Spin Diffusion and Relaxation in a Nonuniform Magnetic Field.

Transcription

1 Sin Diffusion and Relaxation in a Nonuniform Magnetic Field. G.P. Berman, B. M. Chernobrod, V.N. Gorshkov, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico V.I. Tsifrinovich IDS Deartment, Polytechnic University, Brooklyn, NY Abstract We consider a quasiclassical model that allows us to simulate the rocess of sin diffusion and relaxation in the resence of a highly nonuniform magnetic field. The energy of the slow relaxing sins flows to the fast relaxing sins due to the diole-diole interaction between the sins. The magnetic field gradient suresses sin diffusion and increases the overall relaxation time in the system. The results of our numerical simulations are in a good agreement with the available exerimental data.. Introduction Alication of any quantum sin device crucially deends on the relaxation rate in the sin system. Sin diffusion is recognized as one of the most imortant factors in a sin relaxation rocess (see, for examle, [-6]). The idea of sin diffusion originated from Bloembergen who exlained nuclear sin-lattice relaxation in insulating crystals []. He demonstrated that the transort of magnetization from fast relaxing sins (FRS) to slow relaxing sins (SRS) can be described as a diffusion rocess. Due to the sin diffusion, a small amount of FRS (e.g. located near the imurities) can greatly accelerate the sin-lattice relaxation in the whole sin system. A number of theoretical aroaches have been used to describe the sin diffusion. Redfield [] has found the connection between the diffusion constant and the moments of the NMR line. Yang and Honig [] alied the sin diffusion theory to electron aramagnetic system. Rhim, Pines, and Waugh [4] showed that the sin diffusion can be reversed using rf-ulses which change the sign of the diole-diole interaction. Vugmeister [5] has develoed

2 the general theory of the sin diffusion in a delute electron aramagnetic system in terms of the Zeeman sin temeratures for FRS and SRS. Tang and Waugh [6] simulated the nuclear sin diffusion in insulated crystals using classical equations of motion for the nuclear sins. Recently Budakian, Mamin, and Rugar [7] demonstrated effective maniulation with the electron sin relaxation of E - centers in silica using the high gradient of the magnetic field roduced by a ferromagnetic article in the magnetic resonance force microscoy. This effect oens the way for effective suression of the sin diffusion and relaxation, which could be extremely imortant for the sin device alications. Insired by the exeriments [7] we have suggested a comutational model, which could be alied to the simulation of the sin relaxation in a highly nonuniform magnetic field. According to the general theory of the sin diffusion in the electron sin resonance [5] the relaxation rocess deends on the relation between two arameters: the sin-lattice relaxation time for FRS T FL and the cross relaxation time T FS, which is the characteristic time of the energy transfer from FRS to SRS. T If FS FL T then FRS and SRS quickly come to the state of thermal equilibrium, and the bottleneck of the relaxation rocess in the energy transfer from FRS to the lattice. In this case the overall relaxation rate T is given by the arameter T - : F S FL T T = ( n / n ) T, where n F and n S are the concentrations of FRS and SRS. (The sin-lattice relaxation of SRS with the characteristic time T SL is ignored in [5].) In the oosite case TFS TFL, FRS quickly come to the thermal equilibrium with the lattice, and the bottleneck of the relaxation rocess is the energy transfer from SRS to FRS with the characteristic cross-relaxation time T SF : TSF = ( ns / nf ) TFS. In this case the relaxation rate T is, aroximately, equal to T SF. For sins / the value of T FS is given by T µ FS = q ns γ 4π, where q 9.6 / δ for the inhomogeneously broadened resonance line with the width δ, and q 4/ Λ for the homogeneously broadened resonance line with the width Λ of the crossrelaxation form function. Taking arameters from [7] ns 4 m =, 4 δ / γ = T,

