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The two phonon spin-lattice relaxation processes in high magnetic fields
~Solid State Communications, Printed in Great Britain.
Vol. 77, NO. i, pp. 23-27, 1991.
THE TWO PHONON SPIN-LA'I'FICE RELAXATION P R O C ~ E S
0038-i098/9153.00+.00 Pergamon Press plc
IN HIGH MAGNETIC FIELDS
A.M.Witowski *
Max-Planck Institut ff~r Festk~rperforschung Hochfeld Magnetlabor Grenoble, BP 166X F-38042 Grenoble Cedex, France (Received October 4th, 1990 by P.Burlet) The simple calculations are presented which show behavior of relaxation rate due to the two phonon spin-lattice relaxation processes at high magnetic field as function of temperature and magnetic fields. It is shown that neglected at low fields process with emission of two phonons become important above about 20T. It is also shown that this process for certain materials could be effective at lower fields.
In the "classical" experimental investigations of Spin-Lattice Relaxation (SLR) radio frequency or microwave radiation has been used [1,2] to disturb the system from thermal equilibrium. Thus, in all theoretical models describing SLR, it has been justified to make the final assumption that the spin splitting is much smaller than the thermal energy. There are three basic processes involved in SLR. At very low temperatures (2K) process with emission of one phonon is dominant. At higher temperatures (4K) the processes involving two phonous become more important. The so called Raman one consists of absorption of one phonon and emission of another one with higher energy. The intermediate state is a virtual one. If the intermediate state is a real one, the process is called Orbach process. The SLR process with emission of two phonous is neglected at low magnetic fields because of very small density of phonon states for low energy phonons. The assumption AE << kT allows to make very useful numerical approximations in calculations of the relaxation rate for two phonon SLR processes. Recently, the results of investigations of SLR processes at high magnetic fields have been reported [3,4,5]. In such a case the spin splitting is larger than the thermal energy (AE > kT). The purpose of the present paper is to study theoretically the effect of this new condition on the two phonon spin-lattice relaxation processes.
IW12
I
for
#~) = 0
for ~ > ~ ,
•Hlatt +Hd +Hint
N~ ffi F~ =- 21g~B B
The diagram of electronic levels for a system with spin 1/2 in an external magnetic field B. Transitions discussed in the text are denoted by arrows.
where: Hintt is the lattice part, H d describes the electronic system and Hint interaction between these subsystems. The energy level of the simple electronic system with spin 1/z tt in an external magnetic field 8 is split into two levels. These two levels are separated by the energy AE = g~BB (g is the effective g-factor and B is oriented along the z-axis). Let N 1 be the population of the ground state (index 1) and N 2 be the population of the excited state (index 2) (see Fig.l). At any given moment t they fulfill the rate equations (N x + Na - const): N1(t) ffi N2(t)wjl
- N1(t)wlz
,
N:(t) ffi N1(t)wl,
- Nz(t)wjl
,
(2)
where wt2 and wzl are transitionprobabilitiesfrom the state I to the state 2 (up) and from the state 2 to the state I (down), respectively. The thermal equilibrium values for these populations are N:" and N2". If the system at the time t = 0 has the populations Nto and N2°, which are not the thermal equilibrium ones, then the population difference relaxes toward the equilibrium as described by the solutionsof (2):
(1)
where v is the velocity of sound and V volume of the crystal (see e.g. [6]). The physical system under consideration can be described by the Hamiltonian of the form: H
Iw21
Fig.l
To show the consequences of application of high magnetic field on the spin-lattice relaxation, we limit ourselves to the simplest case of a two level system with spin S = 1/2 8 and g-factor of 2 (see Fig.l), and a phonon spectrum described by the Debye model. We also assume that the splitting of levels is not exceeding the acoustic phonon energy (splitting smaller than ~wD , where COD is Debay frequency). Additionally we assume that the crystal is isotropic so the density of phonon states can be described as: 3Vca2 p(o~) ffi 21r2v2
N2(t) E~= 21g,uBB
2
[N,(t)-N,(t)]-[N,'-NI"] = [(N,O-NIO)-(N,'-NI")] exp[- ~1, with a relaxation time r defined by:.
I/r = w1~ + w~1 .
,
(3)
The probability of transitionbetween two states is related to the matrix element of Hint by the golden rule [6]: 2~v I0'I Hintli>l ~ ffi~~ p(E,fief), (4)
* on leave from Institute of Experimental Physics, Warsaw University
23
TWO P H O N O N
24
SPIN-LATTICE
RELb~XATION
PROCESSES
IN HIGH M A G N E T I C
FIELDS
VOI.
77, No. i
with the wave functions of the given state j expressed in the
form [6]:
our
p(Ef=E i ) = I ~(~if_~Or)P(wk)P(Wl)6.[%f_(O:l_(Ok)]d%fdcokdW1 , (9)
case as:
I J) = l ~ ) l n , ..... nk,'"),
where •
described the electronic wave function (eigenfunction of Hal) and n k is the occupation number of the kth phonon mode in the global state j. The energy of the state described by [j) is:
I Usually the interaction Hamiltonian is expressed as a power series in displacement of the investigated center or in the local strains ~ [2,6,7]. Both type of variables are then expressed by the phonon creation (a ÷) and annihilation (a) operators. In the simplest case one could write [6]:.
