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Nuclear relaxation in one-dimensional triplet-exciton systems
J.P. Travers, M. Gugliehni, M. Nechtschein
To cite this version:
J.P. Travers, M. Gugliehni, M. Nechtschein. Nuclear relaxation in one-dimensional triplet-exciton
systems. Journal de Physique, 1982, 43 (4), pp.663-673. �10.1051/jphys:01982004304066300�. �jpa-
00209437�
Nuclear relaxation in one-dimensional triplet-exciton systems
J. P. Travers (*), M. Gugliehni and M. Nechtschein (*)
Centre d’Etudes Nucléaires de Grenoble, DRF, Section de Résonance Magnétique,
85 X, 38041 Grenoble Cedex, France
(Reçu le 20 mai 1981, révisé le 27 novembre, accepti le 22 dicembre 1981 )
Résumé.
2014Le temps de relaxation nucléaire (T1) des protons a été mesuré à différentes températures dans deux
sels semi-conducteurs de tétracyanoquinodiméthane (TCNQ) : triphénylméthylarsonium (TCNQ)2 et Cs2(TCNQ)3.
La variation de T1 en fonction de la fréquence indique, à chaque température, l’existence d’une diffusion de spin
unidimensionnelle (1d), dans les deux composés. Dans le sel d’arsonium, la variation thermique du coefficient de diffusion obtenue entre 70 et 300 K, est plus forte qu’une loi thermiquement activée. Pour expliquer ce résultat,
on propose un modèle qui tient compte à la fois de la marche aléatoire 1d, thermiquement activée, des excitons, et des interactions d’échange qui existent entre excitons lors des collisions.
Abstract.
2014Proton nuclear relaxation time (T1) measurements have been performed at different temperatures in two semi-conducting salts of tetracyanoquinodimethane (TCNQ) : triphenylmethylarsonium (TCNQ)2 and Cs2(TCNQ)3. At every temperature, a T1 frequency dependence is reported, which provides evidence for a one-
dimensional (1d) spin diffusion in both compounds. In the arsonium salt, the diffusion coefficient is obtained as a function of temperature between 70 and 300 K, the observed dependence being steeper than a thermally activated
law. In order to explain such a result, a model is proposed, taking into account both the thermally activated ld random walk of the triplet excitons, and the exciton-exciton exchange interactions which take place during the
collisions.
Classification
Physics Abstracts
76.60E - 71.35
-71.70G
1. Introduction.
-The salts of tetracyanoquinodi-
methane (TCNQ) have attracted considerable atten-
tion during the last twenty years. Besides the highly conducting salts which have a large physical interest
connected with the properties of one-dimensional ( 1 d) electron-gas (Peierls instability, three-dimensional
effects, impurities effect, conduction mechanism, ...)
there exists a class of semi-conducting salts, which exhibit unusual magnetic properties. Most of them
are characterized by the existence of triplet excited
states (excitons) thermally accessible from a singlet ground state. A number of studies on the triplet
exciton ESR spectra have been reported [1-15].
While the ESR fine structure splitting appears to be rather well understood, the understanding of the dynamical effects related to the exciton motion
remains poor. In particular it is noteworthy that twenty years after the first work by Chesnut and
Phillips [1] there is still not quantitative explanation
for the temperature dependence of the ESR linewidth and the dipolar splittings observed in some
TCNQ salts, such as triphenylmethylarsonium 03AsCH3(TCNQ)2 and Cs2(TCNQ)3.
The present work is devoted to a reinvestigation
of the exciton motion in the two semi-conducting salts
of TCNQ : 03AsCH3(TCNQ)2 and Cs2(TCNQ)3.
We take advantage of nuclear spin relaxation time
( T 1 ) measurements, which have been developed as
a new powerful tool for studying the spin dynamics
in low dimensional systems. The frequency dependence
of the nuclear relaxation rate Ti1 is directly connected
to the spectral density of the local spin fluctuations
r(m). The method has been first successfully tested
in Id insulating magnetic systems (Heisenberg chains) [16]. Then, it has been used in highly conducting TCNQ salts [17] and, more recently, in conducting polymers [18]. Very interesting results have been achieved about the conduction electron motion in the first case, and about the soliton motion in poly- acetylene in the second case.
