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The energy mismatch in multiphonon relaxation
CHEMICAL PHYSICS LEti-ERS
Voiume $5, number 2
THE ENERGY
MISMATCH IN MULTlPHONON
15 January 1977
RELAXATION
R. ENGLMAN* Laboraroire de Spectrome’trie Physique. Universit@ Scientifiue 3804I Grenoble-Cedex, France
ef Mkdicale.
Received 30 July 1976 Revised manuscript received 15 September 1976
Modifications are presented for zero temperature to the energy gap law which take into account the energy mismatch (the difference between the energy gap and the nearest excited level of the effective vibrator)_ The results for the decay rate of rare earth ions in fd f excitation show a sub-linear dependence on the square width of the spectrum of the electronphonon interaction for small width.
The so-called energy gap law fo.r multiphonon relax&ion explains a variety of decay rates ( Wnr), in particuIar of rareearths ions in crystals [l-3 ] and glasses [4] _ Of the parameters appearing in the formula (whose simplest but not the only version [3] is shown) w,
oc EAE~fiwo(
1 _ e-AEi~fcrj-~
(T = temperature)
(1)
the energy gap AE is frequently well known, whereas w. the frequency of effective vibrational mode is chosen by some rather imprecise empirical criterion (e.g. that AE/fiwo is that integer p which fits best the temperature data). E is another phenomeno!ogical constant; according to theory it also depends on AE. The theoretical justification for the choice of a single frequency w. has been reexamined recently [5] in some detail and the conclusion of Miyakawa and Dexter [6] was to a large extent confirmed. Namely, given an effective spectral density of the electron-phonon coupling (which is identical to the experimentally observed one phonon side band) then [6]
M2 = <(w -
(w))~ 4 lw12
(2)
* On leave of absence from Soreq Nuclear Research Centre, Yavne, Israel.
370
is the condition to be able to choose a single effective frequency_ In the above ( ) means averaging over the effective spectral density and the inequality will be satisfied if this density is sufficiently narrow. Wnen (2) holds, then clearly the most rational choice is w,=(w).
(3)
(If the &equality (2) is invalid, then either a time inor, equivalently, repeated convolutions 183 are required.) Let us now suppose that the choice has been made in accordance with (3), and the inequality (2) is being satisfied. One could then expect that for any integer tegration [7]
P AE-pmw)ap
#O.
i.e. that there is an energy mismatch
which will make the use of eq. (1) inapplicable with integer p_ In physical terms, the energy gap is not exactly bridged by an integer number of effective phonons and one needs additional, low-energy phonons to bring about the energy transfer. What is the effect of these phonons on the non-radiative decay rate? The answer to this question at zero temperature is shown in eq. (6), below. To derive it we recall that the decay rate may be expressed through the generating function g(t),
i.e.
15 January 1977
CHEMICAL PHYSICS LETTERS
Volume 45, number 2
5 x lo-
wnra--m _f dt eg(‘),
-2
(4)
where -
4_
g(t) = S(eiw ‘) - iAEt/fi .
(5)
Here Sstands for an average (over all normal modes in the medium) Huang-Rhys number (called iK2 in ref. [S] ), being a measure of the electron-phonon coup!ing strength. (Typically for f + fin rare earths s< 0.1.) Introducing in (4) trivially ei(wjt and expand-
3s v) 2-
ing
expk(Ol
sf
jjo$
X e?ip[g((eiwt)
I-
P w(--iS,tiW
- ei(w)t)]
1 O0
;
Gpi
here 6, = AE - mfi(o). Each term in the sum may be substituted into (4) and evaluated for 6, > 0 by a saddle point method. For 6, < 0 the integral (4) can be seen to vanish (since, at zero temperature phonons can only be absorbed, not emitted). The answer is then, valid for 6, (01% ii.?Mz, [AE/&w)]
X(m!)-1(e~2/ZS,(u))sml’w’
Fig. 1. The sum in eq. (6) versus the energy mismatch G#z
the factor sUb(w) in eq. (6) for further dependence the energy.) p = 5 for aJl curves in the figure.
