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The mechanism of thermally stimulated delayed fluorescence
CHEhllCAL
3 (1971) 103-109.0
PHYSICS
NORTH-HOLLAND
THE MECHANIShl OF THERMALLY
Stratdi-lnstht.
Rapid delayed
hearing
of isotropic
fluorescence
(DF).
miscd
crystalline
This behaviour
and J. hlARQUARD
BerliwWest, German)
Teclrt~ische Uniwrsitd. Received
an
COMPANY
STlhlULATED DELAYED FLUORESCENCE
F. AURICH I.N.
PUBLISHING
13 September
1973
htzudeutcrobenzene be explained
at 4.1”#
uuses
by the rapid depopulation
XI intent
peak in the emission of
of triplet cxciton tnpsduring
thcrnul stimulation. which is followed by annihilation of excitons. Rapid haling was performed cilhcr by IR radiation or by electrical hating of the ample. The pcsk height of rhe DT: emission WJS found IO be dependent on the initial lempenturc via the square of occupied-trap concsntrarion and proportional
to the hating
this only one type of trap were nqlccred.
I.
rate during ~3s
slimulalion.
This agrees with
3
kinetic analysis of thermally
stimulated
DF. For
assumed. Since weak excitation only was presumed, nonlinear terms in Ihe rrte equations
Introduction
The steady state delayed fluorescence (DF) intensity of crystalline benzene is enhanced by a rapid healing of the crystal [I]. Trapped triplet excitons are thereby liberated, and the subsequent mutual annihilation of the excitons causesan intense fluorescence emission. A maximum emission intensity of IO6 times that of steady state DF was observed [l] at heating rates of about 10SoKsec-l. In this paper we restrict ourselves to relatively low emission intensities (about IO’ times that of DF intensity) in particular with regard to the kinetic analysis of the thermal stimulation of DF (vice versa). Moreover, annihilation is investigated under small excitation intensities. This means that the annihilation
intensity is underectable due lo the heavy quenching of the triplet excitons. In the isotopic mixed (crystalline) benzene system it is possible to produce traps of definite concentn-
r;on and energy depth by doping perdeuterobenzene (B-de) with partly deuterated benzenes (B-d3,Bd4) or with benzene (B-h,).
2. Experimental 2.1. Apparahrs The optical arrangement was similar to that described previously [I]. Exciting light was obtained from
a HBO 100 lamp, an argon plasma arc source
[41
an argon ion laser (CRL 52) with a frequency doubler (I mWat 257 nm).
process is not considered to have an influence on the occupation density of the triplet state. Under this con-
or
dition DF intensity
Emission was analysed by 0.5 m Ebert monochromator (Jarrell Ash) and detected by means of an EM 6356 S photomultiplier with PAR I22 amplifier. A fast rotating disc phosphoroscope was used to separate
shows a second-order
dependence
on the phosphorescence
intensity and the lifetime of DF is half that of the phosphorescence decay time [3]. The temperature dependence of the steady state DF intensity is due to the trapping and detnpping mecha-
nism of the triplet cxcitons and gives information
about Traps
trap properties as was shown by Siebrand [3]. become operative at low temperatures (200 -4°K).
They cause a prolongation
of the lifetime of the excitons and thereby an increase in the exciton density as well as the DF intensity. At room temperature DF
delayed emission from prompt fluorescence. The sample was cooled by a helium cryostat [S]. Constant rate heating of the crystal was performed by heating the sample cell electrically. Heat contact between cell and crystal was achieved using helium gas at different pressures. Rapid heating was performed by irradiating the crystal with IR light (1.7 -2.3,~).
104
F. Auriclr and
3000
2950
3. Marquard, The nzechanism of lhennal!v rrimulated DF
2900
2050
2803
2750
2650
2700
A
-A
Fig. I. Fluorescence
spectrumof crystallinehexadeuterobenzene(B-d,)99%.