The ESR spectra of these compounds depend strongly on temperature. Usually, the triplet exciton spectrum at low temperature consists of two lines which are due to the intra-exciton electron-electron
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004304066300
dipolar coupling. As the temperature increases, the
lines broaden and shift towards each other until
eventually they merge into a single broad line. At higher temperatures the single line narrows. The
absence of hyperfine structure and the narrowness
of the lines at low temperature are generally attributed
to a « motional » narrowing due to a rather fast motion of the triplet excitons. The temperature depen-
dence of the ESR line shape is qualitatively under-
stood. As their concentration increases, the triplet
excitons collide more and more frequently. They undergo spin-spin exchange interactions which give
rise to an « exchange » narrowing of the fine struc- ture. From structural consideration the triplet exciton
motion is supposed to be Id. Moreover, because of exciton-phonon interactions, it has been shown that the triplet excitons tend to be localized, and to move
in a diffusional manner [9, 19]. The two last points
have been confirmed by nuclear relaxation time (T1)
measurements [10, 21]. In this paper we shall consider both the effect of the exciton hopping motion and of the exciton-exciton exchange interactions.
Our experimental work is based upon T1 measure-
ments as a function of both frequency and tempe-
rature in 3 3(TCNQ)2 and Cs2(TCNQ)3. It
is a matter of fact that, in this class of compounds,
the nuclear relaxation is dominated by the triplet
exciton motion. Evidence is given for a one-dimen-
sional spin motion in both materials. The diffusion coefficient is determined as a function of tempe-
rature. In the arsonium salt, an interpretation of the
results taking into account the effect of the exchange
interaction is proposed. It leads to a determination of the activation energy for the hopping rate and to
an estimate of the exciton-exciton exchange integral.
2. Experimental.
-In order to avoid any ambiguity
on the origin of the nuclear relaxation in the arsonium salt (methyl group rotation for examplex we have performed the measurements on the salt whose cation
was deuterated : O*ASCD 3 3(TCNQ)2’ It was pre-
pared by reaction of triphenylmethylarsonium iodide
with TCNQ in acetonitrile [22]. The iodide of deute- rated cation was synthetized in two steps : prepa- ration of triphenylarsine di s and methylation accord- ing to Michaelis’s method [23]. The cesium salt was
prepared by the same method, but the solvent was a mixture of acetonitril and methylic alcohol. The
total deuteration of O*ASCD as determined by
mass spectroscopy was 99 %.
The proton T, has been measured on powder samples by conventional 7C-7T/2 sequences on a Bruker SXP spectrometer over the frequency range 4-200 MHz. Measurements have been performed
on Cs2(TCNQ)3 at very low frequency (between 0.2
and 4 MHz) by Minier (Laboratoire de Spectrom6trie Physique de l’U.S.M.G.) using a field cycling method.
In O*ASCD 3 3(TCNQ)2 measurements have been achieved at 10 different temperatures, between 70 and 293 K. Room temperature (293 K) data have
Fig.1.
-Proton relaxation rate as a function of the inverse square root of the nuclear Larmor frequency F-1/2 for
different temperatures in ØAsCD3(TCNQ)2 : a) T = 293, 250.5, 179.5, 161.5, 144 and 125 K ; b) T = 100, 90, 80 and
70 K.
Fig. 2.
-Proton relaxation rate as a function of the inverse square root of the nuclear Larmor frequency F -1/2 for
different temperatures in Cs2TCNQ3 : a) T = 293, 254
and 220 K; b) at very low frequencies (F > 0.21 MHz).
The straight lines are obtained from the fit of the experi-
mental data with a ld diffusion at both nuclear and elec- tronic Larmor frequencies for F > 55 MHz, and with a
ld diffusion at electronic Larmor frequency only for
F 55 MHz.
been taken from Avalos et al. [21]. In the cesium salt, T, has been measured at three temperatures, namely : 293, 254 and 220 K. Measurements have not been carried out at lower temperatures in both salts because the nuclear relaxation time is already longer
than n-, 1 s. It is, thus, no longer certain that the exciton motion is the main source of nuclear relaxa- tion. Other relaxation mechanisms, such as para-
magnetic impurities, should be probably taken into
account. We shall come back more quantitatively
to this point later on in the discussion (§ 3.4).
