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THESIS

am

- LIBRARY 7
[l Michigan State
University I

This is to certify that the
dissertation entitled

Magnetic Scattering in Dilute AgFe Kondo Wires Below the
Kondo Temperature

presented by
Gassem Mohammad Alzoubi

has been accepted towards fulfillment
of the requirements for the

Ph.D degree in Physics and Astronomy

 

 

flmaw

Major Professor’s Signature

0% /?/ 2007

Date

MSU is an affirmative-action. equal-opportunity employer

 

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Magnetic Scattering in Dilute AgFe Kondo Wires Below the Kondo Temperature
By

Gassem Mohammad Alzoubi

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

2007

ABSTRACT

Magnetic Scattering in Dilute AgFe Kondo Wires Below the Kondo Temperature
By

Gassem Mohammad Alzoubi

The scattering of conduction electrons by magnetic impurities is known as the
Kondo effect. In metals, the resistivity normally decreases monotonically with de-
creasing temperature and reaches a constant residual value at zero temperature. In
some cases, a minimum was observed in the resistivity at low temperature. This
minimum was attributed to the presence of dilute magnetic impurities in the metal
host. In this thesis, we discuss the effect of scattering of conduction electrons by Fe
magnetic impurities in AgFe Kondo wires.

In a metal containing a small amount of magnetic impurities, the resistivity is
proportional to the total scattering rate, which increases with decreasing temperature
and then saturates at very low T, where the inelastic scattering events are frozen out.
The inelastic scattering rate, 7'; 1, is important for dephasing of electrons, which is
important in quantum transport and mesoscopic physics. Until very recently, there
was neither experimental data nor theoretical expression to describe the temperature
dependence of the inelastic scattering rate, due to dilute magnetic impurities, valid
for temperatures T not too far below the Kondo temperature, TK. In this work,
our goal is to measure the magnetic inelastic scattering rate over a broad range of
temperature, and to stimulate theoretical work in this direction.

In metals, the conduction electron can be scattered in two ways, elastically or
inelastically. It is well known that elastic scattering preserves the phase coherence,
since there is no energy exchange between the conduction electron and the scattering
center. On the other hand, the inelastic scattering destroys the phase coherence of

the conduction electron. At low temperature, there are three main sources for the

inelastic scattering in metals; the electron-electron, electron-phonon, and electron-
magnetic impurity interactions.

The weak localization magetoresistance (MR) method is used to determine the
value of the electron dephasing time 743. We have measured 1'4, for both types of
samples, pure and implanted samples. By subtracting the total dephasing rate of
the pure sample from the total dephasing rate of the implanted ones, we extract
the magnetic scattering rate, 7",, of the conduction electrons off Fe impurities. We
then compare our data with a recent theory of electron dephasing by dilute magnetic
impurities.

The high field magnetoresistance is used in this work to obtain an independent
estimate of the Kondo temperature, TK. At an intermediate field scale, the implanted
samples show a negative MR, whereas in pure films it is positive and proportional to
82. The negative MR observed in the implanted samples is due to the presence of Fe
magnetic impurities. From the high field MR data, we find that below 200 mK the
magnetoresistivity saturates towards its unitary limit (its value at T = 0), indicating
that the Fe spin is completely screened by the surrounding conduction electrons.
We then compare our high field data with the Numerical Renormalization Group
calculations of the MR for spin 1/2 magnetic impurities, provided to us by Theo

Costi.

To my family

iv

Acknowledgments

Finally, after being done with this thesis, it is time to thank many people who
have contributed directly or indirectly to this work.

I would like to start with a big thanks to my supervisor, Norman Birge, from whom
I learned many things during my stay here at Michigan State University. Norman, you
are amongst the very few people that truly impressed me, not only in the professional
point of view, but also as a person. I have always enjoyed our discussions in the
field of physics and in other aspects of life. I learned from you many things about
American culture and its values. I would like also to thank you for your financial
support and for being so approachable whenever I needed help.

I would like also to thank Achirn Rosch and T. A. Costi for providing us with the
results of their Numerical Renormalization Group calculations, which were crucial to
fit our experimental data for both phase coherence and high field magnetoresistance.

The encouragement I have received from my entire family is really appreciated. I
really have to thank all of them individually: my father, my mother, my brother Iyad,
who has dedicated his life to support our nuclear family, all of the rest of my brothers;
Mustafa, Ali, Abdelah, my sisters : Abeer, who took care of us when we were kids,
Amal and here husband, Jawaher, Amani and here husband, Amneh, Eman, Asma,
Asrah, my sister and brother I lost Tahani and Omar, and my step mother, Khadeeja.
Big thanks go to all my uncles, specifically my uncle Fawoor and his wife, relatives,
my ant Dalal and her husband, and my grandmother, who always wants me to finish
my PhD and come back as soon as I can. I would like also to thank my father and
mother in law for giving me the best gift of my life, my wife Basma. And finally I
would like to thank my lovely daughter Abeer for her nice interactions.

Special thanks go to Michael Crosser and Ion Moraru, who have taught me much
during the first two years of my work with Norman.

I would like also to thank Baokang Bi for teaching me how to use properly the

critical instruments in the clean room, and Reza Loloee, for his technical support and
for always being there to help. I also thank Prof. Pratt for the beneficial discussions
we had when I was helping him running the advanced lab, and Prof. Mahanti for
helping me understand the physics of local moments in metals and for teaching me
two advanced courses in condensed matter physics.

And finally, I would like to thank my best friend Mustafa Alhaj-Darweesh, the
one who recommended me to the graduate chairperson to be accepted here at MSU.
Mustafa, you are the best friend of mine so far. I will never forget the best period of

my life, the period we lived together for three years in Cherry Lane.

vi

Contents

Introduction
1.1 Introduction ................................
1.1.1 Transport in metals ........................
1.1.2 Scattering mechanisms and phase coherence in metals .....
1.2 Magnetic impurities and Kondo effect ..................
1.2.1 Early observations of resistivity minimum and Kondo problem
1.2.2 Solution of the Kondo problem .................
1.2.3 Previous measurements of Kondo contribution to decoherence
rate ................................
1.3 Motivation for current study .......................
1.4 Outline of thesis ..............................

Theory I: Quantum Transport and Phase Coherence
2.1 Mesosc0pic physics and length scales ..................
2.2 Electron weak localization ........................
2.2.1 Weak localization in disordered metals .............
2.2.2 Spin-Orbit scattering (weak antilocalization) ..........
2.2.3 Low-field magnetoresistance ...................
2.3 Decoherence mechanisms in metals ...................
2.3.1 Electron-Electron scattering contribution to decoherence rate .
2.3.2 Electron-Phonon scattering contribution to decoherence rate .
2.3.3 Electron-Magnetic impurity scattering contribution to decoher-
ence rate ..............................
2.4 Electron-Electron interaction contribution to resistivity at low T . . .

Theory II: Kondo Scattering in Dilute Magnetic Alloys
3.1 Magnetic moments in metals .......................
3.2 The Anderson model ...........................
3.3 The s-d exchange model .........................
3.4 The Kondo model .............................
3.5 Adding the decoherence rates ......................
3.6 The theories of Zarand and Micklitz (Numerical Renormalization Group
technique) .................................
3.7 Impurity-Impurity interaction ......................

vii

30
30

33
33
34
39
43
47

49
53

4 Experimental Methods 55

4.1 Introduction ................................ 55
4.2 Sample fabrication ............................ 55
4.2.1 Wafer processing ......................... 55

4.2.2 Electron beam lithography; preparing the electron microscope
for writing ............................. 57
4.2.3 pattern development, evaporation and lift-off .......... 63
4.2.4 Ion implantation ......................... 66
4.3 Measurement methods .......................... 69
4.3.1 Mounting and testing samples at room temperature ...... 69
4.3.2 H e3 — H e4 Dilution refrigeration ................ 72
4.3.3 Sample characterization ..................... 73
4.3.4 Low field magnetoresistance ................... 73
5 Results and Discussions I: Phase Coherence 80
5.1 Introduction ................................ 80
5.2 Weak localization and phase coherence ................. 80
5.2.1 Total dephasing rate : pure samples ............... 81
5.2.2 Total dephasing rate : 2 ppm sample .............. 88
5.2.3 Total dephasing rate : 10 ppm sample ............. 89
5.3 Extracting the magnetic scattering rate from the total dephasing rate 92
5.4 Fitting the magnetic scattering rate to theory of Micklitz et al. . . . . 95
6 Results and Discussions II: High Field Magnetoresistance 99
6.1 Introduction ................................ 99
6.2 Resistivity vs temperature ........................ 99
6.3 High field magnetoresistance and NRG calculations .......... 105
6.3.1 High field MR data ........................ 105
6.3.2 Theory of Costi for high field MR in dilute Kondo alloys . . . 110
6.3.3 Fitting the 83 mK high field data set to theory of Costi . . . . 113
6.3.4 Fitting the high temperature MR data sets to theory of Costi 114
7 Conclusions 119
7.1 Overview .................................. 119
7.2 Summary of Results ........................... 120
7.2.1 Phase Coherence Results ..................... 120
7.2.2 High Field Results ........................ 121
7.3 Future work ................................ 122
REFERENCES 123

viii

List of Figures

1.1
1.2
1.3

1.4
1.5

2.1
2.2
2.3

3.1

3.2
3.3

4.1
4.2

4.3
4.4
4.5
4.6
4.7
4.8

4.9
5.1

5.2

Schematic of resistivity of metals as a function of temperature . . . . 4
Schematic of the energy configurations of two spin 1/2 system 8
Schematic of resistivity vs temperature for metal containing magnetic

impurities ................................. 10
Schematic of scattering rates in pure and diluted metals ........ 13
Schematic of the theoretical situation of scattering rates as a function

of temperature (at the time we started this work) ........... 15
Schematic of the origin of weak localization with two time-reversed paths 21

