Photoluminescence studies of isotopically enriched - Max Planck ...

phys. stat. sol. (b) 235, No. 1, 63–74 (2003) / DOI 10.1002/pssb.200301540 

Editor’s Choice 

Photoluminescence studies of isotopically enriched silicon 

D. Karaiskaj *; 1 , M. L. W. Thewalt 1 ,T.Ruf 2 , and M. Cardona 2 

1 Department of Physics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada 

2 Max-Planck-Institut fçr Festkærperforschung, 70569 Stuttgart, Germany 

Received 24 July 2002, revised 19 August 2002, accepted 19 August 2002 

Published online 30 December 2002 

PACS 63.20.Dj, 63.20.Ry, 78.55.Ap 

We report the first high resolution photoluminescence studies of isotopically pure silicon. New information 

is obtained on isotopic effects on the indirect band gap energy, phonon energies, and phonon 

broadenings, which is in good agreement with previous results obtained in germanium and diamond. 

Remarkably, the line widths of the no-phonon boron and phosphorus bound exciton transitions in the 

28 Si sample (99:896% 28 Si) are much sharper than in natural Si, revealing new fine structure in the 

boron bound exciton luminescence. Most surprisingly, the small splittings of the neutral acceptor 

ground state in natural Si are absent in the photoluminescence spectra of acceptor bound excitons in 

isotopically purified 28 Si, demonstrating conclusively that they result from the randomness of the Si 

isotopic composition. 

1. Introduction The effects of isotopic composition and disorder on the properties of semiconductors 

have been the subject of numerous studies, as detailed in several reviews [1–4]. Isotopically 

purified germanium (Ge) and diamond (C) were the first such semiconductors to become available in 

the form of bulk single crystals with reasonable purity, allowing for the investigation of isotope effects 

on thermal conductivity, phonon energies and broadenings, and direct and indirect electronic band 

gaps [1, 5–16]. Previous studies on isotopic effects in silicon (Si) have been limited to Raman scattering 

[17, 18], thermal conductivity [19, 20], and the direct gap due to a lack of bulk samples of 

sufficient purity for a study of the indirect band gap via photoluminescence (PL) until the recent 

growth of the samples for the aforementioned thermal conductivity studies. In this article, we report 

on the differences between the PL spectra of natural Si (92:23% 28 Si þ 4:67% 29 Si þ 3:10% 30 Si) and 

isotopically enriched 28 Si, 29 Si, and 30 Si, obtaining new experimental data on the isotope effect on the 

indirect band gap of Si, together with new information on changes in phonon energies and lifetimes [21]. 

Remarkably, the no-phonon bound exciton (BE) PL lines, which have already been shown to be extremely 

sharp in natural Si, are much narrower in the 28 Si sample. Thus, the limiting broadening mechanism 

for the no-phonon BE PL in high purity Si is found to result from the isotopic randomness. The 

removal of this broadening in fact reveals new fine structure in the boron (B) BE [22]. Furthermore, the 

small but sample-independent splittings of the neutral acceptor (A 0 ) ground state in natural Si, observed 

by different experimental techniques [23–39], was absent in the isotopically pure 28 Si sample. This 

surprising result suggests that the A 0 splittings in natural Si result from the isotopic disorder [40]. 

* Corresponding author: e-mail: dkaraisa@sfu.ca 

# 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0370-1972/03/23501-0063 $ 17.50þ.50/0

64 D. Karaiskaj et al.: Photoluminescence studies of isotopically enriched silicon 

2. Experimental methods The PL spectra were collected from samples mounted in a strain-free 

manner and immersed in liquid helium at a bath temperature of either 4.2 or 1.3 K, with excitation 

provided either by the 514.5 nm line of the Ar þ laser or, in order to ensure bulk excitation, 1020 nm 

from a Ti : sapphire laser. The natural Si sample used for comparison was a float-zone crystal selected 

to have approximately the same B and phosphorus (P) concentration as the 28 Si sample, which was 

cut from the SI283 crystal. This crystal was used previously for thermal conductivity studies and Hall 

measurements [19, 20, 41]. Spectra were collected with the Bomem DA8 Fourier transform interferometer 

using a cooled Ge detector (North Coast) as described in previous studies of ultrahigh resolution 

PL spectroscopy on natural Si [36–39, 42, 43]. The maximum instrumental resolution achieved in 

this study is estimated to be 0.014 cm 1 from the observed FWHM of the line-narrowed Ti-sapphire 

laser line, using an intracavity etalon, and collected under the same conditions as the PL spectra. 

