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Quantum diffusion and depolarization of muons in superconductors
P H YSI CS LETTERS A
Physics Letters A 159 (1991) 289—294 North-Holland
Quantum diffusion and depolarization of muons in superconductors Yu. Kagan and N.y. Prokof’ev IV. Kurchatov Inslitute ofAto,nic Energy, 123 /82 Moscow, USSR Received 15 July 1991; accepted for publication 2 August 1991 Communicated by V.M. Agranovich
A theoretical analysis of the muon localization and delocalization in a superconductor with defects is presented. It is shown that suppression of inelastic scattering on the normal electron excitations results in a complete muon localization in a crystal volume or in the appearance (at low defect content) of free band motion regions. The depolarization rate has a remarkable nonmonotonic temperature dependence below T~.The exponential increase of the depolarization rate starts with a significant shift from T~to the low-temperature region and theexponent has an unusual dependence on the muon—defect interaction potential.
1. Recent study [1,2] of the muon depolarization process in superconducting Al was a great stimulus for us for a more elaborate analysis of quantum diffusion in a superconductor (SC) with defects. In connection with the muon depolarization problem it was first discussed by the authors in ref. [3]. In a doped metal the tunneling diffusion kinetics and, consequently, the general picture of the particle localization and delocalization changes drastically upon the SC transition. The study of the l.t~-muon depolarization in a superconductor provides a unique method for revealing all the basic aspects of the phenomena. The ~pin relaxation in a diamagnetic metal is due to the random local hyperfine interactions between the muon and surrounding nuclear magnetic moments. During the particle motion the random fields are averaged, and the temporary decline of the polarization proves to be dependent on the particle lifetime in the unit cell, and, hence, on the diffusion coefficient. The hyperfine interaction is charactertzed by the second moment of the local field distribution, ~ A typical value of 5 is ô~5x 106 K. In a perfect crystal the width of the coherent band, A, exceeds ô by many orders of magnitude. Therefore, almost no depolarization occurs during the free bandlike motion. Actually, the depolarization can take place only in ~,
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regions of dynamic or static destruction of the band. The former case implies a muon—electron interaclion in the metal, the latter case implies an interaction between the muon and crystal defects. The inhomogeneity of the problem, when the static energy disorder prevails, requires a solution of the kinetic equation taking into account both the local diffusion and local depolarization of the particles. This is most clearly seen in the case of a superconductor, when due to the freezing up of the normal electron cxcitations the interaction with static defects turns out to be very strong, even if in the normal state its role is restricted. 2. In the regions where bandlike motion is suppressed the transition between two equivalent wells is of incoherent character, and site representation is the most adequate solution to the problem. Due to the small cä value the particle diffusion is independent of the hyperfine interactions, and the spin relaxation process can be considered with the given ~t~muon motion. Thus, the kinetic equation for the spin polarization can be written in the form ~(R t) a +L(s) = [Aw(R), s(R, t)] (1) ‘
—
.
Here we introduce the local spin density, s(R, t), and the local magnetic field, ~w(R), at the muon site, R. 289
Volume 159. number 4,5
PHYSICS LETTERS A
The exact form of the lattice diffusion operator is L(s(R)) =
>
[WI~R±gS(R)
4~+gRs(R+g)1
—
J
(2) The structure of this equation has a clear physical meaning: the change of the spin density in site R is connected, on the one hand, with the particle jump to (from) the given site and, on the other hand, with a spin rotation under the local magnetic field, The muon polarization is defined as the quantity averaged over the sample volume. P(i)= ~s(R,t).
(3)
14 October 1991
For short times 1 z from eqs. (4) and (5) we get the well-known “motional narrowing” effect. Therefore, the kinetic equation (4) turns out to be valid with reasonable accuracy for an arbitrary relation between 1, ô and r. Under zero external magnetic field the expression for the depolarization rate in the r.h.s. of eq. (4) can be given as /
T(R.t)=— Jdt’G(R,R.t’)
a2In[g(t’)] at’2
(7)
R
It is conventional to consider ji~-muonsbeing stopped at an arbitrary point of the crystal at an initial moment of time. Therefore, the problem is actually reduced to the determination of the P(t) value being the integral over the volume with the initial conditions(R, 0)=s 0=const. Obviously, in this case averaging over the random configurations of local fields and defects takes place. We substitute the formal solution of eq. (1) into the r.h.s. and average it over the random field distribution assuming no correlations between different sites, <&o~(R)~w~(R’)> =~ôRR’~(~’ and <&o> = 0. Then, the following simple approximation can be derived [3], ~-+L(s)=—T(R.t)s.
(4)
In the case toofthe a strong magnetic field, HIIz, transverse initial external polarization s 0~~x, one has to consider only thewrite random field the z-direction and down eq.fluctuations (4) for thealong s,, component in the rotational frame of coordinate. Then, T(R,
ô2
1)
J
dt’ G(R, R,
t’).
