- Email: [email protected]
Nonuniform spin relaxation mechanisms
Journal of Magnetism and Magnetic Materials 140-144 (1995) 1733-1734
Journal of magnetism and magnetic materials
ELSEVIER
Nonuniform spin relaxation mechanisms Z. Wilamowski a,*, H. Przybylinska a, W. Jantsch b a Institute of Physics, Polish Academy of Sciences, AI. Lotnikow 32 / 46, PL 02-668 Warsaw, Poland b Johannes Kepler Universitiit, A-4040 Linz, Austria
Abstract In many cases, the spin relaxation time of paramagnetic impurities at different magnetic sites differs by orders of magnitude. We show that instead of resonance line broadening, a decay of the line amplitude is expected. We analyze various types of spin coupling that can lead to this 'nonuniform' relaxation mechanism.
The relaxation rate of localized spins may be strongly influenced by other nearby magnetic impurities. Depending on the local configuration, it can differ by orders of magnitude. Here we introduce the term nonuniform to describe such spin relaxation mechanisms as well as the corresponding probability distribution of the relaxation rates. In the nonuniform case, line broadening of magnetic resonance is not the main effect observed. Instead, the total resonance amplitude changes since resonances occurring at centers that relax very rapidly are so broad, that they do not contribute to the observed line amplitude. This effect is observed, for example, in metals [1] if the relaxation proceeds mainly via RKKY coupling to other local magnetic moments, or in semiconductors, if the energy of the local spin is transferred to electrons bound at shallow donor states, or in systems with mixed valence, if the lifetime of an impurity in a given valence state is nonuniform due to fluctuations in the ionization energy [2]. In these cases, the signal decays without line broadening, with a characteristic dependence on the temperature and concentration of magnetic centers. Another effect of a nonuniform longitudinal relaxation time is the following: For a given microwave power some fraction of the spins is completely saturated and the corresponding EPR spectrum is totally quenched, while other spins still contribute to the spectrum. As a consequence, we observe an extended plateau in the dependence of the line amplitude versus microwave power. This plateau is very characteristic of nonuniform longitudinal spin relaxation, and the width of the plateau corresponds to the dispersion of the relaxation times.
* Corresponding author. Fax: +48 (22) 430926; email: [email protected].
In this paper we discuss several types of spin coupling characterized by different distance dependences, F ( r ) , which can lead to nonuniform spin relaxation. We consider a system of centers to which the relaxation may occur, distributed at random positions, rj, and sum over all the contributions from every such center to a random spin position R. We assume that the sum of the F ( R - rj) is proportional to the relaxation rate, v ( R ) , and calculate the probability distribution of v ( R i) by numerical simulation. We analyze the logarithmic probability distribution d P / d ( l n v) = v d P / d v treating the standard deviation of this distribution, O'h~,as a measure of the nonuniformity of the distribution. A logarithmic standard deviation larger than one indicates that the relaxation times are distributed by one order of magnitude. One can treat such distributions as nonuniform.
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Fig. 1. Logarithmic probability distribution, dP(v)/d(ln v)= v d P ( v ) / d v , of the relaxation rate, which results from a sum of contributions from randomly distributed centers to which the spin is coupled by the power dependence of the distance, F(r) = r -k. The inset shows the dependence of the logarithmic standard deviation, o-i,, on the exponent, k.
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 1 1 8 5 - 0
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Fig. 2. Dependence of the logarithmic standard deviation on the characteristic length A, normalized to the mean center distance (r > = n-1/3. Circles correspond to an exponential dependence of the spin coupling, F(r) = = (l/(,irA3)) exp( - 2 r / A ) ; triangles to the absolute value of the Ruderman-Kittel function; and crosses to the square of this function, where ARK= 1/2kF, and k v is the Fermi momentum vector.
