The theory of the measurement of the London penetration depth in uniaxial type II superconductors by muon spin rotation

Physica C 156 (1988) 515-522 North-Holland, Amsterdam

THE THEORY OF THE M E A S U R E M E N T OF THE L O N D O N PENETRATION DEPTH IN UNIAXIAL TYPE II SUPERCONDUCTORS BY M U O N SPIN ROTATION

W. BARFORD and J.M.F. G U N N Rutherford Appleton Laboratory, Chilton, Oxford.shire, OXI 10QX, United Kingdom

Received 3 August 1988 Revised manuscript received ! 9 September 1988

The effect of anisotropy on the measurement, by muon spin rotation, of the London penetration depth in the high-T¢ uniaxial type II superconductors is considered in detail.

Expression are derived which will allow the principal penetration depths, A m (2 2_ ) and 2., (2 i), to be determined using measurements of the laSR line width from single crystals.

For polycrystalline, powder or sintered, samples an expression is derived which will allow an effective penetration depth, 2err, to be determined from the measured laSR line width. Further, it is shown that for all anisotropy ratios, 2_,/2 ,, greater than five 21 ~ 0.812off.

1. Introduction A = M c 2 / 4 n n s e 2 , (1)

Muon spin rotation (~tSR) has been extensively used over the last year in an attempt to measure the London magnetic field penetration depth in the new high-T~ superconductors [ 1-4 ]. Since polycrystalline, sintered or powder, samples have been used the results are open to ambiguous interpretation as the strong anisotropy of the new materials has not been taken into account. Other experimental techniques, such as magnetisation experiments [ 5 ] and critical neutron reflection [6,7] have reported widely dif- fering results.

In this paper we consider carefully the ~tSR experiments in uniaxial type II superconductors and derive an expression for the frequency line width broadening in single crystals and for polycrystalline samples which will allow a more accurate interpretation of the data to be carried out and perhaps re- solve the discrepancies between the various experimental results.

Since the high-To materials are uniaxial the con- duction of the superconducting currents are not isotropic. This anisotropy is conveniently represented by the tensor A defined by

where M is the effective mass and ns is the number of superconducting electrons. (In general one may consider both to be tensors). A has a degenerate ei- genvalue A l associated with super currents flowing in the a-b plane and A2 associated with super currents flowing along the c-axis. Typically the anisotropy is large and A:~ (25-100)A,. The London penetration depth tensor, A, is equal to A ,/2. In the literature 22(=A9 I/2) is often called 2pa r and 21(=AI/2) is called 2oerp as currents flowing in the

a-b plane are responsible for screening magnetic fields perpendicular to the planes [ 8 ].

Using single crystals 2, and 22 can be independently determined from expressions which will be derived later. For polycrystalline samples, however, an effective penetration depth is measured. We will calculate how 2err is related to 2, given the anisotropy ratio 22/21. it will be shown that for most anisotropy ratios 2, can be determined independently.

Throughout only intermediate fields will be considered: Hcl << H<< He_, for which the average inter- vortex spacing, d, obeys ~<< d<~: 2 (where ~ is the co-

0921-4534/88/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

516 w. Barford and J.M.F. Gunn / #SR measurements of the London penetration depth

herence length). In practice this condition is met in most experiments on high-T¢ materials as He, is very small and He_, is very large.

2. Muon spin rotation line width

In the following we give a brief review of the aspects of p, SR theory which are relevant to our object and calculate some new results. For more details about general aspects of ~tSR the reader should con- sult the work by Schenck or Cox [9,10].

Consider a muon implanted into a specimen at r with its initial polarisation, /, in the x direction. There is a local field h( r ) ; this is to be distinguished from the magnetic induction, B, which is the spatial average of h over one flux cell: B= ( h ) . This experimental geometry is illustrated in fig. 1. Muons precess about the magnetic field with the Larmor frequency (which is equal to 7h/2n where y is the gy- romagnetic ratio for the muon and h is the field intensity). When muons decay (their half-life is 2.2 las) they emit positrons. The intensity of positrons in the direction p, due to muons sitting at r, Pp(h(r) , t), is

Pp(h(r) , t) ocI .p+constant (2)

and the total positron intensity is the integral over r o f P p ( h ( r ) , t).

