האוניברסיטה העברית בירושליםTHE HEBREW UNIVERSITY OF JERUSALEM
The Racah Institute of Physics
AC & DC PROPERTIES OF NIOBIUM IN A MAGNETIC FIELD
by
Gilad Masri
A thesis submitted in partial fulfillment of the requirements for the
degree of
M.Sc. in Physics
Advisors: Prof. Israel Felner , Dr. Menachem Tsindlecht
The Hebrew University of Jerusalem
January 2011
ABSTRACT
In Part I, the real and imaginary part of the magnetic susceptibility of a Niobium sample was
measured, in an AC magnetic field superimposed on a DC magnetic field, at various frequencies,
amplitudes and sweep rates. The results were compared to Fink’s model, which was found incorrect.
Key parameters of the susceptibility such as skin penetration depth and surface current were
calculated and their dependence on sweep rate was investigated.
In Part II, the magnetic field inside an electropolished and annealed Niobium tube was measured
when external DC and AC magnetic fields were applied. The dependence of the internal magnetic
field on temperature and sweep rate was investigated, as was the effect of the electropolishing and
annealing. The variation in time of internal magnetic field at constant external magnetic field was
measured and compared to previous experiments.
תקציר
בתדירויות , AC עם רכיב DCנמדד החלק הממשי והמדומה של הסוספטיביליות המגנטית של דגם מניוביום בשדה , בחלק הראשון
חושבו פרמטרים . ולא נמצאה התאמה, התוצאות הושוו למודל של פינק. באמפליטודות אקסיטציה שונות ובקצב סריקה שונה, שונות
. ונחקרה התלות שלהם בקצב הסריקה, סוספטיביליות המגנטית כגון עומק החדירה וזרם השפהחשובים של ה
צילינדר זה עבר . AC ו DCשלו רכיבי , נחקרה התלות של השדה המגנטי בתוך צילינדר חלול מניוביום בשדה החיצוני, בחלק השני
והשינוי בתלות זו , שדה המגנטי הפנימי בשדה המגנטי החיצוניונחקר האפקט של עיבוד זה על התלות של ה, חישולליטוש אלקטרוכימי ו
וזו הושוותה לניסויים , נחקרה התלות בזמן של השדה המגנטי הפנימי בשדה מגנטי חיצוני קבוע, כמו כן. עם השינוי בטמפרטורה
.קודמים
ii
TABLE OF CONTENTS
PART I.......................................................................................................................................................................................1 THEORETICAL BACKGROUND..........................................................................................................................................1
PHENOMENOLOGY..........................................................................................................................................................1 THEORY............................................................................................................................................................................3
SAMPLES AND EXPERIMENTAL METHODS ..................................................................................................................17 EXPERIMENTAL SETUP.................................................................................................................................................17 SAMPLE ..........................................................................................................................................................................18 MEASUREMENT TECHNIQUE........................................................................................................................................19
RESULTS ............................................................................................................................................................................19 DISCUSSION.......................................................................................................................................................................29 CONCLUSIONS...................................................................................................................................................................37
PART II ...................................................................................................................................................................................38 THEORETICAL BACKGROUND........................................................................................................................................38
PHENOMENOLOGY........................................................................................................................................................38 THEORY..........................................................................................................................................................................39
SAMPLES AND EXPERIMENTAL METHODS ..................................................................................................................45 RESULTS ............................................................................................................................................................................48
RESULTS FOR UNPOLISHED SAMPLE ..........................................................................................................................48 RESULTS FOR POLISHED AND ANNEALED SAMPLE ...................................................................................................53
DISCUSSION.......................................................................................................................................................................61 CONCLUSION.....................................................................................................................................................................66
iii
ACKNOWLEDGMENT
The author wishes to thank his supervisors, Prof. Israel Felner & Dr. Menachem Tsindlecht for their
assistance and economical support; to Hezi Shlussel for his assistance with the Matlab software; to
Esteban Malel for his help during the electrochemical polishing; and to his parents Aida Babor &
Roberto Masri for their encouragement and economical support.
PART I
THEORETICAL BACKGROUND
PHENOMENOLOGY
Superconductivity is a physical phenomenon, which was investigated for the last 100 years. In 1911, H.
Kamerlingh-Onnes discovered that by cooling a mercury (Hg) sample below a certain critical
temperature TC, its resistance dropped dramatically. This was found to happen in many other metals
such as Al, Cd, Nb. In 1933, W. Meissner & R. Ochsenfeld discovered that besides being an ideal
conductor, a superconducting material has interesting magnetic properties: if a sample was cooled
below TC and then put in an external magnetic field H, the internal
magnetic field B would be zero. The same result would be obtained if the
sample would be put in an external magnetic field and then cooled below
TC. The diagram on the right shows this effect. Below CT , the sample will
be superconducting, given that the magnetic field is below CH . This
critical field decreases with rising temperature, and reaches zero at TC, as
shown on the diagram below. This type of superconductors is named Type I superconductors, or soft
superconductors.
Type II, or hard superconductors, have a different response to external magnetic fields for CT T< . This
is shown on the diagram on the next page. These diagrams describe the internal magnetic field B as
function of the external magnetic field H, below the critical temperature TC. Above this temperature, the
behavior of the metal is normal, as the dashed lines in the diagrams show. In Type II superconductors,
in the range 1 2C CH H H< < , the metal is in the “Vortex State” or “Abrikosov State”. In this range, the
metal is still in a superconducting state, but it is penetrated by magnetic field lines, which are
surrounded by small circulating electrical currents. These currents mask the magnetic field lines, so the
rest of the metal is superconducting.
H<HC H>HC
2
Above HC2 , the metal enters another state – the
Superconducting Sheath State, denoted SSS. In this state, the
bulk of the metal is normal, and magnetic field lines can
penetrate it. However, on the surface of the sample there is a
sheath that is still superconducting. This sheath is a region of
the metal that has opposing circular electrical currents, whose
sum is zero. The diagram on the right describes this for a
cylindrical specimen.
The existence of this third critical field is supported by
experiments, which compare the magnetization of the sample
and its resistivity ratio, as the diagram below shows.
Many other experiments, such as susceptibility measurements, also support the existence of HC3 . The
ratio HC3/ HC2 has been measured for different metals to be of the order of 1.69, which is the theoretical
value.
H
B
Type I
HC
B
H
Type II
HC1 HC2
H
T
Meissne
Abrikosov’
H
T
SSS
NormalNormal
HC1
HC2
HC3
HC
Meissne
H
N S
3
THEORY
The London Model
The earliest model describing superconductivity is the London Model (1935), which is summarized by
the following two equations:
2
22
2
( )
0 4
SCS
S
d mE Jdt n e
mcH Hn e
λ λπ
= Λ Λ ≡
+ ∇×∇× = ≡
ur uuur
uur ur ur uur
In these equations nS is the density of superconducting electrons, m is the electron mass and JSC is the
superconducting current density. If we choose the Coulomb gauge for the magnetic vector potential
( A H∇× =ur ur uur
), nr
being the unit vector normal to the sample’s surface:
0 0A A n∇ ⋅ = ⋅ =ur ur ur r
than the second London equation will be:
24SCcJ A
πλ= −
uuur ur
The parameter is the London penetration depth, and it is a measure for the penetration of the external
magnetic field into the specimen. It varies with temperature through the empirical formula, 0 being the
zero-temperature London penetration depth:
( )0
4( )
1 / C
TT T
λλ =−
The London equations are purely classical, and do not take into account the electron’s quantum nature.
To correct this, the generalized London equation assumes that the electrons are described by a wave
function
4
( ) ( )2
i rSnr e θψ =rr
A particle with mass * 2 em m= and charge * 2e e= moving in a magnetic potential Aur
has a momentum
24 e sep m v A
c= +
ur ur ur
In the absence of magnetic potential, the particle’s flow density is given by
( )* *
2 4s s
e
v n i em
ψ ψ ψ ψ= ∇ − ∇ur
ur urh
If we insert the expression for the particle’s wave function, we get
2 e sm vθ∇ =ur urh
If now add the magnetic potential, we can see that the superconducting current is:
22
sSC s s
e
n e eJ n ev Am c
θ = = ∇ −
uuur ur ur urh
If we now insert the constants and 0 the equation can be rewritten as:
002
1 2 2
eSC
s
m hcJ Ac n e e
θπ
Φ = ∇ − Λ ≡ Φ ≡ Λ
uuur ur ur
The constant 0, termed “fluxon”, has the units of magnetic flux. As will be seen later on, it is the
smallest amount of flux that can enter a superconducting sample.
The charged particles in this model have twice the electron charge and mass because they are pairs of
electrons, as was shown later by Bardeen, Cooper and Schreiffer. In a normal metal, the charge carriers
5
are single electrons, and therefore the flux quantum in the normal state is twice that of the flux quantum
in the superconducting state.
