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This is to certify that the
dissertation entitled
Proximity Induced Superconductivity

in Mult ilayered Metallic Systems

presented by
Michael L. Wilson

has been accepted towards fulfillment
of the requirements for

 

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MSU is an Affirmative Action/Equal Opportunity Institution O~ 1 2771

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PROXIMITY INDUCED SUPERCONDUCTIVITY IN
MULTILAYERED METALLIC SYSTEMS

by

Michael L. Wilson

A DIS SERTATION

submitted to
Michigan State University
in partial fiilfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1994

 

ABSTRACT

PROXIMITY INDUCED SUPERCONDUCTIVITY IN
MULTILAYERED METALLIC SYSTEMS

by

Michael L. Wilson

The superconducting transition temperature TC and superconducting upper critical
field Hc2(T) have been measured in the proximity coupled superconductor/normal-metal
multilayered system Nb/CuX, where CuX represents Cu, Cu1_yMny, or Cu0.95Ge0_05. This
represents the first study in which a single system has been used to examine the proximity
effect through independent variation of the superconductor layer thickness ds, and the
normal-metal layer thickness (in, concentration of magnetic impurities y, and resistivity pn.
The dependence of Tc on (In reveals that Ge atoms elastically scatter proximity induced
superconducting pairs in the normal metal layer N, while Mn impurities predominantly act
as pair breakers, destroying the induced superconducting state. At a fixed (in, the variation
of T0 with changing dS exhibits scaling consistent with the equation
(ch —TC)/Tcb 5(ds/do)-p where do is a scale length and T: is Tc for bulk Nb. The
exponent p increases systematically as either (In or y increase, but is unchanged by the
addition of Ge to the Cu layer. This scaling form is consistent with the limiting forms of
two proximity effect theories. Quantitative agreement between the theories and the
experimental data, however, is only achieved if Tcb is allowed to decrease with decreasing

ds. The temperature dependence of the upper critical field Hc2(T) was measured for

A!

 

 

applied fields directed parallel and perpendicular to the plane of the layers Hall and chi,
respectively. The dependence of ch(T/Tc) follows trends similar to those of the Tc data,
but Hc2J. appears to measure a slightly different superconducting pair penetration depth in
N than does Tc. The behavior of Hc2“(T) changes from that indicating a 3D-coupled state
near To, to a 2D-decoupled state below a temperature T‘. This dimensional crossover is
described by a model in which the application of a field parallel to the layers predominantly
destroys the induced superconductivity in the CuX layers without destroying the
superconductivity in Nb. This model is opposed to earlier models of dimensional
crossover which attribute crossover to the temperature dependence of the

superconducting coherence length.

 

 

to my parents

iv

 

 

ACKNOWLEDGEMENTS

I would like to thank Dr. Jerry Cowen for his guidance and support these last few
years. I especially thank him for his patience during the writing and editing of this
dissertation.

I also thank Dr. Carl Foiles for his many hours of discussions about structural
analyses.

For their invaluable aid in maintaining my sanity during this project, I thank all the
graduate students and stafi‘ at MSU. Special thanks, though, need to go out to Mike
Jaeger, Dan Koppel, Roseann Miller, Diandra Leslie-Pelecky, and Reza Loloee. Their
discussions have helped more than they probably know.

My parents also deserve special recognition since, without their support for the last
28 years, I could never have made it this far.

My greatest appreciation and thanks must extend to my wife Jennifer for keeping
me going these last few years. Without her support, help, and even some needed
prodding, I would probably still be writing.

Finally, I would like to thank the Department of Physics and Astronomy, the
Center for Fundamental Materials Research, and the National Science Foundation for their

financial support throughout my studies.

 

TABLE OF CONTENTS

 

LIST OF TABLES ..................................................................................................... viii
LIST OF FIGURES ................................................................................................... ix
1. INTRODUCTION ................................................................................................ 1
1.1 Previous Studies of Tc and Hc2 in Multilayered Superconductors ........ 4
2. SAMPLE PREPARATION AND FABRICATION ............................................ 7
2.1 The Substrates .......................................................................................... 7
2.2 Sample Mounting .................................................................................... 8

2.3 The Sputtering Guns ............................................................................... 10

2.4 Target Production and Preparation ....................................................... 13

2.5 Cleanliness ............................................................................................... 14

3. STRUCTURAL ANALYSES ............................................................................... 16

3.1 Elemental Compositions ......................................................................... 17

3.2 X-ray Diffraction ..................................................................................... 18

3.2.1 The Diffractometer ...................................................................... 18

3.2.2 Reflection Scans .......................................................................... 21

3.2.2.1 Small-Angle X-Ray Diffraction ..................................... 23

3.2.2.2 First-order Region ........................................................ 25

3.2.2.3 Second-order Region .................................................... 30

vi

 

3.2.3 Off-axis Scans ............................................................................. 32

 

3.2.4 Rocking Curves .......................................................................... 40
3.3 XAFS Measurements .............................................................................. 42
3.4 Studies of Cross-Sectioned Samples ....................................................... 49
3.5 Structure Summary ................................................................................. 53
4. EXPERMENTAL PROCEDURES .................................................................... 56
4.1 Magnetic Measurements ......................................................................... 57
4.2 Resistivity Measurements ....................................................................... 58
5. THE SUPERCONDUCTIN G TRANSITION TEMPERATURE ....................... 62
5.1 Tc Experiments ........................................................................................ 62
5.1.1 The Dependence of Tc on the N Layer Thickness ........................ 62
5.1.2 The Dependence of Tc on the S Layer Thickness ......................... 68
5.2 Tc Theory ................................................................................................. 73
5.2.1 The KZ Model ............................................................................ 74
5.2.2 The HTW Model ........................................................................ 81
5.3 Model Applications ................................................................................. 85
6. THE SUPERCONDUCTING UPPER CRITICAL FIELD ................................ 89
6.1 Introduction ............................................................................................ 89
6.2 The Perpendicular Upper Critical Field ................................................. 99
6.2.1 Hc2_L Experiments ........................................................................ 100
6.2.1.1 Positive Curvature ........................................................ 108
6.2.1.2 The Dependence of Hc2l on dn ...................................... 111
6.2.1.3 The Dependence of H‘,2 i on (18 ....... _ ............................... 114
vii

 

6.2.3 HCu(T) Theory ........................................................................... 117

6.3 The Parallel Upper Critical Field ........................................................... 123

6.3.1 Ha“ Experiments ......................................................................... 123

6.3.2 H62“(T) Models ............................................................................ 140

6.3.2.1 The TQ Model ............................................................. 140

6.3.2.2 The FQ Model .............................................................. 144

7. SPIN-FREEZING NEAR Tc ................................................................................ 152
8. CONCLUSIONS ................................................................................................... 159
8.1 Future Directions of Study ..................................................................... 161

LIST OF REFERENCES .......................................................................................... 163

viii

 

 

LIST OF TABLES

 

Table 2.1: Sputtering gun settings. .............................................................................. 12
Table 3.1: Off-axis angles, planer spacings, and diffraction angles for bcc Nb and

fee Cu .......................................................................................................................... 37
Table 3.2: Off-axis lines observable using Cu K“ radiation ........................................... 38
Table 3.3: Ofilaxis peak positions recorded for Nb crystallites. ................................... 40
Table 3.4: XAFS derived parameters for Nb/CuMn(1%) samples. ............................... 45
Table 5.1: Scaling parameters for Tc(ds). ‘Data from Wong et all“,41 ......................... 72
Table 5.2: Nb and CuX length scales ........................................................................... 86
Table 6.1: d; and d; for various CuX materials ........................................................ 112

ix

 

LIST OF FIGURES

 

Figure 2.1: Sample holder ........................................................................................... 9

Figure 2.2: The sample positioning and masking apparatus (SPAMA) plate ................. 9

Figure 2.3: The substrate cooling system ..................................................................... 10
Figure 2.4: Cross section of a triode magnetron sputtering gun. .................................. 11
Figure 2.5: Overall view of the sputtering configuration. ............................................. 13
Figure 3.1: The x-ray difi‘ractometer. .......................................................................... 19
Figure 3.2: The x-ray diffraction sample holder. .......................................................... 19
Figure 3.3: Definitions of scattering and goniometer angles. ........................................ 20

Figure 3.4: Full scattering spectra for a Nb/CuMn(13%) 2.0 urn/2.0 nm
multilayer ..................................................................................................................... 22

Figure 3.5: Low angle x-ray scattering in a metal film. ................................................ 24

Figure 3.6: Small-angle diffraction scan and peak analysis used to determine A for
Nb/ CuGe(5%) 5.0 nm/2.0 nm. .................................................................................... 26

Figure 3.7: First order diffraction peaks for Nb/CuMn(0.3%) 70.0nm/70.0nm (a)
and 10.0mn/13.0nm (b). ............................................................................................... 27

Figure 3.8: First order spectrum and peak analysis used to determine A for
Nb/CuMn(10%) 15.0nm/7.0nm .................................................................................... 29

Figure 3.9: Second order peaks for a Nb/Cu 2.0 nm/2.0 nm (a) and a 3.0 urn/3.0
nm (b) multilayer. Error bars represent a 2% change in dNb, and a 3% change in

dCuX. ............................................................................................................................ 31

Figure 3.10: Transmission scan for a 500 nm thick Nb film .......................................... 33

 

Figure 3.11: Off-axis scattering geometry. .................................................................. 35

Figure 3.12: CuMn (311) peaks observed in a 500 nm thick CuMn(13%) film, and

a Nb/CuMn(13%) 3.0 urn/3.0 nm multilayer ................................................................. 39
Figure 3.13: Off-axis peaks observed in a 500 nm thick Nb film: (a) (321) peak,

(b) (312) peak and (c) (123) peak. ............................................................................... 41
Figure 3.14: Rocking curve for the (110) line of a bulk Nb film. .................................. 43
Figure 3.15: Raw Cu absorption data for a Nb/CuMn(1%) 5.2 nm/4.6 nm

multilayer ..................................................................................................................... 46
Figure 3.16: Chi plots for a Nb/CuMn(1%) 5.2 urn/4.6 nm multilayer .......................... 47

Figure 3.17: Radial mass distribution fimction surrounding Cu in a Nb/CuMn(1%)
5.2 urn/4.6 nm multilayer. ............................................................................................ 48

Figure 3.18: Dark field electron image of a cross-section of a Nb/CuMn(7%) 28
nm/ 7.0 nm multilayered sample. .................................................................................. 51

Figure 3.19: Electron difi‘raction pattern taken from a cross sectioned
Nb/CuMn(7%) 28 nm/ 7.0 nm multilayered sample. Miller indicies for the
observed Nb planes and the CuX (200) plane are labeled. Thin lines denote the

direction parallel and perpendicular to the plane of the Nb and CuX layers. .................. 52
Figure 4.1: Diagram of the sample pattern ................................................................... 57
Figure 4.2: Zero-field-cooled (zfc) and field-cooled (fc) magnetization data for a

500 nm thick CuMn(1%) film. ..................................................................................... 59
Figure 5.1: Qualitative dependence of Tc on (In for fixed ds. ......................................... 63

Figure 5.2: The dependence of Tc on (1H for (1s = 28.0 nm and various CuX
materials. ..................................................................................................................... 64

Figure 5.3: The dependence of Tc on dn in Nb/Cu and Nb/CuGe(5%) multilayers
for Nb layer thicknesses of 10 nm and 28 nm. .............................................................. 65

Figure 5.4: The three dominant processes for a phase coherent electron entering a
normal metal layer. ...................................................................................................... 67

Figure 5.5: Normalized superconducting transition temperature Tc/T‘,b vs. Nb layer
thickness. Solid lines are the results of fits to Equation (5.2). ...................................... 70

 

 

Figure 5.6: Reduced superconducting transition temperature t vs. Nb layer
thickness for the same data series as in Figure 5.5. Solid lines are the results of fits

to Equation (5.2). ........................................................................................................ 71
Figure 5.7: Multilayer to bilayer deconvolution. .......................................................... 75
Figure 6.1: Hc2(T) for a 500 nm thick Nb film. The line is a linear fit to the data. ........ 92
Figure 6.2: A schematic diagram of the vorticies in a multilayer showing how fix},

a”, and i, are defined. ................................................................................................... 93
Figure 6.3: Han and Hc2l for Nb/CuGe(5%) 10 nm/ 10 nm ........................................... 94
Figure 6.4: Han and chi for Nb/CuGe(5%) 70 nm/3OO nm multilayer (filled

symbols), and a bulk Nb sample (open symbol). ........................................................... 97
Figure 6.5: chu and chi in a Nb/CuMn(O.3%) 4O nm/20 nm multilayer. ..................... 98
Figure 6.6: HC2L(T) for Nb/CuGe(5°/o) multilayers with (1n = 20 nm. ............................ 101
Figure 6.7: ch(T) for Nb/CuGe(5%) multilayers with (15 = 28 nm. ............................ 102
Figure 6.8: ch(T) for Nb/CuMn(O.3%) multilayers having (in = 20 nm. ..................... 103
Figure 6.9: ch(T) for Nb/CuMn(0.3%) multilayers with ds = 28 nm .......................... 104
Figure 6.10: Hc2i(T) for Nb/CuMn(2.2%) multilayers with cln = 6.0 nm. ..................... 105
Figure 6.11: Hc2i(T) for Nb/Cu multilayers with'dn = 40 nm. ...................................... 106
Figure 6.12: chCl‘) for Nb/Cu multilayers with (18 = 28 nm ........................................ 107
Figure 6.13: ch(T) fit used to extract H'i(T). ........................................................... 108

Figure 6.14: H'i(T) for Nb/Cu multilayers with (1n = 40 nm. H; is defined as the
slope of these data. ...................................................................................................... 109

Figure 6.15: H‘ i(T) for Nb/Cu multilayers with (18 = 28 nm. H"i is defined as the
slope of these data. ...................................................................................................... 110

Figure 6.16: Hc2i(0.5) for d5 = 28 nm and various CuX materials and layer
thicknesses ................................................................................................................... 113

xii

 

 

Figure 6.17: Dependence of Hc2 i(0.5) on ds for various tin and CuX ............................ 115
Figure 6.18: Dependence of h i(0.5) on (15 for various tin and CuX. .............................. l 16
Figure 6.19: Sketch of H" i(Tc) for varying ds and cln ................................................... 122

Figure 6.20: Qualitative dependence of E,Z(T) showing how T21) and T“ are
defined ......................................................................................................................... 125

Figure 6.21: Qualitative dependence of aw“) showing how TZD, T‘, and T: are

 

defined ......................................................................................................................... 126
Figure 6.22: Hc2II(T) resulting from the §Z(T) and §,y(T) data sets shown above. .......... 127
Figure 6.23: Hc2"(T) in Nb/Cu multilayers with dn = 40 nm. ......................................... 128
Figure 6.24: Hc2“(T) in Nb/Cu multilayers with ds = 28 nm. ......................................... 129
Figure 6.25: Hc2"(T) in Nb/CuGe(5%) multilayers with ds = 10 nm. ............................. 130
Figure 6.26: H.2..(T) in Nb/CuGe(5%) multilayers with cln = 20 nm .............................. 131
Figure 6.27: H02”(T) for Nb/CuGe(5%) multilayers with ds = 28 nm. ........................... 132
Figure 6.28: H02"(T) in Nb/CuMn(0.3%) multilayers with dn = 20 nm. ......................... 133
Figure 6.29: Hc2II(T) in Nb/CuMn(O.3%) multilayers with (IS = 28 nm. ......................... 134
Figure 6.30: Hc2H(T) in Nb/CuMn(2.2%) multilayers with cln =6.0 nm. ......................... 135
Figure 6.31: H02"(T) for Nb/Cu multilayers with (in = 40 nm (filled symbols) and

Cu/Nb/Cu sandwiches with (in = 300 nm (open symbols). ............................................. 138
Figure 6.32: Expanded view of H62"(T) for Nb/Cu multilayers with ds = 28 nm. ........... 139

Figure 6.33: H02"(T) for Nb/Cu 28 nm/40 nm and the extension of the 2D state
predicted by the FQ model. .......................................................................................... 142

Figure 6.34: E,‘ as a function of A for various CuX layers and Nb and CuX layer
thicknesses ................................................................................................................... 143

Figure 6.35: Schematic diagram of F and E as a fiinction of position perpendicular
to the layers of an S/N multilayer. ................................................................................ 145

xiii

 

Figure 6.36: Approximate vortex cores for external fields directed parallel to the
layers. .......................................................................................................................... 146

Figure 6.37: Approximate vortex structure in a S/N multilayer. ................................... 148

Figure 6.38: Hc2||(T) for Nb/Cu 28 nm/4O nm and the extension of the 2D state
given by the F Q model. ................................................................................................ 149

Figure 7.]: Tc (open symbols) and Tf (filled symbols) as functions of dCuMn for
various constant values of dNb. ..................................................................................... 156

 

Figure 7 .2: Tc (open symbols) and Tf (filled symbols) as fiinctions of dNb for
various constant values of dCuMn ................................................................................... 157

Figure 7.3: Zero-field-cooled magnetization data for three Nb/CuMn(1%) samples

having dCuMn = 6.0 nm. Tfis defined as the temperature at the peak of the zero-
field-cooled magnetization. .......................................................................................... 158

xiv

 

 

 

 

CHAPTER 1
INTRODUCTION

A superconducting metal (S), placed in physical contact with a normal, non-super-
conducting metal (N), can induce a superconducting pair density into N. This
phenomenon, known as the proximity effect, is useful because it can be used to study the
superconducting properties of traditionally non-superconducting materials.

One class of materials particularly suited to investigation using the proximity effect
are magnetic materials. In typical bulk superconductors the addition of a few atomic
percent of magnetic impurities destroys the superconducting state. This occurs because
the superconducting state is composed of electrons whose spins are opposite to one
another. When such a pair scatters from a magnetic ion, the electron spins are scattered
into two unrelated spin states, thereby breaking the spin pair. This interaction is so strong
that only in a few rare earth (RE) alloys and compounds can the magnetic impurity
concentration be made high enough to observe effects related to interactions between the
magnetic ions.1 The properties of these RE systems, however, appear to be uniquely
related to the f-band magnetic state of RE ions. Since the proximity effect can induce
superconductivity into any normal metal, one can use the proximity effect to study the
effect of magnetism on the superconducting state in any magnetic system.

Superconductor/ferromagnetic multilayers have been studied as candidate prox-
imity effect systems in which to examine the behavior of coexistent magnetism and super-
conductivityfl’l“,5 Although the superconducting state for these materials has been
observed at temperatures well below the magnetic phase transition temperature Tf, it has
not been determined if the ferromagnetic state still exists below Tc.4 In addition, one
cannot easily examine the regime where Tf is of order To since the bulk Tf for most ferro—
magnets is two orders of magnitude greater than the Tc's of most conventional super-

conductors. Another difiiculty of using these systems is that ferromagnets strongly break

1

 

2

superconducting pairs both because of their high concentration of magnetic spins and their
spontaneous magnetization below If. Due to this effect, superconductivity can only be
induced a short distance into the magnetic layer. Hence, to examine these systems in the
layer thickness regime where superconductivity is induced within the entire magnetic layer
one must use very thin layers where problems related to layer thickness uniformity and
connectivity become severe.

To avoid some of these difficulties this thesis entails a study to determine if super-
conducting/spin—glass (SC/SG) multilayers can be used to study the interplay between
magnetic and superconducting order. In particular, the multilayer system Nb/CuMn has
been examined where Nb is a bulk SC and CuMn is a bulk SG. Due to the large spin of
Mn in a Cu host (SMn = 5/2), one can shift smoothly from the non-magnetic limit to the
strongly magnetic limit simply by increasing the Mn concentration y. Increasing y leads to
an increase in the spin-freezing transition temperature of CuMn Tf and to a decrease in the
electron mean-free-path Kn. To correct for 8,, induced changes in the measured super-
conducting properties of these multilayers, the results were compared with those measured
in Nb/CuGe samples. In a Cu host, Ge impurities are non-magnetic and lead to roughly
the same increase in 6,, with impurity concentration as occurs in CuMn alloys.6 CuGe
layers are, therefore, expected to correctly model changes in the superconducting prop-
erties of these systems which are related to changes in En. For simplicity in the following
discussions, CuMn and CuGe will generically be denoted CuX.

There are several advantages to using spin-glasses over ferromagnets in these
studies:

1. CuMn is a dilute magnetic alloy that undergoes a magnetic phase transition
into a state having no net magnetization. Therefore, the superconducting pair
penetration depth should be much larger for CuMn than it was for ferromag-
netic layers and more easily used deposition techniques can be used to fabricate

the samples.

 

 

 

3

2. The dependence of Tf on the CuMn layer thickness dn, the Nb thickness ds, and
the Mn concentration y is fairly well known.7=8 Therefore, Tf can easily be
adjusted to fall near or below Tc.

The primary disadvantage to using spin-glass layers is that any interactions
between the superconducting and magnetic states are likely to be weak. This is the result
of the weakly-coupled magnetic state and the dilute magnetic ion concentration of spin-
glass alloys. Hence, the main problem in establishing if SC/SG multilayers can be used to
study interactions between superconductive and magnetic order will be understanding the
superconducting properties of these materials well enough so that changes in the
superconducting behavior of the samples can be related to a magnetic state, or to some
other phenomenon—changes in crystal structure, effects due to the paramagnetic nature of
the spins, effects related to 8“, etc.

Few studies have examined the proximity effect in dilute magnetic N
layers,9:1°’11'12913 and none have studied these effects in multilayered systems. Therefore,
this thesis focuses on the study of the dependence of Tc and the superconducting upper
critical field, ch, on ds, (1", y, and En.” It represents the first study in which such a wide
range of material properties have been systematically varied within the same multilayered
system.

To set the stage for the rest of this thesis, the remainder of this introduction
discusses what is currently known about the superconducting state in proximity effect
multilayers. Chapter 2 covers the details of sample preparation, and Chapter 3 covers
sample characterization. The experimental section begins with Chapter 4 and a discussion
of the experimental procedures used to take the Tc and H02 data sets. Chapter 5 then
presents the analyses of Tc, while Chapter 6 presents a discussion of the Hc2 results.
Chapter 7, then discusses what has been learned about the interplay between Tf and Tc.

Finally, Chapter 8 summarizes the findings of this project.

 

4

1.1 Previous Studies of Tc and ch in Multilayered Superconductors

Even though the superconducting transition temperature is the easiest experiment
with which to study multilayered superconductors, only three significant experiments on
the dependence of Tc in multilayered superconductors have been performed: Bannerjee et
a1.15 studied Nb/Cu multilayers; Missert and Beasley16 studied S/N multilayers of the
amorphous superconductor M0796e21 and the amorphous metallic alloy Mol,xGex; and
Wong and Ketterson?»4 studied superconducting/ferromagnetic V/Fe multilayers. The
main findings of these studies are summarized below.

As dn increases in V/Fe and Mo—,9Ge21/Mol_xGex multilayers Tc decreases mono-

tonically until it saturates to a minimum value Tcm for (in > dLC. For layer thicknesses

.

m, pairs cannot cross N without being broken. Therefore, d; is propor-

greater than (1
tional to the superconducting-pair penetration depth in N. Wong and Ketterson indicate
that d; may decrease slightly with decreased ds. Missert and Beasley found that in
MoHGex alloys d;c increases as the electron mean free path 6,, increases. Therefore, the
pair penetration depth appears to increase with increased electron mean free path, and may
depend slightly on the S layer thickness.

