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Terahertz-assisted excitation of the 1.5mm photoluminescence of Er in crystalline Si

Moskalenko, A.S.; Yassievich, I.N.; Forcales Fernandez, M.; Klik, M.A.J.; Gregorkiewicz, T.

Publication date 2004 Document Version Final published version Published in Physical Review B

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Citation for published version (APA): Moskalenko, A. S., Yassievich, I. N., Forcales Fernandez, M., Klik, M. A. J., & Gregorkiewicz, T. (2004). Terahertz-assisted excitation of the 1.5mm photoluminescence of Er in crystalline Si. Physical Review B, 70, 155201.

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Download date:29 Sep 2021 PHYSICAL REVIEW B 70, 155201 (2004)

Terahertz-assisted excitation of the 1.5-␮m photoluminescence of Er in crystalline Si

A. S. Moskalenko and I. N. Yassievich A. F. Ioffe Physico-Technical Institute, Politekhnicheskaya 26, St. Petersburg, 194021 Russia

M. Forcales, M. Klik, and T. Gregorkiewicz Van der Waals-Zeeman Institute, University of Amsterdam, Valckenierstraat 65, NL-1018 XE Amsterdanm, The Netherlands (Received 25 February 2003; revised manuscript received 8 September 2003; published 5 October 2004)

We show that the 1.5-␮m emission from Si:Er generated by continuous-mode band-to-band optical excitation can be dramatically enhanced by terahertz radiation from a free-electron laser. The effect is observed at cryogenic temperatures in samples prepared from FZ-Si by high-temperature implantation with Er ions and requires a high density of infrared photons ͑ប⍀Ϸ100 meV͒. Based on experimental characteristics of this effect, we argue that the excitation mechanism responsible for the enhancement is, in this case, different from the previously discussed free-electron-laser-induced optical ionization of trapped carriers. A theoretical model of the energy transfer path is developed. It involves participation of a higher-lying conduction c2 band of the 4 3+ Si host and excitation into the I11/2 second excited state of Er ion. Since formation of the Er-related level is not necessary in this mechanism, it opens a possibility to excite a large fraction of Er3+ ions, including also these which are not linked to a recombination level in the band gap. Possible implications of the proposed model toward realization of a true three-level scheme in Si:Er are pointed out. Finally, further experiments necessary for testing and conﬁrmation of the theoretically developed model are proposed.

DOI: 10.1103/PhysRevB.70.155201 PACS number(s): 78.66.Db, 61.72.Tt, 41.60.Cr

I. INTRODUCTION =4.2 K under conditions of cw band-to-band pumping and high-power THz illumination, for samples prepared by high- Importance of Er doping as a method of circumventing temperature implantation of Er ions into n-type FZ-Si sub- the lack of optical activity of crystalline silicon ͑c-Si͒ is strates. Such a sample choice follows from consideration of growing and practical Si:Er-based devices are successfully the crystalline quality of the material. Although in FZ mate- 3+ developed—see, e.g., Ref. 1 for an overview. The currently rial emission from an Er ion is known to be weaker than in recognized energy transfer mechanisms responsible for acti- optimized oxygen-rich samples, the high substrate quality combined with the low implantation damage related to the vation Er photo- (PL) and electroluminescence—electron- elevated temperature of implantation is likely to result in a hole recombination at an Er-related shallow level and a hot superior crystalline quality of the samples, making them carrier impact2—excite Er3+ ion into the 4I first excited 13/2 ideal for investigations of energy transfers. In contrast to the state. In this way a true three-level excitation scheme en- earlier discussed one, this new excitation path is character- abling population inversion and lasing is not realized. ized by a linear dependence on the FEL radiation density. Previous investigations revealed that, at cryogenic tem- Triggered by these findings, we propose a theoretical model ␮ peratures, the 1.5- m Er-related emission from Si:Er can be for this new excitation mechanism which accounts for the induced by a mid-infrared (THz frequency range) pulse from experimental data. It is similar to that discussed in Ref. 7 and a free-electron laser (FEL) applied shortly after initial band- involves participation of a higher-lying c2 conduction band to-band excitation.3 The microscopic mechanism of the FEL- 4 of the silicon host and the I11/2 second excited state of the induced Er emission has been identified as recombination of Er3+ ion. The excitation does not proceed via a level in the Si a hole optically ionized from a shallow trap populated by the bandgap. In that way, this mechanism is very different from pump pulse with an electron localized at an Er-related level the excitonic Auger mediated energy transfer which can be in the bandgap.4 Subsequently, it has been found that this realized only for Er-related centers forming such levels— phenomenon is related to the afterglow of slowly decaying typically for ϳ1% of the total Er concentration. The process emission5 appearing as a result of thermalization of shallow requires high-energy free carriers, and therefore, its effi- traps. In addition, it was shown6 that a FEL-induced en- ciency depends on the lifetime of free carriers generated op- hancement of the Er-related emission appears also under tically by band-to-band illumination. In the proposed excita- conditions of continuous (cw) band-to-band pumping, when tion scheme, a true three-level system is realized (presently a steady-state population of carriers at shallow traps is possible only in nanocrystalline Si but not in crystalline bulk 4 3+ reached. In such a case, the magnitude of the PL enhance- Si) where the second I11/2 excited state of Er ion becomes ment was measured to be proportional to the square root of populated. While the present study is clearly of fundamental excitation density, over 4 orders of magnitude of FEL power: character, it is interesting to note that eliminating the band- ϳͱ IPL PFEL. gap level from the energy transfer should disable the so- In the current contribution, we report observation of an called back-transfer process of excitation reversal. Since the experimental fingerprint of another FEL-induced excitation back-transfer is generally held responsible for thermal mechanism of Er3+ ions in Si. It has been observed at T quenching of Si:Er emission, the proposed excitation mecha-

nism gives new hopes for realization of efﬁcient room- temperature PL based on Si:Er material, a crucially important aspect in view of device applications.

