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Numerical studies of magnetization relaxation of S-type ions with spin 52 in cubic crystals
0038-1098/93$6.00+.00 Pergamon Press Ltd
Solid State Communications, Vol. 86, No. 6, pp. 347-350, 1993. Printed in Great Britain.
NUMERICAL STUDIESOF MAGNETIZATION RELAXATIONOF S-TYPE IONS WITH SPIN 5/2IN CUSIC CRYSTALS institute
of Experimental
Physics,
A.M. Witowski Warsaw University, Poland
00-681 -Warsaw
ul. Hoza 69,
(received by M. Grynberg 4 March 1993)
The numerical solutions of rate equations for S-type ions with spin quantum number 5/2 (e.g. Mn*+ centers in zinc blende crystals) are used for the description of magnetization relaxation. Two types of relaxation mechanisms are taken into account, namely modulation of the crystal field and/or magnetic interaction by lattice vibrations. The first one allows transitions with ASz = tl, t2, and the second one only those with AS= = +l. In both cases the magnetization relaxes exponentially over many orders of magnitude, in some cases showing two characteristic time constants. At higher magnetic fields these constants are related to transition probabilities by a simple formula.
The temporal evolution of magnetization relaxing toward its thermal equilibrium after an excitation is strictly exponential only in two level systems. In such a case the relaxation rate (RR) is given by the sum of up and down transition probabilities. For an n - level system the temporal behavior is described by a linear combination of n-l exponential functions (solutions of rate equations like Eqs 1). An analytical solution for the relaxation rates in such a combination exists for n 5 5 [3], but these relaxation rates can be easily expressed by transition probabilities only for n 5 3. Therefore, only numerical solutions are possible for Mn*+ ions. One can show [4,5], that when the thermal energy (kT) is much larger than the separation of the levels, an approximate analytical solution exists, giving one relaxation time. In this paper we would like to find numerically the temporal evolution of Mn*+ ion magnetization relaxing toward its thermal equilibrium with the lattice. The assumed conditions are almost opposite to those mentioned above. We studied the system at low temperature (2K) at different magnetic field up to high fields (30T), moving towards conditions for which the splitting is much larger than the thermal energy. We would like to check when the relaxation can be well ap-
We recently investigated experimentally the magnetization relaxation in CdMnTe at high magnetic fields [l]. The studied materials are Diluted Magnetic Semiconductors based on CdTe with Mn as a magnetic ion. The Mn*+ ground state in this case is an orbital singlet with S = 5/2 [2], leading, in a magnetic field, to six spin-split levels in the ground state (Fig. 1). In the experiment the sample was placed in the pick-up coil. The thermal equilibrium between the spins and the lattice was disturbed with a nonresonant laser pulse (increase of the lattice temperature). The changes of the magnetization M toward a new thermal equilibrium with the lattice excite a voltage in the coil (see Fig. 2). After a short transition period the signal from the coil is proportional to the time derivative of magnetization [l]. Assuming an exponential time dependence of M(t), a single exponential function {P exp(-Rt)} was fitted to the decaying part of the signal. The parameter R was taken as the relaxation rate. A good fit was obtained for data measured at different magnetic fields B and temperatures T (B up to 25T and 3K < T < 1OK). Due to the signal to noise ratio it was possible to follow the decay of the signal over two orders of magnitude [1] (in the best case). 347
348
MAGNETIZATION RELAXATION OF S-TYPE IONS
Vol. 86, No. 6
s/2
a)
b)
Fig. 2 The temporal behavior of the coil signal after the laser heat pulse (after T. Strutz, PhD Thesis). The broken line corresponds to the titted exponential function.
