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Submitted on 1 Jan 1993
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Electromagnetic response of a conductor with complex conductivity
L. Leylekian, M. Ocio, J. Hammann
To cite this version:
L. Leylekian, M. Ocio, J. Hammann. Electromagnetic response of a conductor with complex con- ductivity. Journal de Physique III, EDP Sciences, 1993, 3 (2), pp.139-165. �10.1051/jp3:1993123�.
�jpa-00248911�
Classification Physics Abstracts
07.55 41. IO 74.30G
Electromagnetic response of a conductor with complex conductivity
L Leylekian, M~ Ocio and L Hammann
Service de Physique de l'Etat Condensd, C-E- Saclay, 0rnle des Merisiers, 91191 Gif-sur-Yvette Cedex, France
(Received 22 September 1992, accepted 24 November 1992)
Rdsumd. Cet article a pour but de ddcrbe la rdponse dlectromagndtique d'un conducteur muni d'une conductivit6 complexe. Nous montrerons comment la gdomdtrie du dispositif de mesure peut modifier l'amplitude de cette rdponse. Nous insisterons particulikrement sur le rble que joue une
conductivitd complexe, comme nous pouvons en trouver dans les supraconducteurs granuIaires,
sur la susceptibilitd magndtique mesurde de l'dchqntillon.
Abstract. The aim of this paper is to describe the electromagnetic response of a conductor with
complex conductivity. We will show how the geometry of the measuring apparatus can modify the
amplitude of this response. We will particularly emphasize the role that plays a complex conductivity, as we can find in granular superconductors, on the mesured magnetic susceptibility
of the sample.
1. Introduction.
We recently performed measurements of the dynamic magnetic response on ceramics of the
high T~ superconductor Laj ~Sr_~Cu04 [1]. Our set of experiments was divided into two parts :
First, we measured AC susceptibility classically. The sensitivity of our DC SQUID device allowed us to use very low field excitations (less than one moe). By this way, we intented to
probe the linear susceptibility x (w
= x'(w ) + ix "(w of the system.
On the other hand, we performed noise experiments, I.e. measurementj of the spontaneous flux fluctuations of the sample in the absence of any excitation. The typical information that
comes from this kind of experiment is the power spectrum S(w ) of the flux fluctuations in the
sample~
In the general frame of the linear response theory, both quantities x"(w and S(w are
related by the fluctuations-dissipation theorem (FDT) [2]
x"(w)cc s(w)w/T. (i)
Experimentally however, in spite of the general agreement between noise and susceptibility
measurements (Fig. I), one can see that both ways of probing the dynamics of the system are
not strictly equivalent.
0.1
,~
O tHz
o o D 10Hz
i t h t00Hz
3
~
o G3
,,
X
Tet~pe~~tu~©
o
1@ 1> 14 1' m >@ ii
O 0.17 Hz
~
Q 1.7 Hz
d 17 Hz
10 IS 20 25 30 35
Temperature
Fig. I. Actual imaginary susceptibility and imaginary susceptibility derived from noise measurements
by the FDT for different frequencies.
As far as this problem was already met in spin glasses [3], we know that this difference is
mainly due to the difference between the geometrical configurations of the relevant flux Iines in both kinds of experiments. However, in contrast with the spin glass case, the dynamic magnetization one records in superconductors is not due to localized spins but rather to current
loops in the sample~ We shall treat our superconducting ceramics as a continuous medium [4]
with a permeability and a macroscopic conductivity. After the predictions of several theories
on the glassy behaviour of granular superconductors (The so-called Superconducting Glass State or SGS) [5], it appears that this conductivity should be complex. We will emphasize here the influence of its imaginary pan. Parameters describing the coupling of intergrain currents to a measuring set of coils will be derived using classical electromagnetic theory.