The uncertainty of the results is about 3 %, except
at low frequency (F 10 MHz) where it may reach 8 %. The recovery of the magnetization in powder samples is not strictly exponential because of the
anisotropy of T,. In O*ASCD 3 3(TCNQ),2 the aniso-
tropy, as measured by Avalos et al. [21] at room tem- perature, is about 2. Actually, for both compounds
the departure from exponential was smaller than the uncertainty on T 1 measurements, except in the
arsonium salt at the lowest temperature (70 K). Since
this effect appears only at 70 K, it cannot be attributed
to TB anisotropy. An explanation could exist in an
inhomogeneity of the proton relaxation. In fact,
all the protons have the same T, only if the excitons
move fast enough to visit %very site many times in a
time smaller than T,. This condition may not be completely fulfilled at 70 K.
The proton relaxation rates T11 measured on O*ASCD 3 3(TCNQ)2 and Cs2(TCNQ)3 are given in figures 1 and 2. The data are defined from the recovery of magnetization at short times, namely between
t = 0 and t = Tl/2.
3. Analysis of results.
-3.1 EXPRESSION FOR NUCLEAR RELAXATION RATE.
-The nuclear relaxation rate due to the electronic spin motion in a poly- crystalline sample is given as follows [21, 32] :
where ron, and roe, are the nuclear, and electronic,
Larmor frequencies; f((D) is the two-spin corre-
lation function :
where sf is the z component of the spin operator associated to the unit cell i. C(T) is the effective electronic spin concentration per unit cell, which is supposed to follow the Curie law. It corresponds
either to the effective number of electronic spins
(S = 1/2) in a metal-like compound, kT q(EF) where
il(EF) is the density of states at the Fermi level, or,
as in the present case, to the exciton (S = 1) concen-
Table I.
-Anisotropic spin diffusion.
tration. Last, a, and d, are the scalar, and dipolar, parts of the hyperfine couplings between the nuclei and the effective electronic spin.
In the general case of anisotropic diffusive motion,
one has to define three diffusion coefficients D,,, Dy and Dx (assuming Dz > Dy > Dx). A good Id system is characterized by D,, >> Dy, Dx, and a 2d
system is such as D., -- Dy >> D,. In the most general anisotropic system the spin motion may appear to be Id, 2d, or 3d according to the observing frequency
range. The theoretical expressions of f((o) are given
in table I as a function of frequency.
3.2 DIMENSIONALITY OF THE SPIN DIFFUSION.
-Thus, the dimensionality of a diffusive spin motion
can be determined from the frequency dependence of
the nuclear relaxation rate T11.
For the two compounds that we have investigated
the T11 data are reported in figures 1, and 2, as a
function of the inverse square root of the nuclear Larmor frequency. In such a plot, a Id diffusive process gives a straight line. In order to test the hypothesis of a 2d spin diffusion, T11 has been plotted
in a logarithmic frequency scale in figure 3, and figure 4, for the arsonium, and for the cesium salt, respectively.
For 0!AsCD3(TCNQ)2 the comparison between figures 1 and 3 gives strong evidence for a Id spin
diffusion. On the one hand, within the error bars,
the T11 data plotted versus F -1/2 consists of straight lines, except in the low frequency-room temperature domain. On the other hand, the T11 variation pre- sents a general curvature over the whole frequency
range in a logarithmic scale (Fig. 3). As concerns
the room temperature spin dynamics, a more detailed analysis [21] has concluded in favour of a ld spin
diffusion in the electronic Larmor frequency range, and of a 2d diffusion in the nuclear Larmor frequency
domain. Thus, the T11 variation results in a super-
imposition of two different contributions. Below room
temperature, the spin diffusion appears to be more ld in the experimental frequency range, which implies
that the ld behaviour extends down to the (On domain.
This can be explained in terms of an overall narrowing
of the spectrum motion with decreasing temperature.
In the case of Cs2(TCNQ)3, from figures 2 and 4,
it is clear that the spin niotion is diffusive and one-
dimensional at 220 and 254 K. At room temperature, the spin diffusion is neither purely Id nor purely
2d. In fact, the T, 1 variation versus F -112 is composed
Fig. 3.
-Proton relaxation rate as a function of the nuclear Larmor frequency F for different temperatures in
3 3(TCNQ), : a) T = 293, 250.5, 179.5, 161.5,
144 and 125 K; b) T = 100, 90, 80 and 70 K.
of two straight lines. The straight line corresponding
to the high frequency range has a zero intercept.
This suggests that the spin motion is a ld diffusion
both in the We and Wn frequency ranges, at high fre-
Fig. 4.