Fig. 1 The energy levels of the h4n2+ ground state in a magnetic field B and possible transitions between the levels. a) Transitions for the magnetic relaxation. b) Transitions allowed by modulation of the quadrupole interaction or the crystal field by lattice vibrations.
proximated by a single exponential function and find when the RR can be easily related to transition probabilities. In our model we take into account two types of relaxation mechanisms. In the first one we assumed magnetic relaxation [4,5] (modulation of magnetic interactions by lattice vibrations) leading to transitions with a change of S, equal to il (Wl transitions) shown in Fig. la. The second one, involving modulation of the crystal field [4-61 or quadrupole interaction [4,5] by lattice vibrations, leads to transitions (see Fig lb) with a change of Sx of tl and *2 (W2 transitions). In the first case the temporal evolution of the system follows the following six rate equations describing the occupancy ni of the i-th level: dn1 dt=
-8W1un1
+ BW1dn2
dnn = 5W1 u tl1 - (5W1d + 8W ‘1”) n2 + 8Wld n3 dt dns dt= 8Wl,n2-(8Wld +9W 1,)ns t 8Wld n4 !!I4= 9Wlu n3 - (8Wld t 8Wlu) tl4 + 8W1d tl5 dt dn5 = 8Wlu tl4 - (8W1d + 5W1u) t-t5 t BW1 d 86 dt dns = 8W1 u ns - SW1 d ns dt
(1)
where subscripts u and d denote transition up and down, respectively. To find numerically the solutions of equations (1) one should know the behavior
of the transition probabilities Wl, and Wld as functions of magnetic field and temperature. For further calculations we assume that the relaxation is a direct process (with emission (down) and absorption (up) of one phonon with an energy equal to the spin splitting energy AE). It is also assumed that the phonon density of states is described by the Debye model. Therefore, the transition probabilities can be written in the form [e.g. 3,4]: wl"= Wl d =
C (AE)3 (1 t ll(exp(AE/kT)-1)) C (AE)3 /(exp(AE/kT) -1)
(2)
C is treated as a parameter (in our calculations taken as 1) proportional to the matrix element of electronic wave functions [e.g. 3961. We assume that at t = 0 the spins have a temperature T and the lattice has a higher temperature, namely TtO.lK. The fourth order Runge-Kutta method was used to solve Eq. (1) with such conditions. In our system the magnetization is given by the sum M(t) = Zi n;(t) Szi. In Fig. 3a the difference AM(t) = M(t) - M(0) is plotted for B = 10T. As one can see the decay, after a short transition period, is exponential over many orders of magnitude. To this part a function F(t) = P exp(-Rt) (where P and R are fitting parameters) was fitted and in Fig. 4a we compare the obtained values of R with the relaxation rate RRM = 5Wlu + 5Wld. Such a value of the relaxation rate (RR) can be obtained for a two level system or, in our case, in a very high field approximation. This means that the lowest level is occupied and only the two lowest levels play a role in the relaxation process (AE/kT >> 1). As one can
Vol. 86, No. 6
MAGNETIZATION RELAXATION OF S-TYPE IONS
349
10-l
g I
10-6
0 e
10 _,
lo+
10-3
3 'jt Q 10-16
10-12
0
0.601
0.002
0.003
0.004
0.00
1.00
2.00
0.004
0.006
3.00
4.00
5.00
0.01
0.02
0.02
lo-'
10-4 g
10-6
I d
10-c
k
10-6
B I
10-6 lo-'O
a 10-16
lo-l2 0.00
6.01
0.02
6.03 TIME
0.04
0.03
0.06
0
Ttme
($1
Is1
Fig. 3 Typical time dependence of magnetization difference M = M(t) - M(0) after a heat pulse at 2K for: (a) magnetic relaxation; @,c,d) relaxation due to the modulation of the quadrupole interaction or the crystal field by lattice vibrations for different D/C ratio. The lines show fits of exponential dependence.
103
103
: = :
103
10'
= zi
b) 10-l
10' 106
102
10'
100
10'
102
D/C - 1.0
10' Magnotlc
Flold (T)
102
106
10' Mqjnatlc
102 Field Ul
Fig. 4 The fitted relaxation rates compared with calculated RR1 and RR2 (see text) for: a) magnetic relaxation; b,c,d) crystal field or quadrupole interaction modulation by lattice vibrations for different D/C ratio.
