Physica 116B (1983) 7-17
Paper presented at ICDS-12 Amsterdam, August 31 - September 3, 1982
North-HoUand Publishing Company
ELECTRON INDUCED LATTICE RELAXATIONS AND DEFECT REACTIONS
Institute for Solid State Physics, University of Tokyo Roppongi 7, Minato-ku, Tokyo 106, Japan Stability of defect-bound electrons in the phonon fields is discussed with particular attention to the different roles of short and long range parts of electron-phonon, electron-defect and defect-phonon interactions. The short range electron-phonon interaction triggers shallow-deep instability with stable and metastable states separated by a potential barrier. Extrinsic self-trapping, persistent photoconductivity and negative U effect are discussed in this context. With repulsive short range and attractive long range electron-defect or electron-hole interaction consisting of phonon= mediated as well as direct ones, there can occur separated self-trapping with broken parity. Dynamics of nonradiative recombination via a deep state and recombination induced defect reactions is described in terms of the accepting and reaction modes.
retical argument for the shallow-deep intability the large-small (orbital radius) discontinuityof a defect-bound electron(s) in the phonon field, and to demonstrate how a variety of phenomena can be explained in a consistent way on the basis of it. In Section 2 we introduce the notion of "extrinsic self-trapping" as an extension of (intrinsic) self-trapping, and emphasize the importance of short-range electron= p h o n o n interaction as a trigger for the shallow= deep instability and its modification discussed in Section 3. The argument will be extended to a system of two electrons in Section 4, whereby we study the phonon-mediated electron-electron attraction and find again the essential role of the short range electron-phonon interaction for the "negative U" situation. Instabilities of the similar and the opposite types in the electron-hole system will be described in Section 5, and be compared with those of the preceding sections. Sections 6 and 7 are devoted to the qualitative description of various dynamical processes caused by strong electron-phonon coupling, such as the formation of a deep state after electron capture, the electron-hole recombination with multiphonon emission, and the recombination-induced defect reactions. -
The electron-phonon interaction is usually considered to be a small perturbation for the behavior of electrons in typical semiconductors such as the IV-th column elements, IK-V and ~ compounds. Namely, the mean free path of carriers limited by phonon scattering is much larger than the lattice constant even at high temperatures, and the lattice distortion due to the capture of a carrier at a shallow impurity state is negligibly small. On the other hand, it has been pointed out that the capture of a carrier at the deep level lattice defect sometimes causes a strong distortion of the surrounding lattice [ I ~ 1 2 ] . It has also been found that nonradiative recombination via the deep level gives rise to defect reactions which play important roles in the degradation of opto-electronic devices [13~17]. These facts, characteristic of strong electron-phonon coupling, may be somewhat disconcerting in view of the aforementioned weak coupling situation prevailing with free and loosely bound eJectrons in tbe same host crystals. Of course, these variable situations are consistent with the general trend  that the effective electron-phonon coupling strength (e.g. the magnitude of capture induced lattice distortion) increases as the orbital radius of the bound electron decreases, and hence, as the binding energy increases. However, the remarkable difference between the weak and strong coupling situations as found in varioud observations seems to indicate the existence of a sort of shallow-deep instability which, in the present case, results from the cooperative effect of the electron and the lattice [6,9,19]. The bound electron would induce lattice distortion around itself, which in turn would help to bind the e l e c t r o n - - the mechanism similar to the self= trapping in perfect lattice . The purpose of this paper is to present a theo-
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The most transparent way of deriving the shallow -deep instability and clarifying the role of the electron-phonon force range is to make use of the continuum model for the phonon fields. We consider an electron in the conduction band with effective mass m around a point defect (impurity , vacancy or interstitial atom) at r = 0, both interacting with the elastic and dielectric medium described by the dilation A(r) and the electrostatic potential ~(r) due to the ionic displacement polarization P(r) ( ~ V ~ ( r ) ) which represent respectively the acoustic and the optical phonon fields. The total energy of this electron-phonon system will be given by a functional
Y. Toyozawa / Electron induced lattice relaxations
~ ( r ) L. -~2V2 ~m +v0~(r) _ Z e ~r