(5)
where g(I) is the static Kubo—Toyabe function [5], g(t)= 4 + 4(1 _r5212) exp( _~52l2/2) (8) These equations are exact for both immobile and rapidly diffusing (ót<< I) particles. However, in the .
case of slow motion (5r>> 1) they fail to reproduce the correct 4 exp( 4t/t) decay of the polarization [6]. Making use of eq. (4) with (5) and (7) we can study the depolarization process in inhomogeneous media. To proceed we need to know the transition probability between neighbouring wells, W, as a function of the static and dynamic level shifts. The general —
expression for W has the form [7] j~Q W(~.T)=2 2+Q2 ~ 2’ >< e~~
Here
~
1(1 + 2b) (9) is the energy level shift:
2=2ithT (10) is the damping frequency of the phase correlations
0
The function G(R, R, t’) defines the probability of finding a particle at site R at time t’ if it was there at t’=O. For long times this function gives the par-
between neighbouring sites due to the interaction with electrons; 1, (11) ~ ( T) =j ‘Iwo!
tide lifetime in the well [4], T(R)=
J
dt’ G(R, R,
i’).
(6)
is the tunneling amplitude renormalized by the electron polaron effect (the initial amplitude, J, takes
0
into account both the usual phonon polaron effect 290
Volume l59,number4,5
PHYSICSLETTERSA
l4October 199!
and the effect of barrier preparation [8]); m/w0 is the quasiclassical time of the particle passage under the barrier (w0 is of the same order as the zero-point vibration frequency of the muon in the well); y is the Euler constant; F is the gamma function. The dimensionless coupling b describes the interaction between the muon and electrons. In a typical case b~4, although there are no special reasons for small b. Eq. (9) holds true provided the parameters ~or Q remain large as compared with the inverse particle lifetime in the well, P —‘ 12 ~)max>> ~
In a perfect crystal the muon depolarization is only due to dynamic destruction of the band, Q5r>> 1. From eq. (9) with ~=0 one can easily deduce the homogeneous diffusion rate. Upon the SC transition the diffusion coefficient, D, as defined by eq. (15), exponentially increases with decreasing temperature, and the depolarization rate goes to zero. Thus, the quasilocalization of particles in the normal state due to the muon—electron interaction is removed in the SC state. Of course, to observe this picture the muon bandwidth mustbe very narrow in order to get a reasonable depolarization rate at T>~T~. If in the regime of dynamic destruction of the band
In the opposite limit the particle motion is of band character. As we have already mentioned, in this case the spin polarization remains essentially undamped.
the muon diffusion remains very rapid, then the only possibility for the depolarization process is connected with the regions close to the defects where static level shifts are large. In a sample with low defed concentration, n, we introduce a trapping region of radius RT. This radius is given by
.
3. The appearance of the energy gap, A. ( T), in the electron spectrum in a SC metal has two important effects. The damping of phase correlations due to the interaction with normal excitations drops exponentially with decreasing temperature. For the SC state one should replace Q with [9,10] 4itbT I + exp(A / T)
=
(13)
‘
~“(R-~-)I = T, (17) where ‘11(R) is the interaction between the muon and the defect. At low temperature RT>> a, and long-range interactions are the most important. In a metal there are with an oscilR~ tail. two The basic first long-range results frominteractions the Kohn—Friedel
On the other hand, with A~larger than the coherent bandwidth, A, the limiting value ofA0 ( T) is achieved,
lations of the electron density. The second one is connected with the elastic strain field. The attraction
/ eA \“ A~.=J(~—~-)
potential which is necessary for trapping of the former case arises automatically and in the latter case it is a consequence of the alternating-sign depen-
(14)
-
With the energy transfer to the electron subsystem transition probability is close to its value in the normal state, eq. (9). With ~< T< 7’. provided the inequality (12) holds true the transition probability is given as 2Q WRR+g 2 ~2± Q2 (15) (A~) c~>2A~the
In the intermediate region T<<~<2A the specific
/
(A~)2Q, ~2
—
—
,,
i/2
5))
where a—~l.In a normal metal D(RT)’-~T~~2~° [12,13] and the corresponding K( T) dependence has the form ‘
)
19 ‘
In the SC state the picture is changed drastically. With T/T~—+0in the entire region of large level shifts, ~> A, where A 2zA~ (z is the coordination num-
~/T exp ( ~/ T)
ItT \l6i~(1+i~/2A
,
K~T ~T_(l~/3.2h)
s,
expression arises from the singular electron density of states near the gap [111 W= 2
dence of the strain field as a function of the direction in a crystal. The fraction ofuntrapped muons is given by the well-known relation ~ e”. K=4iraRTnD(RT) (18)
(16)
ber), the particle motion slows down exponentially. Thatisanobviousresultofeqs. (15) and (16). We introduce the radius of this region as 291
Volume 159. number 4,5
~(R) =zl
.
