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TEMPORAL DECAY OF THE OPTICALLY
INDUCED PROPERTIES OF A SMALL-POLARONIC SOLID
D. Emin
To cite this version:
D. Emin. TEMPORAL DECAY OF THE OPTICALLY INDUCED PROPERTIES OF A SMALL-POLARONIC SOLID. Journal de Physique Colloques, 1981, 42 (C4), pp.C4-535-C4-538.
�10.1051/jphyscol:19814115�. �jpa-00220732�
JOURNAL DE PHYSIQUE
CoZZoque C4, s?~ppZ&ment au nolO, Tome 4 2 , octobre 1982 page C4-535
TEMPORAL DECAY OF THE OPTICALLY INDUCED PROPERTIES OF A SMALL- POLARONI C SOLID*
Sandia National Laboratoriest, AZbuquerque, New Mexico 87285, U . S . A.
Abstract
-
The essential physics of the photoluminescence of a small-polaronic solid (one in which electrons, holes and ex- citons all find it energetically favorable to self-trap) is described. Emphasis is placed on the dependence of the decay rates associated with the various luminescence bands on excita- tion energy and temperature. The results are in general accord with observations on chalcogenide glasses.Introduction.- An electronic excitation, be it a charge carrier or an exciton, in a material characterized by a short-range electron- lattice interaction can exist in one of two distinct types of
~ t a t e s . l - ~ The excitation can be spread out over a very large num- ber of atomic sites with the displacements of the equilibrium posi- tions of the atoms in its vicinity being small compared with their zero-point amplitudes. This type of excitation is nonpolaronic.
Alternatively, the excitation can be localized severely, with a spatial extent less than or comparable to an interatomic separation.
In the latter situation, the atoms surrounding this excitation will experience substantial alterations of their equilibrium positions:
the excitation is small-polaronic. For an excitation to exist without being small-polaronic it must be able to move between sites suf-
ficiently rapidly so as to preclude the atoms in its vicinity from suffering significant displacements in response to its presence.3 Although the properties of small-polaronic and nonpolaronic excita- tions are qualitatively distinct from one another, the parameters in many semiconductors and insulators are such that they are near the borderline which determines whether the excitations are polaronic or nonpolaronic. 2 * Since disorder fosters localization, some non- polaronic excitations in an ordered crystal become polaronic in their disordered c o u n t e r ~ a r t . ~ This has been verified in (magnetic and mixed) crystals in which disorder is imposed in a controlled manner. In what follows, it will be presumed that electrons,
holes and excitons all find it energetically favorable to self-trap, forming small polarons and self-trapped excitons, respectively.
Self-trapping Times.- Even if it is energetically favorable for an excitation to self-trap, the time required for an injected excitation to self-trap may be significantly in excess of the minimal time for an atomic displacement, a vibrational period. 6 r The self-trapping time decreases as the coupling of the excitation to the atomic dis- placements increases.6 Thus, since, by virtue of their chargk, charge carriers typically interact more strongly with the atomic displacements than do excitons, they are associated with shorter self-trapping times than excitons. Furthermore, in that disorder facilitates localizationl. it reduces self-trapping times. Here, anticipating application to the chalcogenide glasses (in which there is evidence of near minimal self-trapping time for holes and no observation of the motion of electrons5), it is assumed that both
*Supported by the U. S-; Department of Energy under contract DE-AC0476-DP00789.
t A U. S. Department of Energy facility.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19814115
JOURNAL DE PHYSIQUE
carriers self-trap in a minimal time. However, the possibility of lengthy self-trapping times for at least some excitons will not be excluded.
With the absorption of a super-band-gap photon an electronic excitation is created. If, after initial thermalization, the wave- functions associated with the electron and hole do not overlap appreciably with one another, they are regarded as separated charges which readily self-trap to form small polarons. However, an excitonic excitation, characterized by overlapping electron and hole wave- functions, may manifest an appreciable self-trapping time. Although the time for self-trapping is minimal for spatially compact excitons, for more extended excitons (be they intrinsic or impurit related) the lifetime against self -trapping can be rather long. ql' Excitons in molecules often possess lifetimes of the order of
microsecond^.^
Distributions of electron-hole separations and exciton radii are asso- ciated with the absorption of super-band-gap photons of a given
energy. With increasing photon energy the average el-ectron-hole separation and the average exciton radius associated with these respective distributions are both presumed to increase.
