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Determination of the Penetration Depth λ(T) in Superconductors by Reflection Pulse Method
Voicu Dolocan
To cite this version:
Voicu Dolocan. Determination of the Penetration Depthλ(T) in Superconductors by Reflection Pulse Method. Journal de Physique III, EDP Sciences, 1995, 5 (7), pp.925-936. �10.1051/jp3:1995102�.
�jpa-00249363�
Classification Physics Abstracts
75.50K
Determination of the Penetration Depth I(T) in Superconductors by Reflection Pulse Method
Voicu Dolocan
Faculty of Physics, University of Bucharest, Romania
(Received 27 October 1993, revised 31 May1994 and 3 November1994, accepted 13 April 1995)
Abstract. By using a new method, based on the reflection pulse technique, we measured the temperature dependence of the penetration depth in YBa2Cu307-~. Also, we present the
calculus of the impedance of a coil which contains a sample of cylindrical form and a plate form, both homogeneous and granular, respectively.
1. Introduction
The magnetic penetration depth I(T) is an important parameter of a superconductor. The
zero field I(T) is a direct measure of the superfluid density. Its temperature dependence as
T - 0 reflects the low-lying single-particle excitations of a superconductor and therefore yields
information regarding the pairing state in the material ill. Several methods are proposed in the literature for the determination of the penetration depth. On bulk specimens were performed
~SR measurements [2,3], magnetization measurements [4] and microwave measurements [5].
On superconducting films were performed measurement of the mutual inductance of a pair of coils on opposite sides of the specimen [6], measurement of the inductance, kinetic and
magnetic, of a long meander line [7] and measurement of the velocity of microwaves confined in the transmissionline composed of a ground plane and a superconducting [5,8,9].
In this paper we describe how by using a new method, the measurement of the inductance of a coil in which the sample is introduced, can be used to determine 1.
2. The Impedance of a Coil Which Contains a Solid Cylindrical Sample
Let us consider a superconductor solid cylinder inserted in a coil. The total current density in
a superconductor is the sum of the normal current density in, plus the supercurrent Ja~
1=In+lsc Jn is still determined by Ohm's law:
in = ae (1)
© Les Editions de Physique 1995
and Ja~ is governed by equations:
bloc e
$ ~12
i7 x Jac = -~
(2)
In order to derive the impedance of the coil filled with superconductor salnple, we will proceed
as follows. The differential equation for magnetic field can be obtained from Maxwell's curl equations:
i7 x H
= Jac + in + icJee (3)
v x ~ = -(l
= -itdJ1H (4)
where it is assumed a sinusoidal time variation of the form exp(iwt) and that B
= ~H.
Taking the curl of both sides of equation (3)
i7xi7xH=i7(i7H)-i7~H=i7xJac+i7xJn+icJei7xe (5) Substituting equations (2) and (4) into equation (5) for the appropriate terms, and using the
well-known fact that i7 B = 0 gives
~~~ 2
~ ~i~~ ~~i~~~~ ~~~
In polar coordinates equation (6) becomes:
W~iW"~~~ ~~~
where, if the displacement current represented by the term is neglected, r is given by the
equation:
r~
=
j + ) (8)
I = 1 and 6 is the skin depth
~ i/2 6
= (9)
cJ~a
cJ is the frequency and a is the normal conductivity of the sample just above the transition temperature Tc. The solution of equation (7) is:
~~~~ Jo~ir0)~~~~~~ ~~~~
where Hs
= NI Ii is the magnetic field at the field at the cylinder surface (parallel with its
generatrix), ro is the cylinder radius, N is the number of turns, I is the electric current and I is the length of the coil; Jo(ix) is the Bessel function (not current density) of zero order and
first kind. Also, Jo(ix) is denoted by lo(x), the modified Bessel function. The voltage induced
according to Faraday's law -d#/dt = e where the flux # is given by
ro
(ii) j = N2x B(r)rdr
Therefore:
~ ro Ii(Fro
~ " ~~°~'~ ~~h
lo(rro)
where Ii(x) is the modified Bessel function of first order. Series representation of the Bessel functions is
~°~~~ ~ ~
~~ ~ 2~)2 ~~~ ~ 3~)2
~~ ~
~~~~~ ~ l~~~~
~ ~31~~~~
~
By letting 6 very large it results r
= ill and e
= iCJLI where
By using asymptotical expansion for x » 1
one obtains
~r2~
L = 2xrol- (13)
In this limiting case, as it appears from equations (12) and (13) the superconductor looks like a pure inductor. In this discussion we neglected the appearance of the vortex lines.