The Ginzburg-Landau Theory
Albeit successful in describing the electromagnetic fields in a superconductor, the London model is not a
quantum model. It does not take into account that the superconducting state is a more ordered than the
normal state, and does not explain the evidence showing that the superconducting to normal transition
is a second-order phase transition. The Ginzburg-Landau Theory is a more suitable theory, and it
assumes that the electron wave function is related to the superconducting electron density by the
equation:
2( )
2Snrψ =
r
This theory is semi-empirical, and assumes that the free energy can b described by:
2 40 2S NF F βα ψ ψ= + +
The constants and are phenomenological parameters that depend on the metal considered. They
can be calculated by minimizing the free energy and comparing it with the free energy difference at the
transition ( CMH being the critical magnetic field at a given temperature below CT ):
2
0 22
2
0
48
2
CMN S
CM
N S
HF FH
F F
παπβα
β
− = ⇒ =
− =
The GL equations are a set of two quantum mechanical equations for the electronic wave function ψ
and the vector potential Aur
. The GL model’s starting point is the Gibbs free energy of a superconductor:
6
( ) ( ) ( )22
02 4 1 2,2 4 8 4S N
A A HeĞ A Ğ i A dVm c
βψ α ψ ψ ψ ψπ π
∇ × ∇× ⋅ = + + + − ∇ − + − ∫
ur ur ur urur ur ur
h
The first equation is derived from the minimization of the Gibbs free energy with respect to ψ :
( )22 1 04
i Am
αψ βψ ψ ψ+ + ∇ + =ur urh
This equation is restricted to the following boundary condition:
2 0ei A nc
ψ ψ ∇ + ⋅ =
ur ur rh
This is a physical requirement, meaning no resistance-free electrical current (or supercurrent) pass
through the superconductor’s surface. The second GL equation emanates from the minimization of the
Gibbs free energy with respect to Aur
:
( )2
2* * 22SCi e eJ A
m mcψ ψ ψ ψ ψ= − ∇ − ∇ −
uuur ur ur urh
By setting the following definitions:
220 2
0
2 44S
S
n mcn em
ψψ ψ ξ λψ πα
≡ → ≡ ≡h
we arrive at the standard formulation of the GL equations:
( )
222
0 0
2* *0
2 2
2 20 0
4
i A i A n
A i A
π πξ ψ ψ ψ ψ ψ ψ
ψψ ψ ψ ψ
πλ λ
∇ + − + = ∇ + ⋅ = Φ Φ
Φ∇×∇× = − ∇ − ∇ −
ur ur ur ur r
ur ur ur ur ur ur
7
These equations are gauge invariant, as are Maxwell’s equations, and they add another important
parameter – the coherence lengthξ . This parameter is a measure of the variation of the superconducting
electron wave function ψ inside the specimen. The GL theory assigns ψ another important role: it is the
order parameter of the theory:
0 Normal State
0 1 Superconducting State
ψψ=
< ≤
The ratio between these two key parameters is defined as /κ λ ξ≡ , and it marks the distinction
between Type I and Type II superconductors. For Type I superconductorsλ ξ< , while for Type II
superconductorsλ ξ> . This difference is closely related to the surface energy of the Normal-
Superconductor interface in a given sample. To show this, let us assume that the interface is
perpendicular to the x-axis and that the magnetic field is parallel to the z-axis. In addition, due to the
simple geometry of the problem, all variables vary with x and the vector potential Aur
is assumed parallel
to the z-axis. For these assumptions, and if the origin lies on the interface, the GL equations can be
written as:
22 2 22 3
2 2 20
2 0 d d AA Adx dx
ψ πξ ψξ ψ ψ ψλ
− + − + = = Φ
urur
If we integrate with respect to x and combine the two equations, we arrive at the following equation:
2 2 242
0 0
2 212
A dA d Cdx dx
πξ ψ πλξ ψψ ξ − − + + = Φ Φ
Here, C is an integration constant that can be found from the boundary conditions. Since we assume the
material is superconducting to the left of the interface, i.e. on the negative part of the x-axis, the
boundary conditions are:
1 0 0x x xd Adxψψ →−∞ →−∞ →−∞ → → →
ur
8
With these boundary conditions, the constant is 12
C = .
From the definition of the Gibbs free energy we can arrive at the Gibbs free energy deep inside the
superconducting domain:
0 / 4s s cmG F HH π= −
Here, Fs0 is the free energy density of the superconductor without magnetic field. Since at the far left
0H = , we get 0s sG F= . Far to the right of the interface, the free energy density is:
2 / 8n cmF F H π= +
In the normal domain, the Gibbs free energy density is:
2 2 2
04 8 4 8cm cm cm cm
n n n sH H H HG F H F F F
π π π π= − = + − = − =
Here, we have used the relation between Fn and Fs0 written above. This result means that in equilibrium,
the Gibbs free energy density of the normal domain is equal to the free energy in the superconducting
state (both far from the interface).
At the interface, the two must vary, and we can define the surface energy of the interface as:
( )ns sH nG G dxσ∞
−∞
= −∫
If we substitute the expressions for the Gibbs free energy density in the superconducting state with
presence of a magnetic field H and the Gibbs free energy density for the normal state, we arrive at the
following expression for the surface energy of the interface:
( )22
22 2cmcm
nscm
H H HH d dxdx Hψσ ξ
π
∞
−∞
− = +
∫
9
If we analyze this result we can see that, since the magnetic field in the superconducting region is
always smaller than Hcm , the second term inside the integral is negative. To evaluate the first term we
must make some evaluations. When going from the normal domain to the superconducting domain, the
order parameter changes from 0 to 1 in a distance of the order of . Therefore:
221 1d d
dx dxψ ψξ
ξ ⇒
This term is nonzero over a distance x ξ and therefore:
( )22 /d dx dxξ ψ ξ∞
−∞∫
The second term in the integral for ns is approximately -1 at the interface and approaches zero both
deep inside the normal domain and deep inside the superconducting domain. The area where it is
nonzero extends over a distance the order of . Hence its contribution to the integral is roughly – .
Hence, we conclude that
( )2
cmns
Hσ ξ λπ
−
Now we can see that for ξ λ , the surface energy must be positive, while for ξ λ , it must be
negative. If we define the parameter κ as the ratio between and , we can summarize this conclusion
as:
1 01 0
NS
NS
κ σκ σ
⇒ >⇒ <
The diagrams below show how this variation in the surface energy affects the interface region of the
superconductor and the normal metal:
10
1κ 1κ
Since the order parameter is related to the superconducting electron density (and this density increases
as the temperature increases), and since experiment shows that for Type I superconductors the magnetic
field abruptly penetrates the metal when CT T> , the conclusion must be:
1 0 for Type I1 0 for Type II
NS
NS
κ σκ σ
><
Calculations that are more exact show that the value of κ at which the surface energy is zero is:
1 02 nsκ σ= ⇔ =
Therefore the correct relation between κ and ns to Type I and Type II superconductors is:
1 / 2 0 for Type I
1/ 2 0 for Type IINS
NS
κ σ
κ σ
< >
> <
Surface Superconductivity
11
Consider a semi-infinite Type II superconductor, whose surface coincides with the y-z plane. If we
assume that 2CH H≈ , than we can assume that 1ψ , and therefore 2ψ ψ ψ . The first GL
equation (in C.G.S. units) will
then be
The vector potential can be chosen to be (0, ,0)A Hx=ur
, and then the first GL equation is:
22 2 2 2
2 2
1 2 04 4 4
ei Hxm x m y c m z
ψ ψαψ ψ ∂ ∂ ∂− + + − = ∂ ∂ ∂
h hh
D. Saint-James & P.G. de Gennes showed that from this equation, the following relation could be
derived:
3 1.695 2C cmH Hκ=
Here, cmH is the critical thermodynamic field, meaning the field for which superconductivity is
destroyed. For Type II superconductors (12
κ > ), 2 2C cmH Hκ= . Hence, we arrive at the relation:
3 21.695C CH H=
Surface superconductivity also exists for Type I superconductors (12
κ < ).
AC Response of Type II superconductors: The Critical State Model
21 2 04
ei Am c
αψ ψ + ∇ + =
ur urh
12
The superconducting sheath state (SSS) was investigated theoretically and experimentally during the
1960’s, and it was found that this sheath is capable of supporting lossless currents up to a critical value.
In 1964, Strongin, Schweitzer, Paskin & Craig1 investigated the real and imaginary susceptibility, ’and ’’
at an AC field superimposed on a DC field, for Type I and Type II superconductors. Their Type II
sample was copper plated 2% Bi-In-Pb. They were the first to find that the real part of the susceptibility
strongly depends on the sweep rate.
In their article from 1967, Rollins & Silcox2 proposed a model for the complex susceptibility of the
specimen (Pb-2wt% In in their experiment). They applied a varying magnetic field with the form
( ) 0 cosH t h tω=
They proposed that the induced magnetic field inside the specimen should be:
( ) ( ) ( )00
' cos '' sinn nn
B t H n t n tµ ω µ ω∞
=
= + ∑
Here, 'µ is the real part of the permeability and ''µ is the imaginary part of the permeability. The
permeability is related to the susceptibility through1
4µχ
π−= . In other words, the permeability is defined
as BH
µ ≡ . From this definition, it is easy to see why the imaginary part of permeability (and hence the
imaginary part of the susceptibility) is related to the energy loss in the metal:
( )( )
( )0 0 0( )
0 00
cos sini t
ii t
H t H e B Be iH HB t B e
ωδ
ω δµ δ δ
+
= = = +=
1 Strongin M., Schweitzer D.G., Paskin A., Craig P.P., Phys. Rev. , 136(4A) , A926-A928 (1964)
2 R.W. Rollins & J. Silcox, Phys. Rev. 155(2), 404-418 (1967)
13
Referring to mh as the AC critical field amplitude, the induced magnetic field would be zero for 0 mh h< .
When 0 mh h> , a temporary induced magnetic field would emerge. They arrived at the theoretical
conclusion that the permeability should be related to the ratio 0/mh h through:
( ) ( ) ( )
( ) ( )
1
21
10
2 sin 2 sin 1 sin 11' ' n odd2 1 1
1 cos 1 1 cos 1sin 1'' '' 2sin1 1
n
mn
n nn n n
n n hn n n h
π θ θ θ θµ µ
π π
θ θθµ µ θπ π
−
− + + −= = − + −
− + − −= = − ≡ + −
In this model, the even harmonics are all zero, because of the cylindrical symmetry of the model. This
model is termed “Critical State Model”.
