First-principle study of excitonic self-trapping in diamond

We present a first-principles study of excitonic self-trapping in diamond. Our calculation provides evidence for self-trapping of the $1s$ core exciton and gives a coherent interpretation of recent experimental X-ray absorption and emission data. Self-trapping does not occur in the case of a single valence exciton. We predict, however, that self-trapping should occur in the case of a valence biexciton. This process is accompanied by a large local relaxation of the lattice which could be observed experimentally.

Diamond presents an unusually favorable combination of characteristics that, in connection with the recent development of techniques for the deposition of thin diamond films, make this material a good candidate for many technological applications. Particularly appealing is the use of diamond in electronic or in opto-electronic devices, as e.g. UV-light emitting devices. Moreover, diamond is an ideal material for the construction of windows that operate under high power laser radiation or/and in adverse environments. It is therefore interesting to study radiation induced defects with deep electronic levels in the gap, since these can have important implications in many of these applications.

Excitonic self-trapping is a possible mechanism for the formation of deep levels in the gap. The study of such processes in a purely covalent material, like diamond, is interesting also from a fundamental point of view. Indeed, excitonic self-trapping has been studied so far mostly in the context of ionic compounds, where it is always associated with, and often driven by, charge transfer effects. In a covalent material the driving mechanism for self-trapping is instead related to the difference in the bonding character between the valence and the conduction band states.

Both experimental data and theoretical arguments suggest the occurrence of self-trapping processes in diamond. In particular, a nitrogen (N) substitutional impurity induces a strong local deformation of the lattice @NitrogenEPR [@NitrogenDRT; @Nitrogentheo] that can be interpreted as a self-trapping of the donor electron. The structure of a $1s$ core exciton is more controversial @M84 [@JP91; @N92; @B93; @MSWGMGN; @MS94]. Indeed the similarity between an excited core of carbon and a ground-state core of nitrogen suggests that the core exciton should behave like a N impurity. However, the position of the core exciton peak in the diamond K-edge absorption spectra is only 0.2 eV lower than the conduction band minimum @M84 [@B93; @MSWGMGN], while a N impurity originates a deep level 1.7 eV below the conduction band clarkteresa@example.net. On the other hand, emission spectra @MSWGMGN suggest that a $1s$ core exciton should self-trap like a N impurity. Finally, we consider valence excitations. In this case experimental evidence indicates that a single valence exciton is of the Wannier type, i.e. there is no self-trapping. To our knowledge, neither experimental nor theoretical investigations on the behavior of a valence biexciton in diamond have been performed, although simple scaling arguments suggest that the tendency to self-trap should be stronger for biexcitons than for single excitons.

In this letter, we present a detailed theoretical study of excitonic self-trapping effects in diamond. In particular, we have investigated the Born-Oppenheimer (BO) potential energy surfaces corresponding to a core exciton, a valence exciton and a valence biexciton in the context of density functional theory (DFT), within the local density approximation (LDA) for exchange and correlation. Our calculation indicates that the $1s$ core exciton is on a different BO surface in absorption and in emission experiments. Indeed X-ray absorption creates excitons in a $p$-like state as required by dipole selection rules. Subsequently the system makes a transition to an $s$-like state associated to a self-trapping distortion of the atomic lattice, similar to that found in the N impurity case. These results provide a coherent interpretation of the experimental data. In addition, our calculation suggests that self-trapping should also occur for a valence biexciton. This is a prediction that could be verified experimentally.