If, on the other hand, I becomes very large compared with 6 then the latter becomes the
limiting factor and reduces to that of a normal conductor r
= @/6. Since the argument of the Bessel function is complex, it is desirable to separate the expression of Jo into its real and
imaginary parts. This can be done easily with the aid of the following identities:
Jo(xiii)
= ber x + ibeix (14)
where for our case x
=
roli16 and the real and imaginary parts of the Bessel function are
~~~ ~
~
(x/2)~
~
(x/2)~
(2')~ (4')~
(~/~)2 (~/~)6 (~ /~)10
~~~ ~
i ~ (3!)~ ~ (5!)~
These function are well tabulated. Introducing equation (13) in equation (10) and by using
relations
)To x ber xdx = To bei' To
0 20
x bei xdx
= To ber' To (15)
°~~ °~~~~~~
2KN~I~rod bei' To I b~~' ~°
= I(R + iLW)
e = itd / ber zo + ibeix0
Elimination of the complex quantity in the denominator yields
~
2xN~~ rod berxo bei' xo beixo ber' zo
l @ ber To)~ + (beixo)~ ~~~~
~
2xN~~cJ rod beixo bei' To + ber To ber' zo
l @ ber zo)~ + (beizo)~ ~~~~
As the frequency approaches zero, 6 goes to infinity or z approaches zero. In this case, the Bessel functions can be represented by the first terms of their series expansions for small
arguments, that is
~2 berxo"1 beixo" °
~
4 (18)
ber'xo
"
~° bei' xo
"
~°
16 2
Substitutions of these approximations for L and R gives
j~~2 L
= Jt~7rr]
a well-known relation, and
because (ro/6) - o.
At sufficiently high frequencies, 6 becomes very small or x is very large. For such a case, the Bessel functions can be approximated by simplified expressions [10] and it can be easily be
shown that for the resistance term
ber xo)~ + (beixo)~ v5 ~~~~
Thus, at high frequencies the surface resistance becomes Rhf j~~2
" cJ~-xro6 (20)
The same result as in equation (19) is obtained for the Bessel function part of the reactive term and it results
L =
~~~
=
~~ xro6 (21)
cJ
~
The resistive and reactive terms thus become equal at high frequencies and are given by greatly simplified expression.
3. The Impedance of the Coil Which Contains a Plate Sample
By using equation (6) the inductance of the coil which contains a plate sample is given by the
expression [11]
L'
=
~~~ ~~
tanh(rd/2) (22)
The plate introduced into the coil has a thickness d, width W and length I. When the super-
conducting penetration depth is very small compared with 6 this limits the penetration of the magnetic field in the conductor. In other words, by letting 6 very large, from equation (8) one
obtains r
= ill and
L
=
~) Wl~tanh(d/21) (23)
which, for d much larger than I, reduces to
L = )~ Wl~ (24)
It can be seen from this equation that for this limiting case the superconductor looks like a pure inductor and dissipates no energy. If in the superconductor, the normal conduction current is small but not completely negligible, r may be rewritten as
j2 j2
r~~
= I(1+ 2ip)~~/~ Cf I(i ip)
where we use the series expansion and retain only two terms of the expansion. One obtains icJL'= icJL + Rhf
where L is given by equations (23), (24) and
Rhf = ~°~ (25)
represents normal conduction losses which if d » I, is very small.
