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Observability of conduction electron spin resonance (CESR) in superconductors
Solid State Communications, Vol. 38, pp. 1095 - 1097. Pergamon Press Ltd. 198 I. Printed in Great Britain.
0038-1098/81/231095-03502.00/0
OBSERVABILITY OF CONDUCTION ELECTRON SPIN RESONANCE (CESR) IN SUPERCONDUCTORS R.E. Amritkar, N. Kumar and R. Srinivasan Department of Physics, Indian Institute of Science, Bangalore - 560 012, India (Received 2 December 1980 by M. Cardona) The condition for the observability of CESR in superconducting thin films is analysed taking into account the finiteness of the flux penetration depth. We have explicitly evaluated the path-dependent phase mixing factor occurring in the expression for power absorption. The calculated line width turns out to be of the order of, or larger than, the nominal resonance frequency for the experimentally realisable choice of parameters.
CONDUCTION ELECTRON SPIN RESONANCE (CESR) has been observed in simple metals [1 ]. In particular it has been observed in aluminium in the normal state [2]. Here the smallness of the critical field makes CESR measurement in the superconducting phase very unfavourable. These normal state CESR measurements are generally consistant with the theory developed by Dyson in his classic 1955 paper [3]. The possibility of observing CESR in superconductors was considered theoretically by Kaplan [4] and Aoi and Swihart [5]. The later authors have essentially extended the treatment of Dyson to the superconducting case. The problem isbasically different from the case of normal state CESR in that the static magnetic field Ho varies over the sample thickness due to ffmiteness of the London penetration depth ~. Since in experiments [6] the film thickness L is of the order ~, the variation of the static magnetic field AH is Of the order of H0 itself. Thus some form of motional narrowing is absolutely essential in order to neutralise the otherwise large inhomogeneous line broadening. The earlier treatments have implicitly assumed that this indeed happens and, accordingly, they have replaced the static field by some average value. On the experimental side, in a recent paper Ekbote et at [6] have reported observation of pronounced CESR in thin films of some A15 compounds of niobium in the normal state. The signal, however, disappeared completely when the sample was cooled to the superconducting state at 4.2 K. The explanation of this negative result is not quite clear in the light of the following experimental conditions [7]. The film thickness (L) used is ~ 2000 A, the penetration depth is 4000 A, the estimated time for the electron to diffuse through the sample thickness Tr. = L2/2D ~ 10 -12 see. Here D = t3 V~-A is the diffusion constant, and VF and A are, respectively, the Fermi velocity and the mean free path. The spin-flip relaxation time Tt ~ 10 -s sec. The
nominal resonance frequency Wo = 2 # H o / h " 10 l° Hz. Thus, naively one would expect motional narrowing. This question was indeed discussed by Aoi and Swihart [5]. Following the general treatment of Dyson [3] they had incorporated the spatial variation of the static mag. netic field through the phase mixing factor;
x'(t') -F(x,t,x',t') They have not evaluated this expression in any detail. They do conclude generally that the ~ondition for the observability of CESR is 2/~z~t/Tt ~ 1, where AH is the total variation of the static field across the sample thickness. Hence they require L < ~. This condition is loosely related to the Riemann ~besgue theorem [8]. It is not detailed enough to relate to any specific experimental situation. This has motivated us to evaluate this phase mixing factor more explicitly. In the following treatment we shall treat the translatory motion of the conduction electron as diffusive, i.e. A "~ L. Then the spatially varying static magnetic field will be translated by the diffusive motion of the electron into an implicitly time-dependent field as seen by the conduction-electron spin. This implicitly timedependent field will be in general a stochastic variable with certain autocorrelation. We have found, however, that for any simple minded assumption about this stochastic variable, e.g. assuming it to be a gaussian process or a Kubo-Anderson Process (KAP), the phase mixing factor can be evaluated exactly and it gives pronounced motioual narrowing effect. This, taken in conjunction with the negative result of Ekbote et al. [6] indicates that the stochastic process is presumably far from being a simple one. Hence we proceed to evaluate the phase mixing factor directly.