Let us start by discussing a simple model @HS [@MC2]. In diamond, the occupied valence and the lower conduction band states derive from superpositions of atomic $sp^3$ hybrids having bonding and antibonding character, respectively. Thus, when an electron, or a hole, or an electron-hole pair is added to the system, this can gain in deformation energy by relaxing the atomic lattice. Scaling arguments suggest that the deformation energy gain $E_{ def}\propto - 1/N_{ b}$, where $N_{b}$ is the number of bonds over which the perturbation is localized. This localization, in turn, has a kinetic energy cost $E_{ kin}\propto + 1/N_{ b}^{2/3}$ due to quantum confinement. The behavior of the system is then governed by the value of $N_{ b}$ that minimizes the total energy $E_{ sum}=E_{ def}+E_{ kin}$. Since the only stationary point of $E_{ sum}$ is a maximum, $E_{ sum}$ attains its minimum value at either one of the two extrema $N_{ b}=1$ or $N_{ b}=\infty$. If the minimum occurs for $N_{ b}=1$, the perturbation is self-trapped on a single bond which is therefore stretched. If the minimum occurs for $N_{ b}=\infty$, there is no self-trapping and the perturbation is delocalized. When $N_{p}$ particles (quasi-particles) are added to the system, one can show that, for a given $N_{ b}$, $E_{ def}$ scales as $N_{ p}^2$, while $E_{ kin}$ scales as $N_{ p}$. As a consequence, the probability of self-trapping is enhanced when $N_{p}$ is larger. This suggests that biexcitons should have a stronger tendency to self-trap than single excitons@MC2 [@nomorethan2].

In order to get a more quantitative understanding of self-trapping phenomena in diamond, we performed self-consistent electronic structure calculations, using norm-conserving pseudopotentials @bhs82 to describe core-valence interactions. The wave-functions and the electronic density were expanded in plane-waves with a cutoff of 35 and of 140 Ry, respectively. We used a periodically repeated simple cubic supercell containing 64 atoms at the experimental equilibrium lattice constant. Only the wave-functions at the $\Gamma$ point were considered. Since the self-trapped states are almost completely localized on one bond, they are only weakly affected by the boundary conditions in a 64 atom supercell. The effect of the $k$-point sampling was analysed in Ref. @Nitrogentheo where similar calculations for a N impurity were performed using the same supercell. It was found that a more accurate $k$-point sampling does not change the qualitative physics of the distortion but only increases the self-trapping energy by 20$\%$ compared to calculations based on the $\Gamma$-point clarkteresa@example.net.

In order to describe a core exciton we adopted the method of Ref. @PS93, i.e. we generated a norm conserving pseudopotential for an excited carbon atom with one electron in the $1s$ core level and five electrons in the valence $2s$-$2p$ levels. In our calculations for a valence exciton or biexciton we promoted one or two electrons, respectively, from the highest valence band state to the lowest conduction band state. Clearly, our single-particle approach cannot account for the (small) binding energy of delocalized Wannier excitons. However our approach should account for the most important contribution to the binding energy in the case of localized excitations. Structural relaxation studies were based on the Car-Parrinello (CP) clarkteresa@example.net. We used a standard CP scheme for both the core and the valence exciton, while a modified CP dynamics, in which the electrons are forced to stay in an arbitrary excited eigenstate@MC2 [@MCT93], was necessary to study the BO surfaces corresponding to a valence biexciton. All the calculations were made more efficient by the acceleration methods of Ref. @TMC94.

We first computed the electronic structure of the core exciton with the atoms in the ideal lattice positions. In this case the excited-core atom induces two defect states in the gap: a non-degenerate level belonging to the A$_1$ representation of the T$_{\rm d}$ point group, 0.4 eV below the conduction band edge, and a 3-fold degenerate level with T$_2$ character, 0.2 eV below the conduction band edge. By letting the atomic coordinates free to relax, we found that the absolute minimum of the A$_1$ potential energy surface correponds to an asymmetric self-trapping distortion of the lattice similar to that found for the N clarkteresa@example.net. In particular, the excited-core atom and its nearest-neighbor, labeled <span>*a*</span> and <span>*b*</span>, respectively, in Fig. \[IS\], move away from each other on the (111) direction. The corresponding displacements from the ideal sites are equal to 10.4$\%$ and to 11.5$\%$ of the bond length, respectively, so that the <span>*(a,b)*</span>-bond is stretched by 21.9$\%$. The other atoms move very little: for instance the nearest-neighbor atoms labeled <span>*c*</span> move by 2.4$\%$ of the bond length only. This strong localization of the distortion is consistent with the simple scaling arguments discussed above.