Measuring thermally stimulated DF versus IR wavelength an IR spectrum of the host crystal was obtained [61, 2.2. Materirrls Materials were Merck uvasol hexadeuterobenzene (Bd6) 99% and benzene (B-&). BdG contained isotopic impurities in the following approximate mol fractions: Bd, = 5%; Bd, * 0.1%: B-d, = 0.01%; Bdz, B-d,, B-h,
3. R&ts In the first example the Bd, host crystal was doped with 1% B-d, to produce a trap with an energy depth
Temperature
42°K.
of about 100 cm-l [7]. This trap causes a maximum in the steady state DF versus temperature curve, shown in fig. 3. The maximum was the same regardless of whether the temperature was being increased or decreased. This only applies to a temperature rate sufficiently small as to preserve the steady state conditions. Under non-steady-state conditions DF is shown in fig. 4. The cryostat was held at a constant temperature of 4.7, “K. At r, (fig. 4)a temperature rise of about 2°K was effected by irradiating the sample with IR radiation. DF passes through a narrow peaked maximum, the intensity of which is twice that of the steady state DF maximum (fig. 3). In the subsequent part of the diagram, DF reaches a steady state intensity which corresponds to a temperature of about 2°K higher than just before the beginning of the temperature rise at rl. At ‘1 IR radiation is cut off, the drop in temperature causes the DF to pass through a minimum and to return to the original intensity. In the second experiment the temperature rise was performed by heating the crystal electrically. The sample was Bd6 containing the isotopic impurities mentioned above; this host material was doped with 0.1% B-he. There are three operative traps at energy depths of 66 cm-l (Bd4), 100 cm-l (Bd3) and 200 cm-t (B-Q which were occupied by excitons at 41°K. The Bd5 does not operate as a trap as its
F. Aurich and J. Marqlrard,
The mechanism of thermally
5
slimubfed
IO5
DF
IO
T-
15
K
Fig. 3. Temperature dependence of steady stale delayed Iluorescence (DF) intensity. B-d6 doped with I% I1d3.
Moreover, the heights of the peaks are dependent on the initial temperatures as shown in fig. 6 and decrease with increasing initial temperature. The points give the maximum intensity of thermally stimulated DF which was obtained with the same heating rates but with different initial temperatures. In addition, steady state DF and phosphorescence are plotted in fig. 6. A comparison between the points and the (!ph)2 curve show that the maximum intensity
1 Fig. 2. Temperalure dependence of fluorescence spccva of B-d6 and B-&-8-116 (O.OSC’L)mixed crystals in the rqion of first vibration band.
high concentration produces rapid trap-trap energy transfer and the excitons migrate to lower energy traps. Beginning at 4.2’K the sample was heated using various heating rates. The DF versus time curves registered on an oscilloscope are shown in fig. 5. The three intensity peaks in the DF versus time curves are caused by subsequent depletion of the traps during the heating period, whereby the B-h, trap is depleted finally. The different heights of the three peaks are effected by the different concentration of the traps. The heights of the peaks are proportional to the heating rates, whereas time intervals between the peaks and the halfwidths are inversely proportional to the heating rates.
Fig. 4. Delayed fluorescence (DF) as a function of time during rapid temperature changes. tr : risein temperature of 2°K; 12: drop in temperature of 2°K. IId6
doped with
1% Bdl.
106
E .&rich
und 1. Xlarquard.
The mechanrrm of thermally
stimulated
DF
b
_I
0 Fig. 5. Thermally function of time, 12.2. (b) 9.4, (c) by the B-116 trap.
2
I
s
tstimulated dehyed fluorescence (DF) as 3 obtained with different heating rates: (a) 6.3. (d) 3.6”R se?. Broad pak is cnsed B-d.5 doped with 0.1% B46.
of thermally stimulated DF decreases with increasing initial temperature in the same way as the square of
phosphorescenceintensity wirh increasingtemperature.
TFig. 6. Steady state delayed fIuorescencc (DF) and phosphorescence (Ph) intensity 3s a function of temperature. Points give peak heights of thermally stimulated DF 3s P function of initial tempenlure. Heating rates conslant. k+, doped with 0.15 BA6.