-Proton relaxation rate as a function of the nuclear Larmor frequency F in CS2(TCNQ)3 at T = 293 K, 254 K
and 220 K.
quencies, and only in the We frequency range, at low frequencies; consequently, there is a cross-over from
a Id to a 3d diffusion in the nuclear Larmor frequency
range (see Fig. 5). A quantitative analysis will confirm
this hypothesis.
Fig. 5.
-Spin diffusion spectrum of Cs,(TCNQ)3 at room temperature. It displays a cross-over from a ld diffusion
to a 3d diffusion in the nuclear Larmor frequency range.
3.3 SPIN DIFFUSION COEFFICIENT.
-A quanti-
tative analysis of experimental data requires the knowledge of both the exciton concentration and the hyperfine couplings. The exciton concentration is given by :
where J is the singlet-triplet separation, which can
be determined from the temperature dependence of
the spin susceptibility. As in most of the TCNQ compounds, J is temperature dependent in 0jAsCH3(TCNQ)2 [8-9] and CS2(TCNQ)3 [24].
The hyperfme couplings have been experimentally
determined in ØAsCD3(TCNQ)2 by Devreux et
al. [25]. They have obtained :
These values are normalized; they correspond to
the couplings between one TCNQ proton and one electron spin delocalized over this TCNQ only.
We will take the same values for the hyperfine couplings in Cs2(TCNQ)3, although the intermole-
cular contribution to the dipolar coupling may be somewhat different in the two salts.
In the arsonium salt, the unit cell for a given chain
is composed of four TCNQ molecules. The two electrons coming from the cations (and consequently
the triplet exciton) are delocalized over the four
TCNQ [26, 27]. It is shown in the appendix that the hyperfine couplings between one proton and one exciton are I a/Ye = 0.3 gauss and
In Cs2(TCNQ)3 the TCNQ chain unit cell is
composed of three molecules. The two electrons of a unit cell are located on two TCNQ only [27, 29] ;
thus two proton species exist in this compound : i) A protons located on TCNQ- ;
ii) B protons located on TCNQ’.
The relative number of A, and B, protons are 2/3,
and 1/3, and their direct electron induced relaxation times will be denoted as T 1 and TB, respectively.
The resulting nuclear relaxation rate is expressed
as follows [30] :
This expression holds either for strong nuclear spin- spin coupling between A and B, giving rise to a single
nuclear spin temperature, or for weak A-B coupling
if one considers the resulting T, at short time (t TI). It is well-founded to suppose that the direct
couplings between the B protons and the electronic
spins are very small and consequently the relaxation rate T-’ may be neglected in expression (4). One then
obtains :
In the expression of T11 one must take the effective hyperfine couplings for one exciton delocalized over two TCNQ, i.e.
The diffusion coefficient depends on the fourth power of the couplings, therefore, the knowledge of
their exact value is important. Actually, the uncer- tainty of the room temperature experimental deter-
mination [25] is about 8 %, which might give a syste- matic error larger than 30 % on D,,. Another error
could arise from a temperature dependence of the
electron distribution on the TCNQ molecules. Let a
be the difference between the electron charges of the
two TCNQ of a dimer. The effective scalar coupling
between one proton of a diad and one electron deloca- lized on the diad is :
where a is the coupling between one proton of a TCNQ
and one electron on the TCNQ. The square of the
couplings depends on the charge asymmetry a, at the second order only. The temperature dependence of a is
unknown in our compounds, but, in the TEA(TCNQ)2
a similar compound, this variation is definitely less
than 0.2 between 40 and 340 K. [A. Filhol, private communication.] Thus, it is reasonable to suppose that this effect is negligible on the temperature dependence of Dz. The systematic error which might be introduced
on Dz does not affect its temperature dependence,
which is the main interest of this paper. It will be
neglected in the following parts, and on figure 7 for clarity.
Now one can determine the ld diffusion coefficient
Dz from the experimental data. At this stage, three
Fig. 6.
-Different hypothesis of Id spin diffusion : a) dif-
fusion at nuclear Larmor frequency only :
b) diffusion at both nuclear and electronic Larmor fre-
quencies : Dy WN We Dz ; c) diffusion at electronic
Larmor frequency only : (ON D., Dy Cùe D,,.
hypotheses remain possible since the relaxation rate
T11 is composed of two contributions :
a) the diffusive contribution to T11 comes from the
(Un range, the contribution from We being equal to
zero : Dy (On D,, (De ;
b) the two contributions are diffusive :
c) only the contribution from We is diffusive, the
contribution from Wn being constant :
These three situations are illustrated in figure 6.
