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Spectroscopic manifestations of the exciton self-trapping barrier
Solid State Communications, Vol. 37, pp. 165—169. Pergamon Press Ltd. 1981. Printed in Great Britain. SPECTROSCOPIC MANIFESTATIONS OF THE EXCITON SELF-TRAPPING BARRIER V. V. Hizhnyakov and A. V. Sherman Institute of Physics, Estonian SSR Acad. Sci., Tartu 202400, USSR (Received 16 June 1980 byA. A. Maradudin) Severalpeculiarities in the spectra of absorption, excitation of self-trapped exciton luminescence and hot luminescence are shown to be related with the self-trapping barrier. In absorption and excitation spectra these are the maxima, which at low temperatures are located to the blue from free exciton maxima. In the hot luminescence spectrum at high temperatures an intensive anti-Stokes component arises when the excitation frequency approaches a value corresponding to the difference of adiabatic potentials for the barrier. 1. AS IS KNOWN [1, 2], in crystals with broad exciton bands and sufficiently strong exciton-phonon coupling free (FE) and self-trapped (STE) excitons may coexist, which are separated by the adiabatic potential barrier, The existence of the barrier is the main characteristic feature of the case under consideration. Therefore the manifestations of the barriers are of definite interest to be investigated. One of such peculiarities the activational temperature dependence of the reciprocal lifetime of FE and the creation probabilities of STE are well known [2]. The aim of this communication is to show that the barrier must have a number of purely spectroscopic manifestations like new maxima in the spectra of absorption, hot luminescence (HL) and the excitation of STE luminescence. The cause of these effects is evident: the top of the barrier is a special point on the adiabatic surface of STE. Near this point the velocity and acceleration, corresponding to the configurational coordinate, are small and the contribution of the corresponding states to optical transitions are great [3]. The conditions for the Frenkel excitons, whose fulfilment allows the coexistence of FE and STE, are the following [1,41: (1) B ~ B/2, E~(l+ 2h) ~B2,
2. Let us first regard the HL spectrum. We are taking interest in multiphonon secondary radiation of the crystal, appearing in the region of the exciton band bottom. The main contribution into this process should be made by the vibrational states of STE as interstitials, since according to equation (1) STE interact strongly with phonons*. Here a mechanism is considered when STE are created directly by the exciting light near the top of the self-trapping barrier (see Fig. 1). A self-trapping model proposed in [6] is used. In accordance with this model a crystal is regarded, which possesses a selected band of the Frenkel excitons selftrapping on one site. In this case for the states close to the self-trapped one, in the exciton—phonon interaction Hamiltonian, only the terms corresponding to the scattering of an exciton by phonons on the self-trapping site can be considered. As was shown in [3] this model is applicable also for describing the top of the barrier at B~ In this model the exciton—phonon interaction Hamiltonian depends but on one configurational coordinate Q, and the Hamiltonian of the exciton—phonon system is written as follows:
where B is the exciton bandwidth, ~) and ii are the average frequency and number of phonons (h = 1), and ~ is the Stoke’s shift (energy gain at self-trapping, see below). The first of the inequalities (1) is the applicability condition of the adiabatic approximation, the second one the self-trapping condition indicates together with the first inequality that the parameter ~ characterizing the interaction degree between an exciton trapped on one site and phonons, is great. The coupling between delocalized excitons (FE) and phonons, however, is weak due to the third inequality (1).
Here a~(ak)is the creation (annihilation) operator of an exciton with a wave vector k and l~ its energy (minimum at k = 0); a~(ao)is the creation (annihilation) operator of an exciton at the localization site 0; = ~ rI Vk rI/~kr, Vk, exciton—phonon interaction constant depending, as assumed, only on the wave vector k and on the number of the phonon branch r; Wk,. is
—
—
~
H
~,
—
—
—
k
—
~kekakak
+
++~P2+
—
2’I~Qao
2
Q.
—
*
165
Contributions into resonant secondary emission of excitons from processes with small phonon number were considered in [5].