3 γ /( π ) =.8 Hz/ T, we estimate the value of T FS : T 5 FS. s =. It follows from the exeriments [7] that the sin-lattice relaxation rate for SRS T SL, which can be revealed under the high magnetic field gradient is about.4s. It is reasonable to assume that the ratio T / T does SL FL not exceed 5, i.e. T FL s 4. In this case TFS TFL, and the bottleneck of the relaxation rocess is the energy transfer from FRS to the lattice. For this situation, we simulated the electron sin dynamics in the resence of the nonuniform magnetic field. We consider the motion of electron sins couled via the diole-diole interaction using the classical equations of motion. In the second section we describe our model and comutational algorithm, and in the third section resent the results of our numerical simulations and comarison with the exerimental data.. Descrition of the model and the comutational algorithm. We consider a quasiclassical electron sin with magnetic moment µ and negative electron gyromagnetic ratio ( γ ). The magnitude of the magnetic moment, which is equal to the Bohr s magnetron ( µ = µ B ), conserves in the rocess of the sin relaxation. The motion of a magnetic moment µ satisfies to the quasiclassical equation of motion with the relaxation term R : µ = γ [ µ B ] + R, ξ () R = [ µ µ ]. µ B Here B is the magnetic field on sin " ", which includes the uniform external magnetic field B ext, the nonuniform external magnetic field roduced, for examle, by a ferromagnetic article the diole field roduced by other sins of a samle that B ext oints in the ositive z direction. The diole field B is given by d B f, and B d ; ξ is the relaxation arameter. We assume B d ( µ n ) µ, µπ k k k = 4 k rk ()

4 where n k is the unit vector, which oints from the sin k to the sin, r k is the distance between the two sins. Below we use two conditions, which were satisfied in exeriments [7]: B, B B B + B d f f ext fz where... means the average over the sin system. Note that according to the Maxwell s equation divb f = there must be the nonuniform transversal comonent of the magnetic field B f. However because of B f B f B ext + B fz the transversal comonent of B f in the first aroximation does not influence the Larmor frequency. The average recession frequency ω in the sin system can be found aroximately as ω = γb, B = B + B, () ext fz We transfer to the system of coordinates, which rotates with the frequencyω. Ignoring the fast oscillating terms and assuming ξ we obtain the following equations of motion: dm =Ω [ m], dτ χ χ Ω = Akmkx + ξ my, Akmky ξ mx, δ + χ Ak mkz. k k k (4) Here we use the following dimensionless notations: m = µ / µ, m = dm / dτ, τ = ξ ω t, ξ = ξ / ξ, B Bfz Bfz µ cos µ θk B δ =, χ=, A, k = ξb 4 π ξa B r / a ( k ) (5) 4

5 where a is the average distance between the sins, θ k is the olar angle of the unit vector n k, ξω =. If we ut origin inside the sin system at the oint where δ = then the value of δ s can be aroximated as δ ag B = ( z / a ), G =. (6) fz ξ B z z= The arameter χ is the characteristic constant of the diole-diole interaction in our system, while the ratio G = ag/( ξb) describes the characteristic Larmor frequency difference caused by the magnetic field gradient. The aroximations leading to (4) from () maintain, as it should be, the conservation of the magnetic moment: m + m + m =. (7) x y z In the absence of relaxation ( ξ = ) the z comonent of the total magnetic moment also conserves: m z =. In exeriments [7] the values of arameters are the following: B =.6 T, ω / π =.96 GHz, a 8 nm, 4 5 χ =.. When G changes from T / m to G T / m.6 T / m, T 6.5 s 5 s. The value of χ is 4.6 / T m the dimensionless arameter G raises from.44χ to 56χ (for a = 8 nm). Thus, the Larmor frequency difference for the neighboring sins becomes greater than the diole-diole interaction constant. The comutational scheme for eq. (4) must satisfy to the conditions of conservation absence of relaxation) m Equations (4) are aroximated by the difference equations z m and (in the. Below we describe the suggested comutational algorithm. j+ j j+ j m m ( av) ( av) m ( av) + m = Ω ( m, m,... mn ) τ (8) 5

6 where m j+ m j + j ( av) k + k k = j m τ = τ τ, j - counts the time instant. Equation (8) can be written as, k =,,... N, N - number of magnetic moments in a system, j+ j ( m ) = fil( Ω) ( m) i l l (9) Here il, = xy,, z, fil ( Ω ) - are nonlinear functions of Ω. As arameters Ω deend on j m + k the solution of equations of motion was found by iterations. In our simulations, we consider the system of identical cells (see Fig.). Every cells contains N = n sins located near the oints of the cubic lattice: r = a( l i + l j + l k) +δ r. Here n is an odd number, l - are integers, ( ncell )/ l,, ( ncell )/, δ x, δy, δz- are the random numbers, which do not exceed.5a, i, j, k - are the unit vectors in the ositive x, y, and z directions. At τ = the z comonents of m take random values between.8 and with the random direction of the transversal comonents. In every cell, the central sin is the only FRS. In our simulations, we take the same value ξ = β for all FRS and the same value ξ = α for all SRS. The main (central) cell is surrounded by 44 identical auxiliary cells in order to eliminate the boundary effects, which influence the relaxation rocess. We comuted the dynamics of the sins in the central cell taking into consideration their interaction with the sins of all the cells and assuming that the corresonding sins in all the cells have the same direction. The outcome of our comutation is the function Mz( τ) = mz( τ) mz () () 6