4. Granular Superconductors
Let us consider a granular material with weakly coupled superconducting grains. The Joseph-
son coupling energy is Ej
= (#o/2xc)Io where #o = hc/2e is the flux unit and lo is the maximum Josephson current flowing between adjacent grains. The grains themselves are con- sidered superconducting with condensation energy Eg = Hj~~/8x which is much larger than the coupling energy Ej between the grains (H~g is the thermodynamic critical field of the grains
and ~ is the volume of a grain). In this limit, which applies well to the oxid superconductors,
the current in the ceramic is limited by the coupling Ej and not by the suppression of the
order parameter in the grains [12]. At temperatures T < Tcj, when the grains are phase- locked, application of a small magnetic field will induce screening currents that flow around
the outer surface of the sample. T~j is the Josephson phase locking temperature. Since the
Ambegaokar-Baratoff theory [13] yields
~~
= (6.35 x lo~K/A)Io(o)(I TIT
KB
near Tc, one obtains
Tc T~j = (1.57 x 10~~A/K)Tf /Io(0) (26)
which yields T~ Tcj = IA K for Tc
= 95 K and lo(0) = 10~~ A [14].
The depth to which the applied field penetrates into a grain is lg, which for the present we treat as a scalar. Because of the weakness of the intergranular coupling, the applied magnetic field penetrates more deeply between grains, to a depth lj, along the grain boundaries. The formulas for the intergrain penetration depth lj, the intergrain coherence length (j, and for the lower and upper Josephson critical fields HciJ and Hc2J> respectively, derived in terms of lj, and (j were given by Clem [14] and by Tinkham and Lobb [15].
Let us first consider a cylindrical sample of radius R with zero coupling between the grains,
in the form of the array of cylindrical grains of radius ro. Assuming a fraction in of the cross- sectional area of the sample to be permeable (this is intergranular region), on its cross-sectional
area can be arranged a number of grains equal to
(1- fn)S
no "
xro2
In this situation the inductance of the coil, which is filled by the sample, is given by the
expression
L'
= ~~l~~~ + noL (27)
where L is given by one of the expression given in Section 2 and I e lg.
As the coupling Ej between the grains is turned on, coherence between the grains is estab- lished and screening currents start to flow through the sample. We consider in this paper only
the case of sufficiently weak coupling that lj » lg. Moreover, to derive a continuum theory,
we further assume that lj » ro. If the sample initially contains no trapped (intergranular or intragranular) vortices, in an applied field lower than HciJ, the inductance of the coil is given by relation (27) where S is substituted by
2xR fi (riR)
~ lo (riR)
where ri is given by equation (8) with I e lj. Therefore, the inductance may be writen
~' ~i~j~~~~~ ~~~ ~ ~ xr/ ~i~j~~~~~ ~~~~
By letting 6 very large it results r = 1fig, ri " 1/lj so that
If lj » R, this equation reduces to equation (27). If in equation (27) in may be neglected,
for ro » lg one obtains
L'
=
~~~° ~
~ 2xrolg (30)
xro When R » lg, equation (29) becomes
which for ro » lg, becomes
L'
=
~ ~~ 2xroljlg (33)
~ ~i
Other models for the geometry of the grain and grain boundary change these expressions.