As a consequence of the atomic relaxation, the non-degenerate level ends up in the clarkteresa@example.net eV below the conduction band edge, while the corresponding wavefunction localizes on the stretched bond. The 3-fold degenerate level remains close to the conduction band edge, but since the distortion lowers the symmetry from T$_{\rm d}$ to C$_{\rm 3v}$, the 3-fold degenerate level splits into a 2-fold degenerate E level and a non-degenerate A$_1$ level.

In Fig. \[CDCE\] we report the behavior of the potential energy surfaces corresponding to the ground-state, the A$_1$ and the T$_2$ core exciton states as a function of the self-trapping distortion. Notice that the distortion gives a total energy gain of 0.43 eV on the A$_1$ potential energy surface. The same distortion causes an increase of the ground-state energy of 1.29 eV.

Our calculation indicates that the core-exciton behaves like the N impurity@Nitrogentheo, supporting, at least qualitatively, the validity of the equivalent core approximation. The similar behavior of the A$_1$ level in the core exciton and in the N impurity case was also pointed out recently in the context of semi-empirical CNDO clarkteresa@example.net. The differences between the core exciton and the impurity@Nitrogentheo are only quantitative: in particular, the relaxation energy and especially the distance of the A$_1$ level from the conduction band edge are smaller for the core exciton than for the N impurity.

Our results suggest the following interpretation of the experimental data of Refs. @M84 [@MSWGMGN]: (i) During X-ray absorption the atoms are in the ideal lattice positions. Dipole transitions from a $1s$ core level to a A$_1$ valence level are forbidden, but transitions to the T$_2$ level are allowed. In our calculation the T$_2$ level is 0.2 eV lower than the conduction band edge, in good agreement with the core exciton peak observed in X-ray absorption spectra @M84 [@MSWGMGN]. (ii) On the T$_2$ BO potential energy surface the lattice undergoes a Jahn-Teller distortion which lowers its energy (see Fig. \[CDCE\]). (iii) Since the LO phonon energy in diamond (0.16 eV) is comparable to the energy spacing between the A$_1$ and the T$_2$ surfaces, which is less than 0.2 eV after the Jahn-Teller distortion, the probability of a non-adiabatic transition from the T$_2$ to the A$_1$ surface is large. (iv) On the A$_1$ level the system undergoes a strong lattice relaxation resulting in a localization of the exciton on a single bond. (v) The self-trapping distortion induces a Stokes shift in the emitted photon energy. If the atomic relaxation were complete the Stokes shift would be equal to 1.9 eV, which correponds (see Fig. \[CDCE\]) to the energy dissipated in the T$_2$-A$_1$ transition (0.2 eV), plus the energy gained by self trapping on the A$_1$ surface (0.43 eV), plus the energy cost of the self-trapping distortion on the ground-state energy surface (1.29 eV).

The data reported in Ref.@MSWGMGN show a shift of about 1 eV in the positions of the peaks associated to the $1s$ core exciton in X-ray absorption and emission spectra. The emission peak is very broad, with a large sideband that corresponds to Stokes shifts of up to 5 eV. As pointed out in Ref. @MSWGMGN, this large sideband is likely to be the effect of incomplete relaxation. This is to be expected since the core exciton lifetime should be comparable to the phonon clarkteresa@example.net. As a consequence, the atomic lattice would be able to perform only a few damped oscillations around the distorted minimum structure during the lifetime of the core exciton.

We now present our results for the valence excitations. While in the case of a single exciton the energy is minimum for the undistorted crystalline lattice, in the case of a biexciton we find that the energy is minimized in correspondence of a localized distortion of the atomic lattice. This is characterized by a large outward symmetric displacement along the (111) direction of the atoms <span>*a*</span> and <span>*b*</span> in Fig. \[IS\]. As a result the <span>*(a,b)*</span>-bond is broken since the distance between the atoms <span>*a*</span> and <span>*b*</span> is increased by 51.2$\%$ compared to the crystalline bondlength. This distortion can be viewed as a kind of local graphitization in which the atoms <span>*a*</span> and <span>*b*</span> change from fourfold to threefold coordination and the corresponding hybridized orbitals change from $sp^3$ to $sp^2$ character. Again, in agreement with the model based on simple scaling arguments, the distortion is strongly localized on a single bond. As a matter of fact and with reference to the Fig. \[IS\], the atoms <span>*c*</span> and <span>*d*</span> move by 1.2$\%$ of the bondlength, the atoms <span>*e*</span> and <span>*f*</span> move by 2.3$\%$, and the atoms not shown in the figure by less than 0.9$\%$.