Weak localization and spin orbit coupling ................ 24
Schematic of the contribution of electron-electron interactions to resis-
tivity at low temperatures ........................ 32
Schematic of Anderson model for metal host containing one impurity

level .................................... 36
Schematic of spin-flip and non spin-flip processes ........... 41
Magnetic scattering rate due to diluted magnetic impurities embedded

in weakly disordered 1D-metal host calculated via NRG ........ 51
Schematic showing the basic steps in EBL ................ 58
Schematic diagram showing the basic components of the J EOL micro-

scope column ................................ 60
Optical microscope picture of the pattern after development. ..... 64
SEM picture of silver wire after evaporation and lift-off ......... 67
The distribution of the implanted Fe Ions in the Ag target ....... 69
Schematic diagram showing the sample holder and the circuit used to

test samples at room temperature. ................... 70
Schematic diagram showing the circuit used to measure the change in

the resistance due to weak localization. ..... h ............ 75
Schematic diagram showing the circuit used to generate magnetic fields

in the z-direction .............................. 78
Typical magnetoresistance trace of the pure sample taken at 1.6 K. . 79

Magnetoresistance data of the puresamplel measured in the dilution
refrigerator. ................................ 82
Magnetoresistance data of the puresample2 measured in the 1K-cryostat. 83

ix

5.3

5.4

5.5

5.6

5.7

5.8

5.9

5.10

5.11

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

6.10

6.11

6.12

The total dephasing rate, '74,, as a function of temperature for both
puresamplel and puresample2 with the fit to the expression 7¢(T) =
AT‘V3 + BT3 ...............................
The total dephasing rate, 7,1,, as a function of temperature for both
puresamplel and puresample2 with the fit to the expression 7¢(T) =
AT2/3 + 3T3 + 7sf(T) ..........................
Magnetoresistance data of the 2 ppm sample measured in the dilution
refrigerator. ................................
The total dephasing rate, ’Y¢,, as a function of temperature for the 2
ppm sample .................................
Magnetoresistance data of the 10 ppm sample measured in the dilution
refrigerator. ................................
Magnetoresistance data of the 10 ppm sample measured in the 1K-
cryostat ...................................
The total dephasing rate, 7,5, as a function of temperature for the 10
ppm sample .................................
The total dephasing rates, 74,, and the magnetic scattering rates as a
function of temperature for all samples ..................
Magnetic scattering rate as a function of temperature for the implanted
samples. ..................................

Resistivity vs temperature of puresample2 measured at a finite mag-
netic field of 0.1T. ............................
Resistivity vs temperature for 10 ppm Fe magnetic impurities in Ag
measured at a finite magnetic field of 0.1T ................
The e-e and magnetic impurities contributions to the resistivity for the

86

87

89

90

91

92

93

94

97

100

101

10 ppm sample measured in the dilution refrigerator between 40 K - 4 K.102

The Kondo contribution to resistivity for 10 ppm Fe magnetic impuri-
ties in Ag measured in the dilution refrigerator between 40 K - 4 K. .
High field magnetoresistance of a pure sample measured in the dilution
refrigerator. ................................
High field magnetoresistance for 10 ppm Fe impurities in Ag measured
in the dilution refrigerator .........................
Classical magnetoresistance for 10 ppm Fe impurities in Ag measured
in the dilution refrigerator at 10 K. ...................
High field magnetoresistance for 10 ppm Fe impurities in Ag measured
in the dilution refrigerator at 83 mK. ..................
The spin up spectral density function, A, (0, 0, B), vs magnetic field
B calculated at T = 0. ..........................
Kondo contribution to magnetoresistivity for 10 ppm Fe impurities in
Ag measured in the dilution refrigerator at 83 mK. ..........
Kondo contribution to magnetoresistivity for 10 ppm Fe impurities in
Ag with fits to theory of Costi ......................
Magnetoresistance of a system of magnetic impurities calculated using
the Brillouin function. ..........................

103

106

108

109

110

112

114

115

List of Tables

4.1 The parameters used to program the dicing Saw to cut 4 inch Silicon

wafer into 5 mm squares ......................... 56
4.2 Area dose calculations of the implanted ions .............. 68
4.3 Geometrical and electrical characteristics of the measured samples at

1.2 K. ................................... 74

Chapter 1

Introduction

1.1 Introduction

The electron decoherence time (also called dephasing time), 14,, is a quantity of fun-
damental interest in mesoscopic systems. This time scale has been the subject of the-
oretical and experimental investigations for several decades. In mesoscopic physics,
7}), is defined by the specific measurements one performs. In the context of quantum-
transport phenomena such as weak localization, 7'4, determines the energy and length
scales at which quantum behavior is seen. In the last two decades, many theoretical
and experimental works have been directed toward understanding the mechanisms
responsible for losing phase coherence in mesoscopic systems.

The phase coherence time, 74,, depends on temperature, disorder, and dimension-
ality. In quasi-1D wires, 7,, was predicted to have the form 7'4, 0t T"2/3 [1]. That
prediction was proven experimentally on Al wires (at temperatures down to 2 K)
in 1986 [2] and on Au wires (down to 100 mK) in 1993 [3]. In 1997, Mohanty et
al. [4] reported measurements of T¢ in six Au wires. In their study, they showed
that 71,, saturated at low temperature, inconsistent with the theoretical prediction
mentioned above, where 73;, diverges with decreasing temperature. They attributed

this saturation in T,» to an intrinsic unavoidable source of decoherence, coming from

electron-electron interactions.

In 2003, Pierre et al. [5] ruled out the universal dephasing proposed by Mohanty
et al.. They reported measurements of 7'43 for five samples fabricated from Ag and
Au source material of 99.9999% purity. In those samples 7'4, did not saturate, but
rather continued to increase down to 40 mK. Nevertheless, Pierre et al. found small
saturation in some pure samples below 100 mK. They speculated that this saturation
was caused by small amounts of magnetic impurities with low Kondo temperature.
This debate of the zero-temperature dephasing has remained a puzzle, which attracted
many theoretical and experimental works over the last decade [4, 5, 6, 7, 8, 9, 10, 11,
12]. Since then, people want to know what causes the saturation of 7'4, at very low
T. One result of that work is that the scattering of conduction electrons by magnetic
impurities play a major rule in electronic transport at low T. In this work, I will
discuss the temperature dependence of 73,, and resistivity of weakly disordered quasi-
1D Ag wires that are intentionally implanted with Fe impurities. Ion implantation
of the magnetic impurities allows us to study the Kondo physics in samples with a

known concentration of impurities.

1.1.1 Transport in metals

The simple classical model that describes transport in metals is the Drude model.
This model of electronic conduction was developed in the 19003 by Paul Drude to
explain the transport properties of electrons in materials (especially metals). The
Drude model is the application of kinetic theory to electrons in a solid. It assumes
that the metal contains immobile positive ions and an electron gas of classical, non-
interacting electrons of density n, each of whose motion is damped by a frictional

force, due to collisions of the electrons with various scattering sites. These collisions

lead to a finite resistivity which can be expressed as [13]

m

 

p = (1-1)

ne27'e

where m is the electron mass, 6 is the charge, 72 is the electron density, and Te is the
average time between collisions. The quantity, T8, is related to the mean free path 1..
by 16 = vp're, where vp is the Fermi velocity.

The electrical resistivity of most metals is dominated at room temperature (300 K)
by collisions of the conduction electrons with lattice vibrations (phonons) and at
liquid helium temperature (4.2 K) by collisions with static disorder such as grain
boundaries, impurities, and lattice defects. Assuming that the two rates of these
collisions are independent, the total resistivity could be represented by the sum of

the two contributions (Matthiessen’s rule) [13]

ptotal = P0 + pphonon (12)

where p0 is the temperature independent contribution due to static impurities and
boundary scattering and the second term pphmon is the contribution due to phonons,
which is temperature dependent. For temperature much less than the Debye temper-
ature, ppm“, takes the form pphonon = aT5, where a is constant.