+ f [Ed0(r)2 +EdD6(r)]A(r)dr + f [(-e)~(r) 2 +Ze6(r)]~(r)dr
(1) plus the lattice kinetic energy which will be neglected in our adiabatic approximation. The electron and the defect interact with A(r) through their respective deformation potentials Ed and Ed D, and with ~(r) through their respective charges -e and Ze, in addition to their mutual interaction consisting of short and long range parts: v06(r) and -Ze2/6~r. The last line of eq.(1) denotes for the lattice potential energy, where C is an elastic constant and ~ defined by 6-i = 6 ~ I _ 6o--1 with C 0 and 6 ~ being respectively the static and the high frequency dielectric constants. Minimization of the functional (I) with respect to ~ would give the adiabatic potential E[@(A,~) ,A,~] ~ E[A,%] for the lowest electronic state. If one is interested only in the extrema of this adiabatic potential, one can take an alternative but simpler way by inverting the order of extremizations. Firstly, minimization of eq.(1) with respect to A and ~ gives the "field-source" relations: -CA(r) =EdO(r) 2 +EdD6(r)
4 ~ V 2 ~ = (-e)0(r) 2 +Ze6(r)
It can be seen from eq.(2) that an isoelectronic substitutional impurity with radius smaller or larger than the host atom has positive or negative EdD , respectively, that an interstitial atom has negative EdD and that the sign is not definite for a vacancy. Putting eqs.(2) and (2') into eq.(1) and removing the defect self~ energies (o¢(EdD)2,~(Ze)2) which do not depend on the electronic state, one obtains the functional E[*] =
~2 0(r) [-~-~mV2 + (v 0
-~e210(r)d r Or - ~1 ffdrdr'~(r) 2 "Ed2 [--~--"
') *(r') 2
As the second step, one extremizes E[}] with respect to @ to obtain the extrema of the adiabatic potential. It is to be noted that the product of the first and the second terms on the right hand side of
eq.(2) contributes to e~.(3) the short range impurity potential (-EdEdD/C)6(r). The corresponding quantity from eq.(2') contributes the long range screening potential (i/6~- 1/60)Ze2/r which reduces the impurity Coulomb potential - Z e 2 / ~ r into -Ze2/~0r. The factor -1/2 in the second term of eq.(3) comes from two contributions - - -i from the electron-phonon interaction and +1/2 from the phonon potential energy (virial theorem) as is obvious from the derivation. This second term representing the self= energy of the e l e c t r o n - - the effect of self= induced lattice distortion upon itself - - a g a i n consists of short and long range parts. It has the effect of localizing the electron; in particular, the short range part triggers the shallow-deep instability in the defect system and the self-trapping in the host lattice as will be shown below. In order to find the extremum values of the adiabatic potential, we put a trial function @(r) = (/2/a) 3/2exp[-~(r/a)2] into eq.(3), and obtain
a0 2 -Es(~--) a0 3 - S [ ( ~ E=S(a)=S(~-) where B - 3 ~ 2 / 2 m a o Es
EdEd D 23/2 Ed2 F (-v 0 + - - ~ - - ) z 2Ca03 a03
e2 E~ = ~
We will assume E s > 0 and E~ > 0 in the present section. It is convenient to introduce the short and long range coupling constants gs and g~ by gs,% ~ Es,z/B ~ gs,~ L + g s ,D~
where the decomposition into the lattice (L) and defect (D) parts corresponds to the first and second terms in both of eqs.(5) and (5'). In eq.(4), the parameter a is to be varied between a 0 and ~ since the orbital radius a of the envelope function @(r) should not be smaller than the lattice constant a 0. As a function of : a0/a , eq.(4) has a minimum F and a maximum M at I F , M = [i$ i/~gsgA]/3gs, respectively, with energy difference E M - E F z ~* = (4B/27gs2)(l3gsgA) 3/2, provided that 3gsg ~ < i. It should also be noted, however, that the maximum IM is within the proper range [0,i] only when the inequalities 3g s + g K > 2 and 3g s > 1 are both fulfilled, and that in this case the end point IS : 1 gives another minimum ES( = B - E s - Et). When either i) 3gsg ~ > 1 or ii) 3g s + g A > 2 with gg > 1 is satisfied, the only minimum is %g - i. In the remaining region: 3g s + g A > 2, %F is the only minimum. Thus we obtain the phase diagram of Fig.l, wbich indicates, by simbols F or S, the stable state, and if any, the metastable state in parenthesis, on each region of (gs,gg)-space [6,9]. Across (towards right) the solid line which corresponds
Y. Toyozawa / Electron induced lattice relaxations
from the electron-phonon part of it as is obvious from the fact that the instability is connected with the two minima of the adiabatic potential in the configuration coordinate space. The discontinuity which remains even with the static electron-defect part (of gs) alone is an artefact of our trial function; in fact, a more exact theory gives discontinuity which vanishes with E d . The configuration coordinate model of the defect state is shown schematically in Fi~.2(a) for various values of gs D with fixed gs < 1 and g£ =0, while Fig.2(b) is the phase diagram for the stable state shown on the (gs D, gsL)-plane. The discontinuity across the solid line in Fig.2(b) disappears as one approaches the gsD-axis.