PHYSICS LETTERS A
(20)
In a typical case 4~> RT. Suppose that the impurity concentration is low enough and the regions of radius R~do not overlap. Then there exists a nonzero fraction of the crystal volume where bandlike muon motion is conserved. Note that the band particles have no chance to get into the defect regions of radius R~because such a transition involves inelastic scattering on the normal electron excitations. Muons stopped in the regions (R ~(RT).Actually, for the energy level shift we have ~—aIV1/(RT)~,while Q and Tare close in magnitude (see eq. (10)). In this case immediately below T~an increase in the diffusion rate at radius R~(see eq. (15)) and, consequently. in the trapping rate, eq. (18). takes place. At slightly lower temperature the width Q. becomes smaller than ~(RT). The diffusion rate increase is replaced with the diffusion decrease at the trapping radius. Since that moment the trapping rate goes through a maximum and decreases exponentially. Such a behaviour reflects the interplay between inhomogeneous and thermal broadening of the energy levels in eq. (1 5). Actually, the trapping picture loses its meaning with relatively small deviation of T from T~.At lower temperature the depolarization rate increases once again, this time is due to the muon quasilocalization in the region between the trapping, RT, and defect,
14 October 1991
distance where the inverse particle lifetime in the well is close to the hyperfine interaction (5r(R~) I ). The position of the depolarization rate minimum can be estimated from the condition R~~RT. ‘S.’
To solve the problem quantitatively we make use of the general equation (4). Because we are inter-
ested in large distances R>>a the diffusion operator (2) can be written in the differential form as
a L=
—
div J.
.1,, =
—
D,~13
+ 1’!,S.
(21)
with local diffusion tensor I
D,!JJ(R) =
~ g,~g~1 ~ RR+g
and “hydrodynamic” velocity D (R) ae(R) l’!,(R) = —
In the region R> RT (see (17)) we can neglect the “hydrodynamic” flow. On the other hand, we can use a simple representation. 2r(R) (22) T(R, 1) ~/3ó for studying the depolarization during the process of muon motion. Here /3=~2 for the zero field and /3= I for the transverse external field. Here we assume spherical symmetry of the prob1cm. Then, in the quasistatistical approximation eq. (4) can be written as .
a
a R
2
R2D(R)
s(R) =/3c~2t(R)s(R).
(23)
The muon polarization is defined by the same law (18) with the effective. Keff( 1’). value I .
,
K~ff=4mn/Jö
I
y(R)s(R)RdR.
(24)
With decreasing R eq. (23) describes a continuous slowing down of the particles along with the increasing depolarization rate. The radius R~is just the distance where the muon polarization effectively damps. s(R) is negligibly small at R~R~<
R 1, radii. To describe the evolution of the particle localization as a function of temperature we have to introduce the radius R~(T) which is defined as the
292
the solution. At R~<
Volume 159, number 4,5
PHYSICS LETTERS A
Let the impurity potential tail obey the power law and with ~(R)=~(a/RY. ~> Q,~we have Then, according to eq. (15) r( R)
~0
(aIR) 2w a2 4i(R)’
D(R) = ~za2W(R)
(25)
The solution of eq. (23) in this case is found to be 2K,, s(R) =c(z/rn)”” Here
12 (z)
.
(26)
‘
c
R~ a[~,/7~oi0(rn—1)/rn] (m— 1)/2
(27)
Substituting the solution into eq. (24) we arrive at K~11=~/~5X4ItC1 R~an,
.f’(l—rn/2)
(28)
It is noteworthy that K~1~ depends on the particle lifetime in the well only via the radius R~(T)which increases with decreasing temperature. Comparing eq. (25) with eq. (15) we have TO
3fl
(~
2zA~)’~]}_I
(30)
teresting that the temperature T~is appreciably lower than T~ Below T 4 a distinct two-component decomposition of the muon polarization appears. With the
count static depolarization in the regions of volume V( R~)is defined as
2 ~+ 2 v— 1
and K,,I/,(z) is the McDonald function. The explicit value ofR,, is
,,
11ö
the quasilocalization of a muon takes place in the entire defect region and the radius R,, acquires its limiting value RA. Here all muons in the defect region are subjected to a static depolarization, g(t). It is in-
P(1)
c1 =(rn/2)’
2itb(2v+ l)TA
Ti~As1ln[
typical ratio T~/c5.-.l0~and A~/~0”~ iO~we obtain muon spin relaxation function taking into adtheThe estimate T,,=0.25T~ (t’=3).
2”~’”i),
z=rn(R~/R) 2 (rn/2 ) !!l/2 C= T( rn / 2)
14 October 1991
‘-~-:t~~ ~
which yields the temperature dependence of Keff as / 2A 2R ~..~exP(~( 5 “ Keiç~Q~ 2 I )T) (29) 2~’-I)
The exponential increase of K~ffwith Tis much slower than expected from the law exp ( 4~/ T). We consider this as a specific feature of the depolarization mechanism described. Note that the law (29) is independent of the impurity potential sign. For the Kohn— Friedel potential v=3 and for the strain field v=4. Considering the large distances from the defect R~we argue that the asymptotic behaviour of the former potential governs the temperature dependence of Kei~. Below the temperature
[1 —nV(R~)] exp( —K~ 11t)
+nV(R~)g(1).