A C

X

T ~ P

h(r)

/ Fig. 1. The coordinate axes showing Hapo, the local field h(r), the crystallographic axis, i, and the direction of the muon spin, I, at t=0.

Since the field intensity, h, varies in the medium muons precess at different frequencies and consequently dephase. This dephasing, which is measured experimentally by determining the line width of the positron intensity, P, will allow the London penetration depth to be determined since the penetration depth governs the spatial variation of the magnetic field.

We need to calculate l ( t ) to deduce the measured positron intensity. I f the c-axis is neither perpendicular nor parallel to the applied field, Happ, then there will be non-zero transverse field components. (Par- ticularly intriguing is that for fields close to He, the flux lattice, and hence B, does not lie parallel to Happ but distorts towards the a-b plane. This is because in the Gibbs free energy there is a competition between the magnetic pressure which tries to align the flux lattice parallel to the applied field and the magnetic energy density which is minimised if the flux lattice is parallel to the a-b plane [ l 1 ], see also de Gennes [12,ch.3]. (This limit will not concern us here, however. )

Consider fig. 1. The z-axis is defined by the applied field, Hapw the muon spin angular momentum, I, is composed of a static part, L , parallel to h and a rotating part, I z (t), perpendicular to h. We write

I ( t ) = l ~ +I± (t) , (3)

where

I± = IIx/~I (4)

and

L =I./~ (5)

and h = h ( r ) / h ( r ) . Henceforth L will be ignored as it is of no conse- quence to the dephasing.

Let fi and b be unit vectors in the plane perpendicular to h. Then

I± (t) =1± (t~ cos(?ht)+t) sin(Tht) ) , (6)

where

# = { i - h x ~ } / 1 4 x i l (7)

and

~,=/ix..i/I t l x . i l . (8)

W. Barford and J.M.F. Gunn / IzSR measurements of the London penetration depth 517

Note that the phase has been defined such that at t=O, I±(0)=I±~.

It is now possible to calculate the time dependent positron intensity in a given direction using eqs. (2) to (8). For example, in the "forward direction" (the

direction ):

P,,(r, t) oclx ( t).~oc ( 1 -•x (r) 2 cos (Th (r) t ) , (9)

whereas in the "transverse direction" (the direction)

P~(r, t)oc -hx ( r ) ~y(r) cos(Th(r) t)

+h , ( r ) sin(Th(r) t ) . (10)

Notice that the intensity in the ~ direction has cos(?ht) and sin(Tht) components of different am- plitudes. The total intensity in the x direction is the integral of (9) over a flux cell:

Px( t )=S -l j d2r ( 1 - ~ x ( r ) 2) cos(Th(r) t) .

( l l )

As noted earlier, the quantity readily related to the London penetration depth is the line wid th (variance) of the positron intensity frequency spectrum defined by

(Aid 2 ) = (O9 2 ) -- ( ( 0 ) 2 , (12)

where the average is performed over the local magnetic fields. To obtain this we Fourier transform ( 11 ) with respect to time, and calculating moments we find

(O.) 2) m~y2S -1 J h(r)2( 1 ~ ~x(r) 2 ) d2r

=72S - l ~ (hz(r)2+hy(r) 2) d2r. (13)

Here the average over local field is embodied in the integral over a flux lattice cell, of area S, is performed over the plane perpendicular to the vortices. Since h(r) is periodic eq. (13) can be rewritten in terms of the spatial Fourier components:

(to z) =7 2 ~ (h~(G)2+hy(G)2), (14) G

where G are the reciprocal lattice vectors in the plane perpendicular to the vortices.

