Theory of Spin Lattice Relaxation Review Article phys stat sol

Theory of Spin Lattice Relaxation Review Article phys stat sol, (b) 117, 437 (1983) Subject classification 6 and 13; 20 1; 22 5; 22 8 1 School of Physics, University of Ryderabad) Theory of Spin Latt.

Review Article phys stat sol, (b) 117, 437 (1983) Subject classification: and 13; 20.1; 22.5; 22.8.1 School of Physics, University of Ryderabad') Theory of Spin-Lattice Relaxation BY K N SHRIYASTAVA Contents lnti*oduction The spin-lattice interaction The dii*ectprocess 3.1 3.2 3.3 3.4 The direct process for non-Kramers systems by acoustic phonons Special case of direct process The direct process by optical phonons The direct process for Kramers systems The Raman process 4.1 The Ranian process by spin-two-phonon interaction 4.2 The Raman process b y spin-one-phonon interaction in second order 4.3 Raman process a t short-wavelength phonons 4.4 Optical acoustic Raman-like process The s u m process The Orbach process The three-phonon process The local mode process The collision process 10 Conclueions References Introduction The phenomena of exchange of energy between the spin system and the lattice is of importance to study the establishment of thermodynamic equilibrium, once the spin populations are disturbed by external forces Since both the upward and the downward transition probabilities are important, the phenomenon of spin-lattice relaxation is the zero-energy case of the phenomenon of resonance Very often the lifetimes of the upper states are so large that the spin system need not be in thermal equilibrium 1) P.O Central University, Hyderabad 500 134, India 438 K N SHRWASTAVA with the lattice Besides the thermal conductivity of the lattice near the spin may be so small that energy cannot be carried away fast enough The populations in the upper level can even be larger than those on the lower levels so that if the lifetimes and the energy separations of various levels are favourable, there may be population inversion giving rise to the phenomenon of maser (or laser) The first detailed account of nonresonant paramagnetic relaxation was given by Gorter [I] The problem of the lack of thermodynamic equilibrium between the spin and the lattice systems has been discussed [2 to 41 The equilibrium spin-lattice relaxation has been discussed by Manenkov and Orbach [ ] , Standley and Vaughan [6], Stevens [7], Abragam and Bleaney [8],and others Though such accounts of the spin-lattice relaxation exist, they deal with only those experimental data which can be explained on the basis of the known relaxation mechanisms The first calculation of the relaxation times was given by Waller [9] who modulated the dipole-dipole interaction by the lattice vibrations and discussed the processes of transfer of energy by the spin to the lattice by the one-phonon direct process as well as the two-phonon Raman process Until this date these two processes form the basis of any investigation of the spin-lattice relaxation The two processes are fundamental although the driving forces can be altered Instead of modulating the dipole-dipole interaction Van Vleck [lo] modulated the crystalline field, which is essentially a Taylor expansion of the attractive Coulomb interaction between the electrons and the nuclei in terms of the thermal displacements of atoms which are described by a generalization of normal coordinates t o take into account the phonon continuum which is most conveniently treated by the Debye model The blame of the failure of the experimental data on spin-lattice relaxation time to agree with the Van Vleck type direct and Raman process is often thrown on the use of the Debye type density of states for the phonons The dependence of the relaxation times on the structure of the spin levels is not too well studied There is considerable amount of experimental data which cannot be explained by any of the known processes A systematic study of the spin-lattice relaxation is therefore undertaken in the hope of understanding the behaviour of the relaxation t,ime on such parameters as the temperature and the magnetic field which are accessible to the experimentalists The Spin-Lattice Interaction The crystal field is normally written as the expansion of the attractive nuclear potential C (2,e) ef/rij in terms of spherical harmonics which are functions of the elecij