In Fig. 1, the logarithmic probability distribution of the relaxation rate is plotted for a coupling described by a power dependence, F(r) = r -k. The logarithmic standard deviation is insignificant for small k. For a dipolar interaction, the coupling decreases as r -3 with trt, = 0.6. Therefore, relaxation via dipolar coupling can be regarded as uniform. With increasing power exponent k, however, the standard deviation increases, as shown in the inset of Fig. 1. For a spin coupling described by distance dependences decreasing with a higher power exponent, the relaxation is no more uniform. In Fig. 2, the logarithmic standard deviation is plotted as a function of a length parameter A, normalized to the mean center distance = n -1/3, for various types of distance dependences, F(r). For the coupling described by the square of the Ruderman-Kittel (RK) function (see crosses in Fig. 1) the logarithmic standard deviation is large, o'ln = 2.5, and almost independent of the impurity concentration, n, as well as of the RK length, ARt,:= 1 / 2 k F (where k F is the k-vector at the Fermi level) Such a probability distribution of longitudinal relaxation rates is expected in mixed valence HgSe:Fe [1], where the dominant relaxation mechanism is proportional to the square of the indirect RKKY coupling between Fe + 2
and F e 3 + centers. Since it is not uniform, it is easy to explain why the EPR spectrum of the Fe 3÷ ions decays without appreciable broadening with both increasing temperature and Fe concentration. Cross relaxation among localized spins in metals, which is caused by RKKY exchange coupling, is characterized by a more uniform probability distribution as shown by triangles in Fig. 2. The logarithmic standard deviation in this case is also independent of the characteristic length. (The same weakly nonuniform distribution describes the statistics of local molecular fields in RKKY ferromagnets.) The most spectacular case is the distribution resulting from a sum of exponential contributions (see circles in the Fig. 2). It is very uniform for high impurity concentrations, where the product ABnl/3 is large, but O'ln increases sharply at lower concentrations. The evident border between uniform and nonuniform distribution corresponds to the metal-insulator transition in semiconductors doped with hydrogen-like impurities. Here the logarithmic standard deviation is even more sensitive to the parameter ABn1/3 than the mean value of the distribution. Such a distribution describes the relaxation of local spins in diluted magnetic semiconductors. It can be applied (i) to describe of the longitudinal relaxation, where local spins flip due to sp-d or s p - f exchange coupling. The spin flips are caused by the time-dependent wave function of the neighboring donor electron. This case also pertains (ii) double, d - s p - d , exchange coupling (and corresponding cross-relaxation) between local spins. The experimental evidence for nonuniform relaxation can be found by measuring the dependence of the EPR amplitude on microwave power. A wide plateau is commonly observed in diluted magnetic semiconductors. Acknowledgements: This work was partially supported by the Committee for Scientific Research in Poland and by the Fonds zur FSrderung der Wissenschaftlichen Forschung in Austria. References:
[1] Z. Wilamowski, W. Jantsch and G. Hendorfer, Semicond. Sci. Technol. 5 (1990) $266. [2] Z. Wilamowski, A. Mycielski, W. Jantsch and G. Hendorfer, Phys. Rev. B 38 (1988) 3621.
Journal of magnetism and magnetic materials
ELSEVIER
Nonuniform spin relaxation mechanisms Z. Wilamowski a,*, H. Przybylinska a, W. Jantsch b a Institute of Physics, Polish Academy of Sciences, AI. Lotnikow 32 / 46, PL 02-668 Warsaw, Poland b Johannes Kepler Universitiit, A-4040 Linz, Austria
Abstract In many cases, the spin relaxation time of paramagnetic impurities at different magnetic sites differs by orders of magnitude. We show that instead of resonance line broadening, a decay of the line amplitude is expected. We analyze various types of spin coupling that can lead to this 'nonuniform' relaxation mechanism.
The relaxation rate of localized spins may be strongly influenced by other nearby magnetic impurities. Depending on the local configuration, it can differ by orders of magnitude. Here we introduce the term nonuniform to describe such spin relaxation mechanisms as well as the corresponding probability distribution of the relaxation rates. In the nonuniform case, line broadening of magnetic resonance is not the main effect observed. Instead, the total resonance amplitude changes since resonances occurring at centers that relax very rapidly are so broad, that they do not contribute to the observed line amplitude. This effect is observed, for example, in metals [1] if the relaxation proceeds mainly via RKKY coupling to other local magnetic moments, or in semiconductors, if the energy of the local spin is transferred to electrons bound at shallow donor states, or in systems with mixed valence, if the lifetime of an impurity in a given valence state is nonuniform due to fluctuations in the ionization energy [2]. In these cases, the signal decays without line broadening, with a characteristic dependence on the temperature and concentration of magnetic centers. Another effect of a nonuniform longitudinal relaxation time is the following: For a given microwave power some fraction of the spins is completely saturated and the corresponding EPR spectrum is totally quenched, while other spins still contribute to the spectrum. As a consequence, we observe an extended plateau in the dependence of the line amplitude versus microwave power. This plateau is very characteristic of nonuniform longitudinal spin relaxation, and the width of the plateau corresponds to the dispersion of the relaxation times.
* Corresponding author. Fax: +48 (22) 430926; email: [email protected].