Similarly the average frequency:

((o) = 7 S - ' j h(r) ( l - h x ( r ) 2) d2r. (15)

Thus the frequency variance becomes

( A t 0 2 ) = 7 2 ( B 2 + G~0 ~ (hz(G)2+hy(G)2)) -(tO)2' (16)

since the zeroth Fourier component of h is ( h ) = B the magnetic induction in the sample. Notice that if there is no anisotropy, so that hx=O for an applied field in the £ direction, then the frequency variance is proportional to the variance of the magnetic field intensity. This result, true for isotropic superconductors, has been incorrectly assumed in the ~tSR literature for new high-Tc uniaxial superconductors.

The London penetration depth can be calculated from eq. (16) in terms of the three experimental pa- rameters, namely, the line width, (Ao92), the average frequency, ( to ) and the magnetic induction, B, which for intermediate fields is equal to the applied field, Happ-O(H~, ) [l 1,13]. This is because the second term on the right-hand side is some function of it o which will be calculated in the next section from the London equation.

3. Calculation of the line width

Throughout we will only consider intermediate field regimes for which the flux lattice is parallel to the applied field. Consider a uniaxial type II superconductor with its c-axis oriented arbitrarily with respect to the applied field. As before, the x-axis represents the "forward direction" and the z-axis the direction of the applied field and hence the flux lattice.

The electrodynamics of extreme (x = 2 / ~ >> 1 ) type II superconductors for Hc, < H<< He_, are sufficiently well described by the London equation [ 14 ]:

h+VX (A-VXh)=¢~o ~ ~ ( x - x i ) ~,, (17) i

where 0o is the flux quantum and the sum is over the lattice cells. The right-hand side therefore represents the periodic flux quanta in the z-direction. The be- haviour of the magnetic field in the flux cores is ex- plicitly ignored; this is a reasonable assumption when it >> ~ and H<< Hc 2 because the flux cores make a negligible contribution to the area of the flux cell.

518 1,14. Barford and 3.M.F. Gunn / ItSR measurements of the London penetration depth

Magnetisation experiments [ 5 ] suggest that K is in the range 10-40.

For uniaxial superconductnrs

A~, = A , 6 a + Aa6,(, , (18)

where 6, is the projection of ~ onto the ith (x, y or z) axis, and in terms of the principal eigenvalues:

A , = A I , A /~=A2-AI . (19)

Fourier transforming and inverting ( 17 ) the Fourier transform of the field, ~, is

~ ( G ) = ( I + A , GZ)_ , ( ~ _ ApQ:Q ~NOo 1 +A,G2+ABQ2} '

(2o)

where

Q = G x ~ , (21)

G are the reciprocal lattice vectors of the flux cell and N is the number of flux quanta per unit area. A der- ivation of (20) is shown in the Appendix.

For the line width to be calculated from eqs. ( 16 ) and (20) a sum over the reciprocal lattice vectors must be performed. The distortion of the flux lattice in an uniaxial type II superconductor has been elu- cidated by various authors [ 11,13,15 ] so we make a brief digression here to summarise the results.

3.1. The flux lattice in a uniaxial type H superconductor

In an isotropic superconductor the flux lattice is composed of equilateral triangles which may be oriented arbitrarily in the plane normal to the vortices. In an uniaxial superconductor, however, the equilateral triangles are distorted to iscoceles ones and the degeneracy, with respect to orientation, is lifted.