(2)
166
MANIFESTATIONS OF THE EXCITON SELF-TRAPPING BARRIER A(t, r1, r2)
__________________
Fl g. 1 Fig. 1. Adiabatic surfaces (U= adiabatic potential; Q = configurational coordinate) of the ground electronic state (1), exciton band bottom (2) and STE (3). The upward arrows symbolize the creation process of STE by light (ac atO ~ v; be atO v); the downward arrow denotes HL. ~‘
the frequency of the phonon; v ~ denotes the vibration frequency, and F, the momentum conjugated to Q. In case the polariton effect is neglected the probability of the transformation of the primary photon with 2~into the seconthe wave vector x1, and frequency & dary one (ic 2, ~2) may be expressed in the form [5] w = dt dr 1 dr2exp {i~TZ2p—i~l1t}A(t, T1, r2), (3)
J JJ’
-iHr
r1,r2)
=
however, SFCP (t[H1,H2] 0) justifies itself only at 0 ~ v when the main contribution as the initial states is made also by highly excited vibrational states [7]. In case of 0 ~ v the so-called s-function approximation 2+w th~ tUmay be used [7,8], where U= Q 1 is the adiabatic potential of STE. Note that SFCP and t5-function approximation are applicable only for those STE vibrational states for which tunneffing can be neglected. Expanding the adiabatic potential U into series near the barrier top and preserving only the quadratic terms 2, (6) U = a(Q Qb) it can be seen easily [9] that the energy E of such states —
—
must satisfy the condition
o
—0~
(O)
2P2f4 + Q2,H
where H1 = v 2=H1 + co1(Q)is the vibrational Hamiltonian corresponding to STE. From conditions (1) it follows that in the optical transitions with an exciton located near the barrier top in an intermediate state, vibrational states of STE with large quantum numbers are involved. At A> v (A is the barrier height from the FE side) the final states as well are the highly excited vibrational states of the exciton vacuum. Therefore in the part of correlator (5) describing radiation the approximation ~.L[H1,H2] 0 may be used, that is the semi-classical Frank—Condon principle (SFCP) [7]. In the part describing absorption,
/
A(t,
(5)
eiH1 e_uht2nl e_t~~1t,
(eiH2
=
Vol. 37, No.2
=
(a~eWT2a,~2euI~taKe
Sp {(0Ie_k~bOOI0)}/Sp{(OIemo O)}
+
Ub
-iHt>
~
(7)
‘aK,e
(4)
where 10) is the vacuum state of the exciton subsystem; o is the temperature in energetic units and p = t + r1 r2. As far as we are interested in the spectrum of hot (not ordinary) luminescence, the operator of radiational decay in equation (4) is omitted. In equations (3) and (4) the slowly-changing multiplier is omitted as well. Let B7 = ~k c,k(Q)a~be the creation operator of excitation, 2F2/4 —Q2 with corresponding the eigenvalue to ~ the Hamiltonian For the H—v given Hamiltonian (2) the latter can easily be found by the Green’s function method [3,6]. Using the new denotations in equation (4), in line with what was said about, we can keep only the term containing the —
Here Ub and Qb are the adiabatic potential and configurational coordinate of the barrier top, whereas the parameter a may change over a wide range from zero to values considerably exceeding unity [3]. Let us regard two temperature regions separately. (a) 0 ~ v. Using SFCP, we get from equations (3) and (5),
w
=
f f J dt
ds
~ xA(t, s, s’), A(t, s, s’)
=
ds’ exp {i(~l2 ~21)t+ if~2s’} —
2s
(8)
where c~ = exp (1H 2s)w1 exp (— iH2s), s
operators B1 and Bj~,where f = 1 corresponds to the split-off (in this model at a fixed self-trapping site the only one) STE adiabatic surface. Note that not only near the minimum of the surface but already in the region near the barrier top at ~ ~ B, c1(Q) const. (the exciton almost entirely trapped at one site [3]). Omitting these values, instead of equation (4) we shall find
(9)
(e~~1t/2 e_~~~)js(t+8) ei~~~1~2),
S
=
T1
—
=
(T1
+ r2)/2,
T2.
In the regarded exciton isincreated and evolves nearprocess the barrier top. an Therefore, equation (9) for the adiabatic potential of H2 approximation (6) is used and intop the only expansion w1(Q) = U(Q) — near the barrier the term linear in Q is preserved. The correlator (9) was found using the Bloch—de Dominicis theorem [10]. Consider that in equation (8) to HL corresponds the
Vol. 37, No.2
MANIFESTATIONS OF THE EXCITON SELF-TRAPP1NG BARRIER
region of integration over s, which considerably exceeds the region Is’ I. As a result the limits of integration over s’ may be substituted by ±0o~Such substitution is
a)
secondary with theto growth number justified violation participating at ofphotons ll2~ the phonons — phase I correlation and v, that the corresponds transformation of of thethe primary to a gradual and ofof the
b)
0.
~
I
equations Raman scattering (8) and to (9)HL we [11]. get As a result, from
W
=
Ov
J
exp
L
(~i — Wb — 2~)2) ~
4Q0
(10)
(11) 2}. i = 0 dz exp k(f1 ch z —f2 sh z) Here ~b = ~Jb— Q~,z = — in th (~/~vs/2), k = a(4Q~O)’,fI( 2/a).The 2) = (11) Wb at ~i(2) divergence in formula &2~= ~222Qb’~’(P results from the substitution of the integration limits. (—
—
~Q~OIa)(~b
while at ~ I
—
~22),
—
~2)’.