The samples used to collect the aluminum (Al), gallium (Ga), and indium (In) A 0 X spectra were 

cut from the same crystal (SI283) as used in the study of the B and P A 0 X luminescence, but Al, Ga, 

and In, were introduced via variable-energy ion implantation, providing a roughly uniform concentration 

of the aforementioned impurities of 2 10 15 cm 3 to a depth of 3 mm followed by an anneal at 

1000 C for 20 min in order to activate the implant. 

3. Results and discussion The most remarkable features of the PL data presented in this work are 

related to the impurity BE no-phonon PL transitions in the isotopically enriched silicon [21, 22]. 

While phonon assisted transitions are broadened through the wave-vector conserving phonons, these 

PL transitions are very sharp in natural Si and can only be resolved by Fourier transform spectroscopy 

[44]. Our investigations in isotopically purified 28 Si (SI283, enriched to 99.896% 28 Si) revealed that 

these transitions become even sharper in this material. We start our discussion with the P donor and B 

acceptor BE transitions in isotopically enriched 28 Si. These impurities were unintentionally incorporated 

during the growth process, but their concentrations were sufficient to collect the no-phonon 

transitions. 

The PL spectra of natural Si, isotopically enriched 28 Si, 29 Si, and 30 Si in the TO replica region of 

the BE and bound multi-exciton complexes (BMEC) are shown in Fig. 1. The spectra of the natural 

Photoluminescence Intensity 

T= 4.2 K 

(a) 

(b) 

(c) 

(d) 

α 1 TO 

B 1 TO 

α 1 TO 

B 1 TO 

(a) 28 Si 

(b) Natural Si 

(c) 29 Si 

(d) 30 Si 

B 1 TO 

8800 8850 

Photon Energy (cm -1 ) 

Fig. 1 Photoluminescence spectra of natural Si, isotopically enriched 28 Si, 29 Si, and 30 Si in the TO 

phonon replica region are superimposed. The natural Si, 28 Si, and 30 Si spectra are dominated by the B BE 

(B 1 TO ) and BMEC lines, while the 29 Si spectrum shows stronger P BE (a 1 TO ) line. The energy shift between 

the spectra is due to the indirect electronic band gap and wave-vector conserving phonon shifts [40].

phys. stat. sol. (b) 235, No. 1 (2003) 65 

Si, 28 Si and 30 Si samples are dominated by the B impurity showing strong B BE (B 1 TO ) and BMEC 

transition lines with a weaker P BE (a 1 TO ) line [52]. The natural Si sample was intentionally lightly 

doped with 1 10 15 cm 3 B impurities and, using the method introduced by Tajima [45], we estimate 

the impurity concentration in the 28 Si sample to be [B] ¼ 7 10 14 cm 3 and [P] ¼ 7 10 13 cm 3 ,in 

good agreement with the previous Hall measurements [19, 41], and in the 30 Si sample, the B concentration 

is approximately [B] ¼ 2 10 15 cm 3 . The uncertainty in estimating the impurity concentration 

using the method of Tajima is larger for the 30 Si sample due to the much weaker TO phonon 

assisted transition of the free exciton. The 29 Si crystal is dominated by the P impurity showing a 

strong P BE (a 1 TO ) (Fig. 1c) transition and the P concentration is estimated to be approximately 

[P] ¼ 1:5 10 15 cm 3 . 

The BE and BMEC TO replicas are energetically shifted due to the different isotopic composition 

of the samples. This energy shift is composed of two contributions: the shift of the electronic band 

gap and the shift of the TO phonon energy. The band gap shift can be measured directly from the nophonon 

spectra, which are shown in Fig. 2 at a spectral resolution of 0.014 cm 1 , with the 28 Si spectrum 

shifted up in energy by 0.92 cm 1 so as to align the features in both spectra. The most remarkable 

differences between the PL spectra of the 28 Si and natural Si samples are the much sharper nophonon 