In metals, as seen from equation (1.2), the resistivity normally decreases mono-
tonically with decreasing temperature and reaches a constant residual value at zero
temperature (see figure 1.1). This general picture of low temperature transport is in
good agreement with most experiments done on very pure metals. In some cases, a
minimum was observed in the resistivity at low temperature, as shown in figure (1.1).
This was attributed to the presence of magnetic impurities and will be discussed in

section (1.2).

 

p (arb.units)

 

‘O
o

l

pure metal

 

 

 

Tmin T

Figure 1.1: Schematic of resistivity vs temperature for pure metal (lower curve) and
metal containing residual magnetic impurities (upper curve). The minimum in the
upper curve is caused by the presence of magnetic impurities in a metal host.

1.1.2 Scattering mechanisms and phase coherence in metals

The conduction electron can be scattered in two ways, elastically or inelastically.
It is well known that elastic scattering preserves phase coherence, since there is no
energy exchange between the conduction electron and the scattering center. Typical
elastic scattering events are scattering of conduction electron by static impurity atoms
or film boundaries. To a good approximation, the elastic scattering is temperature
independent and depends only on the amount of disorder in the system. The amount
of disorder in a system is usually measured by the diffusion constant D or alternatively
by the mean free path la. The higher the value of D ( or the longer la) is , the less
the amount of disorder.

On the other hand, the inelastic scattering destroys the phase coherence of the

conduction electron due to the energy exchange between the conduction electron and
the scattering center. There are three main sources for the low temperature dephasing
in metals; the electron-electron , electron-phonon, and electron-magnetic impurity
interactions. It is known that the above three dephasing sources may coexist in real
systems, with one or two sources typically dominating, depending on the system
dimensionality, the amount of disorder, and the measurement temperature.

In many cases, quantum interference phenomena are observed in weakly disordered
systems, in which electrons are able to undergo multiple elastic scattering at low
temperatures, before the coherence of their wave function is randomized. The electron
dephasing time 11,, (also called phase breaking time) depends on both disorder and
temperature. In the diffusive regime, the quantity 1‘4, is related to the dephasing
length L4, (also called phase breaking length) by L¢ = \/D—'r¢,. The electron diffusion
constant is given by D = vfi're/d, where Up is the Fermi velocity, Te is the elastic mean
free time, and d is the effective dimensionality of the system under study. Usually,

L,» can be several microns at liquid-helium temperatures in disordered metals.

1.2 Magnetic impurities and Kondo effect

When the spin degree of freedom of both the conduction electrons and the magnetic
impurity atoms is taken into consideration, the quantum transport effects become
sensitive to the magnetic spin-spin scattering. The spin-spin scattering can originate
from scattering of conduction electrons off magnetic impurity atoms. The scattering
of conduction electrons by magnetic impurity is know as Kondo effect. In the following

three subsections, I will discuss briefly the history of this effect.

1.2.1 Early observations of resistivity minimum and Kondo

problem

The minimum in resistivity of some metals as a function of temperature was first
observed in 1934 in 3. Au sample[14]. Other observations [15], with a controlled
concentrations of magnetic Fe impurities in Cu samples, have shown that this effect
was related to the presence of magnetic impurities in the non-magnetic host metal.

It was not until 1964 that this minimum was shown by J. Kondo [16] to arise
from some unexpected features of the scattering of conduction electrons that arise
only when the scattering center has a magnetic moment. In such a case the exchange
interaction between the conduction electrons and the local moment leads to scat-
tering events in which the electronic spin is flipped (with a compensation change of
spin on the local moment). The essential facts that Kondo proposed to explain the
resistivity minimum are as follows [16]. First, the low temperature anomalies result
from magnetic impurities. Second, the depth of the minimum (Ap = pT=0 — pm...)
is proportional to the impurity concentration. Third, the value of pm," itself is also
proportional to the impurity concentration; therefore Ap/pmin is concentration inde-
pendent. This is true only if the magnetic impurities dominate the residual resistivity.
Fourth, the temperature of the resistivity minimum is almost independent of the im-
purity concentration.

The Kondo Hamiltonian takes the form
H = H0 + Had (1.3)

where H0 is the unperturbed Hamiltonian of the conduction electrons and the pertur-
bation H ,d describes the interactions between the local moment and the conduction

electrons, which is given by

H... = — Z .1515 (1.4)

where J is the coupling constant, de is the spin of the localized moment and g is the
conduction electron spin.

Let us first discuss the physics of two spins S1 and S; interacting via the Hamilto-
nian H = -J.571.S-'.2, where J is the interaction strength. The total spin is S = 87.1 +3.2,

and using S2 = $2 + S2 + 2.571.572, the total energy can be written as
1 2

E = —-:—[S(S +1) — 51(31 +1) — 52(32 +1)] (1.5)

consider the special case 51 = 82 = 1 / 2, the total energy becomes
J 3
E = —§[S(S + 1) — 2] (1.6)

Now for two a spin 1 / 2 system, the total spin could be 0 (singlet) or 1 (triplet). In
the singlet state, the spins favor antiparallel alignment, therefore it is non magnetic,
while in the triplet state the spins favor parallel alignment, and hence it is magnetic.
The energy difference between parallel and antiparallel configurations is the exchange

energy, J. So from equation (1.6) we have

§J ifS=0
—-j- ifS=1

From this result, one can see that if J < 0, the ground state is an antiferromagnetic
singlet. On the other hand, if J > 0, the ground state is a ferromagnetic triplet
(see figure 1.2). Kondo effect is relevant to the first case, J < 0, where the coupling
between the conduction electron spin and the local moment spin is antiferromagnetic.
The other limit J > 0, for ferromagnetic coupling, the perturbation theory converges
and the physics is much simpler.

In the first Born approximation (second order in J) it had been shown that Hsd

Ex.S (s=1) —-l—l—— F (E= J/4) (3:0) -—[-l-— AF (E= J/4)

G.S (s=0) __l-l__ AF (E: -3J/4) (3:1) —]—l—- F (E= -3J/4)
J<0 J>o
Gound state is antiferromagnetic (AF) Ground state is ferromagnetic (F)
(Kondo effect)

Figure 1.2: Schematic of the energy configurations of two spin 1/2 system. Left :
energy levels of the two spins if they are coupled antiferromagnetically, J < 0. Right
: energy levels of the two spins if they are coupled ferromagnetically, J > 0.

does not lead to any anomaly in the resistivity, merely a dependence on J 2. Kondo,
however, calculated the resistivity in the second Born approximation (third order
in J) and found, for a negative J, a low-temperature correction to the resistivity
proportional to —cp110g(T), where c is the concentration of magnetic impurities and
p1 is constant. The negative value of J means that the ground state of the conduction
electron and the localized moment is a singlet, which favors antiparallel alignments (i.e
antiferromagnetic coupling). When this new term is added to the other contributions

to the resistivity, we find

k T
p = Cimp pa + (1T5 — Cimp p1 log (-%-) (1.7)

with amp = Ns/N, where N, is the number of magnetic impurities and N is the
number of sites, and D = 25;: is the bandwidth, where the electronic density of states
(DOS) was assumed to be constant. By setting gg = 0, we have
Tmin = (fl)l/5Cl/5 (18)
5a ""1”
it is easily seen that equation (1.7) has a minimum, and thus Kondo was able to

explain the resistivity minimum.

Kondo’s theory was successful in explaining the resistivity minimum. But looking
carefully at equation (1.7), one can see that as T —> 0, the resistivity correction di-
verges. This divergence in Kondo’s theory is termed as the ”Kondo problem”. On the
other hand, the experimental data show that the resistivity increases logarithmically
down to some temperature, but it then levels off at some point as shown schematically
in figure (1.3). The temperature at which the resistivity is observed to deviate from
equation (1.7) is referred to as Kondo temperature, TX, and in term of the exchange

constant, J, can be written as [17]

1

|J|N(EF“)) (1'9)

TK=TF exp(—

where TF is the Fermi temperature of the host and N (Ep) is the conduction electron
density of states at the Fermi level. The characteristic temperature TK varies wildly
from 300 K for Vanadium in Au to 40 mK for Mn in Au or Ag [17]. For a compre-
hensive review about Kondo effect and related topics, I refer the reader to reference

[18].

1.2.2 Solution of the Kondo problem

In the late 603 Anderson was trying to solve the Kondo problem at T < TK using the
ideas of scaling. He showed that at temperatures far below the Kondo temperature
(T << T K), the effective interaction strength between the conduction electrons and
local moments becomes infinitely large [19]. Thus as T —) 0, the conduction electrons
become localized around the moment forming a spin compensated (i.e nonmagnetic)
ground state. Quantitatively, Anderson’s approach broke down at low temperatures
(T << TK) because of the infinite coupling between the conduction electrons and the
local moment.