The straight lines have been fitted to the experi-
mental data in the F- 1/2 plot by least mean squares.
From the slopes of the straight line one obtains values
for the diffusion coefficient D,, in each of the three
hypotheses. Some of them turn out not to be consistent with the value obtained for the diffusion coefficient and therefore can be excluded. For instance, in the case of
arsonium salt at 80 K the calculated values for the dif- fusion coefficient are smaller than We in any of the three
hypotheses. One can thus conclude that hypothesis a)
is the only one to be physically valid. Moreover, from continuity considerations over the temperature range,
some other hypotheses appear to be, unrealistic. The
resulting values are given in tables II, and III, for
3AsCD3(TCNQ)2, and Cs2(TCNQ)3, respectively.
It turns out that the diffusion coefficient increases
drastically with temperature in both salts.
In the arsonium salt at low temperature, the spin
motion is diffusive only at the nuclear Larmor fre- quency. As the temperature increases, the motion spectrum broadens progressively, and at high tempe-
rature it becomes diffusive both at (On and roe, At
Fig. 7.
-Spin diffusion coefficient D. as a function of the
inverse temperature in O*ASCD 3 3(TCNQ),. The solid curve
is calculated from a theoretical model (see text § 4) with
the following pararheters :
-the hopping frequency Wh = 9.4 x 109 exp( - 185/T) rad./s ;
-the exciton-exci-
ton exchange integral J’/4 cr 0.13 K. The dotted-dashed
line represents the contribution of the hopping motion only.
Table II.
-Spin diffusion coefficient as a function of temperature in Ø;AsCD3(TCNQ)2’
Table III.
-Spin diffusion coefficient as a function of temperature in CS2(TCNQ)3’
intermediate temperature it exists a cross-over range where the determination of Dz is rather difficult. We
give in table II values obtained from two different
ways, i) from the high frequency data assuming
diffusion at cow ii) from the low frequency data assuming diffusion at roe and (On
.The D. values are only in qualitative agreement.
In the cesium salt the quantitative analysis at 293 K
confirms the existence of a cross-over of the spin
motion from a ld to a 3d diffusion in the nuclear Larmor frequency range. So, in Cs2(TCNQ)3 one has Dx Dy Dz (Fig. 5). The T11 experimental varia-
tion has been fitted with two straight lines (see Fig. 2b),
each of them giving rise to a determination of Dz. Both
values agree quite perfectly (see table III). From the crossing-point of these straight lines one gets the
cross-over frequency Fco which is related to the
transverse diffusion coefficient Dy by [31] :
At room temperature one obtains Dy = 2 x lU8 rad./s,
which gives a spin diffusion anisotropy of
D,, .IDY = 3 x 101. Since the conductivity anisotropy
is only 25 [32] this result gives a new evidence that spin
motion and charge motion are completely discon-
nected in this class of materials.
In order to test the results of our analysis in Cs2(TCNQ)3 we have measured T,,, the nuclear
relaxation time in the rotating frame. The measure- ment has been performed at 45 MHz; we have obtained
T iP = 24 s - 1. The expression for T 1 p is given by [33] :
where mi = yHl, H, being the amplitude of the rotat- ing magnetic field. In our experiment, ro1 is of the order of 105 rad. s- -’, which is much smaller than Dy.
By means of equation (7), expressions of r(ro) in table I,
and values of DZ and D. the calculation yields
TiP = 26 s-1, which is in excellent agreement with experiment.