BE lines seen in the isotopically pure sample and the resulting extra fine structure revealed in 

the B BE transition. It must be remembered that the no-phonon BE and BMEC transitions are already 

very narrow in natural Si, and the detailed study of their linewidths and fine structure only became 

practical with the advent of ultra high resolution Fourier transform PL spectroscopy using Michelson 

interferometers [44]. We discuss first the P BE (a 1 ) transition, which has no electronic substructure, 

and in the natural Si sample, was found to have a FWHM of 0.041 cm 1 , in agreement with previous 

measurements [42, 43]. The observed FWHM in the 28 Si sample is only 0.014 cm 1 , much less than 


α 1 NP 

natural Si 

28 

Si 

B 1 NP 

T=1.3 K 

9275.0 9280 9281 

Photon Energy for natural Si (cm -1 ) 

Fig. 2 A high resolution spectrum in the no-phonon region is shown for the natural Si sample, together 

with the same spectrum obtained from the 28 Si sample shifted up in energy by 0.92 cm 1 (the length of 

the horizontal arrow) so as to align the transitions. The P BE (a 1 ) line is seen to consist of a single 

component, while the B BE (B 1 ) line has considerable fine structure. The vertical tick marks below the B 

BE show the predicted locations of the fine structure components obtained from a theoretical fit to the 

behavior of the spectrum under uniaxial stress [42]. An identical instrumental resolution of 0:014 cm 1 

FWHM was used for both spectra. The oscillations on either side of the P BE line are not noise, but rather 

the sincðxÞ lineshape resulting from transforming an interferogram using boxcar apodization when there is 

still strong modulation at the maximum optical path difference of 50 cm [21].


that observed in the best natural Si samples, even though the 28 Si sample is of only moderate chemical 

purity. It must also be stressed that the observed FWHM of 0.014 cm 1 is dominated by the 

instrumental resolution, which at our maximum optical path length difference of 50 cm, and using 

boxcar apodization and ignoring resolution degradation from the finite size of the collimating aperture, 

would give a FWHM of 0.012 cm 1 . At present, we can thus only set an upper limit of 

0.005 cm 1 on the true linewidth of the P bound exciton no-phonon transition in 28 Si. The broadening 

seen in Si is consistent with a simple calculation combining the isotopic dependence of the band gap 

energy determined here with statistical fluctuations of the isotopic composition in an ‘‘effective” excitonic 

volume having a radius of approximately 3.5 nm [53]. 

A similar reduction in the no-phonon linewidths is seen in Fig. 2 for the components of the B BE 

line, which had been shown to have 9 resolved components in natural Si, explained in terms of a 

threefold splitting of the electron levels of the BE due to the valley-orbit effect, coupled with a threefold 

splitting of the hole states due to hole–hole coupling. More detailed studies of the B BE in 

natural Si followed, involving both uniaxial stress [42] and Zeeman [43] perturbations revealing that a 

complete description required the inclusion of additional terms, such as the electron–hole coupling. 

The linewidths of the components of the B BE are again much narrower in the 28 Si sample than in 

natural Si, with the narrowest components having an observed FWHM of 0.017 cm 1 , slightly larger 

than that of the P BE. As a result, the B BE spectrum in the 28 Si sample now reveals 13 resolved 

components and one partially resolved doublet [21]. This new fine structure is well accounted for by 

the earlier investigations of the effects of uniaxial stress and magnetic fields on the B BE [42, 43]. 

The vertical tick marks along the bottom of the B BE spectrum, which are seen to be in quite good 

agreement with the newly observed fine structure, are the predicted zero-stress energies of the various 

components obtained from the theoretical fit to the observed energies of the many stress-split components 

as a function of the applied uniaxial stress [42]. 

The evidence that the residual broadening of the no-phonon BE transition lines in natural Si is 

mostly due to local band gap fluctuations resulting from isotopic randomness also raised further ques- 


T=1.8 K 

Si (Al) 

natural Si 

28 

Si 

Γ 5 

Γ 1 

A O X 

Γ 

Γ 5 

3 

Γ 3 

A O 

Γ 1 

9271 9272 


Fig. 3 A high resolution PL spectrum of the no-phonon region of the aluminum BE in natural Si is compared 

with the aluminum BE spectrum in a 28 Si sample, shifted up in energy (for ease of comparison) by the 