1095
1096
Vol. 38, No. 11
OBSERVABILITY OF (CESR) IN SUPERCONDUCTORS
LetHo be parallel to the plane (YZ) of the film (a preferred geometry) [5]. The magnetic field variation along X-axis is then given by [4]
H(x) = Ho cosh(x/X) cosh (LI2X)'
L__
(2)
The diffusion of the spin-bearing quasi-particles is given by
Of(x, t) = D 02f(x' t) a---7ax ,
r . x(t)
's
exp
X
[X(U)
- ~ - + hco -- 2/a/-/[x(u)l du
x' )
)
We notice from the exponent occurring in equation (4) that one can def'me an effective Lagrangian 2
'
so that the eigenvaluesE n and eigenfunctions ~,t of the associated Hamiltonian determine the path average
as F = ~1 Z en(x)¢*(x')exp
{_i-ff E n(t--t') } •
(5)
The Lagrangian L describes the motion of a particle of mass m and an effective potential Ven (x) = -- h(w -- 2tJ-/(x)/h), 0%
2uHo h[ 2pHo I "2 Eo = --h¢o + cosh (L/2X) F ~- m cosh(L/2X)X z
(8) SubstitutingD for ih[2m we get for the imaginary part of the frequency (i.e. relaxation frequency) as
-- L/2 < x < L/2; Ix
I>
L/2.
(6)
The infinite potential barrier is mathematically equivalent to the boundary condition on the eigenfunctions, namely that ¢,, (+- L/2) = 0. Thus we have a bounded anharmonic oscillator problem. The eigenvalues determine the relaxation and hence the line width. This is readily seen if we substitute for the phase-mixing factor from our equation (5) into the equation (21) of Aoi and Swihart [5]. The normalisation factor G cancels out and
t112
= [ 2tlY'/oD . [ h cosh (L/2X)X =
1 [ cooTx = T--x 2 c o s h ~
(4)
(x, t) to (x', t').
(7)
The lowest eigenvalue of the resulting approximately unbounded harmonic oscillator is
I r
where G [= G(x, t;x', t')] is the normalization factor and is nothing but the free particle propagator from
=
cosh (x/X) cosh (L/2 X) 1 + x2/2X 2 "" Ho cosh (L/2X)"
n ( x ) = Ho
(3)
where f(x, t) is the probability density of occurrence. Thus the path dependent averaging involved in equation (1) is to be carried out with respect to the probability measure (the Wiener measure) def'med by the diffusion equation. Now we take advantage of the well known fact that the Wiener measure is formally the same as the Feynman path integral measure except that ihl2m is to be replaced b y D [9]. The path average can now be written as F = ~
we are left essentially with the time-dependent factor of the type exp {- i/h)En(t - t')}. Thus non-zero imaginary part of En should imply relaxation. In order to estimate the damping (the line width) it turns out to be sufficient to study the lowest eigenvalue. To this end we make a harmonic approximation and replace
~
(9)
where Td and TL are times required for a normal electron to diffuse through the lengths X and L respectively. Thus we expect to see a line broadening corresponding to a relaxation frequency = (1/r) + (1/Ti ). Higher eigenvalues of course give terms with larger relaxation frequencies. In point of fact the confining effect of the infinite barrier will further raise the eigenvalues and enhance the relaxation frequency. Thus, the above result is a lower bound. For the experimental values [7] of the parameters of [6], we find a line width ~ 1200 Oe. Also for the choice of parameters for aluminium as in [5], the line width turns out to be ~ 600 Oe. This certainly rules out the possibility of observing CESR except when very close to Tc when the variation of magnetic field across the sample is very small, i.e. X -+ ~. We have recalculated the line width for another choice of the effective potential which simulates the conditions of equation (6) accurately enough for L ~ X. Here the eigenvalue problem is solvable exactly. The choice is (co v;.(x)
=
-h
2p.Ho -
1
))
cos(L/2x) ×
x Ix l
'
(10)
The calculated relaxation frequency turns out to be
1
r'
,-v4
T;
(11)
Vol. 38, No. 11
OBSERVABILITY OF (CESR) IN SUPERCONDUCTORS
Thus in all cases the linewidth roughly corresponds to the reciprocal of the diffusion time T~. The proportionality factor involves some dimensionless ratios as seen from equation (9). To conclude we have found that the simple minded expectation of motional narrowing is not realised and the replacement of rio(x) by the average value is not justified. A posteriori, the stochastic field seen by the conduction electron spin may not have simple autocorrelation structure one normally assumes in the discussions of motional narrowing. We believe that our calculations explain the negative result of the experiments by Ekbote etal. [6].