indicate, that the excitons are trapped already in the singlet state. k, to k4 are rate constants; the corree sponding processes are shown in fig. 7. Only k, , the thermal release rate constant of trapped excitons,
4. Kinetic analysis
In this section
the kinetics of thermal!y stimulated
delayed fluorescence will be analysed in detail. We consider a crystal containing one type of trap with depth ET= 200 cm-t and with a concentration of 17~ (B-d, doped with 1% BI,). In the case of low excitation intensity, implying low exciton density, the kinetic equations read &/dr
= arl- ktx + k,y - k4x,
dy/dr=k,x.-k2y-k,y;
(1) (2)
wherex is the triplet exciton concentration in the traps,y the triplet cxciton concentration in the band. al is the concentration of the absorbed photons per unit time, which are transferred to the triplet state by intersystem crossing. This term al was added to eq. (I) since fluorescence spectra at low temperature
is temperature dependent. k, depends on the collision raIL .:z:stant Z and the Boltzmann factor, k, = Z X where K is Boltzmann’s constant and exp (-ET/U) 2 = 10’1 xc-1 for benzene (81. k, is the rate constsnt
for collisionsbetween free excitons and traps and is given by the product of collision rate constant and trap concentration. In the case we are concerned with we have k2 = Zc, = log set-l , where cT is the trap concentration. k3 is the rate constant for quenching of excitons in the band; this constant being unknown, the value kj = 10’ SK-~ has been estimated according to the position of the maximum in the DF versus temperature curve 131. k4 is the monomolecular decay constant of trapped excitons and can be determined from the phosphorescence lifetime at low temperatures: for benzene k4 = lo- 1 see-t . The nonlinear expressions (bimolecular terms) representing the annihilation process are omitted in eqs. (I) and (2) since we consider weak excitation only, not exceeding I = lOIs photons/seccm3, i.e.,
a negligible proportion only
decays
of the free or trapped excitons
by triplet-triplet
annihilation.
107
E .-lurich and J. Marquard, The mechanism of rhermally srimulared DF
I
aI
Fig. 7. Transition klp.
rates concerning triplet exciton band and
The delayed fluorescence intensity (fDF) portional to the annihilation probability I,,
-)‘2
is pro-
+ fir,, .
(3)
The annihilation consists of two processes: (I) the homogeneous annihilation of free excitons being proportional 10 the square of band exciton concentration and (2) the heterogeneous annihilation of free cxcitons and traps. The latter is proportional to the free exciton concentration and occupied-trap concentration. The factor fl< 1 takes into account that the trapped exciton taking part in the annihilation process is immobile. At low temperatures where traps are operative heterogeneous annihilation is predominant. This is the case with Bd6 crystals. For this reason homogeneous annihilation is disregarded and eq. (3) reduces to IDF -xy
(4)
Under steady state conditions, &/dr
this means
= dvldr = 0
Equations
(5)
(1) and (2) can be solved with respect
to
xand) x = IicZl(k&
+kz”4)j
(YI 1
(6)
Y = [k,/(k,$
+k$q)l ffl1
(7)
where k, WE, neglected compared with k2, the latter being greater by several orders of magnitude. From (4), (6) and (7) we obtain for the DF intensity $,F - [k,k2/(k,k,
+ kzk4)*3
(aO*
.
Cakulated curves ofx, y andxy for a Bdg doped with 1% 9.11, ate shown in fig. 8. The concentration of the occupied traps portional to the phosphorescence intensity decreases with increasing temperature. The
(8)
IL
I2
16
18
20
K
TFig. 8. Calculrfed tentpeaturc variation (under study state conditions) of population ol’ triplet cxciton tnps (x). popul~. lion of triplet exciton band (_I,)and DF intensity (xf).
sity curve passes through a maximum when the temperature is either raised or lowered. This applies to small hearing rates only. Under rapid heating conditions, which ate realised during thermal stimulation k, becomes time dependent and eq. (5) is no longer valid. The rate equations a.~ a (I) and (2) must be solved with temperature function of time, which in general, can be performed with the aid of a computer. It is possible, however, to obtain information concerning the mechanism of stimulated DF by solving the rate equations approximately. Addition of eqs. (I) and (2) yields
dx/dt t dyjdt = crl - k,y - k4x . We now introduce a fixed population band and traps which is proportional sponding transition rates
(9) ratio
between
lo the corre-
crystal
u/x=k,lk,
is prowhich DF inten-
The physical meaning of this approximation is as follows: The population of traps from the band (k,) is a process which is much faster than the other processes considered. Consequently a pzrtial equilib-
.
(10)
108
F. Auricb and .I. hlarquard. The mechanism of thermally
stimulared DF
rium is established which may be assumed to be dependent on k, and k2 only (thermal equilibrium). It is now possible to solve the rate equations in closed form. The temperature dependence of k, is expressed by k, =Z@
.
