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Spin-echo decay in a glassy system at low temperature
Volume 138, number9
PHYSICS LETTERSA
17 July 1989
SPIN-ECHO DECAY IN A GLASSY SYSTEM AT LOW TEMPERATURE S. DEL SORDO and F.M. GELARDI Istitutodi Fisica deli’ Università and Sezione CISM-GNSM, Via Archirafi 36, 1-90123 Palermo, Italy Received 1 March 1989; revised manuscript received 13 April 1989; accepted for publication 10 May 1989 Communicated by D. Bloch
The decay of the spin-echo signal has been experimentally investigated in a sample of E’ centers in glassy silica at low temperature. The experimental results indicate that the phase-coherence loss of the spins is affected by interactions with the tunneling systems ofthe glassy matrix. Some analogies with previous results in optical systems are discussed.
Relaxation mechanisms which determine the homogeneous line width ~ of optical transitions of active centers are widely investigated by various experimental techniques, both in crystalline and in amorphous host matrices [1—6].These experiments show that, for a given active center, the homogeneous width in an amorphous system is much larger than in crystalline ones. Moreover, at very low ternperature, also the temperature dependence of 4,,. is found to be different in amorphous and in crystalline matrices; in particular, a homogeneous line width increasing as T13 is measured in several glassy systems in a wide range of temperatures between T—~0.1 K and T—~40K [4,5 1. The anomalous broadening of the homogeneous line in amorphous systems is ascribed to the interaction of the optical center with the ensemble of the two-level systems (TLS), which represent the model generally and successfully used to explain several properties of glasses [7,81. In this frame, because of the interaction between the optical center and the TLS, the on energy level splitting depends the occupied state, of upthe or active down, center of the TLS, and is modulated by transitions of the TLS induced by thermal phonons. Experiments in glasses containing nuclear [9] and electronic [101 magnetic impurities have pointed out the role played by TLSs in affecting the relaxation mechanisms also in spin systems. Even if the hypothesized interaction between a spin center and the TLS ensemble is obviously different from the optical 526
case, the measured spin—lattice relaxation rate shows the same dependence on the temperature as the optical homogeneous line width. Several models of the interactions between active centers and the TLS ensemble have been proposed to explain the experimental results on the temperature dependence of the homogeneous line width of the optical transitions in glasses [11—131.In spite of many formal analogies, all these models differ from each other as regards either the hypothesized interaction mechanism or the average procedure of the tunneling parameters over the TLS ensemble. In particular, the model proposed by Huber et al. [13] proved to be successful in explaining the experimental results of optical-echo decay obtained in Nd3~:SiO 2, in a range of temperatures between T—~0.1 K and T~1 K [4]. In the frame of Huber’s model, in which elastic dipole—dipole interactions among active centers and TLS are hypothesized, both the single exponential decay of the echo signal and the13temperature dependence can be accounted for. of its decay time as T Huber’s model is independent of the particular (optical or spin) active center considered and its formalism is essentially the same as that previously devised to describe spectral diffusion processes in magnetic resonances [14,151. Due to these characteristics, it seems reasonable that experiments in glassy systems with magnetic resonance impurities may yield valuable information on the validity limits of these models.