Q1 ~Qz surface near the Fig. 2. (a) The part of the adiabatic barrier top. of In an theactivated region above the dashed the probabdity overgoing of theline barrier (wavy line with an arrow) is great. The arrows denote the relaxation of an exciton. (b) HL spectra in two cases Wb — ~2i ~‘i~J(AO/a)(1); Wb — ~i ~sJ(A0/a)(2). the divergence at fl~= fZ 2 the exciting light is assumed to have a finite spectral width and is described by normal distribution). (b) v ~>0. In this case
(12)
A(t, s, p) = eW’t/2 e”t# e~t~~2 e_iHit). (14) Like above, let us use equation (6) and the corresponding approximation for w1(Q), linear in Q, and substitute in
(13)
equation (8) the limits of integration over s’ with ±oo. It is easy to calculate the HL spectrum when it is considered that due to conditions (1) the motion of the sys-
~ i
2\/(irQ~O/a)(wb
___
~‘
Let us first consider the part of the spectrum lying in the region f2 ~>f1. It appears at the radiational decay of an exciton whose adiabatic potential is less that at the moment of creation. At ~/7~f~ ~ 1 (but -.Jkf2 ~ 1) we get from (11) I
167
The doubling of the intensity of the spectrum in case of ~ 1 may be interpreted as follows: near the top of the barrier a region exists in which there is a great probability for an exciton to occur in the opposite side of the barrier due to activation transitions. If such transition has taken place, the emitted light will appear in another spectral region, whereas in the region regarded the ~/i~f1
intensity decreases. Note that in the event of transition the emission appears in the anti-Stokes region (hot and ordinary luminescence of FE). Thus, as the frequency ~ approaches Wb from the red side, the intensity of the HL spectrum in the Stokes region decreases, while in the anti-Stokes region it increases. The interval Wb — where such changes become essential is determined.by 2O/a) ~/(AO/ctXQ~ A [3]). the value ~1/2 = .~/(4Q Assuming that similar to a Xe crystal A = 70meV [12], 2= 25meV. O/kBAs = follows 100 K, afrom = 1,formula we obtain k’~’ (11) the value I in the region f2 ~f1 and s~f1~ 1 is exponentially small and, consequently, the spectrum is exponentially weakened as compared with the region f2 ~ f1. This is understandable, as in case of f~~ f1 at the moment of radiational decay the exciton is located on the adiabatic surface higher that at the moment of creation. A qualitative shape of the spectrum is shown in Fig. 2 (to eliminate
tern along the adiabatic surface of STE may be regarded as a classical one. Proceeding in equation (14) to the classical limit we find 2 dQ W = 4ir 1 dQ2 p(Q1, O)~[f11 + v/2 U(Q 3ir3 x~[fl /2 2—w1(Q2)][v(Q2)]~ = —i-—
JJ
—
~)]
,,J
x (Ub
—
x exp (—
—
~-
{~,,+
U~
—
— P12]
2)
~ [4Q~(f~, + v/2 — Ub) + a(w,, — ~2)]. (15) Here are configurational coordinates of the system atand theQ2 creation and radiational decay of the exciton, v(Q 2) = dQ2/ds is the velocity of the system (in accordance with 2 = U(Qthe energy conservation law [v(Q2)/v] 1) — U(Q2)). p(Q1, 0) is the diagonal element of the density matrix of the oscillator in the coordinate representation at zero temperature [131. Formula (15) is applicable in the region U,, 1b — ~ V~v[see (7)] and — ~2>— v/2 ~/[4Q2,,(U,, — equation — v/2)/ct]. Q
1
~-‘-
168
MANIFESTATIONS OF THE EXCITON SELF-TRAPPING BARRIER
It can be seen from this formula that the spectrum as the function of f12 towards the red from the maximum is of the same shape as in a high-temperature case [seeequations (12) and (13)]. In the short-wave region, however, the spectra may differ drastically; in case of 0 <~v the anti-Stokes component of the emission is absent. Besides, the excitation of the low-temperature HL spectrum with the creation of STE near the top of the barrier occurs in an other spectral region than the high-temperature one (see Fig. 1). This is related with the circumstance that in the case of 0 v the motion of the system in the ground electronic state may be considered classical, whereas at 0 <~v quantum effects are strong. ~‘