Hint = V~ + V~ ~ + ....
with hw r = AE. Thus, the transition probability w after integration over coit is: 2~ j'.f IM~I ~ p(~,) p(~) q%-(~-~k)] cl~d~o~. (1o) w~ = ~Once more we would like to stress that, if one assumes that AE << kT, then ft~ ~- f i ~ will follow and consequently wij would become independent on the electronic levels splitting. Now we can write the expression for relaxation time taking equations (7, 8 and 10) and the density of phonon states in the Debye model (1):
2# if(Ig,,l'+lg,,I ') p(~,) p(~) 61%-(~-~,)]d ' ~ d ' , ' k ;! = ~-
=
(5)
with: 2~r3,/2vlO~T
q where M is mass of the crystal. As one can see, the one phonon process is described by the term linear in ~. It leads to the well known formula [2,6,7] giving the inverse of relaxation time (also at high magnetic fields) proportional to: I°cr I(~xlVII~'}I'(AE)S coth[2~T 1 "
=
l¢,)In~,...,nk
.....
n~,..).
(6)
In the case of the Raman process the final state then is:
I1~ =
I~,) I", ......~-I ......:l,...).
At small B one can assume that AE << kT and most of the phonons taking part in the process have energies comparable with kT, thus fi~zk ~ / ~ and w ~ ~ w~x . This is not true at high magnetic fields and further we will show the consequences of going beyond mentioned approximation. The matrix element of Hint between the involved states is:
where "~ = M / V is density of the material and n(w) is the occupation of the phonon states with energy h~. Taking for n(~) the Bose-Einstein distribution function [exp(hta/kT)-1]-1, we obtain that I/r is proportional to:
! ~, I(~',lv,l¢,)l' (kr)' 1(~,T),
~
<.k-l,.,+l I ~ : t
+ a~.k
(12)
,,OD/T-A I x'~Cx+A'O.' eX(l+eA) fF(x,A) dx [(B, T) = Jo " ' ( : - I)(:+a_ I) dx =
with: x = h~,/l~T and h The dependence of presented in Fig.2. Note tion is small, and give a
= 5E/kT. the function F(x,A) on x and A is that for x > I0 the value of the func-
negligible contribution to the integral I(B,T). Thus, for OD/T-A>IO, I(B,T) is independent on its upper limit. It also means that these results will not depend on the Debye temperature. In the approximation used at low fields, I(B,T) is independent on the splitting of levels (magnetic field). With additional assumption that kOD >> k T it is also temperature independent and can be calculated analytically. At high fields the integral I(B,T) should be calculated numerically for given values of T and B in order to find its dependence on these parameters. There are two interesting cases. First, when the Debye temperature is very high, the level splitting in available non pulsed magnetic fields (about 30T) is much smaller than kOD .
Uif = (flHint li) =
=~
( 1 1)
where:
In the first order perturbation theory the two phonon processes are described by the term containing ¢~. Using the second order perturbation calculations one could also obtain results in the same order of magnitude with the term linear in strain. For sake of simplicity we restrict ourselves only to the former case. The later case will give basically the same phonon part as the former one assuming all other states far from discussed one [6]. The reader interested in the all possible terms appearing in such calculations could find the details in [7]. Let us take as the initial state:
Ii)
(n(tak+wr)ln(o.~)+ I ]+[n(wk+t~r)+ 1]n(wk) ) .
~
I.k,.,)(¢,lv,l¢,)
In this point we do not assume neither any properties of the electronic system but non zero value of the matrix element (~]V~[¢~) for given conditions (magnetic field and temerature)," nor the kind of mechanism giving rise to the Hin t. The matrix element (~/~[V~I~) is of course independent on phonon energies. Finally, the matrix element M~: which is related to the W~l is: For the probability w~2 describing "up" transition from electronic state 1 to electronic state 2 in which one phonon from lth mode is absorbed and one from k th mode is excited, the matrix element Mlt is: g,, = ~ 2h (~,lv, l*,) ~ ~ , ~/~k +l • (8) In order to evaluate the transition probability of the Raman process, we take the global density of states in the
.~~
/
/
~--10~
/
/
~
/
/
/
x
I
10
8
F(x,~ 1.5~10~ 1¢ 0.5.10"
i ~
6
~.
Fig.2 The dependence of the function F ( x , A ) (eq.(12)) describing two phonon relaxation process of the Raman type, on x = h w / k T and A = A E / k T .
VO1. 77, NO, 1
TWO PItONON SPIN-LATTICE RELAXATION PROCESSES IN HJGlt MAGNETIC FH':I,DS
Thus, at high enough temperatures I(B,T) will be magnetic field and temperatures independent because the limit AE << kT will be obtained. One could think that this will be the case, for example, for diamond (0o ~ 2000K [8]) or in more limited range for silicon (0D • 450K [9]). The second case exists for materials where 0D is smaller than in the first case, like for example in InSb (0D • 170K [10]), or CdTe (0D • 125K Ill]). In this situation above certain temperatures the upper limit of integral I(B,T) will be small enough (below I0 - see Fig.2) to influence (decrease) the value of this integral. It will lead to the additional strong decrease of I(B, 7") with increasing magnetic field and/or temperature. Our model calculations have been done with ~D = 100K to show also this dependence of upper limit changes. Obtained results are presented in Fig.3. At higher fields (30T) and lower temperatures (4/0 the temperature variation can be approximated by T -1. Then, at higher T this variation is weaker. Finally, when the upper limit in I(B, 7") becomes smaller than about 10, because of rising T, the changes are more rapid and l(B--const, T) could be approximated as proportional to T -(.s (see Fig.3a). Thus, the inverse of relaxation time (relxation rate) as given by eq. (12) is no more rising like T ~, but much slower (between T s and T ~.~). The acoustic part of the phonon spectrum of the real crystals has two maxima in the Density of Phonon States (POPS) coming from phonons at boundaries of Brillouin Zone. The low energy one is given by TA phonons and a second one at higher energy is connected with LA phonons. The Debye energy is rather close to the LA phonon energy (see Tab.I). So in the real situation one could expect the total change of temperature behavior of the SLR process, when kT is close to the energy o f the zone boundary TA phonons (substantially smaller than kOD). One can say that this effect is due to the deviation of the real DoPS from the Debye model, but it should be stressed that it involves the limited energy of acoustic phonons and not the certain energy dependence of DoPS in the part of the spectrum where they exist. Table I The comparison of Debye energy with estimated energy of peaks of density of phonon states for acoustic phonons.