350
MAGNETIZATION RELAXATION OF S-TYPE IONS
see from Fig. 4a, such an approximation (T = 2K) can be reasonable above 7 T. Of course this limit is strongly temperature dependent and at 4K is above 10 T. In the case of relaxation by phonon modulation of the crystal field or quadrupole interaction, the problem is more complicated. In the rate equations (1) one should not only adjust the transition probabilities for AS, = *l, but also, as shown in Fig. lb, add the that transitions with AS z = *2. We assume these transitions can be described by similar formulas as the AS z = tl transitions, but with energy 2AE, which leads to the following expressions for transition probabilities: W2” = D (2 AE)~ (1 + ll(exp(2AElkT) W2d = D (2 AE)3 /(exp(2AE/kT) -1)
-1)) (3)
where D is treated as a parameter. What is really important is the ratio D/C showing the relative strength of the W2 transitions. Please note that for D = C and large AE/kT, W2 is at least eight times larger than Wl. The calculations were done for T = 2K, for selected values of magnetic field strength and a few values of D/C ratios. The most interesting results are obtained for D/C around 0.01 (W2 smaller than Wl) - see Figs 3b,c. At low fields (below 10T) the relaxation is mainly governed by transitions with AS, = a2 (Fig. 3b). After quite a short transition period the relaxation is exponential over many orders of magnitude (with a relaxation rate RR2 = W2, t W2d - Fig. 4b). At higher fields the relaxation with the rate due to transitions with AS, = +l begins to be more important and nonexponential behavior is more
Vol. 86, No. 6
pronounced. First, for a few orders of magnitude the relaxation is described by the relaxation rate RR1 = Wl, t Wld and later, after a nonexponential transition period, by RR2 (Fig.3c). At very high fields the basic part of the relaxation is exponential with a relaxation rate RR1 (AS, = *l). Our results strongly support the method used for the interpretation of experimental data [l]. The idea that studies of magnetization relaxation in a system with a multilevel ground state should be performed at high magnetic fields (above 10T) and low temperatures is also confirmed. Only under such conditions do the experimental results have a simple theoretical description, and the measured relaxation rates can be related to transition probabilities. It should be pointed out that in the case of a heat pulse method of disturbing the thermal equilibrium between lattice and spins, the recovery is in all cases exDonentia1, sometimes showing two relaxation rates in more complex systems. This is not related to the different possible solutions of rate equations but to the different kinds of transitions (with ASZ = tl and AS, = *2) between spin levels. The signal to noise ratio in the mentioned experiments [l] does not allow to check which of the discussed relaxation mechanisms is involved in the relaxation of magnetization in CdMnTe. Acknowledgements: The author would like to acknowledge the encouraging discussions with Dr. M. Potemski. I also am grateful to Dr. M. L. Sadowski for a critical reading of the manuscript. This work was supported by the State Committee for Scientific Research through Grant #224109203.
References [l] T. Strut& A.M. Witowski, and P. Wyder, Phys. Rev. Lett. 66, 3912 (1992) & T. “High Magnetic Field Electron Strutz, Spin-Lattice Relaxation in a Diluted Magnetic Semiconductor: CdMnTe” (PhD Thesis, Hartung-Gorre Verlag, Konstanz 1991). [2] e.g. J.K. Furdyna, J. Appl. Phys. 64, R29 (1988) and references therein.
[3] K.W.H. Stevens, Rep. Prog. Phys. 30, 189 (1967). [4] K.J. Standley and R.A. Vaughan, Electron Spin Relaxation Phenomena in Solids (A. Hilger, London, 1989). [5] E.R. Andrew and D.P. Tunstall, Proc. Phys. Sot. 76, 1 (1981). [S] M. Blume and R. Orbach, Phys. Rev. 127, 1587 (1982).