3. Let us now regard the peculiarities of exciton absorption spectra, which are conditioned by the existence of the self-trapping barrier. Here two factors should be allowed for: (a) the wave functions of the excited vibrational states of FE (trapped in the left-hand hole in Fig. I) with the energy E < U~are practically orthogonal with respect to the ground vibrational state of the exciton vacuum: non-orthogonality is essential only in case ofE U,,; (b) the overlapping integrals of the vibrational functions of STE with the ground state wave function of the exciton vacuum increase with the energy of these functions approaching the value Ub = eo + A. It follows that at 0 ~ v a maximum must appear in the spectrum whose energy is higher than that of the FE maximum by a value approximately equal to the hight of the barrier. An approximate shape of the long-wave part of this maximum can be found if in the formula for absorption spectrum [14] only the contribution by STE vibrational states are allowed for and if the s-function approximation is used. At 0 = 0 we get
rra
— \/(21r) — .~/[av(Ub — v/2 — ~~Z1)]
2 x exp Ub
—;
/U,, Qb
+
—
~/2 a
2
—
,
(16)
v/2 ~ ~ ~./av[seeequation (7)]. In this connection note [15], where exciton absorption spectra at 0 = 0 were calculated, specifically, for the values of parameters satisfying conditions (1). In these spectra the growth of the envelope can be found on the frequency exceeding that of the maximum of FE absorption by about the height of the barrier (see —
—
-
-
Fig. S from [15], the spectrum corresponding to the parametersS=40,B=30;A~2—3 [3]intheunits used in [15]). One may therefore think that this growth of the absorption intensity is related with self-trapping barrier corresponds to the effect described by formulaand (16).
As is known [3, 6] at 0
~‘
v
Vol. 37, No.2
the contribution of the
vibrational states of STE into the absorption spectrum is entirely located towards the long-wave region from e0, and for the given model it is structureless. Therefore, with the increase of temperature from the values 0 ~ v up to 0 ~ i.’ the maximum smooths out, being displaced towards the red. Such maxima in the spectra of reflection and the imaginary part of dielectric constant may have been already observed in a number of alkali halide crystals [16]. Such supposition is supported by the fact that these maxima are observable in the spectra of the crystals where the coexistence of STE and FE and, consequently, the self-trapping barrier is the most manifest [2]. An analogous maximum must be observable at low temperatures in the excitation spectra of STE luminescence [see equation (15)]. Here also an experiment exists, which gives hope that such maximum has already been found in Nal crystal [171(maximum at 5.64 eV). Thus, the spectra of HL, absorption and excitation of STE luminescense reveal a number of peculiarities related with the self-trapping barrier. An investigation of these spectra allows to determine, alongside with other parameters, the characteristics of this barrier by several independent methods. These effects are essential not only for self-trapping excitons but also in all the cases when the adiabatic surface of the excited state has several maxima separated by barriers (impurity centres, molecules in solutions, etc.).
REFERENCES 1.
E. I. Rashba, Izv. Akacl. Nauk SSSR, Ser. Fiz. 40, 1793 (1976). 2. I. L. Kuusmann, G. G. Liidja & Ch. B. Lushchik, Trudy Inst. Fiz. Akad. Nauk Est. SSR 46, 5 (1976); I. Ya. Fugol’,Adv. Phys. 27,1(1978). 3. V. V. Hizhnyakov & A. V. Sherman, Trudy Inst. Fiz. A/cad. NaukEst. SSR 46,120(1976). 4. V. V. Hizhnyakov & A. V. Sherman, Phys. Status Solidi (b) 92, 177 (1979). s~ V. V Hizh ‘ako &A8.)V~S~erma~i, 1~ys1Satus Lang & S. T. Pavlov Phys. Status Solidi(b) 85, 81 (1978); A. Sumi, Techn. Report of ISSP, Ser. A, No. 971(1979); A. A. Klochikhin, S. A. Permogorov & A. N. Reznitsky, Zh. Eksp. Teor. Fiz. 71,2230 (1976). 6. H. Sumi & Y. Toyozawa, J. Phys. Soc. Japan 31, 342 (1971) ~t. ~ Chem.Phys. 20,1752(1952). 8. V. Hizhnyakov, Light Scattering in Solids (Edited by29. J. L. Plenum Birman,Press, H. Z.New Cummins K.London K. Rebane), York & and p ). ‘~.
(
Vol. 37, No.2 9. 10. 11.
12.
MANIFESTATIONS OF THE EXCITON SELF-TRAPPING BARRIER
L. D. Landau & E. M. Lifshits, Quantum Mechanics, p. 220. Nauka, Moscow (1974). C. Bloch & C. de Dominicis,Nucl. Phys. 7,459 (1958). K. K. Rebane, I. J. Tehver & V. V. Hizhnyakov, Theory ofLight Scattering in Condensed Matter (Edited by B. Bendow, J. L. Birman & V. M. Agranovich), p. 393. Plenum Press, New York and London (1976). R. A. Kink & A. E. Lohmus, Izv. Akad. Nauk SSSR, Ser. Fiz. 42,466 (1978).