04 8~ ~i S 0,6 I
,,, I r
," 2/:5 /
i . . . . .
I gs --~ 0
Fig.l The phase diagram for the stable (metastable, when inside a parenthesis) state of an electron subject to short range (gs) and long range (g£) interactions with a lattice defect and phonon fields [6,9,52]. F and S mean a shallow (unbound, when the defect charge Z E 0) state and a deep state, respectively• The configuration coordinate model is also shown schematically for each region, where the lowest line should be replaced by the broken line when Z i 0.
to the condition: E F =Es, the orbital radius of the stable state shrinks discontinuously from a F ~A~la0 > a 0 to a S - A~la 0 = a 0 which amounts to an increase in electron binding energy of gsL(As 3 %F 3) + g ~ L ( A s - %F) as is seen from the virial theorem mentioned before. In the regions on the both sides of this discontinuity line, the shallow state F and the deep state S coexist as stable and metastable states (or vice versa). This is nothing but the "shallow-deep instability" alluded in §I. As far as we know, this terminology was first used by Altarelli  in his study of the binding energy of donors and core excitons which is a purely electronic problem. Here we will use it for this electron-phonon system which shows the bistability as described above. As seen from Fig.l, the discontinuity originates from the short range coupling gs (no discontinuity along the gE axis), or to be more exact,
Fig.2 (a) The configuration coordinate model of an electron subject to short range defect attractive potential with gs D and phonon interaction with gs L. (b) The corresponding phase diagram indicating the stable state on the (gsD,gsL)-plane .
The variational method for the localized electron-phonon system, as presented above, was first introduced by Kubo  to study the strong effect of electron-optical phonon interaction on color centers in alkali halides. Consideration of short range electron-acoustic phonon interaction thus introduces a new aspect. The phase diagram in Fig.l was first derived for an electron in deformable lattice without a defect . In that case, F and S represent the free polaron state and the self-trapped state, respectively. With a defect, we can use the same phase diagram provided we redefine gs and g~ by adding contributions gs,~ D from the defect potentials (see eqs.(5), (5') and (6)) and reinterpret F and S as the shallow (when Z > 0) and the deep states, respectively (F remains to be a free state if Z = 0). In the coexistence region (F(S) or S(F) in Fig.l), the two states are separated by a potential barrier, with activation energy e* (given before) needed for F + S transition. With g~D = Z = 0 and small g~L, one can conceive the situation that neither the defect potential nor the self-induced lattice distortion is enough to bind the electron (gs L < i, gs D < i) but their cooperation can bind it
Y. Toyozawa / Electron induced lattice relaxations
strongly (gs L + g s D > i), as is represented by the curves for gs D = 0.4 and 0.6 in Fig.2(a). Such a situation is called extrinsic self-trapping or defect (impurity)-assisted self-trapping [6,19]. The adiabatic potentials corresponding to these variable situations are shown schematically in Fig.l. There are a number of observations on the electron-induced large lattice relaxation in semiconductors, as already mentioned briefly in §i and as described in more detail in review articles [ 9 ~ 1 2 ] . We refer here only to a few typical examples of extrinsic self-trapping or shallow-deep instability. Firstly, Lang and Logan  introduced the configuration coordinate model shown in Fig.3 for the DX-center in the alloy AlxGal_xAs , in order to explain the persistent photoconductivity. Once the trapped electron is optically excited into the conduction band by absorbing a photon with energy > l.leV, the lattice around the center relaxes to the new equilibrium configuration (QR ÷ 0 ) for which the center has no bound state (this means Z = 0 so that the F state is a free polaron). Thermal activation energy: ~* = 0.2eV is established from the temperature dependence of the electron recapture cross-section; thus, at low temperature the conductivity lasts for many hours after photoionization of the trapped electrons.
N z I
(stable) which can be converted to the former through optical ionization (see Fig.4). This system, like the preceding one, corresponds to the S(F) region of our phase diagram, Fig.l.
CdF2:In 0.05%mol In3++e;'~./
o° inz+ Fig.4 The configuration coordinate model for CdF2:In, introduced by Langer et al.[4,11].