In 1975 [20], Wilson solved the Kondo problem using the numerical renormaliza-

 

unitary limit

 

p (arb.units)

 

min

 

 

 

T

Figure 1.3: Schematic of resistivity vs temperature for metal containing magnetic
impurities. At high T, the resistivity is dominated by electron-phonon scattering; at
low T by electron-magnetic impurity scattering; and at temperatures far below TK,
the magnetic impurity is completely screened and behaves like static impurity.

tion group technique and ruled out the zero-temperature divergence of the resistivity
for S = 1 / 2 magnetic impurity. He showed that at temperatures far below the Kondo
temperature (T < TK), the spin of the magnetic impurity is totally screened by the
conduction electrons. For conduction electrons, the screened impurity appears then
as a potential scattering center. Using this non-perturbative approach, Wilson was
able to predict the low temperature thermodynamic behavior of dilute Kondo alloys.
For the resistivity at T < TK, the predicted form is [18]

-(T)— 1—7r4 2 TY (110)
pimp —,00 16w TK '

 

10

where w as 0.4128 is Wilson’s number, and p0 corresponds to the unitary limit, which

is given by
4m
m—QWE%NEB

(1.11)
with m and N (EF) being the electron mass and the conduction electron total density
of states at the Fermi level, and cimp = N8 / N = n, / n is the concentration of magnetic
impurities. The unitary limit is the value of the electrical resistivity at T = 0. This
value is related to the total scattering cross section at T = 0. At zero temperature,
the conduction electron scatters only elastically (both energy and spin unchanged)
with a scattering cross section 08 = 47r/kfiw, where [CF is the Fermi wave number of the

host. The elastic scattering is due to scattering of the conduction electron by static

disorder with a mean free path L. = 1/noe. The residual resistivity then reads

m

 

po — (1.12)

71827“e

where Te is the elastic mean free time. Using le = vae and 19;: = hkp, where up and

pp are the Fermi velocity and Fermi momentum, p0 becomes

47rfi

=.____ 113
esz ( )

P0

using N (e p) = mkp/W2h2 in the last equation, we arrive back to equation (1.11).
After this great work of Wilson, the Kondo problem was considered to be solved.
His contribution was recognized later in the award of the Nobel prize in 1982. It is
time here to mention that all these models developed by Kondo, Anderson, Wilson,
and others are valid only for S = 1/2 magnetic impurities. For S > 1 / 2, the problem

becomes more complicated.

11

1.2.3 Previous measurements of Kondo contribution to de-

coherence rate

In a pure but disordered quasi-1D metal, the decoherence scattering rate of the con-

duction electrons is given by [1, 11]
7,,(T) = AT2/3 + 3T3 (1.14)

where the first term describes the electron-electron scattering rate, dominating at
lower T, and the second term describes the electron-phonon scattering rate, dominat-
ing at higher T. In a metal containing a small amount of magnetic impurities, there
is an additional contribution to 7¢(T) from the scattering of the conduction electrons
by the magnetic impurities, 7m(T). Experimentally, the temperature dependence of
the magnetic scattering rate, 7m(T), is usually obtained by subtraction of 7¢,pu,.e(T)
values for the pure sample from 7¢,W(T) values for the doped sample as shown
schematically in figure (1.4). The figure shows that 7m(T) peaks at T = TX. The
magnetic scattering rate, 7m, is sometimes called the spin-flip scattering rate, 73f. In
this thesis I will use both terms interchangeably.

The magnetic impurity contribution to the conduction electron phase decoherence
rate, 7.;1, was first measured explicitly by two groups in 1987 [21, 22]. C. Van
Haesendonck et al. [21] measured the spin-flip scattering rate, 73f(T), in very thin
Cu films (t a: 5 nm), doped with magnetic Cr atoms (with two concentrations of
13, and 40 ppm, respectively). They found a maximum in 78f(T) which is in a good
agreement with the Suhl-Nagaoka approximation (see below) near the vicinity of the
Kondo temperature TX 2 2 K. In their work, no conclusions can be drawn about
the behavior of 7,f(T) below the Kondo temperature, since they only have four data
points below TK down to 0.5 K.

R. P. Peters et al. [22] measured the spin-flip scattering rate, 73f(T), in Au films

12

 

y¢ (arb.units)

 

 

 

 

Log(T/TK)

Figure 1.4: Schematic of the total scattering rates as a function of temperature for
pure metal (dashed line) and metal containing magnetic impurities (dashed-dotted
line). The solid line is the magnetic scattering rate, cym(T), which is the difference
between the other two. At temperatures far below TK, 7m(T) vanishes, indicating
that the magnetic impurity is completely screened and behaves like static impurity.

containing Fe magnetic impurities in the range between 0.08 and 4 K. They found a
strong temperature dependence of 73f(T) with a broad maximum at the characteristic
temperature TX 2 1 K. The same system, AuFe, has been studied by P. Mohanty et
al. [23] in 2000 and F. Schopfer et al. [24] in 2003, but this time in quasi-1D wires.
They both found a maximum near 0.3 K. Again, because of the relatively low Kondo

temperature of this system, it was hard to explore the regime where T < TX.

13

1.3 Motivation for current study

The inelastic scattering problem of the conduction electrons in a pure metal was
completely solved by Altshuler-Aronov—Khmelntzky (AAK) [1] theory, where there
are only two sources of the decoherence, electron-phonon and electron-electron. But
adding a small a mount of magnetic impurities to the pure metal makes things com-
plicated and until very recently, no complete solution was available for the complete
temperature dependence of the magnetic scattering rate. At T > TX, the experimen-

tal data could be described well by the Suhl-Nagaoka expression [17, 21].

i _ n, «25(5 +1) (115)
1,, " win/F «25(3 + 1) + ln2(T/TK) '

 

with S, TX, and up respectively the spin and the Kondo Temperature of the magnetic
impurity, and the total DOS of the conduction electrons at the Fermi level. On the
other hand, for (T << T K), Fermi liquid theory predicts a T2 dependence of 7m [25],

which, however, has never been observed experimentally. This takes the form [26]

 

1 n 2

:37 cc hr; (%) (1.16)
At the time we started this work, there was neither experimental data nor theory
to describe the intermediate regime for T not too far below TK (see figure 1.5). Our
goal was to measure the magnetic scattering rate over a broad range of temperature,

and to stimulate theoretical work on the problem.
While we carried out our experiments, a theory [27] was proposed to explain the
complete temperature dependence of 7m(T) for conduction electrons scattered by di-
lute spin- 1/2 magnetic impurities. The theory relates 7m(T) to an earlier calculation

[28] of the inelastic scattering cross section, aim; (w), by the numerical renormal-

ization group method. The main conclusion of these works is that the magnetic

14

 

Suhl-Nagaoka

sf

7 (arb.units)

   

— Fermi-liquid

 

 

1E-3 0.01 0.1 1 10 100

 

Figure 1.5: Schematic of the theoretical situation of the magnetic scattering rate
as a function of temperature (at the time we started this work). At T > TX, the
experimental data could be described well by the Suhl-Nagaoka expression. For T <
TK, Fermi liquid theory predicts a T2 dependence of 7m, which falls far below the
Suhl—Nagaoka expression.

scattering rate, 7m(T), has only a very weak (logarithmic) temperature dependence
above TX, has a broad maximum around TK, scales approximately linearly with T
for 0.1TK < T < TX, and scales as ~ T2 for T < 0.1TK. These theories can be
considered as a major breakthrough for the decoherence problem, as they allow one
to compare experimental data with theoretical results for all temperatures, ranging
from well above TK down to T < TK.

In addition to the main conclusion, mentioned above, the theory of Micklitz et
al. [27] predicts that if the density n, of magnetic impurities is sufficiently low, the

magnetic scattering rate, 7m(T), is a universal function of temperature, 7m(T) o<

15

n, f (T/ TX), and depends only on two parameters, 718 and TX.

What is the best candidate Kondo system to start with? In the presence of
magnetic impurities 7'51, contains the additional contribution 7m, which peaks at
T = TK. In order to observe this peak in 7m while keeping the magnetic impurity
concentration low enough to avoid interactions between impurities, one must choose
a system with TK below about 10 K; otherwise T¢_l is dominated by electron-phonon
scattering. In order to acquire data far below TK, however, it is important to keep
TK as high as possible. The optimal range for TX is a few Kelvins, which is achieved

with Fe impurities in Ag [17, 29].

1.4 Outline of thesis

The outline of this thesis is as follows. Chapter 2 covers the theoretical techniques
necessary to understand the physics of phase decoherence in pure metals. It includes a
brief description of length scales related to our experiment, electron weak localization,
spin-orbit coupling, and decoherence mechanisms in pure metals.

Chapter 3 describes the Kondo effect in dilute magnetic alloys. Here we give more
details about the scattering of conduction electrons from magnetic impurities.

Chapter 4 discusses the experimental techniques used. First, fabrication of the
samples will be presented. This includes wafer processing, e-beam writing and devel-
opment, evaporation and lift-off, and Ion implantation. Second, the methods used to
make measurements on these samples will be shown.