Let the c-axis lie in the x - z plane. Then, according to refs. [ 1 l, 13,15 ], if the primitive cell is described by the vectors a~ and a2, a~ lies parallel to the x-axis and (see fig. 2)

a2/ a, = ½ ( 1 + 3 A 2 / A-- ) I /2,

cos X=d, "d2 =Ay_2 / ( 3A2 + A_-_-) ~/2 (22)

We may form the reciprocal lattice vectors in the usual manner (from ai'b,=2n~ij):

Z X ta._~

Fig. 2. The triangular flux lattice under an anisotropic distortion. The applied field and hence ~ is normal to the page. The crystallographic axis, ~, lies in the x-z plane.

b~ = 2x{a2 X~/la , Xa2 I},

b2 =2x{£Xa, / la l Xa2l} • (23)

Consequently the reciprocal lattice vectors are

G ...... =mbj +nb2 . (24)

Furthermore the area of the real primitive cell is 1/N= la~ Xa21 =aja2 sin Z.

3.2. Single crystals

Before considering the line width obtained from a polycrystaUine sample we first consider the line width from a single crystal when the applied field, Happ, is in two particular orientations with respect to the c- axis: parallel and perpendicular to it.

3.2.1. Applied field parallel to the c-axis For the applied field parallel to the c-axis there are

no transverse field components so h = h-~ and the frequency variance is simply

( A t o 2 ) = y 2 ~ h:(G) 2, (25) ~¢o

which is 72 multiplied by the variance of the magnetic field intensity.

From eq. (20) with 6i=t~,3, eq. (25) becomes

1 (ACD(Cl)2)=72N2~jo2 E (26)

G~0 ( I T A IG 2 ) 2 '

where < A~o(~ )2) denotes the line width measured for ~ parallel to Harp.

Anisotropic effects do not arise so the flux lattice is composed of equilateral triangles and Gmax = 4/t/

W. Barford and J.M.F. Gunn / gSR measurements of the London penetration depth 519

aj3 t/2. In the intermediate field regime AIG 2 is much greater than unity for G > 0 so (26) is easily summed and we obtain

(Ato(i H )2) =72 3.706× 10-302A ~-2 • (27)

Thus A, can be easily obtained by measuring (Aw 2) for an applied field parallel to the c-axis. We have not converted (26) into an integral ~t la Pincus [ 16 ] as this introduces unecessary errors into the calculation. For example, in this simple case one would get (Ao9(~11)2)=y2(3/(4n)3)02Ai-2 which is 2.45 smaller than (27).

3.2.2. Applied field perpendicular to the c-axis Now the applied field lies in the a-b plane and

there are also no transverse field components so h=h_-£ still. Since h~ is independent of the direction of the applied field in the a-b plane we may then choose i=a?.

Thus using eq. (20) eq. (25) becomes

1 ( A o ) ( ~ ± ) 2 ) (28). . .72N2~o2 ~ 2 )2 ,

G*O ( I + A2G2x + AI Gy

(28)

where, similarly, (Am(~±) 2) represents the line width measured for the c-axis perpendicular to the applied field.

Now the anisotropy has to be taken into account and using (24) with a, parallel to ~,

Gx=2xn/al , (29)

G r = 2x{ m/a2 - n (cos X)/al }/sin X,

X and a2/at are given by (22) with Azz=Al. In the intermediate field regime (28) can be as-

sumed numerically and expressed in the form

(A07(~) 2 ) ----7 2 3.706 X 10-3~2Ai-2Fl ( , 1 2 / , 1 1 )

---- ( A f o ( ~ ) 2)F1 (,12/,11 ) , (30)

where F, is a function of the anisotropy ratio ,12/,1 ,. We find by plotting log(F, ) versus 1og(,12/2, ) that F, = (,12/,1,)- ~ where u = 2.00; this is plotted in fig. 3. Thus

(Aog(~)2)=y23.706xIO-3(b2(A~A2) -I . (31)

Equation (31 ) shows that for an applied field perpendicular to the c-axis the effective penetration

10-1

F) (~,/~,

100

\ \

10-

1 0 - 3 , 100

\ \ \ \ \ \

\ , \ \

\ \ \

\ \ \, \ \

\ \ .