In this paper we discuss several types of spin coupling characterized by different distance dependences, F ( r ) , which can lead to nonuniform spin relaxation. We consider a system of centers to which the relaxation may occur, distributed at random positions, rj, and sum over all the contributions from every such center to a random spin position R. We assume that the sum of the F ( R - rj) is proportional to the relaxation rate, v ( R ) , and calculate the probability distribution of v ( R i) by numerical simulation. We analyze the logarithmic probability distribution d P / d ( l n v) = v d P / d v treating the standard deviation of this distribution, O'h~,as a measure of the nonuniformity of the distribution. A logarithmic standard deviation larger than one indicates that the relaxation times are distributed by one order of magnitude. One can treat such distributions as nonuniform.
2.5
~ 2.0
°tl
/
_=._/I 1.~h I "0
r4
II
~
/i~,~
~ i~ / ~ 05L " ~
PowerExponent. k
0.5
O0
1~
lo~
lo 6
lo'
lo ~
9
~o
lo
10
lo
11
V
Fig. 1. Logarithmic probability distribution, dP(v)/d(ln v)= v d P ( v ) / d v , of the relaxation rate, which results from a sum of contributions from randomly distributed centers to which the spin is coupled by the power dependence of the distance, F(r) = r -k. The inset shows the dependence of the logarithmic standard deviation, o-i,, on the exponent, k.
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 1 1 8 5 - 0
r3
1734
Z. Wilamowskiet aL /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1733-1734 ~I0
0
N 8 t21
.o\
.-
..........+ ...........+
0
o,
0 N--e~ ~
lo:3 io:2 io:1 fro lo, ~, n lt3
Fig. 2. Dependence of the logarithmic standard deviation on the characteristic length A, normalized to the mean center distance (r > = n-1/3. Circles correspond to an exponential dependence of the spin coupling, F(r) = = (l/(,irA3)) exp( - 2 r / A ) ; triangles to the absolute value of the Ruderman-Kittel function; and crosses to the square of this function, where ARK= 1/2kF, and k v is the Fermi momentum vector.
In Fig. 1, the logarithmic probability distribution of the relaxation rate is plotted for a coupling described by a power dependence, F(r) = r -k. The logarithmic standard deviation is insignificant for small k. For a dipolar interaction, the coupling decreases as r -3 with trt, = 0.6. Therefore, relaxation via dipolar coupling can be regarded as uniform. With increasing power exponent k, however, the standard deviation increases, as shown in the inset of Fig. 1. For a spin coupling described by distance dependences decreasing with a higher power exponent, the relaxation is no more uniform. In Fig. 2, the logarithmic standard deviation is plotted as a function of a length parameter A, normalized to the mean center distance
and F e 3 + centers. Since it is not uniform, it is easy to explain why the EPR spectrum of the Fe 3÷ ions decays without appreciable broadening with both increasing temperature and Fe concentration. Cross relaxation among localized spins in metals, which is caused by RKKY exchange coupling, is characterized by a more uniform probability distribution as shown by triangles in Fig. 2. The logarithmic standard deviation in this case is also independent of the characteristic length. (The same weakly nonuniform distribution describes the statistics of local molecular fields in RKKY ferromagnets.) The most spectacular case is the distribution resulting from a sum of exponential contributions (see circles in the Fig. 2). It is very uniform for high impurity concentrations, where the product ABnl/3 is large, but O'ln increases sharply at lower concentrations. The evident border between uniform and nonuniform distribution corresponds to the metal-insulator transition in semiconductors doped with hydrogen-like impurities. Here the logarithmic standard deviation is even more sensitive to the parameter ABn1/3 than the mean value of the distribution. Such a distribution describes the relaxation of local spins in diluted magnetic semiconductors. It can be applied (i) to describe of the longitudinal relaxation, where local spins flip due to sp-d or s p - f exchange coupling. The spin flips are caused by the time-dependent wave function of the neighboring donor electron. This case also pertains (ii) double, d - s p - d , exchange coupling (and corresponding cross-relaxation) between local spins. The experimental evidence for nonuniform relaxation can be found by measuring the dependence of the EPR amplitude on microwave power. A wide plateau is commonly observed in diluted magnetic semiconductors. Acknowledgements: This work was partially supported by the Committee for Scientific Research in Poland and by the Fonds zur FSrderung der Wissenschaftlichen Forschung in Austria. References:
[1] Z. Wilamowski, W. Jantsch and G. Hendorfer, Semicond. Sci. Technol. 5 (1990) $266. [2] Z. Wilamowski, A. Mycielski, W. Jantsch and G. Hendorfer, Phys. Rev. B 38 (1988) 3621.