Chapters 5 talks about our phase coherence results for both pure and implanted
samples. Specifically, it describes how L4,, and hence 7'45, can be determined from mag-
netoresistance measurements. Then, we show how the magnetic scattering rateam,
due to dilute magnetic impurities, can be extracted from the measured total dephas-
ing rates, 7,». And finally, we compare our experimental 7m data with recent theory

[27] of scattering of conduction electrons by dilute magnetic impurities in quasi-1D

16

wires.

Chapters 6 discusses the high field magnetoresistance measurements for both pure
and implanted samples. Specifically, it describes how the Kondo temperature, TK can
be determined independently from analyzing the high field magnetoresistance data in
the context of Numerical Renormalization Group (N RG) calculations.

Chapter 7 summarizes this work.

17

Chapter 2

Theory I: Quantum Transport and

Phase Coherence

2.1 Mesosc0pic physics and length scales

Mesosc0pic physics refers to systems which are somewhere in between macroscopic
(i.e. classical) and microscopic (i.e. quantum). Macroscopic systems are considered
big enough to be described well by classical mechanics, whereas mesoscopic systems
are small enough such that their properties can be described quantum mechanically.
In a macroscopic system, any physical quantity can be represented by its average,
where fluctuations around the average are negligibly small. On the other hand, the
fluctuations in a mesoscopic system become important and sometimes they can be
as big as the average itself. In the mesoscopic regime, scattering from static disorder
induces quantum interference effects which affect the flow of electrons. A typical
example of these quantum effects is the electron weak localization which I will discuss
in the next section. For more details about mesoscopic physics, I refer the reader to
references [30, 31]. The length scales play a major rule in mesoscopic physics. Next,
I will briefly define some of the length, energy, and time scales that are relevant to

this work. The system dimensions will be denoted by L, w, and t, where L, w, and

18

tare

the length, the width, and the thickness, respectively. Below I list the most

commonly used length, time, and energy scales:

/\F

: Fermi wavelength; wavelength of electrons that carry electrical transport. The

corresponding energy scale is the Fermi energy, Ep, which is the energy of the
highest occupied electron energy level. In metals, typical values of AF and EF

are of order several A and several eV, respectively.

1., : electron elastic mean free path; the distance an electron travels before it is elas-

Te—e

Te—ph

Tsfi

tically scattered by static disorder.The corresponding time scale is the electron
mean free time, Te. le and Te are related by le = vp're, where 1),: is the Fermi
velocity. I8 is considered temperature independent and depends only on the
amount of disorder in the system. In weakly disordered metals, la is of order 50

nm.

electron phase coherence length; the distance an electron travels before its
phase memory is lost due to inelastic scattering event.The corresponding time
scale is the phase coherence time, T45. L, and T¢ are related by L¢ = \fD—Td”
where D = rural,3 / 3 is the diffusion constant. Both L4, and 7,), are temperature

dependent. In metals, L, can be several microns at liquid He temperature.

: electron-electron scattering time; the time it takes for an electron to be scat-

tered inelastically by another electron. This time scale is temperature depen-

dent. At T 2 1 K, 76-8 can be around 10 ns in pure metals.

: electron-phonon scattering time; the time it takes for an electron to be scat—
tered inelastically by phonon. This time scale is temperature dependent. At

T m 1 K, “re-” can be around 30 us in pure metals.

the spin-flip scattering time; the time between scattering events involving an

electron and a magnetic impurity such that the electron’s spin is flipped. This

19

time scale is temperature dependent; it decreases near the Kondo temperature,
then increases with reducing temperature. It also depends on concentration of

magnetic impurities.

kBT : electron thermal energy; at room temperature, kBT z 26 meV. At 40 mK,

kBT R: 3.4 peV.

By comparing the mean free path 1., with one of the system dimensions, say L,
one can discriminate between two regimes of transport, diffusive and ballistic. The
diffusive regime is defined by AF << l3 << L. On the other hand, le > L defines the

ballistic regime.

2.2 Electron weak localization

2.2.1 Weak localization in disordered metals

Weak localization (WL) is a quantum effect, which occurs in disordered electronic
systems at very low temperatures. In weakly disordered systems (kple >> 1) and at
low temperatures, this effect gives rise to higher resistivity than that predicted by the
Drude model due to the enhanced backscattering of the electron partial waves. This
effect is referred to as weak localization to distinguish it from the strong localization
that occurs in highly disordered systems.

In weakly disordered metals, the electron motion is diffusive rather than ballistic.
That means an electron does not move along a straight line, but experiences a series of
random scatterings off static impurities which results in a random walk. Consider two
electron partial waves (1, 2) starting at the origin and traveling diffusively along two
different paths in the opposite directions and finally arriving back to the origin (see
figure 2.1). In principle there are many paths the two partial waves can follow, but

for simplicity we will take only two of them. Along the paths, the two partial waves

20

 

 

 

¢

Figure 2.1: Two electron partial waves starting at the origin, moving diffusively along
two time-reversed paths, and arriving back to origin coherently. The size of the path
is of order L¢. The crosses are the static impurities.

are scattered elastically by static impurities. Let A1 = [Allei‘f’1 and A2 = [Agle'f’
be the wave functions of the two partial waves, where (1)1 and (112 are their phases.
Classically, the total return probability is the sum of the individual probabilities along
the two paths (Wdassm) = [All2 + [A2]?) However, quantum mechanically, to find
the total probability we have to sum up the quantum-mechanical amplitudes of the
two paths rather than the corresponding probabilities. Therefore, the correct formula
for return probability, Wmntum, includes the classical part (individual probabilities

of the two diffusive paths) and the interference term (product of the two amplitudes

21

corresponding to the two different paths). This can be written as [30, 31]

W = |A1 + A212
= (A1 + A2)( 1+ A3)

= 1.41? + |.42|2 + 2|A1||A2|cos(¢1-— a.) (2.1)

The first two terms are the classical probability and the third one is the quantum
interference contribution to the backscattering. In the absence of magnetic field, spin-
orbit scattering, and dephasing sources along the paths, the classical and quantum
interference contributions are equal. Let us assume that the two paths are time re
versed; this means the two partial waves traverse the same path in opposite directions.
For any two time-reversed paths, A1 = A2, and since the two waves leave the origin at
the same time, cover the same path length, and see exactly the same elastic scattering
events along the paths, they arrive back to the origin coherently (i.e $1 = (152). Then

the total quantum probability reads
unantum = 4lAll2 = 2Wclassz'cal (2.2)

Therefore, the quantum probability is twice the classical one. Because the probability
of finding the electron at the origin has been enhanced, its chances for diffusion are

reduced thereby increasing the resistivity.

2.2.2 Spin-Orbit scattering (weak antilocalization)

In the absence of spin-orbit coupling the wave functions of the two complementary
partial waves are in phase and interfere constructively once they arrive back to origin.
The spin-orbit scattering originates from the interaction of the conduction electron’s

spin with the angular momentum of the host atoms. This interaction rotates the

22

spin direction of the partial waves. However, quantum theory tells us that spin 1 / 2
particles have to be rotated by 47r to transfer the spin state back into itself. On the
other hand, the rotation of 27r changes the sign of the wave function [32]. The two
partial waves, discussed above, experience the same elastic scattering events, but in
opposite directions. The spin of the first partial wave is rotated during each scattering
process, and when the wave has reached its final state (3’), all rotations add up to

finite rotation R. This can be written as [32]
s’ = Rs (2.3)

where the spin rotation R is usually described by three angles , (6, (15, 1b).The second
complementary wave experiences the same infinitismal rotations in opposite direction

and opposite sequence. This reads
3” = R'1.s (2.4)

The two rotations are not equivalent, therefore the final spin states of the two wave

functions have diflerent orientations and their scalar product can be written as
(8"IS') = (8|R2|S> (2-5)

This factor was calculated in references [32, 33, 34, 35], and it was found that in the
presence of spin-orbit coupling, the weak localization magnetoresistance is modified

by a factor a which is given by [34]
— 3 _ 1
Q50“) : 56 475/3730 _ 5 (2.6)

This results shows that there are two important limits: weak and strong spin-orbit

coupling. In the weak spin-orbit limit, where 71,0 / 7}), >> 1, equation (2.6) gives a = 1.

23

 

AR/R

 

 

 

 

0.005

 

—»L[=100Lso
-15 -10 -5 o 5 1o 15
B(mT)

Figure 2.2: The calculated magnetoresistance curves using equation (2.8) of 1 KO
wire with L = 100 pm, 11) = 100 nm, and L4, = 20 pm, for different spin-orbit
coupling lengths. From top to bottom: L¢/L,o = 0.1, 1, 5, and 100. Notice that
for strong spin-orbit coupling, the magnetoresistance is positive (dip), while for weak
spin-orbit coupling, it is negative (peak).

 

 

 

This has no effect on the WL magnetoresistance lineshape. In this limit the WL
magnetoresistance is expected to be negative (peak at B = 0) as shown in figure (2.2).
On the other hand, for the strong spin-orbit limit, where Tso/T¢ << 1, equation (2.6)
gives m = —1/2. This means that the spin-orbit coupling reverses the sign of the
weak localization as well as reduces the magnitude of the signal by factor 2. Therefore,
the WL magnetoresistance is expected to be positive (dip at B = 0). The latter
effect is called weak antilocalization, since it changes the sign of the backscattering
interference.