kaD (meV)" LA (meV) b TA (meV)
Si
GaAs
InSb
CdTe
55.2 48. 18.5
20.7 25. 10.5
20.7 15. 5.
15. 18.5 4.
=for Si, GaAs, InSb, and CdTe see ref. [9], [12], [10], and [l I], respectively bfor Si, GaAs, InSb, and CdTe see ref. [13], [14], [15], and [16], respectively The calculations also show that the relaxation rate given by the Raman process is magnetic field dependent through the integral I(B,T) (eq.(12)). In the first approximation at lower temperatures (about 4/0 one could compare B dependence with a parabolic one (see Fig.3b). For higher temperatures the dependence is weaker. It is worth to notice the sharp decrease of I(B, T) at high fields where the splitting strongly affects the integral's upper limit (g#BB,,, kOD, it means lack of high energy phonous which could be emitted). We would like to stress once more that the case presented here is the simplest model in which we assume that the electronic matrix element is magnetic field independent. At low magnetic fields the process involving emission of two ohonons (TPE) can be neglected. Will be this process more important at high fields? In the TPE process for "down" transition the initial state is the same as in the Raman process and is described by function (5). This time the final state is:
25
a) 10~'
10~
10~
........ ;ii 2
4
6 8 10 TFMPERATURE [K]
20
40
1¢ 10-~
10; -
4K
-
",,.
8K ........ 20K
----
\
2
2b
i0
iT}
MAGNEV,
Fig.3 The value of the integral I(B,T) (eq. (12)) calculated as: a) the function of temperature for given magnetic field; b) the function of magnetic field at given temperature. The light lines show the behavior described by the power laws with powers marked in the figure. Note double logarithmic scale.
I/) = I%)ln~......k+l . . . . . .,+l,...).
(13)
Using the same formalism as for the Raman process one can obtain the matrix element in the form:
8
= M:/~,lv~l%) ~
~
.,~;T.
O4)
In the case of "up" transitions (absorption of two phonons) the matrix element will be: Jt g,,
ffi ~-~ (%1v, i%> ~
~
V~l
.
(is)
Putting the sum wI + wk instead of difference u~ - wk in the expression for density of states (8) one can obtain an expression for the relaxation rate given by the TPE process: 1 2x
; - ~- lf(Ig~,l'+lM,,l') P(~) P(~) 6[~,-(~+~)]d~,a~, = 9 2x*~*vlO~7 {[n(w k +~, )+ I ] [n(w k )+ i ] + n ( ~ +o), )n(w k )} .
( ] 6)
One can see, that with the assumption AE < hwD the relaxation process with emission of two phonous is independent on the Debye temperature. The integral in eq.(16) can be divided into two terms: I e
TWO PHONON SPIN-LATTICE RELAXATION PROCESSES IN HIGH MAGNETIC FIELDS
26
and [zx, from which the first one depends only on the magnetic field and the second is dependent on magnetic field and temperature (through A = A E / k T ) . These integrals are:
I , : J0
d(~i%') (htak)a (h%-/%ht)a : ] -1~ (h%) 7
Vol. 77, No. 1
I0 s
/
(17)
,"/ /
o
and
// / I
Its = l
d(h°~k) (h°ht)s (h%-h°)k)a {[2n(%t+%)n(%) +
/
J0
/
10~
+ n(Wk+~Or) + n((~t)} =
,//
/. A = (kTy
J0 xS(A-x)3
e~(+ets dx = f f ( x , A ) dx . ( e~- l )(ets -e x )
(lg)
The integral Its is calculated numerically. Its dependence on A as well as the integrated function f ( x , A ) vs. x and A, are presented in Fig.4. It is clearly seen that for A between 2 and 7 the behavior of the integral could be approximated by A 4-7 whereas for higher values of A the power is 4.0. Because A is linear in B, the dependence of l/r on magnetic field for constant T will be like mentioned above. The temperature
'
f(x.~)
i jjj.
4
.~ 4
J
10s
b}
-q
~ . ~
10" 103 102 101 100 I~ I
2
4
,
,
6
8 10
20
....
-" .- //'"/
Ii.......... i........