Earlier, in 1965 Maxwell & Robbins3, and in 1966 Schwartz & Maxwell4, found that the complex
susceptibility ' ''iχ χ χ= + is a function of the parameter
0hdH
dt
ωγ =
Here, the applied field has the following form:
( ) 00 0 cosdHH t H t h t
dtω= + +
Their results showed that γ is independent of the field ( )H t in the range 1 2C CH H H< < . Another
result was that 1
4χ
π= − , in this range of fields, but
only for 1γ < . For 1γ > , 'χ decreased while ''χ
increased (see picture on right). This continued until
3 E. Maxwell & W.P. Robbins, Phys. Lett. 19(8), 629-631 (1965)
4 B.B Schwartz & E. Maxwell, Phys. Lett. 22(1), 46-47 (1966)
14
a certain value of γ , at which both parts decreased. Theoretically, for any metal the susceptibility is
defined by:
( ) 4 1 4M H B H M Hχ π πχ= = + = +
For a superconductor 0B = , and therefore1χπ
= −4
. Comparing this to the experimental results,
Schwartz & Maxwell deduced that the imaginary part of the susceptibility ''χ is related to the energetic
losses due to normal currents in the superconductor which arise from the fast change in external
magnetic field. This may be seen as the metal’s inability to compensate for the external magnetic field.
Schwartz & Maxwell showed that the complex 1st harmonic susceptibility is given by:
( ) ( ) ( )
( ) ( )
1
3
2
1 320
1 41' '' 2
4 4
cos cos
H t i i
n n n n n
dMdH
i e d e d i
t
τ πτ τ
τ
πχ χ β τ τ β τ τ π
π γ π
τ ω β τ τ τ γ τ τ
+
+ = + + −
≡ ≡ − + −
∫ ∫
Here, the magnetic field cycle is divided as follows:
1. 10 t t≤ ≤ , where the magnetic field is increasing and the system follows the magnetization curve
( )M H .
2. 1 2t t t≤ ≤ , where the magnetic field is decreasing and the system follows a diamagnetic line with
slope 1
4dMdH π
= −
3. 2 3t t t≤ ≤ , where ( ) ( )1 3H t H t= and
0dMdH
> again.
15
4. 32t t πω
≤ ≤ , where the system continues along the magnetization curve ( )M H .
The diagram on the right, from the original article, shows this time division. It shows how the
magnetization ( )M t will behave (solid bold line) and how the induced field ( )B t will behave
according to this model.
AC Response of Type II superconductors: Fink’s Model
In 1967, H.J. Fink5 offered his own theoretical model, which applies for low frequencies. He found that
in addition to the parameter 1qγ
≡ , there is another parameter ( )0
0
Ch Hp
h≡ which defines the response
of the superconductor in a swept field. Here, Ch is a critical AC amplitude which depends on the swept
magnetic field 0H . He numerically calculated the first, second and third harmonics of the permeability as
function of p and q . Fink’s model assumes that relaxation effects can be ignored, meaning that 1ωτ ,
when τ is the relaxation time of the
superconductor.
In Fink’s model, there are 6 time-points, defined 1x
to 6x ( i ix tω= ). These points are shown on the
diagram on the right. In this model, B is the
5 H.J. Fink, Phys. Rev., 161(2), 417-422
16
average induced magnetic field, and Cb is the deviation from it during the cycle.
Fink was able to construct a function which describes ( )B t . It is similar to a sinus wave superimposed
on the increasing DC field. However, when the external field decreases, at point 2x , the surface sheath
will screen this change. It will continue to screen only if the deviation in the external field 0h is not too
large. If it is too large, and if 1q < , it may cause the induced field to decrease too. This is shown on the
upper-right side of the diagram, where the curve ( )B t changes from the points 1, 2, 3, 5A, 6 to 1, 2, 3,
4, 5, 6. The value of 0h which defines if the curve will change is given in the condition:
10 0 2 2 2 cos sin
2p p p x q x x qπ − = ≡ + − ≡
If 0p p> , than 0h is small enough and the screening will succeed (curve 1, 2, 5A, 6). On the contrary, if
0p p< the screening will not suffice (curve 1, 2, 3, 4, 5, 6). The point 2x is the point at which the total
external field is at its local maximum. The point 4x is the contrary one, where the external field is at its
local minimum. The point 3x is the point at which the screening currents in the sheath cease to screen
the external field. It is defined as:
( ) ( )3 4 2 CB x B x b− =
The function ( )B t was calculated using a numerical method and then decomposed to its Fourier
components. Assuming the following form for ( )B t :
( ) ( ) ( ) ( )0 0 0 01 1
cos sin ' cos '' sinn n n nn n
B t b a b n t a n t h c n t n t H tω ω µ ω µ ω µ∞ ∞
= =
= + + = + + =
∑ ∑
The results were plotted for the 1st, 2nd and 3rd harmonics (real and imaginary part), and are shown in the
original paper.
17
In this thesis, I will compare the results obtained for Niobium to the existing models.
SAMPLES AND EXPERIMENTAL METHODS
EXPERIMENTAL SETUP
The measurement system, as described in the diagram, is composed of a commercial MPMS SQUID
magnetometer, which has two superconducting coils capable of generating magnetic field. The DC coil
generates the DC field, and reaches a maximal field of 5 Tesla. There were two experimental setups
used:
1. In point-by-point setup, the magnetometer and the DC coil were controlled by the MPMS5
software.
2. In swept-field setup, the magnetic field was measured through a voltmeter connected in series
with a load resistor to the SQUID superconducting solenoid, and the DC coil was connected to a
power supply controlled by the Matlab R2008b software.
18
In DC measurements in point-by-point mode, the
sample was inserted to the sample chamber. There,
the pick-up coil inside the SQUID measured the
induced field inside the sample.
In AC measurements (point-by-point or swept field),
the samples were put in one of two pick-up coils
(made in the laboratory) that are connected in series
and have opposing screw-direction. These coils are
balanced so that the potential drop measured by
both of them, in the absence of a sample, is almost
zero. When a sample was inserted in one of these
coils and the driving AC coil was connected to a waveform generator (LFG), the magnetic flux variation
in this sample causes an induced voltage drop in the coils:
1 dc dt
ε Φ= −
This voltage drop was measured by three lock-in amplifiers (LCK), that were synchronized with the
waveform generator. One of them measured the first and third harmonics, and the other measured the
second harmonic. The waveform generator and the lock-in amplifiers, as well as the voltmeter
measuring the DC magnetic field and the power supply were controlled by the Matlab R2008b software.
The temperature in all measurements was 4.5ºK.
SAMPLE
The sample used in this part is a Niobium (Nb) single
crystal sample, with a measured Residual Resistivity Ratio
(RRR) of 200. This sample was mechanically polished, and
has the form of a parallelogram of sizes 1 mm by 2.4 mm
by 10 mm. The sample was put in a way that the AC and
10 mm
1 mm
2.4
H(t)
19
DC magnetic fields were parallel to its longest side, as shown in the picture.
MEASUREMENT TECHNIQUE
In order to get a broad picture of the behavior of the 1st, 2nd and 3rd harmonics (real and imaginary), the
waveform generator was set to give frequencies from 146.5Hz to 1465Hz, with excitation amplitudes
0.05Oe to 0.2Oe. The DC magnetic field was raised from slightly below zero to above HC3. The sweep-
rate of the DC field, dH/dt, was also varied from 0 Oe/sec (point-to-point measurements) to 30 Oe/sec.
RESULTS
The results will be divided to three parts:
1. The results for varying sweep rate at constant frequency and amplitude.
2. The results for varying frequency at constant sweep rate and amplitude.
3. The results for varying amplitude at constant sweep rate and frequency.
For each section, the two variables that remain constant were chosen at two different values. The results
shown here were normalized so that the real and imaginary parts of the susceptibility (first harmonic) at
zero DC field would be:
20
1 11' '' 0 at 0
4 DCHχ χπ
= − = =
Regarding the second and third harmonics, the absolute value of them will be plotted against the
magnetic DC field.