It was found that 7331 depends on Z, the atomic number of the host atom, and

24

can be written as [36]
1 _ Z404

Tao Te

 

(2.7)

where a = 62/47reoch z 1/ 137 is the fine structure constant and Te is the elastic
mean free path. Equation(2.7) predicts that spin-orbit coupling is stronger in high Z
metals such as An, Z = 79, and it can be neglected in low Z ones. In Ag, Z = 47,

the spin-orbit strength has a moderate value.

2.2.3 Low-field magnetoresistance

In this section I briefly review the use of weak localization magnetoresistance to de—
termine the low-temperature value of the phase coherence time, 7,1,. Physically, 7'45 sets
the time scale over which the two partial waves, discussed in the last subsection, inter—
fere constructively once they arrive back to the origin (see figure 2.1). In the presence
of magnetic field and any time-reversal breakers such as e-e, e—ph, and e—magnetic
impurity interactions, the weak localization magnetoresistance will be suppressed. In
1980, Hikami et al. [33] showed how the weak localization might be suppressed by an
external magnetic field.

In a mesoscopic one dimensional conductor, the weak localization magnetoresis-

tance at a given temperature, AR(B) = R(B) — R(0), is given by [1, 5, 11]

1+1(w)2
Lg 3L3,r

where R is the resistance of a wire, L and w are its length and width, RK = h/e2 is

1/2

am3y_2R a i%_4 +1(3)2_ _1
12 ‘TRKL 2 1% 3L; 3 .Lg 2

—1/2

 

 

mm

the resistance quantum, L4, = (/D'r¢ is the phase coherence length, D is the diffusion
coefficient of electrons, L H = (Ni/eB is the magnetic length, B is the magnetic
field applied perpendicularly to the sample plane, and Lao = \/ D730 is the spin-orbit

length. The quantity under the square root of the first term in equation (2.8) is called

25

the triplet term and the quantity under the square of the second term is the singlet
one. In the limit of weak spin-orbit coupling (L80 ——> 00), both quantities are equal,
whereas in the limit of strong spin-orbit coupling (L30 ——1 0), the triplet term has
a small contribution and can be ignored (see figure 2.2). Equation (2.8) is valid for
metallic wires in the diffusive regime; where 1.. << w,t << LH, L¢,Lso < L, with t
the sample thickness [5]. By fitting the low-field magnetoresistance data to equation
(2.8), the value of the decoherence length, L45, can be extracted. The value of the
decoherence time, 74,, is then calculated using the equation, L, = \/D_'r¢.

Figure (2.2) shows typical magnetoresistance curves calculated via equation (2.8).
The figure shows how the weak localization can be destroyed by a small magnetic
field. We have mentioned before that the two partial waves interfere constructively
once they arrive back to origin to produce the enhanced backscattering. The effect
of the magnetic field is to create a phase difference between the two partial waves.
The phase difference created between the two waves is proportional to the magnetic
flux penetrating the area enclosed by the two waves. As the magnetic flux changes,
the transmission probability oscillates with a periodicity of hc/e; hence making the
magnetoresistance of the two trajectories oscillate with the same periodicity [37]. The
oscillatory behavior of the magnetoresistance observed near the zero field at low T
is known as the Aharonov-Bohm effect. Summing over all electron trajectories with
different areas causes the oscillations to average out everywhere except near B = 0,
where they are all in phase. The result is a peak or dip in the resistance near zero

field, depending on the strength of the spin-orbit coupling.

26

2.3 Decoherence mechanisms in metals

2.3.1 Electron-Electron scattering contribution to decoher-

ence rate

In this section we discuss briefly the effect of Coulomb electron-electron interaction
on the phase coherence time, u, in weakly disordered lD-systems. Physically, the
electron-electron scattering is equivalent to the interaction of a conduction electron
with the fluctuating electromagnetic field produced by all other surrounding electrons
in the system. Due to the statistical nature of the fluctuating field, the e-e scattering
is different for each electron, thus the conduction electron loses its coherence The
electron-electron scattering time, Tee, is defined as the time it takes for an electron
to be scattered inelastically by another electron. This time scale is temperature
dependent. In clean metals, Fermi liquid theory predicts that for electrons sufficiently
close to the Fermi surface and at low T, the e-e scattering rate has a T2 dependence.
On the other hand, in weakly disordered lD-metals, where screening is less effective,
the temperature dependence of the electron-electron scattering rate, 786 was found to
be [1, 11]

7ee(T) = AT”3 (2-9)

with A = h‘1[7rk§3R/4VFLthK]l/3, where V}? is the total DOS at the Fermi energy.
Equation (2.9) predicts that the e-e scattering rate vanishes as T —> 0. In pure metals
and at low T (roughly below 1 K), the electron-electron scattering dominates other

scattering processes (see for example reference [5]).

27

2.3.2 Electron-Phonon scattering contribution to decoher-

ence rate

When the conduction electron is scattered inelastically by phonons, it loses its co-
herence due to the energy exchange between the electron and the phonon. At high
temperatures, the electron-phonon scattering is the dominating source for the deco-
herence of conduction electrons. The electron-phonon scattering time, 76-”, sets the
time scale over which the conduction electron is inelastically scattered by lattice vi-
bration (phonon). Each phonon is characterized by its wave vector q. The magnitude

of the wave vector of a typical thermal phonon at temperature T is

_ kBT
QT — fw,

 

(2.10)

where v, is the sound velocity in the metal (a: 2700 m/s in silver). Based on the
quantity, que, the electron-phonon scattering is usually divided into three different
regimes; clean, dirty, and intermediate [38]. In the case of clean limit, que > 1. On
the other hand, qu6 << 1 for the dirty limit. In the intermediate regime, que ~ 1.

The phonon wave length is defined by AT = 271' / qT. In silver, this reads

_ 130
T “ T<K>

 

nm (2.11)

At liquid Helium temperature, /\T w 31 nm. In our experiments on silver wires,
18 z 32 nm. That means our experiment falls into the intermediate regime in the
vicinity of 4.2 K. The clean limit might be reached at much higher temperatures,
where AT << 16. On the other hand, at 40 mK (the base temperature of the dilution
refrigerator used in this experiment), AT z 3250 11111 >> 13, indicating that there is
cross over to the dirty limit. It should be emphasized here that there is no sharp

boundary for crossing over from one regime to another.

28

The temperature dependence of the e—ph scattering rate, 78.1,}, takes the form
7e_ph(T) = BT” (2.12)

with the exponent of T , p z 2 — 4, depending on the specific system and dimen-
sionality. Experimentally, the temperature dependence of 78.1,). reported by various
measurements on different material systems are not always in agreement with one
another. This issue becomes even more controversial when the dependence of 76-”,
on disorder is concerned [38].

The phonon has two polarization modes: longitudinal and traversal. For metals,
above 0.5 K, the longitudinal phonons dominate transversal ones. In the case of clean

limit, with no traverse phonons, the decoherence rate of the e—ph scattering is given

by [35, 38,39]

77TC(3)E12;~VFIC% 3 3
T E BT 2.1
963pv3k} ( 3)

 

7e—ph(T) =

where E p is the Fermi energy (5.6 eV for silver), p is the mass density ( 10.5 x 103kg/m3
for silver), and C (2) = 2k If” is the zeta function (C (3) z 1.2). For Ag wires, equation
(2.13) yields Bthy x 0.002 ns’lK‘3. In our experiments on Ag wires and over a wide
range of temperatures (40 mK - 18 K), Bap m 0.026 ns'lK‘3, which is higher than
the predicted value of equation (2.13). The relatively big difference between Bthy and
Bap, indicates that the e-ph scattering is not limited to one regime, instead it crosses
over between them. At low temperatures, the clean limit is no longer valid, and one
has to consider the complete expression of 'ye_ph(T), given in reference [39], where the

temperature dependence is between T2 and T4 instead of T3.

29

2.3.3 Electron-Magnetic impurity scattering contribution to

decoherence rate

In a metal containing a small amount of magnetic impurities, there is an additional
contribution to 7¢(T) from the scattering of the conduction electrons by the mag-
netic impurities, 7m(T) (also called 7,;(T)). This is known as the Kondo effect. The
conduction electron interacts with the magnetic impurity via the exchange Hamilto-
nian H = —J.S-';..§', where 57;, and 3' are the spins of the magnetic impurity and the
conduction electron, respectively. This exchange interaction between the conduction
electron and the magnetic impurity leads to scattering events in which the electronic
spin is flipped (with a compensating change of spin on the magnetic impurity). Since
the spin of the conduction electron is modified in this inelastic process, the conduc-
tion electron loses its coherence. The spin-flip scattering time, 73f, is defined as the
time between scattering events involving an electron and a magnetic impurity such
that the electron’s spin is flipped. The temperature dependence of 7,,(T) has been

discussed briefly in Chapter 1, and we will discuss it in more detail in Chapter 3.