Consider the xv plane parallel to the cross-section of the sample (and also of the coil) and z direction along the coil length (this direction is parallel to the applied magnetic field). If we model the grain boundaries as a periodic array of parallel insulating barriers all parallel to the
xz plane, with periodicity ao, and thickness di in the y direction, equation (28) is modified
for a cylindrical sample of radius R, where d
= ao d~. For a plate sample of thickness D and width W this expression becomes
L'
=
~~~ ~~'~°
tanh(riD/2) in
+ ~~ ~~ ~~)~~~~~~ l(35)
ri d
When D » lj and ao » lg we obtain from equations (34) and (35) L'
=
~~~° 2xRlj in
+
~"2dlgj
(36)
d and
L'= ~~~°2Wlj in + ~~2dlgj
(37)
respectively. The temperature dependence of the sample impedance is determined by the temperature dependence of lj and lg. With the help of the dirty-limit result [16]
lg(0) ~ A(T) A(T)
lg(T) A(0) ~~~~ A(0)
we obtain
i~(T) =1[ 1 ~ ~~~
(38) Tc
near Tc, where11
= 0.613 lg(0) and A(T) is the temperature-dependent gap parameter. When
ro> ao < lg one obtains [17]
lj = (c#o/8x~aoJo)~~~ (39)
and when ro> ao > lg one obtains [14]
>j
= (cjo/87r2aoJoJtew)~~~ (40)
U~
U~
O o
t t~
O O
a b
Fig. I. Reflected pulse by a series inductance and resistance: a) r > to, b) r < to-
where ~~w is the term contained in the square parantheses from the above equations, e-g. (28)
and (35), that is
~ ~ ~~
~~~ ~" ~ xr) ~~ ~(r~~~
and
iLeW " in +
~~~ ~~ ~~)~~~~~~
respectively. Jo is the maximum current density in the Josephson junction. The temperature dependence of lj is determined by the temperature dependence ofJo and of ~~w which depends
on lg. According to the Ambegackar-Baratoff theory [13]
where Rn is the junction normal-state tunneling resistance and lo
" Joa(. Near T~ it results
Jo(T)
= c(1- T/Tc).
When lg < ao one obtains
lj(T)
= ci(lgJo)~~~~
" lj(o) 1 ~
Tc (41)
~~~
in the case where T~j is nearly T~. In this treatment we have neglected the apparition of the
intergranular or intragranular vortices.
5. Experimental Results
In a previous paper ii ii we have presented a new method for the inductance measurement. The method is based on the reflection nanosecond pulse by a coil which contains the superconducting
material. A nanosecond pulse generator and a sampling oscilloscope are used. The amplitude of the reflected pulse at his first step (shown in Fig. 1) is given by the relation
u~ = u~ j~~-(t-2T)/r ij
L (42)
T #
~ ~
~ fi
rd
~
84 85 88 87 88 89 00 91
T, K
Fig. 2. Temperature dependence of L~~
Relation (42) was writen under the conditions R < Z, where Z is the impedance of the coaxial cable, R and L are the resistance and the inductance of the coil, respectively. Uo and Ui are
defined in Figure la. From equation (42) it results that
T " ~~ (43)
~~ ((ui/uo) + lj
2T is the propagation delay of the pulse through the coaxial cable. For t 2T
= to, the direct
pulse width, we measure Ui and Uo, determine T from equation (43) and also
L = TZ (44)
In Figure la is presented the pulse form for the case T > to- If Ui " 0 then, from equation (43) we obtain
~
ti (~~)
ln2
where t[ is defined in Figure 16. This situation appears when T < to We used this method to determine the temperature dependence of the penetration depth in polycrystalline YBa2Cu3 07-~ material prepared by sintering procedure [18]. The samples have a fine-grained mi-
crostructure with average grain size of 3 ~m. For g of the order of 3 ~flm just above the
transition temperature, and a repetition time of 8 ~s, a skin depth of the order of 2.4 mm is obtained, a larger value than the grain size. Therefore, the skin effect does not limit our
experimental results. The coil in which is inserted the sample has N
= 24 turns, = 10 mm and diameter D
= 5 mm.