0375-960l1891$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 138, number 9
PHYSICS LETTERS A
17 July 1989
In this Letter we report, for the first time to our knowledge, experimental results on the temperature dependence of the homogeneous width of a spin resonance line in a glassy sample in the range 1.75—17 K. In our experiments we measure the decay time of the echo signal produced by means of a standard twopulse sequence. As is known, the echo signal decay is related to the phase-coherence loss of the resonant spins, caused by their interactions or by the time
eter is compared with the nominal one in a resistor bridge, whose unbalance is used to feed a warming resistance fixed to the cavity. For the measurements at T~4.2K the cavity is embedded in liquid helium and its temperature is controlled by stabilizing the helium vapour pressure. Temperature stability is better than 0.1 K in the whole range investigated. Typical results of the echo signal measured at various pulse distances r1, are reported in fig. 1. In par-
fluctuations of their local fields. The characteristic time ‘r of the echo decay is inversely proportional to 4h, i.e. sib theOur homogeneous width experimentsline have been carried out in samples of glassy silica, in which paramagnetic E’ centers (S= ~) were created by y-irradiation. The concentration of E’ centers is estimated, on the basis of the total irradiation dose, to be c~~1017 cm3. The EPR line of the E’ centers in glassy silica is inhomogeneously broadened because of anisotropy of the g tensor, and its total width is ~ —~2G. The experimental set-up has been described in previous reports [16,17] and here we limit ourselves to give the parameters relevant for spin-echo measurements. The spin system is tuned to v 0=5.9 GHz by the static fieldand H0=2irv0/y, is the gyromagnetic ratio, is excitedwhere by a ystandard sequence oftwo equal-intensity equal-width pulses with a time distance r 0. We detect the echo signal at frequency v0 emitted by the system at t~2r~ and we measure its maximum at various values of r0, I(z,~), for r~,varying from 10 to 300 1t5 typically. The width w~of the exciting pulses is so adjusted as to maximize the echo signal, which occurs for a pulse-area of 1200; the repetition frequency of the sequence is low enough to ensure complete relaxation of the system to thermal equilibrium between successive sequences. We recall that a peculiarity of our experimental set-up is that the excitation is carried out by pulses at frequency t.’~/2,namely by means of double-quantum transitions; this experimental condition is particularly convenient in the investigation of coherent transients, due to the large spectral distance between excitation and response. We measured the echo decay in the temperature range 1.75—17 K. For the measurements at T> 4.2 K, the microwave cavity is cooled by helium vapour. Its temperature is measured by a calibrated carbonglass thermometer. The voltage across the thermom-
ticular, the data of fig. 1 were measured at T= 1.75 K with w~= ~.ts.Dtootspoint are out experimental and a semilog plot3.3 is used the extent data to which the decay is well described by the single exponential law (—~20 dB) The straight line is the exponential law that best fits the experimental points. The echo decay time determined by this curve is r= 160±10 us. Experimental curves similar to the one plotted were detected at other operating temperatures and in all the cases the echo decay curves were fitted by a single-exponential law. In fig. 2 the measured decay time r is reported as a function of the temperature, which range from T= 1.75 K to T= 17 K. In the logarithmic scale of
—
the figure, theshown, straightthis linepower represents the fits power 3. As is law well the law exT—”
-40
2
-7°
I
0
100
200
300
400
pulse distance (pa) Fig. 1. Spin-echo decay as a function of the pulse distance r~at T= 1.75 K. Dots are the maxima of the echo signal experimentally detected and the straight hue represents the exponential law that best fits the experimental points. The characteristic time of this exponential decay is r= 160 ±10 ~ss.