EXPERIMENTAL FLUX MEASUREMENT DEVICE. The detailed setup is fully described
elsewhere [6]. Schematically, our cylindrical sample takes place into a purely inductive third- order gradiometric coil of niobium. This pick-up coil and the input coil of the SQUID are
connected by a twisted pair of niobium wires. The whole circuit is superconducting and acts as a flux transformer so the measured flux A45~ is proportional to the flux A45 seen by the
gradiometer (Fig. 2a). The variation of flux A45 in the gradiometer generates a current
I such that
A45
= (Lo+L,)I +M~i~. (2)
We neglect the inductance of the twisted wires. The flux seen by the SQUID is then
A45~ = M~I + L~ i~ (3)
~ ~
~
@
~pick-up coil ~
~ tS
ii~
(a)
~~~°
p ~§] r
Q Q
L~
Lj Lj
(b>
Fig. 2. a) Setup of the susceptibility experiment: the sample is excited by an AC field H and the flux detected by the coil Lo is directly sent to tile SQUID with the help of the coupled coil L,. b) Setup of the noise experiment : the set « coil Lo + sample » is seen by the SQUID as an equivalent
inductance L and a equivalent resistance R.
so that
M, M)
A45~
=
A45 + L~ i~ (4)
Lo + Li Lo + L;
Then, neglecting the small current i~, which can be represented by a small modification of the effective mutual inductance M,, we find :
M, k fi
A45~ =
A45
= A45
= f A45 (5)
Lo + L> Lo + L,
the best coupling is obtained for L, = Lo and one then has f~~~ =
~. In our setup, 2
L, = 2 ~LH, M,
= 18 nH and Lo = 1.8 ~LH so that f
=
1/211.
2. Classical susceptibility measurements.
In the classical case, the sample is submitted to an uniform sinusoidal magnetic field. A current I
= lo e~'~~ flowing in the excitation coil produces a uniform magnetic field H
=
nI where
n is the number of loops per meter of excitation coil. The induction in the sample is then
responsible for eddy currents j which obey the Maxwell relation
VXB=pj. (6)
On the other hand, linear response implies j
=
«E (7)
where p is the permeability of the medium and « its conductivity. Finally as V x E
= wB and
V B
=
0, the induction in the sample obeys the classical Laplace equation V~B + k~ B
=
0 (8)
with
k~=ip«w. (9)
In (8) and (9), p
= po p~ represents the static magnetic permeability. One must notice that this way of treating the problem makes a distinction between the magnetic response due to
eddy currents wich is the aim of this calculation, and the static magnetic response which is
supposed to come from microscopic properties of the sample (for a superconductor, it would be the Meissner effect) and which represents for us the initial field lines density that the eddy
currents are going to screen. Then, the magnetic flux we are calculating is only the dynamic
one and does not take into account the static field p H. This general consideration remains valid for the whole paper.
For an infinite cylinder of radius b, the solution of (8) is, with respect to the boundary
conditions
Jo(kr)
~~~~ (lo)
" ~~~
Jo(kb)
or
Ji (kr) j (r)
=
V x B/p
= nlk (I I)
Jo(kb)
where J~ is the n-th Bessel function of first kind. Equation (I I describes the cylindrical sheet of current which is responsible for a magnetic flux
d45
= pj (r) wr~ dr (12)
and the total flux seen by the pick-up coil is
6
45
= pj (r) wr~ dr (13)
o
Combination of (11), (12) and (13) gives the solution
~J~(kb)
45
= pnlwb (14)
Jo(kb)
where the dimensionless parameter
J~(kb 2 Ji(kb
~ Jo(kb) ~ kbJo(kb) ~~~~
can be identified [4] as the susceptibility of an infinite cylinder. We finally obtain
~P
=
vniwb2
x (16)