2.4 Electron-Electron interaction contribution to
resistivity at low T

In a pure metal and at low T (where e-ph interactions are negligible), there are two
contributions to resistance; e-e interactions and weak localization. Experimentally,
the weak localization can be suppressed by applying a small magnetic field (~ 30 mT)
perpendicular to the film. On the other hand, the electron-electron interaction (EEI)
effect has a very small magnetic field dependence, so EEI can not be subtracted

out simply by applying a small magnetic field. The contribution of electron-electron

30

interactions to resistance is given by [11]

R2 LT_ Ch,
ARee(T) ~ 3.126 ——L—: — V: (2.14)

where LT = W is the thermal length.

Experimentally, the measurement of the electron-electron interaction contribution
to resistance at low T can be used as a thermometer to check the electron’s tempera-
ture. This is very important at very low temperatures to avoid heating the electrons.
At very low T, the drive, V, must be restricted to eV 5, kBT, to avoid significant heat-
ing of electrons above their equilibrium temperature. If the measured AR” follows
equation (2.14), then this is a good sign of the absence of electron heating (see figure
(2.3)). On the other hand, if the measured ARee deviates strongly from equation
(2.14) and saturates early, then this means that the electron temperature is higher

than the bath temperature, which is a sign of electron heating.

31

 

AP ~T-1/2

ee

 

(arb.units)

ee

 

AP

saturation(heating l!)

 

 

 

Temperature

Figure 2.3: Schematic of the contribution of electron-electron interactions to resistiv-
ity at low temperatures. The solid line represents equation (2.14), where the electron
temperature is the same as the bath one. The dashed line shows the effect of electron
heating on the e-e interactions contribution to resistivity.

32

Chapter 3

Theory II: Kondo Scattering in
Dilute Magnetic Alloys

3.1 Magnetic moments in metals

In a pure metal and in the absence of a magnetic field, the ground state of a free
electron gas has equal numbers of spin-up (NT) and spin-down (N l) electrons. A
metal is then nonmagnetic. Introducing a small magnetic field makes the NT 31$ N1,

hence creating net magnetization given by
M = XPB (3.1)

with xp = gp‘gN (5 F) / 2 is the temperature independent Pauli susceptibility of the
host metal; where g = 2 for electrons. Therefore, measurements of the Pauli sus-
ceptibility lead to information about the electronic density of states at the Fermi
energy.

Experimentally, however, it is often observed that some metals containing a dilute
concentration of magnetic impurities show a Curie-Weiss component to the suscep-

tibility, indicating the formation of a local moment. If the system has a number

33

of magnetic impurities with a concentration of 713, this component is temperature

dependent and has the form [18]

n 92u233(3 + 1)
s 3k3(T+9)

 

Ximp(T) : (3.2)

where g is the Landau g-factor, 113 is the Bohr magneton, k3 is Boltzman constant,
and 0 is a constant with a value in the thermal energy range (0 < 0 < 300 K).

In general, the local moment does not always develop, depending on the type
of the magnetic impurity and on the metallic host in which the magnetic impurity
is embedded. In any Kondo system, the existing models try to answer two basic
questions. First, how can a magnetic impurity keep its moment when dissolved in the
host metal? Second, if the magnetic moment develops, how does it afl’ect the transport
properties of the host? In the next three sections I will use the Anderson model to

address the first question and the Kondo model to answer the second question.

3.2 The Anderson model

The Anderson model provides us with a simple picture to understand the mechanism
by which it is possible for a 3d transition element to retain a magnetic moment when
dissolved in a host metal. For simplicity, consider a S = 1 / 2 magnetic impurity with
a single energy level 5,, embedded in a metal host, where 5,; < SF. The Anderson

model can be written as [17, 18]

HA = Hc+Hd+Hu+Hmix (3.3)

Now let us go through the details of each term of the above Hamiltonian.

The first term, He, describes the conduction electrons in the host, and using the

34

second quantization notation, can be written as

He = Zeknka = Z Ekclmcka (3.4)
k0 ka

where 5,, is the electron energy in state It, mm is the occupation number of the
conduction electrons in state (19,0), and cjw and ck, are the creation and annihilation
operators for electrons in momentum state It and spin state 0. Using this Hamiltonian,
one can easily find the density of states (DOS) of the electrons in the metal host,
DOS (5) oc 61/2. This is shown schematically in figure (3.1). By integrating the
quantity, 5N(e), over all filled k and 0 states one arrives at the total energy of the
conduction electrons in the host.

The second term in the Anderson Hamiltonian, Hd, accounts for the kinetic energy

of the d-electrons of the magnetic impurity added to the host and takes the form

Hd = Ed 2: Tlda = 8d 2: CLOCda (3.5)

where ed is the energy of the d-orbital, nda is the occupation number of the magnetic
impurity in state ((1, a), and CL, and odd are the creation and annihilation operators
that create or destroy an electron with energy 5,, and spin a. The number of electrons
in the impurity orbital nd obeys Fermi-Dirac statistics, 71,; = Eda ndd, (i.e. 71,, =
0, 1, or 2).

The third term in the Anderson Hamiltonian, Hy, describes the interactions be-
tween the d-electrons of the impurity. This term exists only when the impurity is

doubly occupied, 71,, = 2. This term takes the form
HU = UndTndl = U(Cfljcdl)(cjucdl) (3-6)

where U is the strength of interaction between the d-electrons.

35

 

 

003(8)
pose)

 

 

 

 

 

 

p----—-—
------

 

 

 

 

non-mag

magnetic

008(8)
<nd>

 

non-mag
sd-l-U 8,, s +U
8 8':

 

 

 

 

 

 

 

co [----—-

Q.
11

Figure 3.1: Schematic of Anderson model for metal host containing one impurity
level, 5d: (a) DOS for a host metal and an isolated singly occupied impurity level, 54.
(b) DOS for a host metal and an isolated doubly occupied impurity level, 50;, with an
on site interaction, U. (c) same as in (b), but with mixing between the conduction
electrons of the host and the d-electrons of the impurity; this mixing results in an
energy broadening of the impurity levels at ed and ed + U. ((1) the average occupation
number of the impurity, (714) = (12,11) + (“(11).

The second and the third terms of the Anderson model, Hd and Hg, describe
an isolated magnetic impurity. For an isolated magnetic impurity, there are three
energy configurations for the impurity where Fermi-Dirac statistics are obeyed. These
configurations depend on the position of the impurity level ed relative to the Fermi
level 5F. One can go from one configuration to another by tuning the Fermi energy 5F
around the impurity energy ed (see figure 3.1d). First, if 5;: < ed, then the impurity is
not occupied with a total energy E = 0, since the highest occupied level must be the

Fermi level. Second, if 6,; < 85‘ < 5,; + U, then the impurity is singly occupied with

36

either spin up or spin down electron with a total energy E = 5d. Third, if 6.1+ U < 8F,
then the impurity is doubly occupied with a total energy E = 2501 + U. Due to partial
occupation, the second configuration has two-fold degeneracy corresponding to spin
%, and hence it develops a magnetic moment which will give a Curie law contribution
to the susceptibility. The other two configurations are trivial and have no magnetic
moments. In this simple picture of the Anderson model, the condition for a local
moment to exist is that ed < 5F, so that it is favourable to add one electron and
ed + U > ASP so that it is unfavourable to add a second. There is one special case
where the single occupation can be non- magnetic. This happens when 54 = 5F,
where we have (ndT) = (11,11) = 1/[81‘p((€d—Ep)/k3T) + 1] = 1/2. We will not discuss
this case here since we already assumed from the beginning that ed < 5p.

The last term in the Anderson Hamiltonian, Hm“, is the mixing term, which

provides a mechanism for electrons in the conduction band to hop onto and off of the

impurity site. It can be written as

Hm“: = Z de(C;erda + Clack”) (3.7)
ka

where de is the strength of the interaction. For simplicity, let us assume that de is
constant, V. This mixing results in an energy broadening of the impurity level 5,, as
shown in figure (3.1c).
The transition rate of an electron to hop into and off of the magnetic impurity is
given by the Fermi golden rule
1 27r
— = — V 2N 5 3.8
T ,, |< >| ( .) ( )
where N (Ed) is the DOS of the conduction electrons per spin at the scattering center

(i.e. magnetic impurity center). Assuming that N (ed) = N (5;), the total scattering

37

rate from a number of magnetic impurities with an atomic concentration of Gimp is

1 27r
r = — = .m —V2N .
7' C P h (5F) (3 9)

using N(5p) = 1/2 (3n/25p), this becomes

I‘=1

.. —__—- . —— 2 .
T c,mp2h£FV (3 10)

if I‘ < U, it is possible to predict the existence of a local moment in a metal.
The contribution of the magnetic impurities to the resistivity can be calculated
from

pimp =

m (i) (3.11)

n82

substituting for 1 / 7' from equation (3.10) we find

p... = Wig—:12 (3.12)
This is just a form of potential scattering. It is temperature independent. This result
is usually referred as the first Born approximation (second order in the potential
strength, V).
In summary, we see that the Anderson model can in principle explain the existence
of a local moment for a singly occupied S = 1/2 magnetic impurity embedded in a
host metal (see figure 3.1d). This model was developed for a single impurity level,
6,1. In the real world, the impurity in principle can have more than one level and
furthermore those levels can be split more by the crystal field of the host. Therefore,
it seems to be hard to apply this model to real experiments. Nevertheless, it is
believed that the Anderson model contains the basic physics one needs to understand

other models of transport in Kondo alloys.