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Volume 138, number 9
PHYSICS LETTERS A
7 July 1989
tribution ofthe TLSs. The normalized echo intensity is calculated to be E(2r~)=exp(—cn0T’ ~r~’) (I) where c is a constant, n0 is the spatial density of the
200
,
E
100 80
5)60
N, 40
• 20
10 1
2
3
4
5
6 7 89
10
temperature (K
20
)
Fig. 2. Characteristic decay time t of the spin-echo signal as a function of the temperature. Dots are the values ofT determined by best fitting experimental curves like the one shown in fig. 1. The straight line represents the power law 1~-
tunneling centers, ~i is related to the density of states of the TLSs, and n characterizes the hypothesized multipolar interaction. Eq. (1) is derived under two main approximations. The former is that the resonance line of the active centers is highly inhomogeneous so that only a narrow part of the whole line is directly involved in the echo formation. The latter is that the maximum tunneling rate Rmax, within the TLS ensemble, is much larger than the dephasing rate r~ of the generic active center, i.e. R,,.00r>> 1. Both conditions are well fulfilled in our experimental conditions. As regards the former one, we have measured at the used input power level the Rabi frequency during the cx-
perimental values of r in the temperature range below T=4.2 K, within the experimental uncertainties. At T> 4.2 K, the measured values of r decrease less rapidly on increasing T, gradually approaching a limit value of r= 30 us. As regards the temperature range below T=4.2 K, the similarity between the experimental results on spin echo reported above andSi0 those obtained by pho3~in glassy ton echo of Nd 2 [4] is evident, in both cases the echo decays follow a single-exponenhal law whose decay time depends on T as T ‘~ at low temperature. It is worth noting that the amorphous matrix which the activewhich centerssuggests are diluted is the same in in both experiments, that both the form of the decay and the temperature dependence of its characteristic time are to be related to properties of the glassy matrix rather than to the peculiarity of the active center. Now we wish to discuss our experimental results in the light of the predictions of Huber’s stochastic
citation pulse to be of the order of 1 00 kHz, to be compared with the inhomogeneous line width, 4,.~10 MHz. As regards the tunneling rate, by deriving from ref. [13] the values of ~ at T=~I .75 K and T=4.2 K and by using the corresponding cxperimental values of r, we obtain Rma~t’.~~ 3.9 X I O~ and ~ 2 x I 0~ at the two temperatures, respectively. Eq. (1) fits our experimental results if~u=O.3and ii= 3. The value ~i=0.3 is in agreement with the energy dependence of the state density of the tunneling systems, derived from specific heat measurements in glasses [18]. The value n = 3 indicates that the interaction between the spin centers and the TLS systern has a 3. distance dependence like the dipolar one, that is A/r be noted that, according to Huber’s calIt should culations, two different time scales can be distinguished, in which the interaction of the active centers with the tunneling systems is characterized in a different way. In fact, at short times (Rn~axl‘~x 1) the dephasing interactions originate mainly in the TLSs close to the active center; at longer times (R,.,,.,
model [131. In this model, a generic multipolar interaction is assumed between the active centers and the TLS system and the echo decay function is obtamed by calculating the effect of the TLS tunneling on the time evolution of the center, and then averaging the tunneling parameters over the spatial dis-
far TLSs mainly interact with the latter. From this point of view, by comparing the typical observation times of the spin and optical echo experiments, we can infer that the loss of phase coherence among the spins, in our experimental system, is almost entirely produced by interactions with tunneling systems, on
—
01>> I
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Volume 138, number 9
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the average, very further with respect to the analogous optical system. This fact, on the one hand, implies the TLS—spin interaction to be weaker than the TLS—optical center one; on the other hand, it seems to rule out that the decrease of the interaction intensity with the TLS—spin center distance is more pronounced than in a dipolar interaction law. The experimental values of r, as shown in fig. 2, are in agreement with Huber’s model in the ternperature range below T-.