Solid State Communications, Vol. 86, No. 6, pp. 347-350, 1993. Printed in Great Britain.
NUMERICAL STUDIESOF MAGNETIZATION RELAXATIONOF S-TYPE IONS WITH SPIN 5/2IN CUSIC CRYSTALS institute
of Experimental
Physics,
A.M. Witowski Warsaw University, Poland
00-681 -Warsaw
ul. Hoza 69,
(received by M. Grynberg 4 March 1993)
The numerical solutions of rate equations for S-type ions with spin quantum number 5/2 (e.g. Mn*+ centers in zinc blende crystals) are used for the description of magnetization relaxation. Two types of relaxation mechanisms are taken into account, namely modulation of the crystal field and/or magnetic interaction by lattice vibrations. The first one allows transitions with ASz = tl, t2, and the second one only those with AS= = +l. In both cases the magnetization relaxes exponentially over many orders of magnitude, in some cases showing two characteristic time constants. At higher magnetic fields these constants are related to transition probabilities by a simple formula.
The temporal evolution of magnetization relaxing toward its thermal equilibrium after an excitation is strictly exponential only in two level systems. In such a case the relaxation rate (RR) is given by the sum of up and down transition probabilities. For an n - level system the temporal behavior is described by a linear combination of n-l exponential functions (solutions of rate equations like Eqs 1). An analytical solution for the relaxation rates in such a combination exists for n 5 5 [3], but these relaxation rates can be easily expressed by transition probabilities only for n 5 3. Therefore, only numerical solutions are possible for Mn*+ ions. One can show [4,5], that when the thermal energy (kT) is much larger than the separation of the levels, an approximate analytical solution exists, giving one relaxation time. In this paper we would like to find numerically the temporal evolution of Mn*+ ion magnetization relaxing toward its thermal equilibrium with the lattice. The assumed conditions are almost opposite to those mentioned above. We studied the system at low temperature (2K) at different magnetic field up to high fields (30T), moving towards conditions for which the splitting is much larger than the thermal energy. We would like to check when the relaxation can be well ap-
We recently investigated experimentally the magnetization relaxation in CdMnTe at high magnetic fields [l]. The studied materials are Diluted Magnetic Semiconductors based on CdTe with Mn as a magnetic ion. The Mn*+ ground state in this case is an orbital singlet with S = 5/2 [2], leading, in a magnetic field, to six spin-split levels in the ground state (Fig. 1). In the experiment the sample was placed in the pick-up coil. The thermal equilibrium between the spins and the lattice was disturbed with a nonresonant laser pulse (increase of the lattice temperature). The changes of the magnetization M toward a new thermal equilibrium with the lattice excite a voltage in the coil (see Fig. 2). After a short transition period the signal from the coil is proportional to the time derivative of magnetization [l]. Assuming an exponential time dependence of M(t), a single exponential function {P exp(-Rt)} was fitted to the decaying part of the signal. The parameter R was taken as the relaxation rate. A good fit was obtained for data measured at different magnetic fields B and temperatures T (B up to 25T and 3K < T < 1OK). Due to the signal to noise ratio it was possible to follow the decay of the signal over two orders of magnitude [1] (in the best case). 347
348
MAGNETIZATION RELAXATION OF S-TYPE IONS
Vol. 86, No. 6
s/2
a)
b)
Fig. 2 The temporal behavior of the coil signal after the laser heat pulse (after T. Strutz, PhD Thesis). The broken line corresponds to the titted exponential function.