(Note on
e.g., to f + d in rareearth
(6)
([x] is the largest integer in x.) This formula is remarkable on several counts. First, since it retains in its first factor the energy gap law for g Furthermore, it exhibits in the sum the dependence of Won the energy mismatches. Thus, when the inequality (2) is amply satisfied only the last term in the sum, with 6, the least, is important. Then we find an essentially exponential dependence of iU on the (least) energy mismatch. The dependence of W on M2 (the square width of the effective spectral distribution) is weak, sublinear. Still for M2 + 0 (a very hypothetical situation) we have W + 0; i.e. if the lowenergy phonons are not affected by the electronphonon coupling then conservation of energy prevents the multiphonon process (1) to operate. (Higher order processes in the non-adiabatic operator may yet operate)_ We remark that the limit !%M2%-&.A2 (appropriate,
ions) shows a gaussian de-
pendence on the mismatch [S] _ To summarize, the criterion (2) for the use of a single frequency impels the choice (w) to be made. But then there may be an energy mismatch sP f 0, which enters the decay rate EVnras in eq. (6). The dominant weakness in this expression is probably the neglect of anharmonicity, which is important at high m. This will not only change eq. (6) quantitativehi, but by level broadening will introduce new features. However, the energy gap law eq. (l), from which we started, was also in the harmonic approximation! The temperature dependence and experimental applications will be discussed in a more extended communication.
My thanks are due to professors D. Curie and H. Thomas for suggestions.
References 111 L.A. [2] M.J.
Riseberg and J.W. Moos, Phys. Rev. 174 (1968) 429. Weber, Pbys. Rev. B8 (1973) 53.
371
Vab~~e 45; number 2
CREMICAL
f31 F.k. Fang, XV. Lauer, CR. Cfdiver and M-M. Mifier, 1.
Wi’TERS
15 January 1977
Cbem. Phys. 63 (1975) 366.
(61 ‘F. Mivakawa and D-t. flexter, Phys. Rev. BI 0970) 2961. [73: R. Kubo and Y. Toyozawa, Progr. Thearet. phys. 13 (1955)
Luminescence LO (1975)
[8I M.H.L. lWc%
f4] R. Reisfeld, I._ Baehm, Y, E&stein and N_ Lieblich, 1. 193.
f.51 R. Engiman, J. Chem. Phys. (19771, to be published.
372
RiY~ICS
162. in: Phonons in perfect lattices (Oliver and Boyd, Edinburgh, 1966) p. 431.
Voiume $5, number 2
THE ENERGY
MISMATCH IN MULTlPHONON
15 January 1977
RELAXATION
R. ENGLMAN* Laboraroire de Spectrome’trie Physique. Universit@ Scientifiue 3804I Grenoble-Cedex, France
ef Mkdicale.
Received 30 July 1976 Revised manuscript received 15 September 1976
Modifications are presented for zero temperature to the energy gap law which take into account the energy mismatch (the difference between the energy gap and the nearest excited level of the effective vibrator)_ The results for the decay rate of rare earth ions in fd f excitation show a sub-linear dependence on the square width of the spectrum of the electronphonon interaction for small width.
The so-called energy gap law fo.r multiphonon relax&ion explains a variety of decay rates ( Wnr), in particuIar of rareearths ions in crystals [l-3 ] and glasses [4] _ Of the parameters appearing in the formula (whose simplest but not the only version [3] is shown) w,
oc EAE~fiwo(
1 _ e-AEi~fcrj-~
(T = temperature)
(1)
the energy gap AE is frequently well known, whereas w. the frequency of effective vibrational mode is chosen by some rather imprecise empirical criterion (e.g. that AE/fiwo is that integer p which fits best the temperature data). E is another phenomeno!ogical constant; according to theory it also depends on AE. The theoretical justification for the choice of a single frequency w. has been reexamined recently [5] in some detail and the conclusion of Miyakawa and Dexter [6] was to a large extent confirmed. Namely, given an effective spectral density of the electron-phonon coupling (which is identical to the experimentally observed one phonon side band) then [6]
M2 = <(w -
(w))~ 4 lw12
(2)
* On leave of absence from Soreq Nuclear Research Centre, Yavne, Israel.
370
is the condition to be able to choose a single effective frequency_ In the above ( ) means averaging over the effective spectral density and the inequality will be satisfied if this density is sufficiently narrow. Wnen (2) holds, then clearly the most rational choice is w,=(w).