~4 K, but move away from the power law T — on increasing temperature, tending to a constant value up to T~20 K. This behaviour can be explained by hypothesizing a saturation of the TLS—spin interactions. We note that a similar “plateau”, in the same range of temperatures, has been experimentally detected also in the temperature dependence ofthe thermal conductivity of various amorphous materials [19,20]. In particular, Zaitlin and Anderson [20] have shown that this anomalous behaviour of the thermal conductivity occurs in the same temperature range, independently of the particular amorphous system, and it can be ascribed to an abrupt decrease of the mean free path lofphonons, on increasing phonon frequencies (/xw4). These authors have also pointed out that such an effect is not incompatible with the tunneling systems model and could be caused by a density of TLS which increases quadratically with the temperature. On the basis of the above considerations, we ascribe the decrease of the experimental values of r to the fact that a larger number of TLSs are involved in the interaction with the spin system on increasing temperature. Nevertheless, in the the temperature range T>~4 K, in spite of the increased number of TLS transitions, the TLSs with higher energy gaps are expected to be ineffective in dephasing the spin system, owing to the abrupt decrease of the mean free path of the high frequency phonons. So, analogously to the case of the thermal conductivity, it is the low frequency phonons that mainly interact with the spin centers, so producing the saturation effect experimentally observed. The nature of the TLS—spin interaction is still an open problem and, admittedly, no information on the detailed mechanism of this interaction can be deduced by the above results. Nevertheless, some hypothesized interaction mechanisms can be dis‘~
17 July 1989
cussed. A possible mechanism could be established through the nuclear spins surrounding the electron spin center, in the sense that tunneling transitions of TLSs may induce nuclear spin flipping, which modulate the electron spin frequency via the hyperfine interaction. This mechanism, proposed by Kurtz and Stapleton [10] for explaining experimental data on the spin—lattice relaxation of F~centers in ~-alumina, does not seem to be supported by our results. In fact, as noted above, the interactions with the spin center, in our experimental case, involve mainly distant tunneling systems, in contrast both with the “short range” action of the hyperfine interaction and with the strong localization of the E’ center nearby a Si atom [211. A pure elastic dipolar interaction, as suggested by Huber et al. [13], is an alternative mechanism. However, as noted by the same authors, this interaction is effective only if the active center itself is localized in a TLS. By comparing the local density ofthe TLSs and of the E’ centers in our sample, both of the same order (‘~10’~cm 3), we note that the effectiveness ofsuch an interaction should need both an E’ center and a TLS to exist at the same lattice site. In our opinion, such a condition does not seem very probable, even though it cannot be ruled out at all. A third mechanism that can be considered is a modulation of the energy splitting of the spin levels of the E’ centers, produced by TLS transitions via the spin—orbit coupling. We are not able, at this time, to support quantitatively this hypothesis, nevertheless we note that it seems to be consistent both with the structure of the investigated spin system and with the experimental results. In conclusion, our experimental results confirm that, in glasses at low temperature, the phase-coherence loss both in spin and in optical systems is caused mainly by the interactions of the active centers with the TLS ensemble. Moreover our data, obtained for the first time from spin-echo measurements, mdicate that these interactions are effective in a relatively short time scale (t 10 us), with respect to the time scale (t ~ 10 ms) explored by previously reported measurements of the spin—lattice relaxation [9,10]. Both the form of the echo decay and the temperature dependence of the decay time are consistent with the predictions of Huber’s model, pro“-j
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vided that a dipolar interaction is assumed. Finally, as regards in particular the theoretical temperature dependence of the echo decay time, proportional to T— the range of validity of this law, according to our results, can be extended up to a temperature T~4.2K. ‘~,
The authors wish to thank R. Boscaino and RN. Mantegna for stimulating discussions and precious suggestions related to this work. Technical assistance in cryogenic work by Mr. G. Lapis is also acknowledged. Partial financial support was provided by Comitato Regionale Ricerche, Palermo, Italy.
References [I] P.M. Selzer, D.L. Huber, D.S. Hamilton, W.M. Yen and M.J. Weber, Phys. Rev. Lett. 36 (1976) 813. [2] J. Hegarty and MW. Yen, Phys. Rev. Lett. 43(1979)1126. [3] R.M. Macfarlane and R.M. Shelby, Opt. Common. 45 (1983)46.
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[41J. Hegarty, M.M. Broer, B. Golding, J.R. Simpson
and J.B. MacChesney, Phys. Rev. Leit. 51(1983) 2033. [5] R.T. Brundage and M.W. Yen, Phys. Rev. B 33 (1986) 4436. [61G. Mariotto, M. Montagna and F. Rossi, Phys. Rev. B 38 (1988) 1459. 17] P.W. Anderson, B.I. Halperin and C.M. Varma, Philos. Mag. 25(1972)1. [81 W.A. Phillips, J. Low Temp. Phys. 7 (1972) 351. [9] J. Szeftel and H. Alloul, Phys. Rev. Lett. 34 (1975) 657. [101 S.R. Kurtz and H.J. Stapleton, Phys. Rev. B 22 (1980) 2195. [11]S.K. Lyo, Phys. Rev. Lett. 48 (1982) 688. 112] SolidBroer Stale and Commun. 32 (1979) [13] T.L. D.L. Reinecke, Huber, M.M. B. Golding, Phys. 1103. Rev. Lett. 52 (1984) 2281; DL. Huber, J. Lumin. 36 (1987) 307. [14]J.R. Klauder and P.W. Anderson, Phys. Rev. 125 (1962)
912. [15] P. Hu and L.R. Walker, Phys. Rev. B 18 (1978)1300. [16] R. Boscaino and F.M. Gelardi, J. Phys. C 13 (1980) 3737. [171 R. Boscaino, F.M. Gelardi and G. Messina, Phys. Lett. A
97 (1983) 413.