13. 14. 15. 16.
17.
169
R. P. Feynman, Statistical Mechanics, p. 62, Mir, Moscow (1978). Y. Toyozawa,1~ogr.Theor. Phys. 20, 53 (1958). H. Sumi,J. Phys. Soc. Japan 38, 825 (1975). G. Baldini & B. Bosacchi, Phys. Status Solidi 38, 325 (1970); Mémoires de Ia Soclété Royale des Sciences de Liege 20, 305 (1970); T. Miyata,J. Phys. Soc. Japan 31, 529 (1971). V. G. Plekhanov & A. A. O’Connel-Bronin, Ultrafast Relaxation and Secondary Emission (Edited by 0. Sild), p. 55. Tallinn (1979).
2. Let us first regard the HL spectrum. We are taking interest in multiphonon secondary radiation of the crystal, appearing in the region of the exciton band bottom. The main contribution into this process should be made by the vibrational states of STE as interstitials, since according to equation (1) STE interact strongly with phonons*. Here a mechanism is considered when STE are created directly by the exciting light near the top of the self-trapping barrier (see Fig. 1). A self-trapping model proposed in [6] is used. In accordance with this model a crystal is regarded, which possesses a selected band of the Frenkel excitons selftrapping on one site. In this case for the states close to the self-trapped one, in the exciton—phonon interaction Hamiltonian, only the terms corresponding to the scattering of an exciton by phonons on the self-trapping site can be considered. As was shown in [3] this model is applicable also for describing the top of the barrier at B~ In this model the exciton—phonon interaction Hamiltonian depends but on one configurational coordinate Q, and the Hamiltonian of the exciton—phonon system is written as follows:
where B is the exciton bandwidth, ~) and ii are the average frequency and number of phonons (h = 1), and ~ is the Stoke’s shift (energy gain at self-trapping, see below). The first of the inequalities (1) is the applicability condition of the adiabatic approximation, the second one the self-trapping condition indicates together with the first inequality that the parameter ~ characterizing the interaction degree between an exciton trapped on one site and phonons, is great. The coupling between delocalized excitons (FE) and phonons, however, is weak due to the third inequality (1).
Here a~(ak)is the creation (annihilation) operator of an exciton with a wave vector k and l~ its energy (minimum at k = 0); a~(ao)is the creation (annihilation) operator of an exciton at the localization site 0; = ~ rI Vk rI/~kr, Vk, exciton—phonon interaction constant depending, as assumed, only on the wave vector k and on the number of the phonon branch r; Wk,. is
—
—
~
H
~,
—
—
—
k
—
~kekakak
+
++~P2+
—
2’I~Qao
2
Q.
—
*
165
Contributions into resonant secondary emission of excitons from processes with small phonon number were considered in [5].
(2)
166
MANIFESTATIONS OF THE EXCITON SELF-TRAPPING BARRIER A(t, r1, r2)
__________________
Fl g. 1 Fig. 1. Adiabatic surfaces (U= adiabatic potential; Q = configurational coordinate) of the ground electronic state (1), exciton band bottom (2) and STE (3). The upward arrows symbolize the creation process of STE by light (ac atO ~ v; be atO v); the downward arrow denotes HL. ~‘
the frequency of the phonon; v ~ denotes the vibration frequency, and F, the momentum conjugated to Q. In case the polariton effect is neglected the probability of the transformation of the primary photon with 2~into the seconthe wave vector x1, and frequency & dary one (ic 2, ~2) may be expressed in the form [5] w = dt dr 1 dr2exp {i~TZ2p—i~l1t}A(t, T1, r2), (3)
J JJ’
-iHr
r1,r2)
=
however, SFCP (t[H1,H2] 0) justifies itself only at 0 ~ v when the main contribution as the initial states is made also by highly excited vibrational states [7]. In case of 0 ~ v the so-called s-function approximation 2+w th~ tUmay be used [7,8], where U= Q 1 is the adiabatic potential of STE. Note that SFCP and t5-function approximation are applicable only for those STE vibrational states for which tunneffing can be neglected. Expanding the adiabatic potential U into series near the barrier top and preserving only the quadratic terms 2, (6) U = a(Q Qb) it can be seen easily [9] that the energy E of such states —