7 We have erformed comutations for n = 5, 7,. The corresonding ratio n / n = N = 4, 4,. s f. Results of numerical simulations First, we have studied the relaxation rocess in the absence of magnetic field gradient ( G = ). We define the effective dimensionless relaxation time τ using the relation τ = [ M z ( τ) ] dτ, () which is derived from the exonential decay M z ( τ ) = ex( τ / τ). As s ξω =, the numerical value of τ is equal to the value T in seconds. For the sin system with no FRS the relaxation time τ = τ. / α. In exeriment [7] τ = 5s, and α =.88. We have simulated SL SL the relaxation rocess for various values 4 χ >, but all our figures below corresond to the exerimental value 5 χ =.. Our comutations show that the overall relaxation time τ can be described by the relation. τ = α + β / N. () Putting N n / n and transferring to the dimensional relaxation rate, we obtain from () s F SL T = T + T. () The first term in this exression describes the sin-lattice relaxation rate for SRS, and the second term F s FL T = ( n / n ) T, first derived in [5] (see Introduction), describes the effect of the sin diffusion. From the exerimental data [7] T =.6s, T SL.4s = we can estimate the value of T : T =.s. The corresonding value β / N.6. 7

8 Fig. demonstrate the deendence M z ( τ ) for various values of the ratio β / N. Fig. shows the deendence of the relaxation time τ on the ratio β / N for α = and α =. One can see the excellent agreement between the numerical data and formula (). Next, we transfer to the main objective of the aer: analysis of the influence of the magnetic field gradient on the relaxation rocess. Our simulations show that the suression of the relaxation rate deends on the single arameter K = G / χ, which is the ratio of the characteristic Larmor frequency difference to the diole-diole interaction constant. The two other results of our simulations are the following: ) The significant increase of the relaxation time τ aears in the region 4.4 K 44, 4 which corresonds to the values T / m G T / m in a good agreement with exeriments [7]. ) For K > 44, the ratio RK ( ) = τ( K)/ τ( K= ) is aroaching to the value + β /( α N). It means that the overall relaxation time τ ( K) is aroaching to the exected value τ SL. As an examle, Fig. 4 demonstrates deendence M ( τ ) for 5 values of K, and Fig. 5 shows the function RK ( ) = τ( K)/ τ( K= ). Finally, we have studied the dynamics of the satial distribution m ( r, t). We have found that in the absence of the magnetic field gradient ( K = ) the relaxation rocess sreads randomly in all directions from FRS to SRS. In the resence of the magnetic field gradient the sin diffusion rocess becomes anisotroic. Fig. 6 demonstrates the relaxation of a slice magnetic moment z z M zi for all the slices, ( n )/ i ( n )/, in the central cell. One can see that the relaxation rocess first develos in the slice containing FRS ( i = ), then it sreads to the slices i < below the central slice, then it sreads to the uer slices i >. This henomenon can be exlained as following. The sin frequency in the rotating frame is given by eq. (4). We have Ω K χ( z / a) +Ω Ω = χ A m, (4) d, d k kz k 8