1
2
, 4
/
'/] / / /
/
/
() 8 10 20 HAGNETIE FIELD {T]
4o 6o C
Fig .5 The comparison of all types of the two phonon relaxation processes discussed in the text. The solid line represents IB/(kT)7, the broken line Its/(kT) 7, and the dashed line l(B, T). Presented data are calculated for T = 4K. Note double logarithmic scale. dependence contains also factor T r, thus the global behavior could be approximated by power law with powers 2 (higher temperatures) and 3 (lower temperatures). The precise temperature regions for which these dependencies will dominate depend on magnetic fields. For higher fields the border between these regions will move toward higher temperatures. For A>3 1B dominates over I~. It means that for g#B8 > 3kT the TPE process becomes temperature independent and the relaxation rate will be proportional to B7 up to the field for which g # a B = h ~ . Above this field it becomes magnetic field and temperature independent. To find for what conditions the TPE process will be competitive with the Raman one, the integrals I(B,T) (eq.(12)), I a (kT)7 (eq.(17)) and Its (kT) 7 (eq.(18)) are plotted in Fig.5 as function of B. As one can see only the temperature independent part of TPE process (given by IB) is important and at 4K will dominate the Raman process at fields above 22T. At higher temperatures this limit will move to the higher fields (for example at 8K will be at 40/3. It should be stress that for systems with larger splitting (higher g-factor like in MgO:Co [17] or zero field splitting) the relaxation process with emission of two phonons can be effective at lower fields easier to obtain in laboratories.
0
IJlkT) 7
/
=
DELTA Fig.4 a) The dependence of the function f ( x , A ) (eq.(lg)) describing the relaxation process with emission of two phonons, on x = tlta/kT and A = A E / k T . b) Integral lts/(kT) ~ (eq.(18)) versus A = A E / k T . The light lines show the behavior described by the power laws with powers market in the figure. Note double logarithmic scale.
In conclusions, it should be pointed out that the two phonon relaxation processes at high magnetic fields are expected to behave differently as functions of temperature and external magnetic field than at low fields where g # s B is smaller than kT. The temperature variation in some regions can still be described by the power laws but with power smaller than 7 (namely 5 or even 2.5). At very high magnetic field (above 22T for spin 1/2 h and g = 2 electronic system) the process with emission of two phonons becomes dominant giving rise to a relaxation time independent on temperature. This process has previously been neglected for low fields. The magnetic field dependence of the relaxation rate is given not only by the dependence of the electronic matrix element on B but also contains factors coming from the phonon part. In the
Vo]. 77, NO. 1
TWO PHONON SPIN-LATTICE
RELAXATION
case of the Raman process at high fields and low temperatures (4/0 the additional nonmonotonic factor is introduced to the relaxation rate. At very high fields (where g/JsB is comparable with the Debye energy) the rate strongly decreases. At lower fields it can be approximated by B=. The relaxation rate, given by the process with emission of two phonons, at high fields (gI,B B > 3kT) is temperature independent and behaves like
PROCESSES
IN HIGH MAGNETIC
FIELDS
27
peaks in the acoustic phonon density of states in very low energies, the behavior of relaxation rate as discussed above, can be found at lower magnetic fields which are easier to obtain in laboratories. Acknowledgemenls The author would like to thank Prof. P.Wyder for continanus interest in this work, Prof M.Grynberg for illuminating discussions and Dr. T.Strutz for fruitful cooperation.
B ? .
It should be also pointed out that for systems with: larger g-factor additional zero field splitting
References [1] R. Orbach and H.J. Stapleton in "Electron Paramagnetic Resonance" ed. S. Geschwind, Plenum Press 1972, oh.2, p.12], and "Time Domain Electron Spin Resonance" ed. L. Kevan and R.N. Schwartz, John Wiley & Sons Inc., 1979 [2] K.W.H. Stevens, Rep. Prog. Phys. 30, 189 (1967) [3] T. Strutz, A.M. Witowski, and P. Wyder, Proc. 20th Int. Conf. Phys. Scmicond., Thessaloniki 90 (in press) and Proc. XIX Int. School Phys. Semicond. Compounds, Jaszowiec 90, Acta Phys. Polon. (in press) [4] T. Strutz, A.M. Witowski, R.E.M. de Becker, and P. Wyder, Appl. Phys. Lett. 57, 831 (1990) [5] W. Knap, L.C. Brunel, A. Witowski, and G. Martinez, Proc. 20 th Int. Conf. Phys. Semicond., Thessaloniki 90 (in press) [6] Baldassare di Bartolo "Optical Interactions in Solids", ed. John Wiley & Sons Inc., New York 1968 [7] R.D. Mattuck and M.W.P. Strandberg, Phys. Rev. 119, 1204 (1960)
[8] A.C. Victor, J. Chem. Phys. 36, 1903 (1962) [9] P. Flubacher, A.J. Leadbetter, and J.A. Morison, Philos. Mag. 4, 273 (1959) [10] U. Piesbergen, Z. Naturforsch. lga, 141 (1963) [11] R.R. Galazka, Shoichi Nagata, and P.H. Keesom, Phys. Rev. B22, 3344 (1980) and ref. therein [12] H.R. Philipp and H. Ehrenreich, Phys. Rev. 129, 1550 0963) [13] W. Weber, Phys. Rev. BI5, 4789 (1977) [14] G. Dolling and R.A. Cowley, Proc. Phys. SOC. 88, 463 (1963) [15] K. Kunc, M. Balkanski, and M.A. Nusimovici, phys. star. sol. (b), 72, 229 (1975) [16] J.M. Rowe, R.M. Nicklow, D.L. Price, and K. Zanino, Phys. Rev. BI0, 671 (1974) [17] M.H.L Pryce, Proc. Roy. Soc. 283A, 433 (1965)
Vol. 77, NO. i, pp. 23-27, 1991.