(31)
With T< TA one has to replace n V(R~) with nV(RA)~ ~m, and ~ with the exponentially decreasing value of the transition probability between bandlike and localized states, Kei~=Keii(R~i) exP[_As(~
—
After that, eq. (31) explicitly describes the decomposition of muons in two independent components. The fraction of the undamped muons is that of the band motion volume, l—~ItR~n=l—x/x~1, where
3/w.
(32)
x4= (3/4m)(A/~~o) With the defect concentration x>xA overlapping of the spheres of radius R4 takes place. Obviously, in this case the above analysis remains valid with the replacement RA R. a ( 4itx/ 3) I where R~ is the average distance between impurities. Consequently, TA is replaced with —~
~,
ç rsItZ(2P+l)bT(As)2 Ts~=Astln[ 3/3I/2~5~2 \(2P
x(~)
I
.
(33)
Now, with T< T~the fraction nV(R~) I and the 293
Volume (59, number 4.5
PHYSICS LETTERS A
14 October 1991
first term in the r.h.s. of eq. (31 ) is equal to zero so that all muons are subjected to the static depolarization, Fig. I presents the results of the numerical calculation of the effective depolarization rate from eq. (31). To minimize the number of parameters we restrict ourselves to the typical parameters for the interaction potential due to the Kohn—Friedel charge oscillations in a metal, ~= 1000 K, v= 3. For the isotropic attractive potential the coefficient a in eq. (18) equals to a=1( 1 + 1 /~)=f’( ~) 0.89. We
experiment clearly demonstrated that the exponential increase of the depolarization rate was shifted significantly to the low-temperature region from ‘I’. in agreement with the T, estimate. The fitted energy gap ( 1.0 K) [2] for the exponential increase ot’
take into account the oscillating nature of the potential by assuming a=rO.5, that is, only half of the transitions are effective for trapping (with ~>0). Using the experimental data [2]. we get b~0.3from the known temperature dependence of the trapping rate in the normal state AL 114] K( 1’)—?’ -°~ and expression (19). After that the only free parameter (for a given defect content) is J~. Recent experimental results [2] for muon spin relaxation in Li-doped aluminium revealed a picture
vanadium (7~ 5 K).
which is in qualitative agreement with the picture of localization and delocalization presented here. The
A
Depolarization rate
(T)
0.3
025
~SC 0.2
normal 0.16
•
\
-
I
0.01
0.02
0.05
I
0.1
0.2
I
I
I
0.5
1
/
C
Fig. I. Depolarization rate in a superconductor with defects (the defect content 2K). is 75 ppm and the tunneling amplitude J~ is l.7xl0
294
We wish to thank E. Karlsson for stimulating discussions and detailed information about the expcrimental studies in the field and some preliminary unpublished data.
References
[110.
Hartmann. E. Karisson, S. Harris, R. Wàppling and TO. Niinikoski, Phys. Lett.A 142 (1989) 504. [2) R.F. Kiefi, R. Kadono, S.R. Krcitzman, Q. Li. T. Niz, TM. Riscman, H. Zhou, R. Wäppling. S. Harris. 0. Hartmann. E. Karisson, R. Hempelmann, D. Richier, TO. Niinikoski, L.P. Lee. G.M. Luke, B. Stcrnlieb and E.J. Ansaldo, I-hp. mt. 64 (1990) 737. [3] Yu. Kagan and N.V. Prokofev, preprint IAE-446l/9. 1987.
[8] Yu. Kagan and MI. Klingcr. Zh. Eksp. Tcor. Phys. 70 ((976)255 [Soy. Phys.JETP43 (1976) 32). [9] Al. Morosoy. Zh. Eksp. Teor. Phys. 77 (1979)1471 [Soy. Phys. JETP 50 (1979) 738]. 110] J.L. Black and P. Fulde, Phys. Rev. LetI. 43 (1979) 453. [11] Yu. Kagan and N.V. Prokofev. Zh. Eksp. Teor. Phys. 97
~
X
0.05
0
(42 K). This nontrivial fact is in obvious correlation with the law (29). For the case v = 3 we get the effective activation energy ~zi,. Ii is worth to note here that an analogous picture can be observed in the superconducting state of pure
[4] T. McMullenandE.Zaremba,Phys. Rev. B 18(1978) 3026. [5] R. Kubo and T. Toyabe, in: Magnetic resonance and relaxation. cd. R. Blinc (North-Holland, Amsterdam. 1967). [6[Y.J. Ejemura. R.S. Hayano, J. Imazato. N. Nishida and T. Yamazaki, Solid State Commun.31 (1979) 731. 171 Yu. Kagan and NV. Prokofev, Zh. Eksp. Teor. Phys. 90 ((986) 2176 [Soy. Phys.JETP63 (1986) 12761.