—
must satisfy the condition
o
—0~
(O)
2P2f4 + Q2,H
where H1 = v 2=H1 + co1(Q)is the vibrational Hamiltonian corresponding to STE. From conditions (1) it follows that in the optical transitions with an exciton located near the barrier top in an intermediate state, vibrational states of STE with large quantum numbers are involved. At A> v (A is the barrier height from the FE side) the final states as well are the highly excited vibrational states of the exciton vacuum. Therefore in the part of correlator (5) describing radiation the approximation ~.L[H1,H2] 0 may be used, that is the semi-classical Frank—Condon principle (SFCP) [7]. In the part describing absorption,
/
A(t,
(5)
eiH1 e_uht2nl e_t~~1t,
(eiH2
=
Vol. 37, No.2
=
(a~eWT2a,~2euI~taKe
Sp {(0Ie_k~bOOI0)}/Sp{(OIemo O)}
+
Ub
-iHt>
~
(7)
‘aK,e
(4)
where 10) is the vacuum state of the exciton subsystem; o is the temperature in energetic units and p = t + r1 r2. As far as we are interested in the spectrum of hot (not ordinary) luminescence, the operator of radiational decay in equation (4) is omitted. In equations (3) and (4) the slowly-changing multiplier is omitted as well. Let B7 = ~k c,k(Q)a~be the creation operator of excitation, 2F2/4 —Q2 with corresponding the eigenvalue to ~ the Hamiltonian For the H—v given Hamiltonian (2) the latter can easily be found by the Green’s function method [3,6]. Using the new denotations in equation (4), in line with what was said about, we can keep only the term containing the —
Here Ub and Qb are the adiabatic potential and configurational coordinate of the barrier top, whereas the parameter a may change over a wide range from zero to values considerably exceeding unity [3]. Let us regard two temperature regions separately. (a) 0 ~ v. Using SFCP, we get from equations (3) and (5),
w
=
f f J dt
ds
~ xA(t, s, s’), A(t, s, s’)
=
ds’ exp {i(~l2 ~21)t+ if~2s’} —
2s
(8)
where c~ = exp (1H 2s)w1 exp (— iH2s), s
operators B1 and Bj~,where f = 1 corresponds to the split-off (in this model at a fixed self-trapping site the only one) STE adiabatic surface. Note that not only near the minimum of the surface but already in the region near the barrier top at ~ ~ B, c1(Q) const. (the exciton almost entirely trapped at one site [3]). Omitting these values, instead of equation (4) we shall find
(9)
(e~~1t/2 e_~~~)js(t+8) ei~~~1~2),
S
=
T1
—
=
(T1
+ r2)/2,
T2.
In the regarded exciton isincreated and evolves nearprocess the barrier top. an Therefore, equation (9) for the adiabatic potential of H2 approximation (6) is used and intop the only expansion w1(Q) = U(Q) — near the barrier the term linear in Q is preserved. The correlator (9) was found using the Bloch—de Dominicis theorem [10]. Consider that in equation (8) to HL corresponds the
Vol. 37, No.2
MANIFESTATIONS OF THE EXCITON SELF-TRAPP1NG BARRIER
region of integration over s, which considerably exceeds the region Is’ I. As a result the limits of integration over s’ may be substituted by ±0o~Such substitution is
a)
secondary with theto growth number justified violation participating at ofphotons ll2~ the phonons — phase I correlation and v, that the corresponds transformation of of thethe primary to a gradual and ofof the
b)
0.
~
I
equations Raman scattering (8) and to (9)HL we [11]. get As a result, from
W
=
Ov
J
exp
L
(~i — Wb — 2~)2) ~
4Q0
(10)
(11) 2}. i = 0 dz exp k(f1 ch z —f2 sh z) Here ~b = ~Jb— Q~,z = — in th (~/~vs/2), k = a(4Q~O)’,fI( 2/a).The 2) = (11) Wb at ~i(2) divergence in formula &2~= ~222Qb’~’(P results from the substitution of the integration limits. (—
—
~Q~OIa)(~b
while at ~ I
—
~22),
—
~2)’.