First, we show the magnetization graph for the sample:
0 1 2 3- 2 . 0
- 1 . 5
- 1 . 0
- 0 . 5
0 . 0
HC 1
Mag
netic
mom
ent (
emu)
M a g n e t i c f i e l d ( k O e )
Hc 2
M a g n e t ic m o m e n t V s . H D C , T = 4 . 5 K
Varying Sweep Rate
21
0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0
- 0 .0 8
- 0 .0 6
- 0 .0 4
- 0 .0 2
0 .0 0
0 .0 2
0 .0 4
χ 1' & χ
1''
H D C [O e ]
0 O e /s5 O e /s1 0 O e /s 2 0 O e /s 3 0 O e /s
χ1' & χ
1' ' V s . H D C fo r f = 1 4 6 .5 H z h A C = 0 .2 O e
H C 2H C 1
H C 3
0 1000 2000 3000 4000 5000 6000
0.000
0.002
0.004
0.006
0 Oe/sec 5 Oe/sec 10 Oe/sec 20 Oe/sec 30 Oe/sec
|χ2| Vs. HDC , hAC = 0.2 Oe, f = 146.5 Hz
|χ2 |
HDC
[Oe]0 1000 2000 3000 4000 5000 6000
0.0000
0.0005
0.0010
0.0015
|χ3| Vs. HDC , hac= 0.2 Oe, f = 146 Hz
0 Oe/sec 5 Oe/sec 10 Oe/sec 20 Oe/sec 30 Oe/sec
HDC [Oe]
|χ3|
22
0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0
-0 .0 8
-0 .0 6
-0 .0 4
-0 .0 2
0 .0 0
0 .0 2
0 .0 4
0 .0 6
H D C [O e ]
χ 1' & χ
1''
χ1' & χ
1' ' V s . H D C fo r f = 1 4 6 .5 H z h A C = 0 .0 2 O e
0 O e /s 5 O e /s 1 0 O e /s 2 0 O e /s 3 0 O e /s
0 1000 2000 3000 4000 5000 6000
0.000
0.001
0.002
0.003
|χ
2| Vs. H
DC , h
AC = 0.02 Oe, f = 146.5 Hz
0 Oe/s 5 Oe/ s 10 Oe/s 20 Oe/s 30 Oe/s
|χ2|
HDC [Oe]0 1000 2000 3000 4000 5000 6000
0.0000
0.0005
0.0010
0 Oe/s 5 Oe/s 10 Oe/s 20 Oe/s 30 Oe/s
|χ3| Vs. H
DC , f = 146.5 Hz h
AC = 0.02 Oe
|χ3|
HDC
[Oe]
23
0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0
-0 .0 8
-0 .0 7
-0 .0 6
-0 .0 5
-0 .0 4
-0 .0 3
-0 .0 2
-0 .0 1
0 .0 0
0 .0 1
0 .0 2
0 .0 3
0 .0 4
H D C [O e ]
χ 1' & χ
1''
0 O e /s5 O e /s1 0 O e /s 2 0 O e /s 30 O e /s
χ1' & χ
1' ' V s . H D C fo r f = 1 4 6 5 H z h A C = 0 .0 2 O e
0 1000 2000 3000 4000 5000 6000
0.0000
0.0002
0.0004
0.0006
0.0008
|χ2|
HDC [Oe]
0 Oe/s 5 Oe/s 10 Oe/s 20 Oe/s 30 Oe/s
|χ2| Vs. HDC , f=1465 Hz hAC=0.02 Oe
0 1000 2000 3000 4000 5000 6000
0.00000
0.00005
0.00010
0.00015
0.00020
0.00025
|χ3|
HDC [Oe]
0 Oe/s 5 Oe/s 10 Oe/s 20 Oe/s 30 Oe/s
|χ3| Vs. HDC , f=1465 Hz hAC=0.02 Oe
24
0 1000 2000 3000 4000 5000 6000
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
H DC [O e]
χ 1' & χ
1''
χ1' & χ
1'' Vs. H DC for f = 1465 Hz hAC = 0.05 O e
0 Oe/s10 Oe/s 20 Oe/s30 Oe/s
0 1000 2000 3000 4000 5000 6000
0.0000
0.0005
0.0010
|χ2|
0 Oe/s 10 Oe/s 20 Oe/s 30 Oe/s
HDC [Oe]
|χ2| Vs. HDC , f = 1465 Hz hAC = 0.05 Oe
0 1000 2000 3000 4000 5000 6000
0.0000
0.0001
0.0002
0.0003
0.0004
HDC [Oe]
|χ3|
|χ3| Vs. HDC , f = 1465 Hz hAC = 0.05 Oe
0 Oe/s 10 Oe/s 20 Oe/s 30 Oe/s
25
Varying Frequency
0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0
-0 .0 8
-0 .0 6
-0 .0 4
-0 .0 2
0 .0 0
0 .0 2
0 .0 4
H D C [O e ]
χ 1' & χ
1''
χ1' & χ
1' ' V s . H D C , S R = 1 0 O e /s h A C = 0 .2 O e
1 4 6 .5 H z 2 9 3 H z 5 8 6 H z 8 7 9 H z 1 1 7 2 H z 1 4 6 5 H z
0 1000 2000 3000 4000 5000 6000
0
1
2
3
4
5
6
|χ2| Vs. H
DC , SR = 10 Oe/s h
AC=0.2 O e
146.5 Hz 293 Hz586 Hz 879 Hz1172 Hz 1465 Hz
|χ2| [
mkV
olt]
HDC [Oe]0 1000 2000 3000 4000 5000 6000
0
2
4
6
HDC [Oe]
|χ3| [
mkV
olt]
|χ3| Vs. HDC , SR = 10 Oe/s hAC=0.2 Oe
146.5 Hz 293 Hz586 Hz 879 Hz1172 Hz 1465 Hz
26
0 1000 2000 3000 4000 5000 6000
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
H DC [Oe]
χ 1' & χ
1''
χ1' & χ
1'' V s. H DC , SR = 30 O e/s hAC=0.1 O e
146.5 H z 293 H z 586 H z 879 H z 1172 H z 1465 H z
0 1000 2000 3000 4000 5000 6000
0
1
2
3
4
146.5 Hz 293 Hz 586 Hz 879 Hz 1172 Hz 1465 Hz
HDC [Oe]
|χ2| [
mkV
olt]
|χ2| Vs. HDC , SR = 30 Oe/s hAC=0.1 Oe
0 1000 2000 3000 4000 5000 6000
0
1
2
3
HDC [Oe]
|χ3| [
mkV
olt]
146.5 Hz 293 Hz 586 Hz 879 Hz 1172 Hz 1465 Hz
|χ3| Vs. H
DC , SR = 30 Oe/s h
AC=0.1 Oe
27
Varying Amplitude
0 1000 2000 3000 4000 5000 6000
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
χ 1' & χ
1''
χ1' & χ
1'' V s . H D C , S R = 10 O e/s f = 146 .5 H z
H DC [O e]
0 .05 O e 0 .1 O e 0 .15 O e0 .2 O e
0 1000 2000 3000 4000 5000 6000
0
1
2
3
4
5
H D C [O e]
|χ2| [
mkV
olt]
|χ2| Vs. H DC , SR = 10 O e/s f = 146.5 Hz
0.05 Oe0.1 O e 0 .15 Oe0.2 O e
0 1000 2000 3000 4000 5000 6000
0.0
0.5
1.0
1.5
2.0 0.05 Oe0.1 Oe 0.15 Oe0.2 Oe
HDC [Oe]
|χ3| [
mkV
olt]
|χ3| Vs. HDC , SR = 10 Oe/s f = 146.5 Hz
28
0 1000 2000 3000 4000 5000 6000
-0 .08
-0 .07
-0 .06
-0 .05
-0 .04
-0 .03
-0 .02
-0 .01
0 .00
0 .01
0 .02
0 .03
0 .04
0 .0 5 O e 0 .1 O e 0 .1 5 O e0 .2 O e
χ1' & χ 1'' V s . H D C , S R = 3 0 O e/s f = 8 79 H z
χ 1' & χ
1''
H D C [O e]
0 1000 2000 3000 4000 5000 6000
0
2
4
6
8
HDC [Oe]
|χ2| [
mkV
olt]
0.05 Oe0.1 Oe 0.15 Oe0.2 Oe
|χ2| Vs. H
DC , SR = 30 Oe/s f = 879 Hz
0 1000 2000 3000 4000 5000 6000
0
1
2
3
4
5
|χ3| [
mkV
olt]
H DC [Oe]
0.05 Oe0.1 Oe 0.15 Oe0.2 Oe
|χ3| Vs. HDC , SR = 30 Oe/s f = 879 Hz
29
DISCUSSION
Determination of the Critical Fields
From the DC magnetization curve and from the swept field measurements we can determine HC1 and HC2
:
1
2
1000 50 Oe2600 50 Oe
C
C
HH
= ±= ±
From the swept-field graphs we can determine HC3 :
3 4850 20 OeCH = ±
The swept-field graphs coincide with the magnetization graph, and show a rise at HC1 and a peak at HC2 .
From these critical field we can calculate the parameter κ , through the formula6 for the magnetization at
a field close to HC2:
( )2 0
24
1.16 2 1CH HMπ
κ−− =
−
From a fit to the magnetization curve close to HC2 , the slope was extracted and κ was calculated to be:
1.6 0.1κ = ±
This value is in accordance with the well-known value from literature. Another parameter that can be
calculated is the ratio 3 2/C CH H :
3
2
1.87 0.04C
C
HH
= ±
6 See Schmidt V.V., “The Physics of Superconductors: Introduction to Fundamentals and Applications” , Springer (1997), pp 110 - 113
30
This ratio is close to the theoretical value, which is 1.695.
The effect of swept DC field on measurements7
From the results for different sweep-rates, we can see that 1’’ exhibits a minimum at HC2, which means
that there is a change in the underlying mechanism that is responsible for dissipation in the vortex state
and in the surface superconducting state. This minimum is also apparent at 1’ and at the higher
harmonics. We notice that as the sweep rate rises, the signal from the first and second harmonics rises
also. However, the signal from the third harmonic decreases with increasing sweep rate.
To better see the effect that swept-field has on the susceptibility, the susceptibility at zero sweep rate
(point to point measurements) was subtracted from the susceptibility at other sweep-rates:
( ) ( )( ) ( )
1 1 1
1 1 1
' ' 0 ' 0
'' '' 0 '' 0
SR SR
SR SR
χ χ χχ χ χ
∆ = ≠ − =
∆ = ≠ − =
0 1 2 3 4 5
0 . 0 0
0 . 0 1
0 . 0 2
0 . 0 3
0 . 0 4
0 . 0 5
H D C [ k O e ]
∆ χ ' ; 5 O e / s ∆ χ ' ; 3 0 O e / s ∆ χ ' ' ; 5 O e / s ∆ χ ' ' ; 3 0 O e / s
H D C [ k O e ]
1 4 6 H z
∆ χ'
& ∆
χ''
0 1 2 3 4 5
0 . 0 0 0
0 . 0 0 5
0 . 0 1 0
8 7 9 H z
∆ χ1' & ∆ χ 1 ' ' V s . H D C , f = 1 4 6 . 5 H z & 8 7 9 H z , h A C = 0 . 1 O e
7 The results shown here and in the following pages are after the results and discussion in the article: Tsindlekht M.I., Genkin V.M, Leviev
G.I., Schlussel Y., Masri G., Tulin V.A., Berezin V.A., “Measurement of the AC conductivity of a Nb single crystal in a swept magnetic field”. (Submitted to Europhysics Letters)
31
These graphs show that as the frequency increases, the difference between point-to-point and swept-
field measurements decreases, both in 1’ as in 1’’. Near HC3 there is a region where the susceptibility at
swept-field is lower than in point-to-point measurements. This difference increases at increasing
frequency, and this is seen in the results. The range of magnetic field in which this happens is smaller
for higher amplitude. . We also note that there is a certain DC field above which this difference
decreases as the sweep rate increases. This phenomenon might be related to the sheath state
Comparison with Fink’s model
According to Fink’s model, the AC response of the superconducting sheath state in a swept field is a
function of only two parameters, p and q. To compare the results to this model, we compared data for
two different sets of frequencies, amplitudes and sweep-rates. We calculated the surface current, derived
in the following way.