~
01
4(T) was considerably lower than the BCS value
(1990)1698 [Soy. Phys. JETP 70 (1990) 990). [12]J.Kondo,PhysicaB+C 126 (1984) 377. [13]K. Yamada, Prog. Theor. Phys. 72 (1984) 195. [14]0. t-Iartmann, F. Karlsson, E. Wàckelg~rd,R. Wappling, D. Richter. R. Hempelmann and T.O. Niinikoski. Phys. Rev. B37(1988)4425.
Physics Letters A 159 (1991) 289—294 North-Holland
Quantum diffusion and depolarization of muons in superconductors Yu. Kagan and N.y. Prokof’ev IV. Kurchatov Inslitute ofAto,nic Energy, 123 /82 Moscow, USSR Received 15 July 1991; accepted for publication 2 August 1991 Communicated by V.M. Agranovich
A theoretical analysis of the muon localization and delocalization in a superconductor with defects is presented. It is shown that suppression of inelastic scattering on the normal electron excitations results in a complete muon localization in a crystal volume or in the appearance (at low defect content) of free band motion regions. The depolarization rate has a remarkable nonmonotonic temperature dependence below T~.The exponential increase of the depolarization rate starts with a significant shift from T~to the low-temperature region and theexponent has an unusual dependence on the muon—defect interaction potential.
1. Recent study [1,2] of the muon depolarization process in superconducting Al was a great stimulus for us for a more elaborate analysis of quantum diffusion in a superconductor (SC) with defects. In connection with the muon depolarization problem it was first discussed by the authors in ref. [3]. In a doped metal the tunneling diffusion kinetics and, consequently, the general picture of the particle localization and delocalization changes drastically upon the SC transition. The study of the l.t~-muon depolarization in a superconductor provides a unique method for revealing all the basic aspects of the phenomena. The ~pin relaxation in a diamagnetic metal is due to the random local hyperfine interactions between the muon and surrounding nuclear magnetic moments. During the particle motion the random fields are averaged, and the temporary decline of the polarization proves to be dependent on the particle lifetime in the unit cell, and, hence, on the diffusion coefficient. The hyperfine interaction is charactertzed by the second moment of the local field distribution, ~ A typical value of 5 is ô~5x 106 K. In a perfect crystal the width of the coherent band, A, exceeds ô by many orders of magnitude. Therefore, almost no depolarization occurs during the free bandlike motion. Actually, the depolarization can take place only in ~,
Elsevier Science Publishers B.V.
regions of dynamic or static destruction of the band. The former case implies a muon—electron interaclion in the metal, the latter case implies an interaction between the muon and crystal defects. The inhomogeneity of the problem, when the static energy disorder prevails, requires a solution of the kinetic equation taking into account both the local diffusion and local depolarization of the particles. This is most clearly seen in the case of a superconductor, when due to the freezing up of the normal electron cxcitations the interaction with static defects turns out to be very strong, even if in the normal state its role is restricted. 2. In the regions where bandlike motion is suppressed the transition between two equivalent wells is of incoherent character, and site representation is the most adequate solution to the problem. Due to the small cä value the particle diffusion is independent of the hyperfine interactions, and the spin relaxation process can be considered with the given ~t~muon motion. Thus, the kinetic equation for the spin polarization can be written in the form ~(R t) a +L(s) = [Aw(R), s(R, t)] (1) ‘
—
.
Here we introduce the local spin density, s(R, t), and the local magnetic field, ~w(R), at the muon site, R. 289
Volume 159. number 4,5
PHYSICS LETTERS A
The exact form of the lattice diffusion operator is L(s(R)) =
>
[WI~R±gS(R)
4~+gRs(R+g)1
—
J
(2) The structure of this equation has a clear physical meaning: the change of the spin density in site R is connected, on the one hand, with the particle jump to (from) the given site and, on the other hand, with a spin rotation under the local magnetic field, The muon polarization is defined as the quantity averaged over the sample volume. P(i)= ~s(R,t).
(3)
14 October 1991
For short times 1
T(R.t)=— Jdt’G(R,R.t’)
a2In[g(t’)] at’2
(7)
R
It is conventional to consider ji~-muonsbeing stopped at an arbitrary point of the crystal at an initial moment of time. Therefore, the problem is actually reduced to the determination of the P(t) value being the integral over the volume with the initial conditions(R, 0)=s 0=const. Obviously, in this case averaging over the random configurations of local fields and defects takes place. We substitute the formal solution of eq. (1) into the r.h.s. and average it over the random field distribution assuming no correlations between different sites, <&o~(R)~w~(R’)> =~ôRR’~(~’ and <&o> = 0. Then, the following simple approximation can be derived [3], ~-+L(s)=—T(R.t)s.