Tcm was observed to decrease with increasing 8“ in the M079Ge21/1\/Iol_xGex series.
While this trend is readily apparent in the data, Missert and Beasley did not speculate on a
cause. One possibility, however, is that increased 6,, leads to a greater ability for super-
conducting pairs to enter N and subsequently be broken. This idea will be discussed again
in Chapter 5.

In Nb/Cu multilayers having (18 = do = d, Bannerjee et al. found that Tc decreased
more rapidly with decreasing d than could be explained using the available proximity efl‘ect
models”,l7 They attributed this discrepancy to a decrease in the effective bulk super-
conducting transition temperature of Nb, Tc", with decreased d,. In thin Nb films, the

electron mean free path 88 is dependent on the mean crystallite size and both are typically 3

 

 

d5. Hence, the reduction in T: is related to a reduction in 65. Such a dependence of TC on

65 is known to occur in radiation damaged Nb samples,”"“”20 and has been associated with
broadening of a peak in the electronic density of states of Nb near its Fermi energy. As
this peak broadens it leads to a decrease in the density of states at the Fermi level, which in
turn leads to a decrease in T5.

In the limit that (In —> 0, Tc should approach the bulk superconducting transition
temperature of S. If however, Tc was reduced by structural disorder within the S layer,
this disorder would remain constant as dn decreased, and TC would approach the effective
bulk Tc discussed above. In V/Fe multilayers, Tc(dn—>O) is independent of (Is as is
expected since V does not have a sharp peak in its density of states near the Fermi energy.
In M079Ge2 1/1\/Iol_xGex multilayers, however, Tc(dn—)O) appears to increase as 6“ increases.
While noting that this effect existed, Missert and Beasley did not explain the observed
difference between Tc(dn—)O) and the true bulk transition temperature of M079Ge21.

One of the reasons that the Tc experiments have received so little attention may be
because studies of the superconducting upper critical field ch have located a unique
dimensional crossover transition. HC2(T) provides a measure of the superconducting
coherence length in a plane perpendicular to the direction of the applied field, so that in
multilayers, Hc2 is strongly dependent on the angle between the field and the plane of the
layers. There are two limiting values of Hc2(0): ch for external fields directed perpen-
dicular to the plane of the layers, and Hall for external fields directed parallel to the layers.
Hc2_L has been related to the superconducting coherence length in the plane of the film a.“
while H02” has been related to both gin and to the coherence length perpendicular to the
layers 51.21

In most multilayer experiments, chi(T) 0c (Tc - T), as is found in typical bulk
superconductors.432333435 This indicates that for most multilayered systems, the tempera-
ture dependence of a,” is not strongly affected by the layering. In some V/Ag multilayers,

however, Kanoda et al. observed a significant positive curvature of Hc2l near T,25 which

 

 

 

6

became more noticeable as (In and (15 were increased in tandem. They did not, however,
study the curvature as tin or (18 was varied independently. A curvature similar to this is
predicted by the theoretical models of Biagi, Kogan, and Clem,” and Takahashi, and
Tachiki (TT).27 TT argue that the curvature is related to the variation of the pair density
along a vortex line. As dB or (18 increase, the pair density becomes more localized within
the S layers and leads to the observed curvature. This is a very rough conjecture, and it
will be reexamined in Chapter 6.

Hc2|l(T) has proven particularly interesting. As the temperature T decreases below
Tc, the temperature dependence of H02” changes from being indicative of a 3D coupled
superconductor to that indicative of an isolated 2D superconductor. This behavioral
change is related to a decoupling of the superconducting layers in a multilayered stack
and has been argued to occur when £2 became approximately equal to a physical length
scale E,‘ which is related to the two layer thicknesses“,25 The model used to interpret
both these experiments begins near Tc, where 5,, diverges as (Tc - T)“2 and is larger than
any physical length scale in the multilayer. Therefore, g, is large enough to average over
the superconducting properties of the S and N layers with behavior similar to that
observed for §(T) in bulk superconductors. As the temperature decreases, 5,, decreases
until it becomes approximately equal to E‘. At lower temperatures, £2 is too short to
effectively couple superconductivity between two S layers, and the superconductive
coupling and the 3D state are lost. While this model is in qualitative agreement with the
previously published data?“25 there are some subtle problems with it which will be
elucidated in the H62" section of Chapter 6.

To summarize, a good deal is known about the qualitative dependence of T‘: and
Hc2 on such parameters as dn, ds, 6,, and (is, but high quality quantitative information is still
lacking. To provide this quantitative level of understanding, most of this thesis is spent in

examining Tc and H02 in S/N multilayers.

 

 

 

 

CHAPTER 2
SAMPLE PREPARATION AND FABRICATION

All of the samples used in these studies were created using the MSU condensed
matter physics group sputtering system designed primarily by Dr. William Pratt Jr. and
built by Simard Inc.28 This system consists of an ultra-high-vacuum compatible chamber
housing 4 sputtering guns and up to 16 substrates. The base pressure of this system, prior
to sputtering, was 2.0x10'8 torr. This, along with the ultra high purity gases used in the
sputtering process, limited background contamination of the gas in the chamber to less
than 10 ppm. Most samples were deposited onto polished, single-crystal, (001)-oriented,
Si substrates. Additional samples were made on glass, mica, and single-crystal KCl
substrates which were used in a variety of analytical tests to be discussed in the structure
section.

The remainder of this section will detail the substrates used, the general aspects of
sputter deposition using this system, the temperature control systems for the substrates,
the preparation of the initial sputtering targets, and the cleaning procedures used to insure

a low level of contamination in the system.

2.1 The Substrates

The typical substrates used were of polished, boron-doped, single-crystal, (001)-
oriented, Si. The dopant levels were such that the room temperature resistivity was at
least 10 Q-cm/El. A rather high resistivity substrate was chosen to avoid problems due to
the possibility of measuring currents shorting out through the substrates.

Substrates were purchased as 3 inch diameter wafers from Silicon Quest Inter-

national.29 The wafers were cleaved into 1/2"x1/2" squares for use. The cleaving process

 

 

 

 

determined the use of (001)-oriented Si, other orientations ((111) in particular) proved

difficult to cleave reliably.

2.2 Sample Mounting

The substrates were mounted in an aluminum holder as shown in Figure 2.1. The
substrate temperature was stabilized during sample deposition by using a Cu block and foil
to thermally link the substrate to the Al holder. Details of the temperature control scheme
will be discussed below. The holder also employed a rotatable shutter so that only one of
the substrates was used at any one time.

The 8 substrate holders, each holding 2 substrates, were arranged on the SPAMA
(Substrate Positioning And Masking Apparatus) plate as shown in Figure 2.2. The
position and orientation of the sample holders ensured that each substrate saw the same
sputtering conditions. In addition to the substrate holders, the SPAMA plate also held
two film thickness monitors, one each for Nb and CuX.

Substrates were cooled by a small liquid N2 reservoir located near the SPAMA
plate. This reservoir was thermally linked to the sample holders as shown in Figure 2.3.
Incoming high pressure (1000 psi) N2 gas passed through a thin stainless steel tube which
was cooled by a liquid N2 bath inside the sputtering chamber. This caused the high
pressure gas to liquefy. The liquefied N2 was then forced through a fine stainless-steel
tube into a small reservoir near the SPAMA plate. Heat, deposited in the substrate by the
incoming sputtered atoms was conducted off the sample by Cu foils and into the sample
holder. Heat was then progressively transferred to the SPAMA plate, to a series of OFHC
(Oxygen Free High Conductivity) Cu pieces and finally to the liquid N2 reservoir.

Evaporation of liquid N2 then carried the heat energy out of the system. With this cooling

 

Substrate pressure screw

    
  
 

Screw

Bridge assembly

/ Cu foil

~O.25 cm

 

 

 

' ~1cm '
Shutter / \ Substrate

Figure 2.1: Sample holder

   
    
 

/ Al plate

/ Substrate radius

I~5 cm
\ Film Thickness
Monitor (FTM)

\ Sample Holder (SH)

\ Substrate positions

Figure 2.2: The sample positioning and masking apparatus (SPAMA) plate.

 

Aluminum

    

Liquid N2 resovoir

Figure 2.3: The substrate cooling system.

system in place, substrate temperatures were kept between -20 and +20°C for

of a sputtering run (typically 4-10 hours). Without cooli

the duration

ng, the temperature of the

2.3 The Sputtering Guns

The CFMR sputtering system contained four triode, magnetron, sputtering guns.

A schematic of one gun is shown in Figure 2.4. In the sputtering process, ionized Argon

atoms were accelerated toward the target. On impact with the target, the ion's kinet‘c

. t
energy was transferred to target atoms. These atoms were then ejected from the targe

surface resulting in a beam of target atoms.

The Argon atoms were iomzed by stabrlrzrng a highly energetic electric Cline"

between the anode and the filament. Typically, this current operated at 500 to 700 kV and

 

 

 

Magnetic
onfinement
Shield

    
    

Aluminum
Gun

Housing

Tantalum
Filament

Cooling Water

Anode

Figure 2.4: Cross section of a triode magnetron sputtering gun.

 

 
 

 

 

12

confined the Ar plasma to the region just above the target. A high negative potential was
then applied between the Ar plasma and the target to accelerate the ions toward to target.
Finally, a collimation shield limited the ejected material to a beam approximately 5 cm
across.

Although material was readily ejected from the target surface, most of the kinetic
energy of the Ar ions was deposited in the target as heat. The target was water cooled to
avoid melting. Once the Argon ions struck the target, they acquired an electron from the
metal surface and were reemitted as neutral Argon atoms. To provide these electrons one
must use electrically conducting targets. This made it difficult to deposit insulating inter-
layers, hence the only 'insulator' used was doped Si, where the dopant level and target
temperature were high enough to allow sputtering to proceed.

A schematic of the sputtering system is shown in Figure 2.5, with typical gun
settings listed in Table 2.1. The target to sample separation distance was 12 cm, the
targets were 5.72 cm in diameter, and the collimator hole was ~5 cm across. These
resulted in a atomic beam whose intensity was uniform to within 5% over a region of
about 4 cm2 on the SPAMA plate. The angular divergence of this beam was about 25° at

the sample position.

Table 2.1: Sputtering gun settings.

 

 

 

Argon flow rate 35 cm3/min
Anode-to-Filament potential 50-60 V
Anode-to-Filament current 5-7 Amp

 

 

Egon Plasma-to-Target potential 500-700 kV
. Argon plasma-to-target current 0.5-1.1 mAmp

 

 

 

 

Cu E E/ Substrate holder
_
Substrate /f/

I
—— i\ i f —— Beam collimator
r \ E 3 metal /
r s
l ,

 

 

    

— Gun body

 

 

 

 

Target

Figure 2.5: Overall view of the sputtering configuration.

All the activities of the sputtering system were coordinated by an external
computer which directly controlled the angular position of the SPAMA plate, and
performed all calculations and timing needed to deposit layers of the desired thicknesses.
The computer system also monitored, but did not interactively control, the voltage applied
to the sputtering target and the current passing through the target. These parameters were
adjusted manually during each run to provide a constant sputter deposition rate. The
actual deposition rate was measured a few times during each run using the film thickness
monitors and manually entered into the computer. With this system, run-to-run variability

of the deposited layer thicknesses were typically :5%.

2.4 Target Production and Preparation

Sputtering targets were cut to the desired cylindrical shape, 5.72 cm diameter by

0.64 cm thick, by the staff of the Physics Shop at MSU. Pure Nband Cu targets were cut

14

from 0.64 cm thick plate obtained from Angstrom Sciences”. The purity of these bulk
materials was 99.95% for Nb and 99.999% for Cu. The CuX alloy targets were cut from
alloyed ingots fashioned here using an rf-induction furnace.

The alloy targets were made from 99.9999% pure Cu, 99.9% pure Mn, and 99.999
pure Ge, obtained from Aesar/Johnson Matthey inc.31 The raw materials were cleaned in
dilute nitric acid before being placed in a boron nitride lined graphite crucible. The boron
nitride was needed to avoid carbon contamination of the alloy. This crucible was then
placed in a vacuum chamber which was evacuated to 5x1045 torr before alloying began.
The chamber was then back filled to a pressure of ~250 torr using a 90 % Ar and 10 % H2
gas mixture. The H2 gas helped to remove residual oxygen from the system. The mixture
was heated to a temperature between 1100 and 1150°C and the alloy was kept molten for
~15 minutes. It was then cooled slowly to avoid formation of cracks in the alloy block.
To gain the maximum homogeneity, the block was typically inverted in the crucible and
remelted at least one additional time. The final Ge and Mn concentrations were deter-
mined from EDX and magnetic measurements on thick sputtered films. These measure-

ments will be detailed in the structural analysis section.

2.5 Cleanliness

Substrates were cleaned ultrasonically in acetone, and then alcohol. If residues
remained on substrates after this procedure, the substrates were re—cleaned in hexane,
followed by acetone, and then alcohol. A more lengthy cleaning procedure was avoided
by keeping the substrates as clean as possible fi'om the time the full wafers were removed
from their factory container.

An acid etch consisting of a mixture of 50% I-INO3 and 50% water was used to
clean most of the other parts of the sputtering system. These included the sample holders,

sample shutters, SPAMA plate, sample masks, gun body, A] target confinement ring, and

 

 

 

15

film-thickness-monitor covers. Deposits not readily removed by the acid were scrubbed
vigorously with a iron wire brush and then re—etched. Nb proved very difficult to remove
from the gun parts because the A] metal of the gun parts etched away nearly as fast as the
Nb deposits. A razor blade was used to scrape away most of the Nb deposits, after which
approximately 5% HF was added to the acid solution to help remove the remaining Nb.
The stainless steel sample masks are unaffected by HF, and were always etched with a
HNO3/water/HF solution. After the acid etch, all these parts were cleaned ultrasonically,
using a process similar to that used for the substrates. The SPAMA plate is too large to
fit in the ultrasound bath, and so was rinsed by hand with acetone and alcohol.

The shutters for the sputtered beams were never cleaned directly. Before each run,
the collimating chimneys were covered with Al foil which was discarded at the end of the
run. In addition, the iron magnetic confinement plates, sample bridge assemblies, Cu heat
sinks and thermocouples were not cleaned before each run, but were simply kept clean
between runs. These components were not cleaned because the were either, never in
contact with the sputtered metal beam (and therefore have no significant deposits on
them), were not in close proximity with the polished face of the substrate, or, in the case
of the magnetic shield, were highly reactive in all acids, and so were discarded when their

deposits became severe.

 

CHAPTER 3
STRUCTURAL ANALYSES

The purpose of structural analysis was to determine the internal structure of each
sample and to identify structural trends that may effect its magnetic and conductive

properties. To this end, the primary questions of interest were

. Were the samples layered?

. Were the layer thicknesses as expected fiom the sputtering conditions?

. Were the layer thicknesses uniform?

. Was the Nb/CuX interface sharp?

. What were the concentrations of desired and undesired impurities in the Nb
and CuX layers?

. Were the Nb and CuX layers crystalline or amorphous?

. What were their crystallite sizes?

. Were there any changes in the Nb or CuX lattice spacings as the layer thick-
nesses were reduced?

. Were there strains in the Nb or CuX lattices near the Nb-CuX interface?

0 Were the Nb or CuX crystallites preferentially oriented?

All these questions, with the exception of chemical impurity concentrations, were
addressed by analyzing x-ray diffraction data taken on as-produced samples. Many of the
x-ray diffraction results were also checked by studying x-ray absorption fine structure
(XAF S) studies and cross-sectioned multilayer films. The cross-section experiments
focused on imaging and electron diffraction. The chemical concentrations were deter-
mined using energy dispersive x-ray (EDX) analysis and, in the case of CuMn, by
measuring the spin-fieezing transition temperature.

16

 

17

3.1 Elemental Compositions

EDX and magnetic measurements were used to determine the concentrations of
the various desired and undesired impurities in the layers. EDX measures the elemental
compositions by exciting a sample with a high energy electron beam (typically 100-300
keV) and comparing the relative fluorescent x-ray line intensities from the host, Cu or Nb,
and impurity atoms. These intensities were suitably normalized and compared with those
expected from each pure material. For the CuGe alloy, EDX data provided the only
measure of the Ge concentration, which was 4.9 d: 0.3 atomic percent (at. %) for the
target used in these experiments. An analysis of EDX data also found no evidence for any
unwanted impurities in either the Nb or CuX films above the detection limit (~0.1 at. %)
of the method .

CuMn(y) is a spin-glass alloy with a magnetic phase transition which is sensitively
dependent on the Mn impurity concentration y. The spin-glass freezing temperature is

roughly given by

Tf :Tfpyv (31)

where y is the Mn impurity concentration in atomic percent (at. %) and T£0 and u were
derived from a least squares fit to data for LG») taken from the Landolt-B'omstein
Tables”,33 A fit to these data for y s 11 at. % resulted in the values: T£0 = 9.89 i 0.27 K
and u = 0.659 i 0.014. Hence, a measurement of Tf provides values of y. For y > 3 at.
%, magnetization and EDX measurements agreed to within 10%. Larger disagreements

were observed at low y due to the increased difficulty of quantitative x-ray peak detection.

 

 

 

3.2 X—ray Diffraction

X-ray diffraction was used to study the spacing of atomic planes within the Nb and
CuX crystal structures. An x-ray beam incident on a crystal will diffract from sets of
atomic planes within that crystal because the x-ray wavelength is of the order of the planar

spacings. The diffraction condition, known as Bragg's law, is

ml, = 2d sin(€) (3.2)

where d is the spacing between two planes of atoms within a crystal, 9., is the wavelength
of the incident x-ray beam, m is the order number of the diffracted beam, and 29 is the

angle between the incident and diffracted x-ray beams.

3.2.1 The Diffractometer

All the x-ray data sets discussed in this section were taken on a Rigaku powder x-
ray diffractometer. The general layout of this system is shown in Figure 3.1. The x-rays
were emitted from a rotating Cu anode and collimated by a set of divergence slits. The
divergence angle of the x-ray beam was set to (1/6)0 for most scans. After the beam
interacts with the sample, it was recollimated by a second set of slits and passed into a
graphite monochrometer. This selected out Cu—K0L radiation from the total emission
spectrum of the Cu source. The wavelength of Cu-K0L is 0.1542 nm.

The samples were mounted on the sample goniometer using an aluminum holder
consisting of a flat Al sheet with four protruding feet on one side and a stainless steel
spring clip on the back, as shown in Figure 3.2. This arrangement insured that the front
surface of the sample was parallel to the front surface of the sample holder, the position

required to ensure that the sample was in the proper alignment for the goniometer.

  
  
    
   
 

 

Sample Sample
gomometer \ D t t
‘ e ec or
Circle ’
\\ j ,. /gomometer
X-ray <1“ // circle
| I I I / Scattering slit
' x-ray
Cu x-ra Divergence
source y slit ‘/ detector
Graphite //
monochrometer , /

Monochrometer slit

Figure 3.1: The x-ray diffractometer.

     

Front view Back view Cut-away view

Figure 3.2: The x-ray diffraction sample holder.

 

20

Due to the fabrication tolerance of this holder (i 0.025 mm on each foot) the sample was
parallel to the surface to within only i 0.l°. This possible misalignment did not effect
spectra for scanning at angles greater than ~10°, but may have induced small errors in the
peak positions observed at smaller angles.

The sample and detector goniometer angles, or and 29 respectively, are driven in-
dependently by computer. In the reflection geometry or was set equal to 9, hence, the
incident and diffracted beams appeared to reflect fiom the sample surface. For the other
common drive geometries the detector and sample angles were set so that or = 9 - d), as
shown in Figure 3.3. In the off-axis geometry (1) was fixed and 9 and 29 were varied in
tandem, while in the rocking curve geometry (1) was varied for fixed 9 and 29. These scans
will be described in detail below.

The instrumental resolution of the apparatus was limited by two effects. The first
was the inherent minimum line width measurable by the system. This width is primarily
related to the widths of the various collimation slits used in the system. This led to an
apparent x-ray line width ranging from 005° for 29 = 10° to 011° for 29 = 120°. The
second resolution limitation arose because the monochrometer was not sensitive enough

to separate the narrowly spaced Cu-Ka doublet (K0ll and K00). The effective peak

Sampl

  
 
   

Incident beam

Diffracted
beam

Figure 3.3: Definitions of scattering and goniometer angles.

—é~“

2]
separation of the Kal and Kaz lines was 0.025° at 29 = 10° and 049° at 29 = 120°.
Therefore, analysis of peak widths, to be described below, was not performed on peaks

whose widths were comparable to either of these two instrumental widths.
3.2.2 Reflection Scans

In the reflection geometry, diffraction only occurs from planes of atoms parallel to
the substrate, this results in a set of x-ray diffraction peaks related only to structure along
a line normal to the sample. Such data measures the Nb and CuX lattice spacings, crystal
structures, orientational texture of the metal film perpendicular to the layers, and the
thicknesses of the two layers.

A reflection diffraction pattern for a Nb/CuMn(l3%) 2.0 nm/2.0 nm sample is
shown in Figure 3.4. Aside from the broad peak near 25° due to scattering from the
amorphous glass substrate used for this sample, there are three regions of immediate
interest. At small angles 0° < 29 < 9°, a series of closely spaced peaks appear which are
due to Bragg reflection from the bilayer periodicity A. This was the region of small-angle
x-ray diffraction (SAXD). The peaks near 40° are first order (m = 1) while those near 90°
are second order (m = 2) diffraction peaks from the Nb (110) and CuMn (111) planes.
These three regions will be discussed in detail below following discussion of one broad
feature typical of all spectra.

In Figure 3.4, no peaks were observed which could be related to scattering from
any Nb or CuX crystal planes except for Nb (110) and (220) and CuX (111) and (222).
This indicates that the Nb and CuX crystallites were oriented with these planes parallel to
the substrate. This behavior was related to the type of film growth which occurs in sputter

deposited films.

 

103

102

Intensity (arb. units)

101

22

 

 

104,

 

Nb(110)| lCuMn(111)

llllll

 

 

Nb (220) CuMn (222)
l l
_— SAXD 1at order , ,r
: 211d order
20 I 40 l 60 80 100 120

29 (deg.)

Figure 3.4: Full scattering spectra for a Nb/CuMn(13%) 2.0 nm/2.0 nm multilayer.

23

As each atom arrives on the film surface during film growth, it has enough energy
to wander about on the surface before settling into a stable site. Due to this motion, the
atom seeks out a position that maximizes the number of nearest neighbor bonds, resulting
in the most densely packed atomic planes being oriented parallel to the surface of the
sample. For the Nb bee and CuX fcc lattices, the densest planes are the (110) and (1 l 1)
planes respectively. Therefore, the deposition process lead to polycrystalline films with
their Nb bcc (110) and CuX fcc (111) planes parallel to the sample surface. While these
analyses imply that the growth process leads to oriented crystallites, it does not provide

any information about the extent of orientational ordering within the film plane.