II. EXPERIMENTAL Intense mid-infrared radiation ͑␯Ϸ30 THz͒ for the current experiment was provided by a free-electron laser. There- fore, the experiment has been conducted at the Free Electron Laser for Infrared eXperiments (FELIX) users facility of the FOM Institute for Plasma Physics “Rijnhuizen” in the Neth- erlands. The FELIX delivers infrared macropulses with a repetition rate of 5 Hz and duration of approximately 5 ␮s. Each macropulse contains a train of micropulses of Ϸ1.7 ps Ϸ duration and repetition rate of either 25 MHz or 1 GHz. The FIG. 1. Dynamics of the Er PL at a temperature T 5 K under energy carried by a macropulse is chosen by selecting the illumination with a Nd:YAG and a FEL with a delay of 5 ms re- repetition rate of micropulses and adjusting their amplitude spect the Nd:YAG. The dashed line is the steady-state level of the 1.5-␮m emission generated by cw illumination from a diode by attenuators (up to 30 dB) inserted into the infrared beam. ͑820 nm͒. The inset shows the dependence of the THz-assisted ex- In the correlated two-beam experiment the Si:Er sample citation (PL enhancement) of Er3+ ions as a function of the FEL (FZ-Si, n-type, implanted at 500°C with 1100 keV Er ions 13 −2 power (normalized). The solid line is a guide for the eye. In the to a total dose of 10 cm —Ref. 8 for details on sample second panel, temperature dependence of PL intensity observed in preparation and PL spectrum) is placed in a variable tem- the investigated material is given. perature cryostat and excited with above bandgap light and, simultaneously, exposed to the THz beam from the FEL. The the sample to a short ͑ns͒ pulse from the Nd:YAG laser ͑␭ ͒ primary optical excitation is provided by a cw operating YAG=532 nm . As can be seen, Er-related emission in- ␭ solid state laser at a wavelength of cw=820 nm with creases following the additional band-to-band excitation and Ϸ10 mW power. In the experiment, we followed the influ- then decays towards the equilibrium value with a time con- ence of the FEL beam tuned to the wavelength around stant of ␶Ϸ0.80 ms characteristic for the lifetime of the ex- ␭ Ϸ ␮ 3+ FEL 10 m, for which the maximal dynamic range is cited state of Er ion, as expected. The sample response to a available on the most intense part of the Er-related PL spec- high energy FEL pulse, represented by the second peak, is trum selected by a 20 nm band-pass filter centered at the quite different; its magnitude is bigger (comparable to the strongest spectral component at 1549 nm. equilibrium PL level generated by pumping in the visible), and its kinetics is much slower than that of the PL signal induced by Nd:YAG. In order to check whether the en- III. EXPERIMENTAL RESULTS hancement is not due to a small increase of sample temperature, which might be induced by the powerful FEL pulse, we Upon cw illumination by the pump laser an equilibrium investigated thermal quenching of PL intensity in the sample situation is created. That means that a steady-state concen- used in the present study. The result, depicted in the right trations at all stages of the excitation process are reached. panel of Fig. 1, shows that only a monotonous lowering of Under conditions of this experiment—low temperature, high the amplitude of PL signal is seen at higher temperatures. density pumping—these will include free carriers in the In the inset to the figure, the integrated value of the addi- bands, carriers bound at available traps, free and bound ex- tional PL signal appearing upon FEL pulse is plotted as a 3+ citons, and Er ions in the ground and the excited states. In function of THz excitation density (normalized). As can be that way a steady-state Er-related emission at ␭Ϸ1.5 ␮mis concluded, the enhancement effect varies linearly with power achieved. This equilibrium emission is perturbed by addi- and does not show saturation within the studied FEL power tional laser pulse and restores to the equilibrium value when range. The different dependence on the macropulse power ͑ ϳ ͒ the pulse is terminated. Experiments performed with low IPL PFEL indicates that this enhancement effect is due to a power FEL pulses showed quenching of PL intensity. Such a different physical mechanism than the trap-ionization-related behavior is consistent with our earlier findings for a pulsed enhancement identified and discussed in Ref. 4 for the more band-to-band excitation regime (pumping with the second “standard” Cz-Si:Er material. We propose that the additional harmonics of the Nd:YAG laser) where quenching of PL PL appearing for large FEL powers represents a manifesta- intensity was observed when the FEL pulse of 25 MHz rep- tion of a thus far unknown excitation path which can be etition rate was applied coincident with the Nd:YAG.9 The induced by high-power THz radiation and active exclusively present experiments were performed with the 1 GHz FEL at cryogenic temperatures in high-quality FZ material. (We repetition rate, corresponding to the 40 times higher energy note that oxygen-lean material is not usually used for prepa- of the FEL macropulse (up to ϳ60 mJ). ration of Si:Er photonic structures, as high oxygen content is Figure 1 summarizes the most important experimental re- known to improve efficiency and thermal stability of Er sults of the current study. The zero level of the vertical scale emission in crystalline Si matrix). corresponds to the steady-state Er-related PL, as obtained We have also looked how the observed enhancement ef- under cw illumination. The first peak represents response of fect was changing upon temperature increase. We have ob-

155201-2 TERAHERTZ-ASSISTED EXCITATION OF THE... PHYSICAL REVIEW B 70, 155201 (2004) served that magnitude of the additional luminescence was gradually decreasing, and no enhancement could be observed at TϾ40 K. At the same time, PL quenching4 visible also at low temperatures for small FEL ﬂuxes was becoming dominant. Therefore, it cannot be decided at the moment whether the observed thermal reduction of PL intensity represents an inherent feature of the MIR-induced excitation, or results from competition with the quenching. It is not clear whether this question can be resolved experimentally, as the two effects are entangled.