(4)
In the case toofthe a strong magnetic field, HIIz, transverse initial external polarization s 0~~x, one has to consider only thewrite random field the z-direction and down eq.fluctuations (4) for thealong s,, component in the rotational frame of coordinate. Then, T(R,
ô2
1)
J
dt’ G(R, R,
t’).
(5)
where g(I) is the static Kubo—Toyabe function [5], g(t)= 4 + 4(1 _r5212) exp( _~52l2/2) (8) These equations are exact for both immobile and rapidly diffusing (ót<< I) particles. However, in the .
case of slow motion (5r>> 1) they fail to reproduce the correct 4 exp( 4t/t) decay of the polarization [6]. Making use of eq. (4) with (5) and (7) we can study the depolarization process in inhomogeneous media. To proceed we need to know the transition probability between neighbouring wells, W, as a function of the static and dynamic level shifts. The general —
expression for W has the form [7] j~Q W(~.T)=2 2+Q2 ~ 2’ >< e~~
Here
~
1(1 + 2b) (9) is the energy level shift:
2=2ithT (10) is the damping frequency of the phase correlations
0
The function G(R, R, t’) defines the probability of finding a particle at site R at time t’ if it was there at t’=O. For long times this function gives the par-
between neighbouring sites due to the interaction with electrons; 1, (11) ~ ( T) =j ‘Iwo!
tide lifetime in the well [4], T(R)=
J
dt’ G(R, R,
i’).
(6)
is the tunneling amplitude renormalized by the electron polaron effect (the initial amplitude, J, takes
0
into account both the usual phonon polaron effect 290
Volume l59,number4,5
PHYSICSLETTERSA
l4October 199!
and the effect of barrier preparation [8]); m/w0 is the quasiclassical time of the particle passage under the barrier (w0 is of the same order as the zero-point vibration frequency of the muon in the well); y is the Euler constant; F is the gamma function. The dimensionless coupling b describes the interaction between the muon and electrons. In a typical case b~4, although there are no special reasons for small b. Eq. (9) holds true provided the parameters ~or Q remain large as compared with the inverse particle lifetime in the well, P —‘ 12 ~)max>> ~
In a perfect crystal the muon depolarization is only due to dynamic destruction of the band, Q5r>> 1. From eq. (9) with ~=0 one can easily deduce the homogeneous diffusion rate. Upon the SC transition the diffusion coefficient, D, as defined by eq. (15), exponentially increases with decreasing temperature, and the depolarization rate goes to zero. Thus, the quasilocalization of particles in the normal state due to the muon—electron interaction is removed in the SC state. Of course, to observe this picture the muon bandwidth mustbe very narrow in order to get a reasonable depolarization rate at T>~T~. If in the regime of dynamic destruction of the band
In the opposite limit the particle motion is of band character. As we have already mentioned, in this case the spin polarization remains essentially undamped.
the muon diffusion remains very rapid, then the only possibility for the depolarization process is connected with the regions close to the defects where static level shifts are large. In a sample with low defed concentration, n, we introduce a trapping region of radius RT. This radius is given by
.
3. The appearance of the energy gap, A. ( T), in the electron spectrum in a SC metal has two important effects. The damping of phase correlations due to the interaction with normal excitations drops exponentially with decreasing temperature. For the SC state one should replace Q with [9,10] 4itbT I + exp(A / T)
=
(13)
‘
~“(R-~-)I = T, (17) where ‘11(R) is the interaction between the muon and the defect. At low temperature RT>> a, and long-range interactions are the most important. In a metal there are with an oscilR~ tail. two The basic first long-range results frominteractions the Kohn—Friedel
On the other hand, with A~larger than the coherent bandwidth, A, the limiting value ofA0 ( T) is achieved,
lations of the electron density. The second one is connected with the elastic strain field. The attraction
/ eA \“ A~.=J(~—~-)
potential which is necessary for trapping of the former case arises automatically and in the latter case it is a consequence of the alternating-sign depen-
(14)
-
With the energy transfer to the electron subsystem transition probability is close to its value in the normal state, eq. (9). With ~< T< 7’. provided the inequality (12) holds true the transition probability is given as 2Q WRR+g 2 ~2± Q2 (15) (A~) c~>2A~the
In the intermediate region T<<~<2A the specific
/
(A~)2Q, ~2
—
—
,,
i/2
5))
where a—~l.In a normal metal D(RT)’-~T~~2~° [12,13] and the corresponding K( T) dependence has the form ‘
)
19 ‘
In the SC state the picture is changed drastically. With T/T~—+0in the entire region of large level shifts, ~> A, where A 2zA~ (z is the coordination num-
~/T exp ( ~/ T)
ItT \l6i~(1+i~/2A
,
K~T ~T_(l~/3.2h)
s,
expression arises from the singular electron density of states near the gap [111 W= 2
dence of the strain field as a function of the direction in a crystal. The fraction ofuntrapped muons is given by the well-known relation ~ e”. K=4iraRTnD(RT) (18)
(16)
ber), the particle motion slows down exponentially. Thatisanobviousresultofeqs. (15) and (16). We introduce the radius of this region as 291
Volume 159. number 4,5
~(R) =zl
.