3.2.2.1 Small-Angle X-Ray Diffraction

In addition to the Nb (110) and CuX (111) lattice spacings there is a third struc-
tural periodicity perpendicular to the layers, the bilayer period: A = and110 + nCuxdm,
where an and nCuX are the number of atomic planes within the Nb and CuX layers. This
period is much larger than either (1110 or d1“, therefore this period should produce diffrac-
tion peaks at very low angles. Furthermore, the Bragg condition is slightly modified in
this regime because at low scattering angles (< 5°), the difference between the x-ray index
of refraction in air and in a metal must be taken into account. In most metals, the index of
refraction at x-ray wavelengths nm is slightly less than that of air na, resulting in x-ray

scattering as depicted in Figure 3.5. Based on this figure, Bragg's law was written

j7t = 2A sin(9m)

where j is the order number of the reflection.

 

 

 

 

 

Figure 3.5: Low angle x-ray scattering in a metal film.

Snell's law was then used to relate 9a to 9,n
nm sine...) = n. sin(¢.) (3.3)

where 9 and d) are related by sin(¢) = cos(9). Then, na and nm were set to 1 and 1-5

respectively, and d) was replaced by 9 in Snell's law to obtain (in the limit that 5 << 1):
sin2(6la)Esin2(6m)+25 (3.4)

which was used to replace 9m by 93 in Bragg's law. This finally lead to an equation

relating the measured angle 92, to A and 6:

sin2(6,)s[%) j2+25 (3.5)

A was then extracted from a least squares fit of sin2(9,) to j2. A low angle data set and
resulting fit are depicted in Figure 3.6.
The derived A's were typically within 7% of those expected from the fabrication

conditions. Agreement between measured and expected values for A suffered in part

“f4

25

because of the small number (typically 2 to 4) of low angle peaks observed for each
sample.

Reliable values for 5, the deviation of nm from 1, were unobtainable from these fits
because the internal accuracy of the low angle measurement in the Rigaku system was not

high enough for such an analysis. Therefore, 5 was only used as a fitting parameter.

3.2.2.2 First-order Region

Figure 3.7 shows x-ray diffraction data for two Nb/ CuMn(0.3%) multilayered
samples, 70.0 nm/70.0 nm and 10.0 nm/13.0 nm. For large layer thicknesses [Figure
3.7(a)], the Nb and CuX bulk lines were virtually identical to those measured in bulk films.
The widths of these lines were related to the average crystallite size perpendicular to the

layers through Scherrer's equation:

S = mi; (3.6)

 

where S is the crystallite size and B is the width of the difiraction peak in 29 radians.34
Using Equation (3.6), the crystallite sizes of thick Nb and CuX films were found to be
approximately 25 nm .

As the layer thicknesses decreased, the Nb and CuX peaks broadened, but their
positions remain unchanged [Figure 3.7(b)]. The crystal sizes obtained from the line width
showed that the crystallites extended across the entire Nb or CuX layers. In addition, for
A below about 30 nm, small satellite peaks appeared flanking the bulk peaks. The spacing

of these peaks was directly related to A through Bragg's law with m ~ 80.

 

 

26

 

 

 

 

 

 

 

 

    

 

 

 

i=1 '
A _ l
a 2'- .
'a 104: g
:3 " .
.ci - 2
3‘1 2:?
>. 103E Z
+3 :
'8 Z 3
£1 .
«3 2f I 3
1 2 3 4 5 6
29(deg).
0.0025frj........,.,.,,,
0 Data
0.0020— Fit
8 0.0015~
NV F
.E
m 0.0010—
A=6.78i.01nm 1
= + —5
0.0005 5 (2.48_.06)x10 _
0.0000:'-'+llieltn.r.u.i
0 l 2 3 4 5 6 ’7 8 9 10
j2

 

 

 

Figure 3.6: Small-angle diffraction scan and peak analysis used to determine A for Nb/
CuGe(5%) 5.0 nm/2.0 nm.

 

27

 

 

 

 

 

[— T l 1 l l l T l T l 4
2: l
’5; 104’: A 2‘
:23 E l
G :
a 2 r
..o :
3 _ _
:3 10 5
I _ 1
a - 1
a 2 ~ (b)
5 102 ~ ;
+2 CuMn '
5: (110) (111)
2 ' (a) E
101 . 1 . 1 . 1 . 1 1 1 . 1 . 1 . 1 . 1 l I
35 36 37 38 39 40 41 42 43 44 45

29 (deg.)

Figure 3.7: First order diffraction peaks for Nb/CuMn(0.3%) 70.0nm/70.0nm (a) and
10.0nm/13.0nm (b).

 

28

A was extracted from a least squares fit to Bragg's law as shown in Figure 3.8 for
a Nb/CuMn(10%) 15.0 nm/7.0 nm multilayer. Due to the large number of satellites
typically observed (5 to 12), the statistical uncertainty of the resulting A's was much lower
than was the case with A derived from low angle data. The statistical and systematic
errors of the x-ray determination were considered low enough that most of the observed
deviations between fabricated and measured A's (~ 5%) were attributed to errors made
during the fabrication process. This lead to the 5% uncertainty in deposited layer thick-
nesses quoted in the sample preparation chapter.

The peak widths are related to a Fraunhofer type diffraction pattern due to there
being coherent diffraction within a single layer thickness. This results in there being an
overall Fraunhofer type diffraction envelope which modulates the intensity of the satellites.
This is clearly observed in Figure 3.8: the missing peaks (m = 3 and 6) occur at the
minimum positions for a Fraunhofer pattern due to the 7.0 nm thick CuX layer. The Nb
layer does not show a significant Fraunhofer pattern because its thickness is significantly
larger than that of CuX.

For very small layer spacings (< 4 nm), the peak broadening due to decreased layer
thickness became of order the separation between the bulk Nb and CuX peaks, and the
bulk Nb and CuX lines were entirely washed out by the satellite peaks. Hence, for these
small layer thicknesses, data taken at the first order peaks could not be used to determine

the lattice parameters of the individual constituents.

 

 

29

 

 

Intensity (arb. units)

 

 

 

 

35‘36 37'38'39'40‘41 42 43‘44'45
29(deg.)

 

0.38 ' I ' l ' l ' I ’ ‘I l ' l | ' i ' l

 

0 Data
0-37 " Fit: A = 21.013 nm

 

sm(e)

 

 

 

 

0.331111111111
01234567891011

order number, m

 

 

 

Figure 3.8: First order spectrum and peak analysis used to determine A for
Nb/CuMn(10%) 15.0nm/7.0nm.

 

3O

3 2.2.3 Second-order Region

The difiiculty of resolving the bulk Nb and CuX peaks was avoided by examining
the second order spectra. Here the enhanced resolution due to scattering at large angles
allowed the lattice spacing of Nb and CuX to be determined on layer thicknesses down to
2.0 nm (Figure 3.9). By estimating an envelope for the Nb and CuX peaks sets, the lattice
spacing can be calculated. The resulting spacings agreed to within 2% and 3% with the
expected Nb and CuX spacings, respectively. In addition, the crystallite sizes determined
fi'om the Nb and CuX peak widths confirmed that the layer thicknesses were roughly equal
to their expected values.

Fullerton et al.35 recently developed a recursive x-ray diffraction modeling program
to study layer thickness fluctuations in Nb/Cu multilayers. Their analysis of sputtered
Nb/Cu 2.6 nm/2.6 nm multilayers showed that experimental observation of satellites asso-
ciated with the Nb (220) and Cu (222) peaks placed an upper limit of 0.3 nm on the rrns
fluctuations in the two layer thicknesses. On the basis of the similarity in layer thicknesses
and satellite structure of their published Nb/Cu data and the above Nb/CuX 2.0 uni/2.0 nm
and 3.0 uni/3.0 nm data (Figure 3.9) we inferred that the rms fluctuations in the Nb and
CuX layer thicknesses in our samples were no more than 0.3 nm. Satellites were observed
near the Nb (220) and CuX (222) peaks for all samples with Nb thicknesses s 6.0 nm and
CuX thicknesses s 5.0 nm. Since satellites could be resolved for these samples, there
could be no large changes in the magnitude of layer thickness fluctuations for multilayers

having layer thicknesses greater than 3.0 nm.

31

 

 

 

 

 

120 ~ ~
”a? i 4
fig 100; Cu ‘
5
. 80 _ (111) -
e _ .
3 60 — (a) i ‘ ~
>3 -
.4: .
g 40 Nb
.3 * 110 ‘
E: 20 - (b) ( ) _
O ’— 1 I L l 1 1 IM. l 1 i l 1 1
80 85 90 95 100 105
29 (deg.)

Figure 3.9: Second order peaks for a Nb/Cu 2.0 urn/2.0 nm (a) and a 3.0 urn/3.0 nm (b)
multilayer. Error bars are bounds for a i- 2% change in dNb, and a i 3% change in doux-

 

32

Analysis of the reflection x-ray diffraction data shows that the Nb and CuX layers
crystallized in their bulk crystal structures (bcc for Nb and fee for CuX) with their bulk
lattice spacings. The data showed that the Nb and CuX crystallites were preferentially
oriented with their Nb (110) and CuX (l l 1) planes roughly parallel to the substrate. This
growth was a direct result of the sputtering process. Analyses of the SAXD, First, and
Second-order data showed that the samples were layered with individual layer spacings
and bilayer repeat distances approximately equal to those expected from the sputtering
conditions. The second-order spectra were also used to show that the rms deviations of
the Nb and CuX layer thicknesses were less than about 0.3 nm for layer thicknesses

greater than 2.0 nm.

3.2.3 Off-axis Scans

While a great deal of information was gained from the reflection scans described
above, they provided no information about in-plane ordering between crystallites or about
changes in atomic spacing within the layer planes. In multilayered materials, in—plane
strain may be induced in the crystallites due to the occurrence of epitaxial growth at the
metal-metal interface. Epitaxial growth occurs when material B attempts to grow in
atomic registry with the existing atomic order of material A. This registry typically
requires one or both of the crystal lattices to strain to improve the registry. Hence, these
strains will lead to the most significant changes in the A and B lattice constants for
spacings parallel to the interface.

The most straight-forward means of measuring these in-plane parameters is to
rotate the sample by 90° and then perform a normal 9-29 scan. Figure 3.10 shows such a

scan taken on a 500 nm thick Nb film deposited on a 120 pm thick glass substrate.

33

 

 

Intensity (arb. units)

 

 

 

0120‘40L60 80 100‘120
29(deg.)

Figure 3.10: Transmission scan for a 500 nm thick Nb film.

34

The measured intensities were very low because the diffracted beam passed through the
substrate. Glass substrates were used in these experiments because, for a thickness of 120
pm, a transmission of about 10% of the incident x-ray intensity was expected. Although
diffraction peaks were observed, the peaks were anomalously broadened. In addition,
some of the peaks in bulk films were shifted from their expected positions. These efi‘ects
were attributed to inelastic scattering of the diffracted beam within the glass substrate.
Attempts to remove the sample from the substrate resulted in samples that crinkled or
rolled up into tiny tubes and were nearly impossible to flatten and measure. For these
reasons, transmission experiments were abandoned.

A more subtle means of determining in-plane structure is to examine x-ray lines
that contain information about both the in-plane and out-of-plane lattice spacings. By ex-
amining several such peak positions, the in-plane lattice spacing can be extracted. For
instance, in the 2—D square lattice shown in Figure 3.11, the (01) planes are aligned paral-
lel to the substrate. Therefore a normal reflection scan will measure only ax. By rotating
the entire sample by 45° the (11) planes are brought into the proper alignment for a

diffraction scan. This scan then measures d2 given by

1 1 l
_:_+_

3.7
d: a: a ( )

2
Y

where ay is the in-plane lattice spacing. Hence, by combining the results of the two

measurements, both lattice spacings can be determined.

 

 

35

 

 

Normal Scan

 

7., = 2 d1 sin(91)

Off-Axis Scan

 

It, = 2 d2 sin(92)

 

Figure 3.11: Off-axis scattering geometry.

 

 

36

The off-set angles (1) are related to the geometry of the crystal structure and, for a

cubic lattice, are calculable using the equation36

<1) : hlhz + klkz +111, (38)

\/(h12+k12 +lf)(h§ +k§ +122)

 

where (hlklll) and (hzkzlz) are the Miller indices for the two sets of planes involved (i.e.
(01) and (11) for the case in Figure 11) and d) is the angle between the planes, (hlklll) is
the plane from which the off set angles are measured, for Nb and CuX these were the
(110) and (111) planes, respectively. (1) is sensitive to the absolute order hzkzlz, therefore,
several values of (1) can be calculated for each family of hzkzl2 indices. For instance, for a
bee crystal, the angle between the (110) and (211) planes is 30°, while the angle between
the (110) and (112) planes is 547°. All possible (1) for the (hzkzlz) families up to Nb (440)
and CuX (620) are listed in Table 3.1.

Diffraction from most of these planes still resulted in diffracted beams which
passed through the substrate. For a small subset of these lines the diffracted beam was
observed above the substrate when d) < 9. These lines are listed in Table 3.2. Off-axis
diffraction was been observed for all these lines using Nb and CuX films and Nb/CuX
multilayers. From an analysis of these peak positions and the peak positions observed at (I)
= 0, the in-plane lattice spacing was determined.

Figure 3.12 shows the off axis (311) peaks observed for a CuMn(13%) film and a
3.0 nm thick CuMn(13%) layer. Within the accuracy of the data, there was no change in
the in-plane atomic spacing for CuX thicknesses down to 3.0 nm. In addition, analyses of
the off-angle CuX peak widths showed that the CuX crystallites had the same dimension

within and across the layer both for bulk CuX films and for thin CuX layers.

 

37
Table 3.1: Off-axis angles, planer spacings, and diffraction angles for bcc Nb and fee Cu.

 

 

 

 

Line Off set angles, 4) (degrees) (1 (nm) 29 (deg) # of (11's
Nb 110 0 60 90 0.237 37.9 3
(110) 200 45 90 0.1675 54.8 2
211 30 54.7 73.2 90 0.137 68.4 4
220 0 6O 90 0.118 81.5 3
310 26.6 47.9 63.4 77.1 0.106 93.2 4
222 35.3 90 0.0970 105.2 2
321 19.1 40.9 55.5 67.8 79.1 0.0895 118.8 5
400 45 90 0.0838 133.7 2
330 0 60 90 0.0790 154.5 5
411 33.6 60 70.5 90
420 18.4 50.8 71.6 0.0749 3
332 25.2 41.1 81.3 90 0.0714 4
422 30 54.7 73.2 90 0.0684 4
431 13.9 46.1 56.3 65.4 73.9 82.0 0.0657 7

510 33.7 46.1 56.3

 

 

 

 

520 25.4 39.2 58.9 67.2 0.0612 4
440 0 60 90 0.0592 3
Cu 111 0 70.5 0.2090 43.3 2
(111) 200 54.7 0.1810 50.4 1
220 35.3 90 0.1280 74.0 2
311 29.5 58.5 80.3 0.1090 90.0 3
222 0 70.5 0.1045 95.0 2
400 54.7 0.0905 116.7 1
331 22.0 48.5 82.4 0.0830 136.3 3
420 39.2 75.0 0.0809 144. 2
422 19.5 61.9 90 0.0739 3
511 38.9 56.2 70.5 0.0697 4
333 0 70.5
440 35.3 90 0.0640 2
531 28.6 46.9 73.0 84.4 0.0612 4
600 54.7 0.0603 3

442 15.8 54.7 78.9
620 43.1 68.6 0.0572 2

 

38

Table 3.2: Off-axis lines observable using Cu K0L radiation.

 

 

 

Line (1) (deg) 29 (deg)
Nb 321 19.1 119
(110) 310 26.6 93.2
211 30.0 68,4
222 35.3 105
312 40.9 119
213 55.5 119
Cu 331 22.0 136.
(111) 311 29.5 90
220 35.3 74
331 48.5 136
400 54.7 117

 

 

 

Nb proved to be more interesting. As 4) was increased, the diffraction peaks
broadened, an effect which was related to a decrease in the effective crystallite size. This
implied that in bulk sputtered Nb, the crystallites were not isotropic, rather they were rod-
like with length ~ 25 nm and diameter ~ 6.0 nm. In addition, bulk Nb was expected to be
cubic with a,( = ay = a2. However, analyses of the bulk Nb (321), (312) and (123) peaks,
shown in Figure 3.13, showed that the peak shifted to a larger angle (and smaller planer
spacing) with increasing (1). Table 3.3 lists the planar spacings determined from these
peaks. When all 6 lines listed in the table are taken into account, the peak shift
corresponds to a (1.1 i0.2)% change in the in-plane lattice spacing in plane spacing az
relative to the normal lattice spacing ax. Although this effect appears real, it must be
related to an instrumental artifact since there are no driving forces in a pure Nb film that

might cause a change in g relative to ax.

39

 

 

 

 

. 1 u 1 r l T T 3 1 ‘ I ’ l '
100 - f. 7
,.. 90 - \ —
m * I ‘ x ‘
.t’. 80 ‘ .A. A. 4 K 7
a _ i x x A\ , \ "
:3 .70 H X l A _
F9. 60 h— f], ‘ A\k—A j
:3 50 — ray" -
r /
:2} 40 P (A, A‘ 'A‘Ad
g 30 " All ’9‘ I
Q.) _ ‘ I; A /1 ‘1, I
g 20 ~ x,» x 4
H 10 if“. —0— Bulk CuMn(137.)
/ ‘1' "-4-- 3.0 nm CuMn(137.)
O _ _
L I 1 l L 1 1 1 L l 1 1 L 4 L
84 85 86 8'7 88 89 9O 91 92
29 (deg)

Figure 3.12: CuMn (311) peaks observed in a 500 nm thick CuMn(13%) film, and a
Nb/CuMn( 13%) 3.0 nm/3.0 nm multilayer.

 

40

The Nb (321) family of peaks for a Nb/CuMn(l3%) 3.0 nm/3.0 nm multilayer
displayed a similar shift in peak position with increasing 4). This shift resulted in an
apparent change in the in-plane lattice spacing of (1.9 i 1.0)% relative to ax. Since this
shift was significantly larger than the bulk Nb lattice shift there may be a real change in the
in-plane Nb lattice spacing in these thin Nb layers. Additional studies need to be

performed to definitively test if the Nb lattice is really strained in thin Nb layers.

Table 3.3: Off-axis peak positions recorded for Nb crystallites.

 

 

 

 

 

 

 

 

bulk Nb 3.0 nm Nb layer

Line 4) (deg) 29 (deg) (1 (nm) S (nm) 29 (deg) d (nm) S (nm)
110 0 38.1i0.1 0.2359 26. -- -- --

220 0 -- -- -- 82.2:t1.5 0.1172 --

310 26.6 94.4d:0.3 0.1050 9.9 94.83:].2 0.1047 3.3
222 35.3 107.2:tO.4 0.0957 7.5 110.0:l:1.5 0.0942 2.8
321 19 120.0i0.3 0.0890 7.9 120.9:I:l.5 0.0886 3.4
312 41 121.0:l:0.4 0.0885 6.3 122.3i1.5 0.0880 3.4
123 55.5 121.6:t0.6 0.0883 6.0 123.6i1.5 0.0874 3.5

 

3 .2.4 Rocking Curves

Rocking curves were used to determine the approximate angular distribution of
crystallite orientations normal to the metal film plane. For the rocking curves, the detector
angle 29 was held fixed, thereby fixing the planar spacing examined, and the sample angle
or was swept through the diffraction peak at or = 9. For a sample composed of randomly
oriented crystallites, the rocking curve is expected to show little change in intensity as or is
changed. If, however, the crystallites are preferentially ordered along the direction normal
to the layers, the rocking curve is expected to show a peak at or = 9 whose width is pro-

portional to the distribution of normals to the crystallites.

 

 

41

 

100
90 '
80 ”
'70 '
60 h
50 "
40
30
20 '
10

O . 1 1 l 1 1 1 1 . 1 . 1 1 1

115 117 119 121 123 125 127 129

29 (deg)

(321)
(312) j
(123) _

Intensity (arb. units)

 

 

 

 

Figure 3.13: Off-axis peaks observed in a 500 nm thick Nb film: (a) (321) peak, (b) (312)
peak and (c) (123) peak.

 

42

Figure 3.14, shows a rocking curve for a 500 nm Nb film. On the basis of the
width of this peak, the (110) planes were interpreted as being parallel to the substrate to
within ~ 5°. Similar data were recorded for thin Nb and Cu layers. From these data one
can infer that in all the multilayers of this study, the Nb (110) and CuX (111) planes lay

parallel to the substrate to within ~5°.

3.3 XAFS Measurements

X-ray absorption fine structure (XAFS) analysis was used to check the lattice
parameters and crystal structure in very thin CuX layers (< 4.0 nm thick). On the high
energy side of the x-ray absorption edge of an element, resonances occur between the
excited core electron and electrons bound to neighboring atoms. These resonances lead to
oscillations of the absorption spectrum of the element whose period is proportional to the
fourier transform of the radial electron distribution near the absorbing atom. One of the
main drawbacks to XAFS measurements is the sensitivity of the extracted radial distri—
bution function to systematic effects To get meaningful measurements of lattice spacings
and crystal structures using XAFS, a standard sample must always be run whose XAFS
spectra are very similar to that of the measured sample. This sample is then used as an
internal standard and lattice and crystal structure changes are then determined relative to
the standard. The XAFS analyses software package used for these comparisons was
developed at the University of Washington.37

The XAFS data were taken using beam line X23b at the National Synchrotron
Light Source at Brookhaven National Laboratory. A monochromatic x-ray beam was sent
through an Ar filled proportional counter to measure the incident beam intensity. The

beam then passed through the sample, and into a second proportional counter. The

43

 

100

I

70?
601
50!
401
301
201
10»

O ' 1 1 . I . I

Intensity (arb. units)

 

 

 

 

10 15 20 25 ‘ 30
9 (deg)

Figure 3.14: Rocking curve for the (110) line of a bulk Nb film.

44

absorption was defined as the ratio of the intensities measured in the two counters. The
recorded absorption spectrum for the Cu edge in a Nb/CuMn(1%) 5.2 nm/4.6 nm sample
is shown in Figure 3.15.

The linear background was removed from these data and the intensity of the oscil—

lations was normalized to the height of the absorption edge. In addition, the horizontal

 

scale was converted fi'om energy to k-space using k = J26.3(E —E°) , where k is in run:1

and E is in eV. E0 was always defined as the energy at the midpoint of the absorption
step. The extracted oscillations for the above Nb/CuMn(1%) sample, now called a Chi
plot, are shown in Figure 3.16.

These k—space data were then fourier transformed into real space to create the
radial distribution function. The end points in k-space for the back transform were picked
to include as large a span of k-values as possible, but still be usable for all the data sets
measured. The resulting picks for the maximum and minimum k's are shown in Figure
3.16. The resulting radial distribution function for the Nb/CuMn(1%) sample is shown in
Figure 3.17.