(11)
From eqs. (9), (10) and (11) follows
These equations give information as to the temperature dependence of the population of bands and traps. To simplify the rale equations further, only low temperatures (T < 40°K) are considered, i.e., k, is neglected with respect to kz. In addition the excitation term is omitted. The previous results also hold good when the exciting ligh! is cut off before the heating process begins [I]. With these omissions and k,/k2 = cyleAt) ,
(14
eqs. (12) and (13) read dx/dr = - [ ~5’ efi’) (k3t df;df) t k4] x ,
These equations can be integrated in the case when we assume a particular dependence of temperature on time during the heating process: (17)
where n~is the parameter of temperature rate and II is given by n=ET/KTo,
population x, band popuhtion y and DF intensity xy under thermal stimulation. xo, ho initial values. At I = 0 excitation is shut off. Heating begins
at I = 1 set (1%
dy/dl = - [c;t eAf) (k3 t df/dr) t k,- d/ldr] y . (16)
flr)=mr--n,
Fig. 9. Calculatedvariation of trap
(1%
with To as the initial temperature. Certainly there arises no serious restriction by this particular choice since we are only interested in the qualitative behaviour. The integration of eqs. (15) and (16) together with (I 7) yields
m = 2
with initial temperature To = 14°K. Heating rate
see-’
x=xoexp
{-[r;.‘(exp[mt-n]-exp
x (k31ttt+I)+k4t] I, y=yo exp {-[I$’ (exp[mr+r]-exp x (k#n + 1) t (k4 -nz)t] ) . Fig. 9 shows the population of traps and of the band, in addition the product of both as a function of temperature [eqs. (17), (19), (20)]. The initial temperature is TO = l4’K. The heating interval is 14-25OK; this interval is passed in 4.5 set (III= 2 set-l). At r = 0 the exciting light is cut off; DF and phosphorescence begin to decay. After one second the heating begins. The increase of temperature causes an accelerated decrease of the trap population and therefore
E Aurich and J. Marquard.
The mechanism of thermally
a decrease ofthe phosphorescence intensity. Contrary to this, the exciton concen:ration in the band and consequently the DF intensity are increased and pass through maxima at different times. The position of the DF maximum is determined by I rn3x=~7-‘{rr-ln
[2(k
,+m)/+-X,)l~l
and the corresponding temperature
(21)
is
T mLX= ET/K In (2(k3+ m)/c-&nz - 2 k,)]
.
(22)
Restricting m to the interval (23)
and with eqs. (17-22) the maximum intensity of thermally stimulated DF is calculated as =(,nx$k,)exp[(Z!k,/rnc~)exp
perimental results (fig. 6). Moreover, eq. (24) shows the maximum intensity being proportional to the heating rate which holds true in the interval given by (23) and at constant initial temperature. This too agrees with the experimental results (see fig. 5). The preceding results were obtained with reduced rate equations and under numerous restrictions as, e.g., one type of trap only was considered. For a sequence of narrow peaks, as shown in fig. 5, caused by subse-
quent depletion of severaltypes of traps with different
k4 Qnt
WrnLX
109
stimulated DF
depth, an analysis might be possible using rate equations which include Ihe nonlinear terms for the
energy
annihilation future
process. This shall be considered in a
publication.
(-11) - I], (24)
x0 being the occupied-trap concentration at the initial temperature. Equation (21) shows, that a maximum only exists for the logarithmic term being smaller than II. From this a relation can be derived yielding information as to which value the heating rate at least should have (in function of the initial temperature To) in
Acknowledgement We wish to thank F. Hofelich for helpful discussions. This work
was supported
Wirtschaft
als Verwalter
by the Bunderrninister
fiir
des ERP Sondervermogens.
References
order to guarantee the existence of a maximum in the DF curve. Taking into account the restriction (23), eq. (22) shows an increase of Tmax with increasing heating rate.
For an initial temperature lower than the temperature, at which the steady state DF maximumoccurs, the exponent in eq. (24) can be assumed to be -1, )I being sufficiently high. At constant heating rates the maximum DF intensity is dependent on the initial temperature
via the square of the occupied-trap
con.
centration, the latter being proportional to the steady state phosphorescence. This is consistent with the ex-
[ 11 F. Aurich and J. hlarquard, Chem. Phys. Letters6 (1971) 91; 11 (1971) 167. [2] T. Azumi and S.P. h!cClynn. J. Chem. Phys. 38 (1963) 2773. [3] W. Siebnnd. J. Chem. Phys. 42 (1965) 3951. 141 F. Aurich and E.-P. Resewitz, J. Sri. Instr. 3 (1970) 899. [S] F. Aurich, L. Helbing and J. hfarquard, Z. Angew. Phys. 29 (1970) 365. 161 F. Aurich and J. hlarqusrd. unpublished results. 171 D.hl. Burland, C. Castro and C.W. Robinson, 1. Chem. Phys. 52 (1970) 4100. 181 J.B. Birks, Photophysics of aromatic molecules (WileyInterscience, New York, 1970).