THE TWO PHONON SPIN-LA'I'FICE RELAXATION P R O C ~ E S
0038-i098/9153.00+.00 Pergamon Press plc
IN HIGH MAGNETIC FIELDS
A.M.Witowski *
Max-Planck Institut ff~r Festk~rperforschung Hochfeld Magnetlabor Grenoble, BP 166X F-38042 Grenoble Cedex, France (Received October 4th, 1990 by P.Burlet) The simple calculations are presented which show behavior of relaxation rate due to the two phonon spin-lattice relaxation processes at high magnetic field as function of temperature and magnetic fields. It is shown that neglected at low fields process with emission of two phonons become important above about 20T. It is also shown that this process for certain materials could be effective at lower fields.
In the "classical" experimental investigations of Spin-Lattice Relaxation (SLR) radio frequency or microwave radiation has been used [1,2] to disturb the system from thermal equilibrium. Thus, in all theoretical models describing SLR, it has been justified to make the final assumption that the spin splitting is much smaller than the thermal energy. There are three basic processes involved in SLR. At very low temperatures (2K) process with emission of one phonon is dominant. At higher temperatures (4K) the processes involving two phonous become more important. The so called Raman one consists of absorption of one phonon and emission of another one with higher energy. The intermediate state is a virtual one. If the intermediate state is a real one, the process is called Orbach process. The SLR process with emission of two phonous is neglected at low magnetic fields because of very small density of phonon states for low energy phonons. The assumption AE << kT allows to make very useful numerical approximations in calculations of the relaxation rate for two phonon SLR processes. Recently, the results of investigations of SLR processes at high magnetic fields have been reported [3,4,5]. In such a case the spin splitting is larger than the thermal energy (AE > kT). The purpose of the present paper is to study theoretically the effect of this new condition on the two phonon spin-lattice relaxation processes.
IW12
I
for
#~) = 0
for ~ > ~ ,
•Hlatt +Hd +Hint
N~ ffi F~ =- 21g~B B
The diagram of electronic levels for a system with spin 1/2 in an external magnetic field B. Transitions discussed in the text are denoted by arrows.
where: Hintt is the lattice part, H d describes the electronic system and Hint interaction between these subsystems. The energy level of the simple electronic system with spin 1/z tt in an external magnetic field 8 is split into two levels. These two levels are separated by the energy AE = g~BB (g is the effective g-factor and B is oriented along the z-axis). Let N 1 be the population of the ground state (index 1) and N 2 be the population of the excited state (index 2) (see Fig.l). At any given moment t they fulfill the rate equations (N x + Na - const): N1(t) ffi N2(t)wjl
- N1(t)wlz
,
N:(t) ffi N1(t)wl,
- Nz(t)wjl
,
(2)
where wt2 and wzl are transitionprobabilitiesfrom the state I to the state 2 (up) and from the state 2 to the state I (down), respectively. The thermal equilibrium values for these populations are N:" and N2". If the system at the time t = 0 has the populations Nto and N2°, which are not the thermal equilibrium ones, then the population difference relaxes toward the equilibrium as described by the solutionsof (2):
(1)
where v is the velocity of sound and V volume of the crystal (see e.g. [6]). The physical system under consideration can be described by the Hamiltonian of the form: H
Iw21
Fig.l
To show the consequences of application of high magnetic field on the spin-lattice relaxation, we limit ourselves to the simplest case of a two level system with spin S = 1/2 8 and g-factor of 2 (see Fig.l), and a phonon spectrum described by the Debye model. We also assume that the splitting of levels is not exceeding the acoustic phonon energy (splitting smaller than ~wD , where COD is Debay frequency). Additionally we assume that the crystal is isotropic so the density of phonon states can be described as: 3Vca2 p(o~) ffi 21r2v2
N2(t) E~= 21g,uBB
2
[N,(t)-N,(t)]-[N,'-NI"] = [(N,O-NIO)-(N,'-NI")] exp[- ~1, with a relaxation time r defined by:.
I/r = w1~ + w~1 .
,
(3)
The probability of transitionbetween two states is related to the matrix element of Hint by the golden rule [6]: 2~v I0'I Hintli>l ~ ffi~~ p(E,fief), (4)
* on leave from Institute of Experimental Physics, Warsaw University
23
TWO P H O N O N
24
SPIN-LATTICE
RELb~XATION
PROCESSES
IN HIGH M A G N E T I C
FIELDS
VOI.
77, No. i
with the wave functions of the given state j expressed in the
form [6]:
our
p(Ef=E i ) = I ~(~if_~Or)P(wk)P(Wl)6.[%f_(O:l_(Ok)]d%fdcokdW1 , (9)
case as:
I J) = l ~ ) l n , ..... nk,'"),
where •
described the electronic wave function (eigenfunction of Hal) and n k is the occupation number of the kth phonon mode in the global state j. The energy of the state described by [j) is:
I Usually the interaction Hamiltonian is expressed as a power series in displacement of the investigated center or in the local strains ~ [2,6,7]. Both type of variables are then expressed by the phonon creation (a ÷) and annihilation (a) operators. In the simplest case one could write [6]:.