As the field H is uniform, we can define an effectively-measured susceptiblity i
~p ~ 2
~~ ~
HoHSp.u ~ ~~ i ~ ~~~~
3. Behaviour of the classical susceptibility in the case of a complex conductivity.
As shown by equation (15), the complex quantity x depends only on the product
kb where b is the radius of the sample and k
= (ip«w )~'~
Low frequency behaviour : )k~b)
= VW )« b « I, the field penetrates into the whole sample and we obtain x =
~)~~ (l +
~(~~
(18)
Thus, with «
= «~ + i«; and x
= x'+ ix " we find
pm, wb~
~ ~
ml ml
X' = ~
~ l + ~ b (19)
">
and
x" =
~ ~~ ~°~
l I
«, b~ (20)
For a classical ohmic conductor («, = O), equations (19) and (20) show that x'cc w~ and
x" cc w. Within these low frequency conditions, the introduction of an imaginary conductivity
has just a perturbative influence on x", as the out-of-phase susceptibility keeps its sign and is
smoothly modified by a second ordre term. In contrast, the low frequency behaviour of x' is completely different in the presence of an imaginary conductivity : the leading term
becomes proportional to «, w, whereas the previous one was proportional to («~ w )~.
High frequency behaviour : k~ b
= VW « b ~ l, the field is confined into a skin of thickness
= I/ k at the edge of the
cylinder.
Then,
~~ ~~~
=
~°~ ~~ ~ "~~
~ l ~
Jo(z) cos (z mm I + tg (z) (21)
As tg (z) - I for )z) - cc, Arg (z) # 0, we find
x = -1+ ~
(22)
If we take «
= «o e'~ we finally obtain :
« w
~ ~' 4 2
~ ~ ~ ~~~~
p «o wb
The representative curves of x'(w and x "(w for different arguments of « are displayed in
figures 3a and 3b.
0~4
4
0~3
7«/20
4«/10
j 0~2 ,,
~
' ~~~~~
'
/ ~, , «/2
~°~ / '~ l',
~~'°"".,~~__~~_j _ /~~'-- -_
0
"~""""'~"''
0 4 8 12 16 20
(k(b
(a)
0.8
/4
0.6 /10
'?
7«/20 0.4
4n/10
~ ~
9n /20
n /2
~
0 4 8 16 20
)k)b
(b)
Fig. 3. a) Imaginary part of the classical susceptibility of an infinite cylinder versus k b for different
Arg k. The case Arg (k) = w/4 is the normal conductor case whereas the case Arg k
= w/2 is for a purely imaginary conductivity. In this last case, the dissipation is zeTo whateveT k. b) Same as figure 3a but for the real part of the susceptibility.
4. Noise measurements.
In this experiment, the system is free of any excitation and the pick-up coil just detects the spontaneous fluctuations of flux due to the spontaneous fluctuations of current in the sample.
From the Nyquist theorem, we know that the power spectrum of the voltage in the pick-up coil
is related to the resistance R of this coil
i7~(w
=
~ k~ TR (24)
We know that the circuit is resistive in the presence of a dissipating medium and the impedance
of the pick-up coil becomes (Fig. 2b) Z
= iL~(I + qx w (25)
so that
R
= Lo qx " w (26)
and &L
= Lo q' X'
where q and q' are filling factors.
Then the current flowing in the circuit is
~~~~° ~~~
(Lo + L, ~~~~
W~
+ R~ ~~~~
We can neglect &L
= Lo q' x' when q'x' is small compared to I, which corresponds to the
penetration regime or to a small q'. Nevertheless, q' cannot be neglected in the skin regime
with good filling factor. The effect of &L is through the coupling factor f.
Then, the flux seen by the SQUID is
@[(w
=
M) 12(w)
= M) ~ ~~ ~~
~
i ~
~ ~
l(28)
" (Lo + L,) w (Lo +
,)
w
2 k~ TR
~ (29)
@((w
= -~ f
We then see that the calculation of the power spectrum of the flux seen by the pick-up coil is
equivalent to the calculation of the resistance of this coil. To deterrnine this resistance, we will consider a current I e~'~~ flowing in the pick-up coil. It generates a vector potential in the
sample Aj(r). This vector potential generates itself eddy currents in the sample and finally J(r)
= i~rwA~(r) (30)
where A~(r) is the total vector potential at r, I-e- the sum of Aj(r) and of Aj(r) the vector
potential due to the eddy currents in the sample. Then, we can derive the flux measured by the coil and its impedance Z= ~
=
~~°~ The resistance of the coil is the real part of
I I
Z. Thus with the help of relation (28), we can derive the noise power density that the whole system forrned by the coil and the sample would produce if it were free of any excitation.