IV. THEORETICAL CONCEPT OF A NEW EXCITATION MECHANISM Upon cw diode excitation we have stationary concentra- ͑ ͒ ͑ ͒ tion of free electrons n and holes p , which results in FIG. 2. Scheme of the energy bands of Si. Two lowest conduc- 3+ stationary concentration of excited Er ions and the time- tion bands c1 and c2, and two upper valence bands h,l are shown. ⌬ independent PL intensity. We propose that application of The band gap is Eg =1.17 eV, c =0.5 eV, a=5.43 Å is the lattice 3+ ͑ ␲ ͒ ͑ ␲ ͒ THz radiation opens an additional excitation channel for Er constant of Si, k0 =0.85 2 /a , kX =0.15 2 /a . ions embedded in c-Si. The energy quantum ប⍀ of THz radiation is absorbed by an electron and then the recombination of the electron-hole pair assisted by Auger excitation of 3+ 4 4 Phonons will also enable a fast relaxation from the second to Er from state I15/2 to state I11/2 takes place. The excitation ប⍀ the first excited state. is possible only if +Eg (Eg is the band gap energy of ⌬ Ϸ In fact, there could be two possible realizations of such an silicon) is larger than the energy difference ffЈ 1.24 eV between these states. Such a process can be described by the Auger process induced by THz radiation: second order perturbation theory and involves transition 1. An electron from conduction band c1 [see Fig. 3(a)] ប⍀ through a virtual electron state. gains energy by absorbing a THz photon and goes to The proposed mechanism is based on a model of c-Si conduction band c2, from where it recombines with a heavy 3+ 4 band structure adopted in Refs. 10 and 11—see Appendix A hole and excites the Er ion into the I11/2 state via Coulomb for details. We use a simplified approach, where the spin- interaction. At this stage phonons take part in the process. orbit interaction is neglected and the heavy and the light hole The electron state in the c2 subband is virtual. bands coincide, therewith forming a heavy hole band h, and 2. A heavy hole in band h absorbs a photon and goes into the split-off band plays a role of a light hole band l. There- a virtual state in band l. Then it recombines with an electron ជ 3+ fore, the heavy hole band h is doubly degenerated in k and from band c1 transferring energy to an Er ion and to ជ phonons see Fig. 3 b . the light hole band l is nondegenerated in k. The double [ ( )] degeneration in spin should be also taken into account. The Because of zero overlap integral between the top of va- ជ lence band and the bottom of c1, the second recombination wave functions of band h are polarized perpendicular to k, process occurs due to admixture of c2 states into c1, which whereas the wave functions of band l are polarized parallel ជ appears only for nonzero electron energies. This leads to an to k. For conduction bands it is enough to consider only the additional vanishing factor in the expression for the probabil- first two subbands c1 and c2. The resulting band structure is ity of such a process. Therefore, the second excitation path shown in Fig. 2. The introduced simplifications should not can be neglected in comparison to the first one. considerably affect the results of the following calculations. The probability of excitation of an Er3+ ion can be written The energy quantum of the terahertz radiation is not large as enough for a direct transition of an electron from the bottom of the first c1 to the second c2 conduction band. However, if the excited electron immediately recombines with a hole in an Auger process with participation of an Er3+ ion, the exact energy conservation at the intermediate stage is not required. The momentum and the energy released by the electron-hole recombination are then transferred to an Er3+ ion. Such a transfer is effective if the released energy is sufficient for excitation of an Er3+ ion, in this case into the second excited state. It is known that transitions between the ground and the 3+ second excited states of Er ions in insulating hosts involve FIG. 3. Feynman diagrams for two possible processes which 12 ͑4 ͒ phonons, in contrast to excitations into the first exited state. lead to transition of erbium ions to the second excited state I11/2 . Therefore, we may involve phonons to compensate for the Both transitions are stimulated by means of terahertz radiation. See energy difference appearing in the excitation process. text for details.

155201-3 A. S. MOSKALENKO et al. PHYSICAL REVIEW B 70, 155201 (2004)