The position and intensity of the peaks in Figure 3.17 are related to the number of
nearest neighbors and the distance to each shell of neighbors. Here one shell, at 0.226 nm,
and three weaker shells at 0.35 nm, 0.42 nm, and 0.48 nm were observed, the other small
peaks appearing in Figure 6.17 were attributed to artifacts of the fourier transform
because they were not observed in all samples. For the CuMn(1%) fcc lattice there should
be 12 nearest neighbors appearing at a distance of 0.2556 nm. The observed peaks in the
r-space plots did not coincide with the expected distance because of systematic changes
related to the analyses and choice of k—space cut points. For this reason, accurate meas-
urements of atomic spacing and numbers of neighbors were made by comparing the data
fi'om the unknown sample with data taken on a standard 2 pm thick pure Cu foil. By
analyzing the standard and sample using the same apparatus and choice of k-space cut

points, the systematic effects altered both data sets in the same manner. Therefore, by

 

45

comparing the standard and sample radial distribution functions, the true lattice spacings
and nearest neighbor numbers were determined.

To compare the Cu foil and Nb/CuMn(l%) data, the data for the first atomic shell
(near R = 0.226 nm) were back transformed into k-space. An analytic routine was then
used to fit the standard data set to the multilayer data. To systematize these back
transformed data, the same R-space end-points (0.182 nm and 0.255 nm) were used both
for the Nb/CuMn sample data and the standard Cu film data. Table 3.4 shows the
resulting values for the number of nearest neighbors N, distance to the nearest shell of
atoms, Rm, and a parameter related to the thermal broadening of the XAFS oscillations oz.
The table shows that a slight change of about 0.02 nm in the CuX lattice may occur in
very thin (2.0 nm) CuX layers. This was consistent with the previously discussed x-ray
studies on Nb/Cu 2.0 nm/2.0 nm multilayers which showed that the CuX lattice spacing
was within $0.070 nm of its bulk value. Therefore, XAFS analyses provided a
significantly more accurate measure of the CuX lattice constant than was possible using x-
ray diffraction.

The observed trend of decreasing N with decreased CuX layer thickness is not
understood but is consistent with Cu XAFS data taken on Co/Cu multilayers by Dr. Carl
Foiles of Michigan State University.38

XAFS measurements were attempted on thin Nb layers. The observed signal to

noise ratio, however, was sufficiently low that the data could not be analyzed.

Table 3.4: XAF S derived parameters for Nb/CuMn(1%) samples.

 

Sample N Rm(nm) (:2

 

 

Cu standard 12 0.2556 0

7.6 nm CuMn(1%) 10.6:l:2 0.2550i0.0020 0.0003
4.6 nm CuMn(1%) 9.93:2 0.2554i0.0020 0.0007
2.1 nm CuMn(1%) 8.2i2 02534300020 0.0012

 

 

 

 

46
—1.0 1 I | 1 T 1 ' T r n r r
23 —1.1 r 1
\
O
C" —1.2— ~
5 . .
Q —1.3 ‘ Eo “
O
33
-£ —1.4~
O
.8
< -1.5 —

 

 

 

8800 8900 9000 9100 9200 9300 9400 9500
Energy (eV)

Figure 3.15: Raw Cu absorption data for a Nb/CuMn(l%) 5.2 nm/4.6 nm multilayer.

 

 

47

 

0.06 - ' 1 T Y ' ' i
0.05 } -
0.04;
0.03;
0.02 _—
0.01 —

/\ A
is \V/Vvvw
—0.02_—

L L 1

11+lnl

 

L11

1

-0.03 f
-0.04j k km“ ‘
-0.05 - j

_0.06 1 1 1 1 . 1 1 1 1 1 1 1
O 20 40 60 80 100 120

k (1/nm)

.1

 

—.1

Chi (normalized units)

 

 

 

Figure 3.16: Chi plots for a Nb/CuMn(1%) 5.2 nm/4.6 nm multilayer.

48

 

20 WW
189
16~
14;
12-
10;

Amplitude

 

 

 

WNW

.0 0.1 1 0.2 1 0.3 1 0.4 0.5 0.6 0.7 0.8
Radius (nm)

 

come.

Figure 3.17: Radial mass distribution function surrounding Cu in a Nb/CuMn(1%) 5.2
uni/4.6 nm multilayer.

 

49

3.4 Studies of Cross-Sectioned Samples

In principle, a thorough study of cross-sectioned multilayer films using electron
diffraction and electron imaging can provide information on most of the structural
parameters described above. The primary problem with this as a generally applied tech-
nique is the difliculty of preparing satisfactory cross—sections. David Howell of the Mate-
rials Science and Engineering Department of Michigan State University prepared the
cross-sections used and took the electron micrographs and electron difli'action scans used
in the following analysis.

The cross-sections were prepared by removing a multilayered sample from its
substrate, embedding it in an epoxy block and then sectioning the sample and epoxy using
an ultramicrotome. The resulting sections were typically 60-70 nm thick, just thin enough
for good electron transmission. The primary preparation problem was the lack of
adhesion of the multilayer film to the epoxy. Usually, the films detached from the epoxy
as they were sectioned, thereby mining the section. The above preparation dificulties
limited the study to only one sample: Nb/CuMn(7%) 28.0 nm/4.0 nm with 25 bilayers.

Figures 3.18 and 3.19 show a dark field image and the corresponding electron
difl‘raction pattern, respectively, for the Nb/CuMn(7%) 28.0 nm/4.0 nm sample. A dark
field image highlights individual crystallites whose orientations are such that they diflract
electrons into one of the spots on the electron diffraction pattern. These images provided
better contrast between the Nb and CuMn layers than was obtained from normal trans—
mission (bright field) images.

In the dark field image, (Figure 3.18), the Nb and CuMn layers appear continuous
with no large scale fluctuations in their layer thicknesses. In addition, the Nb crystallites
were observed to extend across the entire thickness of each Nb layer. This agrees with
the previously discussed x-ray diffraction results. As the electrons pass through the

section, they are inelastically scattered. This scattering broadens the sample features to

50

such an extent that good images of the CuMn layers or Nb-CuX interfaces were not
achievable.

The electron diffraction pattern (Figure 3.19) showed a great deal of detail about
the crystallite orientations within the Nb and CuX layers. The difi‘raction spots lie on
concentric circles. These circles correspond to diffraction from the Nb and CuX planes
listed in Table 3.1. The angular spread of the difliaction spots provided a measure of the
angular spread in orientations of the Nb and CuMn crystallites. These spreads were found
to be ~5°, in agreement with the rocking curve data. Note that for the (110) circle, spots
appear at 0°, 60°, and 90° away from the line normal to the film. These positions concur
with the expected off-set angles listed in Table 3.1. By carefully indexing these diffraction
spots with the expected off-angle peak positions, an absolute identification was made of all
the diffraction spots in this, and other, diffraction patterns. Most of the observed difl’rac-
tion spots were indexed to scattering from Nb. The only clearly resolved CuMn spots
observable in Figure 3.19 were the (200) spots occurring at ~55° from the Nb(110)

grth direction.

 

 

51

3983 353538 E: o

is: mm Axtczsobz a Mo 883

880 a mo owafim 55020 Eon ciao

M:

m 2&5

 

 

 

 

 

 

 

 

Figure 3.19: Electron diffraction pattern taken from a cross sectioned Nb/CuMn(7%) 28

nm/ 4.0 nm multilayered sample. Miller indices for the observed Nb planes and the CuX
(200) plane are labeled. Thin lines denote the direction parallel and perpendicular to the

plane of the Nb and CuX layers.

 

53

3.5 Structure Summary

Returning to the questions posed in the introduction to this chapter, the following

answers have been found:

. Were the samples layered?
Yes. Reflection x-ray diffraction and electron micrographs showed that the samples

were layered.

. Were the layer thicknesses as expected from the sputtering conditions?
Yes. Analyses of the reflection scans indicated that the bilayer separation A was

within 5% of its expected value.

. Were the layer thicknesses uniform?
Yes. Qualitative analyses of the second-order reflection x-ray diffraction data indi-
cated that the rms deviations in the Nb and CuX layer thicknesses were no larger than

0.3 nm.

. Was the Nb-CuX interface sharp?
Unresolved. The resolution of the electron micrographs was not high enough to study

the interfaces. More studies need to be done to answer this question.

. What were the concentrations of desired and undesired impurities in the Nb and CuX
layers?
Analyses of EDX and magnetization data showed that the Ge concentration of CuGe

was 4.9 at. % and the Mn concentrations of the primary CuMn layers used were 0.3

54

at. %, 1.0 at. % and 2.2 at. %. No other stray impurities were observed in either the

Nb or CuX materials.

. Were the Nb and CuX layers crystalline?
Yes. The x-ray diffraction, XAFS, and electron difli'action experiments all indicated

that the Nb and CuX layers were crystalline for all layer thicknesses studied.

. Were there any changes in the Nb or CuX lattice spacings as the layer thicknesses
were reduced?
Perhaps. The XAFS data indicated that a slight (~l%) change may occur in the CuX
lattice spacing for CuX thicknesses 2.0 nm thick, while the x-ray results concluded
that the CuX lattice changes by no more than 3%. The Nb data indicated that the in-
plane lattice spacing of Nb may decrease by ~1% for Nb layers 3.0 nm thick, but this

change was very tentative.

. Were there strains in the Nb or CuX layers near the Nb-CuX interface?
Perhaps. Such a strain would provide the driving force for the small shifts in lattice
spacing described above. More studies need to be made, especially electron diffraction

on very thin Nb and Cu layers, to be sure that these observed shifts were real.

. Were the Nb or CuX crystallites preferentially oriented?
Yes. Analyses of the reflection x-ray diffraction and the electron diffraction patterns
showed that the Nb (110) and (111) planes were parallel to the sample substrate to

within approximately 5°.

Other than a decrease in the mean crystallite size, these studies have indicated that

there were no changes in the Nb or CuX crystal structures for Nb and CuX layer thick-

55

nesses greater than about 4.0 nm. For thinner layers, there were some indications that
both the Nb and CuX lattice were slightly strained, possibly due to ordering at the Nb-
CuX interface. Further studies need to be performed to determine if the changes

described above for thin Nb and CuX layers were real or simply systematic effects.

CHAPTER 4
EXPERIMENTAL PROCEDURES

All of the experimental results reported in this thesis were obtained using electrical
resistivity and magnetization measurements. Resistivity was used to determine the super-
conducting transition temperature Tc, and the superconducting upper critical field ch(T),
while magnetization was used to determine the spin-glass freezing temperature Tf. Both
these experiments were carried out utilizing a Quantum Design Magnetic Property
Measurement System (MPMS). The magnetization experiments used the system's SQUID
magnetometer circuits, while the resistivity measurements used the MPMS system as a
precision cryostat and for computer control of a Keithley K181 nanovoltmeter and a K224
current source.

All samples used in these studies were sputtered through a patterned mask during
fabrication to produce both a 5-probe resistivity sample and a separate magnetization
sample on each substrate. Figure 5.1 shows the resulting sample pattern. The right side
was used in a four probe resistance geometry and was used for measurements of Tc and
H02(T) with the magnetic field parallel to the sample surface Hc2I|(T). The left side was

used for magnetization measurements and for Hc2(T) measurements with the magnetic

field applied perpendicular to the sample surface Hc2i(T). The magnetization measure-

ments used the entire left sample. However, the ch(T) measurements were made using

only a small, 3 mm x 4 mm, piece, the maximum size which would fit into the MPMS

sample holder.

56

 

57

 

Si substrate

 

\ Patterned
C Sample

Left Sample

 

 

 

 

 

 

 

 

Figure 4.1: Diagram of the sample pattern.

4.1 Magnetic Measurements

Magnetization measurements were used to determine the spin-glass freezing tem-
perature Tf. For each sample, the left side (Figure 4.1) was cleaved from the remainder of
the substrate, then all uncoated regions of Si were also cleaved away from the sample.
This was done to limit the mass of Si that was measured along with the sample. The
remaining sample was then cleaved into three 3 mm x 5 mm pieces which were stacked
together, sealed in a small close-fitting plastic bag and mounted on the MPMS sample
holder with the long axis of the sample perpendicular to the long axis of the system.

In the MPMS system, M(T) was measured using a difference technique. The
sample is raised vertically through a set of three pickup coils attached to the MPMS
SQUID circuit. The resulting SQUID response was then used to calculate M. The
response of this system was sensitive to the length of the sample along the axis of the
pickup coils. By positioning the sample as described above, a slight improvement in signal

to noise was found compared to orienting the sample with its plane along the coil axis.

 

58

In a spin-glass, Tf is typically defined by the temperature of a cusp in M(T), as
shown in Figure 4.2. To obtain the zero-field-cooled (zfc) data, the sample was cooled
fi'om well above Tf to well below Tf in zero applied magnetic field. A measuring field of
50 or 100 gauss was then applied to the sample and the magnetization was measured as
the temperature was increased. Here, Tf was defined as the temperature at the maximum
in M(T). For field-cooled (fc) data, the sample was cooled to below Tf with the
measuring field (50 or 100 gauss) already on. Then M(T) was measured as the
temperature was increased, similar to the zfc data. For the fc data, Tf was defined as the
temperature at a kink in M(T), as shown in Figure 4.2. Typically the fc and zfc
measurements of Tf agreed to within 0.3 K. This combined with the sample-to-sample
variability, measurement-to-measurement reproducibility, and natural width of the peak at

Tf limited the final systematic error in Tf to $0.5 K.

4.2 Resistivity Measurements

The current and voltage leads were attached to each sample using silver paint. The
paint was applied and allowed to air dry for 30 minutes prior to use in the cryostat. For
the Tc and H62" measurements, the current leads were attached to the ends of the sample,
while the voltage leads were attached to the tabs along the right side of the sample. For

the Hc2i(T) sample chip, the two current leads were attached to two corners on one side

of the sample while the voltage contacts were attached to the remaining comers.

 

59

 

62E—6 . r 1 T
6.0E—6
5.8E—6
5.6E—6 F
5.4E—6
5.2E—6
5.0E-6
4.8E—6
4.6E-6
4AE—6

4.2E_6 . r . 1 1 l .
O 10 20 3O 40

Temperature (K)

I

r r

l l l l l

Magnetization (emu)

l

 

 

 

 

Figure 4.2: Zero-field-cooled (zfc) and field-cooled (fc) magnetization data for a 500 nm
thick CuMn(1%) film.

60

The resistance measurements were performed by decreasing the temperature
stepwise through the superconducting transition at a fixed external magnetic field. The
standard temperature interval used, 0.02 K, was the smallest reliable increment produced
by the MPMS system. Once the temperature was stabilized, the voltmeter was set to read
zero for zero applied current, a positive current of 1.00 mA was applied and the voltage
was measured 20 times. The current was then reversed and 20 additional data points
taken. The absolute values of these voltages were averaged and divided by the current
(1.0 mA) to yield the resistance R. The current was then turned off to limit sample
heating and the temperature changed to the next value. The resultant R's were typically in
the range 10‘3 to 1.0 0, depending on the layer thicknesses used, and the noise level (i.e.
the voltage measured for T << Tc) was ~5 x 10'6 Q for all samples. The 1 mA measuring
current corresponded to a current density of 40 to 120 A/cm2, well below the critical
current of Nb ( ~105 A/cmz).39

Measuring the (H, T) phase diagram for a single sample is a very time consuming
process, typically taking 12 to 24 hours per sample. To minimize this time and use the
MPMS more efficiently, the following procedures were derived. During each temperature
series at constant field H0, the computer monitored the decreasing resistance until it
dropped to less than 10'5 Q, at which point the run was stopped. The computer then cal-
culated the transition temperature TC(HO) and began a new run at a field Ho + AH and a
starting temperature Tc(Ho) + 0.06 K. Typically, small field increments 250 G < AH < 500
G were used when H0 was less than 2000 G while larger increments 1 kG < AH < 2 kG
were used at larger fields.

The patterning process used to fabricate the 5-probe samples lead to rather broad
sample edges (sometimes up to 35 % of the total sample width). To check that these
edges were not affecting the Tc and Hc2||(T) measurements, data was compared with TC
and Hc2H(T) measured in portions of the unpattemed left side of the sample. The unpat-

temed samples were prepared by cleaving all the sputtered edges away from the left-hand

 

 

 

61
sample and then attaching leads as was done for the ch(T) samples. No significant or

systematic change in either Tc or HeZH(T) was observed when the data for these two sam-
ples were compared.

Based on the sample-to-sample and measurement-to-measurement variability of TC
the total error in TO was approximately i 0.10 K. For the Hc2(T) experiments, most of the
sample-to-sample reproducibility error was related to errors in Tc. When HC2(T) data were
normalized to Tc, the variability in Hc2(T) data sets was found to be of order i 2 % of
Hc2(T). In the temperature regime from 4.10 K to 4.40 K however, the inherent error in
temperature was greatly increased due to temperature instability of the MPMS system in
this regime over the long times needed to measure H02(T). Hence, most Hc2(T) data sets
display small glitches in this temperature regime. For samples in which Tc was in this
region the increased error became rather serious, therefore, several Hc2(T) data sets were

averaged to create a single Hc2(T) set.

CHAPTER 5
THE SUPERCONDUCTING TRANSITION TEMPERATURE

In S/N multilayers, the superconducting transition temperature will be affected
both by the extent of proximity effect induced superconductivity in N and by the influence
of reduced S layer thickness on the superconducting properties of S. In this chapter,
measurements and models for the behavior of the zero field superconducting transition
temperature Tc of an S/N multilayer will be used to study these phenomena. First, the
experimental results will be discussed along with a qualitative model for some of the
results. Then, two theoretical models for proximity effect superconductivity will be pre-

sented. Finally, the results of these models will be compared to the experimental data.
5.1 T0 Experiments
5.1.1 The Dependence of Tc on the N Layer Thickness

In this section studies of how the thickness and impurities in the N layer (at fixed S
layer thickness) modify Tc through the scattering of coherent electrons induced in N by the
proximity effect will be examined. For small (in, the superconducting pair density in N is
large, few pairs are destroyed in N, leading to strong coupling between S layers and Tc E
T: the bulk superconducting transition temperature of S, as shown in Figure 5.1. As dn
increases, Tc decreases monotonically due to the breakup of superconducting pairs in the
N layer and a decrease in proximity effect coupling between S layers. Eventually, for very
large dn, all superconducting pairs entering N are destroyed before reaching the next S
layer, hence the S layers are decoupled from one another. In this large dn regime, Tc
reaches a minimum value T,m determined by the maximum loss of superconducting pairs

into N. Since there is no longer any superconductive coupling, Tc'“ should be the same for

62

 

 

63

 

 

Tc/TB

 

 

 

 

 

“100 2 34 1012 34 102 2 34 103
dn

 

 

 

Figure 5.1: Qualitative dependence of Tc on (1D for fixed ds.

multilayers, where coupling can occur, and for bilayer films, where coupling cannot occur.
Therefore, d;, the N layer thickness beyond which Tc = Tam, is representative of the
penetration depth of induced superconductivity into the N layer.

Figure 5.2 presents experimental data for the dependence of Tc on dn for multilayer
series having fixed Nb thickness (28.0 nm) and Cu, CuGe(5%), and CuMn(y) interlayers.
All series exhibited a decrease of Tc with increasing dn, consistent with the penetration of
superconductivity into N. For thin non-magnetic layers, Tc was independent of whether
Cu or CuGe(5%) layers are used, while for thick non-magnetic layers (dn > 20 nm), TC
was larger for Nb/CuGe(5%) samples than for Nb/Cu. Associated with this enhancement

of Tc there was also a reduction in d;c- These differences between Cu and CuGe for thick

N layers also held for other S layer thickness, as shown in Figure 5.3.

 

64

 

1.0 ' I'I'III'I'I l llll] 'I'T" II'II ITTII’

  
 
 
  

Tc/Tr",

0.5 ~

V

l 1 i

 

4.00

y ffrfiy v

CuGe(57.)

Cu
CuMn 0.37.
CuMn 2.27.

    

 

 

111

 

 

4 l 1 llLJL 1 11111
100 2 34 101 2 34

deux (nm)

102 I

234

Figure 5.2: The dependence of Tc on dn for d, = 28.0 nm and various CuX materials.

64

 

 

   

CuGe(57.)

Cu

CuMn

Cu

 

 

(0.37.
Mn 2.27.

l

TIII

l

 

 

2
0; v
E —J
\o
5" 1
. ..
l
4 . _
V
0.4 or1.111111111111111.1111411211,1,1L1111
10234 10234 10 234
deux(nm)

Figure 5.2: The dependence of Tc on dn for (1s = 28.0 nm and various CuX materials.

103

 

65

 

100 V'vI"'IV“'ITIITIYY'IIYII‘T‘TIIITII r rvrv.rrrrrrr

 

 

     
  
 

 

 

 
     

 

 

 

0.0
B
\. 1
B * .
0.5 — CuX -
' 0 10nm Cu
0'4 " O 28nm Cu ~
“ O 10nm CuGe25Z;
0.3 — <> Zflnm CuGe 52 . _
0.2 . . .1...1.r.r L 1 rrrl .1 .r.r.r...1 11 lLl . 1.1.1.L.r.r 14 111
100 2 3 4 101 2 3 4 102 2 3 4 103

dCuX (nm)

Figure 5.3: The dependence of Tc on dn in Nb/Cu and Nb/CuGe(5%) multilayers for Nb
layer thicknesses of 10 nm and 28 nm.

 

66

At small (1n and ds, Tc did not tend toward ch, rather, it saturated to a value well
below ch. This may be an effect of the depression of the superconducting properties of S
observed in isolated Nb films.15 Such a depression of ch in Nb will be discussed more
fully below.

Series having magnetic CuMn(y) interlayers displayed the same general behavior.
For magnetic interlayers, however, Tc and d; decreased with increasing impurity concen-
tration while Tcm remained fairly constant for y _>_ 0.3 at. %. There was a magnetic effect
on Tom, however, since it decreased significantly when the N layer was changed from Cu to
CuMn(0.3%).

These results can be qualitatively understood using a scattering model for the
behavior of electrons in the N layer. In a lowest order approximation, electrons traveling
in N can be treated as traveling by way of a random walk. Electrons entering N from S
are phase coherent with the superconducting layer from which they originate. As phase

coherent electrons traverse the N layer one of three things can occur:

. Transmission: The electron can remain phase coherent and traverse the entire
thickness of the N layer to enter the next S layer thereby coupling the phases of the
two S layers.

. Spin-Scattering: The electron can spin-scatter from an impurity in N and lose its
phase memory. This removes the electron from its phase coherent state.

. Reflection: The electron can scatter back into the S layer from which it originated.

This increases the superconducting pair density in S as 8,, decreases.

These three processes are sketched in Figure 5.4.
For thin normal metal layers, the electron mean free path 3,, is greater than (in, so
little scattering occurs in N and the transmission coefficient T is approximately unity.

Hence, in the thin N limit, Tc should be independent of the N material used. Qualitatively,

67

this is the behavior observed for very thin Cu, CuGe(5%), and CuMn(0.3%) N layer

materials.
0
0
fl/ . .
Transmrssron
. Spin-Scattering
/
o
/ . Reflection
o
o
e

 

 

S N S

Figure 5.4: The three dominant processes for a phase coherent electron entering a normal
metal layer.