[18] J.C. Lasjaunias, A. Ravex, M. Vandorpe and S. Hunklinger, Solid State Commun. 17 (1975) 1045. [191 R.C. Zeller and R.O. Pohl, Phys. Rev. B 4 (1971) 2029. [201 M.P. Zaitlin and A.C. Anderson, Phys. Rev. B 12 (1975) 4475. [2l]R.H.Silsbee,J.Appl.Phys.32(196l) 1459.
PHYSICS LETTERSA
17 July 1989
SPIN-ECHO DECAY IN A GLASSY SYSTEM AT LOW TEMPERATURE S. DEL SORDO and F.M. GELARDI Istitutodi Fisica deli’ Università and Sezione CISM-GNSM, Via Archirafi 36, 1-90123 Palermo, Italy Received 1 March 1989; revised manuscript received 13 April 1989; accepted for publication 10 May 1989 Communicated by D. Bloch
The decay of the spin-echo signal has been experimentally investigated in a sample of E’ centers in glassy silica at low temperature. The experimental results indicate that the phase-coherence loss of the spins is affected by interactions with the tunneling systems ofthe glassy matrix. Some analogies with previous results in optical systems are discussed.
Relaxation mechanisms which determine the homogeneous line width ~ of optical transitions of active centers are widely investigated by various experimental techniques, both in crystalline and in amorphous host matrices [1—6].These experiments show that, for a given active center, the homogeneous width in an amorphous system is much larger than in crystalline ones. Moreover, at very low ternperature, also the temperature dependence of 4,,. is found to be different in amorphous and in crystalline matrices; in particular, a homogeneous line width increasing as T13 is measured in several glassy systems in a wide range of temperatures between T—~0.1 K and T—~40K [4,5 1. The anomalous broadening of the homogeneous line in amorphous systems is ascribed to the interaction of the optical center with the ensemble of the two-level systems (TLS), which represent the model generally and successfully used to explain several properties of glasses [7,81. In this frame, because of the interaction between the optical center and the TLS, the on energy level splitting depends the occupied state, of upthe or active down, center of the TLS, and is modulated by transitions of the TLS induced by thermal phonons. Experiments in glasses containing nuclear [9] and electronic [101 magnetic impurities have pointed out the role played by TLSs in affecting the relaxation mechanisms also in spin systems. Even if the hypothesized interaction between a spin center and the TLS ensemble is obviously different from the optical 526
case, the measured spin—lattice relaxation rate shows the same dependence on the temperature as the optical homogeneous line width. Several models of the interactions between active centers and the TLS ensemble have been proposed to explain the experimental results on the temperature dependence of the homogeneous line width of the optical transitions in glasses [11—131.In spite of many formal analogies, all these models differ from each other as regards either the hypothesized interaction mechanism or the average procedure of the tunneling parameters over the TLS ensemble. In particular, the model proposed by Huber et al. [13] proved to be successful in explaining the experimental results of optical-echo decay obtained in Nd3~:SiO 2, in a range of temperatures between T—~0.1 K and T~1 K [4]. In the frame of Huber’s model, in which elastic dipole—dipole interactions among active centers and TLS are hypothesized, both the single exponential decay of the echo signal and the13temperature dependence can be accounted for. of its decay time as T Huber’s model is independent of the particular (optical or spin) active center considered and its formalism is essentially the same as that previously devised to describe spectral diffusion processes in magnetic resonances [14,151. Due to these characteristics, it seems reasonable that experiments in glassy systems with magnetic resonance impurities may yield valuable information on the validity limits of these models.
