Giant oscillator strength

Giant oscillator strength is inherent in excitons that are weakly bound to impurities or defects in crystals.

The spectrum of fundamental absorption of direct-gap semiconductors such as Gallium arsenide (GaAs) and Cadmium sulfide (CdS) is continuous and corresponds to band-to-band transitions. It begins with transitions at the center of the Brillouin zone, \boldsymbol{k}=0. In a perfect crystal, this spectrum is preceded by a hydrogen-like series of the transitions to s-states of Wannier-Mott excitons.[1] In addition to the exciton lines, there are surprisingly strong additional absorption lines in the same spectral region.[2] They belong to excitons weakly bound to impurities and defects and are termed 'impurity excitons'. Anomalously high intensity of the impurity-exciton lines indicate their giant oscillator strength of about f_i\sim10 per impurity center while the oscillator strength of free excitons is only of about f_{\rm ex}\sim10^{-4} per unit cell. Shallow impurity-exciton states are working as antennas borrowing their giant oscillator strength from wast areas of the crystal around them. They were predicted by Emmanuel Rashba first for molecular excitons[3] and afterwards for excitons in semiconductors.[4] Giant oscillator strengths of impurity excitons endow them with ultra-short radiational life-times \tau_i\sim1 ns.

Bound excitons in semiconductors: Theory

Interband optical transitions happen at the scale of the lattice constant which is small compared to the exciton radius. Therefore for large excitons in direct-gap crystals the oscillator strength f_{\rm ex} of exciton absorption is proportional to |\Phi_{\rm ex}(0)|^2 which is the value of the square of the wave function of the internal motion inside the exciton \Phi_{\rm ex}(\boldsymbol{r}_e-\boldsymbol{r}_h) at coinciding values of the electron \boldsymbol{r}_e and hole \boldsymbol{r}_h coordinates. For large excitons |\Phi_{\rm ex}(0)|^2\approx 1/a^3_{\rm ex} where a_{\rm ex} is the exciton radius, hence, f_{\rm ex}\approx v/a^3_{\rm ex}\ll1, here v is the unit cell volume. The oscillator strength f_i for producing a bound exciton can be expressed through its wave function \Psi_i(\boldsymbol{r}_e,\boldsymbol{r}_h) and f_{\rm ex} as

f_i=\frac{1}{v}\frac{(\int d\boldsymbol{r}_e\Psi_i(\boldsymbol{r}_e,\boldsymbol{r}_e))^2}{|\Phi_{\rm ex}(0)|^2}f_{\rm ex} .

Coinciding coordinates in the numerator, \boldsymbol{r}_e=\boldsymbol{r}_h, reflect the fact the exciton is created at a spatial scale small compared with its radius. The integral in the numerator can only be performed for specific models of impurity excitons. However, if the exciton is weakly bound to impurity, hence, the radius of the bound exciton a_i satisfies the condition a_ia_{\rm ex} and its wave function of the internal motion \Phi_{\rm ex}(\boldsymbol{r}_e-\boldsymbol{r}_h) is only slightly distorted, then the integral in the numerator can be evaluated as (a_i/a_{\rm ex})^{3/2}. This immediately results in an estimate for f_i

f_i\approx\frac{a_i^3}{v}f_{\rm ex} .

This simple result reflects physics of the phenomenon of giant oscillator strength: coherent oscillation of electron polarization in the volume of about a_i^3 >> v.

If the exciton is bound to a defect by a weak short-range potential, a more accurate estimate holds

f_i=8\left(\frac{\mu}{m}\frac{E_{\rm ex}}{E_i}\right)^{3/2}\frac{\pi a^3_{\rm ex}}{v}f_{\rm ex}.

Here m=m_e+m_h is the exciton effective mass, \mu=(m_e^{-1}+m_h^{-1})^{-1} is its reduced mass, E_{\rm ex} is the exciton ionization energy, E_i is the binding energy of the exciton to impurity, and m_e and m_h are the electron and hole effective masses.

Giant oscillator strength for shallow trapped excitons results in their short radiative lifetimes

\tau_i\approx\frac{3m_0c^3}{2e^2n\omega_i^2f_i} .

Here m_0 is the electron mass in vacuum, c is the speed of light, n is the refraction index, and \omega_i is the frequency of emitted light. Typical values of \tau_i are about nanoseconds, and these short radiative lifetimes favor the radiative recombination of excitons over the non-radiative one.[5] When quantum yield of radiative emission is high, the process can be considered as resonance fluorescence.

Similar effects exist for optical transitions between exciton and biexciton states.

An alternative description of the same phenomenon is in terms of polaritons: giant cross-sections of the resonance scattering of electronic polaritons on impurities and lattice defects.