As dn is increased for fixed 6“, spin-scattering and reflection begin to be important.
If a small amount of Mn is added to Cu, the spin-scattering probability increases and the
probability for a superconducting pair to be destroyed in N also increases. This should
result in a decrease in the superconducting pair density in both 8 and N, and hence a
decrease in Tc. This was the behavior observed as Mn ions were added to Cu. This
simple picture, however, does not explain why Tcm is independent of the Mn concentration
at very large CuMn layer thicknesses. This phenomenon will be discussed in the theory
section.

If the spin-scattering probability is low, as occurs for non-magnetic N materials,
then reflection should dominate. This leads to an enhancement in the superconducting pair
density in S, and hence a higher Tc as 6,, is decreased. This increase was observed for

thick N layers in which 5 at. % Ge was added to the Cu.

68

5.1.2 The Dependence of Tc on the S Layer Thickness

An analysis of how Tc depends on (18 at fixed dn provides information on how the
superconducting properties of S are altered by the influence of the N layer. In general, as
ds is decreased, for fixed dn, Tc should decrease toward zero. In bulk Nb, Tc is strongly
related to the electron mean free path €5- As 68 decreases, a peak in Nb's density of states
near the Fermi energy broadens resulting in a decrease in Nb's density of states at the
Fermi energy N(0), and hence a decrease in Tc. In thin films, 68 is often proportional to
the mean crystallite size, which, as was discussed in the structure chapter, is proportional
to the sample layer thickness d8. Tc should, therefore, decrease with decreasing ds. For

Nb, Cooper has theoretically modeled this process and found that Tc is roughly given by:40

1
b_ f r

T. ch E[9£] (5.1)
T d

C S

 

where T: is Tc in the Nb film, and d: is a scale factor for the layer thickness. The prox-
imity effect may alter this relationship depending on how the superconducting pairs are
scattered in N and on how the S layers are coupled.

Figure 5.5 shows the dependence of Te as a function of ds, Tc(ds), for two strongly
coupled ((1n = 2.0 nm of CuGe(5%) and CuMn(0.3%)) and two weakly coupled (dn = 300
nm of CuGe(5%) and 70 nm of CuMn(0.3%)) multilayered series. Tc was observed to
decrease with increased Mn concentration and increased N layer thickness for all ds. On
the basis of this figure, however, it was difficult to determine if there were any quantitative
changes in the behavior of Tc(ds) as either (in or the composition of the N layer were
changed.

In Figure 5.6, the reduced temperature t2 (ch -— Tc)/ Tcb is plotted as a fiinction of ds for

the same data as in Figure 5.5. All four series were found to scale, according to:

69

p
t: (%] (5.2)

with 1 s p < 1.7. All Nb/CuX series studied show this behavior. Values for p and the
scale factor (10 for the series in Figure 5.6, for all other measured series, and for data on
V/F e multilayers taken from Wong et al.,”41 are listed in Table 5.1.

Thin, non—magnetic N interlayer series (2.0 nm of Cu or CuGe(5%)) were best fit
using p = 1. As the non-magnetic interlayer thickness was increased, p increased system-
atically reaching an average of 1.24 for samples having decoupled S layers. Series having
CuMn layers showed a rapid increase of both p and do as y was increased. These changes
were related to the destruction of pairs by the Mn spins. In the decoupled limit (dn > d; ~
60 nm) p was not significantly dependent on y, which showed that Tcm was nearly inde-
pendent of y for all ds. That p was larger for CuMn interlayered series than for similar Cu
or CuGe series indicated that the Mn ions did significantly alter the superconducting state

of the multilayer.

70

10
0.9
0.a~
0.7L
0.6;
0.5L
0.4;
0.3
0.2— —
0.1: J
0.0' ,

 

I

     
   

2.0nm CuGe(57.)
300nm CuGe .) ‘
2.0nm CuMn 37.; ‘

37. ‘

b
c

0139.

5
0.
70.nm CuMn 0.

Tc/T

 

 

 

     

O 10 20 3O 4O 5O 60 7O 80 90 100 110
dNb (nm)

1_

 

Figure 5.5: Normalized superconducting transition temperature Tc/Tcb vs. Nb layer thick-
ness. Solid lines are the results of fits to Equation (5.2).

70

 

10
0.9;
0.31
0.7:
0.6
0.5-
0.41
0.3;
0.21
0.1".
0.0b

 

 

 

 

     
  

l

2.0nm CuGe(57.)
300nm CuGe 57.) ‘
2.0nm CuMn 0.37.) ‘
70.nm CuMn 0.37. ‘

Tc/TB
<> [1 O I

 

—

     

 

 

O 10 20 30 4O 5O 60 ’70 80 90 100 110
dNb (nm)

l

J_

 

Figure 5.5: Normalized superconducting transition temperature Tc/Tcb vs. Nb layer thick-
ness. Solid lines are the results of fits to Equation (5.2).

71

 

    
 

 

 

 

 

10O _ r
5 \\
4:
3E
as: 2.-
; :
53’ 10-1:
' E
’00 5 4
E; 4 r I 2.0mm CuGe(5Z)
3; O 300nm CuGe 57.) *
: D 2.0mm CuMn 0.37. :
2 O 70.nm CuMn 0.37. 1
10-2+11.1|llllllll . 111‘1‘1'1111 ‘
2 3456101 2 3456 102 2
dub (nm)

Figure 5.6: Reduced superconducting transition temperature t vs. Nb layer thickness for
the same data series as in Figure 5.5. Solid lines are the results of fits to Equation (5.2).

72

Table 5.1: Scaling parameters for To(ds). ‘Data from Wong et all“,41

 

 

 

 

 

 

 

 

Interlayer do (run) do (nm) 1)

CuGe(5%) 2.0 1.70 i 0.03 1.00 i 0.02
CuGe(5%) 20. 4.6 i 0.1 0.99 i 0.03
CuGe(5%) 300. 8.3 i 0.2 1.21 i: 0.04
Cu 2.0 1.76 i 0.04 0.98 i 0.02
Cu 40. 7.4 i- 0.4 1.08 i 0.08
Cu 300. 13.6 i 0.4 1.27 i 0.06
CuMn(0.3%) 2.0 4.45 i 0.04 1.18 i 0.02
CuMn(0.3%) 20.0 14.3 i 0.2 1.35 i 0.02
CuMn(0.3%) 70.0 18.3 i 0.6 1.50 i 0.10
CuMn(1.5%) 2.0 8.6 i 0.1 1.42 i 0.03
CuMn(2.2%) 2.0 9.4 i 0.2 1.45 i 0.04
CuMn(2.2%) 6.0 15.2 i 0.2 1.38 i 0.02
CuMn(2.2%) 70.0 20.0 .4: 0.5 1.64 i 0.10
V/Fe" 0.27 13.1 i 0.1 1.26 i 0.02
V/Fe‘ 0.41 21.8 i 0.3 1.42 i 0.05
Fe/V/Fe‘ ~5.0 12.5 i0.6 1.10i0.04

 

 

73

To provide a check of Equation (5.2) in the extreme magnetic limit, data published
by Wong et al. on To in V/Fe multilayersz'4 and Fe/V/F e sandwiches41 have been
reanalyzed. Their data on V/Fe multilayers was published as To as a function of do for
various constant values of do. These data have been interpolated to extract sets of To as a
function of ds at two values of do, 0.27 nm and 0.41 nm. These extracted data sets could
be scaled according to Equation (5.2) indicating that this relationship holds for strongly
magnetic N. In addition, similar trends in the dependence of p and do on the N layer
thickness were found (Table 5.1).

The Fer/Fe sandwich data, however, did not agree with either the V/Fe multi-
layer results or with those observed in the Nb/CuX studies. In V/Fe/V sandwiches, p z 1
while p > 1.4 would have been more consistent with the multilayer data. Wong et al.
noted that the To‘s of multilayers having thick Fe layers (approximately equal to those used
in the sandwich films) deviated from the behaviors they had observed in samples having
smaller Fe thicknesses.4 They proposed that their observed deviations at large do were
related to a change in the internal structure of the multilayer but were unable to validate
this proposal.

All these experimental data indicate that the superconducting state of the S
material is altered in a different fashion by magnetic N layers than by non-magnetic N
layers. A qualitative model for the mechanism behind the dependence of To on do for
varying Mn concentration has been included above, but an explanation of the dependence

of p and do on y and do requires a more careful theoretical treatment.

5.2 To Theory

Most models of proximity effect superconductivity can be classified into two
general forms: those based on McMillan's model;42 and those based on de Gennes and

Guyon's model.43 Both these forms were originally derived in the early 1960's, but much

74

work has been done on refining them. Of particular interest are extensions of these models
to the case of magnetic normal layers. The appropriate extension of McMillan's model
was developed by Kaiser and Zuckerrnann (KZ)44, while de Gennes and Guyon's model
was extended by Hauser, Theuerer, and Werthamer (I-ITW).14’45’46

These models are similar in that they both require that the electron mean-free-path
of the S and N material 6? is less than its respective coherence length if where i stands
for the S or N material. This assumption, known as the dirty limit approximation, has
proven vital in making the proximity effect problem tractable. There are other models that
relax this constraint but their final forms have proven too complex to be applied here.28

The primary difference between these two model classes lies in how they treat the
boundary conditions at the S-N interface and the spatial dependence of the super-
conducting pair function within each layer F(z). F(z) is defined as A(z)/V(z), where A(z)
is the position dependent superconducting gap parameter, and V(z) is the position
dependent electron-phonon coupling parameter. The importance of F(z) is that it is
directly proportional to the superconducting pair density at each position, 2, within the
multilayer. Due to the nature of the differences between these models, each will be

discussed in turn, then both models will be compared to the experimental data.

5.2.1 The KZ Model

The KZ model will be introduced by developing the basic concepts and assump-
tions needed for the model, then the results will be written down and expanded upon in a
few limiting cases.

KZ's first assumption was that a multilayered sample can be described as a series of
bilayer films as shown in Figure 5.7, where the thickness of each layer in the bilayer is half

the thickness of that layer in the multilayer. This assumption is justified because the

(rs/2
dn/2

   

Figure 5.7: Multilayer to bilayer deconvolution.

superconducting boundary conditions on the metallic side of a superconductor-vacuum
interface are the same as those appearing at the center of a symmetric S or N layer.

The next, and perhaps most important assumption was that both layers were much
thinner than their respective coherence lengths, but they are thicker than their respective
mean-free-paths: E"? < di << éf. Since di << éf, F(z) is expected to vary only slightly

across each layer which leads to the condition that F(z) is approximately given by:

Fo in the N layer

F(z) = (5.3)

F5 in the S layer

Therefore, F(z) is discontinuous at the S-N boundaries. This may seem to be a rather
severe constraint; however, as will be shown below, the KZ model is applicable for many
of the systems studied here.

Although F(z) is discontinuous at the S-N interface, there are still matching condi-
tions to be taken into account. Kaiser and Zuckermann created boundary conditions at the
S-N interface by treating the interface as a superconducting-pair tunneling barrier. The

basic energy scales arising from this barrier are:

76

ro = grzAdsN', (0) : h/2to (5.4)

I; = gTzAdoN'o (0) = h/zr, (5.5)

where T is the tunneling matrix element, A, the area of the interface, N'AO) is the density
of states per unit volume at the Fermi energy, ‘Ci is an associated response time, and i

stands for the S or N material. They also showed that To can be given by:

r = “ (5.6)

 

 

where 2Bdo is the distance traveled by an electron in N between collisions with the S-N
interface, B is a parameter related to the relative magnitudes of E: and do and is expected
to be approximately 2 for a clean layer, 0 is the transmission probability across the S-N
interface, and vf’o is the Femri velocity in N. This form for to is simply the average time an
electron spends in N before tunneling back into S. Since Equation (5.6) predicted that To
was proportional to 1/do, KZ also assumed that I”s was proportional to 1/ds.

With these approximations, two new length scales, co and cs, were defined which

characterize the effect of superconductivity in the N and S materials:

(5.7)

where i represents the S or N material, and AI, is the bulk gap parameter in S. The ratio
co/cs = N'o(0)/N'o(0), hence c8 and co are not independent parameters.
The final approximation was the inclusion of magnetic impurity effects within the

N layer in terms of a magnetic coupling parameter F:

77

r = gyN'o (0)JZS(S +1) (5.8)

where J is the s-d exchange coupling constant, S is the spin of the magnetic ion, and y is
the concentration of magnetic impurities. This form treats magnetic spins in the Born
approximation and does not include effects due to impurity-impurity interactions or the
Kondo state.

With all the terms defined and approximations clear, the equations used for calcu-

lating To(ds, do, y) are:

 

 

 

 

 

In T: if: 1 — 1 (5 9)
Tc n=0 n+1/2 n+1/2+gs '
gs =% (2n+1)t +g 2 (5.10)
s 2t‘((2n+l)t'+g+ dc")
t‘ = ka° (5.11)
Ab
F
=_ 5.12
6 Ab ( )

where ko is Boltzmann's constant, To is the superconducting transition temperature of the
multilayer, and Tob is the superconducting transition temperature in bulk S. In most super-
conductors, the bulk gap parameter Ab is related to Tob by Ab = BkoTob with B approxi-

mately equal to 1.67 (in Nb, [3 = 1.90).47 The uppermost integer nmox contained in the sum

is set by including only energies below the Debye energy of S and hence (2nmoo +1)Tc <

OD where OD is the Debye temperature. This typically results in an nmoo > 20.

78

Due to the complexity of these equations, it is difficult to get a qualitative under-

standing of how Tob behaves for varying d d and y. Therefore, they have been simplified

S’ n’

in two limits: very large magnetic impurity concentration, i.e., g >> c /d ' and small do,

n n?

i.e. d /d <<1 and g << co/do.

’ n S

In the strongly magnetic limit, go is given by

9.5 s . (5.13)

 

 

Since go is no longer dependent on n, and assuming that nmoo. is large, the summations were

approximated by a digamma function ‘I’(x).

 

Z 1 _ 1 :‘I’(l+go)—‘P(l) (5.14)
n+1/2 n+1/2+go 2 2

n=0

One can further simplify the digamma functions using:
‘I’(l+ )—‘P(—l-)=ln 1+"—2 (515)
2 Gs 2 — 2 Cs -

which is valid for go < 6.47 Unfortunately, this approximation cannot be checked since
there is no good way to determine the magnitude of co. Without this assumption,
however, Equation (5.14) cannot be simplified. Justification for use of Equation (5.15)
here will be addressed when the resulting equation for To is compared to the experimental

data. Combining Equations (5.13), (5.14), and (5.15) lead to a single equation for To:

T_°b:1+.n_2203 1 : £53ch

_ ,_+ —
T 2dszt 2doTo

C

 

 

B (5.16)

79

where B = Ao/koTo". This was fithher simplified to

 

71:
t: C CE_ _5 5.17
2 B ( )

Note here that all effects of the N layer are removed. This may seem counter-intuitive,
but is markedly similar to the behavior observed in Nb/CuMn multilayers in the limit of
thick CuMn layers (see Figure 5.2).

The other limit of interest is that of very small do. In this case:

1 .. c d
E—: 2n+1t + —s——"— <<1 5.18
g. 21 (( ) g)o o < >

8 fl

Unlike the large g case, the resulting go is dependent on n. To obtain summations in the
form of a digamma fiinction requires a shuffling of the various terms appearing within the

sum. After some lengthy algebra one obtains:

p- -I

(1 cd
4 1i 05/ 1— /
T cd cd
ln ° E “ s — “ s 5.19
[1"] Z n+1/2 g Csdn ( )
2t'cods_

4% 4(1)? 148—.3an +0432 (5.20)
cods 2 2 2t cods cods

 

 

 

n=0

n+1/2+

 

 

 

 

where C is defined by

 

6;“: 1 (5.21)

o=0n+l/2

80

For typical values of To, and OD, nmax E 20 and C E 2.5. Hence, for do/co <<1, one also

finds that Cdo / co << 1. Equation (5.15) is again used to simplify the digamma functions

yielding:

 

 

Making use of the relation between the ci's results in:

t_=_Ad;l

where

n

i:—,6cs for g>> :—"
14:1

 

Ns(0)

n

[égfl+§)wdo for $1<<l and g<<:—“
c

(5.22)

(5.23)

(5 .24a)

(5.24b)

for the two limits of the model. Numerical analysis utilizing the full equations for values

of g and do between these limits shows that this model can always be approximated by t at

6,-1.

81

5.2.2 The HTW Model

A more complete theoretical model than McMillan's was developed by de Gennes
and Guyon,43 and refined to include magnetic efl‘ects by Hauser, Theuerer, and Werthamer
(I-ITW).1“~46’47 This model treats the explicit position dependent pair fiinction F (2) within
the S and N layers. Although HTW also assumes the dirty superconducting limit
approximation and that a multilayer can be approximated by a half-thickness bilayer, they
place no limitations on do or do. In addition, rather than describing the S-N interfacial
boundary conditions by a tunneling barrier as done by McMillan and KZ, HTW have

developed more precise microscopic continuity conditions for the interface.

 

 

They approximate F(Z) by
'coik.(Z-ds/2)) 0<z<d /2
cos(kd /2) — _ s i
F Z 0C . s 5.25
() icosh(ko(2+dn/2)) -d /2<z<0 ( )
cosh(k,,dn /2) n .- —

 

where ko and k5 are inverse length scales to be defined below. The boundary conditions on
F at the S-N interface are that both
F (2) d

N'(z) and d—zln(F(z))

are continuous across the N—S interface where N'(z) is the position dependent density of
states in the bilayer. When these boundary conditions are combined with the assumption

that N'(z) is given by No in N and N5 in S one finds:

82

kSN's n3 tan(ksds /2) = koN'o nfi tanh(kodo /2) (5.26)

where 711 is the effective superconducting coherence length in the S and N materials. Here
11 is used for the coherence length to avoid confusion in the upper critical field chapter
where there will be two quite different forms for the coherence length. In the HTW
model, r)i is defined as

_ 71v)! . zzhko 1

.2 _ a 5.27
"I 6flkac 662']; yipi ( )

 

where vfii is the Femii velocity, e is the electron charge, 7, is the coemcient of the linear
term in the low temperature specific heat, pi is the resistivity and i represents either the N
or S material. The approximate term in Equation (5.27) was derived by HTW and is

based on use of Pippard's expressions relating V121 and E, to the measurable quantities y, and

po48
Utilizing these expressions, the final equations for To in a superconducting multi-
layer are:
koN’o n: tan(ksds /2) = koN’o n2 tanh(kodo /2) (5.23)
ch 2 2
ln ? = X(nsks) (5.29)
where,

 

 

b
k:=-—1—1+crTc a: h bi
ni T 27zkoTc to

C

83

24x)=\r(1/2+x/2)-AP(1/2) (5.30)

and too is the magnetic scattering time in the N material.
An additional simplification is made by noting that y, is linearly proportional to
N'i.49 This allows the ratio N'o/N's to be approximated by yo/ys. Equation (5.28) then

becomes:

k, tan(ksds /2) = k, fitanhaodo /2) (5.31)
on

As with the KZ model, a visual inspection of the HT W model does not lead to an
easy understanding of its underlying physics. To make a comparison of the similarities and
differences between the two models, the HTW model will be examined in the same limits
as the KZ model: strong magnetism (or >> 1), and small do (kodo << 1).

In the limit of large or :

b
k: = -1—o[1+aT—°J >>io (5.32)
nn C Tin
and, therefore,
kodo >>1 (5.33)

provided that do is not very small. This implies that tanh(kodo/2) E l, and that,

tan(kodo / 2) ; [1:41-32- >>1 (5.34)

S n

84

and, hence, that,

ff

kodo/2;
2

(5.35)

Substituting this into Equation (5.29) and using Equation (5. '15) for the digamma function,
one arrives at a single equation for To:
Tb—T

7‘4 2 2
———° ° 5— d’ 5.36
T? 4nms ( )

where noo is no evaluated at To = Tob. This form is similar to that derived from KZ's model,

but it contains a different scaling exponent.

The limit of small do follows similar lines:

 

kodo <<l
therefore,
ks tan(kods/2) Ekfifigfi <<1 (5.37)
2.2
and,
k2
.d. 51.343 (5.38)
2 po 2

Since k8 is small, Equation (5.15) is valid leading to

 

b 2 2 b
T—°21+lfli"—;[1+aT—°) (5.39)
TC 4 pr) n" TC
and
ch_Tc 2 n dn
Tb 27: (1+01 )— (5.40)

which has the same functional form as that derived from the K2 model in the thin N limit.

Therefore, the HTW model predicts that To is given approximately by:

 

Ziryfodgz for a>>l
ta , (5.41)
”2 —N—n———:O)(1+ 006,4;l for kodo<<1
4 N' (0)

where N'o(0)/N's(0) E yo/ys.

5.3 Model Applications

Before comparing these models to the experimental data, the validity of the

assumptions should be verified for both the S and N materials. The primary constraint

common to both models was the dirty superconducting limit approximation: (‘3 << «5?.
EM, was determined from the temperature dependence of the superconducting upper
critical field measured in a thin Nb film and is listed in Table 5.2.

The normal metal coherence length cannot be directly measured. Instead, it was

estimated using a form similar to HTW's expression for 11s:50

Mk, 1
262T 7.006)..

 

‘5: .2. (5.42)

86

where the temperature T used in estimating 5°

11

was the highest temperature of physical
interest, in the present case T = Tob for Nb. (p€)o was assumed to be unchanged by the
small concentrations of impurities used in the CuX alloys, and therefore was approxi-
mately 6.60x10'l6 sz.51 yo for Cu and CuGe have been measured};53 and are listed in
Table 5.2. For CuMn, yo can not be simply measured because magnetic contributions to
the specific heat mask the electronic contribution. Therefore, yo for CuMn(y) was

assumed to be the same as for CuGe(y), these values are also listed in Table 5.2. The

resulting fi's are listed in Table 5.2.

In any material, 6‘; is approximately given by:

 

gt: E (p€)i

(5.43)
A
where p: is the bulk resistivity of N, and (p€)o is the finite-size correction to the resistivity.
For Nb (of) = 3.2x10'16 9m?“ The measured pz's and the resulting €‘f’s are also listed in
Table 5.2.

Thus, the dirty limit approximation (£?<<§f) is valid for all the CuX materials
used, with the exception of pure Cu. For Cu, as dCu is decreased, (Co also decreases

according to:

Table 5.2: Nb and CuX length scales.

 

Nb Cu CuMn(0.3%) CuMn(2.2%) CuGe(5%) Units

 

 

y, 709.4 96.8 98.4 102.1 108.8 J/m3K2
pf 7.2 0.5 3.4 9.4 19.4 10-89. m
49 4.4 130 19 7.0 3.4 nm

4‘? 33 160 160 150 140 nm

 

 

87

—1— 1 +i (5.44)
(o m: d

 

~
—
—

n

where B is a constant of order 3/8 for thin films.51 Therefore, sufficiently thin Cu layers
should also be described by the dirty limit approximation.