\

I L t L I

101

Fig. 3. Iog(F~ (2_,/2,) ) versus log(2 J) . , ). The slope is - 2 which confirms eq. (30).

depth is the geometric mean of,1, and ,12. This result would have been expected if we had evaluated (28) as an integral and used elliptical polar coordinates in the spirit of Pincus [ 16 ] but it is not necessarily expected when a triangular lattice is used and is only true for the orientation of the triangular lattice as determined by refs. [ 11,13].

3.3. Polycrystalline samples

Unfortunately good single crystals are not easy to obtain so most oSR experiments have been performed on polycrystalline, sintered or powder, samples. For these samples only one measurement can be made so unless the anisotropy ratio is known a priori, in general 2, and ,l 2 cannot be determined independently. However it will be shown that for most practical purposes ,l, can be determined independently.

We will firstly consider the line which would be obtained from a single crystal with its c-axis at some arbitrary angle with respect to the applied field. This line width will then be integrated over the solid angle to get the effective line width which will be a function of,1, and ,12/,11.

Remember from eq. (16) that

(Am2) = ~ + ~u, (32)

520 W. Barford and J.M.F. Gunn / ~SR measurements of the London penetration depth

where

~ = y 2 B 2 - ( 0 ) ) 2 , (33)

which need not be zero but which can be experimentally determined and

Q=y2 y~ (h_.(G)2+h, , (G)2) . (34) G#O

Using (20) this becomes

+ O y ) ] + A a O : O .... X Y. [ I+Ac~GZ+Aa(Q~ 2 2 2 2 2

~. .o [ ( I + A , ~ G 2 ) ( I + A , ~ G 2 + A u Q 2 ) ] 2 '

(35)

which must be evaluated for an arbitrary orientation of ~ with respect to the applied field.

The measured line width is the isotropic average, i.e.

I2,.rr = (g2(~))~ = 1/4~ f £2(d) d2~, (36)

where the angular brackets represent the average over all orientations of t . Summing (35) numerically and performing the isotropic average (36) numerically also we find that

ff2eff= y 2 3.706 X 1 0 - 3 q ~ o 2 A / 2 F 2 ( 2 2 / / l i ) (37)

= <AgD(C I ) 2 > F 2 ( 2 2 / ) . 1 ) ,

where F2 is a function of the anisotropy ratio and is plotted in fig. 4. Thus, we may define an effective penetration depth, kerr,

2 - - ' 4 1 / 2 = 2 1 / F t / 4 (38) elf -- ~'*cff

such that

~2e~r = y 2 3.706× 10-302oAg 2 . (39)

Finally we have the result that

},2 3.706 × l 0-3002) . i -4F2 = ( A O ) 2 ) + < O ) > 2 - - y 2 B 2 ,

(40)

where B=H~p o. Notice from fig. 4 that F2 tends to an almost con-

stant value of -~ 0.435-0.440 for an anisotropy ratio of greater than five. This is because (35) becomes independent of the anisotropy ratio, 22/41, when 22/ 2, is large for all directions of the applied field ex-

0.8

0.6

0.4

0.2

o ~ a g g 1'o ~ 1.

Fig. 4. F2(22/2, ) versus the anisotropy ratio, ,t2/,l L. For an anisotropy ratio of greater than five F2 is almost a constant value of 0.44.

cept for applied field directions close to and in the a-b plane. Consequently, when the average over solid angles is performed ~2eff is independent of the anisotropy ratio, too. This result is very convenient, for experiments on polycrystalline samples, because it means that even if the anisotropy ratio is not known for all practical purposes 2, may be determined with the aid ofeq. (40). From (38) it may be seen that for polycrystalline samples, the measured penetration depth, 2cn--~ 1.2321.

When experiments are done on single crystals, however, great care must be taken to ensure that the applied field lies in the a-b plane for the results of section 3.2.2 to be of use.

In practice most experimentalists do not Fourier transform P( t ) to obtain P(to) and hence calculate the moments because of the poor signal to noise ratio. Instead they assume a Gaussian envelope to P (t) and consequently calculate the second moment, variance, of/~(o9).