9 where the first term, Ω d K, is associated with the nonuniform magnetic field, and the second term,, is associated with the diole field roduced by the other sins of the system. Within the slice the value of Ω d changes slightly between 5.6χ and 6.4χ. Thus, the value Ω is, aroximately, Ω = 6χ in the slice i =, Ω = (6 + K) χ in the slice i =+, Ω = (6 K) χ in the slice i = and so on. It is clear that for Ω Ω =Ω Ω Ω (or K > 6 ) the relaxation due to the sin diffusion develos first in the slice containing FRS ( i = ). When the magnetic moment M z in the central slice, i =, changes its direction, Mz = Mz =+, the diole contribution on the neighboring slices i = ± in our disturbed cubic lattice increases as Ω.5 χ[ + M ( τ )], i=± (5) i z The diole term Ω d within the central slice itself changes as Ω 9 χ[ + ( τ )]. (6) M z Thus, after the sin relaxation in the slice i = the frequency difference between the slices decreases for the slices i = and i =, but increases for i = and i =. As a result the sin relaxation rocess due to sin diffusion starts in the slice i =, then in the slice i =, and so on. The characteristic value K = K, which is necessary to suress the sin diffusion between the slices, can be estimated as following. We assume that at K = K the frequency difference between the slices i = and i = (after the sin relaxation in the slice i = ) remains greater than Ω = 6χ. Taking into consideration that the initial frequency difference between these slices is Kχ, the change of the frequency is 8χ for i = and.7χ for i = we obtain K 5, which is in a good agreement with our numerical simulations. 9

10 4. Conclusion We simulated the rocess of electron sin diffusion and relaxation in the resence of a nonuniform magnetic field using the quasiclassical equations of motion. The diole-diole interaction between the sins causes the energy transfer from SRS to FRS greatly increasing the relaxation rate. The Larmor frequency difference caused by the magnetic field gradient suresses the sin diffusion and recovers the long relaxation time in the sin system. The main results of our simulations are the following:. In the absence of the magnetic field gradient we roved the equation () for the rate of the sin relaxation for exeriment [7]. T and comuted the value of arameter ( ) T = n / n T.s F S FL. We have shown that the suression of sin diffusion in the resence of the magnetic field gradient deends on a single arameter, which is the ratio of the characteristic Larmor frequency difference to the diole-diole interaction constant. We have demonstrated a good agreement between the numerical simulations and exeriment [7] on the sin diffusion suression.. We have shown that the sin diffusion becomes highly anisotroic in the resence of the magnetic field gradient. The relaxation rocess first develos in the transversal slice containing FRS, then it sreads in the direction of the smaller magnetic field, and, finally, in the region of higher magnetic field. Acknowledgement We thank Dan Rugar for useful discussions. This work was suorted by the National Security Agency (NSA) and Advanced Research and Develoment Activity (ARDA) under Army Research Office (ARO) contract # 77.

11 References. N. Bloembergen, Physica (Utrecht), 5, 86 (949).. A.G. Redfield, Phys. Rev., 6, 5 (959).. G. Yang and A. Honig, Phys. Rev., 68, 7 (968). 4. W.K. Rhim, A. Pines, and J.S. Waugh, Phys. Rev. B,, 684 (97). 5. B.E. Vugmeister, Phys. Stat. Sol. (b), 9, 7 (978). 6. C. Tang and J.S. Waugh, Phys. Rev. B, 45, 748 (99). 7. R. Budakian, H.J. Mamin, and D. Rugar, Phys. Rev. Lett., 9, 75 (4). Fig. cations Fig. A attern of the x y lane containing FRS for n = 7. The values of x and y are given in units of a. FRS. Fig.. Sin relaxation M z ( τ ) for the various values of the ratio β / N and α =. Curves -7: β / N =,,,, 4, 5, 6. Fig.. The overall relaxation time τ as a function of ratio β / N : -α =, -α = received from the numerical calculations, solid line corresonds to formula ().. Dots are Fig.4. The relaxation M z ( τ ) for various values of K : -5 - K =, 5,,, 5. Curve 6 is the relaxation in the sin system with no FRS (and K = ). The values of other arameters: α =, n = 7, β = 7 ( β / N = 4). Fig.5. Deendence R( K ) on K. at α =, n = 7. Curves, - β = 7, 686 ( β / N = 4, ). Fig.6. Relaxation of the slice magnetic moment M zi ( τ ) in the resence of the magnetic field gradient: a - for the central slice, which contain FRS ( i = ), and for the slices below the central slice; b -for the slices above the central slice. The values of arameters: n = 7, α =, β / N = 4, K =.

12 y x Fig.. M z Fig.. τ

13 τ β / Ν Fig.. M z τ Fig. 4.

14 G, T / m 4 5 R K 44 Fig. 5. M zi i = a τ.8. Fig. 6a. 4

15 M zi - Number of slice - - b grad(b f z ) -.4 τ.8. Fig. 6b. 5