An exciton is a neutral particle which is subject only to the short range interaction with phonons and defects (g~L = g ~ D = 0). A typical example of extrinsic self-trapping of exciton is TIC1 containing isoelectronic Br impurities . In the rigid lattice, a Br impurity cannot bind the exciton (indirect) as is evident from the amalgamation type absorption edge of mixed crystals TICII-xBr x. On the other hand, the luminescence from the relaxed exciton in TICI:Br is a broad Stokes-shifted band characteristic of the S state while that of pure TIC] (as well as TIBr) is a sharp,resonant line characteristic of the F state (see Fig.5). Namely, the cooperation of the impurity potential and the exciton-lattice interaction is necessary to localize the exciton. Reviews of theoretical and experimental studies on the F-S instability in perfect lattice are found in references [ 9 , 2 5 ~ 2 7 ] . 3.
Fig.3 The configuration coordinate model for the DX-center in the alloy AlxGal_xAs, introduced by Lang and Logan .
The second example is the charged center, CdF2: In, which has Z = 1 since In +++ replaces Cd ++. Langer et al [4,11] found that an electron trapped at this center has two relaxed states: the shallow coulombic state F (metastable), and the deep state S with strong lattice distortion
If the short range part of the defect potential in eq.(3) is repulsive: (v 0 - EdEdD/C) > O, the trial wavefunction of Gaussian form which has the greatest amplitude at the defect is clearly inadequate. When the repulsion is strong enough, we can treat it like a hard core. The radial wavefunction of the Coulomb bound state (we assume Z > 0 and neglect the long-range electron= phonon interaction, for simplicity) in rigid lattice is thus suppressed not only at r ÷ ~ but also at r + 0 (or more physically, at r < a0).
Y. Toyozawa / Electron induced lattice relaxations
mobility. On the other hand, it is more reasonable to r e p l a c e ~ 0 in eq.(7) by a smaller value since dielectric screening is not fully effective for the atomic distance a 0. So, these two materials are good candidates to observe the symmetry broken deep state.
exciton band Qn(host) Qo(impurity) S~em. abs. em. JJ(impure)
Fig.5 The configuration coordinate model for the exciton in TICI:Br .
It is then difficult to imagine that the shallowdeep instability takes place as a shrinking of orbital radius, namely, as a change of radial wavefunction only. The spherical shell like distribution of ~(r) 2, and hence of A(r), is not very favorable in stabilizing the self-energy (see the second line of eq.(3)). The short range electron-phonon interaction will favor the three-dimensional self-localization rather than the one-dimensional localization in radial direction only. Thus, the s h a l l o w ÷ d e e p instability will take place as a sudden localization of angular as well as radial wavefunctions, whereby the parity (inversion symmetry) is broken. This state is a modification of the dichotomy of an electron-hole pair  as will be mentioned in Section 5. If the self-trapped electron is stable already in the host crystal and the defect Coulomb potential -Ze2/~0 r is not strong, the deep defect state mentioned above is nothing but the self= trapped electron located at a distance of r ~ a 0 from the defect, We expect this symmetry broken deep state to be stable under the following condition which is less stringent than for the host self-trapping, Ed 2 + Ze 2 B <
3 a0 ) (i
The term with a H ~ 2 ~ 0 / m Z e 2 represents the energy of the shallow state under the Coulomb attraction plus the hard core.
NEGATIVE U SYSTEM
In order to explain the unexpectedly small spin paramagnetism in semiconducting glasses with high density of defects, Anderson  suggested that strong electron-phonon coupling results in the phonon-mediated attraction between the two electrons on the same defect which overcomes the Coulomb repulsion U between them, thus favoring a paired electron ground state. Some of the defects in crystalline semiconductors have also been identified [32,7,8] to belong to this "negative U" system. To study this and related problems more systematically, we extend the theory of electrondefect-phonon interaction presented in 52 to the two electron system. Although more elaborate studies of electron-correlation in similar systems have been presented [33,34],we assume here for simplicity that two electrons with opposite spin occupy the same orbital ~(r) without correlation. Then the total energy K E will be given by simply doubling the terms with ~(r) 2 in eq. (i) and adding the electron-electron (e-e) Coulomb energy in the dielectric medium: U = f/dr I. e2 [email protected]
(rl)2~ jrl_r2j ~(r2) 2 . After minimizing ~E[~,A,~] with respect to A and ~, we obtain the energy expression consisting of 2 times the first term plus 4 times the second term of eq. (3) plus the e-e Coulomb energy. Putting in the same trial function @(r) as before, and writing ~E(a) = 2 B [ ~ 2 - ~ g s ~3 - Kg£~]
in conformity with eqs.(4) and (6), we find ~ g s = g s + (Ed2/2Ca03)/B ' ~g£ =g~-