0375-960l1891$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 138, number 9
PHYSICS LETTERS A
17 July 1989
In this Letter we report, for the first time to our knowledge, experimental results on the temperature dependence of the homogeneous width of a spin resonance line in a glassy sample in the range 1.75—17 K. In our experiments we measure the decay time of the echo signal produced by means of a standard twopulse sequence. As is known, the echo signal decay is related to the phase-coherence loss of the resonant spins, caused by their interactions or by the time
eter is compared with the nominal one in a resistor bridge, whose unbalance is used to feed a warming resistance fixed to the cavity. For the measurements at T~4.2K the cavity is embedded in liquid helium and its temperature is controlled by stabilizing the helium vapour pressure. Temperature stability is better than 0.1 K in the whole range investigated. Typical results of the echo signal measured at various pulse distances r1, are reported in fig. 1. In par-
fluctuations of their local fields. The characteristic time ‘r of the echo decay is inversely proportional to 4h, i.e. sib theOur homogeneous width experimentsline have been carried out in samples of glassy silica, in which paramagnetic E’ centers (S= ~) were created by y-irradiation. The concentration of E’ centers is estimated, on the basis of the total irradiation dose, to be c~~1017 cm3. The EPR line of the E’ centers in glassy silica is inhomogeneously broadened because of anisotropy of the g tensor, and its total width is ~ —~2G. The experimental set-up has been described in previous reports [16,17] and here we limit ourselves to give the parameters relevant for spin-echo measurements. The spin system is tuned to v 0=5.9 GHz by the static fieldand H0=2irv0/y, is the gyromagnetic ratio, is excitedwhere by a ystandard sequence oftwo equal-intensity equal-width pulses with a time distance r 0. We detect the echo signal at frequency v0 emitted by the system at t~2r~ and we measure its maximum at various values of r0, I(z,~), for r~,varying from 10 to 300 1t5 typically. The width w~of the exciting pulses is so adjusted as to maximize the echo signal, which occurs for a pulse-area of 1200; the repetition frequency of the sequence is low enough to ensure complete relaxation of the system to thermal equilibrium between successive sequences. We recall that a peculiarity of our experimental set-up is that the excitation is carried out by pulses at frequency t.’~/2,namely by means of double-quantum transitions; this experimental condition is particularly convenient in the investigation of coherent transients, due to the large spectral distance between excitation and response. We measured the echo decay in the temperature range 1.75—17 K. For the measurements at T> 4.2 K, the microwave cavity is cooled by helium vapour. Its temperature is measured by a calibrated carbonglass thermometer. The voltage across the thermom-
ticular, the data of fig. 1 were measured at T= 1.75 K with w~= ~.ts.Dtootspoint are out experimental and a semilog plot3.3 is used the extent data to which the decay is well described by the single exponential law (—~20 dB) The straight line is the exponential law that best fits the experimental points. The echo decay time determined by this curve is r= 160±10 us. Experimental curves similar to the one plotted were detected at other operating temperatures and in all the cases the echo decay curves were fitted by a single-exponential law. In fig. 2 the measured decay time r is reported as a function of the temperature, which range from T= 1.75 K to T= 17 K. In the logarithmic scale of
—
the figure, theshown, straightthis linepower represents the fits power 3. As is law well the law exT—”
-40
2
-7°
I
0
100
200
300
400
pulse distance (pa) Fig. 1. Spin-echo decay as a function of the pulse distance r~at T= 1.75 K. Dots are the maxima of the echo signal experimentally detected and the straight hue represents the exponential law that best fits the experimental points. The characteristic time of this exponential decay is r= 160 ±10 ~ss.