Acknowledgement - One of us (REA) would like to thank NTPP (DST India) for financial assistance. REFERENCES
1.
See, e.g.J. Winter, Magnetic Resonance in Metals,
2. 3. 4. 5. 6. 7. 8. 9.
1097
Chapter 10. Oxford University Press. S. Schultz, G. Dunifer & C. Latham, Phyg Lett. F.J. Dyson, Phyg Rev. 98,349 (1955). J.I. Kaplan, Phyg Lett. AI9,266 (1965). K. Aoi&J.C. Swihart,Phys. Rev. B2, 2555 (1970). S.N. Ekbote, S.K. Gupta & A.V. Narlikar, Curr. Sci. 49,144 (1980); S.N. Ekbote & A.V. Narlikar, Mat. Re~ Bull 15,827 (1980). A.V. Narlikar (private communication). See, e.g.N.G, van Kampen, Irreversibility in Many Body Problem (Edited by K. Biel and J. Rae), p. 373. Plenum Press, New York (1972). The propagator G(x, t;x', t') of [5] is indeed the Feynman propagator. Note, however, an error in their equation (22) where ~tn occurring in the exponent should be replaced by U2.
0038-1098/81/231095-03502.00/0
OBSERVABILITY OF CONDUCTION ELECTRON SPIN RESONANCE (CESR) IN SUPERCONDUCTORS R.E. Amritkar, N. Kumar and R. Srinivasan Department of Physics, Indian Institute of Science, Bangalore - 560 012, India (Received 2 December 1980 by M. Cardona) The condition for the observability of CESR in superconducting thin films is analysed taking into account the finiteness of the flux penetration depth. We have explicitly evaluated the path-dependent phase mixing factor occurring in the expression for power absorption. The calculated line width turns out to be of the order of, or larger than, the nominal resonance frequency for the experimentally realisable choice of parameters.