PHYSICS LETTERS A
(20)
In a typical case 4~
14 October 1991
distance where the inverse particle lifetime in the well is close to the hyperfine interaction (5r(R~) I ). The position of the depolarization rate minimum can be estimated from the condition R~~RT. ‘S.’
To solve the problem quantitatively we make use of the general equation (4). Because we are inter-
ested in large distances R>>a the diffusion operator (2) can be written in the differential form as
a L=
—
div J.
.1,, =
—
D,~13
+ 1’!,S.
(21)
with local diffusion tensor I
D,!JJ(R) =
~ g,~g~1 ~ RR+g
and “hydrodynamic” velocity D (R) ae(R) l’!,(R) = —
In the region R> RT (see (17)) we can neglect the “hydrodynamic” flow. On the other hand, we can use a simple representation. 2r(R) (22) T(R, 1) ~/3ó for studying the depolarization during the process of muon motion. Here /3=~2 for the zero field and /3= I for the transverse external field. Here we assume spherical symmetry of the prob1cm. Then, in the quasistatistical approximation eq. (4) can be written as .
a
a R
2
R2D(R)
s(R) =/3c~2t(R)s(R).
(23)
The muon polarization is defined by the same law (18) with the effective. Keff( 1’). value I .
,
K~ff=4mn/Jö
I
y(R)s(R)RdR.
(24)
With decreasing R eq. (23) describes a continuous slowing down of the particles along with the increasing depolarization rate. The radius R~is just the distance where the muon polarization effectively damps. s(R) is negligibly small at R~R~<
R 1, radii. To describe the evolution of the particle localization as a function of temperature we have to introduce the radius R~(T) which is defined as the
292
the solution. At R~<
Volume 159, number 4,5
PHYSICS LETTERS A
Let the impurity potential tail obey the power law and with ~(R)=~(a/RY. ~> Q,~we have Then, according to eq. (15) r( R)
~0
(aIR) 2w a2 4i(R)’
D(R) = ~za2W(R)
(25)
The solution of eq. (23) in this case is found to be 2K,, s(R) =c(z/rn)”” Here
12 (z)
.
(26)
‘
c
R~ a[~,/7~oi0(rn—1)/rn] (m— 1)/2
(27)
Substituting the solution into eq. (24) we arrive at K~11=~/~5X4ItC1 R~an,
.f’(l—rn/2)
(28)
It is noteworthy that K~1~ depends on the particle lifetime in the well only via the radius R~(T)which increases with decreasing temperature. Comparing eq. (25) with eq. (15) we have TO
3fl
(~
2zA~)’~]}_I
(30)
teresting that the temperature T~is appreciably lower than T~ Below T 4 a distinct two-component decomposition of the muon polarization appears. With the
count static depolarization in the regions of volume V( R~)is defined as
2 ~+ 2 v— 1
and K,,I/,(z) is the McDonald function. The explicit value ofR,, is
,,
11ö
the quasilocalization of a muon takes place in the entire defect region and the radius R,, acquires its limiting value RA. Here all muons in the defect region are subjected to a static depolarization, g(t). It is in-
P(1)
c1 =(rn/2)’
2itb(2v+ l)TA
Ti~As1ln[
typical ratio T~/c5.-.l0~and A~/~0”~ iO~we obtain muon spin relaxation function taking into adtheThe estimate T,,=0.25T~ (t’=3).
2”~’”i),
z=rn(R~/R) 2 (rn/2 ) !!l/2 C= T( rn / 2)
14 October 1991
‘-~-:t~~ ~
which yields the temperature dependence of Keff as / 2A 2R ~..~exP(~( 5 “ Keiç~Q~ 2 I )T) (29) 2~’-I)
The exponential increase of K~ffwith Tis much slower than expected from the law exp ( 4~/ T). We consider this as a specific feature of the depolarization mechanism described. Note that the law (29) is independent of the impurity potential sign. For the Kohn— Friedel potential v=3 and for the strain field v=4. Considering the large distances from the defect R~we argue that the asymptotic behaviour of the former potential governs the temperature dependence of Kei~. Below the temperature
[1 —nV(R~)] exp( —K~ 11t)
+nV(R~)g(1).
(31)
With T< TA one has to replace n V(R~) with nV(RA)~ ~m, and ~ with the exponentially decreasing value of the transition probability between bandlike and localized states, Kei~=Keii(R~i) exP[_As(~
—
After that, eq. (31) explicitly describes the decomposition of muons in two independent components. The fraction of the undamped muons is that of the band motion volume, l—~ItR~n=l—x/x~1, where
3/w.