Bound excitons in semiconductors: Experiment

While specific values of f_i and \tau_i are not universal and change within collections of specimens, typical values confirm the above regularities. In CdS, with E_i\approx6 meV, were observed impurity-exciton oscillator strengths f_i\approx 10.[6] The value f_i>1 per a single impurity center should not be surprising because the transition is a collective process including many electrons in the region of the volume of about a_i^3>>v. High oscillator strength results in low-power optical saturation and radiative life times \tau_i\approx 500 ps.[7][8] Similarly, radiative life times of about 1 ns were reported for impurity excitons in GaAs.[9] The same mechanism is responsible for short radiative times down to 100 ps for excitons confined in CuCl microcrystallites.[10]

Bound molecular excitons

Similarly, spectra of weakly trapped molecular excitons are also strongly influenced by adjacent exciton bands. It is an important property of typical molecular crystals with two or more symmetrically-equivalent molecules in the elementary cell, such as benzine and naphthalene, that their exciton absorption spectra consist of doublets (or multiplets) of bands strongly polarized along the crystal axes as was demonstrated by Antonina Prikhot'ko. This splitting of strongly polarized absorption bands that originated from the same molecular level and is known as the 'Davydov splitting' is the primary manifestation of molecular excitons. If the low-frequency component of the exciton multiplet is situated at the bottom of the exciton energy spectrum, then the absorption band of an impurity exciton approaching the bottom from below is enhanced in this component of the spectrum and reduced in two other components; in the spectroscopy of molecular excitons this phenomenon is sometimes referred to as the 'Rashba effect'.[11][12][13] As a result, the polarization ratio of an impurity exciton band depends on its spectral position and becomes indicative of the energy spectrum of free excitons.[14] In large organic molecules the energy of impurity excitons can be shifted gradually by changing the isotopic content of guest molecules. Building on this option, Vladimir Broude developed a method of studying the energy spectrum of excitons in the host crystal by changing the isotopic content of guest molecules.[15] Interchanging the host and the guest allows studying energy spectrum of excitons from the top. The isotopic technique has been more recently applied to study the energy transport in biological systems.[16]

See also

References

  1. R. J. Elliott, Intensity of optical absorption by excitons, Phys. Rev. 108, 1384 (1957).
  2. V. L. Broude, V. V. Eremenko and É. I. Rashba, The Absorption of Light by CdS Crystals, Soviet Physics Doklady 2, 239 (1957).
  3. E. I. Rashba, Theory of the impurity absorption of light in molecular crystals, Opt. Spektrosk. 2, 568-577 (1957)
  4. E. I. Rashba and G. E. Gurgenishvili, To the theory of the edge absorption in semiconductors, Sov. Phys. - Solid State 4, 759 - 760 (1962)
  5. E. I. Rashba, Giant Oscillator Strengths Associated with Exciton Complexes, Sov. Phys. Semicond. 8, 807-816 (1975)
  6. V. B. Timofeev and T. N. Yalovets, Anomalous Intensity of Exciton-Impurity Absorption in CdS Crystals, Fiz. Tverd. Tela 14, 481 (1972).
  7. M. Dagenais, Low-power optical saturation of bound excitons with giant oscillator strength, Appl. Phys. Lett. 43, 742 (1983); http://dx.doi.org/10.1063/1.94481.
  8. C. H. Henry and K. Nassau, Lifetimes of Bound Excitons in CdS, Phys. Rev. B 1, 1628 (1970).
  9. E. Finkman, M.D. Sturge, and R. Bhat, Oscillator strength, lifetime and degeneracy of resonantly excited bound excitons in GaAs, Journal of Luminescence 35, 235 (1986)
  10. A. Nakamura, H. Yamada, and T. Tokizaki, Size-dependent radiative decay of excitons in CuCl semiconducting quantum spheres embedded in glasses, Phys. Rev. B 40, 8585(R) (1989)
  11. M. R.Philpott, Theory of the Vibronic Transitions of Substitutional Impurities in Molecular Crystals. The Journal of Chemical Physics, 53, 136 (1970).
  12. H.‐K. Hong and R. Kopelman, Exciton Superexchange, Resonance Pairs, and Complete Exciton Band Structure of {^1}B_{2u} Naphthalene, J. Chem. Phys. 55, 724 (1971); http://dx.doi.org/10.1063/1.1676140
  13. K. P. Meletov and M. F. Shchanov, Rashba effect in a hydrostatically compressed crystal of deuteronaphthalene, Zh. Eksp. Teor. Fiz 89, 2133 (1985)
  14. V. L. Broude, E. I. Rashba, and E.F.Sheka, Anomalous impurity absorption in molecular crystals near exciton bands, Sov. Phys. - Doklady 6, 718 (1962).
  15. V. L. Broude, E. I. Rashba, and E. F. Sheka, Spectroscopy of molecular excitons (Springer, NY) 1985.
  16. C. Paul, J. Wang, W. C. Wimley, R. M. Hochstrasser, and P. H. Axelsen, Vibrational Coupling, Isotopic Editing, and β-Sheet Structure in a Membrane-Bound Polypeptide, J. Amer. Chem. Soc., 126, 5843 (2004).
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