(3)
(If the &equality (2) is invalid, then either a time inor, equivalently, repeated convolutions 183 are required.) Let us now suppose that the choice has been made in accordance with (3), and the inequality (2) is being satisfied. One could then expect that for any integer tegration [7]
P AE-pmw)ap
#O.
i.e. that there is an energy mismatch
which will make the use of eq. (1) inapplicable with integer p_ In physical terms, the energy gap is not exactly bridged by an integer number of effective phonons and one needs additional, low-energy phonons to bring about the energy transfer. What is the effect of these phonons on the non-radiative decay rate? The answer to this question at zero temperature is shown in eq. (6), below. To derive it we recall that the decay rate may be expressed through the generating function g(t),
i.e.
15 January 1977
CHEMICAL PHYSICS LETTERS
Volume 45, number 2
5 x lo-
wnra--m _f dt eg(‘),
-2
(4)
where -
4_
g(t) = S(eiw ‘) - iAEt/fi .
(5)
Here Sstands for an average (over all normal modes in the medium) Huang-Rhys number (called iK2 in ref. [S] ), being a measure of the electron-phonon coup!ing strength. (Typically for f + fin rare earths s< 0.1.) Introducing in (4) trivially ei(wjt and expand-
3s v) 2-
ing
expk(Ol
sf
jjo$
X e?ip[g((eiwt)
I-
P w(--iS,tiW
- ei(w)t)]
1 O0
;
Gpi
here 6, = AE - mfi(o). Each term in the sum may be substituted into (4) and evaluated for 6, > 0 by a saddle point method. For 6, < 0 the integral (4) can be seen to vanish (since, at zero temperature phonons can only be absorbed, not emitted). The answer is then, valid for 6, (01% ii.?Mz, [AE/&w)]
X(m!)-1(e~2/ZS,(u))sml’w’
Fig. 1. The sum in eq. (6) versus the energy mismatch G#z
the factor sUb(w) in eq. (6) for further dependence the energy.) p = 5 for aJl curves in the figure.
(Note on
e.g., to f + d in rareearth
(6)
([x] is the largest integer in x.) This formula is remarkable on several counts. First, since it retains in its first factor the energy gap law for g Furthermore, it exhibits in the sum the dependence of Won the energy mismatches. Thus, when the inequality (2) is amply satisfied only the last term in the sum, with 6, the least, is important. Then we find an essentially exponential dependence of iU on the (least) energy mismatch. The dependence of W on M2 (the square width of the effective spectral distribution) is weak, sublinear. Still for M2 + 0 (a very hypothetical situation) we have W + 0; i.e. if the lowenergy phonons are not affected by the electronphonon coupling then conservation of energy prevents the multiphonon process (1) to operate. (Higher order processes in the non-adiabatic operator may yet operate)_ We remark that the limit !%M2%-&.A2 (appropriate,
ions) shows a gaussian de-
pendence on the mismatch [S] _ To summarize, the criterion (2) for the use of a single frequency impels the choice (w) to be made. But then there may be an energy mismatch sP f 0, which enters the decay rate EVnras in eq. (6). The dominant weakness in this expression is probably the neglect of anharmonicity, which is important at high m. This will not only change eq. (6) quantitativehi, but by level broadening will introduce new features. However, the energy gap law eq. (l), from which we started, was also in the harmonic approximation! The temperature dependence and experimental applications will be discussed in a more extended communication.
My thanks are due to professors D. Curie and H. Thomas for suggestions.
References 111 L.A. [2] M.J.
Riseberg and J.W. Moos, Phys. Rev. 174 (1968) 429. Weber, Pbys. Rev. B8 (1973) 53.
371
Vab~~e 45; number 2
CREMICAL
f31 F.k. Fang, XV. Lauer, CR. Cfdiver and M-M. Mifier, 1.
Wi’TERS
15 January 1977
Cbem. Phys. 63 (1975) 366.
(61 ‘F. Mivakawa and D-t. flexter, Phys. Rev. BI 0970) 2961. [73: R. Kubo and Y. Toyozawa, Progr. Thearet. phys. 13 (1955)
Luminescence LO (1975)
[8I M.H.L. lWc%
f4] R. Reisfeld, I._ Baehm, Y, E&stein and N_ Lieblich, 1. 193.
f.51 R. Engiman, J. Chem. Phys. (19771, to be published.
372
RiY~ICS
162. in: Phonons in perfect lattices (Oliver and Boyd, Edinburgh, 1966) p. 431.