Assuming that the thickness of the surface layer is small in comparison to the sample sizes, we can
separate the measured susceptibility to two parts: the susceptibility for 3DC cH H> is • , and the
susceptibility for 3DC cH H< . To find the surface current, we can write
( ) ( )0 00
41 4 1 4 14
SS
Jh h J
c hχ χππχ πχ
χ π
∞∞
∞
− + = + − ⇒ = +
For the first set, we took sweep-rate 5 Oe/s,
and frequency 146.5 Hz. For the second set,
we took sweep-rate 30 Oe/s, and frequency
879 Hz. The AC amplitude was 0.1 Oe for
bot sets.
The parameter p is equal in both sets since
the critical AC amplitude hC is independent
of the sweep rate and frequency and hAC is
the same. The parameter q is also the same,
since the product of the sweep rate and the
2.0 2.5 3.0 3.5 4.0 4.5 5.0-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
J' s/h
0
J'' s/h
0
HDC (kOe)
146.5 Hz; 5 Oe/s 879 Hz; 30 Oe/s
T = 4.5 Kh0 = 0.1 Oe
JS'/h
0 & J
S''/h
0 Vs. H
DC , T=4.5 K h
AC=0.1 Oe
32
frequency is the same. The result of this comparison is shown on the right. As can be seen, there is a
discrepancy between the two sets, and therefore Fink’s model is incorrect.
Calculation of the skin penetration depth
To find the skin penetration depth we must turn our attention to Maxwell’s equations (neglecting the
displacement current in Ampère’s law) combined with the definition of complex electric conductivity:
( )( )2
1 22
1 2
4 14
0
HH J E HH ic c tc t
J i E H
ππ σ σ
σ σ
∂∇× = ∇× = − ∂ ∇ = − +∂ ∂= + ∇ =
uurur uur ur ur ur uur
uur
ur ur ur uur
Assuming the magnetic field is in the z direction and oscillates in space and time, we arrive at the
following equation
( ) ( )2 2
1 2, , , Z
effi t Z ZZ Z eff
ih hH x y t h x y e h ix y c
ω π σ ωσ σ σ− 4∂ ∂= ⇒ + = ≡ +
∂ ∂
Here we have neglected the dependence of HZ on the z coordinate, since we assume that the sample
size in this direction is much larger than in the x and y directions. Hence, there is a small
demagnetization factor, which we will neglect. If we define the parameters and as:
2 1
2 2
c cλ δπσ ω πσ ω
≡ ≡
we arrive at the equation:
2 2
2 2 2 2
1 2 0Z ZZ
h h i hx y λ δ
∂ ∂ + − − = ∂ ∂
33
Taking the boundary conditions ( ) ( )0 0, ,Z x Z yh L y h h x L hα α± = ± = , when 2LX , 2LY are the sample’s x
and y lengths, we arrive at the following solution for the sample’s susceptibility:
( )
( )( )
22 2
1 2 21,3... 1 2 2
2 2 2
2 22 2 2
1,3...
tanh8 21 1 2
4tanh8
2
m ym
xm m y
m xm
m m x y
mk L k kZ LZ Zm k L ikq L mZ q k
m q L L
παπ
χπ λ δπ
π
=
=
≡ + =
+ − = ≡ − = ≡ +
∑
∑
The complex parameter takes into account the possible existence of the surface layer with
enhanced conductivity. This layer could provide the jump of an AC field at the sample surface. For
example, in the model with thin surface layer of thickness d and conductivity s effσ σ , one could
find ( )1/ 1 Uα = + , where
2 2' '' / 4 /s s sU U iU k d k k i cπ ωσ= + = = −
For the measurements where the sweep rate is zero (point-to-point), there is complete shielding for
0 2cH H< and we could conclude that the penetration depth is very small. In the swept field, we
observe an AC response, which differs from complete shielding, and it permits us to draw some
conclusion about the conductivity of the sample. We found that it is impossible to map adequately
the experimental values for in the swept field onto eff using the solution for :
( )1 2 1 / 4Z Zχ α α π= + −
In general, the two unknown complex quantities eff and could not be found using only one
complex quantity without any additional conditions. Hence, we assumed that the total surface
current has the smallest possible value because the current is localized in a thin surface layer. This
condition with other possible physical restrictions, such as 1 ' 0 '' 0U Uα ≤ ≥ ≤ permit us to find
the bulk eff and . From eff we calculated the skin penetration depth .
34
The results of these calculations allowed us to plot against HDC for different frequencies and
amplitudes:
1 . 0 1 . 5 2 . 0 2 . 5
1 0 - 2
1 0 - 1
M a g n e t i c f i e l d ( k O e )
δ eff (m
m)
5 O e / s e c 1 0 O e / s e c 2 0 O e / s e c 3 0 O e / s e c
h 0 = 0 . 0 2 O e
M a g n e t i c f i e l d ( k O e )
δ V s . H D C , f = 1 4 6 . 5 H z
1 . 0 1 . 5 2 . 0 2 . 5
1 0 - 3
1 0 - 2
1 0 - 1
h 0 = 0 . 2 O e
1 . 0 1 . 5 2 . 0 2 . 5
1 0 - 3
1 0 - 2
1 0 - 1
f = 1 4 6 5 H z
f = 1 4 6 H z δ eff (m
m)
M a g n e t i c f i e l d ( k O e )
δ V s . H D C , h A C = 0 . 2 O e
1 . 0 1 . 5 2 . 0 2 . 5
1 0 - 3
1 0 - 2
M a g n e t i c f i e l d ( k O e )
5 O e / s e c 1 0 O e / s e c 2 0 O e / s e c 3 0 O e / s e c
We can see from these graphs that in the plateau region between HC1 and HC2 the dissipative
component of the bulk conductivity 1 is considerably larger than 2 , i.e.δ λ . We can also see an
35
exponential rise in the range between 1000 Oe to 1500 Oe. This was confirmed by fitting an
exponential to this range, as shown below.
900 1000 1100 1200 1300
0.000
0.002
0.004
0.006
0.008
0.010
E qu atio n: y = A 1*exp(x/t1) + y0 C hi^2/D o F = 1.68 94E -8R ^2 = 0.99 767 y0 0 .000 27 ±0.0 000 6A 1 6 .138 6E -9 ±2.5 647 E-9t1 9 2.89 181 ±2.7 456 4
HDC
[O e ]
δ [m
m]
δ V s. H D C , f = 1465 H z h AC = 0 .2 O e S R = 30 O e/s
900 1000 1100 1200 1300
0 .00
0 .01
0 .02
0 .03
0 .04
0 .05
HDC
[Oe]
δ [m
m]
δ Vs. H DC , f = 1465 Hz hAC = 0.2 Oe SR = 10 Oe/s
Equation: y = A1*exp(x/t1) + y0 Chi^2/DoF = 9.753E-7R^2 = 0.99511 y0 0.00146 ±0.00041A1 2.2037E-9 ±1.4847E-9t1 77.30704 ±3.10353
900 1000 1100 1200 1300 1400
0.000
0.002
0.004
0.006
0.008
0.010
0.012
H DC [O e]
δ [m
m]
δ Vs. H DC , f = 1465 Hz hAC = 0.2 O e SR = 20 Oe/s
Equation: y = A1*exp(x/t1) + y0
Chi^2/DoF = 2.2234E-8R^2 = 0.99745 y0 0.00028 ±0.00007A1 6.0598E-9 ±2.5958E-9t1 92.87377 ±2.79336
900 1000 1100 1200 1300
0.00
0.01
0.02
0.03
0.04
0.05
HDC [Oe]
δ [m
m]
δ Vs. HDC , f = 146.5 Hz hAC = 0.2 Oe SR = 10 Oe/s
Equation: y = A1*exp(x/t1) + y0
Chi^2/DoF = 9.753E-7R^2 = 0.99511 y0 0.00146 ±0.00041A1 2.2037E-9 ±1.4847E-9t1 77.30704 ±3.10353
36
900 1000 1100 1200 1300 1400
0.00
0.02
0.04
0.06
0.08
0.10
0.12
HDC
[O e]
δ [m
m]
δ Vs. HDC
, f = 146.5 Hz hAC
= 0.2 Oe SR = 20 Oe/s
Equation: y = A1*exp(x/t1) + y0 Chi^2 /D oF = 2.8753E-6R^2 = 0.99655 y0 0.0028 ±0.00066A1 2.5485E-9 ±1.4545E-9t1 76.52429 ±2.51793
900 1000 1100 1200 1300 1400
0.00
0.02
0.04
0.06
0.08
0.10
H DC [O e ]
δ [m
m]
δ Vs. H DC , f = 146.5 H z hAC = 0.2 O e SR = 30 O e /s
Equation : y = A 1*exp(x /t1) + y0
C hi^2 /D oF = 1 .6657E-6R ^2 = 0 .9977 y0 0.0017 ±0.00056A1 3.1421E -8 ±1.288E-8t1 89.31202 ±2.47611
As can be seen, there is a good exponential fit,
and the fit parameters are close for the same
frequency. Plotting against the frequency at
different sweep rates, we can see that there is an
exponential decay of with frequency. This is
also confirmed by fitting an exponential to these
frequencies, as shown below. The results
obtained for should to be further investigated
at low sweep rates.
An examination of the relation between and
sweep rate shows a linear dependence, also
shown. This means that the skin penetration depth
is proportional to the sweep rate.
100 10000.01
0.1
δ (
mm
)
Frequency (Hz)
5 Oe/s 10 Oe/s 20 Oe/s 30 Oe/s
δ Vs. f , HDC = 2000 Oe h0 = 0.05 Oe
0 200 400 600 800 1000 1200 1400 1600
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Equation: y = A1*exp(-x/t1) + y0
Chi^2/DoF = 0.00019R^2 = 0.99259 y0 0.03283 ±0.00735A1 0.55636 ±0.0443t1 142.37248 ±16.51002
δ [m
m]
f [Hz]
δ Vs. f , SR = 10 Oe/s hAC = 0.05 Oe HDc = 2 kOe
0 200 400 600 800 1000 1200 1400 1600
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Equation: y = A1*exp(-x/t1) + y0
Chi^2/DoF = 0.0001R^2 = 0.99585 y0 0.05735 ±0.00762A1 0.53687 ±0.03637t1 260.92183 ±27.98557
δ [m
m]
f [Hz]
δ Vs. f , SR = 30 Oe/s hAC= 0.05 Oe HDC = 2 kOe
37
CONCLUSIONS
We have seen that the change in sweep rate
causes a rise in both dissipative and non-
dissipative parts of the susceptibility, an increase
in the skin penetration depth, and an increase in
the surface current, in the mixed state. In the
mixed or vortex state we also note that the
dissipative part of the bulk conductivity 1 is
much larger than the non-dissipative part 2 , since the plateau in is high and stable, implying that
λ δ .