Q1 ~Qz surface near the Fig. 2. (a) The part of the adiabatic barrier top. of In an theactivated region above the dashed the probabdity overgoing of theline barrier (wavy line with an arrow) is great. The arrows denote the relaxation of an exciton. (b) HL spectra in two cases Wb — ~2i ~‘i~J(AO/a)(1); Wb — ~i ~sJ(A0/a)(2). the divergence at fl~= fZ 2 the exciting light is assumed to have a finite spectral width and is described by normal distribution). (b) v ~>0. In this case
(12)
A(t, s, p) = eW’t/2 e”t# e~t~~2 e_iHit). (14) Like above, let us use equation (6) and the corresponding approximation for w1(Q), linear in Q, and substitute in
(13)
equation (8) the limits of integration over s’ with ±oo. It is easy to calculate the HL spectrum when it is considered that due to conditions (1) the motion of the sys-
~ i
2\/(irQ~O/a)(wb
___
~‘
Let us first consider the part of the spectrum lying in the region f2 ~>f1. It appears at the radiational decay of an exciton whose adiabatic potential is less that at the moment of creation. At ~/7~f~ ~ 1 (but -.Jkf2 ~ 1) we get from (11) I
167
The doubling of the intensity of the spectrum in case of ~ 1 may be interpreted as follows: near the top of the barrier a region exists in which there is a great probability for an exciton to occur in the opposite side of the barrier due to activation transitions. If such transition has taken place, the emitted light will appear in another spectral region, whereas in the region regarded the ~/i~f1
intensity decreases. Note that in the event of transition the emission appears in the anti-Stokes region (hot and ordinary luminescence of FE). Thus, as the frequency ~ approaches Wb from the red side, the intensity of the HL spectrum in the Stokes region decreases, while in the anti-Stokes region it increases. The interval Wb — where such changes become essential is determined.by 2O/a) ~/(AO/ctXQ~ A [3]). the value ~1/2 = .~/(4Q Assuming that similar to a Xe crystal A = 70meV [12], 2= 25meV. O/kBAs = follows 100 K, afrom = 1,formula we obtain k’~’ (11) the value I in the region f2 ~f1 and s~f1~ 1 is exponentially small and, consequently, the spectrum is exponentially weakened as compared with the region f2 ~ f1. This is understandable, as in case of f~~ f1 at the moment of radiational decay the exciton is located on the adiabatic surface higher that at the moment of creation. A qualitative shape of the spectrum is shown in Fig. 2 (to eliminate
tern along the adiabatic surface of STE may be regarded as a classical one. Proceeding in equation (14) to the classical limit we find 2 dQ W = 4ir 1 dQ2 p(Q1, O)~[f11 + v/2 U(Q 3ir3 x~[fl /2 2—w1(Q2)][v(Q2)]~ = —i-—
JJ
—
~)]
,,J
x (Ub
—
x exp (—
—
~-
{~,,+
U~
—
— P12]
2)
~ [4Q~(f~, + v/2 — Ub) + a(w,, — ~2)]. (15) Here are configurational coordinates of the system atand theQ2 creation and radiational decay of the exciton, v(Q 2) = dQ2/ds is the velocity of the system (in accordance with 2 = U(Qthe energy conservation law [v(Q2)/v] 1) — U(Q2)). p(Q1, 0) is the diagonal element of the density matrix of the oscillator in the coordinate representation at zero temperature [131. Formula (15) is applicable in the region U,, 1b — ~ V~v[see (7)] and — ~2>— v/2 ~/[4Q2,,(U,, — equation — v/2)/ct]. Q
1
~-‘-
168
MANIFESTATIONS OF THE EXCITON SELF-TRAPPING BARRIER
It can be seen from this formula that the spectrum as the function of f12 towards the red from the maximum is of the same shape as in a high-temperature case [seeequations (12) and (13)]. In the short-wave region, however, the spectra may differ drastically; in case of 0 <~v the anti-Stokes component of the emission is absent. Besides, the excitation of the low-temperature HL spectrum with the creation of STE near the top of the barrier occurs in an other spectral region than the high-temperature one (see Fig. 1). This is related with the circumstance that in the case of 0 v the motion of the system in the ground electronic state may be considered classical, whereas at 0 <~v quantum effects are strong. ~‘
3. Let us now regard the peculiarities of exciton absorption spectra, which are conditioned by the existence of the self-trapping barrier. Here two factors should be allowed for: (a) the wave functions of the excited vibrational states of FE (trapped in the left-hand hole in Fig. I) with the energy E < U~are practically orthogonal with respect to the ground vibrational state of the exciton vacuum: non-orthogonality is essential only in case ofE U,,; (b) the overlapping integrals of the vibrational functions of STE with the ground state wave function of the exciton vacuum increase with the energy of these functions approaching the value Ub = eo + A. It follows that at 0 ~ v a maximum must appear in the spectrum whose energy is higher than that of the FE maximum by a value approximately equal to the hight of the barrier. An approximate shape of the long-wave part of this maximum can be found if in the formula for absorption spectrum [14] only the contribution by STE vibrational states are allowed for and if the s-function approximation is used. At 0 = 0 we get
rra
— \/(21r) — .~/[av(Ub — v/2 — ~~Z1)]
2 x exp Ub
—;
/U,, Qb
+
—
~/2 a
2
—
,
(16)
v/2 ~ ~ ~./av[seeequation (7)]. In this connection note [15], where exciton absorption spectra at 0 = 0 were calculated, specifically, for the values of parameters satisfying conditions (1). In these spectra the growth of the envelope can be found on the frequency exceeding that of the maximum of FE absorption by about the height of the barrier (see —
—
-
-
Fig. S from [15], the spectrum corresponding to the parametersS=40,B=30;A~2—3 [3]intheunits used in [15]). One may therefore think that this growth of the absorption intensity is related with self-trapping barrier corresponds to the effect described by formulaand (16).