In Figure 2 is plotted L~~(T) versus T as were obtained by us. L is the impedance of the coil which contains the superconducting material. It is observed that L~~(T)
-J (1- T/Tc)
near Tc. If we assume that in < 1, lg fro and lj » R then the inductalice of the coil, filled
with the sample, is given by equation (30) where 2ro " 3 ~m. Because in equation (30) L
-J lg
results that the temperature dependence of lg is lj~
-J (1- T/T~) near T~. This behaviour
Pjzi Pt~i
z
z
'izi Pj~i
z z
Fig. 3. The function P(x)
= (2/x)Ii(x)/Io(x).
is in a good agreement with those obtained by the other authors [7]. But, if lj is comparable
with R and lg is also comparable with To is necessary to use the general formula (29) which
now we write:
L = LoPj in + (1 in)Pg]
~ ~
JtON~S,
~ ~
2 Ii(x«) (46)
° ' ~
x~ lo (x~)
where I =J, g, respectively. The general function P(x) is represented in Figure 3. By using lg(0)
= 2100 I
[7] we calculate lg(T) from equation (38), further xg = 1.5 ~m/lg and, therefore, Pg(T). By knowing experimental values of L (Fig. 2) and by using the data obtained for Pg and in = 0.05 [15] we calculate Pj(T) from equation (46) and then lj(T). In Figure 4 is plotted lj~ versus T. It is observed a linear dependence, as it results from equation (41).
We note that the Josephson penetration depth lj is estimated to be 0.31 mm at 84.5 K and 0.47 mm at 89.7 K, in reasonable agreement with other measurements [19].
Equation (46) may be written
L
= ~rLo
where
~lr " Pj in + (I in)Pg]
is an effective relative permeability of the sample.
Studies of the response of zerc-field cooled superconducting ceramics to applied fields revealed that at low magnetic fields up to a certain intergrain field H~ij> grain boundaries are shielded (12]. At higher fields, magnetic field penetrates the grain boundaries in the form of Josephson
vortices.
Intergrain superconductivity in Josephson links is completely extinguished above a grain decoupling field Hc2j For a ceramic YBa2Cu307-z it was estimated that Hcij varies between 1 and 10 G and H~2. varies between 150 and 1000 G. In our experiment the current pulse in the
~ fi
E
84 85 88 87 88 89 00
T, K
Fig. 4. Temperature dependence of Aj~ in granular YBa2Cu307-~ superconductor.
coil is Imax £t 1 mA so that in a coil with n
= I turn /mm a pulse magnetic field of
-J 10 moe
was obtained; this small value indicates that vortex lines are not significant in our samples.
6. Conclusions
We have used a new method, based on the nanosecond pulse reflection, to measure the pene- tration depth in superconductors.
The sensitivity of the method depends on the cable characteristics and also on the sensitivity of the sampling oscilloscope and that of the pulse generator. In granular YBa2Cu307-z we find that lj~ decreases linearly with T increase near T~.
We believe and hope that this method, as being a very rapidly method, offers new possibilities
to study the properties of superconductors.
Also, we have presented the calculus of the impedance of a coil which contains a supercon-
ducting sample of a cylindrical form and of a plate form. Both, homogeneous and granular composition are studied.
References
[Ii Anderson P-W- and Schrielfer R., Phys. Today 44 (1991) 54.
[2] Uemura Y.J., Le L-P-, Luke G-M-, Sternlieb B-J-, Wu W-D-, Brewer J.H., Riseman T-M-, Seaman
C-L-, Maple M-B-, Ishikawa M., Hinks D-J., Jorgensen J-D-, Saito G. and Yamachi H., Phys. Rev.
Lent. 66 (1991) 2665.
[3] Piimpin B., Keller H., Kiindig W., Odermatt W., Savic I-M-, Schneider J-W-, Simmler H., Zim-
merman P., Kaldis E., Rusiecki S., Maeno Y. and Rassel C., Phys. Rev. B 42 (1990) 8019.
[4] Mitra S., Cho J-H-, Lee W-C-, Johnston C-D- and Kogan V-G-, Phys. Rev. B 40 (1989) 2674.
[5] Zhengxiang M., Taber R-C-, Lombardo L-W-, KapituInik A., Beasley M-R-, Merchant P., Elom
C-B-, Hon S-Y- and Philips J-M-, Phys. Rev. Lent. 71 (1993) 781.