The time in excess of a vibrational period required for self- trapping depends exponentially on an energy barrier.
'
This energy barrier decreases (roughly) as the inverse square of the exciton radiusi6 it vanishes for excitons of sufficiently small radius. Due to the strong dependence of the self-trapping time on exciton radius it will be assumed, for simplicity, that excitons of radius, R, less than a critical radius, RC, will self-trap so rapidly compared with their characteristic recombination lifetime and the time scale of the experiments, nanoseconds to microseconds, so as to be presumed to be created as self-trapped excitons. Large-radius excitons, R > RC, will be presumed to enjoy the complementary situation: sufficiently long lifetimes against self-trapping so that their relaxation andrecombination does not involve the self-trapped state.
Luminescence.- Three luminescence processes are associated with recombination in such a polaronic s01id.~ First, at the highest energy, even well above the nominal (absorption) bandgap, there can be radiative recombination of the nonpolaronic molecular-type excitons.
Second, there can be a Stokes-shifted luminescence due to the
recombination of self-trapped excitons. Finally, separated electron and hole small polarons can radiatively recombine to produce another Stokes-shifted luminescence. An important aspect of the recombination in a polaronic solid is that, due to interference between the atomic displacement patterns about oppositely charged small polarons,
theJ
can experience a short range repulsion which impedes their merger.
As a result the recombination of a self-trapped exciton (a self- trapped electron and hole centered at the same location) is distinct from the recombination of electron and hole small polarons (carriers cente d at different sites). For example, interpreting the data on a- As2S3" in terms of this polaronic model indicates that excitation above the 2.4 eV bandgap produces a self-trapped exciton luminescence centered at 1.6 eV and a small-polaron luminescence whose centroid shifts downward in time to that characterizing the minimum energy separation (-5 A), 1.2 eV.
Relaxation Processes.- The principal features of the relaxation of the three types of excitations are summarized in Table 1. Namely, the exciton, by virtue of its large radius, is weakly coupled to the lattice displacements. As a result, the rate characterizing the multiphonon emission required for nonradiative recombination is extremely small. On the other hand, because the wavefunctions of the electron and hole overlap strongly, radiative recombination is
generally expected to occur with an appreciable rate. In addition, there is the possibility of the exciton relaxing via transfer to other potential exciton sites with a small exchange of energy with phonons
.
Since the radius of the self-trapped exciton is of the order of an interatomic spacing, the interaction of the self-trapped exciton with the atomic displacments is strong. Nonetheless, since the magnitude of the Stokes shift of the involved luminescence, 0.8 eV,
is small compared with the difference of the electronic energies (the energy of the emitted light), 1.6 eV, the nonradiative recombin- ation rate, at least for a-As2S3, remains small. On the other hand, due to the strong overlap of the electron and hole of the self- trapped exciton, radiative recombination presumably occurs with an appreciable rate. Finally, the motion of self-trapped excitons is generally very slow at low temperatures due to the polaronic effect:
the atomic displacement pattern must move with the self-trapped exciton.
Electron and hole small polarons interact much more strongly with the atomic displacements than do the self-trapped excitons. In fact, unlike the situations of the two types of excitons, the Stokes- shift of luminescence, -1.2 eV, is comparable to the electronic energy associated with multiphonon non-radiative recombination (the energy of the emitted light), also -1.2 eV. Furthermore, the
radiative recombination rate is reduced from that of the excitons due to the spatial separation of the electron and hole small polarons.
Hence, in accord with observations, it is reasonable to presume that the recombination associated with the lowest energy luminescence band is primarily nonradiative.