tron coordinates The angular parts of this potential are separable and the potential can be described by equivalent operators which operate like the angular momentum operators, the commutators of which are known The operators are equivalent in the sense that the eigenvalues of the equivalent operator are linearly related with those of the actual operator The unknown coefficients of proportionality can be tabulated, once for ever, for all the ions of interest using the Wigner-Eckart theorem The coupling with the lattice is obtained from the Taylor expansion of the crystal potential in terms of the displacements of atoms For this purpose VanVleck suggested the generalization of normal coordinates t o take into account the phonon continuum with the use of the Debye model to preserve the symmetry of the crystal This type of calculation is necessary for the understanding of the proper angular dependence of relaxation times as the magnetic field is rotated with respect t o the crystal axis of symmetry Most of the interest, however, may not lie in the study of the angular dependence as the temperature and the magnetic field are more desirable parameters Besides for the cryst,als of low symmetry, the Van Vleck procedure of calculating the 439 Theory of Spin-Lattice Relaxation normal coordinates is much too complicated without the advantage of obtaining much additional information Therefore, in as much as the study of temperature and field dependence of relaxation rates is concerned, there is not much point in calculating the normal coordinates The derivative of the crystal potential which leads t o the electronic operators of the spin-lattice interaction is equally difficult to calculate for the given normal coordinates Therefore, we avoid the use of normal coordinates and concentrate on the essential structure of the interaction which leads to the correct field and temperature dependence of the experimental parameters The Taylor expansion of the potential looks like where are the derivatives of the crystal potential, the RI are the coordinates of the i-th neighbour of displacement where M is the mass of the crystal, o the frequency of displacement, k the wave vector of the lattice wave, and ak and are the phonon operators I n evaluating the matrix elements, one is faced with the difficulty of handling the exponential operator However, most of the experiments are done a t low temperatures where one can take the long-wavelength approximation 1% R for the lattice waves, so that exp (ik R ) = cos (k R ) i sin (k R ) can be simplified by taking sin (k R)m w k R, cos (k R ) m I n most cases of interest there is a centre of symmetry, so that only the sin (k R ) term survives for which one can take k R and ignore the cosine term However, the cosine term is inore appropriate t o the optical phonons or when there is no centre of symmetry It is of course not necessary to take the longwavelength approximation for the lattice phonons, in which case one is faced with complicated integrands which can be reduced to Bessel functions as demonstrated by Shrivastava [ll] and lead to higher-order corrections to the famous Van Vleck calculations For the present purpose i t is sufficient t o point out that one has t o consider the anisotropy of the wave-vector space with respect to the crystal axes of symmetry This is achieved by using the mathematical techniques developed in connection with the phase-shift analysis of the scattering problem, i.e., - < + - - W exp (ikr cos 0) - a i"21 = 1=1 + 1) j , ( k r ) P,(COS 0) (4) I n the calculation of second-order process, two-phonon wave vectors are involved and hence the integrands are considerably complicated As we are concerned with the need of the experimentalists in this area, we simply write the desired spin-lattice interaction as K N SHRIVASTAVA 440 which is correct up to second-order and contains both the sine and cosine terms The electronic operators V i are independent of the lattice wave vectors to a very good approximation The unperturbed Hamiltonian can be defined as which serves to define the spin states and the phonon field It is convenient to take hS2 = gpBH when there is an external magnetic field The Direct Process We consider a two-spin-level system, [ a ) and Ib), and wish to learn how it interacts with the lattice For direct absorption of energy from the