Hint = V~ + V~ ~ + ....
with hw r = AE. Thus, the transition probability w after integration over coit is: 2~ j'.f IM~I ~ p(~,) p(~) q%-(~-~k)] cl~d~o~. (1o) w~ = ~Once more we would like to stress that, if one assumes that AE << kT, then ft~ ~- f i ~ will follow and consequently wij would become independent on the electronic levels splitting. Now we can write the expression for relaxation time taking equations (7, 8 and 10) and the density of phonon states in the Debye model (1):
2# if(Ig,,l'+lg,,I ') p(~,) p(~) 61%-(~-~,)]d ' ~ d ' , ' k ;! = ~-
=
(5)
with: 2~r3,/2vlO~T
q where M is mass of the crystal. As one can see, the one phonon process is described by the term linear in ~. It leads to the well known formula [2,6,7] giving the inverse of relaxation time (also at high magnetic fields) proportional to: I°cr I(~xlVII~'}I'(AE)S coth[2~T 1 "
=
l¢,)In~,...,nk
.....
n~,..).
(6)
In the case of the Raman process the final state then is:
I1~ =
I~,) I", ......~-I ......:l,...).
At small B one can assume that AE << kT and most of the phonons taking part in the process have energies comparable with kT, thus fi~zk ~ / ~ and w ~ ~ w~x . This is not true at high magnetic fields and further we will show the consequences of going beyond mentioned approximation. The matrix element of Hint between the involved states is:
where "~ = M / V is density of the material and n(w) is the occupation of the phonon states with energy h~. Taking for n(~) the Bose-Einstein distribution function [exp(hta/kT)-1]-1, we obtain that I/r is proportional to:
! ~, I(~',lv,l¢,)l' (kr)' 1(~,T),
~
<.k-l,.,+l I ~ : t
+ a~.k
(12)
,,OD/T-A I x'~Cx+A'O.' eX(l+eA) fF(x,A) dx [(B, T) = Jo " ' ( : - I)(:+a_ I) dx =
with: x = h~,/l~T and h The dependence of presented in Fig.2. Note tion is small, and give a
= 5E/kT. the function F(x,A) on x and A is that for x > I0 the value of the func-
negligible contribution to the integral I(B,T). Thus, for OD/T-A>IO, I(B,T) is independent on its upper limit. It also means that these results will not depend on the Debye temperature. In the approximation used at low fields, I(B,T) is independent on the splitting of levels (magnetic field). With additional assumption that kOD >> k T it is also temperature independent and can be calculated analytically. At high fields the integral I(B,T) should be calculated numerically for given values of T and B in order to find its dependence on these parameters. There are two interesting cases. First, when the Debye temperature is very high, the level splitting in available non pulsed magnetic fields (about 30T) is much smaller than kOD .
Uif = (flHint li) =
=~
( 1 1)
where:
In the first order perturbation theory the two phonon processes are described by the term containing ¢~. Using the second order perturbation calculations one could also obtain results in the same order of magnitude with the term linear in strain. For sake of simplicity we restrict ourselves only to the former case. The later case will give basically the same phonon part as the former one assuming all other states far from discussed one [6]. The reader interested in the all possible terms appearing in such calculations could find the details in [7]. Let us take as the initial state:
Ii)
(n(tak+wr)ln(o.~)+ I ]+[n(wk+t~r)+ 1]n(wk) ) .
~
I.k,.,)(¢,lv,l¢,)
In this point we do not assume neither any properties of the electronic system but non zero value of the matrix element (~]V~[¢~) for given conditions (magnetic field and temerature)," nor the kind of mechanism giving rise to the Hin t. The matrix element (~/~[V~I~) is of course independent on phonon energies. Finally, the matrix element M~: which is related to the W~l is: For the probability w~2 describing "up" transition from electronic state 1 to electronic state 2 in which one phonon from lth mode is absorbed and one from k th mode is excited, the matrix element Mlt is: g,, = ~ 2h (~,lv, l*,) ~ ~ , ~/~k +l • (8) In order to evaluate the transition probability of the Raman process, we take the global density of states in the
.~~
/
/
~--10~
/
/
~
/
/
/
x
I
10
8
F(x,~ 1.5~10~ 1¢ 0.5.10"
i ~
6
~.
Fig.2 The dependence of the function F ( x , A ) (eq.(12)) describing two phonon relaxation process of the Raman type, on x = h w / k T and A = A E / k T .
VO1. 77, NO, 1
TWO PItONON SPIN-LATTICE RELAXATION PROCESSES IN HJGlt MAGNETIC FH':I,DS
Thus, at high enough temperatures I(B,T) will be magnetic field and temperatures independent because the limit AE << kT will be obtained. One could think that this will be the case, for example, for diamond (0o ~ 2000K [8]) or in more limited range for silicon (0D • 450K [9]). The second case exists for materials where 0D is smaller than in the first case, like for example in InSb (0D • 170K [10]), or CdTe (0D • 125K Ill]). In this situation above certain temperatures the upper limit of integral I(B,T) will be small enough (below I0 - see Fig.2) to influence (decrease) the value of this integral. It will lead to the additional strong decrease of I(B, 7") with increasing magnetic field and/or temperature. Our model calculations have been done with ~D = 100K to show also this dependence of upper limit changes. Obtained results are presented in Fig.3. At higher fields (30T) and lower temperatures (4/0 the temperature variation can be approximated by T -1. Then, at higher T this variation is weaker. Finally, when the upper limit in I(B, 7") becomes smaller than about 10, because of rising T, the changes are more rapid and l(B--const, T) could be approximated as proportional to T -(.s (see Fig.3a). Thus, the inverse of relaxation time (relxation rate) as given by eq. (12) is no more rising like T ~, but much slower (between T s and T ~.~). The acoustic part of the phonon spectrum of the real crystals has two maxima in the Density of Phonon States (POPS) coming from phonons at boundaries of Brillouin Zone. The low energy one is given by TA phonons and a second one at higher energy is connected with LA phonons. The Debye energy is rather close to the LA phonon energy (see Tab.I). So in the real situation one could expect the total change of temperature behavior of the SLR process, when kT is close to the energy o f the zone boundary TA phonons (substantially smaller than kOD). One can say that this effect is due to the deviation of the real DoPS from the Debye model, but it should be stressed that it involves the limited energy of acoustic phonons and not the certain energy dependence of DoPS in the part of the spectrum where they exist. Table I The comparison of Debye energy with estimated energy of peaks of density of phonon states for acoustic phonons.