2␲ Јտ ជ ជЈ ͉ ͉2 ͑␧ ͒ ͑␧ ͒ ͉͗ ͉ ͘2 ͑ ͒ tion m 0.1 mЌ (see Appendix A). k1 ,k are counted from Wex = N⍀͚ t21 fe 1 fh 2 ͚ Lf,Ni + N Li,Ni P Ni ␯ ប the bottom of the valley of conduction band c1, and k1Ќ␯ is 1,2 N,Ni ជ ϫ ␦͑ ␧ ␧ ប⍀ ⌬ ប␻͒ ͑ ͒ the length of the transverse component of k1. Eg + 1 + 2 + − ffЈ − N . 1 Calculation of the matrix element corresponding to Cou- ͉ ͘ ͉ជ ͘ lomb interaction is performed following Ref. 14: Here the initial electron state 1 = ks1 is an electron in the ␯th ប͑ជ ͉͗ ជ ͉ ˆ ͉ ␯ ជ ͉͘ valley of the conduction band c1 with momentum k␯0 h,k2p2s2; fЈ Vc c2, ,k1s1; f ជ ជ +k ͒ (wave vector k␯ corresponds to the bottom of the 2 1 0 1 4␲e h c2,␯ ជ ជ ␯-valley of the conduction band) and spin s ͑s =±1/2͒. The = ͩ ͉ͪ͗uជ ͉uជ ͉͉͗͘fЈ͉eik0␯r͉f͉͘, 1 1 ␬͑⌬ ប ͒ 2 k p s k ␯s ជ V fЈf/ ,k0 k0 2 2 2 0 1 final state ͉2͘=͉k p s ͘ is an electron state in the valence 2 2 2 ͑ ͒ បជ 7 band with momentum k2, polarization p2 (there are two h c2,␯ possible polarizations in the heavy hole subband), and spin where uជ and uជ are the Bloch amplitudes of corre- s ͑s =±1/2͒. t is the matrix element of electron transition k2p2s2 k0␯s1 2 2 12 sponding electron states. It has also been considered that in between states ͉1͘ and ͉2͘. ␧ ,␧ are electron and hole kinetic 1 2 the process of Auger excitation in Si a large momentum, of energies in states ͉1͘ and ͉2͘, correspondingly, and order បk , is transferred to the f-electron. In Eq. (7) the time f ͑␧ ͒, f ͑␧ ͒ are distribution functions of electrons and holes 0 e 1 h 2 and spatial dispersion of the dielectric constant ␬͑␻,k͒ is in these states. The sum ͚ is taken over all possible elec- 1,2 also taken into account. Expression (7) is really independent tron and hole states. The second sum ͚ is taken over N,Ni ␯ ជ 3+ of and momentum of hole k2. states of vibrational system of the Er ion. Ni is the number of phonons in initial state, N=N -N is the difference in pho- Because of a relatively small value k0rf =0.4, where rf is f i the radius of the f-shell of an erbium ion, we expand the non number between final Lf and initial Li state of the ion ជ ជ ͑ ͒ ik0␯r vibrational system, and P Ni is the probability of having Ni factor e in Eq. (7) in series and estimate the first three phonons in the initial state. N⍀ is the number of photons, ⍀ terms is the frequency of terahertz radiation. ជ ជ k0␯ik0␯j ͗ Ј͉ ik0␯r͉ ͘ ͗ Ј͉ ͘ ជ ͗ Ј͉ជ͉ ͘ ͗ Ј͉ ͉ ͘ The phonon part in Eq. (1) f e f = f f + ik0␯ f r f − ͚ f rirj f + ¯. ij 2 ͑ ͒ ͉͗ ͉ ͘2 ͑ ͒ ͑ ͒ J N = ͚ Lf,Ni + N Li,Ni P Ni , 2 ͑8͒ Ni The first term at the right hand side of Eq. (8) is equal to zero was calculated in the model of two shifted identical parabolic and in the second term we have dipole matrix element for adiabatic potentials.13 For low temperature kTӶប␻, where ␻ optical transition is the frequency of local vibrations, we have ជ ͗ Ј͉ជ͉ ͘ ͑ ͒ 1 dfЈf = f r f , 9 J͑N͒ = SN exp͑− S͒ . ͑3͒ N! which average value dfЈf can be estimated through the corresponding oscillator strength P for unpolarized light15 Here S is the Huang-Rhys factor ⌬ opt 1 2m0 fЈf ⌬ − h␯ P = ␹ ͚ d2 , ͑10͒ fЈf fЈf ͑ ͒ ប2 fЈf S = , ͑4͒ 3 2J +1 Ј 2ប␻ ff where ͑2J+1͒=16 is the number of states in multiplet where ⌬opt is the energy of optical absorption for transition fЈf 4I ,␹ is the refractive index of the medium, in which er- → Ј͑4 → 4 ͒ ␯ 15/2 f f I15/2 I11/2 and h fЈf is the energy of luminescence. bium ions are embedded. We use values of the oscillator ͉ ͉2 The square of the electron transition matrix element t21 4 → 4 strength for the I15/2 I11/2 transition for different crystals can be written as P=0.2−0.6·10−6,15,16 ␹Ϸ3.5, and we can estimate ជ ជ ជ ជ 2 ͗k p s ; fЈ͉Vˆ ͉kЈsЈ; f͗͘kЈsЈ͉Hˆ ͉k s ͘ d Ӎ r ␤, ͑11͒ ͉t ͉2 = ͚ͯ 2 2 2 c e−r 1 1 ͯ . ͑5͒ fЈf f 21 ជ ␧ ͑ជ ͒ ប⍀ ␧Ј͑ជЈ͒ −3 kЈsЈ 1 k1 + − k where the numerical factor ␤ is of the order of 10 . On the other side, using the symmetry properties of ͗ជЈ Ј͉ ˆ ͉ជ ͘ The optical transition matrix element k s He−r k1s1 is cal- f-states we get for the third term in Eq. (8) culated in Appendix C 2 k0␯ik0␯j k0 2 4 ͚ ͗fЈ͉r r ͉f͘ = ͗fЈ͉z2͉f͘. ͑12͒ ជ ជ 2␲ e ប 1 i j ͉͗ ␯ Ј Ј͉ ˆ ͉ ␯ ͉͘2 2 ␦ជ ជ ␦ i,j 2 2 c1, ,k s He−r c2, ,k1s1 = 2 k1Ќ␯ k ,kЈ s1,sЈ. ␬ϱប⍀V mЈ 3 1 ͑ ͒ We see that in the considered situation the third term is 6 ͗ Ј͉ 2͉ ͘Ӎ 2 ӷ␤ dominant because f z f rf and k0rf . In contrast to Here ␬ϱ is dielectric constant of Si in the high frequency usual radiative transitions, for the process considered just limit, V is normalization volume, eជ is vector of light polar- this term determines the transition probability and not the ization, mЈ is an unknown mass, for which there is a limita- dipole term. In result we have the following estimation for

