Thesis Gilad Masri

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.

66

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