The agreement between the KZ and HTW models in the small do limit implies that
they are approaching a functional form that is independent of the approximations used in
the derivation of each model. Although the two models agree on the form of the equation,
they predict significantly smaller values for do than were found experimentally (do‘Z =
0.93, dzrrw = 1.17, while do0166 = 1.70i.03).54 One possible explanation for this discrep-
ancy is that the superconducting properties of the Nb layers change with ds resulting in an
effective Tob which decreases with decreasing do. Since these changes are related to
changes in the thin Nb layers, we assume that To in the limit that do ~ 0 is given by Tof ,
where Tof was defined in Equation (5.1). Replacing Tob in Equation (5.2) with To{ given by

Equation (5. 1) and recollecting terms, one finds:

 

(Tf—Tc)~ dot-(lg 1
T:[ d ) (5.45)

Therefore, the dependence of To{ on ds changes only the length scaling factor do, and does
not significantly change the scaling exponent p. This increase in the effective do provides a
good rationalization of the discrepancy between the HTW and KZ predictions and the
experimental values of do.

For other limits KZ always predicts t at do'1 which may simply indicate that the KZ
model is always dominated by the thin layer limit. HTW's model, on the other hand,

shows a significant change in p with increased magnetic impurity concentration. In

88

particular the HTW results imply that as or increases from 1 toward 00, p should change
from 1 to 2. This was the trend observed in Nb/CuMn and V/F e multilayers as either the
magnetic ion concentration or magnetic layer thickness was increased. The finding that
the experiments do not show p = 2 may indicate that the strongly magnetic limit of the
theoretical models is not a limit accessible by experiment, or simply that data have not
been taken for sufficiently large or.

To summarize, in S/N multilayers To displays a power-law dependence on the S
layer thickness. This form [Equation (5.2)] holds for all N layer thicknesses, mean-free-
paths, and magnetic ion concentrations studied. In addition, the observed scaling law is
predicted by the limiting forms of the HTW and KZ models. While these models appro-
priately predicted the observed increase of the scaling exponent p with increasing magnetic
concentration, an effective depression of Tob with decreasing Nb thickness (consistent with
behavior observed in single Nb films) was required for the models to adequately predict

values for the scale factor do.

CHAPTER 6
THE SUPERCONDUCTING UPPER CRITICAL FIELD

6.1 Introduction

In this chapter, the superconducting upper critical field Ho2 will be used to gain
insight into the importance of proximity effect coupling and pair penetration in a super-
conducting/normal metal (S/N) multilayer. As was noted in the previous chapter, To
experiments alone could not determine whether the observed behavior of T0 was related to
the coupling strength between S layers or more simply to the penetration, and further loss,
of superconducting pairs into N.

In a bulk superconductor, anisotropy of Ho2(T) is related to the inherent anisotropy
of the superconductor. In a S/N multilayer, the physical layering can induce a periodicity,
and hence an anisotropy, in the superconducting properties. In such a layered anisotropic
superconductor one can define two limiting values of Ho2: chi for external magnetic
fields directed perpendicular to the layers, and chn for external magnetic fields directed
parallel to the layers. The physical processes which determine the magnitude of the
anisotropy of Ho2 are the pair penetration of superconductivity into N, proximity effect
coupling between S layers, how the magnetic fields penetrate into the S and N layers, the
vortex structure of the resulting flux-penetrated state, and how the superconductivity
induced in N is destroyed as H and T are varied.

When a weak magnetic field is applied to a superconducting sample, surface
currents spontaneously form to shield the sample's interior from the applied field. At large
field densities, superconductivity will be destroyed and the sample will behave like a
normal metal in a magnetic field. In type II superconductors, there exists an intermediate
state in which the external field penetrates the interior of the superconductor without
totally destroying the superconducting state. This intermediate state is then characterized

89

90

by two magnetic field strengths, Hcl the lower critical field, where magnetic flux first
penetrates the sample, and Hoz, the upper critical field, where the superconducting state is
finally destroyed.

Magnetic flux penetrates a superconductor in the form of thin flux filaments, or
fluxoids, each of which is surrounded by a circulating superconducting current vortex.
The electrons in a superconductor are quantum mechanically phase coherent. Therefore,
the angular momentum of each electron must be quantized. This leads to quantization of
the magnetic flux enclosed within each vortex. The flux quantum (Do is defined as 72/ 2e
where h is plank's constant and e is the electron charge.

At the center of each vortex, the superconducting pair density pooir is nearly zero.
It grows back toward its bulk value over a length scale E, known as the superconducting
coherence length. Therefore, the core of a vortex is typically regarded as having a radius
of a.

The vortices, which repel one another, crystallize into an ordered array. In most

cases the array is triangular with a vortex-vortex spacing clH given by

$0 1/2
512,...) (.1)

 

where H is the applied magnetic field strength. As H is increased, dH decreases until dH ~
5,. For larger H, the superconducting state is destroyed. This defines the upper critical

field Ho2 as

m0
2 7752

 

H,2 = (6.2)

Hence, a measurement of Ho2 allows one to determine a.

91

The superconducting theory of Bardeen, Cooper, and Schriefer55 shows that i is

related to the superconducting gap parameter A by

J: (6.3)

”A
where VfiS the Fermi velocity of the superconductor. The temperature dependence of a is
then primarily due to the temperature dependence of A(T). The superconducting order
parameter is defined as A(T)/A(0), where A(0) is the bulk gap parameter at T = 0. Mean
field theory predicts that below any second order phase transition (like superconductivity)

the order parameter varies with temperature as56

am: _Tc‘T m (6 4)
11(0) r '

C

This implies that near To, a is given by

 

an) =4<o>[T°T"T) (6.5)

C

and Ho2(T) by

 

H.2(T)=H..(0)[T°T'T) (6.6)

C

where 5(0) and Ho2(0) are the zero temperature values of E and 1182- As shown in Figure

6.1, this is the behavior observed for Ho2(T) near Tc in a 500 nm thick Nb film.

92

 

20000'1v1v1\141VI
18000~

16000: \\\\\
14000-

12000-
100001
60001
6000?
4000i
20001
O P 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1

H11 (G)

H.2 = 31.6 kG :- (Tc — T)/T.

 

 

 

 

O 1 2 3 4 5 6 7 8
Temperature (K)

Figure 6.1: Ho2(T) for a 500 nm thick Nb film. The line is a linear fit to the data.

93

In a multilayered superconductor, i is no longer isotropic, rather it takes on two
primary values, in or éxy for Q lying in the plane of the layers and g, for g directed perpen-
dicular to the layers. This deconvolution of g leads to a similar deconvolution of Ho2 for
two different field orientations. Figure 6.2 shows the approximate vortex cores, and
definitions of ’51”, go, and 5,, for H parallel and perpendicular to the layers. As will be seen
later, F,” is not necessarily equal to go.

In a strongly coupled multilayer (do ~ 0) chn and Ho2i should vary with tempera-

ture similar to the behavior found for an anisotropic bulk superconductor with

~ cpo To—T
H°2"=2rr:1.<0)5.(0)[ T. ] (6'7)

 

and

 

~ (1)0 Tc _T
Hc2_L : 2ߣ(0)[ Tc ] (6.8)

Figure 6.3 shows that this behavior was found in strongly coupled multilayers such as
Nb/CuGe(5°/o) 10 nm/lO nm. Since, Ho," > Hon, 5”? > 2oz,

   

 

H

Figure 6.2: A schematic diagram of the vortices in a multilayer showing how go, g”, and
132 are defined.

94

 

20000 1 1 1 1 1 1 1 1 1 1 1
18000;
16000;
14000;
12000?
100001
60001
60001
4000:
2000;

0 h 1 1 1 1 1 1 1 1 1 I 1
0 1 2 3 4 5 6

Temperature (K)

H.111 (G)

 

 

 

Figure 6.3: chu and Ho2i for Nb/CuGe(5%) 10 nm/ 10 nm.

 

95
In an isolated superconducting film with a thickness do >> Eoz, chn is approximately
equal to H3, (T) the upper critical field for bulk S. Therefore, for such a film at low

temperatures éxy E £2 E 1:, where §o(T) is E in bulk S. As T increases, §o(T) diverges

proportional to (Tob - T)“2 until it becomes approximately equal to do at a temperature

T21). Since the superconductivity cannot extend beyond the edges of the film, above this

temperature £2 saturates to the value do/\/12.56 In addition, at T > T21), the scaling
temperature for go becomes To rather than Tob.

Therefore, for an isolated superconducting film

 

 

 

(I) T —T
H821 E ° [ ° ) (6.9)
2745,2(0) To
and
f b -
(1)20 T‘ bT for T<T2D
2745(0) T1
H1215 q) T —T “2 (6.10)
0d CT for T>T2D
2 O 3 c
1 2:413 > [/5

 

As shown in Figure 6.4, multilayered samples having decoupled S layers show
qualitatively similar behavior. Here, however, the penetration of superconductivity into N
allows superconductivity to extend beyond do which implies that do in Equation 6.10
should be replaced by an effective superconductor layer thickness do".

All of the above variations in Ho2"(T) can be observed in a multilayer having
intermediate coupling strength, as shown in Figure 6.5 for a Nb/CuMn(0.3%) 40 nm/20
nm multilayer. Near To, Hall is proportional to (To-T) as was expected for a strongly
coupled multilayer. Then below a temperature T‘, Ha" varies proportional to (To-T)“2 as
expected for isolated superconducting layers, where T: is the effective zero field super-
conducting transition temperature for the 2D state. Finally, at temperatures below T21),

Ho2"(T) became approximately equal to H22 (T). The transition from 3D (coupled)

 

96

behavior to 2D (decoupled) behavior is called dimensional crossover and is related to the

temperature and field dependence of superconducting coupling between S layers.

To best describe the differences between Hc2 1. and Hozu, these data will be analyzed
separately. Hc2 1(T) will be considered first because it shows no effect due to dimensional
crossover, and is also easier to model theoretically. The analysis of Ha", which follows

will focus on two possible mechanisms for dimensional crossover. Finally both chn and

H1121. will be discussed to help discern which of the proposed dimensional crossover

models provides a more consistent explanation of the observations.

 

97

 

25000 ‘ 1 ' 1 3 1 1 1 ' 1 1 1 ' 1 r 1

I

20000

15000

1

H112 (G)

10000

T

l

5000

 

 

 

 

0‘1‘2‘3L4J5‘6‘7 619
Temperature (K)

Figure 6.4: Ho2” and Ho2i for Nb/CuGe(5%) 70 nm/300 nm multilayer (filled symbols),
and a bulk Nb sample (open symbol).

98

 

30000 T T fir , r , 1 . I 1 T T T

I

25000

I

20000

I

15000

H02 (G)

T

10000

I

5000

 

 

 

 

O

041A243141516 7
Temperature (K)

Figure 6.5: H02” and H.221 in a Nb/CuMn(O.3%) 40 nm/20 nm multilayer.

99

6.2 The Perpendicular Upper Critical Field

In a layered system with a magnetic field applied perpendicular to the layers, each
fluxoid passes through every layer, so ch(T) should be related to the superconducting
properties of the multilayer averaged through the entire thickness of the sample. The
superconducting transition temperature Tc, also provided a measure of the average super-
conducting properties of a superconductor. Therefore, HCu(T) may display some of the
same efi‘ects which were observed in the studies of Tc.

As discussed in the previous chapter, Tc decreased with increasing dn, Mn concen—
tration y, and electron mean-free-path 6", as well as for decreasing d5. For dn greater than

4

a critical scale length d Tc was found to be independent of dn, signifying that super-

conducting pairs penetrate into N roughly a distance d;. d; decreased as either
magnetic or non-magnetic impurities were added to N, although the decrease was much

more rapid for magnetic impurities. With varying ds, Tc was found to scale according to

 

Tab -Tc 2(5) (6.11)

where p was observed to increase with increased dn or y

By analogy with these Tc trends, Hc2_L should decrease with: increasing y;
decreasing d5; decreasing 6n; and increasing dn until dn > d; beyond which it should be
independent of dn. Pushing the analogy, one may expect that Ben scales with the S layer

thickness as ( Hf2 - Hen) oc ds'w, where w depends on (1n and y similar to the behavior of

p.

 

100

6.2.1 HCzi Experiments

As shown in Figures 6.6-6. 10, ch 1(T) was found to be proportional to (Tc - T) for

most of the CuX material studied. Figures 6.6 and 6.7 show data for Nb/CuGe(5%)
samples, while Figures 6.8 and 6.9 present it for Nb/CuMn(O.3%) samples, and Figure
6.10 for Nb/CuMn(2.2%) multilayers. In Nb/Cu multilayers, however, ch displays a
strong positive curvature near Tc. This behavior (Figures 6.11 and 6.12) held for all
Nb/Cu multilayers studied and was more noticeable in samples having thick Nb or Cu
layers. A close inspection of the data for Nb/CuMn(O.3%) and Nb/CuMn(2.2%) multi-
layers reveals that very near To, these curves also show positive curvature. Nb/CuGe(5%)
multilayers are the only samples which show no signs of positive curvature.

The remainder of this section will focus on how Hen varies as (1“, ds, y, and 6,, are
varied. The existence of positive curvature will be discussed first, then the discussion of

the dependence on dn will follow, and finally, the dependence on ds will be discussed.

101

 

 
 
 
   
 

 

 

 

20000 y 1 . 1 . I , T , 1 1 . a r y .
18000? ——~— 70nnLNb
16000¥ 2: $333113};
14000? ‘fiF‘ fififififi?

E; 12000}

a 10000—

m" 8000;
6000;
4000i
2000i

0 i 1
0 1

 

Temperature (K)

Figure 6.6: ch 1(T) for Nb/CuGe(5%) multilayers with cln = 20 nm.

102

 

-—°— 300nm CuGe(57.)
—-—- 20nm CuGe(57.
—-*-— 10nm CuGe 52
—+— 2nm CuGe(57.)

20000 1 1
18000?
16000;
14000?
120001
10000;
8000?
6000?
4000-
2000}

  
     

H021 (G)

 

1 1 1

 

 

0 A 1 I 2 ‘ 3 l 4 I 5 l 6 7 8
Temperature (K)

Figure 6.7: Hc2 1(T) for Nb/CuGe(5%) multilayers with ds = 28 nm.

103

 

 

 

 

 

ZOOOOTVTTfiIfiFrTYI‘T-I
“ —+—— lOOnmNb
18000“ —-——— 70nmNb
" —*— 40nmNb
16000” —+—— 28nmNb
14000_ —"— 20nmNb
:63: 12000:
32' 10000-
:" 8000~
6000-
41000L
2000—
O J

0‘112‘314l5‘6‘7‘8
Temperature (K)

Figure 6.8: Hc2 1(T) for Nb/CuMn(O.3%) multilayers having (1n = 20 nm.

 

104

 

 
   

20000 2+. , . . . . . . . ..2 . , . ,
18000? ——F— LMhunCuMnLBZ):
16000"- : 2832830332)!
14000} + 122:: 8:11: :33:
E; 120005 .
3 10000- -
m" 8000! _
60001 -
4000? -
20001 -
0’ 1 .

 

 

 

0 1 2 3 4 5 6 7 8 9
Temperature (K)

Figure 6.9: Hc2L(T) for Nb/CuMn(O.3%) multilayers with ds = 28 nm.

105

 

20000 , 1 ' 1 . u . 1 . 1

 
 
  
  

. —~! 761.1510, ‘
180001- ——+—-45nn1Nb
i —-*—- 34nmNb
16000: + 24nm Nb
14000 _ 20mm Nb
3; 12000:
a 10000~
m" 3000 —
6000-
4000—
2000-
0 1 1

 

 

 

0 l 1 2 3 4 5 6 7 8
Temperature (K)

Figure 6.10: ch(T) for Nb/CuMn(2.2%) multilayers with dn = 6.0 nm.

106

 

20000

   
 
 
 
 
 

 

 

 

18000:- lOOnnrNb
16000 1 333$ 31'?
:- 12:21::
23 12000f
a 10000—
:" 8000 1
6000!
4000;
20001
0.
0

 

Temperature (K)

Figure 6.11: Hc2l(T) for Nb/Cu multilayers with (1n = 40 nm.

107

 

20000 1 f 1 1 1 1 1 1 1 w 1 1
18000 - —°-—- lOOOnm Cu
’ —e— 600nm Cu
16000 — —1— 300nm Cu
—<>— 140nm Cu
14000 b —r—— 70mm Cu
12000 _ ‘ —'— 40mm Cu

     

20mm Cu

100001
8000;
6000i
40001
20001

H021 (G)

 

 

 

 

0 A 1 L 2 3 I 4 5 ‘6 l 7
Temperature (K)

Figure 6.12: ch J” for Nb/Cu multilayers with (15 = 28 nm.

108

6.2. 1. 1 Positive Curvature

For the purposes of analyses, the curvature of ch(T) was defined as the second
derivative of Hc2_L with respect to T (H 1") The measured data were not accurate enough
for a direct determination of H ", therefore the first derivative of ch (H 1') was extracted
from the data sets and H .1" was defined as the slope of these data sets. H1' was deter-
mined by doing a three point fit of chi to a second order polynomial, and then taking the
first derivative of the fit as shown in Figure 6.13. H1. was then defined as the slope of this

fit at the desired temperature. The resulting values for H i'(T) for the Nb/Cu series are

shown in Figures 6.14 and 6.15.

Four general trends were observed for the curvature as T, dn, ds, y, and 6,, were
varied:

First, H l"(T) increased as T increased toward Tc, reaching a maximum at T = To.
This behavior was observed to occur in all samples showing signs of positive curvature.
In most data sets there appears a sharp negative curvature right at Tc. While this feature
shows up in many data sets it is typically contained within the expected uncertainty of the

data, and therefore, is not significant.

 

1100,
10007
900:
300:
700:
600:
500;
400f
300;
200:
100: ‘
91.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4
Temperature (K)

 

1131(6)

 

 

 

 

 

 

 

Figure 6.13: ch(T) fit used to extract H i'(T).

-1000
Q
\\ —2000
33
. -1
m
—3000
—4000

Figure 6.14: H i'(T) for Nb/Cu multilayers with (1n = 40 nm. H1" is defined as the slope of

these data.

109

 

l

l

l

 

42';

l

14nm Nb
20nm Nb
28nm Nb
50nm Nb
100nm Nb

 

4 1 5 l 6
Temperture (K)

 

110

 

 

 

 

_1000 T T ’ 1 ‘ l ‘ 1' ' 1
—1200 ~ —+— 1000nmCu | _
> -—I— 140nmCu i ‘
—1400 r —--*9— '70 nm Cu ‘. r
- + 40 nm Cu Xi" ‘ Z ‘
'1600 - —+— 20nmCu .. f
32 -1800 ~ ' ,. ’ ~
\ L 1‘I If , i
e -200<)r '1 1 .11 i
E ~2200 — (14f .‘ “ 1
. ' -~ ‘1 1 1 .
—2400— 1 . :v.. - ,
-2600 — a
-2800 - 1
_3000 1 1 1 1 n a 1 1 1 1 1
2 3 4 5 6 '7 8

Temperture (K)

Figure 6.15: H i'(T) for Nb/Cu multilayers with ds = 28 mm. H" is defined as the slope of
these data.

111

Second, H .L"(Tc) increased as dn increased in Nb/Cu and Nb/CuMn(O.3%) multi-
layers. Note that in Figure 6.15 the slopes of H 1' are nearly equal for cln = 140 nm and

1000 nm. This implies that the curvature saturated to a large value when dn became larger

than (1*

nc’

the length scale needed to decouple the S layers. Curvature, therefore,
decreased as the coupling strength between S layer increased.

Third, H .L"(Tc) increased as ds increased. This trend was clear in the Nb/Cu data
(Figure 6.14) and Nb/CuMn(O.3%) data, but may not occur in the Nb/CuMn(2.2%)

samples. As (18 increases, the pair density in S should increases toward its bulk value.

Fourth, H1"(Tc) was large for Nb/Cu samples, smaller for Nb/CuMn(O.3%) and
Nb/CuMn(2.2%) samples, and zero for Nb/CuGe(5%) samples. d; for CuMn(O.3%) and
CuGe(5%) interlayers are nearly equal, however, CuMn shows strong positive curvature
while CuGe does not.

These data indicate that an odd paradox appears when analyzing the curvature.
Curvature increases in magnitude as the layers become more decoupled by increased dn,
but, curvature decreases as the layer become more decoupled by increasing 6“. In

addition, curvature is not simply related to the pair penetration depth in N, since

CuMn(O.3%) and CuGe(5%) layers have the same d;, but CuMn(O.3%) samples show
curvature while CuGe(5%) samples do not. Takahashi and Tachiki predict curvature to
depend on how strongly the superconductivity is localized within the S layers, but they
provide no physical mechanism for the occurence of curvature.28 Another proximity effect

model,27 which is slightly more tractable than the model of Takahashi and Tachiki will be

discussed in the theory section.

6.2.1.2 The Dependence of H02 1 on dn

To systematize the data sets for H1210 ) for the various N layer materials and S

and N layer thicknesses, it has been evaluated for each sample at T = 0.5T, This

112

temperature is low enough to be below any region of positive curvature. Therefore, the

behavior of H121(0.5) should characterize the entire data set. chi at T = O was not used
to systematize the data sets because Hc21 could not be reliably extrapolated to zero
temperature for all data sets.

The dependence of ch 1(0-5) on cln for ds = 28 nm is shown in Figure 6.16. ch
was observed to decrease with increasing dn and to become independent of (in for cln >
d;. The estimated values for d111, however, were much smaller that the d; estimated
from the Tc experiments. Approximate values for both d; and d; are listed in Table

6.1. The difference in these lengths was not due to sample variability since the same

samples were used for both the TC and the Hcz 1(T) studies.

The difference between d; and d; appears to be related to changes in the
inherent spin-scattering rate of N. d; is approximately equal for Nb/CuMn(0.3%) and
Nb/CuGe(5%) samples whereas, d; for these two systems appear to be different.

However, the data are not precise enough to determine why d:1i and d; are difl‘erent.

Table 6.1: d; and d; for various CuX materials

 

 

 

 

 

 

CuX dzc/ nm (1111/ nm
Cu 3 00 70
CuGe(5%) 100 10
CuMn(O.3%) 7o 70

 

113

 

 

 

 

 

 

 

 

16000_1a1vw111a1u1 .12. 1.“.H-.-22,,,jzfl
15000 _— + CuGe(57.)
14000 — ——t— guM 0 37 _
13000 5 + ‘1 “(1 -) p
33 12000_- T l j
A 11000_- 1 1 f
to. 10000 - -
o - .
‘31 9000 _- f
m" 8000:— T .
7000 — _
1 k 4
6000 _— \1 .1 :1
5000 r ‘ :1
10°2423 11101. 23H14L1L62‘ 23 111111103
deux (nm)

Figure 6.16: HC2 1(0-5) for (1s = 28 nm and various CuX materials and layer thicknesses.

114

6.2.1.3 The Dependence of HeZJ. on dS

Figure 6.17 shows the data for H‘,2 1(0-5) as a function of Nb layer thickness.

These curves appear very similar to the raw data for Tc as a function of Nb layer thickness

(Figure (5.5)). Based on this similarity, one can postulate that ch may scale as

 

Hb—H d ‘W
h : c2 c21z s 612
1 11:, (1) ( >

0

and w is a scaling exponent. In addition, ch(T=0) should go to zero only when Tc goes

to zero, therefore, the scale factor (10 used above should be the same as that used for the

scaling of Tc. h i is plotted in Figure 6.18 as a function of ds for the same data as shown in

Figure 6.17. The scatter is too large to definitively state whether or not scaling works.
Studies by Kanoda et al.25 on V/Ag multilayers show that ch 1(0) is described by

 

Hm(0) ; Hf,(0)[1+e:" J (6.13)

S

where s E 0.2 6 11 /€ 3. In the limit that ed,n << d8, this form is equivalent to Equation (6.12).
Hence, scaling of Hc2.L does exist for V/Ag samples.