3 (1971) 103-109.0
PHYSICS
NORTH-HOLLAND
THE MECHANIShl OF THERMALLY
Stratdi-lnstht.
Rapid delayed
hearing
of isotropic
fluorescence
(DF).
miscd
crystalline
This behaviour
and J. hlARQUARD
BerliwWest, German)
Teclrt~ische Uniwrsitd. Received
an
COMPANY
STlhlULATED DELAYED FLUORESCENCE
F. AURICH I.N.
PUBLISHING
13 September
1973
htzudeutcrobenzene be explained
at 4.1”#
uuses
by the rapid depopulation
XI intent
peak in the emission of
of triplet cxciton tnpsduring
thcrnul stimulation. which is followed by annihilation of excitons. Rapid haling was performed cilhcr by IR radiation or by electrical hating of the ample. The pcsk height of rhe DT: emission WJS found IO be dependent on the initial lempenturc via the square of occupied-trap concsntrarion and proportional
to the hating
this only one type of trap were nqlccred.
I.
rate during ~3s
slimulalion.
This agrees with
3
kinetic analysis of thermally
stimulated
DF. For
assumed. Since weak excitation only was presumed, nonlinear terms in Ihe rrte equations
Introduction
The steady state delayed fluorescence (DF) intensity of crystalline benzene is enhanced by a rapid healing of the crystal [I]. Trapped triplet excitons are thereby liberated, and the subsequent mutual annihilation of the excitons causesan intense fluorescence emission. A maximum emission intensity of IO6 times that of steady state DF was observed [l] at heating rates of about 10SoKsec-l. In this paper we restrict ourselves to relatively low emission intensities (about IO’ times that of DF intensity) in particular with regard to the kinetic analysis of the thermal stimulation of DF (vice versa). Moreover, annihilation is investigated under small excitation intensities. This means that the annihilation
intensity is underectable due lo the heavy quenching of the triplet excitons. In the isotopic mixed (crystalline) benzene system it is possible to produce traps of definite concentn-
r;on and energy depth by doping perdeuterobenzene (B-de) with partly deuterated benzenes (B-d3,Bd4) or with benzene (B-h,).
2. Experimental 2.1. Apparahrs The optical arrangement was similar to that described previously [I]. Exciting light was obtained from
a HBO 100 lamp, an argon plasma arc source
[41
an argon ion laser (CRL 52) with a frequency doubler (I mWat 257 nm).
process is not considered to have an influence on the occupation density of the triplet state. Under this con-
or
dition DF intensity
Emission was analysed by 0.5 m Ebert monochromator (Jarrell Ash) and detected by means of an EM 6356 S photomultiplier with PAR I22 amplifier. A fast rotating disc phosphoroscope was used to separate
shows a second-order
dependence
on the phosphorescence
intensity and the lifetime of DF is half that of the phosphorescence decay time [3]. The temperature dependence of the steady state DF intensity is due to the trapping and detnpping mecha-
nism of the triplet cxcitons and gives information
about Traps
trap properties as was shown by Siebrand [3]. become operative at low temperatures (200 -4°K).
They cause a prolongation
of the lifetime of the excitons and thereby an increase in the exciton density as well as the DF intensity. At room temperature DF
delayed emission from prompt fluorescence. The sample was cooled by a helium cryostat [S]. Constant rate heating of the crystal was performed by heating the sample cell electrically. Heat contact between cell and crystal was achieved using helium gas at different pressures. Rapid heating was performed by irradiating the crystal with IR light (1.7 -2.3,~).
104
F. Auriclr and
3000
2950
3. Marquard, The nzechanism of lhennal!v rrimulated DF
2900
2050
2803
2750
2650
2700
A
-A
Fig. I. Fluorescence
spectrumof crystallinehexadeuterobenzene(B-d,)99%.
