, α is the linear absorption coefficient, and I is the intensity of the light. n is proportional to Re (cid:8) χ (3) (cid:9) , and is given by n ( ω ) = Re (cid:8) χ (3) ( − ω ; ω, − ω, ω ) (cid:9) n ǫ c , (2)where c is the speed of light. Similarly, β is given by β ( ω ) = ω Im (cid:8) χ (3) ( − ω ; ω, − ω, ω ) (cid:9) n ǫ c , (3)where ω is the angular frequency of the light. Note that Eqs. (2) and (3) are valid only in the case of negligibleabsorption. The degenerate two-photon absorption cross section σ (2) ( ω ) is given by σ (2) ( ω ) ≡ σ (2) ( ω ) f v = 8 ~ π e n c X i,f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X m r fm r mi ~ ω mi − ~ ω − i ~ Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ( ~ ω fi − ~ ω ) , (4)where Γ is the inverse of the lifetime; the corresponding full width energy broadening of 100 meV is used throughoutthis work. The sum over the intermediate states, m , requires all interband and intraband transitions. As we havementioned previously we compute such expressions without any approximation by including all states that contributeto the chosen energy window. Finally, σ (2) ( ω ) and β are related to each other through β = 2 ~ ωσ (2) ( ω ).Another important factor is the so-called local field effect (LFE) which arise in composite materials of differentoptical properties; the LFEs lead to a correction factor in the third-order nonlinear optical expressions given by, FIG. 2: Optical Kerr index at unity filling factor, n in Ge NCs as a function of the photon energy for different NC sizes. Thevertical labels in the ordinates apply to both plots in the same row and the horizontal labels in the lower and upper abscissasapply to both plots in the same column. L = (cid:16) ǫ h ǫ NC+2 ǫ h (cid:17) (cid:12)(cid:12)(cid:12) ǫ h ǫ NC+2 ǫ h (cid:12)(cid:12)(cid:12) where ǫ h and ǫ NC are the dielectric functions of the host matrix and the NC, respectively.We fix the local field correction at its static value, since when the correction factor is a function of the wavelength itbrings about unphysical negative absorption regions at high energies. Further discussions of our model are availablein our previous work. III. RESULTS AND DISCUSSION
We have performed extensive computations on Ge NCs with six different diameters, namely, D = 1 .
13, 1.47, 1.71,2.25, 3, and 3.5 nm. For D = 1 .
13, and 3.5 nm sizes which correspond to smallest and largest diameters, we havecalculated the nonlinearities at certain important laser wavelengths. As for the rest, the nonlinearities are computedat all frequencies up to 4 eV. For generality, we quote the unity-filling-factor values denoted by an overbar which cantrivially be converted to any specific realization, however, we should caution that the actual amount of Ge atomsforming the NCs is usually a small fraction of the overall excess Ge atoms most of which disperse in the matrixwithout aggregating into a significant NC. In accounting for the LFE the host matrix is assumed to be silica which isthe most common choice (cf., Table I).Under these conditions, n is plotted in Fig. 2 as a function photon energy both in eV (upper abscissas) and inunits of effective gap, E G (lower abscissas). As expected, below half E G (i.e., in the transparency region), there isa monotonous behavior and above this value the resonances take over. An intriguing observation is about the signof n . In general, bulk semiconductors change the sign of n above their half E G values. However, in Fig. 2 weobserve that in Ge NCs this takes place at even beyond E G ; for the D = 2 .