While these results for the dependence of ch 1(0-5) on d“, ds, and N are in quali-
tative agreement with the Tc results, better data are needed to prove the existence of

scaling in Nb/CuX multilayers. A better understanding of the positive curvature, however,

requires a good theoretical model.

18000
16000
14000
12000
10000

8000

6000

Hez1(O-5) (G)

4000’
2000

115

 

l l l

l

I r T

——

.-

1.—

 

 

20 nm CuGe(57.)
40 nm Cu

6.0 nm CuMn(2.22)
20 nm CuMn(O.37.)

—1

—‘

_‘

 

 

dNb(nnn)

Figure 6.17: Dependence of ch 1(0-5) on (15 for various cln and CuX.

O 11111111m1m1211111m1
0 10 20 30 40 50 60 70 80 90 100

116

100 +1511 - 1 1 1 1

 

B

’7

67 ‘
\
4

 

 

 

 

 

1:18 7 \

Cl: : . j
\ f \ f
”:1 .

g .

m 3 ~ _ -

' K
a?" 2 . _.._ 20 nm CuGe(57. 1 __ _
v + 40 nm Cu \
- —l— 6.0 nm CuMn(2.22) 1

+ 20 nm CuMn(O.37I) " 1
1.0—1 LL‘LHJ J 1 1 1 i 1 1 1 J 1 1 11 1 1111mnm . 11.1.1.1
'7 5 9101 2 3 4 5 6 7 a 9102

dNb (nm)

Figure 6.18: Dependence of h i(0.5) on ds for various tin and CuX.

117

6.2.3 ch(T) Theory

In a series of publications, V. Kogan and various collaborators have developed a
model for superconductivity in S/N multilayers.27151157’58 Of particular interest in this series
is the paper by Biagi, Kogan and Clem (BKC)27 where a model predicting the behavior of
ch 1(T) in S/N multilayers was presented. One advantage of this model over previous
models is that it places no limits on the values of the S and N layer thicknesses or on the
electron mean-fi'ee-paths. Therefore the BKC model does not assume the dirty super-
conducting limit and hence unlike the earlier HTW and KZ models, it should be generally
valid for Nb/Cu as well as Nb/CuGe(5%) multilayers. One serious drawback to the model
is that it does not include magnetic scattering in the N layer, and so, cannot directly be
applied to multilayers having CuMn layers.

With these stipulations, BKC formulated the following set of equations to predict

the temperature dependence of Hc2 1- For simplicity in the following discussions Hc2l(T)

has been replaced by H 11
q. 111(q,d,/2)=q,&11111(q,d,/2) (6.14)
911
2 2 H1
= k — 2 — 6.15
611 1 0 <1). ( )
ln[—'I—;;-]= ‘I’(l/2)—‘I’(1/2+k§n§/2) (6.16)
f
k: +21t—i 6,, << £3

 

2 o
q1=( (6.17)
2 nH, 3[2nH,e] ensg"

 

 

f 1 o
F in << an
k2: “ 2 (6.18)
" 1 Hi ,
2 3,, 1—-H— en 3:,
where,
0,, = 1— 1/11n (6.19)
711 = 301/602 (6.20)
1/2
(I) 02 5
H =—°—" — 6.21
° 21 1311..) ‘ )
n? = ’7 a 2" (6.22)
611']th 6e T Yip,
hv
°— 5“ ~ “hkb 1 (6.23)

 

" _ 21111ch" = 262ch y,(p€),

and ‘I’(z) is the digamma function, h is Plank's constant divided by 22, (Do is the flux
quantum, kb is Boltzmann's constant, e is the electron charge, vfii is the Fermi velocity, E, is
the electron mean free path, 7, is the linear coefficient of the low temperature specific heat,
pi is the resistivity, i stands for the S or N material, (p2)n is the finite size correction to the
resistivity in N, T: is the bulk superconducting transition temperature of S, and «if: is the
effective superconducting coherence length in N.

In the limit that H 1 = 0 and 6,, << £3, these equations simplify exactly to the model
derived by Hauser, Theuerer, and Werthamer [Equations (5.29) through (5.31)]. Since it

was already shown that the HTW model does not quantitatively predict Tc, one may not

119
expect the BKC model to quantitatively predict H 1~ Hence, only the general trends

observed in the BKC model will be compared to the data. This model will first be

examined in the limit of small dn, and following this, the fiill model will be used to study

the phenomena of positive curvature of H1-
In the limit that dn is small, tanh(qndn / 2) E qndn / 2 <<1 and therefore,
tan(qsds / 2) E qsds / 2 << 1 also. Hence, Equation (6.14) can be simplified to

qu. s qid. 38— (6.24)
p

n

This is then used to solve for q,2 and hence k3:

k3 ijfid—MZK—HA (6.25)
91 d1 <I>o
Approximating the right hand side of Equation (6. 16) by47
7C2
l11(1/2 + z/2) — 31(1/2) 5 111(1 +72] (6.26)

and combining this with Equation (6.25) leads to the following equation for H 13

Tb 7C2 2 2
—°El+—k 6.27
T 4 Sns ( )

Now, allowing the dirty limit to be applicable, as it should be for very thin N layers (see
Chapter 5) Equation (6.27) becomes

120

 

b 2
L151 7* _2_7mfoHiT_ 11211 ,i 2 £191 (628)
T 4 Q0 T pn dS 77!} p11 d5
where, 1180 = 115(ch). In addition, note that
2
(11;) .2. r1191 (6.29)
11“ YSpS

Therefore, the final equation relating T and H1 in the limit that dn is small is
T 1+1t V“ d—" zfr 1——2——“T‘S°Hi ”Bid—'1 (6.30)
4 vs d_s 4 (D0 pn (1s

The primary difiiculty in applying this equation is that it solves for T as a function of H1
and T: whereas most experiments examine how H 1 depends on (Tc-T).

Note that when T = Tc, H1 = O, and Equation (6.29) collapses to the thin dn limit

of the HTW model

2
T5: T MIL-Lug"- (6.31)
4 78 (1

Using this to replace I? with Tc in Equation (6.30) and reorganizing the terms, one finds:

(1)04 p d T ——T
T _ 1__s __11 __c 6.32
HM Zunson “2) p11 dsii T j ( )

C

 

In addition, in the limit that (18 —> 00, H 1 E Hf, (T), or,

121

1, ~ <1)o f. ch—T
HC,(T)=2m7:ofl,( Tb ) (6.33)

C

 

 

This was then used to remove the explicit temperature dependence from Equation (6.32)

resulting in

b __
._.____Hc2 Hm 5316,11: (6.34)

h
J- HEZ pn

for all temperatures below To. This form is in general agreement with Equation (6.12),

with the data sets for h 1 as a function of ds, and with the experimental results of Kanoda et

al.25 To accurately test the validity of Hc2J~ scaling, however, requires data for sample

series having much smaller N layer thicknesses than have been measured.

Analyzing the fill] numerical solutions to Equations 6.14-6.23, BKC were able to

show that for large dn and large ds, H 1 displays positive curvature near Tc. In particular,

BKC found that as either tin or (18 increase H J."(Tc) increases, eventually passing through a

maximum. These trends are qualitatively shown in Figure 6.19. In the limit of infinite ds,

H1"(Tc) tends toward zero while for large dn, H1"(Tc) remains non-zero.

122

 

 
  
  

—— Hlvs. dn
—-- Hlvs. d,

Hi(Tc)

 

\
.‘. ~—-—_—

2 345 103

 

 

 

Layer thickness (nm)

Figure 6.19: Sketch showing the dependence of H _L"(Tc) on ds and dn.

These trends agree well with the behavior observed in Nb/Cu multilayers,
however, the maximum in H1" was not observed. This may indicate that samples having
sufficiently large (in or dS have not been studied, or that this feature of the theory does not

apply to our data.

The BKC model also predicts that H _L"(Tc) should increase as 8,, increases, as 65
decreases and is greatest when 68 << 6". This also agrees with the data where H1"
decreased systematically as the normal state resistivity of N was increased.

At the level of accuracy which the data allow, comparisons of the BKC model to

the experimental results indicate that both experiment and theory agree on how H621 and
H1" depend on T, although they disagree on the behavior of H1" in the limit of thick N

and S. The model and the results of Kanoda et al. do support the scaling of H621, however

the accuracy of the Nb/CuX data are not high enough to test scaling.

123

6.3 The Parallel Upper Critical Field

6.3.1 Ha“ Experiments

As mentioned in

temperature dependence

the introduction to this chapter, and shown in Figure 6.5, the

of Hall follows one of three trends:

T’ <T<Tc He," oc(T, —T) (6.35)
T2D < T < T‘ He," as (T; —T)” 051/2 (6.36)
T < T2,, Hall ; H2,(T) (6.37)

where T‘ is the dimensional-crossover temperature, T: is the effective Tc for the 2D

regime, T21) is the crossover temperature from 2D to bulk behavior, and HE, (T) is H02 for

bulk S. Since H62"(T) is proportional to l/fixyéz, the temperature dependence of these two

coherence lengths in the same temperature regimes is given by:

52(051

T, — T
TC

T; -T
T.

C

' —1/2
313 '
5w (0)( J T < T < Tc

 

r:<o>(

-l/2
J T2D <T<T'

 

 

124

 

 

 

( _ —1/2

§Z(O)( TCT T) T' < T < Tc
«5.10215 . T2D<T<T*

1112:)

In an ideal case, the behaviors for fixy and 52 are shown in Figures 6.20 and 6.21, and the
resulting Ha” curve based on these simulated data is shown in Figure 6.22. Most samples
studied having dn < d; do show evidence for a dimensional crossover near Tc, but only
samples having (15 > 30 nm show the low temperature bulk behavior.

The data for Hc2”(T) in Nb/Cu multilayers are shown in Figures 6.23 and 6.24. The
Nb/CuGe(5%) data are displayed in Figures 6.25, 6.26 and 6.27, and Figures 6.28 and
6.29 present it for Nb/CuMn(O.3%) multilayers, while Figure 6.30 shows the data for
Nb/CuMn(2.2%) multilayers.

For large d8, all S/N multilayers studied showed evidence of bulk behavior below
T2,). In Nb/Cu, Nb/CuMn(O.3%) and Nb/CuMn(2.2%) multilayers, bulk behavior was
observed for all (15 greater than approximately 30 nm. Nb/CuGe(5%) multilayers,
however, only displayed bulk behavior for (15 equal to 70 nm, as shown in Figure 6.26.
The 2D superconducting state of the isolated Nb layers was more strongly affected by
CuGe than by any other N material. The Tc experiments however, indicated that CuGe
layers affected the superconductivity of the S layers the least of any of the interlayers (i.e.
they lead to the highest Tc's). Hence, it was expected that bulk behavior would be
observed at lower (18 for CuGe interlayers than for other interlayers. This trend is still not

understood.

125

 

(:2 (11m)

 

0
00000000000

0 1 1
00 1 I —
10 000 1 I
I 1
I I

166665

 

 

 

01112131415 6 718 9 10
Temperature (K)

Figure 6.20: Qualitative dependence of §Z(T) showing how T2D and T“ are defined.

126

 

 

 

 

 

A .i 1
a .0 i
a 0' 5
v o :

>5 1 ‘o' E _.
A?) i 11 O '
i 0.0'0 I

.o:'° i _

E b T

_ To ; Tc

0 1 1 1 1 1 1 1 1 1

Temperature (K)

Figure 6.21: Qualitative dependence of §w(T) showing how T21): T‘, and T: are defined.

127

 

H621 (G)

 

 

 

 

213A41516 7‘81
Temperature (K)

Figure 6.22: H02”(T) resulting from the §Z(T) and EMT) data sets shown above.

128

 

   
 
 
 
  

24000? _._ 100nm N117
22000 _- 50nm ij
18000 i 14nm Nb:

A 16000 _ 10nmNb-1
3 14000 3 {
§ 12000; j
a: 10000 5 j
6000} 3
6000 f j
40005 —
20005 —

O 1

 

 

 

0 J 1 A 2 A 3 l 4 l 5 6 I 7 I 8 A 9
Temperature (K)

Figure 6.23: chn(T) in Nb/Cu multilayers with (ln = 40 nm.

129

 

 

40000 -
30000 -
33,
:c 20000 T
10000 ~
0 1 1 1

 

Temperature (K)

Figure 6.24: Hc2H(T) in Nb/Cu multilayers with ds = 28 nm.

 

 

011L21314156

l ' T j

600 nm Cu 1

’70 nm Cu

40 nm Cu .

20 nm Cu
10 nm Cu
4 nm Cu

J

 

130

 

1' IHI‘HIT 1 11'1‘f'

 

 

30000 7 ‘
—*—- 70nm CuGe 57. ,
l —+—- 20nm CuGe 57.;

25000 _ \ ——e—- 10nm CuGe 57. ‘

1 2nm CuGe(57.) 1

A 20000 T 7
8 .

§ 15000 r 7
m .

10000 7 ‘
1

5000 r ‘

0 ' L

 

 

 

0'1'2‘3 415'6 7 3'9
Temperature (K)

Figure 6.25: HC2II(T) in Nb/CuGe(5%) multilayers with (15 = 10 nm.

131

 

 

 

30000 _ I 1 1 r 1 F I I 77
70nm Nb
25000 ’ 28nm Nb
) 18nm Nb
.. 2oooo 12:11::
3 1
$0- 15000 7
E
10000 7
5000 -
O 1 1
0 7 8

Temperature (K)

Figure 6.26: chlI(T) in Nb/CuGe(5%) multilayers with d“ = 20 nm.

 

 

132

 

4000011111111111111

I

30000

T

20000

H0211 (G)

-—e— 300nm CuGe 577:

--— 40mm CuGe 5 .
10000 -—e—— 20nm CuGe 57.
—°—— 10nm CuGe 57.
—°— 2nm CuGe 57.)

O ‘ 1 1 1 1 1 1 1 1 1 1 1 1 “ °“~‘.~ '
0 1 2 3 4 5 6 7

Temperature (K)

T

 

 

 

Figure 6.27: Hc2"(T) for Nb/CuGe(5%) multilayers with (15 = 28 nm.

 

133

 

 
  
  

30000 , T e 1 r i f T T 1 1 r 1 r 1
—°— 100nm Nb
—9—- 70nm Nb
+ 40nm Nb
—-— 28mm Nb
20000 _ 20mm Nb
§
§
E
10000 -
O 1

 

 

 

011L2‘3‘4151617I8
Temperature (K)

Figure 6.28: HC2II(T) in Nb/CuMn(O.3%) multilayers with (1n = 20 nm.

134

 

40000 ' I . I 1 w a r . r . 1 r I * T
x —«+—— 70nm CuMn .37.
\ —+— 40mm CuMn .37.
—¢—-'~ 20nm CuMn .37:
30000 - —-—~ 10nm CuMn .37

4mm CuMn .37:)

 

 

E5,
it} 20000 -
m"
10000-
O 1 1

 

 

011120314 5J6 7MB
Temperature (K)

Figure 6.29: H62"(T) in Nb/CuMn(O.3%) multilayers with ds = 28 nm.

135

30000,I.,,1.,,I.T.T.l

1 —«—— 100nm Nb«

—9— 70nm Nb
25000 . —+— 45mm Nb 0
—¢— 34nm Nb J

 

I

_ ' —-—— 24nmNb

A 20000 " “\ _..... zonmNb‘
a - ‘ f‘\ *
§ 15000— \\\ . \; —
=° — . \ ~
10000~ A \_ —
5000 - L _

 

 

 

‘. |I
O 11.\11 .1.1.41

0 1 2 3 4 5 6 7 Aha ‘ 9
Temperature (K)

Figure 6.30: H.2|.(T) in Nb/CuMn(2.2%) multilayers with dn =6.0 nm.

136

For ds less than the above values, only the 3D-coupled and 2D-decoupled states
were observed. Here, as ds was reduced, the 3D regime expanded rapidly with T‘
decreasing toward zero temperature much more rapidly than Tc. Hence, the 3D coupled
state appears to dominate the superconductivity of multilayers having small d8. This may
indicate that for small ds, there is a larger energy penalty for isolating the supercon-
ductivity within S than is found for larger ds.

Figure 6.31 presents data for Hc2"(T) at various Nb layer thicknesses in Nb/Cu
multilayers with dn = 40 nm and Cu/Nb/Cu sandwiches with (1n = 300 nm. Dimensional
crossover was observed to occur when Hc2II(T) for a multilayered sample became
approximately equal to H02"(T) for a sandwich having the same Nb layer thickness. Below
T", HC2II(T) for the multilayer and sandwich samples did not coincide. This shift was
related to a shift in TC, since Tc in the sandwich films was observed to be slightly less than
Tc observed in Nb/Cu multilayers having the same Cu layer thickness (300 nm). There-
fore, this shift in H62”(T) was not related to dimensional crossover.

Hc2||(T) for the sandwiches also showed strong positive curvature near Tc. This
trend was reminiscent of the positive curvature observed in ch 1(T) (Figure 6.11). Since
there was only a single Nb layer in the sandwich films, the curvature of H02" must be
related to the penetration of superconductivity into the N layer. It is intriguing, though,
that in the current case curvature became more pronounced as ds was decreased, whereas
the curvature of Hc2J_(T) became more pronounced as (19) was increased. This effect is not
understood at any level.

For fixed (18 (28.0 nm) H02"(T) showed a kink at T" for all (1n and N layer materials

Q

studied, provided dn < d as was shown in Figures 6.24, 6.27, 6.29, and 6.30. For small

(in, the kink was barely discerned. As dn was increased, the kink at T‘ became more
pronounced and H62"(O) was observed to decrease.

H02”(T) in the 2D state for non-magnetic interlayered samples was roughly

independent of (in for (in 2 40 nm for Cu (Figure 6.32), and 20 nm for CuGe (Figure 6.27)

137

interlayers. This implied that, in these systems the S layers are fiilly isolated from one
another below T’. Ha” in the 2D state for Nb/CuMn multilayers, however, changes with
increasing (in until (in > d; (Figure 6.29). In addition, in Nb/CuGe multilayers having (18 =
10 nm, Hana) within the 2D state is dependent on (1H for dn up to 70 nm (Figure 6.25).
This implies that the 2D state may not always be indicative of a total decoupling of the S

layers.

138

 

 

 

 

30000 1 1 1 1 . 1 . 1 1 1 1 1 1 1 1 1
—e—*— lOOnmNb
—9—'— 50nmNb
....._._ 28nmNb
20000 — \ .\ ~
A \o \3\‘
8 \x \1
\
.3 )1
:1: \
10000— g -
O 1 1 1 Ah“ 1 1 1

 

0 1 2 3 4
Temperature (K)

Figure 6.31: chn(T) for Nb/Cu multilayers with dn = 40 nm (filled symbols) and
Cu/Nb/Cu sandwiches with (ln = 300 nm (open symbols).

139

 

  
  
   
  

 

 

 

25000 . 1 , , 1 , 1
——+—- 600nm Cu 1
20000 + 140 nm Cu
- —+— 70 nm Cu r
\\ ——*— 40 nm Cu
‘ 20nmCu ‘
E; 15000 .
m" 10000 ~
5000 e
O 1 1 1
2 3 4 5 6 '7

Temperature (K)

Figure 6.32: Expanded view 0fH¢211(T) for Nb/Cu multilayers with (15 = 28 nm.

140

6.3.2 Hc2|l(T) Models

Two qualitative models have been developed to model the occurrence of dimen-
sional crossover. The temperature quenched (TQ) model, introduced by Chun et al.,24
was based on theoretical studies of S/I multilayers where coupling is due to pair tunneling
through the insulating I layer. This S/I model was the only theory available at the time
which predicted dimensional crossover in any superconducting multilayered system. The
field quenched (FQ) model will be developed below to solve some of the problems
encountered when the TQ model was applied to our Hc2"(T) data. The TQ model will be
discussed first, followed by an analysis of the experimental data for Nb/CuX multilayers.
Then, the FQ model will be introduced and its predictions will also be compared to the
same data. Following this section, both models will be compared to recent data taken by
Koorevaar et al. on Nb/NbZr multilayers59 in an efi‘ort to establish which model more
appropriately describes the mechanism behind dimensional crossover in superconducting

multilayers.

6.3.2.1 The TQ Model

This model relates the occurrence of dimensional crossover to the temperature
dependence of £2. Chun et al. argued that when 5,2 became approximately equal to dn,
superconductivity was no longer coherent across the N layer, hence leading to a decou-
pling the S layers.24 Recent experimental studies by Kanoda et al.,25 however, indicate
that crossover occurs when 5,, is approximately equal to the bilayer repeat distance A = ds
+ (in.

Chun et al. originally proposed this model to describe their observed dependence
of dimensional crossover on the S and N layer thicknesses in Nb/Cu multilayers24 while

Kanoda et al. studied V/Ag multilayers.25 Both groups assumed that dimensional

141

crossover behavior in S/N multilayers is similar to that found in S/I multilayers. Klemm,
Luther, and Beasley had previously developed a theoretical model for evaluating HC2II(T) in
S/I multilayers.“o The KLB model shows that near Tc, {:13 (T) varies as (TC - T)“2 until
52D (T) becomes approximately equal to A/J2, T“ was then defined as the temperature at
which 5:13 (T) = A/J2— . As T decreases below T‘, 5,, approaches (15 /\/E , indicative of an
isolated 2D S layer. Since the TQ model links crossover to the temperature dependence
of £1, it predicts that the 2D state extends down to zero applied magnetic field as shown in
Figure 6.33.

This model also predicts that a” .2. fixy and hence that there is no change in the
behavior of fixy below the crossover point. This assumption provides a means to extract
the value of E, at the crossover temperature 6 from the H62”(T) and ch 1(T) data. Chun et
al. performed this extraction and found that for (1s and tin greater than ~15 nm, the data
were roughly consistent with the TQ model. One drawback of their experiment was that
they typically studied multilayers having ds = (in . This made it difiicult to determine
whether i‘ was dependent only on A, independently on ds and dn. In addition, for small A
they found that E was not simply related to A, however, they did not speculate on a
reason for this discrepancy.

Using this model, 8 was extracted from our data for Hc2 in Nb/CuX multilayers
with the results shown in Figure 6.34. Note here that, although [3‘ does depend on A, it is

usually less than A/JE and for the Nb/CuMn(O.3%) series with fixed ds, 5" appears to be

nearly independent of A.

142

 

22000;
200001
10000}
160005
14000~
12000}
10000—
80001
60001
40001
2000}

H.211 (G)

 

 

 

 

0 1 2 3 4 1 5 6 7 8 9
Temperature (K)

Figure 6.33: Hc2H(T) for Nb/Cu 28 nm/4O nm and the extension of the 2D state predicted
by the FQ model.