ExPLiciT cALcuLATioNs. We consider the infinitely long cylindrical sample surrounded by
a loop cawying a current I (Fig. 4). Classical electromagnetism gives us the vector potential
Uz
u
M
_ ~
_
~
7
,
lZ
~''
_r
' -J
v
Fig. 4. Draft of the system of coordinates in which calculations are made.
created by current I. This vector potential has only a 4 component which is
~~~' ~~ ~~ /~ ~~~~ ~~~j
'
~~
(a /r~i
~ ~2
where E(k) and K(k) are respectively the Legendre elliptic integrals of the first and second kind. Due to the cylindrical symmetry of the problem, that expression can be rewritten [7]
A(r, z)
=
~° ~~
j~ ~ Ii (qr) Ki (qa e'~~ dq (31)
" o
where I~ and K~ are the modified Bessel functions of the first and second kind [8]. In fact the
modulating terra in the integral is a cosine but from now, we work in complex notation and
take into account the real part only. In the presence of an infinite cylinder of relative
perrneability p~ inside the coil, it can be shown (see appendix I) that (31) becomes in the different domains (Fig. 5a) :
p ~ ai
I+OO
AI
>n
(r, z)
= Y'(q Ii (qr )Kj (qa e'~~ dq (31. 1)
" 0
A)~~~(r, z)
=
~° ~~
j~ ~ [Ij (qr) + 4l (q) Kj (qr)] Kj (qa)e'~~ dq (31.2)
" o
A)~~~(r, z)
=
~° ~~
l~ ~ [Ij (qa) + ~P(q Ki (qa)] Ki (qr)e'~~dq (31.3)
" o
, , , ,
~°°P °~Eddy current dJ '
,
dA~
,
,
,
,
i
,
,
out
. ~
,
,
.
,
. out
~2
~"front I
,
,
out ,
,
,
,
z
,
,
~~
,
Out
,
, ,
0 0
' ' ' ' '
'
' 0
' 0
0 0
' '
' '
' ' ~ ~
' ' '
~~ in
' ' ' '
a) b)
Fig. 5. a) and b) Sectors of different vector potentials created respectively by the loop of applied
current I and by a loop of eddy current dJ.
In the same spirit, the vector potential created at (r, z) by an orthoradial element of eddy
current at (r', z') in a infinite cylinder of relative perrneability p
~ can be written in the different
regions (Fig. 5b) :
dAj;~(r, z, r', z')
=
~ J(r', z') r'dr'dz' l~ ~ ii (qr)jKi(qr') + 2r(q) ii (qr')I e'~~~~~'~ dq
w ~
(32, I) dA(~~~(r, z, r', z')
=
~ J(r', z') r'dr'dz'
~ ~
Ii (qr')[Kj (qr) + £(q) Ii (qr)] e'~~~ ~~'~ dq
gr
~
(32.2) dA)~~~(r, z, r', z')
=
~ J(r', z') r'dr'dz' l~~ Ij(qr') Kj(qr) ~'~~~ e'~l~~~'~ dq
"
° ~r (32.3)
with
~ ~~~ q~(() i~i~j~)~~~(~
i
qb(p~- I)Ij(qb)I~(qb)
~P(q)
= qb(p~ I)Ii(Qb)Ko(qb) + Y'(q)
=
~~
qb p~ I) I
i(qb)Ko(qb) +
As specified by equation (30) we have
J(r, z)
= iwcr Aj(r, z) + sample dAj(r, z, r', z')1. (33)
Writing ~
= ip«w
=
~
and
j I+
m
J(r, z)
= J(r, q) e'~~ dq (34)
gr ~
we finally obtain I ~~
J (r, q) + all
i (qr) Ki (qa ) ~'~~
+ [Kj (qr) + £ (q ) Ii (qr )] l~ r' J (r', q)Ii (qr') dr' +
2 MT o
b
+ ii(qr) r'J(r', q)jKi(qr') + E(q) ii(qr')j dr'
=
o (35)
This expression gives us implicitly the function J(r, z but cannot be explicitly solved in the
general case. We are jut able to deterrnine the density of current J(r, q) in the two usual
following cases :
Low frequency behaviour » b. The two integrals can be neglected. In other words we
neglect the eddy currents vector potential, and the magnetic field penetrates the whole sample.