It seems logical to relate to Campbell’s penetration length C , which is the parameter for the
penetration of magnetic field due to the deformations in the vortex lattice8: /C CB Jλ ∝ . However,
depends on frequency, while C is frequency independent.
The transition from vortex state to sheath state is related with vortex pinning, and this is related to
the ability of the sheath to shield the bulk from the external field. If the sweep rate increases and the
skin penetration depth increases, we may assume that has to do with the unpinning of vortices,
and their movement in the bulk.
The frequency dependence of ’ and ’’ is also interesting. While ’ increases in absolute value with
rising frequency, ’’ decreases in absolute value. This means that as the frequency increases, the
dissipative part diminishes, and the non-dissipative part is augmented. This may imply that the
surface current shield the AC field better at higher frequencies. However, in the range
8 See: Prozorov R. & Giannetta R.W , Supercond. Sci. Technol. 19 (2006) R41-R67
5 10 15 20 25 30
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.02 Oe 0.2 Oe
δ [m
m]
SR [Oe/sec]
δ Vs. SR, HDc = 2 kOe f=146.5 Hz
38
2 3C CH H H< < , ’’ increases with the frequency above a certain DC field of about 4,500 Oe. This
may imply that there is a DC field above which the surface sheath is so thin, it cannot shield the
external field, and the dissipative currents increase.
Since the surface superconducting state must closely affected by the surface of the sample, we
should mention that the sample’s surface was only mechanically polished, and could be much larger
than the coherence length of Niobium. It is for this reason that the sample investigated in Part II
was electrochemically polished. The sample used here is too small for this process.
PART II
THEORETICAL BACKGROUND
PHENOMENOLOGY
In 1962, Kim, Hempstead & Strnad9 carried out an
experiment on an Nb, powder-pressed long cylinder.
They positioned it in an external magnetic field H,
and measured the magnetic field inside the cylinder
H’ using a resistor. The dependence of H’ on H was
measured, at 4.2°K and 3.2°K. Their results are
shown on the right.
In addition, they heated the Nb sample, along with
other samples (Nb3Sn , 3Nb-Zr), to temperatures from
9 Kim Y.B., Hempstead C.F., Strnad A.R., PRL 7 (9), 306-309 (1962)
39
900°K to 1500°K. This process is termed “Annealing”. Its purpose is to minimize the imperfections and
dislocations of the polycrystalline metal.
Following the model developed by Bean, which will be described in the Theory section, the variable
was calculated for different temperatures for each sample. The results show a linear dependence of on
temperature. Thus, Kim proposed the following empirical relation:
( )1 a bTd
α = −
Here, a and b are constants which appear to satisfy / Ca b T≤ ,
and d is a parameter that depends on the microstructure of
the material.
In parallel with their experiment, Anderson developed a
model (which relied on Bean’s model), to explain this linear
dependence. His model, which will also be described in the
Theory section, predicted that if the external magnetic field H
would be raised to a value in the “shielding” region (i.e. a
value in the rising part of the magnetic field below the 45°
line) and kept constant, then there will be a ”relaxation” of H’ to a H with time. Alternately, when H
will be lowered from a very high field to a field in the “trapping” region (above the 45° line), then the
relaxation will occur in the opposite direction, meaning H’ will decrease to H. Indeed, Kim’s results
show this relaxation, and it is logarithmic in time.
THEORY
Bean’s “Critical Current” Model
40
In his article from 1962, Bean10 proposed a model for hard superconductors which assumes there is a
critical current density JC , which the material can sustain without losses. It depends on the applied
magnetic field, and becomes zero at the critical thermodynamic field HC. He assumed a specimen of
cylindrical form with radius R, and that the magnetic field is parallel to the cylinder’s axis. If the applied
field is less than HC, complete shielding occurs and the field inside the specimen is zero (B=0). If
CH H> , than the current shielding this field will flow in a depth from the surface. This depth is
defined:
( )104
C
C
H HJπ−
∆ =
From this parameter, Bean defined a new field H*:
4*10
CJ RH π=
This field is the field that must be applied in excess of the critical field so that critical currents may be
induced in the entire specimen. In other words, it is the field at which the penetration depth equals the
sample’s radius, R∆ = . In these definitions, Bean made use of “practical units”, where Ampère’s Law is
written:
410
H Jπ∇× =ur uur ur
In these units, H is in Oersteds, J in Ampère/cm2 and V in Volts. After introducing this field, Bean
calculated the induced field B inside the sample for fields above and below HC+H*:
( ) ( )( )
( )( ) ( )
if 0 1 / *0 for *
* 1 / if 1 / *
* 1 / if 0 for *
CC C
C
C
r R H H HB r H H H H
H H r R R H H H r R
B r H H r R r R H H H
≤ ≤ − − = ≤ ≤ + − − − − ≤ ≤ = − − ≤ ≤ + ≤
10 C.P.Bean, PRL 8, 250 (1962)
41
Now that the induced field is known, the magnetization can be obtained through it’s definition:
( )14 M B H dvV
π = −∫
The calculated magnetization is given by:
( ) ( )2 2 2 3
2
0
3 24 *
* 3 ** *
3
C
C C CC C
C
H H H
H H H H H HM H H H H H
H HH H H H
π
− ≤ ≤ − − − = − + + ≤ ≤ + − ≥ +
This magnetization depends on the sample’s radius through H*, which is striking. To compare this
theory to experiment, A. Seybolt measured the magnetization of a Nb3Sn cylinder, ground to two radii.
The cylinder was sintered from a pressed
powder at 1200°K for 8 hours. The results, on
the right, showed a difference between the
magnetization of the two samples, the one
with radius 0.48 cm and the other with radius
0.25 cm. This model, in contrast to earlier
models of Abrikosov and Goodman, does not
assume that the superconductor is in
thermodynamic equilibrium. It introduced for
the first time the notion of “trapped flux”, meaning the superconductor will show hysteresis in high
fields.
Anderson’s Model for a hollow cylinder
42
In Kim’s paper, the internal field H’ is the algebraic sum of the external field H and the field produced
by the current induced in the sample:
( )0
4'10
wH H J B r drπ = + ∫
Here, w is the width of the hollow cylinder wall. Eliminating r, the integral can be transformed into:
( )/2
/2
410
H M
H M
dBwJ B
π +
−
= ∫
Here, H is the mean field in the sample’s wall and 'M H H= − is the field produced by the induced
supercurrents. The general behavior of ( )J B suggested a power series expansion of the following kind:
( )2 3
0 2 3 ...B B a B a BJ B
α = + + + +
Here, and B0 are two constants that depend on the material structure of the sample. If the constants a2
, a3 etc. are considered small enough, then the last equation should be:
( ) 0B BJ B
α = +
Kim substituted H and 1/ M in this equation, and found a linear dependence between them. This gave
an empirical validity to this model.
In Anderson’s model, the critical current is defined as the current above which “flux creep” sets in, and
flux leaks through the material and returns it to the critical state. He defined a parameter that depends
on
( ) ( )0C CT J B Bα = +
43
The parameter varies with temperature, and Anderson explained this by assuming that the motion of
bundles of flux lines because of the Lorentz force. A bundle of flux is a group of flux lines, which are
separated by a distance equal or shorter than the London penetration depth L. This bundle can be
pinned by an inhomogeneity of the material structure, and because of that, there will be a free energy
barrier, which the bundle must overcome in order to get loose. This energy barrier can reach a
maximum of:
( )2
3 3max 8
CBn S
HF F F d dπ
∆ = − =
Here, d is the length of the bundle and HCB is the bulk critical field of the material. Anderson introduced
a new parameter p that is the fractional amount of pinning of the bundle:
maxF p F∆ = ∆
As a result of the penetration of the flux lines into the superconductor, there will be a mean current flow
J and a force f on the flux bundle:
f J Bdτ= ×∫ur ur ur
This force leads to a contribution to the free energy of:
( ) ( )i BF x F x J dx= − Φ
where B is the flux in the bundle. Therefore, the total energy barrier the flux bundle must overcome in
order for him to be unpinned is:
2 32*
8CB
BH dF p J d
π∆ = − Φ
The hopping rate of flux bundles is:
44
0*exp
B
FR Rk T
∆= −
Here, R0 is some frequency factor, which Anderson
estimated of the order 1010 Hz.
To apply this theory to a hollow cylinder, or tube, of radius
a and thickness w, Anderson estimated the number of
bundles in the wall of the tube as *2
B
wHaπΦ
, and the
number of barriers the bundle sees as it traverses the wall
to be cwd
, c being a parameter giving the number of
effective barriers. From these assumptions, he derived the
equation determining the rate of creep:
( ) 021 *' exp* B
dRd FH HH dt ca k T
∆− = − −
Since there is no precise field at which the creep starts, Anderson defined the critical state as that at
which the rate falls below a practically observable limit, which is for Kim’s experiment about 1 hour.
Anderson proceeded to calculate the factor for Kim’s experiment, which is shown on the right.
Regarding the time behavior of the sample, Anderson integrated the last equation over time to obtain:
( ) ( ) ( )
2 304
11
0
2 exp8* exp
'
CB
BB
dR H dK pd d d ac k TH Kdt dH k T
T J t B t B
α α α π
α
≡ −
= − ≡ +
Assuming the total variation of is small this gives the following time dependence of H’:
( ) ( )40' ' / ' / lnBH k w H B k T d tδ = − +
45
This means that the field inside the tube must decay logarithmically with time.