25 and 3 nm NCs, the emergence of a newresonance is seen to develop which can possibly reproduce this sign change at lower than E G values for the largerNCs. This negative sign of n is known to be caused by both the two-photon absorption and the ac Stark effect. In the size range considered in this work, we believe the former to be more dominant in the negative sign of n . For D = 3 nm NC, and below 1 eV, n is of the order of f v × − cm /GW. This value is much larger than the bulkvalue. When compared to the available Ge NC measurements in Table I, this is in very good agreement with Jie etal. given the discrepancy in the NC size and the host matrix. On the other hand, other measurements are aboutthree orders of magnitude lower for the same quantity.The situation is similar in the case of β which is plotted against the photon energy in Fig. 3. An importantobservation is that the two-photon absorption onset lies further beyond the half band gap value which is possibly amanifestation of the indirect band gap nature of the core medium. We should note that β is nonzero (albeit verysmall) down to static values due to band tailing which is represented as in our previous work on Si NCs throughthe Lorentzian energy broadening parameter, ~ Γ, of 100 meV at full width. For D = 3 nm NC, β has a value FIG. 3: Two-photon absorption coefficient at unity filling factor, β in Ge NCs as a function of the photon energy for differentNC sizes. The vertical labels in the ordinates apply to both plots in the same row and the horizontal labels in the lower andupper abscissas apply to both plots in the same column.FIG. 4: The size-scaling of the real part of the third-order susceptibility evaluated at two different wavelengths, 1550 nm and800 nm, for (a) Si and (b) Ge NCs at unity filling factors. The NCs are embedded in silica matrix. The dashed lines are guideto the eye for indicating the overall scaling trend. f v × cm/GW around 2 eV. This value is very high compared to the corresponding bulk value. When typicalvolume filling fraction is taken into account, our values are again in order of magnitude agreement with Gerung etal. and Jie et al. both of which are for somewhat larger NCs. It should be noted that there is an outstandingdisagreement among the experimental data; for instance, the two most recent experimental data measured at veryclose photon energies differ by five orders of magnitude. This emerging picture about the large discrepancy onthe n and β values calls for further experimental investigations, especially probing the ultrafast response.The comparison of the size-scaling trends of the real part of the third-order susceptibility for Si and Ge NCs areshown in Fig. 4. Two different wavelengths are used, 1550 nm and 800 nm both of which fall below the band gap,hence they do not experience any linear absorption. Again unity volume filling factor values are quoted. It can beobserved that for both Si and Ge NCs, there is a common enhancement trend (dashed lines) as the size is reducedespecially below 2.5 nm. The oscillations for certain diameters is common to both materials, however, they are morepronounced in the Ge NCs. This corroborates with the size scaling of the Auger and carrier multiplication lifetimes. Another important finding is that third-order susceptibility of Si NCs are more than 20 times larger compared to GeNCs of the same size embedded in the same host matrix.For optical switching and modulation applications, one needs large tunability of the refractive index, such asthrough the optical Kerr effect without an appreciable change in the attenuation. Hence, as the figure of merit,
FIG. 5: Optical switching parameter, n /βλ in Ge NCs as a function of the photon energy for different NC sizes. The lowerand upper abscissas apply to both plots in the same column. n /βλ is proposed. In our previous work, we have observed that the Si NCs possess much superior figure of meritparameters compared to bulk Si. Fig. 5 shows that Ge NCs closely resemble the results of Si NCs. Both Si and GeNCs benefit from the significant blue shift of the onset of the two-photon absorption from the half E G value, whichenables the design of such switching or modulation elements over an extended wavelength range. A further indirectadvantage of this could be the suppression of the optical loss introduced by two-photon absorption generated carriersat moderately high pump powers which was a major concern in silicon Raman amplifiers. IV. CONCLUSIONS
In summary, we have investigated the wavelength and size dependence of the third-order optical nonlinearities inGe NCs, where our results can serve as a benchmark of the bound-state contribution reflecting the ultrafast responseof an unstrained perfect sample with no size dispersion. Our computed values for n and β are in agreement withsome of the existing experimental data which contain several orders of magnitude disagreement among themselves.We observe that below the band gap, there is a common enhancement trend of both the real and imaginary parts ofthe third-order susceptibility as the NC size is reduced. Another important finding is that third-order susceptibilityof Ge NCs are about an order of magnitude smaller compared to Si NCs of the same size and embedded in thesame dielectric environment. As in the case of Si NCs, the two-photon absorption threshold extends beyond the halfband-gap value. This enables nonlinear refractive index tunability over a much wider wavelength range free fromtwo-photon absorption. As a final remark, our investigation calls for further experimental work especially to probethe ultrafast third-order nonlinear response of Ge NCs. Acknowledgments
This work has been supported by the Turkish Scientific and Technical Council T ¨UB˙ITAK with the project number106T048 and by the European FP6 Project SEMINANO with the contract number NMP4 CT2004 505285. Theauthors would like to thank Dr. Can U˘gur Ayfer for the access to Bilkent University Computer Center facilities. H.Y.acknowledges T ¨UB˙ITAK-B˙IDEB for the financial support. ∗ Electronic address: [email protected] † Electronic address: [email protected] S. V. Gaponenko,
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