The second term of eq.(9') is obtained as the sum of two contributions: +(e 2/~a0) which is the counterpart of the second term of eq.(9), and -(e2/6~a0) which comes from the e-e Coulomb energy. The effective e-e interaction, U*, is defined as the energy difference between a defect with paired electrons (eq.(8)) on the one hand and a pair of separated defects each with a single electron (eq.(4)) on the other hand: U* ~ K E ( a ) m i n ~ 2E(a)mi n
The positive holes in AgBr and TIBr have metastable self-trapped states  at e = B - (Ed2/ 2 C a 0 3 ) ~ O . 1 4 e V higher than the bottom of the conduction band as can be estimated from the anomalous temperature dependence [29,30] of the drift
or U*/2B=[~2 - ~ g s % 3 _
Y. Toyozawa / Electron induced lattice relaxations
where the minimization with respect to ~ (0 5 % 5 i) should be performed separately for each expression [...]. Immediate consequences of eqs.(9), (9') and (i0) are as follows. (i) If the S state is more stable than the F state in the single electron system, it is always the case in the paired electron system (provided Ed2/2CaO0 3 ~,~ e2/~0a0 . Namely, the r e l a t i v e favorability of S against F is greater for the paired electron system under that condition. (ii) If in both systems the F state is stable with the predominance of the long range interactions, U* is always positive due to (9'), as is also obvious from electrostatic argument. (iJi) If in both systems the S state is stable, U* ~ 0 according as Ed2/2Ca03 ~ e2/~0a0 . (iv) When the F state is stable in single electron system and the S state is stable in paired electron system (the opposite situation is excluded by (i)), one must explicitly calculate (i0) in order to find the sign of U*. Here again, a great enough Ed2/2Ca03 is favorable for negative U*. The above argument clearly indicates that the short range electron-phonon interaction is essntial for negative U*, and that it is realized only (when at least the paired electron system is) in the S state. In this connection, two important situations included in the case (iv) should be mentioned. (a) A neutral defect (Z = 0 so that gz = g £ L) cannot bind one electron but can bind two electrons if Ed2 gs + g
L < 1 < gs + g
L + (
e2 ~0a0 )
is satisfied (we have neglected the polaron (F) self-energy- (g~L) 2/4). Similar situation was already discussed by Matsuda and Ohata . (b) In the absence of defect, eq.(ll) with gs replaced by gs L 5 Ed2/2Ca03 gives the condition for "pair self-trapping" despite the absence of single self-trapping. It is different from the so-called "bipolaron" which usually is imagined to be a moving pair of large polarons and hence to be the F state according to our terminology. The bipolaron in this sense is rather improbable according to the above argument (at least in three dimensional system). 5.
SELF-TRAPPING OF AN ELECTRON-HOLE PAIR
In this section, we shall be concerned mainly with defects-free lattice with an electron-hole (e-h) pair, for the following reasons. When one of the two particles is self-trapped, it is essentially like a charged defect, and the behaviors of the other particle are expected to be similar to those studied in Sections 2 and 3. Secondly, it will be instructive to compare the behavior of an e-h pair with that of an e-e pair described in Section 4. Finally, the e-h system has something to do with the formation of defects as will be discussed in Section 7.
Let us replace the electron-defect-phonon system in eq.(1) by an e-h-phonon system with wavefunction ~(re,rh) and write down the adiabatic energy as E [ ~ , £ , ~ ] = f f d r e d r h ~ r t -~ ~2 v~ 2e - ~~v2 ~ Ih e2
+ (dredrh[EdeA(re) + EdhA(rh)]~ 2 + I dredrh[- e~(re) + e~(rh)]~2
After minimization with respect to A(r) and ~(r), we approximate ~(re,rh) by a p r o d u c t ~e(re)~h(rh) neglecting the correlation, and put Pe,h(r) ]~e,h(r) I2. Then we obtain •62 2 1 E[~e,~h] = [ f d r 2 m e ( V ~ e ) - ~ f f d r d r ' P e ( r ) " .(Ede)2 ,) e2 •~ 6 (r-r + 6--~__rST}0 e (r') ] + [the same expression for hi
+ ff drdr'Pe(r) ~"EdeEdh - - ~ - - - % ( r - r ,) e2
The acoustic phonon mediated e-h interaction is short-ranged and is attractive or repulsive acc o r ding as E d e E d h >< 0, while the optical phonon mediated e-h interaction is long-ranged repulsive Coulomb potential: e2/61re-rh[ which reduces the direct attractive p o t e n t i a l - e 2 / ~ ] r e - r h ] into -e2/601re-rhl. If one of the carriers, say, h, is self-trapped so that the charge Ph(r) is almost confined to the central cell, the electron-dependent parts of eq.(13) approximately reduces to eq.(3) (with v 0 = O, Z = 1 and Ed D = E d b ). If both e and h are self-trapped with their own energies being given respectively by the first and second terms of eq.(13), the interaction between these dressed particles is given by the third term. When EdeEd h > 0, both the short and long range parts of the interaction is attractive, so that e and h like to be self-trapped on the same lattice site. Then, after putting P e ( r ) ~ p h ( r ) ~ p(r), the dielectric self-energy (~-I) vanishes due to the charge cancellation while elastic one turns out to be -
The effective deformation potential of this tightly bound pair is given by Ed e + E d h. When E d e E d h % 0 , the self-trapped e and the self-trapped h repel each other at short distance but attract each ether at long distance.