0.92 cm 1 difference in band gap energy between natural Si and 28 Si (the actual energies of the 28 Si transitions 

can be obtained by shifting the spectrum down by the 0.92 cm 1 length of the arrow). The inset shows 

a level scheme explaining the transitions in the natural Si sample. The A 0 X initial state has three populated 

levels corresponding to the three valley-orbit electron states having G 1 , G 3 and G 5 symmetry, and each of 

these initial states produces a doublet in the PL spectrum due to the splitting of the A 0 final state, which is 

clearly absent in the 28 Si sample [22].


tions that needed to be pursued. One of the unexpected discoveries resulting from the previous applications 

of ultrahigh resolution spectroscopic techniques to the PL spectra of natural Si was the observation 

of a small, but reproducible splitting of the ground state of the Al, Ga, and In acceptors in 

natural Si [36, 37]. It was a natural extension of the work on the P and B no-phonon BE transitions to 

investigate the acceptor ground state splitting in isotopically enriched Si, since the broadening of the 

lines should be reduced substantially. 

The acceptor states and acceptor BE transitions have been studied by a variety of different experimental 

techniques such as electron spin resonance (ESR) [23, 24], thermal conductivity studies [32–34], 

phonon absorption spectroscopy [25–31, 46] and, most recently, by photoluminescence [35–39]. 

While electron paramagnetic resonance (EPR) was detected for electrons bound to neutral donor impurities 

(D 0 ) in Si very early in the development of the semiconductor physics [47], careful searches 

for EPR from holes bound to neutral acceptors (A 0 ) in Si were initially unsuccessful, as summarized 

by Kohn [48], who was also the first to point out that the problem likely resulted from the four-fold 

degeneracy of the A 0 ground state. Unlike the D 0 ground state, whose two-fold spin degeneracy can 

only be lifted by a magnetic field, the A 0 ground state in the absence of perturbations is a four-fold 

degenerate state, which can be split into doublets by perturbations such as axial strain or an electric 

field. Thus, a random distribution of perturbations, such as strain fields from dislocations and/or impurities, 

acting on the A 0 ground state produces a distribution of doublet splittings which in turn can 

broaden the EPR transitions to the point where they are unobservable. This conjecture was soon verified 

by the observation of EPR from A 0 in silicon in the presence of externally applied uniaxial 

strain, which produces a large but uniform doublet splitting of the ground state, and thus diminishes 

the effects of the internal random strain fields [49]. While there were several predictions that the 

rapidly improving perfection of single crystal silicon would soon permit the observation of EPR without 

necessity of external strain, the first success in this regard did not occur until considerably later 

[50, 51]. 

However, Neubrand’s pioneering observation of ESR from A 0 in B doped silicon in the absence of 

external strain, with doping concentrations ranging from 10 15 to 10 16 cm 3 , introduced a new problem 

which has remained unresolved until this work [23, 24]. After accounting for all of the possible random 

strain splittings from the known contaminants in the purest, dislocation-free samples, there remained 

a sample-independent distribution of A 0 splittings whose origin could not be explained. 

A similar small (0:10 0:01 cm 1 ) but sample-independent splitting of the Al A 0 ground state in 

silicon was later observed as a final-state splitting of the components of the Al acceptor bound exciton 

(A 0 X) photoluminescence transitions, and it was suggested that this splitting arose from a spontaneous 

lowering of the A 0 symmetry via the Jahn–Teller effect [35, 36]. A subsequent PL study confirmed 

that the doublet splittings arose from the A 0 final state of the A 0 X PL transitions and revealed a 

similar 0:10 0:01 cm 1 doublet splitting for the Ga A 0 ground state, and a 0:15 0:03 cm 1 A 0 

splitting for the deeper In acceptor [37–39]. The 0:15 0:03 cm 1 splitting found for the indium A 0 

state in the PL study [37] agrees well with the phonon spectroscopy [31] result of 0:11 to 0:16 cm 1 , 

and the failure to detect the B splitting (0:043 cm 1 as determined by phonon spectroscopy [28, 29, 

31]) in the PL studies was consistent with the PL observed linewidth of the boron BE components of 

0:08 cm 1 resulting from a combination of insufficient instrumental resolution and PL broadening 

effects. 