527
Volume 138, number 9
PHYSICS LETTERS A
7 July 1989
tribution ofthe TLSs. The normalized echo intensity is calculated to be E(2r~)=exp(—cn0T’ ~r~’) (I) where c is a constant, n0 is the spatial density of the
200
,
E
100 80
5)60
N, 40
• 20
10 1
2
3
4
5
6 7 89
10
temperature (K
20
)
Fig. 2. Characteristic decay time t of the spin-echo signal as a function of the temperature. Dots are the values ofT determined by best fitting experimental curves like the one shown in fig. 1. The straight line represents the power law 1~-
tunneling centers, ~i is related to the density of states of the TLSs, and n characterizes the hypothesized multipolar interaction. Eq. (1) is derived under two main approximations. The former is that the resonance line of the active centers is highly inhomogeneous so that only a narrow part of the whole line is directly involved in the echo formation. The latter is that the maximum tunneling rate Rmax, within the TLS ensemble, is much larger than the dephasing rate r~ of the generic active center, i.e. R,,.00r>> 1. Both conditions are well fulfilled in our experimental conditions. As regards the former one, we have measured at the used input power level the Rabi frequency during the cx-
perimental values of r in the temperature range below T=4.2 K, within the experimental uncertainties. At T> 4.2 K, the measured values of r decrease less rapidly on increasing T, gradually approaching a limit value of r= 30 us. As regards the temperature range below T=4.2 K, the similarity between the experimental results on spin echo reported above andSi0 those obtained by pho3~in glassy ton echo of Nd 2 [4] is evident, in both cases the echo decays follow a single-exponenhal law whose decay time depends on T as T ‘~ at low temperature. It is worth noting that the amorphous matrix which the activewhich centerssuggests are diluted is the same in in both experiments, that both the form of the decay and the temperature dependence of its characteristic time are to be related to properties of the glassy matrix rather than to the peculiarity of the active center. Now we wish to discuss our experimental results in the light of the predictions of Huber’s stochastic
citation pulse to be of the order of 1 00 kHz, to be compared with the inhomogeneous line width, 4,.~10 MHz. As regards the tunneling rate, by deriving from ref. [13] the values of ~ at T=~I .75 K and T=4.2 K and by using the corresponding cxperimental values of r, we obtain Rma~t’.~~ 3.9 X I O~ and ~ 2 x I 0~ at the two temperatures, respectively. Eq. (1) fits our experimental results if~u=O.3and ii= 3. The value ~i=0.3 is in agreement with the energy dependence of the state density of the tunneling systems, derived from specific heat measurements in glasses [18]. The value n = 3 indicates that the interaction between the spin centers and the TLS systern has a 3. distance dependence like the dipolar one, that is A/r be noted that, according to Huber’s calIt should culations, two different time scales can be distinguished, in which the interaction of the active centers with the tunneling systems is characterized in a different way. In fact, at short times (Rn~axl‘~x 1) the dephasing interactions originate mainly in the TLSs close to the active center; at longer times (R,.,,.,
model [131. In this model, a generic multipolar interaction is assumed between the active centers and the TLS system and the echo decay function is obtamed by calculating the effect of the TLS tunneling on the time evolution of the center, and then averaging the tunneling parameters over the spatial dis-
far TLSs mainly interact with the latter. From this point of view, by comparing the typical observation times of the spin and optical echo experiments, we can infer that the loss of phase coherence among the spins, in our experimental system, is almost entirely produced by interactions with tunneling systems, on
—
01>> I
528
Volume 138, number 9
PHYSICS LETTERSA
the average, very further with respect to the analogous optical system. This fact, on the one hand, implies the TLS—spin interaction to be weaker than the TLS—optical center one; on the other hand, it seems to rule out that the decrease of the interaction intensity with the TLS—spin center distance is more pronounced than in a dipolar interaction law. The experimental values of r, as shown in fig. 2, are in agreement with Huber’s model in the ternperature range below T-.