Measuring thermally stimulated DF versus IR wavelength an IR spectrum of the host crystal was obtained [61, 2.2. Materirrls Materials were Merck uvasol hexadeuterobenzene (Bd6) 99% and benzene (B-&). BdG contained isotopic impurities in the following approximate mol fractions: Bd, = 5%; Bd, * 0.1%: B-d, = 0.01%; Bdz, B-d,, B-h,
3. R&ts In the first example the Bd, host crystal was doped with 1% B-d, to produce a trap with an energy depth
Temperature
42°K.
of about 100 cm-l [7]. This trap causes a maximum in the steady state DF versus temperature curve, shown in fig. 3. The maximum was the same regardless of whether the temperature was being increased or decreased. This only applies to a temperature rate sufficiently small as to preserve the steady state conditions. Under non-steady-state conditions DF is shown in fig. 4. The cryostat was held at a constant temperature of 4.7, “K. At r, (fig. 4)a temperature rise of about 2°K was effected by irradiating the sample with IR radiation. DF passes through a narrow peaked maximum, the intensity of which is twice that of the steady state DF maximum (fig. 3). In the subsequent part of the diagram, DF reaches a steady state intensity which corresponds to a temperature of about 2°K higher than just before the beginning of the temperature rise at rl. At ‘1 IR radiation is cut off, the drop in temperature causes the DF to pass through a minimum and to return to the original intensity. In the second experiment the temperature rise was performed by heating the crystal electrically. The sample was Bd6 containing the isotopic impurities mentioned above; this host material was doped with 0.1% B-he. There are three operative traps at energy depths of 66 cm-l (Bd4), 100 cm-l (Bd3) and 200 cm-t (B-Q which were occupied by excitons at 41°K. The Bd5 does not operate as a trap as its
F. Aurich and J. Marqlrard,
The mechanism of thermally
5
slimubfed
IO5
DF
IO
T-
15
K
Fig. 3. Temperature dependence of steady stale delayed Iluorescence (DF) intensity. B-d6 doped with I% I1d3.
Moreover, the heights of the peaks are dependent on the initial temperatures as shown in fig. 6 and decrease with increasing initial temperature. The points give the maximum intensity of thermally stimulated DF which was obtained with the same heating rates but with different initial temperatures. In addition, steady state DF and phosphorescence are plotted in fig. 6. A comparison between the points and the (!ph)2 curve show that the maximum intensity
1 Fig. 2. Temperalure dependence of fluorescence spccva of B-d6 and B-&-8-116 (O.OSC’L)mixed crystals in the rqion of first vibration band.
high concentration produces rapid trap-trap energy transfer and the excitons migrate to lower energy traps. Beginning at 4.2’K the sample was heated using various heating rates. The DF versus time curves registered on an oscilloscope are shown in fig. 5. The three intensity peaks in the DF versus time curves are caused by subsequent depletion of the traps during the heating period, whereby the B-h, trap is depleted finally. The different heights of the three peaks are effected by the different concentration of the traps. The heights of the peaks are proportional to the heating rates, whereas time intervals between the peaks and the halfwidths are inversely proportional to the heating rates.
Fig. 4. Delayed fluorescence (DF) as a function of time during rapid temperature changes. tr : risein temperature of 2°K; 12: drop in temperature of 2°K. IId6
doped with
1% Bdl.
106
E .&rich
und 1. Xlarquard.
The mechanrrm of thermally
stimulated
DF
b
_I
0 Fig. 5. Thermally function of time, 12.2. (b) 9.4, (c) by the B-116 trap.
2
I
s
tstimulated dehyed fluorescence (DF) as 3 obtained with different heating rates: (a) 6.3. (d) 3.6”R se?. Broad pak is cnsed B-d.5 doped with 0.1% B46.
of thermally stimulated DF decreases with increasing initial temperature in the same way as the square of
phosphorescenceintensity wirh increasingtemperature.
TFig. 6. Steady state delayed fIuorescencc (DF) and phosphorescence (Ph) intensity 3s a function of temperature. Points give peak heights of thermally stimulated DF 3s P function of initial tempenlure. Heating rates conslant. k+, doped with 0.15 BA6.