155201-4 TERAHERTZ-ASSISTED EXCITATION OF THE... PHYSICAL REVIEW B 70, 155201 (2004)

1 1 1−xЌ the last factor on the right hand side of Eq. (7): dxʈ ͑␦ ͒ ͵ ͱ ͵ ͵ fN N = dy y dxЌxЌ ͱ ជ 2 0 0 0 xʈ ជ k0 ͗ Ј͉ ik0␯r͉ ͘Ӎ ͗ Ј͉ 2͉ ͘ ͑ ͒ f e f − f z f . 13 ␰ ␦ 2 ϫ␦ͩ h N ͪ ͑ ͒ xЌ + xʈ + y − , 19 ␰ ␰ ͉ ͉2 e e Then t12 can be calculated from Eq. (5) neglecting the difference in electronic kinetic energies in comparison to with ⌬ −ប⍀, and taking into account Eqs. (6) and (7). Substitut- c ␦ = ⌬ + Nប␻ − E − ប⍀. ͑20͒ ing the result in Eq. (1) we get N fЈf g P is the power of FEL, A is the irradiated area and ˜c is the 2␲ V2 ␳ ͵ 3 ͵ 3 ␧͑ជ ͒ ␧͑ជ ͒ light velocity in Si. Wex = ⍀͚ ͚ 6 d k1 d k2 fe k1 fh k2 ប ␯ ͑2␲͒ „ … „ … Fermi energies of electrons ␰ and holes ␰ are found from s1 p2s2 e h concentrations of electrons n and holes p, correspondingly ϫ ͑ ͒␦͑ ␧͑ជ ͒ ␧͑ជ ͒ ប⍀ ⌬ ប␻͒ ͚ J N Eg + k1 + k2 + − fЈf − N ͱ ͱ N 4 2 mЌ mʈ ␰3/2 ͑ ͒ n = 2 3 e , 21 2␲e2ប4 1 1 1 4␲e2 2 ␲ ប ϫ k2 ͩ ͪ ␬ ប⍀ Ј2 1Ќ␯ ͑⌬ ប⍀͒2 2 ␬ 2 ϱ m 3 c − V k0 4ͱ2 m3/2 4͉͗ Ј͉ 2͉ ͉͘2 p = h ␰3/2. ͑22͒ h c2,␯ k0 f z f ␲2 ប3 h ϫ͉͗uជ ͉uជ ͉͘2͚ , ͑14͒ 3 k2p2s2 k0␯s1 4 fЈ The frequency dependence of numerical factor F͑ប⍀͒ has the form of peaks separated from each other by the phonon where ␳⍀ =N⍀ /V is the density of THz photons. In Eq. (14) ␬ϵ␬͑⌬ ប ͒Ϸ energy ប␻. The amplitudes of the peaks decrease for higher we have put ffЈ/ ,k0 1, because the screening ef- frequencies. In an experiment, one can expect that these fect is small for a small distance, of the order of ͑k ͒−1 0 peaks will be broadened due to uncertainty in the phonon Ϸ1 Å, and high frequency (large energy transferred in the energy in the one-mode model, splitting of the multiplets and Auger process)—for details see Refs. 14 and 17. We should other reasons. take sum over the ﬁnal state fЈ of f-electrons and average In order to facilitate an easy comparison with the experi- over the initial state f. The estimation shows that ment, the ﬁnal result for excitation probability can be expressed in a more convenient way as ͉͗ Ј͉ 2͉ ͉͘2 4␥ ͑ ͒ ͚ f z f = rf f , 15 2 fЈ P 1cm n p ϫ 7 Wex =5 10 17 −3 17 −3 ␥ 10 MW A 10 cm 10 cm where the unknown factor f is of the order of 10. mЈ −2 ␥ 1 The averaged value of the overlap integral between bands ϫͩ ͪ f ͫ ͬ ͑ ͒ Fs , 23 h and c2 is calculated in Appendix B and we get 0.1 mЌ 10 s

␯ where P is the THz radiation power in MW. In order to h c2, 2 4 2 ͚ ͉͗uជ ͉uជ ͉͘ = ͉͗Z͉xy͉͘ , ͑16͒ evaluate the frequency dependent numerical factor F , values k2p2s2 k0␯s1 3 s p2s2s1 of the Huang-Rhys factor S and the characteristic energy ប␻ of phonons involved in the process are necessary. Unfortu- where Z is one of the basis wave functions of representation nately, these are not known for the speciﬁc case considered ⌫ having the same symmetry properties as the wave func- 25Ј here—the 4I state of Er3+ ion embedded in c-Si matrix. In tion xy of representation ⌬ . The matrix element ͉͗Z͉xy͉͘ is 11/2 2Ј Fig. 4 we show the wavelength dependence of the F numeri- equal to 0.75 according to Ref. 10. s ␰ ␰ cal factor calculated for the electron/hole concentration of Assuming e and h to be the Fermi energies of electrons n=p=1017 cm−3 and Huang-Rhys and phonon values of Er and holes in bands c1 and h, respectively, we get in the low in ﬂuorozirconate glass: S=0.13 and ប␻=57 meV.12 In order temperature limit to demonstrate the dependence of Fs on these parameters, we 6 ប also show the results of evaluation performed for S=0.5 and 4 P e 1 np mЌ 4 2 ប␻ W =2␲ r ␥ ͉͗Z͉xy͉͘ F͑ប⍀͒. =20 meV. It should be remarked that the shape of Fs ex ␬2␬ ͑⌬ ͒4 Ј2 f f ˜cA ϱ fЈf − Eg m factor is practically independent of free-carrier concentration ͑17͒ while its magnitude depends on the concentrations, for equal electron and hole concentrations approximately like n2/3. Here F͑ប⍀͒ is given by the following expression:

͑⌬ ͒4 V. DISCUSSION fЈf − Eg F͑ប⍀͒ = ͚ J͑N͒f ͑␦ ͒, ͑18͒ ͑ប⍀͒2͑⌬ ប⍀͒2 N N c − N Figure 1 provides a direct experimental comparison of the band-to-band (pumping by the second harmonics of a where Nd:YAG laser) and THz-radiation-assisted excitation

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count that under cw diode illumination (power 10 mW) the stationary concentrations of free electrons and holes are n Ϸ 17 −3 ⌬ Ϸ =p 10 cm , we arrive to the conclusion that NFEL 8 ϫ −5 10 NEr. Since the suggested new mechanism does not require formation of a gap state, we assume that all the Er3+ ions available in the material can be accessed. The fact that FEL-induced Er PL is characterized by a much longer relaxation time supports the idea that, in this case, another kind of erbium centers are (predominantly) activated than under band-to-band pumping. (We note here that introduction of a donor level, a necessary condition for the “classical” excitation path of Er in Si, enables also efficient nonradiative recombination, e.g., due to the so-called “back-transfer” mechanism.21) The PL intensity under pulse excitation should be proportional to the number of excited erbium ions ␶ ␶ ␶ ␶ multiplied by a factor eff/ rad, where eff and rad correspond to the effective and radiative lifetime values of excited Er3+ FIG. 4. Dependence of the numerical factor Fs on radiation wavelength ␭ calculated for n=p=1017 cm−3 and two different sets ions, respectively. As mentioned earlier, in view of the much of parameters: (a) S=0.13 and ប␻=57 meV, or (b) S=0.5 and ប␻ slower decay kinetics one can expect, that in the case of ␶ ␶ Ϸ −2 =20 meV. For detailed explanation see text. In the inset, prelimi- Nd:YAG excitation eff/ rad 10 and for FEL excitation ␶ ␶ Х nary experimental results (the on-going project) on the wavelength eff/ rad 1. We, therefore, conclude that the proposed exci- dependence of the enhancement effect are given. tation mechanism should induce a significantly higher PL intensity than the excitonic one. As can be concluded from mechanisms of the 1.54-␮m Er-related emission from the Fig. 1, this is indeed observed in the experiment thus provid- same Si:Er material. In both cases stationary cw illumination ing support for the credibility of the proposed model. We takes place. One can see that the THz-induced PL enhance- also note that, according to Eq. (23), PL enhancement de- ment is more effective. Moreover, the amplitude of the FEL- pends linearly on FEL power, as verified experimentally— induced emission is comparable to the level of the steady- see inset to Fig. 1. state PL generated by the cw band-to-band excitation with The proposed theoretical model in the presented form is the diode laser. valid for the low temperature case. The limitation is imposed Let us compare the number of erbium ions which can be by the condition that temperature should be lower than excited in both cases. For optical band-to-band pumping chemical potentials corresponding to the equilibrium carrier (Nd:YAG excitation), the effective excitation cross section concentrations. For the parameters of experiment such limi- has been determined as ␴Ϸ10−15 cm2.18 Taking into account tation gives us TՇ10 K. The experiment was carried out for ϳ ␮ the Nd:YAG pulse energy, PYAG 20 J/pulse and duration T=4.2 K that is in the range of validity of our theory. For the ϳ100 ps we get the photon flux of ␾=6ϫ1023 cm−2 s−1 investigated case the temperature dependence of the en- (taking typical area of 1 cm2). Thus the number Er3+ ions hancement effect enters only via dependence of the equilib- ⌬ which will be excited by the Nd:YAG pulse is NYAG rium carrier concentration on temperature. To describe the Ϸ 0.06N1, where N1 is the number of erbium ions eligible for observed temperature quenching of the THz-induced PL en- the band-to-band excitation (i.e., forming a donor level in the hancement effect properly, further development of the model gap, thus enabling the energy transfer19). According to mul- is necessary. 20 tiple reports, N1 corresponds to less than 1% of the total In order to further test the proposed excitation mechanism number NEr of Er ions introduced into the sample. In our as the microscopic origin of the observed enhancement of FZ-Si sample with low oxygen concentration PL intensity is Er PL, dependence of magnitude of the effect on FEL wave- lower than in the Cz-Si:Er. Since we use here the excitation length should be investigated. As given by Eq. (23) and il- cross section ␴ as determined for Cz-Si (the value of ␴ for lustrated in Fig. 4, the THz-assisted excitation mechanism FZ-Si:Er is not known), we will account for the lower PL has a step-like dependence on FEL wavelength and is char- intensity by taking the N1 /NEr ratio to be smaller: N1 /NEr acterized by an energy threshold: It is enabled exclusively by Ϸ10−3. In that way, we estimate the number of Er3+ ions photons with energy quantum ប⍀ larger than the energy dif- ⌬ Շ ⌬ which are excited by the Nd:YAG pulse as NYAG 6 ference between Er excitation and Si bandgap: ffЈ–Eg ϫ −5 10 NEr. Ϸ70 meV. Experimental observation of this threshold is a Now we will estimate the number of Er3+ ions which can major experimental challenge as it lies at the border of FEL ͑⌬ ͒ be excited by the FEL pulse NFEL according to the earlier range available for the current study. The experiments are described mechanism. For that we need to evaluate the FEL currently on the way. The preliminary results for the ␭ Ͻ ␮ photon flux. Typical FEL power during each micropulse is FEL 16.5 m range are shown in the insert to the figure. ϳP=8.6 MW, duration of micropulse is 1 ps and the num- While definitely more experimental effort is necessary, and ber of micropulses in one macropulse is around ϳ5000. the observed behavior differs in detail from the predicted When we neglect Er decay during the macropulse ͑5 ␮s one, a clear variation of the magnitude of the enhancement Ӷ␶ Ϸ ͒ Er 1ms, then following Eq. (23) and taking into ac- with FEL wavelength is evident.