lattice or direct emission of energy into the lattice, i t is important that the energy difference E , - E , be within the upper limit of the Debye cut-off, i.e., E , - E , AmD, where w,, is the Debye cut-off frequency, then only the fundamental requirement of conservation of energy can be satisfied I n the case of a direct process we calculate the transition probability, Pb,,from the spin state [a) to ( b ) with the absorption of a phonon of energy hwk = = E , - E , and the probability, Pab, for the transition from Ib) to la) with the emission of a phonon into the lattice Since t,he frequency of the absorbed phonon is equal to that of the emitted phonon, this is a zero-energy phenomenon The way it causes relaxation is involved in the fact that the probability of emission of a phonon is proportional to nk whereas the probability of absorption is proportional to the Bose factor nk So, as the relaxation occurs the number of phonons in the lattice increases, the spin earlier heated by the rf field, cools (or relaxes), and the lattice is heated I n fact one can collect such extra phonons directly [la] or look a t them through the Brillouin scattering of laser light [13] I n case the thermal conductivity of the lattice < + Fig Spin-lattice relaxation rate in SrF,; Mna+ varying as T;' = 154T x 10-aT6s-l Below a temperature of K a direct process is found as predicted by (12), (20), (39) Above K, the spin-lattice relaxation time describes a 2'6 behaviour as given by (56) and (60) (from [43]) + I 10 100 TIXI- 441 Theory of Spin-Lattice Relaxation is sufficient t o carry away the phonons t o distribute them in the lattice, we can define a relaxation time, T,’ = P a b + Pab which will of course be proportional to (Fig 1) 2nk ho + = coth _2k,T _- ~ 2k,T ho ’ which shows that the relaxation rate varies linearly with temperature a t low temperatures However, depending on the structure of energy, levels and the temperature, the relaxation rate may vary as nk which owing to energy conRervat,ion leads to a n exponential behaviour, 3.1 The direct process jor non-KTamers systems by acoustic phonons Let us see how one may obtain the relaxation time for the direct process from ( ) The states are defined as la, nk,,nk,, , nkD) which are the Born-Oppenheimer products of the spin state and that of the lattice with the cut-off wave vector as k, I n a two-level spin system, the probabilit,y of absorption of the phonon is given by in the long-wavelength approximation The spin matrix element is left undefined and the lattice part is I(nk - 11 ak l?tk)I2 = nk For convenience we have introduced the notation E , - E , = A; Since the lattice spectrum is continuous, we replace the density of states by a Dirac &function which ensures the conservation of energy I n (13) i t is of interest to change the summation over the phonon wave vectors to integration which is appropriate for a phonon continuum from zero to ha, I n this case kn - 3k2 d k , where we have assumed the isotropy of the phonon space, i.e., k, = k, = k, for a three-dimensional solid with k, = w,lv with v as the velocity of sound We replace MIV by Q , the mass density of the solid and obtain where n(d,) is the Bose factor a t db, [ %(A,,) = exp ( k $-l]-lThe reverse probability can be written as which can be reduced to 442 I(.N SHRIVASTAVA so.that the relaxation rate is k,!' in two-level systems, I n this case nk NN ?bk 1, P a b w Pba and we obtain n(Ab) w exp (-Ab/kBT) so that the relaxation rate will appear t o have an exponential dependence, + rather than a linear behaviour The physical situation can often be realized in threelevel systems if, E , - Eb Ec - E b > kBT =deb, Eb - Ea Pab w0 < kBT pCa + PaC = + pbc (23) so that and the relaxation rate will appear to vary as exp ( -Ac,,/kBT) 33 The diiqect process by optical phonons We consider the calculation of the relaxation time using ( ) for optical phonons in the approximation, kR The probability of absorption of one optical phonon is given by < and that of emission in the linear region of the dispersion and for small wave vectors The relaxation rate is then, 443 Theory of Spin-Lattice Relaxation for optical phonons It may be noted that the optical direct process varies as the inverse cubic power of the sound velocity whereas the acoustic phonon direct process varies as the inverse fifth power Since the optical