Y. Toyozawa / Electron induced lattice relaxations
MoTe elaborate study of the e-h pair interacting with the acountic photon field was made byA. Sumi,  with the use of e-h correlated trial wavefunctions as is necessary to describe the excitonic binding. The phase diagram of the stable state on the (Ede,Edh)-plane for a typical case with effective mass ratio mh/m e = 4 and relatively weak Coulombic attraction is reproduced schematically in Fig.6, with rescaling of the abscissa and the ordinate. The free exciton Fex
relative signs of Ed e and Ed h. Recently, Sibille and Mircea  found in InP a deep center with a bound pair of e and h, from which e can be released with 0.25eV activation energy~then h with 0.36eV. They ascribed the small e-b recombination rate of this center to the photon-mediated repulsion between e and h which makes them to be individually self-trapped around the defect (the defect-assisted Se:S h state of Fig.6). An alternative model for this system is proposed by Shinozuka . 6.
I Sh F
El Fig.6 The phase diagram for an electron-hole pair interacting with acoustic photons through the deformation potentials Ed e and Ed h .
unaccompanied by lattice distortion is stable when both of IEdel and IEdhl are small. When one of them, say IEdhl, exceeds a critical value which is proportional to m h - I / z , the hole is self-trapped, and the electron is loosely bound by its Coulomb field as in the shallow defect state (Sh:e in Fig.6). As Ed e increases in absolute value, with the same sign as Ed h (only the ist and 2nd quadrants are shown in Fig.6 since the phase diagram has inversion symmetry), there takes place a discontinuous transition into the Sex state with shrinking of the orbital radius of e-h relative motion, which resembles the shallow-deep instability of the defect state. The lattice distortion accompanying this small radius exciton is governed by the effective deformation potential (Ed e + E d n) as already mentioned and also born out by the 45 ° tilted borderline between Fex and Sex regions in Fig.6. The exciton can be self-trapped even if neither e nor h is, provided Ed e and Ed h have the same sign (to cooperate constructively). This effect is similar to the pair self-trapping mentioned towards the end of Section 4.
MULTIPHONON NONRADIATIVE RECOMBINATION
The deep defect state sometimes plays an important role as an intermediate state in nonradiative recombination of electrons and holes in semiconductors . A great number of photons corresponding to the band gap energy are released at one coup of this process: such is possible only with strong-electron photon coupling, the situation which is appropriately described by the configuration coordinate model . The whole process of recombination, consisting of successive electron and hole captures (or vice versa), can be described by the configuration coordinate model of Fig.7, where the uppermost, the middle and the lowest curves represent the energies of the state with a free electron and a free hole (double continuum), the state with a defect-bound electron and a free hole (single continuum) and the ground state without e-h, respectively, as functions of local distortion around the defect . One can classify the situation into four cases: extrinsic self-trapping (see Section 2) is needed for the first capture in (b), for the second capture (which means annihilation) in (c), and for both in (d). Since the attractive Coulomb potential always has bound states, the case (d) appears only for the defect charge Ze with Z ! - i - - rather unfavorable situation for the electron capture. It is to be noted that the lower one of any two adjacent energy states in Fig.7 represents a discrete bound state split off from the continuum of the upper state, and that always the lower curve is bent down as it approaches and merges tangentially into the upper curve (continuum) at e- or h-point: the upper continuum representing
i:!~:i:i:i:!:~Tiii!,~: In the second quadrant of Fig.6, there appears the region Se:S h in which e and h are self-trapped separately at a distance of the order of a 0 due to the short range repulsion between them as mentioned above. This phenomena is called dichotomy. The defect state of broken directional symmetry which takes place when v 0 - EdeEdD/C is positive (see §3) is an analogue of this Se:Sh state. Thus, the e-h system shows more variable situations than the e-e system because of the two possibilities of the
~ ............. :/,~ i ~ e ~ ~ :
~:.'~~:~'-'~,J . "
Fig.7 Configuration coordinate models for nonradiative recombination of an electron and a hole via a deep defect state .