The self-trapping distortion of the biexciton gives rise to two deep levels in the gap: a doubly occupied antibonding level, at 1.7 eV below the conduction band edge, and an empty bonding level, at 1.6 eV above the valence band edge. Both states are localized on the broken bond.

In Fig. \[CDVE\] we show how different BO potential energy surfaces behave as a function of the self-trapping distortion of the valence biexciton. In particular, from this figure we see that, while for the biexciton there is an energy gain of 1.74 eV in correspondence with the self-trapping distortion, the same distortion has an energy cost of 1.49 eV for the single exciton, and of 4.85 eV for the unexcited crystal. We notice that, while DFT-LDA predicts self-trapping for the valence biexciton, it does not do so for the single exciton, in agreement with experiment.

Similarly to the case of the core exciton the major experimental consequence of the self-trapping of the valence biexciton is a large Stokes shift in the stimulated-absorption spontaneous-emission cycle between the exciton and the biexciton BO surfaces. As it can be seen from Fig. \[CDVE\], this Stokes shift should be equal to 3.23 eV, i.e. to the sum of the energy gain of the biexciton (1.74 eV) and of the energy cost of the exciton (1.49 eV) for the self-trapping relaxation. The fundamental gap of diamond is indirect. Thus the spontaneous decay of a Wannier exciton in an ideal diamond crystal is phonon assisted and the radiative lifetime of the exciton is much longer than in direct gap semiconductors. However, after self-trapping of the biexciton, the translational symmetry is broken and direct spontaneous emission becomes allowed. As a consequence the radiative life time of the self-trapped biexciton is much smaller than that of the Wannier exciton. Using the DFT-LDA wavefunctions, we obtained a value of $\sim 7$ ns for the radiative lifetime of the biexciton within the dipole approximation. This is several orders of magnitude larger than the typical phonon period. Therefore the self-trapping relaxation of the valence biexciton should be completed before the radiative decay.

A self-trapped biexciton is a bound state of two excitons strongly localized on a single bond. Thus the formation of self-trapped biexcitons requires a high excitonic density. To realize this condition it is possible either to excite directly bound states of Wannier excitons, or to create a high density electron-hole plasma, e.g. by strong laser irradiation. In the second case many self-trapped biexcitons could be produced. This raises some interesting implications. If many self-trapped biexcitons are created, they could cluster producing a macroscopic graphitization. Moreover, since the process of self-trapping is associated with a relevant energy transfer from the electronic to the ionic degrees of freedom, in a high density electron hole plasma biexcitonic self-trapping could heat the crystal up to the melting point in fractions of a ps, i.e in the characteristic time of ionic relaxation. Interestingly, melting of a GaAs crystal under high laser irradiation has been observed to occur in fractions of a ps @GaAs91. In Ref. @GaAs91 this phenomenon has been ascribed to the change in the binding properties due to the electronic excitations. Our study on diamond leads one to speculate that in a sub-picosecond melting experiment self-trapping phenomena could play an important role.

In conclusion, we have studied excited-state BO potential energy surfaces of crystalline diamond within DFT-LDA. Our calculation predicts self-trapping of the core exciton and provides a coherent description of the X-ray absorption and emission processes, which compares well with the experimental data. Moreover, we also predict self-trapping of the valence biexciton, a process characterized by a large local lattice relaxation. This implies a strong Stokes shift in the stimulated absorption - spontaneous emission cycle of about 3 eV, which could be observed experimentally.

It is a pleasure to thank F. Tassone for many useful discussions. We acknowledge support from the Swiss National Science Foundation under grant No. 20-39528.93

Present address: Departement of Physics, University of California, Berkeley CA 94720, USA.

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