CONDUCTION ELECTRON SPIN RESONANCE (CESR) has been observed in simple metals [1 ]. In particular it has been observed in aluminium in the normal state [2]. Here the smallness of the critical field makes CESR measurement in the superconducting phase very unfavourable. These normal state CESR measurements are generally consistant with the theory developed by Dyson in his classic 1955 paper [3]. The possibility of observing CESR in superconductors was considered theoretically by Kaplan [4] and Aoi and Swihart [5]. The later authors have essentially extended the treatment of Dyson to the superconducting case. The problem isbasically different from the case of normal state CESR in that the static magnetic field Ho varies over the sample thickness due to ffmiteness of the London penetration depth ~. Since in experiments [6] the film thickness L is of the order ~, the variation of the static magnetic field AH is Of the order of H0 itself. Thus some form of motional narrowing is absolutely essential in order to neutralise the otherwise large inhomogeneous line broadening. The earlier treatments have implicitly assumed that this indeed happens and, accordingly, they have replaced the static field by some average value. On the experimental side, in a recent paper Ekbote et at [6] have reported observation of pronounced CESR in thin films of some A15 compounds of niobium in the normal state. The signal, however, disappeared completely when the sample was cooled to the superconducting state at 4.2 K. The explanation of this negative result is not quite clear in the light of the following experimental conditions [7]. The film thickness (L) used is ~ 2000 A, the penetration depth is 4000 A, the estimated time for the electron to diffuse through the sample thickness Tr. = L2/2D ~ 10 -12 see. Here D = t3 V~-A is the diffusion constant, and VF and A are, respectively, the Fermi velocity and the mean free path. The spin-flip relaxation time Tt ~ 10 -s sec. The
nominal resonance frequency Wo = 2 # H o / h " 10 l° Hz. Thus, naively one would expect motional narrowing. This question was indeed discussed by Aoi and Swihart [5]. Following the general treatment of Dyson [3] they had incorporated the spatial variation of the static mag. netic field through the phase mixing factor;
x'(t') -F(x,t,x',t') They have not evaluated this expression in any detail. They do conclude generally that the ~ondition for the observability of CESR is 2/~z~t/Tt ~ 1, where AH is the total variation of the static field across the sample thickness. Hence they require L < ~. This condition is loosely related to the Riemann ~besgue theorem [8]. It is not detailed enough to relate to any specific experimental situation. This has motivated us to evaluate this phase mixing factor more explicitly. In the following treatment we shall treat the translatory motion of the conduction electron as diffusive, i.e. A "~ L. Then the spatially varying static magnetic field will be translated by the diffusive motion of the electron into an implicitly time-dependent field as seen by the conduction-electron spin. This implicitly timedependent field will be in general a stochastic variable with certain autocorrelation. We have found, however, that for any simple minded assumption about this stochastic variable, e.g. assuming it to be a gaussian process or a Kubo-Anderson Process (KAP), the phase mixing factor can be evaluated exactly and it gives pronounced motioual narrowing effect. This, taken in conjunction with the negative result of Ekbote et al. [6] indicates that the stochastic process is presumably far from being a simple one. Hence we proceed to evaluate the phase mixing factor directly.
1095
1096
Vol. 38, No. 11
OBSERVABILITY OF (CESR) IN SUPERCONDUCTORS
LetHo be parallel to the plane (YZ) of the film (a preferred geometry) [5]. The magnetic field variation along X-axis is then given by [4]
H(x) = Ho cosh(x/X) cosh (LI2X)'
L__
(2)
The diffusion of the spin-bearing quasi-particles is given by
Of(x, t) = D 02f(x' t) a---7ax ,
r . x(t)
's
exp
X
[X(U)
- ~ - + hco -- 2/a/-/[x(u)l du
x' )
)
We notice from the exponent occurring in equation (4) that one can def'me an effective Lagrangian 2
'
so that the eigenvaluesE n and eigenfunctions ~,t of the associated Hamiltonian determine the path average
as F = ~1 Z en(x)¢*(x')exp
{_i-ff E n(t--t') } •
(5)
The Lagrangian L describes the motion of a particle of mass m and an effective potential Ven (x) = -- h(w -- 2tJ-/(x)/h), 0%
2uHo h[ 2pHo I "2 Eo = --h¢o + cosh (L/2X) F ~- m cosh(L/2X)X z
(8) SubstitutingD for ih[2m we get for the imaginary part of the frequency (i.e. relaxation frequency) as
-- L/2 < x < L/2; Ix
I>
L/2.
(6)
The infinite potential barrier is mathematically equivalent to the boundary condition on the eigenfunctions, namely that ¢,, (+- L/2) = 0. Thus we have a bounded anharmonic oscillator problem. The eigenvalues determine the relaxation and hence the line width. This is readily seen if we substitute for the phase-mixing factor from our equation (5) into the equation (21) of Aoi and Swihart [5]. The normalisation factor G cancels out and
t112
= [ 2tlY'/oD . [ h cosh (L/2X)X =
1 [ cooTx = T--x 2 c o s h ~
(4)
(x, t) to (x', t').