(32)
x4= (3/4m)(A/~~o) With the defect concentration x>xA overlapping of the spheres of radius R4 takes place. Obviously, in this case the above analysis remains valid with the replacement RA R. a ( 4itx/ 3) I where R~ is the average distance between impurities. Consequently, TA is replaced with —~
~,
ç rsItZ(2P+l)bT(As)2 Ts~=Astln[ 3/3I/2~5~2 \(2P
x(~)
I
.
(33)
Now, with T< T~the fraction nV(R~) I and the 293
Volume (59, number 4.5
PHYSICS LETTERS A
14 October 1991
first term in the r.h.s. of eq. (31 ) is equal to zero so that all muons are subjected to the static depolarization, Fig. I presents the results of the numerical calculation of the effective depolarization rate from eq. (31). To minimize the number of parameters we restrict ourselves to the typical parameters for the interaction potential due to the Kohn—Friedel charge oscillations in a metal, ~= 1000 K, v= 3. For the isotropic attractive potential the coefficient a in eq. (18) equals to a=1( 1 + 1 /~)=f’( ~) 0.89. We
experiment clearly demonstrated that the exponential increase of the depolarization rate was shifted significantly to the low-temperature region from ‘I’. in agreement with the T, estimate. The fitted energy gap ( 1.0 K) [2] for the exponential increase ot’
take into account the oscillating nature of the potential by assuming a=rO.5, that is, only half of the transitions are effective for trapping (with ~>0). Using the experimental data [2]. we get b~0.3from the known temperature dependence of the trapping rate in the normal state AL 114] K( 1’)—?’ -°~ and expression (19). After that the only free parameter (for a given defect content) is J~. Recent experimental results [2] for muon spin relaxation in Li-doped aluminium revealed a picture
vanadium (7~ 5 K).
which is in qualitative agreement with the picture of localization and delocalization presented here. The
A
Depolarization rate
(T)
0.3
025
~SC 0.2
normal 0.16
•
\
-
I
0.01
0.02
0.05
I
0.1
0.2
I
I
I
0.5
1
/
C
Fig. I. Depolarization rate in a superconductor with defects (the defect content 2K). is 75 ppm and the tunneling amplitude J~ is l.7xl0
294
We wish to thank E. Karlsson for stimulating discussions and detailed information about the expcrimental studies in the field and some preliminary unpublished data.
References
[110.
Hartmann. E. Karisson, S. Harris, R. Wàppling and TO. Niinikoski, Phys. Lett.A 142 (1989) 504. [2) R.F. Kiefi, R. Kadono, S.R. Krcitzman, Q. Li. T. Niz, TM. Riscman, H. Zhou, R. Wäppling. S. Harris. 0. Hartmann. E. Karisson, R. Hempelmann, D. Richier, TO. Niinikoski, L.P. Lee. G.M. Luke, B. Stcrnlieb and E.J. Ansaldo, I-hp. mt. 64 (1990) 737. [3] Yu. Kagan and N.V. Prokofev, preprint IAE-446l/9. 1987.
[8] Yu. Kagan and MI. Klingcr. Zh. Eksp. Tcor. Phys. 70 ((976)255 [Soy. Phys.JETP43 (1976) 32). [9] Al. Morosoy. Zh. Eksp. Teor. Phys. 77 (1979)1471 [Soy. Phys. JETP 50 (1979) 738]. 110] J.L. Black and P. Fulde, Phys. Rev. LetI. 43 (1979) 453. [11] Yu. Kagan and N.V. Prokofev. Zh. Eksp. Teor. Phys. 97
~
X
0.05
0
(42 K). This nontrivial fact is in obvious correlation with the law (29). For the case v = 3 we get the effective activation energy ~zi,. Ii is worth to note here that an analogous picture can be observed in the superconducting state of pure
[4] T. McMullenandE.Zaremba,Phys. Rev. B 18(1978) 3026. [5] R. Kubo and T. Toyabe, in: Magnetic resonance and relaxation. cd. R. Blinc (North-Holland, Amsterdam. 1967). [6[Y.J. Ejemura. R.S. Hayano, J. Imazato. N. Nishida and T. Yamazaki, Solid State Commun.31 (1979) 731. 171 Yu. Kagan and NV. Prokofev, Zh. Eksp. Teor. Phys. 90 ((986) 2176 [Soy. Phys.JETP63 (1986) 12761.
~
01
4(T) was considerably lower than the BCS value
(1990)1698 [Soy. Phys. JETP 70 (1990) 990). [12]J.Kondo,PhysicaB+C 126 (1984) 377. [13]K. Yamada, Prog. Theor. Phys. 72 (1984) 195. [14]0. t-Iartmann, F. Karlsson, E. Wàckelg~rd,R. Wappling, D. Richter. R. Hempelmann and T.O. Niinikoski. Phys. Rev. B37(1988)4425.