Fig. 1 The energy levels of the h4n2+ ground state in a magnetic field B and possible transitions between the levels. a) Transitions for the magnetic relaxation. b) Transitions allowed by modulation of the quadrupole interaction or the crystal field by lattice vibrations.
proximated by a single exponential function and find when the RR can be easily related to transition probabilities. In our model we take into account two types of relaxation mechanisms. In the first one we assumed magnetic relaxation [4,5] (modulation of magnetic interactions by lattice vibrations) leading to transitions with a change of S, equal to il (Wl transitions) shown in Fig. la. The second one, involving modulation of the crystal field [4-61 or quadrupole interaction [4,5] by lattice vibrations, leads to transitions (see Fig lb) with a change of Sx of tl and *2 (W2 transitions). In the first case the temporal evolution of the system follows the following six rate equations describing the occupancy ni of the i-th level: dn1 dt=
-8W1un1
+ BW1dn2
dnn = 5W1 u tl1 - (5W1d + 8W ‘1”) n2 + 8Wld n3 dt dns dt= 8Wl,n2-(8Wld +9W 1,)ns t 8Wld n4 !!I4= 9Wlu n3 - (8Wld t 8Wlu) tl4 + 8W1d tl5 dt dn5 = 8Wlu tl4 - (8W1d + 5W1u) t-t5 t BW1 d 86 dt dns = 8W1 u ns - SW1 d ns dt
(1)
where subscripts u and d denote transition up and down, respectively. To find numerically the solutions of equations (1) one should know the behavior
of the transition probabilities Wl, and Wld as functions of magnetic field and temperature. For further calculations we assume that the relaxation is a direct process (with emission (down) and absorption (up) of one phonon with an energy equal to the spin splitting energy AE). It is also assumed that the phonon density of states is described by the Debye model. Therefore, the transition probabilities can be written in the form [e.g. 3,4]: wl"= Wl d =
C (AE)3 (1 t ll(exp(AE/kT)-1)) C (AE)3 /(exp(AE/kT) -1)
(2)
C is treated as a parameter (in our calculations taken as 1) proportional to the matrix element of electronic wave functions [e.g. 3961. We assume that at t = 0 the spins have a temperature T and the lattice has a higher temperature, namely TtO.lK. The fourth order Runge-Kutta method was used to solve Eq. (1) with such conditions. In our system the magnetization is given by the sum M(t) = Zi n;(t) Szi. In Fig. 3a the difference AM(t) = M(t) - M(0) is plotted for B = 10T. As one can see the decay, after a short transition period, is exponential over many orders of magnitude. To this part a function F(t) = P exp(-Rt) (where P and R are fitting parameters) was fitted and in Fig. 4a we compare the obtained values of R with the relaxation rate RRM = 5Wlu + 5Wld. Such a value of the relaxation rate (RR) can be obtained for a two level system or, in our case, in a very high field approximation. This means that the lowest level is occupied and only the two lowest levels play a role in the relaxation process (AE/kT >> 1). As one can
Vol. 86, No. 6
MAGNETIZATION RELAXATION OF S-TYPE IONS
349
10-l
g I
10-6
0 e
10 _,
lo+
10-3
3 'jt Q 10-16
10-12
0
0.601
0.002
0.003
0.004
0.00
1.00
2.00
0.004
0.006
3.00
4.00
5.00
0.01
0.02
0.02
lo-'
10-4 g
10-6
I d
10-c
k
10-6
B I
10-6 lo-'O
a 10-16
lo-l2 0.00
6.01
0.02
6.03 TIME
0.04
0.03
0.06
0
Ttme
($1
Is1
Fig. 3 Typical time dependence of magnetization difference M = M(t) - M(0) after a heat pulse at 2K for: (a) magnetic relaxation; @,c,d) relaxation due to the modulation of the quadrupole interaction or the crystal field by lattice vibrations for different D/C ratio. The lines show fits of exponential dependence.
103
103
: = :
103
10'
= zi
b) 10-l
10' 106
102
10'
100
10'
102
D/C - 1.0
10' Magnotlc
Flold (T)
102
106
10' Mqjnatlc
102 Field Ul
Fig. 4 The fitted relaxation rates compared with calculated RR1 and RR2 (see text) for: a) magnetic relaxation; b,c,d) crystal field or quadrupole interaction modulation by lattice vibrations for different D/C ratio.