Y. Toyozawa / Electron induced lattice relaxations
the extended state of the relevant carrier should not be influenced by the local distortion. The defect behaves as electron or hole capturing center according as the system is in the neighborhood of e or h point. The electron captured at e will become tightly bound and well incorporated into the defect (the state with different charge) when the lattice is relaxed or overshoots (towards the right) along the middle curve. Once the hole is captured at h, the lattice can relax (towards the left) along the lowest curve; beyond its bending region, the hole is bound tightly enough so that it has practically annihilated the pre-captured electron. The recombination is completed when the lattice relaxes to the same equilibrium position as before the electron capture. Consistent representation of the whole process is possible only with the many body description . The deep defect states observed through their optical and thermal ionization correspond to the minimum point of the middle curve. Most favorable for multiphonon nonradiative recombination is the situation where (i) e-point is close to the minimum of the uppermost curve and (ii) h= point is close to that of the middle curve. (i) is necessary to make large the cross-section of multiphonon electron capture through tunneling (low temperature) or thermally activated (high temperature) process. The condition (ii) may be relaxed significantly because of athermal situation here, namely, the configuration coordinate, starting from e, overshoots not only the equilibrium point but also (even in the cases (a) and (b) of Fig.7), possibly once or more, the h-point where the h o l e - - assumed to be majority carrier - - m a y be captured rather effectively if its density is high enough. In such a situation, the deep state acts only as a transient intermediate state of the recombination process and may not be formed as the relaxed state of the electron capture unless the hole density is low enough. The configuration coordinate represented by the abscissa in Fig.7 (as well as in Figs.2a, 3, 4 and 5) is in fact a sort of interaction mode , which is a linear combination of the normal m o d e s - - local as well as extended m o d e s - with coefficients representing their respective couplings with the electron. Because of the frequency dispersion among these participating modes, the configuration coordinate will make a sort of damped oscillation instead of harmonic oscillation (which is realized only for vanishing dispersion of participating modes), the first turning point being always lower in energy than the starting point , the rest of energy being dissipated into the modes other than the interaction mode which, in general, are extended in space. If the h-point in Fig.7(a) and (b) is beyond the first turning point, the defect state will most probably be formed after the lattice relaxations. The dynamics of the multiphonon nonradiative recombination with its dependence
on the density of majority carriers has been • studied in detail by H.Sumi  taking into account the Landau-Zener effect for the level crossing . 7.
INDUCED DEFECT REACTIONS
The nonradiative electron-hole recombination sometimes enhances or induces defect reactions, as was most extensively studied in ~I-V semiconductors and alkali halides. This is another aspect of strong electron-phonon coupling which can be appropriately described by the configuration coordinate model. However, the reaction coordinate (promoting mode) is usually different from the accepting mode (the abscissa of Fig.7 which we also called the interaction mode) relevant to deep level formation and e-h recombination. So, the problem is how the vibrational energy of the latter mode is transferred to the former. One may well wonder whether or to what extent one can use such a quasimolecular picture with only a few relevant modes to describe the defect reaction which takes place in an extended solid with a huge number of lattice modes. We would argue for this as follows, starting with the recombination process. A great number of phonons, totalling up to the band gap energy, are emitted during the nonradiative recombination which would be completed within a time of the order of picosecond in favorable situations. (We are referring to the duration time of the process which is not to be confused with the inverse of its rate.) Now, the accepting mode in Fig.7 is a sort of interaction mode which is a linear combination of localized (if any) and extended modes, the latter consisting predominantly of short wavelength components near the Brillouin zone edge because of the small orbital radius of the deep level ( a ~ a 0 , see Section 2) and the high density of phonon frequency spectrum. This means that the average group velocity of the participating phonons is significantly smaller than the sound velocity at long wavelength. Thus, most of the released vibrational energy would remain well within the radius of 10A during the recombination process - - the situation which is similar to the de-excitation of a molecule of moderate size and is therefore favorable for defect reactions to take place. A remarkable feature in the recombination enhanced defect reactions observed in III-V semiconductors is that the activation energy E T for thermal reaction is thereby reduced by an energy which is nearly equal to or slightly smaller than the hole transition energy E R (the major part of the recombination energy) and that the reaction becomes athermal when E T < E R [13,16]. The vibrational energy released by the electron-ic transition is thus very efficiently concentrated to the reaction mode without significant dissipation. This might be somewhat embarrassing in v i e w of the fact described in Section 6 that the interaction mode dissipates a fraction