indicate, that the excitons are trapped already in the singlet state. k, to k4 are rate constants; the corree sponding processes are shown in fig. 7. Only k, , the thermal release rate constant of trapped excitons,
4. Kinetic analysis
In this section
the kinetics of thermal!y stimulated
delayed fluorescence will be analysed in detail. We consider a crystal containing one type of trap with depth ET= 200 cm-t and with a concentration of 17~ (B-d, doped with 1% BI,). In the case of low excitation intensity, implying low exciton density, the kinetic equations read &/dr
= arl- ktx + k,y - k4x,
dy/dr=k,x.-k2y-k,y;
(1) (2)
wherex is the triplet exciton concentration in the traps,y the triplet cxciton concentration in the band. al is the concentration of the absorbed photons per unit time, which are transferred to the triplet state by intersystem crossing. This term al was added to eq. (I) since fluorescence spectra at low temperature
is temperature dependent. k, depends on the collision raIL .:z:stant Z and the Boltzmann factor, k, = Z X where K is Boltzmann’s constant and exp (-ET/U) 2 = 10’1 xc-1 for benzene (81. k, is the rate constsnt
for collisionsbetween free excitons and traps and is given by the product of collision rate constant and trap concentration. In the case we are concerned with we have k2 = Zc, = log set-l , where cT is the trap concentration. k3 is the rate constant for quenching of excitons in the band; this constant being unknown, the value kj = 10’ SK-~ has been estimated according to the position of the maximum in the DF versus temperature curve 131. k4 is the monomolecular decay constant of trapped excitons and can be determined from the phosphorescence lifetime at low temperatures: for benzene k4 = lo- 1 see-t . The nonlinear expressions (bimolecular terms) representing the annihilation process are omitted in eqs. (I) and (2) since we consider weak excitation only, not exceeding I = lOIs photons/seccm3, i.e.,
a negligible proportion only
decays
of the free or trapped excitons
by triplet-triplet
annihilation.
107
E .-lurich and J. Marquard, The mechanism of rhermally srimulared DF
I
aI
Fig. 7. Transition klp.
rates concerning triplet exciton band and
The delayed fluorescence intensity (fDF) portional to the annihilation probability I,,
-)‘2
is pro-
+ fir,, .
(3)
The annihilation consists of two processes: (I) the homogeneous annihilation of free excitons being proportional 10 the square of band exciton concentration and (2) the heterogeneous annihilation of free cxcitons and traps. The latter is proportional to the free exciton concentration and occupied-trap concentration. The factor fl< 1 takes into account that the trapped exciton taking part in the annihilation process is immobile. At low temperatures where traps are operative heterogeneous annihilation is predominant. This is the case with Bd6 crystals. For this reason homogeneous annihilation is disregarded and eq. (3) reduces to IDF -xy
(4)
Under steady state conditions, &/dr
this means
= dvldr = 0
Equations
(5)
(1) and (2) can be solved with respect
to
xand) x = IicZl(k&
+kz”4)j
(YI 1
(6)
Y = [k,/(k,$
+k$q)l ffl1
(7)
where k, WE, neglected compared with k2, the latter being greater by several orders of magnitude. From (4), (6) and (7) we obtain for the DF intensity $,F - [k,k2/(k,k,
+ kzk4)*3
(aO*
.
Cakulated curves ofx, y andxy for a Bdg doped with 1% 9.11, ate shown in fig. 8. The concentration of the occupied traps portional to the phosphorescence intensity decreases with increasing temperature. The
(8)
IL
I2
16
18
20
K
TFig. 8. Calculrfed tentpeaturc variation (under study state conditions) of population ol’ triplet cxciton tnps (x). popul~. lion of triplet exciton band (_I,)and DF intensity (xf).
sity curve passes through a maximum when the temperature is either raised or lowered. This applies to small hearing rates only. Under rapid heating conditions, which ate realised during thermal stimulation k, becomes time dependent and eq. (5) is no longer valid. The rate equations a.~ a (I) and (2) must be solved with temperature function of time, which in general, can be performed with the aid of a computer. It is possible, however, to obtain information concerning the mechanism of stimulated DF by solving the rate equations approximately. Addition of eqs. (I) and (2) yields
dx/dt t dyjdt = crl - k,y - k4x . We now introduce a fixed population band and traps which is proportional sponding transition rates
(9) ratio
between
lo the corre-
crystal
u/x=k,lk,
is prowhich DF inten-
The physical meaning of this approximation is as follows: The population of traps from the band (k,) is a process which is much faster than the other processes considered. Consequently a pzrtial equilib-
.
(10)
108
F. Auricb and .I. hlarquard. The mechanism of thermally
stimulared DF
rium is established which may be assumed to be dependent on k, and k2 only (thermal equilibrium). It is now possible to solve the rate equations in closed form. The temperature dependence of k, is expressed by k, =Z@
.