Small-polaron pairs which are closer than the minimum-energy separation will, because of the repulsive nature of their short-range interaction, recombine at a higher energy than those corresponding to the minimum-energy separation. Thus, presuming that the small polarons are created with a spread of separation lengths, closer pairs having an enhanced probability of radiative recombination, the luminescence will shift in time to lower energy as some close pairs recombine and others hop apart toward their minimum-energy separation.
Non-
Electron- Radiative Radiative Spatial Band Radius Coupling Rate Rate Transfer
VerY hergy
Exciton R > R , > > a Weak Small Large Transfer
Self +rapped Polaronic
Exciton R
-
a strong Snall Large H ~ i n gSeparated VerY Luminescence
Small Polarons R
-
a Strong Large Small Shifts in Time TABLE l-
Relaxation Processes ( a - ~ s p ~ )Well-separated pairs must hop together before they achieve an appreciable recombination rate. Since the spatial gradient of the long-range attractive interpolaron interaction falls with separation, as the interpolaron separation increases nearest-neighbor small- polaron hopping involves sites with decreasing energy dis~arities.~
As a result, because the low-temperature small-polaron hopping rate falls dramatically as the energy difference between initial and final
C4-530 JOURNAL DE PHYSIQUE
sites f a l l ~ , l ~ r ~ ~ the recombination rate of pairs declines as their initial separation increases. This yields a recombination rate which falls sublinearly with time for rather long times. In fact, below liquid nitrogen temperatures sufficiently separated pairs (215 A) may remain metastably separated for many minutes.14 Such time dependences have been generally observed in chalcogenide g1asses.l1 Luminescence Decay.- For the nonpolaronic exciton the temperature dependence of the shift of the luminescence results from phonon-
assisted energy transfer involving states with a weak electron-lattice interaction. Since multiphonon processes are rare, the temperature dependence of the decay is weak: however, it increases as the phonon temperature is reached. The quantum efficiency of the self-trapped- exciton luminescence falls as the excitation energy is increased since larger-radius excitons and more small polarons are then pro- duced. However, the decay of the luminescence is not dependent on the excitation energy. Also, since the recombination is primarily radia- tive, its decay has no temperature dependence. The low-temperature decay of the small-polaron luminescence is affected by the excitation energy because the distribution of pair separations is altered. The decay rate falls with excitation energy as the pairs are produced increasingly far apart. At sufficiently high excitation energies the excitation energy dependence of the short-time decay rate saturates since very distant pairs do not contribute to this luminescence.
Since nonradiative recombination dominates, the short-time decay rate increases with temperature above about one third the phonon temperature, where the rates of multiphonon emission are greatly enhanced.13 At sufficiently high temperatures the decay can be slowed when further separated small-polaron pairs can hop to their minimum energy separation and recombine. It should be noted that shift of the centroid of the small-polaron luminescence will have three time regimes. The first (fast decay) is associated with the dominance of close pairs recombining. The second (slower) is asso- ciated with the increased contribution of recombination of pairs near the minimum energy separation. The third (again faster) occurs when distant pairs have sufficient time to hop together and recombine.
These features are in general accord with the observations of M. A.
Bosch, J. Shah and V. T. Nguyen on As2S3, some of which are as yet unpublished.
References 1. RASHBA, E. I., 0pt.Spektrosk.
2
(1957) 75.2. TOYOZAWA, Y., Prog.Theor.Phys. 26, (1961) 29.
3. EMIN, D. Adv.Phys. 22 (1973) 5 7 7
4. EMIN, D., Physics ~ F ~ i s o r d e r e d Solids, S.S. Mitra, ed. (Plenum, New ~ o r k ) 1967, D. 385.
5. EMIN, D., ~ol~crystalline and Amorphous Thin Films and Devices L. L. Kazmerski, ed. (~cademic, New York) 1980, p. 17.
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G.,
J.Phys. C-~'(1977) 3391.8. GUTMANN, F. and LYONS, L. E., Organic Semiconductors (Wiley, New ~ o r k ) 1967.
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