Y. Toyozawa / Electron induced lattice relaxations
of its energy into other modes even during a passage to its first turning point. The situation would be improved, however, if one considers a few coupled modes suitably chosen, including the accepting and the reaction modes; then a much smaller fraction of their energy would be dissipated to the remaining modes before the completion of the defect reaction and the vibrational energy could be effectively concentrated to the reaction mode. This problem was discussed in detail by Kimerling et al. [16,42], with successful explanation of the recombination-reduced activation energy for the defect reaction. Finally, we will describe a more catastrophic p r o c e s s - - the recombination induced defects formation in alkali halides, where the accepting and the promoting modes were introduced in more explicit form to explain the mechanism [43,44]. While this prototype defects formation process has a long history of experimental studies , we confine ourselves here to the recent developments brought about by the short pulse technique. By irradiation of alkali halide crystal with a short pulse of energetic electron beam, pairs of the F-center (an electron bound by an anion vacancy) and the H-center (a positive hole bound by a pair of anions occupying an anion site) are formed as primary products within a time of ten picoseconds [46--48]. It is supposed that things proceed as follows. Irradiation produces free pairs of e and h; h will immediately be self= trapped to form a V K center (X2- m o l e c u l e - two anions come closer to each other by sharing a hole), which will then capture a decelerated e (Fig.8(a)). Defects formation seems to take place while e is cascading down the coulombic levels but certainly before  it becomes relaxed in the lowest level of the self-trapped exciton which has much longer lifetime of radiative annihilation . The most probable path to form the F-H pair starting from the configuration (a) in Fig.8 is the translational motion, denoted by the coordinate Q2, of the X 2- along its molecular axis z
) t~e -
Fig.8 The excitonic mechanism for the formation of the F and H centers in alkali halide [43,
[ii0] towards the configuration (b), since this particular pair of anions are tightly bound together by sharing the hole (h in the antibonding state contributes extra bondage). The electronic-ionic process (a) ÷ (b) in which e and h are decomposed into different sites can be considered as a modification of dichotomy described in Section 5: one has only to extend the configuration coordinate Q2 (up to half the nearest anion-anion distance) so as to include the defects-formed configuration (b). It is, so to speak, extrinsic self-trapping of e and h around the self-induced defects pair. The dichotomy induces defects formation, which in turn encourages the former to be realized. The energetics for this process is as follows. Considering only the anion sublattice which plays the major role here, the electron wants to expand it while the hole to contract it, thus leading to deformation potentials of opposite sign necessary for the dichotomy to take place. In fact, the electron has stabilized itself significantly by pushing out an anion to the neighboring site ((a) + (b)) . Tracing the dynamics of the whole process, one can imagine the following story. As the electron captured by the X2- molecule cascades down its coulombic levels, the hole charge at the central part of the molecule will be more neutralized, the X--X- distance less contracted and the surrounding lattice less polarized. We denote the contraction  of the X--X- distance from its normal value and the polarization  of the surrounding lattice altogether by a single coordinate QI, which is our accepting mode. The adiabatic potentials for the ground, excited (exciton) and,ionized states of this insulating crystal would then be as shown schematically in Fig.9. Thus, the Ql-OScillation will be strongly stimulated during the cascade capture of the electron around the VK-center. The parity breaking mode Q2, which was already introduced and is our reaction (promoting) mode, causes mixing and repulsion between is and 2pz (and higher Pz) states of the electron, especially when QI is larger. This in turn results in the adiabatic instability of the lower state against the Q2 mode, as shown schematically in Fig.10 on the two dimensional configuration coordinate space (QI,Q2). As [Q21 increases and the X 2- molecule is squeezed out, this lowest electronic state with nodeless wavefunction consisting of variable linear combination of is and 2pz wavefunctions transforms to the is state of the F center. If the system relaxes along the Ql-axis instead of being Q2-instabilized, it will finally emit luminescence as is also observed. In any case, the bottleneck after the electron capture at the VK-center seems to be the nonradiative transition from the higher to the lowest adiabatic potential surface of the self-trapped exciton. Recent studies by Itoh and his collab-
Y. Toyozawa /Electron induced lattice relaxations
orators  on the polarization dependent absorption spectra of the lowest state of the self= trapped exciton and on the spectral dependence of the quantum yields for the F-H formation and the luminescence which follow this transient absorption shed a new light on this crucial stage of the recombination induced defect reaction. REFERENCES
Fig. 9 The configuration coordinate model for an electron-hole pair excitation in alkali halide .
therrnal activation ,0oo ,,o0
Fig. i0 The adiabatic instability (Q2) towards the F, H formation versus the relaxation (QI) towards the luminescent state of a self-trapped exciton .
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