As is known [3, 6] at 0
~‘
v
Vol. 37, No.2
the contribution of the
vibrational states of STE into the absorption spectrum is entirely located towards the long-wave region from e0, and for the given model it is structureless. Therefore, with the increase of temperature from the values 0 ~ v up to 0 ~ i.’ the maximum smooths out, being displaced towards the red. Such maxima in the spectra of reflection and the imaginary part of dielectric constant may have been already observed in a number of alkali halide crystals [16]. Such supposition is supported by the fact that these maxima are observable in the spectra of the crystals where the coexistence of STE and FE and, consequently, the self-trapping barrier is the most manifest [2]. An analogous maximum must be observable at low temperatures in the excitation spectra of STE luminescence [see equation (15)]. Here also an experiment exists, which gives hope that such maximum has already been found in Nal crystal [171(maximum at 5.64 eV). Thus, the spectra of HL, absorption and excitation of STE luminescense reveal a number of peculiarities related with the self-trapping barrier. An investigation of these spectra allows to determine, alongside with other parameters, the characteristics of this barrier by several independent methods. These effects are essential not only for self-trapping excitons but also in all the cases when the adiabatic surface of the excited state has several maxima separated by barriers (impurity centres, molecules in solutions, etc.).
REFERENCES 1.
E. I. Rashba, Izv. Akacl. Nauk SSSR, Ser. Fiz. 40, 1793 (1976). 2. I. L. Kuusmann, G. G. Liidja & Ch. B. Lushchik, Trudy Inst. Fiz. Akad. Nauk Est. SSR 46, 5 (1976); I. Ya. Fugol’,Adv. Phys. 27,1(1978). 3. V. V. Hizhnyakov & A. V. Sherman, Trudy Inst. Fiz. A/cad. NaukEst. SSR 46,120(1976). 4. V. V. Hizhnyakov & A. V. Sherman, Phys. Status Solidi (b) 92, 177 (1979). s~ V. V Hizh ‘ako &A8.)V~S~erma~i, 1~ys1Satus Lang & S. T. Pavlov Phys. Status Solidi(b) 85, 81 (1978); A. Sumi, Techn. Report of ISSP, Ser. A, No. 971(1979); A. A. Klochikhin, S. A. Permogorov & A. N. Reznitsky, Zh. Eksp. Teor. Fiz. 71,2230 (1976). 6. H. Sumi & Y. Toyozawa, J. Phys. Soc. Japan 31, 342 (1971) ~t. ~ Chem.Phys. 20,1752(1952). 8. V. Hizhnyakov, Light Scattering in Solids (Edited by29. J. L. Plenum Birman,Press, H. Z.New Cummins K.London K. Rebane), York & and p ). ‘~.
(
Vol. 37, No.2 9. 10. 11.
12.
MANIFESTATIONS OF THE EXCITON SELF-TRAPPING BARRIER
L. D. Landau & E. M. Lifshits, Quantum Mechanics, p. 220. Nauka, Moscow (1974). C. Bloch & C. de Dominicis,Nucl. Phys. 7,459 (1958). K. K. Rebane, I. J. Tehver & V. V. Hizhnyakov, Theory ofLight Scattering in Condensed Matter (Edited by B. Bendow, J. L. Birman & V. M. Agranovich), p. 393. Plenum Press, New York and London (1976). R. A. Kink & A. E. Lohmus, Izv. Akad. Nauk SSSR, Ser. Fiz. 42,466 (1978).
13. 14. 15. 16.
17.
169
R. P. Feynman, Statistical Mechanics, p. 62, Mir, Moscow (1978). Y. Toyozawa,1~ogr.Theor. Phys. 20, 53 (1958). H. Sumi,J. Phys. Soc. Japan 38, 825 (1975). G. Baldini & B. Bosacchi, Phys. Status Solidi 38, 325 (1970); Mémoires de Ia Soclété Royale des Sciences de Liege 20, 305 (1970); T. Miyata,J. Phys. Soc. Japan 31, 529 (1971). V. G. Plekhanov & A. A. O’Connel-Bronin, Ultrafast Relaxation and Secondary Emission (Edited by 0. Sild), p. 55. Tallinn (1979).