kaD (meV)" LA (meV) b TA (meV)
Si
GaAs
InSb
CdTe
55.2 48. 18.5
20.7 25. 10.5
20.7 15. 5.
15. 18.5 4.
=for Si, GaAs, InSb, and CdTe see ref. [9], [12], [10], and [l I], respectively bfor Si, GaAs, InSb, and CdTe see ref. [13], [14], [15], and [16], respectively The calculations also show that the relaxation rate given by the Raman process is magnetic field dependent through the integral I(B,T) (eq.(12)). In the first approximation at lower temperatures (about 4/0 one could compare B dependence with a parabolic one (see Fig.3b). For higher temperatures the dependence is weaker. It is worth to notice the sharp decrease of I(B, T) at high fields where the splitting strongly affects the integral's upper limit (g#BB,,, kOD, it means lack of high energy phonous which could be emitted). We would like to stress once more that the case presented here is the simplest model in which we assume that the electronic matrix element is magnetic field independent. At low magnetic fields the process involving emission of two ohonons (TPE) can be neglected. Will be this process more important at high fields? In the TPE process for "down" transition the initial state is the same as in the Raman process and is described by function (5). This time the final state is:
25
a) 10~'
10~
10~
........ ;ii 2
4
6 8 10 TFMPERATURE [K]
20
40
1¢ 10-~
10; -
4K
-
",,.
8K ........ 20K
----
\
2
2b
i0
iT}
MAGNEV,
Fig.3 The value of the integral I(B,T) (eq. (12)) calculated as: a) the function of temperature for given magnetic field; b) the function of magnetic field at given temperature. The light lines show the behavior described by the power laws with powers marked in the figure. Note double logarithmic scale.
I/) = I%)ln~......k+l . . . . . .,+l,...).
(13)
Using the same formalism as for the Raman process one can obtain the matrix element in the form:
8
= M:/~,lv~l%) ~
~
.,~;T.
O4)
In the case of "up" transitions (absorption of two phonons) the matrix element will be: Jt g,,
ffi ~-~ (%1v, i%> ~
~
V~l
.
(is)
Putting the sum wI + wk instead of difference u~ - wk in the expression for density of states (8) one can obtain an expression for the relaxation rate given by the TPE process: 1 2x
; - ~- lf(Ig~,l'+lM,,l') P(~) P(~) 6[~,-(~+~)]d~,a~, = 9 2x*~*vlO~7 {[n(w k +~, )+ I ] [n(w k )+ i ] + n ( ~ +o), )n(w k )} .
( ] 6)
One can see, that with the assumption AE < hwD the relaxation process with emission of two phonous is independent on the Debye temperature. The integral in eq.(16) can be divided into two terms: I e
TWO PHONON SPIN-LATTICE RELAXATION PROCESSES IN HIGH MAGNETIC FIELDS
26
and [zx, from which the first one depends only on the magnetic field and the second is dependent on magnetic field and temperature (through A = A E / k T ) . These integrals are:
I , : J0
d(~i%') (htak)a (h%-/%ht)a : ] -1~ (h%) 7
Vol. 77, No. 1
I0 s
/
(17)
,"/ /
o
and
// / I
Its = l
d(h°~k) (h°ht)s (h%-h°)k)a {[2n(%t+%)n(%) +
/
J0
/
10~
+ n(Wk+~Or) + n((~t)} =
,//
/. A = (kTy
J0 xS(A-x)3
e~(+ets dx = f f ( x , A ) dx . ( e~- l )(ets -e x )
(lg)
The integral Its is calculated numerically. Its dependence on A as well as the integrated function f ( x , A ) vs. x and A, are presented in Fig.4. It is clearly seen that for A between 2 and 7 the behavior of the integral could be approximated by A 4-7 whereas for higher values of A the power is 4.0. Because A is linear in B, the dependence of l/r on magnetic field for constant T will be like mentioned above. The temperature
'
f(x.~)
i jjj.
4
.~ 4
J
10s
b}
-q
~ . ~
10" 103 102 101 100 I~ I
2
4
,
,
6
8 10
20
....
-" .- //'"/
Ii.......... i........