SAMPLES AND EXPERIMENTAL METHODS
The sample used for this part is a Niobium hollow cylinder (or tube) which was manufactured by
powder pressing. It’s sizes are shown on the diagram on the right. This sample was electrochemically
polished by a method described in a n article by Tian,
Corcoran, Reece & Kelley11. The purpose of this polishing
was to achieve a smoother surface so that the surface effects
taking place at high magnetic fields will not be affected from
the specific surface imperfections of the sample. The
objective of the electrochemical polishing is to achieve a
surface roughness smaller than the coherence length of the
sample.
The electrochemical polishing was done using hydrofluoric
acid (HF) in a concentration of 49% and sulfuric acid (H2SO4)
in a concentration of 96%, in ratio of 1:9 respectively. The
immersed sample was connected to a power supply, at the
11 Tian H., Corcoran S.G., Reece C.E., Kelley M.J., Journal of Electrochemical Society 155 D563-D568 (2008)
4.1 mm
6.3 mm
3.2 cm
46
positive end, and the other electrode was a Platinum electrode. The power supply was set to give 5
Volts and a current of about 0.3 Amperes, for 10 minutes. This was done in a Teflon container, which
was immersed in a bath of cooled water (20°C).
Afterwards, the sample was annealed at a temperature of 1100°C in a low-pressure (about 10-7 Torr or 10-
10 Atm ) chamber. This was done to remove any foreign elements such as hydrogen from the sample,
and to change the sample’s structure so that the grain size would be larger.
This sample was placed so that inside it (in the hollow part) there is a Hall probe. This probe was
calibrated prior to the experiment. It was connected to a Keithley voltmeter and to a Keithley current
supplier, which was set to give a current of 10-3 Ampère. The cylinder along with the Hall probe inside
it were positioned inside the liquid Helium cryostat used in Part I, which also has DC and AC
superconducting coils that can generate a magnetic DC field of up to 5 Tesla, and superimposed on it an
AC field of 23 Hz and amplitude 1 Oersted. The diagram below shows all of these components.
Hall
Current Supply
Voltmeter
( )H tuuuur
( )'H tuuuuuur
Nb Hollow cylinder
47
The measurements made on the Nb hollow cylinder consist in:
1. Point-to-point measurements of H’ Vs. H, where H is a DC magnetic field, for different
temperatures from 5°K to 9°K. These measurements were carried out on the unpolished and
polished sample.
2. Swept field measurements of H’ Vs. H, where H is a DC magnetic field, at 5°K for different sweep
rates. These measurements were made only on the polished sample.
3. Point-to-point measurements of H’ Vs. H, where H is a DC magnetic field with AC field
superimposed on it, at 7.5°K with amplitude of excitation of 1 Oe and frequency 23 Hz. These
measurements were carried out on the unpolished and polished sample.
4. Swept field measurements of H’ Vs. H, where H is a DC magnetic field with AC field
superimposed on it, at 7.5°K with amplitude of excitation of 1 Oe and frequency 23 Hz, at
different sweep rates. These measurements were made only on the polished sample.
5. Relaxation measurements, at the “shielding” region and at the “trapping” region, at 5.5°K. These
measurements were carried out on the unpolished and polished sample.
6. Magnetization measurement at zero-field cooled (ZFC) from 5°K to 9.2°K. These measurements
were made only on the polished sample.
Some of these measurements were made before and after the electrochemical polishing and
annealing so that its effect could be evaluated. After the measurements were made, a roughness
analysis was made, using an Atomic Force Microscope.
The point-by-point measurements were made on a SQUID magnetometer system, controlled by a
computer software (MPMS5). The swept field measurements were made with another computer with
Matlab R2008b software which controlled the waveform generator (Agilent), the coils current supplier
(Stanford Research Systems) and the lock in amplifiers (EG&G and Stanford Research Systems).
The magnetization measurements were carried out by using the two-coil method used in Part I.
48
RESULTS
RESULTS FOR UNPOLISHED SAMPLE
H Hall Vs. H for Unpolished Nb Cylinder, for various temperatures, DC
0
500
1000
1500
2000
2500
3000
0 500 1000 1500 2000 2500
H [Oe]
H H
all [
Oe
6.5K
7K7.5K
8K
8.5K
9K6 K
49
Hc1 & Hc2 Vs. T for unpolished sample
0
500
1000
1500
2000
2500
6 6.5 7 7.5 8 8.5 9 9.5
T [K]
H c1 &
H c2 [O
e] Hc1
Hc2
The first graph in the previous page shows how the internal field H’ varies from zero to H. For the
lowest temperature, 6°K, there is a sudden jump in H’ at H=2380 Oe. This jump is termed “flux jump”,
and happens at lower fields for lower temperatures. The field at which H’ becomes nonzero is HC1 and
the field at which H’ reaches H, is HC2 . The variation of these two critical fields with temperature is
shown in the second graph.
The following graphs show the results for the first, second and third harmonics of the internal field H’
when the external field H is a DC field with AC component, of frequency 23 Hz and amplitude of
excitation of 1 Oe.
50
|H1|Vs. HDC , f = 23 Hz hAC = 1 Oe T = 7.5 K , unpolished cylinder
0
0.5
1
1.5
2
2.5
0 1000 2000 3000 4000 5000 6000
HDC [Oe]
|H1|
[Oe]
This graph shows that the AC field starts penetrating inside the sample at H = 2800 Oe. After this field,
the internal field increases until reaching a maximum. This field will be termed HC3 , and its value is
approximately 3700 Oe. After this field, there is a plateau in H’. There is no difference in H’ when H is
raised or lowered, i.e. there is no hysteresis. The sample is in a normal state in the plateau region.
In the following graphs, for the second and third harmonics, there is a peak, which occurs for an
external field of about 3100 Oe. This field is the field at which the slope of the first harmonic is maximal.
51
|H2| Vs. HDC , f = 23 Hz hAC = 1 Oe T = 7.5 K , unpolished cylinder
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1000 2000 3000 4000 5000 6000
HDC [Oe]
|H2|
[Oe]
|H3| Vs. HDC , f = 23 Hz hAC = 1 Oe T = 7.5 K , unpolished cylinder
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1000 2000 3000 4000 5000 6000
HDC [Oe]
|H3|
[Oe]
52
The following graph shows the results of relaxation measurements at 7.5°K, at external fields of 900 Oe
and 1000 Oe. These fields were chosen because they mark the beginning of the region where H’
becomes nonzero, for this temperature. The internal field was measured for 3600 seconds, i.e. 1 hour.
H' Vs. Time at 7.5 K for different external field H, for unpolished cylinder
120
125
130
135
140
145
0 500 1000 1500 2000 2500 3000 3500 4000
Time [sec]
H' [
Oe] 900 Oe
1000 Oe
This graph shows a slight decay in H’, but does not show relaxation. This is contrary to expected,
since Kim’s experiment shows a logarithmic raise in H’, when H is in the “shielding” part.
53
RESULTS FOR POLISHED AND ANNEALED SAMPLE
H' Vs. H for polished cylinder, at different T
0
500
1000
1500
2000
2500
3000
3500
4000
0 500 1000 1500 2000 2500 3000 3500 4000
H [Oe]
H',
Oe
5K
5.5K
6K
7K
7.5K
8K
8.5K
Hc1 & Hc2 for different temperatures, polished cylinder
0
500
1000
1500
2000
2500
3000
3500
5 5.5 6 6.5 7 7.5 8 8.5 9
T [ K]
H c1 &
H c2 [O
e]
Hc2Hc1
54
The previous graph shows HC1. In the lower temperatures (5°K and 5.5°K), there is a “flux jump” which
is followed by a plateau, after which there is an increase in H’ until it reaches the 45° line. This flux
jump is seen as a vertical line. When the external field is lowered, there is a “trapped flux” inside the
hollow cylinder, which is seen as a departure from the 45° line. This trapped flux remains until the
external field returns to zero.
The following graph shows the results for point-by-point DC field with a superimposed AC field, at a
frequency of 23 Hz and amplitude 1 Oe.
|H1'| Vs. HDC , f = 23 Hz hAC = 1 Oe , T = 7.5K , polished cylinder
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 500 1000 1500 2000 2500 3000 3500 4000 4500
HDC [Oe]
|H1'|
[Oe
In this graph, the critical field HC3 is around 2100 Oe. The values of the internal field H’ are dramatically
lower than its values when H does not have an AC component.
The following graph shows the swept field measurements, at different sweep rates. In this graph, H Hall
is H’, i.e. the field inside the hollow cylinder.
55
-500 0 500 1000 1500 2000 2500 3000 3500 4000 4500-500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
H [Oe]
H H
all [
Oe]
H Hall Vs. H for swept DC field, at T = 5K for different sweep rates
10 Oe/sec
15 Oe/sec
20 Oe/sec
-500 0 500 1000 1500 2000 2500 3000-500
0
500
1000
1500
2000
2500
3000
H [Oe]
H H
all [
Oe]
H Hall Vs. H for swept DC field, at T = 7.5K for different sweep rates
10 Oe/sec
15 Oe/sec
20 Oe/sec
56
-500 0 500 1000 1500 2000 2500 3000 35000
0.5
1
1.5
Absolute value of H'1 Vs. H, at swept AC magnetic field, T = 7.5K f = 23 Hz hAC = 1 Oe
H [Oe]
|H1'
| [O
e]
10 Oe/sec
15 Oe/sec
20 Oe/sec
The second and third harmonics of the signal did not show any meaningful information, so they will not
be showed. The graphs for swept field do not show a variation in H’(H) for different sweep rates. They
show the same critical fields as the ones for point-by-point measurements. The AC measurements at
swept field are very noisy, and we should have set the Lock-in amplifier to use a larger time constant.
The following graph show the relaxation measurement made on the polished sample, at 5.5ºK with an
external field of 2,190 Oe (in the “shielding region”), for 4 hours or 14,400 seconds. Relaxation
measurements were made also at 2,250 Oe and 2,450 Oe, but these did not show any relaxation
whatsoever. These values were chosen because, for the H’ Vs. H curve at 5.5ºK, they are the values of H
just before H’ becomes nonzero.