This approximation, of course, cannot be applied at high frequency as the bracketted terra in (33) is nearly equal to zero (the eddy currents vector potential compensates the applied one
everywhere in the cylinder except on the edges). The equation (35) gives J(r, q)
=
~ aIIj (qr) Kj (qa) ~'~~~
(36)
~ Hr
According to equations (32.2) or (32.3), we know that this density of current creates a vector
potential A(~,j at r = a, z = 0 on the pick- up coil. Using equations (32), (34) and (36) we deterrnine this vector potential :
A(~,, =
~~~~~~
j~ ~'~~~ ~ K((qa)[I((qb) Io(qb) I~(qb)] dq (37)
gr&~ o H,
whereas, following equation (31.2) the vector potential A(~,, created by the coil on itself is
equal to
A(~~, =
~ j~ ~ [Ii (qa + ~P(q Kj (qa )] Ki (qa dq (38)
" o
These vector potentials produce a flux ~P
=
Adi
= 2 wax A seen by the coil and,
co,I
according to Faraday's law, the resultant electromotance is
E=iw2graA=ZI. (39)
By use of (39), A(~,, gives the free inductance Zo
= Lo w of the coil whereas A(~~j is responsible
for the extra-impedance due to the presence of the sample. We find
I+ m
Lo = 2 Ho a~ Iii (qa + ~P(q )Kj(qa )j Kj (qa dq (40)
o
AZ
= (p w ab )~ cr ~ ~' ~Q ~K((qa II (qb) I~ (qb)1~(qb)] dq (41)
o H
r
Then, writing cr
= e", we finally obtain : PO
~~ 2
R
= ~ ~j cos q~ p~ (42)
b0
&L=»rii) )~~isin~
with &( x Mao w
=
I (cro
= I/p~)
and Cli " l~ ~j~ ~K((qa jJ((qb 12(qb lo (qb)i dq
o <
Equation (42) shows that R is proportional to the real part of the complex resistivity of the
sample. The proportionality factor contains the ratio between the penetration depth and the
typical length of the problem which is the algebraic mean of b and a.
High frequency behaviour : 16 « b. The situation is the same as in the susceptibility geometry : the current is confined in the edge of the cylinder on a thickness (. We then write J(r, q
=
J(q) (r b with (r b
= I/ for r
=
b. Then
jr r'J(r', q) Ij(qr') dr' =
0
0
and
b
r'J(r', Q)[Kj(Qr') + l~(Q)ij(Qr')I dr'
=
bJ(Q)[Kj(Qb) + # (Q) ij(Qb)I
r
Hence (35) becomes
I (J(q) b (r b) + a
all, (qr) Ki (qa)
+
~
+ Ii (qr) bJ(q iKj (qb ) + £ (q) Ii (qb)j
=
0 (43) which must -be satisfied for r
= b. The solution of (43) is al Y'(q )Ii (qb Ki (qa)
J(q )
"
~~ bit (qb [Ki (qb + £ (q ) I
i(qb + I /2
~
/
b
~ Kl
qj~~~(~~)(~~~
(qb
~
2 hi
I (qb>jKl(qb'
+ £ (q )ii (qb )j ~~~~
with I
= &~/(&(,