4. Conclusions

In this paper we have considered laSR as a means of measuring the London penetration depth in un-

W. Barford and J.M.F. Gunn / p.SR measurements of the London penetration depth 521

iaxial type II superconductors. We have derived results which allow 2~ and 22 tO be determined if good single crystals are used and a result which will in practice allow 2~ to be determined, even if the anisotropy ratio is unknown, using polycrystalline samples.

We have assumed that the magnetic field inhomogeneities are solely due to the periodic flux lattice. In the high-T~ materials, however, there are likely to be other sources of field inhomogeneities because of the very small coherence length. One obvious source is flux pinning due to the strong twinning. A quali- tative estimate of this effect has recently been given by Brandt [ 17 ] who assumed Gaussian fluctuations in the positions of the vortices from their regular lattice positions. Another more intriguing possibility is that the flux lattice forms a two-dimensional fluid; this has been proposed by Houghton and Moore [18 ], various other exotic possibilities have been considered by Fisher and Lee [ 19 ].

In addition to the inhomogeneities in a single crystal there are also demagnetisation effects in a polycrystalline sample because of the irregularly shaped and sized grains. This will result in a macroscopic field broadening which can be distinguished from the microscopic effects mentioned above [20].

One would expect, therefore, that the effect of other field inhomogeneities would cause the field inhomogeneities due solely to the periodic flux lattice to be overestimated. Since the frequency line width and the variance of the field intensity go as 1/24 this means that 2 should be underestimated. Thus, if the other microscopic field inhomogeneities are present the penetration depths found by ~tSR should be re- garded as lower estimates.

In view of the still unclear picture as to the correct values of 2 t and ~,2 it would be useful if the I~SR data were soon reinterpreted with the anisotropy cor- rectly taken into account.

Note added in proof

Harshman and co-workers [21 ] have recently used our formalism to measure 2~ for a polycrystalline sample of YBa2CuaO7. They find that 2, ~ 1 415 ~k. They also find that for a single crystal of YBa_,Cu306.9, when the magnetic field is applied

parallel to the c-axis, that ;t j = 1 430_+ 30 J~. Bearing in mind the slightly different dopant values these results appear to be in good agreement.

Appendix. Inverting the London equation

The London equation is

h~ + %k ~Ak/etmn Vmh, =J,3 ~ (~oJ(X-X,) . i

(Al)

Fourier transforming, ~ being the Fourier transform of h:

-- ~ijkGjAkl~t,nnG,nf[n -- Ji3N~o . (A2)

where N is the number of flux quanta per unit area. Using

Ao =AaJu + Aa~,~j , (A3)

and substituting into (A2) gives

+Aa(¢ijkGj¢.k ) (enmtGmCl)~n = tJi3N0o, (A4)

which simplifies to

{ ( I + A , G2)J,,+AaQ, Q,}hn=Ji3N~)o, (A5)

where Qi = ~ akGj?-k = ( G × ~) i and GiGfi~ = 0 because V.h=0. Consequently

~n =A £ ~ J~aOoN , (A6)

hence, using the identity that ifA~, = J~o + BQ~Q~, then

B A ~ =J i " - 1 +BQ 2 Q'Qn' (A7)

we have

(A8)

or

f ln=( l + A a G 2 ) - l (~3n- l + ; ~ B Q 2 ) N ¢ o ,

(A9) which is the desired result.

522 14~ Barford and J.M.F. Gunn / gSR measurements of the London penetration depth

Acknowledgements

We thank Prof . S.W. Lovesey and Dr . S.F.J. Cox

for helpful discussions. We are ve ry gra teful to Da le

H a r s h m a n for send ing us a copy o f his p repr in t .

References

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Documents

The theory of the measurement of the London penetration depth in uniaxial type II superconductors by muon spin rotation