3.4 CONTRIBUTION OF NUCLEAR SPIN DIFFUSION. -
We now discuss a possible contribution to the nuclear relaxation due to fixed paramagnetic centres and processing via nuclear spin diffusion. It is a matter of fact that such a contribution is expected to give
rise to a ro-1/2 law [34], similar to the ld electron spin
diffusion process. The problem of a possible non negli- gible fixed spin contribution to the nuclear relaxation should be considered at low temperature since the mobile spin concentration decreases exponentially
with temperature. Namely, in the arsenium salt at 70 K there is only 2.5 x 10- 5 exciton per TCNQ,
while the fixed spin concentration, as estimated from
the intensity of the ESR spectrum central line is about
2 x 10-4 per TCNQ. One can derive the following expression for the contribution of this process to the nuclear relaxation [34] :
where Ys, and y,, are the electronic, and nuclear, gyromagnetic factors, N is the fixed spin concentration per unit volume, D. is the nuclear diffusion coefficient,
and,r is the electron correlation time. D. is related to W,
the nuclear diffusion rate and to t4 the average distance between the protons, by :
We take u = 4.8 A as calculated from the crystallo- graphic parameters [26], and we estimate in the present
case W - lO3 . S-1. As concerns the electron correla- tion time, it is of the order of the electron spin lattice
relaxation time, and we take i ri 10-6 s. Using these parameters, at ron = 100 MHz, we obtain for the relaxation rate due to fixed spins T -’ - 0. 1 S-1.
Since the observed relaxation rate is T11 0.2 S -1,
the former contribution is no longer negligible. In fact, it has been overestimated for the following
reason. We know from crystallographic data [26] that
the distances between protons are very anisotropic
and thus, the nuclear spin diffusion is even more aniso- tropic because the dipolar interaction between two
protons goes as the third power of their inverse dis- tance. In bulk, the anisotropic nuclear spin diffusion
around a fixed spin is slower and much less efficient than an isotropic one. Anyway we can consider 70 K
as a lower temperature limit for our study, in order
the nuclear relaxation by exciton motion not to be hindered by other mechanisms.
4. Discussion.
-Let us now concentrate our
attention to the data obtained for ØtAsCD3(TCNQ)2.
The diffusion coefficient which has been determined has been plotted on figure 7 in a logarithmic scale as a
function of inverse temperature. Actually, one may divide the experimental data into two parts :
-
a low temperature part (T 125 K), which is practically linear, with a rather small slope;
-
a high temperature part (T > 145 K), where the
variation of Dz with temperature is much steeper.
Let us discuss the possible spin diffusion mechanisms
which may account for such a behaviour.
The most straightforward mechanism to consider
is the exciton motion. Theoretical treatment of this
problem has been given earlier [9, 19]. Because of the exciton-phonon coupling, the excitons are self-trapped
and their motion is a Id random walk. According to
Soos and Mc Connell [9], the temperature dependence
of the hopping rate Wh has two limiting behaviours :
-
in the high temperature region, Wh is thermally
activated :
where E is the activation energy, J’/4 the exchange
interaction between excitons, j’ = 6J’16X the deri-
vative of J’ over the intermolecular distance, and g
the elastic force constant between TCNQ molecules;
-
at low temperature, Wh tends towards a finite value which is not temperature dependent.
The temperature of the cross-over between the two
regimes is of the order of the Debye temperature.
Qualitatively, these theoretical predictions could explain the experimental Dz temperature dependence.
Actually, the Debye temperature is between 60 and 100 K in this class of materials [35]. The situation
would be the following :
-
the high temperature part of the experimental
data would correspond to the thermally activated
exciton hopping;
-
the low temperature part should correspond to
the smooth cross-over of the hopping frequency
towards a constant value.
The experimental data for Dz T 1/2 have been fitted with a thermally activated law between 160 and 293 K
by least mean squares. We have obtained :
By identification with equation (9) it is possible to
extract an estimate for J’. We have taken j,21g =
1 600 cm - 1 as in reference [9]. This leads to a very large
value for J’ : 1 500 K. Such a result is physically impossible since J’ is the exchange interaction between two electrons of two contiguous unit cells and
consequently should be much smaller than the intra- cell exchange interaction J - 720 K. Taking one order
of magnitude more for the value j’2/g results in a
small change in J’, namely J ’ ri 1000 K. It is clear that the value obtained for the spin diffusion coefficient is too high to be interpreted by the exciton motion only.
’
We, thus, introduce the spin-spin exchange inter-
action (J’/4) which takes place during a collision
between two excitons. A theoretical treatment of this
effect, which considers the excitons as hard sphere-
like hopping spins, has been given earlier [36]. The main
conclusions of this study can be summarized as follows.
If the exciton concentration C, is such as CJ’/4 Wh,
the resulting local spin fluctuation behaves in a diffu- sive way over the frequency range C2 Wh a) Wh.
The effective spin diffusion coefficient is then the sum
of two contributions : Do, and D,, which arise from the
random walk, and from the exchange interaction,
respectively. They are given by the following expres-
sions :
The hard sphere-like model for the excitons is based upon the fact that two excitons cannot occupy the
same site at the same time.