We return now to a more detailed explanation of the PL spectra shown in Fig. 3. A review and 

further references to the details of PL related to shallow donors and acceptors in Si can be found in 

Ref. [52]. Both spectra are from what is known as the no-phonon region of the Si PL spectrum, in 

which the energy of the emitted photon is exactly equal to the energy difference between the electronic 

initial and final states of the transition. The Al BE no-phonon recombination spectrum, shown in 

Fig. 3 (natural Si), consists of three identical doublets with the same splitting of 0:1 cm 1 . The results 

shown in Fig. 3 are explained by three initial states of the A 0 X in both the natural Si and the 28 Si 

samples, which are labelled by their symmetries of G 1 , G 3 , and G 5 in the tetrahedral point group (T d ) 

appropriate for substitutional impurities in Si. These three A 0 X initial states arise from a splitting of 

the electron states in the A 0 X via the valley-orbit splitting, as also found on a much larger energy


Photoluminscence Intensity 

T=1.8 K 

Si (Ga) 

natural Si 

28 

Si 

Γ 5 

Γ 3 

Γ 1 

9266 9268 


Fig. 4 High resolution no-phonon PL spectra of the gallium A 0 X transitions are compared for natural 

Si and 28 Si samples. For ease of comparison, the 28 Si spectra have been shifted up in energy by the 

0.92 cm 1 difference in band gap energy between 28 Si and natural Si. The inset shows the doublet 

resulting from a singlet A 0 X initial state due to the A 0 final state splitting which is used to fit the 

observed spectra in the natural Si samples [22]. 

scale for the D 0 ground state in indirect band-gap semiconductors such as Si [38, 39]. It should also 

be noted that there is a much larger splitting of the A 0 X initial state due the hole–hole coupling of its 

two holes. At the temperatures used in this study, only the lowest in energy G 1 spin singlet state of 

A 0 X is populated resulting from the reduction of the (G 8 G 8 ¼ G 1 þ G 3 þ G 5 ) two-hole state, and all 

transitions in the Al, Ga, and In A 0 X spectra originate from an initial state of G 1 symmetry. Thus, 

only a triplet due to the valley-orbit splitting of the electron states is expected. The obvious explanation 

is that the extra, identical doublet splittings of the three valley-orbit components are due to a final 

state splitting of the A 0 X (or a ground state splitting of the A 0 ) confirming the already strong evidence 

previously given by ESR, phonon spectroscopy and thermal conductivity studies. A diagram of 

these transitions and the effect of the additional splitting of the A 0 ground state is shown in Fig. 3 

(inset). 

Further investigations on the Ga BE no-phonon recombination spectrum shown in Fig. 4 (natural 

Si), indicated that this spectrum also consists of three doublets. The first doublet originating from the 

G 3 electron state is clearly visible. The second component is thought to result from the overlap of two 

doublets as the one shown in the inset of Fig. 5. This was also confirmed by PL studies in magnetic 

fields [39] and under uniaxial stress [38]. The second component most likely consists of two doublets 

originating from the G 1 and G 5 electron states, which are too close in energy to be fully resolved in 

the natural Si spectrum. Furthermore, the In bound exciton no-phonon spectrum is shown in Fig. 5 

and consists of four components also resulting from the G 1 , G 3 , and G 5 transitions. The fourth component 

is thought to originate from the G 5 component that is split into a doublet by the spin-orbit interaction. 

This has also been confirmed by further investigations in magnetic fields and under uniaxial 

stress [38, 39]. The line shape of the components indicates the presence of underlying structure, which 

is not resolved. These extra unresolved components do appear in the no-phonon absorption spectra 

when the temperature is lowered from 4.2 to 0.4 K (see Ref. [37] for further details). 

The Jahn–Teller theorem [54] states that in any nonlinear configuration of atoms, coupling between 

a degenerate electronic state and asymmetric vibrations of nuclei, must distort this configuration and 

lift the electronic degeneracy by reducing its symmetry. The fact that the maximum of the valence 

band in Si belongs to the G 8 symmetry and is a fourfold degenerate state that could be unstable 

towards vibrations, makes the Jahn–Teller effect the most suitable explanation for the acceptor ground


T=1.8 K 

Si (In) 


natural Si 

28 

Si 

Γ 1 

Γ 3 

Γ 5 

9201 9203 


Fig. 5 High resolution no-phonon PL spectra of the indium A 0 X transitions are compared for natural 

Si and 28 Si samples. For ease of comparison, the 28 Si spectra have been shifted up in energy by the 

0.92 cm 1 difference in band gap energy between 28 Si and natural Si. The inset shows the doublet 

resulting from a singlet A 0 X initial state due to the A 0 final state splitting which is used to fit the 

observed spectra in the natural Si samples [22]. 

state splitting. Vibrations and distortions of the crystal cannot break the time reversal symmetry, and 

as a result, can cause only a splitting into Kramers doublet. 