phonon energies are rather large, these relaxation times are practically independent of the magnetic field as compared with the H dependence in the case of the acoustic phonon process The use of the dispersion relation k = w/v is also not too good for optical phonons Perhaps we should use a flat dispersion for optical phonons with a bandwidth Ao and w, as the frequency at the zone boundary I n this case, we may write w = wo + ek2, w, = wo + cn21a2 and the matrix element part of the probability as and so that the relaxation rate is given by 3.4 The direct process for Kramers systems In the case of atoms containing a n odd number of electrons, the matrix elements of the crystal potential between time-reversed states are zero so that the lifetime given by the calculations of the preceding section will be infinite However, finite lifetimes can be obtained if the magnetic interactions are included Since ( - f I Vl I =0 if the states are pure, we may admix the states by introducing the interaction of the form l p s H J to obtain a non-zero value The perturbed states are + +) An interesting case arises when the field is aligned along the z-direction so that x and y components of the angular momentum operator not occur If the orientation of the field is defined by the unit vectors of the angular momentum as 6, and 61, we can writ,e the matrix element K; N.SERIVASTAVA 444 t"'I '0° L-lA-Uu Pig 2 10 Pig Fig Spin-lattice relaxation rate in CaWO,: Nd3+ Below K a direct process is found a8 in Fig However, abovo K the relaxation rate varies as T10.4*o.z indicating the breakdown of the small-wave-vector approximation for lattice phonom so that the experimentally measured behaviour is in accord with that predicted by (64).The data are taken from [41] Fig Spin-lattice relaxation in CaWO,: Cea+ Below K a direct process is observed Above K a TlO.o*o.~dependence is found a8 in (64) The data am taken from 1411 I Theory of Spin-Lattice Relaxation 445 The form of (19) then for the Kramers systems will be (39) which varies as d s T / d q Since db m g p B H , the dependence of the relaxation rate on the field and temperature is given by d14T (Fig and ) The Raman Process I n the theory of spin-lattice relaxation first by Waller a process was introduced which suggests relaxation of a spin from a n upper level to a lower level by absorption of one phonon which corresponds to a virtual transition of the spin to a n intermediate level and subsequent emission of another phonon which leads t o the spin transition to the lower level The difference between the two phonon frequencies is equal to the spin level separation Therefore, this process is similar t o the Raman scattering of light by molecules except that the photons have been replaced by phonons I n the case of the Raman scattering of light one expands the polarizability whereas for the Raman relaxation process, Waller expanded the dipole-dipole interaction Van Vleck realized that the expansion of the dipole-dipole interaction gives longer relaxation times than are found for spins in insulating systems Therefore, he modulated the crystalline electric field arising from the attractive Coulomb interaction between the electrons and the nuclei, and obtained the correct order of magnitude of the relaxation times Kronig [la]and Fierz [15] also performed the calculation of t,he Raman relaxation process from the modulation of the crystalline field However, Van Vleck obtained full expansion in terms of the normal coordinates whereas Kronig and Fierz did not consider the anisotropy of the crystal field Van Vleck’s calculations are somewhat more detailed for he calculated the relaxation times for both, Kramers and nonKramers systems from the first term in the Taylor expansion of the crystal field, whereas Kronig discussed the importance of the second term 4.1 The Raman process bu spin-two-phonon interaction We consider the calculation of Raman relaxation times from the second term in the Taylor expansion of the crystal potential in the first-order perturbation t,heory The probability of a transition from the Born-Oppenheimer state la, nk, nW), where la> is a spin state and Ink, nk,) describes the phonon state Ink,,nk,, ,nk, , nw, ,nk,), to Ib, nk 1, nk, - 1) with the absorption of a phonon of wave vector k’and emission of a phonon of the wave vector k is given by, + where 3t?,is