At low temperature (T 125 K) the exciton con-
centration is very small (C 1/450). Therefore, the
collisions between excitons are rare and it is quite
natural to neglect the effect of exchange interactions.
Consequently, the experimental diffusion coefficient in the low temperature limit, basically is nothing but
the hopping rate of the intrinsic one-exciton random walk. Then, the cross-over which is observed around 140 K is assumed to arise from the fastly increasing
contribution of exchange interactions. In fact, this model suppose that the 70 to 125 K temperature range
belongs to the exciton motion high temperature regime [9], and therefore, that the Debye temperature is smaller than 70 K. This hypothesis is supported by
the dependence of the Debye temperature with the molecular mass and the volume of the chemical unit cell in this class of materials [35]. According to this relation, the Debye temperature would be around 40 to 50 K in Ø;AsCD3(TCNQ)2. In order to make the analysis of the results, in the framework of the model,
more tractable, without altering its physical content,
we take a simplified version of expression (8)
The parameters of the model have been determined in order to fulhll a twofold condition as follows : i) fit
with the experimental variation of D. at low tempera- ture, ii) account for the cross-over which is observed around 140 K. We have obtained :
The theoretical curve calculated with these values is
given in figure 7. The agreement with the experimental
data is rather good from 70 to 160 K. A departure
arises at higher temperature. Actually, this is not too
much surprising since the theoretical model is valid
only for CJ’14 Wh, which, in the present case, implies the temperature to be lower than about 290 K.
The variation of Do has also been plotted in figure 7 in
order to appreciate the contribution of the exchange
interactions to spin diffusion in the framework of our
interpretation. One can see that D1 is actually much
smaller than Do at the lowest temperatures. However, this is no longer the cage above 100 K, although the
exciton concentration is still as small as 10-2.
Since no other experimental determination for the
exchange integral J’ has never been given, it is difficult
to discuss the validity of our value : 0.5 K. Although
this is a small value, the order of magnitude seems to be
correct for the following reasons. The observation of a fine structure in the ESR spectrum implies that the
two electrons which compose a triplet state are bound and, consequently, that J’ J. From the stacking of TCNQ along the chain, a qualitative analysis of the overlap of the x orbitals, inside a given unit cell, and
between two nearest neighbour unit cells, leads to the
same conclusion. Last, in 03AsCH3(TCNQ)2 the
temperature dependence of the magnetic suscepti- bility may be described in the framework of a strongly alternating antiferromagnet with J/J’ > 7 [10, 37].
As far as the exciton hopping motion is concerned,
some information has been already obtained from the ESR spectrum. Let us, first, report results concerning
the fine structure. Its thermal evolution has been attri- buted to the effect of exchange interactions during the collisions, which give rise to an effective exchange frequency WX. The expression of oB depends on the strength of the exchange coupling J’/4 compare to zp ’
the inverse lifetime of the complex consisting of two coupled excitons [13]. If one assumes that Tp I is equal
to the hopping frequency, thus, in the strong exchange
limit (J’14 >> Wh/3) one obtains :
The fine structure disappears around 200 K. We may
thus infer that, at this temperature, the exchange frequency is of the order of the intra-exciton dipolar interactions, which yields co. t-- 6 x 108 rad./s. From
the Wh and J’ values and equation (12) it can be
verified that the strong exchange limit is valid at
200 K, and one obtains co. = 7 x 10" rad./s which is
about one order of magnitude smaller than the former estimate.
Experimental determinations of the temperature dependence of (J)x have been extracted from both the
splitting and the width of the ESR lines at low tempe- rature, and also from the ESR linewidth at high tempe-
rature. A thermally activated variation is found in each case, but quite different values are found for the acti- vation energy : 580,1280 and 1860 K [1,6]. The discre- pancy between these results is still unexplained. In
our model, from equations (2), (11) and (12), it turns
out, that (J)x is thermally activated, the activation energy being equal to J + E = 900 K.
The absence of hyperfine structure at low tempera-
ture has been attributed to a motional narrowing of the
nuclear hyperfine interaction. Soos has pointed out the specific character of one-dimensional motion in narrowing hyperfme contributions to the line- width [38]. General expressions for the ESR line in the
case of one-dimensional motional narrowing has
been derived by Holczer et at. [39], including both
scalar and dipolar hyperfinc contributions, and also spin-spin contribution.