At this point we are left with two possibilities for explaining the small A 0 splittings: either a Jahn– 

Teller effect as postulated by the optical studies, which would be in conflict with the much higher 

energy Jahn–Teller resonance invoked in the phonon studies [30, 33], or, as postulated in the phonon 

studies, a distribution of splittings from random fields of unknown origin, but essentially identical in 

all high-quality samples. Clearly, the first explanation had some difficulties, at least in Si, where the 

phonon measurements applied, and the second was incomplete in that the origin of the sample-independent 

random fields was left unexplained. 

The Al A 0 X spectrum from the isotopically pure 28 Si sample shown in Fig. 3 ( 28 Si) is shifted up in 

energy by the 0:92 cm 1 reduction in band gap energy of 28 Si relative to natural Si. It is clear that not 

only has the isotopic broadening been reduced in the 28 Si sample, but the A 0 splitting has also vanished. 

The doublets resulting from the splitting of the A 0 ground state, as shown schematically in the 

inset, are now single sharp transitions. The same change is also seen in the Ga A 0 X spectrum collected 

from the 28 Si sample. The G 3 line, which is clearly a doublet in natural Si, is a single line in 

28 Si. The more complicated component, thought to be composed of two doublets, is now reduced to 

two transitions most likely originating from the G 5 and G 1 electron states. The doublet final state 

splittings are less resolved in the natural Si(In) spectrum, although they have been confirmed by careful 

thermalization studies in both PL and absorption spectroscopy, and as mentioned earlier, the In 

A 0 X spectrum is further complicated by a splitting of the A 0 X initial state of G 5 symmetry by spinorbit 

coupling [38, 39]. Nevertheless, it is again clear that the doublet splitting resulting from the A 0 X 

final state, as modelled in the inset, is also absent for the In A 0 Xin 28 Si. 

In conclusion, the linewidths of the no-phonon BE transitions in 28 Si were found to be much narrower 

than those of natural Si, revealing new fine structure in the spectrum of the B bound exciton 

predicted in earlier studies. The residual, sample-independent A 0 ground state splittings in Si are 

absent in isotopically pure samples, and as a result, these splittings must also arise from the effects of 

isotopic randomness in all of the previously studied natural Si samples. This makes good qualitative 

sense – any perturbation which lowers the crystal symmetry from T d at the impurity site can split the


A 0 ground state into a doublet, and the random fluctuations of isotopic composition around individual 

acceptors will provide such a symmetry lowering perturbation, which has the added necessary property 

of having an identical distribution for all samples having the same (natural) isotopic abundances. 

One might therefore ask why such an obvious explanation has never been postulated in the extensive 

literature on these splittings. We believe that the reason is that the effect of isotopic randomness was 

assumed to be too small to explain the observed splitting – this was certainly our assumption before 

obtaining the present results. Previous calculations of the A 0 splittings due to the known perturbations 

of impurities such as oxygen and carbon were based on the long-range strains produced by these 

relatively dilute and highly perturbing impurities, where long-range refers to strains which were essentially 

uniform across the wave function of a given A 0 state. From the point of view of such long-range 

fields, the perturbations arising from isotopic randomness in natural silicon would indeed be much too 

small to explain the observed A 0 splittings. Instead, we believe a new approach is necessary, namely, 

a detailed analysis of how isotopic fluctuations on a scale smaller than the A 0 Bohr radius couple to 

the different A 0 states, and we are currently investigating such effects using realistic A 0 wave functions. 

Isotopic effects on the indirect electronic band gap and phonon energies The electronic states of 

crystals, such as valence and conduction band states, are renormalized by their interaction with phonons. 

Crystal vibrations also exist at zero Kelvin (zero-point vibrations) and interact with electrons 

renormalizing their energy. Changes in the average isotopic mass lead to changes in the lattice vibrations, 

and as a consequence of the electron–phonon interaction, to an energy shift of the electronic 

states. To a lesser extent this energy shift also results from the change in the lattice constant with the 

isotopic composition. The zero-point renormalization energy of a crystal is proportional to the inverse 

square root of the average isotopic mass, and by using this relation and measuring the energy shift of 

the electronic band gap between two crystals of different isotopic composition, the zero-point renormalization 

energy can be estimated. 