(7)
The lowest eigenvalue of the resulting approximately unbounded harmonic oscillator is
I r
where G [= G(x, t;x', t')] is the normalization factor and is nothing but the free particle propagator from
=
cosh (x/X) cosh (L/2 X) 1 + x2/2X 2 "" Ho cosh (L/2X)"
n ( x ) = Ho
(3)
where f(x, t) is the probability density of occurrence. Thus the path dependent averaging involved in equation (1) is to be carried out with respect to the probability measure (the Wiener measure) def'med by the diffusion equation. Now we take advantage of the well known fact that the Wiener measure is formally the same as the Feynman path integral measure except that ihl2m is to be replaced b y D [9]. The path average can now be written as F = ~
we are left essentially with the time-dependent factor of the type exp {- i/h)En(t - t')}. Thus non-zero imaginary part of En should imply relaxation. In order to estimate the damping (the line width) it turns out to be sufficient to study the lowest eigenvalue. To this end we make a harmonic approximation and replace
~
(9)
where Td and TL are times required for a normal electron to diffuse through the lengths X and L respectively. Thus we expect to see a line broadening corresponding to a relaxation frequency = (1/r) + (1/Ti ). Higher eigenvalues of course give terms with larger relaxation frequencies. In point of fact the confining effect of the infinite barrier will further raise the eigenvalues and enhance the relaxation frequency. Thus, the above result is a lower bound. For the experimental values [7] of the parameters of [6], we find a line width ~ 1200 Oe. Also for the choice of parameters for aluminium as in [5], the line width turns out to be ~ 600 Oe. This certainly rules out the possibility of observing CESR except when very close to Tc when the variation of magnetic field across the sample is very small, i.e. X -+ ~. We have recalculated the line width for another choice of the effective potential which simulates the conditions of equation (6) accurately enough for L ~ X. Here the eigenvalue problem is solvable exactly. The choice is (co v;.(x)
=
-h
2p.Ho -
1
))
cos(L/2x) ×
x Ix l
'
(10)
The calculated relaxation frequency turns out to be
1
r'
,-v4
T;
(11)
Vol. 38, No. 11
OBSERVABILITY OF (CESR) IN SUPERCONDUCTORS
Thus in all cases the linewidth roughly corresponds to the reciprocal of the diffusion time T~. The proportionality factor involves some dimensionless ratios as seen from equation (9). To conclude we have found that the simple minded expectation of motional narrowing is not realised and the replacement of rio(x) by the average value is not justified. A posteriori, the stochastic field seen by the conduction electron spin may not have simple autocorrelation structure one normally assumes in the discussions of motional narrowing. We believe that our calculations explain the negative result of the experiments by Ekbote etal. [6].
Acknowledgement - One of us (REA) would like to thank NTPP (DST India) for financial assistance. REFERENCES
1.
See, e.g.J. Winter, Magnetic Resonance in Metals,
2. 3. 4. 5. 6. 7. 8. 9.
1097
Chapter 10. Oxford University Press. S. Schultz, G. Dunifer & C. Latham, Phyg Lett. F.J. Dyson, Phyg Rev. 98,349 (1955). J.I. Kaplan, Phyg Lett. AI9,266 (1965). K. Aoi&J.C. Swihart,Phys. Rev. B2, 2555 (1970). S.N. Ekbote, S.K. Gupta & A.V. Narlikar, Curr. Sci. 49,144 (1980); S.N. Ekbote & A.V. Narlikar, Mat. Re~ Bull 15,827 (1980). A.V. Narlikar (private communication). See, e.g.N.G, van Kampen, Irreversibility in Many Body Problem (Edited by K. Biel and J. Rae), p. 373. Plenum Press, New York (1972). The propagator G(x, t;x', t') of [5] is indeed the Feynman propagator. Note, however, an error in their equation (22) where ~tn occurring in the exponent should be replaced by U2.