They have obtained :
where Am is the ESR half width, a2 and d 2 are the mean
s uare of the scalar and dipolar hyperfine couplings,
CR/ye is an effective spin-spin dipolar field, and co,,,
is the low frequency cutoff frequency of the ld diffusion.
In the present case Wc is the transverse diffusion coeffi- cient. (i2 is defined by :
where Y denotes a summation over all the nuclei of
i
one unit cell ; ai is the scalar hyperfine coupling between
the electron spin and the i-nuclei, and I is the nuclear
spin value. Using the hyperfine coupling values for an electron spin delocalized over one TCNQ (aH =
1.2 G [25], aN = 1.02 G [40]) and taking into account
the actual delocalization over four TCNQ, one obtains a2 -- 4 gauss2. The same expression as (14) holds for 3 2. As for the proton hyperfine couplings, we have
taken for the nitrogen nuclei : aNldN = 2. Last, the
third term in equation (13) has been roughly estimated by the same way as used in reference [39]. At 77 K, it
givesv/-C-R/y,, L-- 1.3 gauss. For cor we have taken the lowest limiting value, which is the linewidth itself. The observed half width at 77 K is ð.w 3 x 106 rad./s [41],
which gives for the diffusion coefficient D., --- 2 x 109 rad./s. This value agrees quite well with that one
obtained from T, analysis at the same temperature D,, - 1.3 x 109 rad/s.
There is a last point which is noteworthy. Dynamic
nuclear polarization has been performed in 03AsCH3(TCNQ)2 as a function of temperature [42].
Below 100 K no effect is observed. At 100 K, a small Overhauser effect takes place. This effect increases
with temperature up to T £nr 230 K. The observation of the Overhauser effect is evidence that components at the electronic Larmor frequency (9.5 GHz) are existing
in the spin motion spectrum. From these results, one
can deduce two interesting informations. First, at
100 K the higher frequency components of the motion- spectrum are of the order of 6 x 1010 rad./s. In the hypothesis of a Lorentzian shape in the high frequency
range spectrum (co > D,, , this agrees with the diffusion coefficient obtained by T 1 measurements. Secondly, as temperature increases, the spin motion spectrum broadens, which confirms also T, results.
As concerns the temperature dependence of the spin
diffusion coefficient in the cesium salt, only three expe- rimental points (in a small temperature range) are
available. Assuming a thermally activated law, an
activation energy of 1 300 K fits rather well with the
experimental data. This could be the activation of the exciton random walk, since their concentration is rather small (between 10- 2 and 10- 3) in the considered
temperature range. This value gives rise to an activa-
tion energy for the exchange frequency of about
3 000 K which has to be compared to values obtained
from ESR [6] : 650 and 2 600 K.
5. Conclusion.
-We have shown that the spin dynamics in two triplet-exciton systems has essentially
a ld diffusive behaviour. The diffusion coefficient which has been calculated in both samples decreases drasti- cally with temperature. In O*ASCD 3 3(TCNQ)2 an interpretation of such a temperature dependence of the
diffusion coefficient has been proposed. It takes into
account two physical processes which contribute to the
spin diffusion : the Id exciton random walk and the
exchange interactions between excitons. The random walk is supposed to be thermally activated. To our
knowledge the interpretation provides the first deter- mination of the exchange integral between excitons in this class of materials. Although, this model is no
longer valid at high temperatures (high exciton concentration), it leads to an important result : the contribution of exchange interactions to spin diffusion
cannot be neglected, even for rather low exciton concentration limit. The exchange interactions pro-
bably play a major role in this problem. Experiments
are in progress in TEA*(TCNQ)i where the singlet- triplet separation J is much smaller.
Acknowledgments.
-It is a pleasure to thank
F. Devreux for many helpful discussions.
Appendix.
-In order to determine the couplings
between one proton and one exciton (S = 1), let us
examine the infinite temperature limit. In this case, two equivalent pictures may be considered for the proton relaxation.
-
In the electron picture, the unit cell is a diad.
For simplicity let us suppose that the dipolar part of the couplings is zero. Then the simplified expression
for T1 I is :
where sjz is the z component of the diad j spin operator and ae, is the coupling between one proton of a diad and the electron spin delocalized over the diad
(I aellYe I = 0.6 gauss).
-