From the relative energy shift of the impurity bound exciton no-phonon transitions, between two Si 

crystals of different isotopic composition, the relative shift of the indirect electronic band gap can be 

determined. Due to the much simpler electronic structure of the donor P BE, these transitions were 

used to obtain the band gap shifts. From this quantity, the zero-point renormalization energies of the 

indirect electronic band gap in Si were estimated using the method of Ref. [3]. The samples used to 

determine the relative band gap shift are the single crystals 28 Si, 29 Si, 30 Si and the solution grown 

isotopic mixture of 28 Si and 30 Si. Their isotopic compositions and estimated average masses are given 

in Table 1. 

The four renormalization energies obtained are listed in Table 2 and are similar to the value obtained 

for germanium ( 51:9 2:1 meV) using the same experimental method [6]. The energy shift 

of the direct electronic band gap with isotopic composition has been measured recently for Si using 

ellipsometry, leading to renormalization energies of 90 meV [55]. 

The simplest approximation, the harmonic virtual crystal approximation (VCA), predicts that the 

phonon energy decreases proportionally to the inverse square root of the average isotopic mass. In 

reality, additional interactions such as anharmonic and disorder effects influence the phonon energies 

and lifetimes. The anharmonic effects to the frequency shift result from the phonon–phonon interac- 

Table 1 

The isotopic composition and average isotopic mass of the Si samples investigated. 

sample isotopic composition average mass (amu) 

natural Si 92.23% ( 28 Si) þ 4.67% ( 29 Si) þ 3.10% ( 30 Si) 28.086 

28 Si 99.896% ( 28 Si) þ 0.09% ( 29 Si) þ 0.014% ( 30 Si) 27.978 

29 Si 2.15% ( 28 Si) þ 97.58% ( 29 Si) þ 0.27% ( 30 Si) 28.958 

30 Si 0.70% ( 28 Si) þ 0.62% ( 29 Si) þ 98.68% ( 30 Si) 29.954 

28 Si þ 30 Si 26% ( 28 Si) þ 74% ( 30 Si) 29.45


Table 2 The indirect electronic band gap shifts relative to the isotopically enriched 28 Si sample. The 

renormalization energies are obtained from these band gap shifts. 

natural Si enriched 29 Si enriched 30 Si mixture of 28Si þ 30 Si 

band gap 0.92 cm 1 8.4 cm 1 16.9 cm 1 12 cm 1 

shift (114 meV) (1.04 meV) (2.09 meV) (1.48 meV) 

band gap –59.2 meV –60.4 meV –61.3 meV –57.7 meV 

renormalization (1 meV) (2 meV) (1 meV) (2 meV) 

tion processes. The coherent potential approximation (CPA) takes into account the random distribution 

of the different isotopes, and as a result, enables us to estimate the contributions of the isotopic 

disorder to the energy shift and broadening of the phonons. 

The frequency shifts for the transverse optical (TO) and transverse acoustical (TA) wavevector-conserving 

phonons were obtained from the phonon replicas of the P and B BE transitions and are listed 

in Table 3. All phonon shifts are given relative to the 28 Si sample, because of its high isotopic purity, 

together with the phonon shifts theoretically predicted by the VCA. The TO(X) and TA(X) phonon 

shifts of natural Si relative to 28 Si are in reasonable agreement with the VCA predicted values. Small 

deviations are mainly due to the isotopic disorder, caused by the random distribution of the 29 Si and 

30 Si isotopes in natural Si, and to a lesser extent, due to anharmonic effects. The TO(X) phonon 

frequency shift of the 29 Si sample agrees also with the VCA. There might be some disorder-induced 

frequency shift due to 2.15% 28 Si still contained in the sample, but it is much smaller than our experimental 

certainty. The TO(X) and TA(X) phonon shifts of the high isotopically enriched 30 Si crystal 

are in good agreement with the value predicted by the VCA within the experimental uncertainty. This 

reconfirms the high isotopic purity of the crystal showing a very weak isotopic disorder induced 

frequency shift [40]. 