~4 K, but move away from the power law T — on increasing temperature, tending to a constant value up to T~20 K. This behaviour can be explained by hypothesizing a saturation of the TLS—spin interactions. We note that a similar “plateau”, in the same range of temperatures, has been experimentally detected also in the temperature dependence ofthe thermal conductivity of various amorphous materials [19,20]. In particular, Zaitlin and Anderson [20] have shown that this anomalous behaviour of the thermal conductivity occurs in the same temperature range, independently of the particular amorphous system, and it can be ascribed to an abrupt decrease of the mean free path lofphonons, on increasing phonon frequencies (/xw4). These authors have also pointed out that such an effect is not incompatible with the tunneling systems model and could be caused by a density of TLS which increases quadratically with the temperature. On the basis of the above considerations, we ascribe the decrease of the experimental values of r to the fact that a larger number of TLSs are involved in the interaction with the spin system on increasing temperature. Nevertheless, in the the temperature range T>~4 K, in spite of the increased number of TLS transitions, the TLSs with higher energy gaps are expected to be ineffective in dephasing the spin system, owing to the abrupt decrease of the mean free path of the high frequency phonons. So, analogously to the case of the thermal conductivity, it is the low frequency phonons that mainly interact with the spin centers, so producing the saturation effect experimentally observed. The nature of the TLS—spin interaction is still an open problem and, admittedly, no information on the detailed mechanism of this interaction can be deduced by the above results. Nevertheless, some hypothesized interaction mechanisms can be dis‘~
17 July 1989
cussed. A possible mechanism could be established through the nuclear spins surrounding the electron spin center, in the sense that tunneling transitions of TLSs may induce nuclear spin flipping, which modulate the electron spin frequency via the hyperfine interaction. This mechanism, proposed by Kurtz and Stapleton [10] for explaining experimental data on the spin—lattice relaxation of F~centers in ~-alumina, does not seem to be supported by our results. In fact, as noted above, the interactions with the spin center, in our experimental case, involve mainly distant tunneling systems, in contrast both with the “short range” action of the hyperfine interaction and with the strong localization of the E’ center nearby a Si atom [211. A pure elastic dipolar interaction, as suggested by Huber et al. [13], is an alternative mechanism. However, as noted by the same authors, this interaction is effective only if the active center itself is localized in a TLS. By comparing the local density ofthe TLSs and of the E’ centers in our sample, both of the same order (‘~10’~cm 3), we note that the effectiveness ofsuch an interaction should need both an E’ center and a TLS to exist at the same lattice site. In our opinion, such a condition does not seem very probable, even though it cannot be ruled out at all. A third mechanism that can be considered is a modulation of the energy splitting of the spin levels of the E’ centers, produced by TLS transitions via the spin—orbit coupling. We are not able, at this time, to support quantitatively this hypothesis, nevertheless we note that it seems to be consistent both with the structure of the investigated spin system and with the experimental results. In conclusion, our experimental results confirm that, in glasses at low temperature, the phase-coherence loss both in spin and in optical systems is caused mainly by the interactions of the active centers with the TLS ensemble. Moreover our data, obtained for the first time from spin-echo measurements, mdicate that these interactions are effective in a relatively short time scale (t 10 us), with respect to the time scale (t ~ 10 ms) explored by previously reported measurements of the spin—lattice relaxation [9,10]. Both the form of the echo decay and the temperature dependence of the decay time are consistent with the predictions of Huber’s model, pro“-j
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PHYSICS LETTERS A
vided that a dipolar interaction is assumed. Finally, as regards in particular the theoretical temperature dependence of the echo decay time, proportional to T— the range of validity of this law, according to our results, can be extended up to a temperature T~4.2K. ‘~,
The authors wish to thank R. Boscaino and RN. Mantegna for stimulating discussions and precious suggestions related to this work. Technical assistance in cryogenic work by Mr. G. Lapis is also acknowledged. Partial financial support was provided by Comitato Regionale Ricerche, Palermo, Italy.
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