the spin-two-phonon interaction given by (7) It is convenient to take the small-wave-vector approximation and change the summation t o integration with the help of the linear dispersion relation o = vk, we obtain, Pba= M, 28 physlea (b) 117/2 1s (n + 1)n’os d o wf3do’6 446 K N SERIVASTAVA Fig Spin-lattice relaxation time of the r5-Fl transition in ZnS: Fe measured by BonvilIe et al [46] and predicted by ~461,~481 T IK) - where 142) Evaluating the 6-function integral we obtain (43) where n' is evaluated a t w + A , / h as Case I: If ho A,, the probability leads t o a well-known temperature dependence as then (41) can be written as 447 Theory of Spin-Lattice Relaxation This type of calculation was done by Kronig but the importance of (48) was not realized It is of interest to note that for very widely spaced energy levels the dominant t.erm does not depend on T7 but rather on T4 The dependence of relaxation times on the structure of energy levels is therefore quite obvious 4.2 The Raman process by spin-one-phmon Ctteractim in second order We can obtain the Raman process by considering the spin-one-phonon interaction given by (5) in the small-wave-vector approximation in the second-order perturbation theory We consider the transition of a spin from a lower level la) to a n intermediate state Ic> by the absorption of a phonon and subsequently a transition from the state Ic) t o the desired state (b> by emission of a phonon The probability of transition from a Born-Oppenheimerst,ate la, %k; nknk'> to Ib, nk - 1, nk, 1) via Ic, nk - 1, nc) is given by + If we use the interaction (5) in the long-wavelength approximation, then (50) can be written as where 4, = E , - E,, and the relaxation time is determined by T-1 - 2c h4E (XB)'fT(eZ - eZdx 1)2' (54) Strictly speaking the integral in the above equation depends on temperature However, co and the relaxation rate vanes as T7 (Fig 5) a t very low temperatures BIT Case 11:At high temperatures x -+ 0, ez = in the numerator of the integrand in (54) and e2 - = x in the denominator leads to so that a t high temperatures the relaxation rate varies as T Case 111: A , < h w , A, with the simultaneous emission of two phonons is calculated to be ( A T1 AbIB AbIn Pab = 2”h 4@Vb7? + J(% ) (n2 + ) M,,W!~;6(0, + 02 - db) dm1do, , (68) where A , == E, - E , and o1and w2 are two phonon frequencies which are such that o1 w2 = A b/h to satisfy the requirement of conservation of energy The matrix element is + + ) (n, + 1) The The reverse probability is proportional to n1n2rather than to (nl two probabilities define the relaxation time where k,B = A , / h = E , - E, As in the case of the Raman process very many cases of the above can be found depending on how the matrix element is treated, particularly the approximations of fast or hard phonons depending on whether the elect,ronic system is retarded or the phonon is retarded The temperature dependence obviously depends on how the denominator in (69) is treated I n order to demonstrate the importance of the sum process we discuss only one case, i.e A , @ hw which indeed is most applicable as only the lowest-energy phonons are required for the sum process Even in this case, in order t o appreciat,e the beauty of the process it is necessary t o evaluabe the integral numerically, because there is a temperature window where the sum process becomes very large I n fact for a temperature of w 0.14(AJkB)the sum process is very important We refer the reader t o [20] for a numerical solution For the present purpose it is sufficient to note that, if the matrix element is taken out of the integral, for A / k , T @ , for the sum process in t,hree levels we can write whereas in the reverse approximation A / F Ti‘ = l2 w!