143

 

l T 1 T ' 1 l l

 

140 ”—o— 28 nm Nb in Nb/Cu
' + 28 nm Nb in Nb/CuGe(57.) ‘
120 r—l— 28 nm Nb in Nb/CuMn(O.37.) // _
—9— 40. nm Cu ( ) /
’ —\—’>—- 20. nm CuGe 57.
100 ~—v£r- 6.0 nm CuMnEZBZ) / / r
A . —B—- 20ng CuMn 0.37. // / .
E, - // .
:1» 60 - / / a
1
g 1
40 ~ //// -
1 4 1
20 ~ ~
1

 

 

O 1 1 m 1 L l 1 1 1 l 1 l 1 l 1 l L 1 1
O 20 40 60 80 100 120 140 160 180 200
Bilayer thickness (nm)

 

Figure 6.34: E“ as a function of A for various CuX layers and Nb and CuX layer thick-
nesses.

144

6.3.2.2 The FQ Model

Another possible decoupling mechanism relates the application of a field parallel to
the layers to the decoupling of the layers. In this model a magnetic field parallel to the
layers will be shown to selectively quench superconductivity in one of the materials in a
superconducting multilayer. To properly develOp a mechanism for field induced decou-
pling, the definition of the superconducting coherence length will first be reexamined.
Then the averaging processes leading to £2, éxy, and g“ will be examined. Finally, it will be
shown that a steadily increasing magnetic field density can destroy the induced super-
conductivity in N without destroying the superconductivity in S.

There are two general ways to estimate i within the S and N layers. In the first, as

and in can be directly calculated using the equation:

Jam, 1

2 =——— 6.38
5' 662T 71101 ( )

where i stands for the S or N material and the other terms are defined in the Hana)
section. T in Equation (6.38) is defined as Tcb for Nb, while for CuX, T is a free variable.
Typically, one uses the highest temperature of interest to provide the smallest possible
estimate of fin, here T: was used. Using values for 1’1 and pi given in the Table 5.2, 5.1.1. is
approximately 11 nm and in varies from 85 nm to 14 nm as the N layer is changed from

Cu to CuGe(5%). Hence, in all cases Em is greater than £8.

Another way to estimate g in the S and N layers is to use Equation (6.3).

: hvf

5 7A— (6-3)

145

In a non-superconducting material A = 0, while the superconducting pair function F (F a A
/V, where V is the electron phonon coupling parameter) is non-zero. Therefore, proximity
effect models, typically replace A by F which implies that a at 1/F. F is proportional to the
phase coherent electron density in the S and N layers and varies with position roughly as
shown in Figure 6.35. Again, E, should always be larger in N than in S independent of the
thickness of the S and N layers.

A A
\_/ \___/

é flvflv

N S N S

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6.35: Schematic diagram of F and i as a function of position perpendicular to the
layers of an S/N multilayer.

For a vortex penetrating an S/N multilayer, éxy, £2, and g“ represent different
averages of fin and is within the multilayer. 15," will always be related to an average of in
and 1; within the entire sample, since each vortex passes through every layer. éxy and E,
on the other hand average only over in and as in the vicinity of the core of the vortex.
When éxy and 1;, are >> A (either near Tc, or in samples with very small A), they will
effectively average over many S and N layers, and hence should be relatively unaffected by
the discreet nature of the N and S layers. Therefore, in this regime éxy ~ 1;“ and £2, Em, and
E" should all be proportional to (Tc-T)“. As fix), and 5,, decrease, or A increases, they will

become distorted by the influence of the layering. :2, which typically averages over a few

146

S and N layers, will not be strongly affected by the layer in which the core of the vortex
resides. éxy, on the other hand, will be dominated by the value of a near the center of the
vortex. For a vortex nucleated in the S and N layers respectively, the approximate cores
are sketched in Figure 6.36. To reduce their condensation energies, the vortices will tend
to nucleate at positions of lowest pair density, hence at the centers of the N layers.

Therefore, éxy will be dominated by Em.

0220220220)
ZmZme/JZ

   

Vortex nucleated in N Vortex nucleated in S

Figure 6.36: Approximate vortex cores for external fields directed parallel to the layers.

At low field densities there are two main factors which regulate the spacing and
structure of the vortices:

First, there is the position dependence of the vortex nucleation energy. As
discussed above, for an isolated vortex, the minimum energy position is at the center of a
N layer. Therefore, the vortex structure will attempt to maximize the fraction of vortices
which sit near the centers of the N layers.

Second, there is the anisotropy of the magnetic penetration depth. Far from the
core of a vortex, the magnetic field density in a superconductor is related to the magnetic
penetration depth 7», therefore, the magnetic repulsion between two vortices will also
depend on 7». The anisotropy of 7» in S/N multilayers has not received a great deal of

experimental or theoretical attention, therefore, good models for predicting the magnitude

147

of the anisotropy do not exist. To obtain some measure of the anisotropy of A, it was
assumed that the anisotropy in a S/N multilayer would be less than the anisotropy of a
single S film. For such a film, the ratio of the in-plane magnetic penetration depth AW to

the perpendicular magnetic penetration depth X, in the limit that Em >> Q, is given by61

1’1 E [iInLflH (6.39)
,1 3 62(0)

In the 3D regime very near Tc, Lin/£2 can be approximated by Han/Hen which is typically

less than 4. This leads to kxy/Xz less than 1.8. Hence, 7. is fairly isotropic. This implies

that the magnetic repulsion should be isotropic and the vortex lattice should also be
isotropic.

Therefore, in small applied external fields, the vortex lattice should be fairly
isotropic provided that the fraction of vortices centered in N layers is kept high. This
leads to a vortex packing arrangement similar to that sketched in Figure 6.37. The
approximate vortex cores are also illustrated in this figure.

As H increases, the vortex-vortex spacing decreases fairly uniformly until the
spacing along the N layers becomes roughly equal to Em. When this occurs, the vortices
will overlap within the N layers thereby destroying the induced superconductivity in N.
Note, however, that when this occurs é, remains less than the vortex spacing perpen-
dicular to the layers, and so, superconductivity will not be destroyed in S. Once the
superconductivity in N is destroyed, the S layers become decoupled and fixy and i, should
change fairly abruptly to values characteristic of isolated S layers. Therefore, the FQ
model predicts that the 3D to 2D crossover occurs as the magnetic field is increased. This

leads to an extension of the 3D state shown approximately in Figure 6.38.

148

 

 

 

 

Figure 6.37: Approximate vortex structure in a S/N multilayer.

 

222222222222

149

 

22000
20000’
16000’
16000’
14000’
12000’
10000'
6000'
6000’
4000
2000:

H.211 (G)

 

 

 

 

0 A 1 2 L 3 1 4 A 5 6 A 7 I 6 i 9
Temperature (K)

Figure 6.38: H02“(T) for Nb/Cu 28 nm/4O nm and the extension of the 2D state given by
the F Q model.

150

The FQ model, predicts that dimensional crossover is predominantly related to
éxy (T') provided that £2 is of the order of A. Therefore, £2 and A are related, but not in
any analytic fashion. This concurs with the observation that E may or may not depend on
A, (Figure 6.34). In addition, this model predicts that near and below crossover §Xy(T)

and &"(T) are unrelated to one another. Hence, the change in the temperature dependence

of §XY(T) will not effect the temperature dependence of §"(T) or Hc2i(T). Therefore, the

data for Hall and Hc2_l. in Nb/CuX multilayers qualitatively supports the validity of the F Q

model over the TQ model in providing a mechanism for the occurrence of dimensional
crossover.

Recent experiments by Koorevaar et al. also support the FQ model.59 Their
experiments focused on studies of the critical current Jc(T, H) for T and H below the
H62"(T) line. Their studies of Nb/NbOJZrOj multilayers (where both the Nb and NbZr

E Tcszr ) located two transitions in the vortex

layers are bulk superconductors with TcNb
pinning potentials as H was increased.

At low field densities, they found that the vortices were not strongly pinned by the
internal layering of the sample for any temperature below Tc. For H greater than H‘, JC
displayed behavior indicating that the vortices had become pinned within the Nb layers.
Above a yet higher field density H” the vortices shifted to lie within the NbZr layers.

The change in pinning sites was related to changes in the nucleation points for
vortices penetrating the sample. Koorevaar argued that below H‘ the vortices were not
strongly pinned since i, was large enough to average over many Nb and NbZr layers.
This state is similar to the 3D-coupled state described above in Nb/CuX multilayers.
Above H‘, the vortices became pinned within the Nb layers since Nb has a larger coher-
ence length than NbZr. Hence, here Nb plays a role similar to the N layer in a S/N multi-
layer. Eventually, at H” the superconductivity of the Nb layers was destroyed, leaving

only isolated vortices in the NbZr layers.

151

By analogy, one may expect that in an S/N multilayer, H’ and H" would roughly
coincide leading to a single crossover line, as was observed in Nb/CuX. The TQ model
cannot explain any of the features found in the Nb/NbZr data and so is not likely to be the

decoupling mechanism operating in S/N multilayers.

 

CHAPTER 7
SPIN-FREEZIN G NEAR Tc

As discussed in the introduction, searches for interplay between superconducting
(SC) and magnetic ordering have been driven by the very fact that magnetism tends to
strongly attenuate superconductivity. Previous studies have definitively shown coexistent
magnetic and superconductive order only in ternary rare earth (RE) alloys.2 Recent
searches for SC and magnetic order in transition metal systems have focused on super—
conducting/ferromagnetic multilayerszv‘l’5 Here, the magnetism and superconductivity are
physically separated enough for both phase transitions to exist. While superconductivity
has been observed at temperatures well below the magnetic phase transition Tf, the ferro-
magnetic state has not been studied at temperatures in the vicinity of To. It is this regime,
where Tf ~ Tc, that the largest interactions between the ordered states are expected. In
addition, coupling between S layers is very weak due to pair breaking by the ferro-
magnetic layers. This required very thin (~ .2 to 2 nm) ferromagnetic layers be used to
observe any significant level of proximity effect coupling between S layers. At these layer
thicknesses, fabrication problems related to layer thickness uniformity and growth become
critically important. In addition, Tf only changes appreciably for very thin ferromagnetic
layers. In Ni, Tf changes from 450 K at dNi = 2.5 nm to O K for dNi = 0.7 nm.62 Hence,
study of the regime where Tc ~ Tf is virtually impossible using ferromagnetic layers.

To avoid some of these problems, superconducting/spin-glass (SC/SG)
Nb/CuMn(1%) multilayers have been examined. Spin-glass magnetic layers offer four
primary advantages over ferromagnetic layers: 1) CuMn(1%) is a dilute magnetic alloy
and so has significantly fewer magnetic spins to scatter and destroy proximity induced
pairs; 2) The spin-glass phase transition has no spontaneous moment, therefore, increased
scattering caused by a spontaneous magnetization (as occurs for ferromagnets) will not
occur; 3) The spin-freezing temperature of CuMn(1%) (11 K) is. approximately equal to

152

153

the bulk superconducting transition temperature of Nb (9 K); 4) The spin-freezing
temperature Tf is strongly dependent on the CuMn layer thickness for dCuMn less than 25
nm,9 therefore spin-glasses offer much more flexibility in setting Tf.

In the RE based systems, two general experiments were used to detemiine if
coexistence occurred. First, Tc and Tf were measured for increasing values of the
magnetic impurity concentration y. As y increased, Tf increased and Tc decreased. When
TC ~ Tf, Tc was often observed to drop precipitously to zero, or in some cases to show a
"reentrant" behavior where at a temperature below Tc superconductivity was destroyed.
These anomalies were shown to be related to the interplay of the magnetic and super-
conductive ordering. Hence, a drop in To when Tc is expected to be near Tf could signal a
coexistent state. Second, the superconducting upper critical field Hc2(T) of the RE alloys
was found to drop in field for temperatures below Tf. Here Tf was measured at tempera-
tures below Tc using neutron difl‘raction or Mossbauer measurements, and was found to
correlate strongly with the observed decrease in Hc2(T). In CuMn, Tf is strongly
depressed and broadened by the application of magnetic fields larger than 1000 gauss.
Since Hc2(T) for Nb and Nb/CuX multilayers was found to be 10,000 to 50,000 gauss at
low temperatures, H62(T) experiments are unlikely to show evidence of coexistent
behavior in Nb/CuMn multilayers.

As discussed in the experimental procedures chapter, Tf was measured
magnetically while Tc was measured resistively. The problem here is that at temperatures
below Tc, superconducting surface currents form which shield the interior of the sample
from weak external fields, such as the Tf probe field. Therefore, Tf could not be measured
for temperatures below Tc. Because of this dilemma, we focused on the layer thickness
regime where Tf was expected to be greater than or equal to Tc. At temperatures above
Tc, superconducting fluctuations occur which can couple to the magnetic state.

Previous studies of Nb/CuMn(y) multilayers have shown that Tf decreases with

decreasing CuMn layer thickness similar to what was observed for Cu/CuMn multilayers.9

154

These studies also found that Tf is nearly independent of dNb for dNb greater than about 7
nm. Therefore, the search for changes in Tf in the vicinity of Tc were made for various
series having fixed CuMn layer thickness, and hence fixed Tf, and varying Nb layer thick-
ness. Any deviation observed would then almost necessarily be related to the super-
conducting state.

Figure 7.1 shows data for Tf and Tc vs. CuMn layer thickness at several values of
the Nb layer thickness. As was expected, the Tf values follow nearly the same general
curve independent of the Nb layer thickness used in the multilayers. The one exception to
this trend is the Nb/CuMn(l%) 28.0 nm/6.0 nrn sample. Tf for this sample is anomalously
high by about 0.7 K. Given that the uncertainty in Tf is 0.5 K, this shifi is just barely
significant.

T6 is also observed to decrease with decreasing CuMn layer thicknesses for dCuMn
< 20 nm. This implies that superconductivity is induced throughout the CuMn layers for
dCuMn < 20 nm. Note, however, that there were no deviations in the Tc data as T0 was
decreased to below Tf. This indicates that the spin-glass freezing transition has no effect
on Tc.

Figure 7.2 shows the same data as in Figure 7.1 replotted to display Tf and Tc as
functions of the Nb layer thickness at various constant values of the CuMn thickness.
Here, Tf was observed to be independent of dNb, except for the dCuMn = 6.0 nm series.
This series shows a slight increase in Tf in the layer thickness regime where one expects Tf
~ Tc. M(T) for the Nb/CuMn(l%) 40.0 nm/6.0 nm, 28.0 nm/6.0 nm and 20.0 nm/6.0 nm
samples are shown in Figure 7.3. The 40 nm sample shows that when Tf << Tc, there is no
rounding of M(T) above Tc, as occurred for the 28 nm sample. Hence, the apparent shift
in Tf of the 28.0 nm sample is not related to weak diamagnetism of the superconducting
state above Tc.

One possible explanation for this shift is that the CuMn layer thickness was larger

than expected. However, the CuMn layer thickness would have to be increased by about

155

30% to account for the observed change in Tf. The structural measurements of these
samples show that such a change did not occur.

Therefore, fluctuation superconductivity for T > TC may be causing an increase in
the effective spin-freezing temperature. In the present case, Tf was observed to increase
by 0.7 K, when Tc was increased to just below the expected value of Tf. Further experi-
ments are needed, however, to prove that fluctuation superconductivity, rather than some

systematic artifact, is the cause of the observed change in Tf.

156

 

 

 

 

12 ‘ I I ’ IFI'W'T'I‘I‘l'I‘I‘I I I ' I I ‘ ' I T I
g 1.1.
W -0—9- lanNb
c1.) 10 — H l5nmNb _
‘5 4—V— 20nmNb /£
.11.: F +4} 28nmNb ‘
2 8 — H 40nmNb 1
<0
2 i V -
a) 6 ~ ~ -
a . ©
a 'L 6
o 4~ . _
-—4
:31
m
1: 2_ i _
cu
L4 1 4
Ed

0 1 r 1 1 1 1 11.111111111111111 1 L 1 1 1 1 . 1 . 1 . 1.

2 3 4 5 6 7 a 101 2 3 4

(11mm (nm)

Figure 7.1: Tc (open symbols) and Tf (filled symbols) as firnctions of dCuMn for various
constant values of dNb.

157

 

  

 

 

 

12 I 1 W 1
E, Tf Tc
(0 10 _ J—EI- 20 nm CuMn -
id 4—47— 10 nm CuMn E
B 4—9- 6.0mm CuMn ‘
ca 8 _ +——A— 4.0 nm CuMn _
3 4—9— 3.0nm CuMn
E‘
111 6 ~ _
E
g 4 L _
.3
ca 1
g: _ _
m 2
5-4 1
E

dNb (nm)

Figure 7.2: Tc (open symbols) and Tf (filled symbols) as functions of dNb for various
constant values of dCuMn'

1.2

1.1

0.7

Magnetization (arb. units)

0.4

158

 

1.0
0.9
0.8

0.6
0.5

l

l

l

l

l

l

T

 

 

 

 

‘r fi fi fi I 1 ! I I l ' I I I l I

17.
17.
17.

—-+—— Nb/CuMn
—o— Nb/CuMn
—-—— Nb/CuMn

20nm/ 6.0nm
28nm/ 6.0nm
40nm/ 6.0nm

 

 

 

I _

 

 

L10112‘14l16l18120L22124

Temperature (K)

Figure 7.3: Zero-field-cooled magnetization data for three Nb/CuMn(1%) samples having
dCuMn = 6.0 nm. Tf is defined as the temperature at the peak of the zero-field-cooled

magnetization.

CHAPTER 8
CONCLUSIONS

In this thesis, samples consisting of alternate layers of Nb and either pure Cu or Cu
with a few percent Mn or Ge have been used to study the effects of magnetic interlayers
on the superconducting properties of proximity coupled multilayers.

Structural analyses of these multilayers have shown that, other than a decrease in
the mean crystallite size, there are no significant changes in the internal structure or the
uniformity of the layering as the Nb and CuX layer thicknesses are decreased. For the
thinnest Nb and CuX layers studied, there are some indications that their lattice spacings
are difl‘erent from those of bulk materials. However, the resolution of these experiments
was not high enough to tell for certain.

The superconducting experiments focused on studies of how the superconducting
transition temperature Tc and the superconducting upper critical field, ch depend on the
Nb layer thickness ds, the CuX layer thickness dn, the impurity concentration, and the
electron mean free path Zn.

Many of the effects observed in the behavior of Tc have been attributed to how the
proximity efi‘ect leads to an increase in the pair density in N, as well as a decrease in the
pair density in S. In N, the type and concentration of impurities determine how the pairs
are scattered. This leads to three limiting cases for a superconducting pair in N:

1. If 6,, > d“, electrons are not strongly scattered in N, and the probability for
superconducting pairs to cross N is high. This leads to strong superconductive
coupling between adjacent S layers and a Tc which approximately equal to the
effective bulk transition temperature of S.

2. If €11 < dn due to scattering from magnetic impurities, superconducting pairs are

broken as they travel in N. Hence, as the magnetic ion concentration is

159

160

increased, both the superconducting pair penetration depth and the super-
conducting transition temperature decrease.

3. If €11 < (1n and there are no magnetic impurities present, the superconducting

pairs are coherently backscattered in to the S layers. Therefore, as 6,,
decreases, the superconducting pair density in S increases, the pair penetration
depth in N decreases, and Tc is observed to increase.
All three of these behaviors have been observed and carefully studied in the Nb/CuX
samples.

For a given N layer thickness and impurity concentration, Tc is observed to scale
with ds. The scaling exponent was found to depend strongly on the N layer thickness and
the concentration of magnetic impurities, but was insensitive to changes in the concen-
tration of nonmagnetic impurities. While scaling was found to be consistent with the
limiting forms of two theoretical models, an additional dependence of the effective bulk
transition temperature of Nb on (18 had to be included for the models to quantitatively
predict the behavior of Tc for thin S and N layers.

The dependence of the upper critical field ch on the layer parameters and on the
relative orientation of the applied field and the sample have proven quite complex and
difficult to interpret. Both H62" and Hc21 are related to a set of superconducting coherence
lengths which are only partially related to one another.

The behavior of Hc21. is very similar to that observed for Tc. There were some
indications, however, that H.121 and Tc measure slightly different length scales in the N

layer. Scaling of H121 with ds, supported by previously published studies, was not well

established in Nb/CuX.

Hc2II(T) was observed to change from behavior indicative of a 3D coupled state
near To, to a 2D decoupled state at a temperature where E, is less than or equal to the
bilayer thickness, to a 'bulk' state at still lower temperatures where both EM and 1:, are

small compared to any dimension of the layers.

161

These Ha“ results were interpreted in terms of two qualitative decoupling models,
the temperature quench (TQ) model, and the field quench (F Q) model. The TQ model
linked the decoupling transition to the temperature dependence of :2. Therefore, it
predicted that crossover occurs even in zero applied magnetic field. In the F Q model, on
the other hand, the application of a magnetic field parallel to the layers was argued to
preferentially destroy the induced superconductivity in N, thereby decoupling the S layers.
Hence, the PO model predicts that in zero applied field, a 3D coupled state exists at all
temperatures. While the present data on Nb/CuX multilayers qualitatively support the
validity of the PO model over that of the TO model, more studies still need to be
performed.

As to the question of whether there are any significant interactions between the
proximity induced superconducting and spin-glass states, the answer appears to be
"possibly". While there are no significant deviations in T6 or Hc2(T) in the vicinity of the
spin freezing temperature Tf, a change in T11 due to fluctuation superconductivity above Tc

may have been found. The results, however, are very speculative at this point, and a much

more through study the spin-glass state in the vicinity of Tc is needed.

8.1 Future Directions of Study

Electron microscopy and electron diffraction studies of cross sections of multi-
layers having thin Nb and Cu X layers are needed to accurately determine if there are

lattice strains in either the Nb or CuX layers at small layer thicknesses.

In order to carefirlly study the dependence of Tc on dn in the limit that dn ——> 0,
experiments over a wider range of (18 are needed. These experiments should be able to
then determine if the dependence of Tc(dn—>O) on ds is due to broadening of the Nb density

of states or to some other structural effect.

162

In this same vein, studies of V/CuX multilayers would also be informative since V
does not have a density of states peak near the Fermi energy and so in not susceptible to a
change in TO due to decreasing 8“. Therefore, V/CuX multilayers would provide a more
accurate test of the validity of the various proximity effect models and of the occurrence

of a scaling of T6 with ds.

Annealing studies of Nb/CuX multilayers can also be used to determine if the
dependence of Tc(dn——)O) on (15 is related to structural disorder. Nb and Cu are mutually
immiscible, therefore annealing should lead to an improvement in their layering, an

increase in their in-plane crystallite size, and hence, an increase in ES.

The FQ model for dimensional crossover needs to be checked by critical current
measurements similar to those used by Koorevaar et al.59 In addition, there is the possi-
bility of examining the critical current of these multilayers using a current directed
perpendicular to the layers. Such a system has been recently developed by Dr. William
Pratt Jr. of Michigan State University to study the perpendicular resistance of a sample at
fields up to 5 kG.63 Such an experiment would provide a much cleaner test of the
decoupling phenomenon since the multilayered sample can then be treated as a set of resis-
tors in series rather than in parallel. Hence, one should be able to infer the resistance of

the center of the N layer which is driven normal by the application of the applied magnetic
field.

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