57
Relaxation for polished sample: T = 5.5 K , H = 2190 Oe: H' Vs. time
1080
1085
1090
1095
1100
1105
1110
1115
1120
0 2000 4000 6000 8000 10000 12000 14000
t [sec]
H' ,
Oe
As seen, this graph shows a decay in H’, but it does not seem logarithmic. We did not observe essential
relaxation in the polished sample, as in the unpolished sample.
The following two graphs show magnetization measurements on a piece of Niobium cut from the
hollow cylinder that was used for the previous measurements (before it was polished and annealed).
The sizes of this piece are 12mm by 3mm by 1.6mm. First, the magnetization was measured for several
temperatures, at ascending and descending DC fields. Between each measurement, the sample was
heated above TC and the cooled again at zero field.
58
M Vs. H for a piece cut from the Nb hollow cylinder at T = 5K
-4
-3
-2
-1
0
1
2
3
4
0 500 1000 1500 2000 2500 3000 3500
H [Oe]
M [e
mu]
5 K
5.5 K
6 K
6.5 K
7 K
7.5 K
8 K
8.5 K
Afterwards, the specimen was heated above TC and then cooled at zero-field to 5°K. Then, a small DC
field of 22 Oe was applied, and the magnetization was measured as the temperature was raised slowly
above TC. . The following graph shows the results.
59
Magnetization Vs. Temperature for a piece cut from the Nb cylinder, ZFC, H = 22 Oe
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
05 5.5 6 6.5 7 7.5 8 8.5 9 9.5
T [K]
M [e
mu
After the measurements on the sample were finished, it was taken to the Nanocenter Laboratories, to
determine its surface roughness through the Atomic Force Microscope. The sample was glued to a
sample holder, and 3 areas of 20µm by 20 µm were scanned. The results are summarized below:
1. Area 1: Rq = 17.9nm, Ra = 14.2 nm, Sqewness = -0.426 , Kurtosis = 3.62
2. Area 2: Rq = 25.5 nm, Ra = 20.2 nm, Sqewness = -0.205 , Kurtosis = 3.05
3. Area 3: Rq = 14.1 nm, Ra = 11.0 nm, Sqewness = -0.128 , Kurtosis = 3.54
The parameters Rq and Ra are related with the surface roughness as follows. Ra is the arithmetic average
of the absolute values of surface height deviations measured from the mean plane. Rq is the root mean
square average of height deviations from the mean plane.
60
2
1
1 N
iq a i
i
zR R z
N N == =∑ ∑
The sqewness and kurtosis are two parameters that measure the distribution of the surface deviations
around Rq.
3 43 4
1 1
1 1 1 1Sqewness Kurtosis N N
i ii i
z zRq N Rq N= =
= =∑ ∑
This roughness must be compared to the coherence length of Nb. To estimate it, we shall use the
formula:
0
22 cHξ
πΦ=
Since HC2 diminishes with temperature, obviously increases until it is infinite at TC. Therefore, we will
estimate it at T=5°K. The critical field HC2 at this temperature can be extracted from the magnetization
graph:
2 3050 50 OeCH = ±
Hence, the coherence length is estimated at:
32.9 0.3 nmξ = ±
This length is not very large compared with Ra and Rq. The surface roughness is 35% to 80% of , so
we must conclude that even at 5°K, the surface roughness is large.
61
DISCUSSION
DC Measurements
The temperature dependence of H’(H) shows two kinds of behavior:
1. For temperatures up to 5.5°K for the polished sample and 6.5°K for the unpolished sample, there
is a “flux jump” at a certain value of H. This flux jump is accompanied by a plateau in the
polished sample, which is not seen in the unpolished sample. This may be because we did not
make measurements below 6.5°K for the unpolished sample. Afterwards, the H’(H) gradually
becomes a 45° line, which means that the sample is normal and H’=H.
2. For temperatures above 5.5°K for the polished sample and 6.5°K for the unpolished sample, there
is no “flux jump” or a plateau, and H’(H) changes gradually from H’=0 to H’=H.
This behavior is described in Parks12 and is due to the critical current that can be carried by the hollow
cylinder. As H increases, the shielding current increases and reaches a value that causes dissipation
around dislocations and imperfections in the lattice. This causes “hot spots” around the imperfections,
and since hard superconductors are usually poor thermal conductors, the heat causes the sample to turn
normal.
The effect of polishing and annealing on H’(H) for different temperatures is seen in the following graph.
12 Parks, R.D.: “Superconductivity”, Vol. 2, pp. 1156-1162.
62
H' Vs. H for different T, before and after polishing and annealing, DC
-500
0
500
1000
1500
2000
2500
0 500 1000 1500 2000 2500
H [Oe]
H' [
Oe
7K polished
7K unpolished
7.5K polished
7.5K unpolished
8K polished
8K unpolished
8.5 K polished
8.5K unpolished
From this graph, it can be seen that the effect of the polishing and annealing is contrary to what was
expected. The superconducting quality of the hollow cylinder was deteriorated, since it becomes normal
at lower H.
Plateau variation with temperature
The H’(H) curve for the polished
sample shows that for temperatures
below 6°K there is a flux jump followed
by a plateau. To see the effect of
temperature on the size of this plateau,
we measured the H’(H) dependence for
5.25°K and 5.75°K. At 5.75°K there was
no plateau. From this data, we can see
Plateau size Vs. T, polished sample
0
100
200
300
400
500
4.8 5 5.2 5.4 5.6 5.8
T [K]
Plat
eau,
Oe
63
the variation of the plateau size with temperature.
The effect of polishing and annealing on the critical fields
As can be seen in the graph below, the polishing and annealing of the sample reduced the critical fields
HC1 and HC2 by an amount of up to 15%, for temperatures below 7.5°K. Above this temperature, HC1 was
raised after polishing and annealing, and for 8.5°K both critical fields are higher for the polished sample.
Comparison btw. polished and unpolished sample: Hc1 & Hc2 for different temperatures
0
500
1000
1500
2000
2500
6 6.5 7 7.5 8 8.5 9
T [K]
H c1 &
H c2 [O
e]
Hc2 polished
Hc1 polished
Hc1 unpolished
Hc2 unpolished
AC field with point-to-point DC field
The results from the AC point-to-point measurements show that the field inside the hollow cylinder is
much lower than in the DC point-to-point measurements. Also, there is an increase of H’ that occurs in
the range 3000 Oe to 3500 Oe for the unpolished sample, and 2000 Oe to 2200 Oe for the polished.
After this increase, there is a plateau of constant AC field, at an internal field H’ of 2 Oe for the
unpolished sample, and 0.6 Oe for the polished sample.
64
To compare the AC response of the unpolished and polished sample in point-to-point measurements,
we plotted the following graph.
Comparison btw. polished & unpolished sample: H1' & Vs. H for AC field in point-to-point measurements, f = 23 Hz hAC = 1 Oe
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
H [Oe]
H 1' [O
e] H1' polishedH1' unpolished
From this graph, we can see that the effect of polishing and annealing on the sample is to lower its
critical field, HC3 (the field H at which H’ reaches the plateau value). This is, again, in contrast to what is
expected. The critical AC field before and after the polishing and annealing are:
3
3
3700 100 Oe
2100 100 Oe
unpolishedc
polishedc
H
H
= ±
= ±
From this we can see that the reduction in HC3 is about 40%.
From the critical field obtained, we can calculate the ratio 3 2/c cH H for the polished and unpolished
sample:
65
3 3
2 2
1.8 0.1 2.6 0.1 polished unpolished
c cpolished unpolished
c c
H HH H
= ± = ±
AC field with swept DC field
The data for this part do not show a difference between constant DC field and swept DC field. The
swept DC field without AC field shows no difference from the point-to-point DC field. The swept DC
field with AC field is a very noisy measurement, but shows the same behavior as the point-to-point DC
field with AC field measurement.
Relaxation Measurements
The relaxation measurements for the unpolished sample show a decline in H’ as time advances. The
decline is in contrast to what was expected, since in Kim’s paper, the internal field H’ rises with time in
the “shielding” part of the H’(H) curve. In the relaxation measurements made on the polished sample,
again, H’ declines in time instead of a rising.
This behavior may be related to the annealing of the sample, which may have degraded the crystal
structure of the grains in the polycrystalline material, or made them smaller in size. Another explanation
is that since the sample used is a polycrystalline material, it may have too many dislocations or an
average grain size that is too small. This may increase the vortex pinning to a degree in which they are
unable to move due to the external magnetic field.
Magnetization Measurements
The data from the magnetization measurements is in accordance to the H’(H) measurements, since they
show a phase transition from the Meissner state (straight line with slope 1/ 4π− ), through to the vortex
state (slope gradually changing to a positive slope), and finally to the normal state ( 0M = ) for the same
values of external magnetic field H.
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CONCLUSION
The effect of surface electrochemical polishing and annealing of the sample is contrary to what was
expected. The sample’s ability to mask external fields was deteriorated, and the relaxation of the sample
did not change significantly. Whether this has to do with the electrochemical polishing or with the
annealing is unknown, but it is likelier that the annealing had more effect. The electrochemical polishing
was satisfactory and decreased the surface roughness, but not to a value which is much smaller than the
coherence length .
The sample used for this part is made of polycrystalline material, which is not a single crystal. This
affects the results. Another factor to be considered is that the SQUID system used here has an initial
non-zero magnetic field, which may also affect the results.
The effect of electrochemical polishing on Niobium superconducting radio-frequency cavities was
investigated by G. Ribeill and C. Reece13. They show that the electrochemical polishing of Nb increases
the cavities’ efficiency and maximum accelerating potential.
13 G. Ribeill & C. Reece, “Roughness analysis of variously polished Niobium surfaces”, Journal of Undergraduate Research, Office of Science,
US Department of Energy, Volume VIII, 2008