Table 3 The experimentally obtained energy shifts of the TO(X), TA(X), and Raman phonons are 

summarized together with the energetic shifts predicted by the VCA for natural Si, 29 Si, 30 Si, 

28 Si þ 30 Si, compared to isotopically enriched 28 Si. 

TO(X) phonon TA(X) phonon Raman phonon 

phonon shift of natural Si measured ( 1.14 0.1) cm 1 ( 0.28 0.1) cm 1 ( 0.81 0.05) cm 1 

with respect to 28 Si 

corresponding shifts predicted 

from the VCA compared to 28 Si 

0.91 cm 1 0.29 cm 1 1.01 cm 1 

phonon shift of 29 Si measured ( 7.8 0.5) cm 1 


corresponding shifts predicted 8cm 1 


phonon shift of 30 Si measured ( 15.7 0.5) cm 1 ( 5.1 0.5) cm 1 


corresponding shifts predicted 15.7 cm 1 5.1 cm 1 


phonon shift of isotopic mixture ( 12.6 0.5) cm 1 

( 28 Si þ 30 Si) measured with 

respect to 28 Si 

corresponding shifts predicted 11.9 cm 1 

from the VCA compared to 28 Si


We were able to obtain accurate values of the zone-center longitudinal optical Raman phonon energies 

in natural Si and 28 Si, since the Stokes–Raman line of the 1020 nm pump laser falls within the 

transparency region of Si, and is in the same spectral region over which the PL spectra were collected. 

The energy shift of the Raman phonon is also given in Table 3. The additional anharmonic 

contributions to this frequency shift have been estimated. These contributions to the phonon frequency 

shift result from phonon–phonon interaction processes and can be estimated using perturbation theory. 

As a result, we obtain an additional correction term, which is proportional to the inverse average 

vibrating mass. By including these contributions, the expected frequency shift for the Raman phonon 

is 0.83 cm 1 , which is very close to the measured one of 0.81 cm 1 [18, 40]. 

A closer examination of Fig. 1 reveals that the TO wavevector-conserving phonon replicas of 

natural Si are noticeably broader than those of 28 Si, indicating that in natural Si the TO wavevectorconserving 

phonon has a substantial lifetime broadening arising from isotopic randomness. Extracting 

this additional broadening parameter is complicated by the fact that the electronic bound exciton 

transitions are themselves not delta functions (see for example the structure of the B bound exciton 

revealed in Fig. 2), and the impurity-related wavevector-conserving phonon replicas have an additional 

broadening which arises from the uncertainty in the center-of-mass electron and hole wavevector 

due to the localization on the impurity, which causes a broadening via the energy dispersion 

of the particular phonon involved. While the detailed analysis of this broadening will be the subject 

of a future study, one can estimate for the surprisingly large difference between Si and 28 Si by 

noting that for the TO phonon the FWHM of the B 3 line in the former is 2.6 cm 1 and in the latter 

only 2.1 cm 1 [21]. 

In addition, we investigated an isotopically disordered alloy with an isotopic mixture of 28 Si and 

30 Si grown by liquid-phase epitaxy from indium solution. The isotopic composition and average atomic 

mass are given in Table 1. The PL spectrum showed a high concentration of indium and phosphorus 

that could not be determined exactly, but it is estimated to be of order 1 10 16 cm 3 or greater. 

In such highly isotopically disordered crystals, the disorder induced phonon frequency shift and 

linewidth broadening become significant. Therefore, the discrepancy between the VCA predicted frequency 

shift and the measured value, given in Table 3, is believed to be primarily due to the isotopic 

disorder induced shift. In fact, recent calculations [18] using the CPA predict for the TO(X) phonon in 

this isotopic composition a disorder-induced frequency shift of 0:5 cm 1 , which is in good agreement 

with our experimental value [18, 40]. 

In conclusion, we have determined the indirect band gap shifts of natural Si, 29 Si, 30 Si, and isotopic 

mixture, relative to the almost single isotope 28 Si crystal. Using these energy shifts of the indirect 

electronic band gap, the zero-point renormalization energies were estimated. We measured the phonon 

frequency shifts of natural Si, 29 Si, 30 Si and an isotopic mixture of 28 Si and 30 Si, relative to the 

isotopically pure 28 Si sample. These shifts were compared with the values predicted by the VCA, and 

the eventual discrepancies due to anharmonic and disorder effects were discussed. 

Acknowledgements D. Karaiskaj and M. L. W. Thewalt acknowledge Natural Sciences and Engineering Council 

of Canada for financial support. 

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