350
MAGNETIZATION RELAXATION OF S-TYPE IONS
see from Fig. 4a, such an approximation (T = 2K) can be reasonable above 7 T. Of course this limit is strongly temperature dependent and at 4K is above 10 T. In the case of relaxation by phonon modulation of the crystal field or quadrupole interaction, the problem is more complicated. In the rate equations (1) one should not only adjust the transition probabilities for AS, = *l, but also, as shown in Fig. lb, add the that transitions with AS z = *2. We assume these transitions can be described by similar formulas as the AS z = tl transitions, but with energy 2AE, which leads to the following expressions for transition probabilities: W2” = D (2 AE)~ (1 + ll(exp(2AElkT) W2d = D (2 AE)3 /(exp(2AE/kT) -1)
-1)) (3)
where D is treated as a parameter. What is really important is the ratio D/C showing the relative strength of the W2 transitions. Please note that for D = C and large AE/kT, W2 is at least eight times larger than Wl. The calculations were done for T = 2K, for selected values of magnetic field strength and a few values of D/C ratios. The most interesting results are obtained for D/C around 0.01 (W2 smaller than Wl) - see Figs 3b,c. At low fields (below 10T) the relaxation is mainly governed by transitions with AS, = a2 (Fig. 3b). After quite a short transition period the relaxation is exponential over many orders of magnitude (with a relaxation rate RR2 = W2, t W2d - Fig. 4b). At higher fields the relaxation with the rate due to transitions with AS, = +l begins to be more important and nonexponential behavior is more
Vol. 86, No. 6
pronounced. First, for a few orders of magnitude the relaxation is described by the relaxation rate RR1 = Wl, t Wld and later, after a nonexponential transition period, by RR2 (Fig.3c). At very high fields the basic part of the relaxation is exponential with a relaxation rate RR1 (AS, = *l). Our results strongly support the method used for the interpretation of experimental data [l]. The idea that studies of magnetization relaxation in a system with a multilevel ground state should be performed at high magnetic fields (above 10T) and low temperatures is also confirmed. Only under such conditions do the experimental results have a simple theoretical description, and the measured relaxation rates can be related to transition probabilities. It should be pointed out that in the case of a heat pulse method of disturbing the thermal equilibrium between lattice and spins, the recovery is in all cases exDonentia1, sometimes showing two relaxation rates in more complex systems. This is not related to the different possible solutions of rate equations but to the different kinds of transitions (with ASZ = tl and AS, = *2) between spin levels. The signal to noise ratio in the mentioned experiments [l] does not allow to check which of the discussed relaxation mechanisms is involved in the relaxation of magnetization in CdMnTe. Acknowledgements: The author would like to acknowledge the encouraging discussions with Dr. M. Potemski. I also am grateful to Dr. M. L. Sadowski for a critical reading of the manuscript. This work was supported by the State Committee for Scientific Research through Grant #224109203.
References [l] T. Strut& A.M. Witowski, and P. Wyder, Phys. Rev. Lett. 66, 3912 (1992) & T. “High Magnetic Field Electron Strutz, Spin-Lattice Relaxation in a Diluted Magnetic Semiconductor: CdMnTe” (PhD Thesis, Hartung-Gorre Verlag, Konstanz 1991). [2] e.g. J.K. Furdyna, J. Appl. Phys. 64, R29 (1988) and references therein.
[3] K.W.H. Stevens, Rep. Prog. Phys. 30, 189 (1967). [4] K.J. Standley and R.A. Vaughan, Electron Spin Relaxation Phenomena in Solids (A. Hilger, London, 1989). [5] E.R. Andrew and D.P. Tunstall, Proc. Phys. Sot. 76, 1 (1981). [S] M. Blume and R. Orbach, Phys. Rev. 127, 1587 (1982).