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VI. CONCLUSION H ͓ 2 ͑ 2 2͔͒ ␴ ␴ ͑ ͒ = A1kz + A2 kx + ky I + A3 xkxky + A4 zkz, A1 We propose a plausible microscopic mechanism for expla- ជ where k is counted from the X point. ␴ , ␴ are Pauli matri- nation of the experimentally observed dramatic increase of x z ␮ ces, and I is identity matrix. Parameters A1 and A2 can be the 1.5- m PL from FZ-Si:Er by THz radiation pulse. We ប2 found through experimental masses m͉͉, mЌ :A1 = /2m͉͉, show that this mechanism accounts for all the currently rec- 2 A =ប /2mЌ. A is defined through A by position of the ognized characteristic features of the effect, namely large 2 4 1 minimum of conduction band c1atk :͉A ͉=2A k . Constant amplitude, linear dependence on THz photon flux, slow de- 0 4 1 X A can be expressed through an unknown mass mЈ:A cay kinetics of the induced emission, and thermal quenching. 3 3 =ប2 /mЈ. We have neglected relativistic effects. Therefore the In contrast to the usually considered gap-state mediated ex- both bands c1 und c2 are twice degenerate due to the spin. citation mechanism, the proposed energy transfer path opens ជ the way to activate possibly all Er3+ ions introduced into the Rewriting Hamiltonian (A1) for k counted from the bottom material. Moreover, since it involves participation of the of conduction band kz =k0 we get 4 3+ higher-lying second I excited state of Er ion, a true ប2 11/2 ជ three-level system is formed, thus opening hopes of realiza- ␧ ͑k͒ kxky 1 mЈ tion of population inversion in Si:Er. Hc1,c2 = . ͑A2͒ Future experiments will test validity of the proposed ΂ ប2 ΃ ជ kxky ␧ ͑k͒ model and explore further potential of the FEL excitation of mЈ 2 Er in FZ-Si for silicon photonics. Here ប2 ប2 ␧ ͑ជ͒ 2 ͑ 2 2͒ ͑ ͒ ACKNOWLEDGMENTS 1 k = kz + kx + ky , A3 2m͉͉ 2mЌ We acknowledge the FELIX staff and, in particular, the expert assistance by Dr. Lex van der Meer. The FZ-Si:Er ប2 ប2 ␧ ͑ជ͒ ⌬ ͑ 2 ͒ ͑ 2 2͒ ͑ ͒ samples were provided by Dr. Frans Widdershoven. The 2 k = c + kz −4kXkz + kx + ky , A4 work received financial support of RBRF and the Dutch Sci- 2m͉͉ 2mЌ entific Organization NWO. The Amsterdam group was also would give energetical spectrum without interaction of the supported by the European Research Office, US Army. The ជ bands. Wave vector k is counted from the bottom of conduc- St. Petersburg group acknowledges support by the grant Rus- tion band c1. The energy gap between conduction bands ⌬ sian Scientific Schools. c at the minimum of the conduction band c1 is given by ប2 ⌬ 2 ͑ ͒ APPENDIX A: MODEL OF ENERGY-BAND SPECTRUM c =2 kX. A5 mʈ Conduction band ⌬ We get c =0.5 eV. To estimate mЈ we calculate spectrum of The space symmetry group of Si belongs to the class Oh (A2) and is nonsymmorphic because it contains elements includ- ជ 1 ជ ជ ing nontrivial translations. This property leads to an obliga- E ͑k͒ = ͓ͫ␧ ͑k͒ + ␧ ͑k͔͒ tory double degeneration of the bands at the X point if the 1,2 2 1 2 11 spin is neglected. It is known that the two lowest conduc- ប2 2 tion bands of Si are nondegenerate at the ⌬ point and corre- ͱ͓␧ ͑ជ͒ ␧ ͑ជ͔͒2 ͩ ͪ ͬ ͑ ͒ ± 1 k − 2 k +4 kxky . A6 ⌬ ⌬ 10,11 mЈ spond to the representations 1 and 2Ј, respectively. At the X point such representations are compatible with repre- Taking into account that for small k band mixing is small sentations X2 and X4, using notations of Ref. 11, which have (there is no mixture is z-direction) we have for small k nonzero projection of the momentum matrix element on the 2 2 axis ⌫−X. Therefore, these doubly degenerated bands have ជ ជ 1 ប E ͑k͒ = ␧ ͑k͒ + ͩ k k ͪ . ͑A7͒ nonzero slope at the X point, which is the reason of the shift 1 1 ⌬ x y c mЈ of the lowest conduction band minimum from the X point towards the ⌫ point, but they certainly have zero slope on Since the corrugation of conduction band c1 is not seen up to 19 −3 average.10,11 More details about the group-theoretical aspects carrier concentrations 10 cm we get a limitation for pa- can be found in the book of Bir and Pikus.11 The minimum rameter mЈ ͑ ͒ of the lowest conduction band c1 lies on the 001 axis at mЈ 2 1 ␰ ⌫ ͩ ͪ տ e , ͑A8͒ distance k0 from the point and at distance kX from the X ⌬ ͑ ␲ ͒ mЌ 5 c point (see Fig. 2). k0 =0.85 2 /a and, therefore, kX =0.15͑2␲/a͒, where a=0.543 nm is the lattice constant of from which we have mЈտ0.1mЌ. Si. Thus the minumum of c1 lies near the X point. According to the group-theoretical considerations11 [Eq. (30.51)] the Valence band Hamiltonian of the conduction band in k-representation can For the valence band structure we use a generalization of be written as the Luttinger Hamiltonian in the spherical approximation17