(11)
From eqs. (9), (10) and (11) follows
These equations give information as to the temperature dependence of the population of bands and traps. To simplify the rale equations further, only low temperatures (T < 40°K) are considered, i.e., k, is neglected with respect to kz. In addition the excitation term is omitted. The previous results also hold good when the exciting ligh! is cut off before the heating process begins [I]. With these omissions and k,/k2 = cyleAt) ,
(14
eqs. (12) and (13) read dx/dr = - [ ~5’ efi’) (k3t df;df) t k4] x ,
These equations can be integrated in the case when we assume a particular dependence of temperature on time during the heating process: (17)
where n~is the parameter of temperature rate and II is given by n=ET/KTo,
population x, band popuhtion y and DF intensity xy under thermal stimulation. xo, ho initial values. At I = 0 excitation is shut off. Heating begins
at I = 1 set (1%
dy/dl = - [c;t eAf) (k3 t df/dr) t k,- d/ldr] y . (16)
flr)=mr--n,
Fig. 9. Calculatedvariation of trap
(1%
with To as the initial temperature. Certainly there arises no serious restriction by this particular choice since we are only interested in the qualitative behaviour. The integration of eqs. (15) and (16) together with (I 7) yields
m = 2
with initial temperature To = 14°K. Heating rate
see-’
x=xoexp
{-[r;.‘(exp[mt-n]-exp
x (k31ttt+I)+k4t] I, y=yo exp {-[I$’ (exp[mr+r]-exp x (k#n + 1) t (k4 -nz)t] ) . Fig. 9 shows the population of traps and of the band, in addition the product of both as a function of temperature [eqs. (17), (19), (20)]. The initial temperature is TO = l4’K. The heating interval is 14-25OK; this interval is passed in 4.5 set (III= 2 set-l). At r = 0 the exciting light is cut off; DF and phosphorescence begin to decay. After one second the heating begins. The increase of temperature causes an accelerated decrease of the trap population and therefore
E Aurich and J. Marquard.
The mechanism of thermally
a decrease ofthe phosphorescence intensity. Contrary to this, the exciton concen:ration in the band and consequently the DF intensity are increased and pass through maxima at different times. The position of the DF maximum is determined by I rn3x=~7-‘{rr-ln
[2(k
,+m)/+-X,)l~l
and the corresponding temperature
(21)
is
T mLX= ET/K In (2(k3+ m)/c-&nz - 2 k,)]
.
(22)
Restricting m to the interval (23)
and with eqs. (17-22) the maximum intensity of thermally stimulated DF is calculated as =(,nx$k,)exp[(Z!k,/rnc~)exp
perimental results (fig. 6). Moreover, eq. (24) shows the maximum intensity being proportional to the heating rate which holds true in the interval given by (23) and at constant initial temperature. This too agrees with the experimental results (see fig. 5). The preceding results were obtained with reduced rate equations and under numerous restrictions as, e.g., one type of trap only was considered. For a sequence of narrow peaks, as shown in fig. 5, caused by subse-
quent depletion of severaltypes of traps with different
k4 Qnt
WrnLX
109
stimulated DF
depth, an analysis might be possible using rate equations which include Ihe nonlinear terms for the
energy
annihilation future
process. This shall be considered in a
publication.
(-11) - I], (24)
x0 being the occupied-trap concentration at the initial temperature. Equation (21) shows, that a maximum only exists for the logarithmic term being smaller than II. From this a relation can be derived yielding information as to which value the heating rate at least should have (in function of the initial temperature To) in
Acknowledgement We wish to thank F. Hofelich for helpful discussions. This work
was supported
Wirtschaft
als Verwalter
by the Bunderrninister
fiir
des ERP Sondervermogens.
References
order to guarantee the existence of a maximum in the DF curve. Taking into account the restriction (23), eq. (22) shows an increase of Tmax with increasing heating rate.
For an initial temperature lower than the temperature, at which the steady state DF maximumoccurs, the exponent in eq. (24) can be assumed to be -1, )I being sufficiently high. At constant heating rates the maximum DF intensity is dependent on the initial temperature
via the square of the occupied-trap
con.
centration, the latter being proportional to the steady state phosphorescence. This is consistent with the ex-
[ 11 F. Aurich and J. hlarquard, Chem. Phys. Letters6 (1971) 91; 11 (1971) 167. [2] T. Azumi and S.P. h!cClynn. J. Chem. Phys. 38 (1963) 2773. [3] W. Siebnnd. J. Chem. Phys. 42 (1965) 3951. 141 F. Aurich and E.-P. Resewitz, J. Sri. Instr. 3 (1970) 899. [S] F. Aurich, L. Helbing and J. hfarquard, Z. Angew. Phys. 29 (1970) 365. 161 F. Aurich and J. hlarqusrd. unpublished results. 171 D.hl. Burland, C. Castro and C.W. Robinson, 1. Chem. Phys. 52 (1970) 4100. 181 J.B. Birks, Photophysics of aromatic molecules (WileyInterscience, New York, 1970).