(A,/h - w1)3 dw, 87z3e2v10 [exp @mllkBT) - 11 {exp [ ( A , - hw,)/k,T] - 1} (75) for h a , > A , and A , < hw, < hwD,the Raman rate P6b = 2n -I(%h n, - 1, n2 can be reduced to Combining (75) and ( 7 ) , + 11 x2 Ib, % ef n2>I2 (76) 453 Theory of Spin-Lattice Relaxation from which the interesting t.emperature dependence emerges as T,' = c1 s exp (-A,/k,T) (kBT/h)' where ex x3(x - ~ l , / k , T dx )~ (ex - 1) [exp (z- A,/k,T) - 11' (79) Le Naour has considered numerical evaluation of the integral and determined the criteria of exp (-AJT) or T7 dependences Thus the temperature dependence is more complicated than either a power law or a n exponential behaviour Considering the interaction (5) in the second-order perturbation theory, we examine the matrix element (c, n, - , n2 - 11 XI It, n, - 1,n,> (t, n, - 1, n z l X l Ib, n1, nz> ho, - A t First we consider the phonon part from to A , which leads to A& z P,1, = 2wv4 ~ 11 (Cl V , It> (tl V , Ib) +(tl Vl Ib) hol - A t ho, - A , x n , ~ z ~ , w z g ( 9(%) ~ z ) dm1 (811 where g(wi) is the density of phonon states a t ot If we consider hw, - At = d, - hw, - A,, hw, A , for the Raman process, we find > The temperature dependence of this process is the same as in (79) I n the case of Kramers systems, one can rewrite the above calculation with proper account of time reversal symmetry of the wave functions The result of this calculation is Ti1 -.r exp [ ( h q - A , ) / k , T ] o$(co, - ~ l , / hdw, )~ [exp (hw,/k,T) - 11 {exp [(ho), - d , )/ k , T ] - ) ~~~ for hru, l2 dEa Ea Pab = I(al VO lb>I2 , - Eb @(En Ea - Eb) [n(db) + '1 - Eh The relaxation rate is determined by which can be written as If we use an infinite square well for V,, the normalized eigenfunction corresponding to the n-th level is I(i + 2j + 11 V , li)I2= 16a2 [(2j + 1)-2 - (27' + 2i + 1)-2]2 (99) We assume that the density of states is proportional to o2substituting (99) into (98), replacinge(Ei+l - E , ) by (Ei+l - E4)2 and setting E, = i28, where k i = 4n2h2/32Ma2 the spin-lattice relaxation rat,e is given by - ( j + 2i + 1)-2]2 exp -9 - A-1 ( (2i ; + 2j + 1) ( j + 1) - The spin-lattice relaxation time of atomic hydrogen in fused silica has been interpreted in terms of the local mode oscillations, essentially T I exp ( / T )where is the local phonon mode, k,0 = ho, 466 K N SHRIVASTAVA The Collision Process The spin-lattice relaxation is studied for paramagnetic ions in insulating materials However, very often as the temperature is varied over a certain cycle, the paramagnetic ions ooze out of the crystal and form fine metallic particles on the surface of the crystal Under these circumstances, these metallic particles give rise t o a conduction electron spin resonance and their relaxation times not follow any of the previously discussed spin-lattice relaxation processes I n fact relaxation times varying as T112have been found [36, 371 for Gd3+ in CaF2 It has been a puzzle to understand these experimental data Therefore, we wish t o mention a collision process which provides a plausible interpretation of these observations We consider the spin dependent Coulomb interaction between electrons which in the case of non-degenerate statistics gives [38] for the electron-electron spin-flip process Here P ( s ) = -1 - (1 + s) e'Ei(-s) + , + + = Ph2 qi/2m*, = 2A(d 2E,)/(A Eg)(2A 3E,), d is the spin-orbit splitting, E , the gap energy, m* the effective mass of the electron, qs the screening wave number, so the state dielectric function of the crysta1,P = l/k,T, Ei( -s) the exponential integral [38, 391 To a very good approximation the relaxation rate given by (101) varies as T112and is of the order of lO-'s for N = 1015 ~ m which - ~ apparently solves the puzzle of relaxation times in CaP,: Gd3f as measured by Bierig et al [36] (Pigs and 8) -Lr01 Fig i 10 20 TNO 60 70 I2 74 I8 ';?fl/K7? 20 Fig Fig Spin-lattice relaxation time in CaF,: Gd3+ showing the T-5 process as predicted by the T -the ~ discussion in theory and T-112 predicted by (101) 8s T,= 1.1 x 10-3T-1/2 + ~ x ~ O ~ See Section The data are taken from [36] Fig Widt,h of two lines in Ag+ in zeolite measured by Textjer et al [37b] and predicted by (101) See the discussion in Section Theory of Spin-Lattice Relaxation 457 10 Conclusions - The spin-lattice relaxation rates are usually explained in terms of the temperature dependences T , T ,exp ( A J T ) ,T7,or P ! which arise from the direct process, the phonon-bottlenecked direct process, the Orbach process, the Raman process, or the Kramers Raman process, respectively I n many cases, the relaxation times are not so simple and complications appear t o arise {36,40, 411 Therefore, it is quite clear that the theories have t o be reexamined to find alternative temperature dependences of the relaxation times The present work achieves just this We have given detailed theories to predict new temperature dependences which will be useful to the experimentalists in 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