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ជ ជˆ 1 ជˆ 1 m m H = ͑A +2B͒ប2k2 −3Bប2͑k · L͒2 + ⌬␴ជˆ · L − ⌬. ͑A9͒ m = 0 , m = 0 , ͑A11͒ 3 3 h ␥ ␥ l ␥ ␥ 1 −2 1 +2 ជˆ Here L is the unit angular momentum operator, ␴ជˆ is an op- ␥ 1 ͑ ␥ ␥ ͒ ͑ ͒ erator of angular momentum 1/2, ⌬ is spin-orbital splitting, = 5 3 3 +2 2 . A12 and ␥ ␥ ␥ The values of constants 1, 2, and 3 for Si are 4.22, 0.53, 22 1 mh + ml 1 mh − ml 1.38, correspondingly. In the limit of a small spin-orbital A =− , B =− , ͑A10͒ splitting ⌬→0, which can be used for Si, we write Hamil- 4 mhml 4 mhml ⌫ tonian (A9) in the basis of 25Ј functions X, Y, Z in where k-representation

͑ ͒ 2 ͑ 2 2͒ A +2B k −3B ky + kz 3Bkxky 3Bkxkz hl ប2 3Bk k ͑ ͒ 2 ͑ 2 2͒ 3Bk k ͑ ͒ H = ΂ x y A +2B k −3B ky + kz y z ΃. A13 ͑ ͒ 2 ͑ 2 2͒ 3Bkxkz 3Bkykz A +2B k −3B ky + kz

␾ ␾ ͑ ͒ The eigenvalues of Hamiltonian (A13) are mx = sin , my = − cos , mz =0, B2 ប2k2 ͑ ͒ប2 2 ͑ ͒ E1,2 = A − B k =− , A14 l = sin ␪ cos ␾, l = cos ␪ sin ␾, l = − cos ␪. 2mh x y z ͑B3͒ with corresponding eigenfunctions ជ Ќ ជ ជ Ќ ជ In this case we have lx mx l k,m k ជ ជ ikrជ ikrជ ␺ = l e , ␺ = m e , ជ Ќ ជ h c2,␯ 2 2 2 1 ΂ y ΃ 2 ΂ y ΃ m l ͚ ͉͗uជ ͉u ͉͘ = cos ␪͉͗Z͉xy͉͘ ␦ , ͑B4͒ k s s ,s k2p2s 0␯ 1 1 2 lz mz l = m =1 p2 ͑A15͒ ⌬ where xy is the wave function of representation 2Ј. The ជ for two times degenerated h band, and averaging of Eq. (B4) over all possible directions of k2 gives

1 3mh − ml ͑ ͒ប2 2 ប2 2 ͑ ͒ h c2,␯ 2 E3 = A +2B k =− k , A16 ͉͗ ͉ ͉͘2 ͉͗ ͉ ͉͘2␦ ͑ ͒ 4 m m ͚ uជ uជ = Z xy s s . B5 h l k2p2s2 k0␯s1 3 1, 2 p2 with eigenfunction

kx/k ជ APPENDIX C: OPTICAL TRANSITION MATRIX ␺ ikrជ ͑ ͒ 3 = ΂ky/k ΃e , A17 ELEMENT k /k z Interaction of electrons with a photon leads to a correction ␺ ␺ ␺ for l band. For all three eigenfunctions 1, 2, 3 there are of Hamiltonian (A2). In the lowest order the correction, two possible orientations of the spin. which induces interband transitions, looks like

APPENDIX B: OVERLAP INTEGRAL BETWEEN h UND បe 0 k A + k A ˆ ͩ x y y x ͪ ͑ ͒ c2 BANDS He−r = , C1 mЈc kxAy + kyAx 0 h The basis for the Bloch amplitude uជ are functions k2p2s2 ជ ជ ជ ជ ⌿ជ ជ⌿ជ ⌿ជ ͑ ͒ ជ where A=A0e, e is the vector of polarization and m and l , where = X,Y ,Z . Suppose that vector k2 has an orientation characterized by angles ␾ and ␪ in spherical c 2␲ប⍀ ͱ ͑ ͒ coordinates so that A0 = . C2 ⍀ ␬ϱV ␪ ␾ ␪ ␾ ␪ k2x = k2 cos cos , k2y = k2 cos sin , k2z = k2 sin . ͑B1͒ is the amplitude of the vector-potential of one-photon elec- tromagnetic ﬁeld. Then we get for the optical transition ma- ជ Then we can choose for mជ , l trix element

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2 2 1 2 2 1 2 2␲ eប ͑k e + k e ͒ = ͑k + k ͒ = k Ќ, ͑C4͒ ͗ជ ͉ ˆ ͉ជ ͘ ͱ ͑ ͒␦ជ ជ ␦ 1x y 1y x 3 1x 1y 3 1 kЈsЈ He−r k1s1 = k1xey + k1yex k ,kЈ s ,sЈ. ␬ ប⍀ Ј 1 1 ϱ V m 2 2 2 where k Ќ ϵ͑k +k ͒, and get ͑C3͒ 1 1x 1y 2␲ e2ប4 1 For unpolarized light we also average over polarization di- ͦ͗ជЈ Ј͉ ˆ ͉ជ ͦ͘2 2 ␦ជ ជ ␦ ͑ ͒ k s He−r k1s1 = 2 k1Ќ k ,kЈ s ,sЈ. C5 rections ␬ϱប⍀V mЈ 3 1 1

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