1
2
, 4
/
'/] / / /
/
/
() 8 10 20 HAGNETIE FIELD {T]
4o 6o C
Fig .5 The comparison of all types of the two phonon relaxation processes discussed in the text. The solid line represents IB/(kT)7, the broken line Its/(kT) 7, and the dashed line l(B, T). Presented data are calculated for T = 4K. Note double logarithmic scale. dependence contains also factor T r, thus the global behavior could be approximated by power law with powers 2 (higher temperatures) and 3 (lower temperatures). The precise temperature regions for which these dependencies will dominate depend on magnetic fields. For higher fields the border between these regions will move toward higher temperatures. For A>3 1B dominates over I~. It means that for g#B8 > 3kT the TPE process becomes temperature independent and the relaxation rate will be proportional to B7 up to the field for which g # a B = h ~ . Above this field it becomes magnetic field and temperature independent. To find for what conditions the TPE process will be competitive with the Raman one, the integrals I(B,T) (eq.(12)), I a (kT)7 (eq.(17)) and Its (kT) 7 (eq.(18)) are plotted in Fig.5 as function of B. As one can see only the temperature independent part of TPE process (given by IB) is important and at 4K will dominate the Raman process at fields above 22T. At higher temperatures this limit will move to the higher fields (for example at 8K will be at 40/3. It should be stress that for systems with larger splitting (higher g-factor like in MgO:Co [17] or zero field splitting) the relaxation process with emission of two phonons can be effective at lower fields easier to obtain in laboratories.
0
IJlkT) 7
/
=
DELTA Fig.4 a) The dependence of the function f ( x , A ) (eq.(lg)) describing the relaxation process with emission of two phonons, on x = tlta/kT and A = A E / k T . b) Integral lts/(kT) ~ (eq.(18)) versus A = A E / k T . The light lines show the behavior described by the power laws with powers market in the figure. Note double logarithmic scale.
In conclusions, it should be pointed out that the two phonon relaxation processes at high magnetic fields are expected to behave differently as functions of temperature and external magnetic field than at low fields where g # s B is smaller than kT. The temperature variation in some regions can still be described by the power laws but with power smaller than 7 (namely 5 or even 2.5). At very high magnetic field (above 22T for spin 1/2 h and g = 2 electronic system) the process with emission of two phonons becomes dominant giving rise to a relaxation time independent on temperature. This process has previously been neglected for low fields. The magnetic field dependence of the relaxation rate is given not only by the dependence of the electronic matrix element on B but also contains factors coming from the phonon part. In the
Vo]. 77, NO. 1
TWO PHONON SPIN-LATTICE
RELAXATION
case of the Raman process at high fields and low temperatures (4/0 the additional nonmonotonic factor is introduced to the relaxation rate. At very high fields (where g/JsB is comparable with the Debye energy) the rate strongly decreases. At lower fields it can be approximated by B=. The relaxation rate, given by the process with emission of two phonons, at high fields (gI,B B > 3kT) is temperature independent and behaves like
PROCESSES
IN HIGH MAGNETIC
FIELDS
27
peaks in the acoustic phonon density of states in very low energies, the behavior of relaxation rate as discussed above, can be found at lower magnetic fields which are easier to obtain in laboratories. Acknowledgemenls The author would like to thank Prof. P.Wyder for continanus interest in this work, Prof M.Grynberg for illuminating discussions and Dr. T.Strutz for fruitful cooperation.
B ? .
It should be also pointed out that for systems with: larger g-factor additional zero field splitting
References [1] R. Orbach and H.J. Stapleton in "Electron Paramagnetic Resonance" ed. S. Geschwind, Plenum Press 1972, oh.2, p.12], and "Time Domain Electron Spin Resonance" ed. L. Kevan and R.N. Schwartz, John Wiley & Sons Inc., 1979 [2] K.W.H. Stevens, Rep. Prog. Phys. 30, 189 (1967) [3] T. Strutz, A.M. Witowski, and P. Wyder, Proc. 20th Int. Conf. Phys. Scmicond., Thessaloniki 90 (in press) and Proc. XIX Int. School Phys. Semicond. Compounds, Jaszowiec 90, Acta Phys. Polon. (in press) [4] T. Strutz, A.M. Witowski, R.E.M. de Becker, and P. Wyder, Appl. Phys. Lett. 57, 831 (1990) [5] W. Knap, L.C. Brunel, A. Witowski, and G. Martinez, Proc. 20 th Int. Conf. Phys. Semicond., Thessaloniki 90 (in press) [6] Baldassare di Bartolo "Optical Interactions in Solids", ed. John Wiley & Sons Inc., New York 1968 [7] R.D. Mattuck and M.W.P. Strandberg, Phys. Rev. 119, 1204 (1960)
[8] A.C. Victor, J. Chem. Phys. 36, 1903 (1962) [9] P. Flubacher, A.J. Leadbetter, and J.A. Morison, Philos. Mag. 4, 273 (1959) [10] U. Piesbergen, Z. Naturforsch. lga, 141 (1963) [11] R.R. Galazka, Shoichi Nagata, and P.H. Keesom, Phys. Rev. B22, 3344 (1980) and ref. therein [12] H.R. Philipp and H. Ehrenreich, Phys. Rev. 129, 1550 0963) [13] W. Weber, Phys. Rev. BI5, 4789 (1977) [14] G. Dolling and R.A. Cowley, Proc. Phys. SOC. 88, 463 (1963) [15] K. Kunc, M. Balkanski, and M.A. Nusimovici, phys. star. sol. (b), 72, 229 (1975) [16] J.M. Rowe, R.M. Nicklow, D.L. Price, and K. Zanino, Phys. Rev. BI0, 671 (1974) [17] M.H.L Pryce, Proc. Roy. Soc. 283A, 433 (1965)