38

3.3 The s-d exchange model

In the previous section and in the context of Anderson model we have discussed the
local moment regime, where the magnetic impurity is singly occupied. The condition
of single occupation was that U + ed > 5;: and ed < 5p. Assuming that the magnetic
impurity is in the local moment regime and if the strength of interaction between
the localized moment and the conduction electron is sufficiently small (i.e de <<
Ed + U), then the Anderson model goes to what is called the s-d exchange model.
The derivation of the s-d exchange model from the Anderson model is referred to
as a Schrieffer and Wolff transformation. In the context of the Anderson model,
the strength of interaction between the local moment and the conduction electron is
described by the parameter de. On the other hand, in the s-d exchange model, the
interaction strength is described by an effective exchange coupling, Jud. The relation
between de and ~1de was first obtained by Schrieffer and Wolff (1966) and takes the
form [18]

 

1 1
J ,=_v 2 3.13
k’k lkdl {U+€d—€k+€k—€d} ( )

where 6k is the energy of the conduction electron. In the local moment regime,
where the magnetic impurity is singly occupied, the effective exchange coupling, Jk,k’,
between the localized spin and the conduction electron is antiferromagnetic (i.e Jug <
O). This can be seen easily from equation (3.13). Since most of the conduction comes
from electrons with energies close to Fermi energy, we can set 61: z 5p. By measuring
the energies relative to Fermi energy and setting 5;: = 0, one can see that Ed < 0 and
U + Ed > 0, hence Jch' < 0 (i.e antiferromagnetic).

The s-d exchange model already assumes that there is a local moment with spin

S which is coupled via an exchange interaction J with the conduction electrons. For

39

a single impurity in a metal, the s-d exchange Hamiltonian takes the form [18]

Had = - 2 Jane (90140163 + g-Clmck’l + 340130163 — 0140134)) (3-14)
k,k’
where S, and S i = S, :l: iSy are the spin operators of the magnetic impurity with
spin S. Jaw represents the strength of the interaction between the local moment
and the conduction electrons, which will be considered a negative constant, J, in the
following derivation. k’ and k represent the initial and final states of the conduction
electron.

The above Hamiltonian contains two types of spin scattering processes; spin-flip
and non spin-flip processes (see figure 3.2). While the first two terms in H 8,), which
includes S+ and S‘, describe spin-flip processes of the conduction electron, the last
term in the parentheses, which includes S2, describes non spin-flip processes.

To obtain the contribution of the magnetic impurity to the total scattering rate,
and hence to the resistivity, we need to calculate the matrix elements of Hsd for the
four processes mentioned above. Those processes are labeled a, b, c, and d as shown

in figure (3.2). For process a, we have

(k l |Hsdlk’ T). = —J[ (k l |S+c]“,ckl,1|k’ T) + (kl [S—CL’TCRI,[]kI 1)

+<kl [SchTckr,1|k'T) — (kl ISch...cw..Ik’ 1)] (3.15)

The last three terms vanish for the following reasons. In the second term, cm destroys
the state Ik', l) which is not present, because the initial state is known, Ik', T). In the
third term, CL, creates the state |k, T) which is not allowed to be created, because
the final state is known , |k, l). The last term has the operator, cm], which destroys

the state |k’, l) which is not present, again because the initial state is known, [k', T).

40

 

 

 

 

k’] kl k’l kl
.4 \_/_.

[ impurity T l impurity l
a) 8+ b)S'

k’ k] k’] kl

\4 \._/_.

[ impurity [ [ impurity ]
c)Sz d)Sz

Figure 3.2: Schematic of spin-flip and non spin-flip processes for a conduction electron
scattered by magnetic impurity.

Therefore, we end up with the matrix element of the first term only
(kl (H,d|k’ T). = —J3+ (3.16)

Similarly for the remaining three processes b, c, and d, we find

0‘ T ledlk’ l>b = —JS'_ (3‘17)
(kl [Hulk] Tlc = —J§z (3-18)
(kl [Hadlk’ 1).: = +JSZ (3'19)

41

The corresponding transition probabilities for the above matrix elements are

l<kl IH...1lk’l>al2 = fishi- (3.20)
|<k T IHalk’ 1).? = JiS‘S‘+ (321)
KM ledlk’ 1).)? = J25”? (3.22)
MM ledlk’ all? = 1232’ (3.23)

where in the last equations I used (S +)"‘ = S ' and (S ")* = S +. The total transition

probability, |(V)|2, is the sum of the individual probabilities
|(V)|2 = J2(§+S‘ + 323+ + 232’) (3.24)

Using S+S‘ + S ‘S + + 2S ‘2 = S (S + 1) and taking matrix elements over the states

of the localized spins, we finally find
|(V)|2 = J2S(S + 1) (3.25)

Substituting this back into equations (3.10) and (3.12), we find for the magnetic

impurities contribution to the scattering rate

1 37m 2
And to the resistivity
37rm
pimp = CimPQ—ezhe—FJ2S(S + 1) (3.27)

Again this is temperature independent. In summary, it is clear that the first Born
approximation is insufficient to explain the anomalies in the resistivity , and hence
in the scattering rate. How about going to third order in J ? This is exactly what

Kondo did in 1964.

42

3.4 The Kondo model

Kondo started with the same Hamiltonian of the s-d exchange model discussed in the
last section [16]. In addition to the first Born approximation mentioned above, Kondo
considered the second Born approximation in his calculations for the scattering rates
and found that the scattering of the conduction electrons by magnetic impurities leads
to temperature dependent terms in the scattering rates, and hence in the resistivity.

In a metal without magnetic impurities, the conduction electrons can be modeled

with the unperturbed Hamiltonian Ho, where the schrodinger equation takes the form
H03. = E23,. (3.23)

where Q5), and E2 are the unperturbed wave function and energy of the conduction
electron in state It. In the presence of the perturbation H’, which is due to dilute

magnetic impurities, the schrodinger equation becomes
(He + H'Wk = Era/J}: (3.29)

where ’t/Jk and E), are the perturbed wave function and energy of the conduction
electron in state It. Kondo considered the Had as the perturbed hamiltonian H’ (i.e.
H’ = Hsd). Using standard time independent perturbation theory, the solution of the

last equation can be written as [40, 41]

.. H’ 1
4'). = as. + Z <<onl __ [’55) 4.. (3.30)
n k n

It is easily seen that the above solution is not an explicit solution because 11);, appears
on both sides of the equation. To have an approximate solution, the above series
solution has to be cut at some order. In the first Born approximation, we stop at the

first term, 45),, the wave function of the unperturbed Hamiltonian H0. This, however,

43

does not lead to temperature dependent scattering rate as mentioned early. In the
second Born approximation, we take the second term into account and in addition
we replace 1,0,, under the sum by (bk, the unperturbed wave function. Then equation

(3.30) becomes

I
2.0;. = (251: + Z ‘ETf’iE—affin (331)
n k n

where H’ M is the transition amplitude (¢n|H’|¢k) of the perturbation H’ taken be-
tween the unperturbed staes k and 71. Now we can use equation (3.31) to calculate
the transition probability for an an electron to be scattered from state It to state k’

and we get [42]

I(¢k'IH'lwk>I2 = (¢k'|H’I1/>k>(¢k'IHIIi/Jk)f;

=(<¢k'IHI¢k) +23%— %(¢kIIHII¢n>)X

((4.13 |¢.I>+ZE—.——H_"":E (3.. >) (3.32)

           

Using H [pk = H [W (i.e. H’ is hermitian), the last equation can be written shortly as

H’, H’ Hm
I<¢k'IH')I2Ii/Jk =(Hl’c’k'I‘ZEo k" _H"——§’8) (Hkk'+ 2 E3130)

H ,an
= H,,,.H,’,.,. + H,’,,,. 2 E——,’;_ E; + C. 0+ 0(H’ ) (3.33)

Now let us relabel the last equation with the same notations used by Kondo in his
original paper [16], where k = a, k’ = b, and n = c, then after ignoring the highest

order term 0(HM), we end up with
I
b H’ a H + —£,——-—Hblibcg)ca +c.c (3.34)
Hab ba

where a, b, and c denote the initial, the final, and the intermediate states of the

44

system and Ea, Eb, and Ec are their total energies. Finally, using the Fermi Golden
rule, the transition rate from the initial state a to the final state b can be written as

W(a —> b) = 2—;—6(Ea — Eb) (Hénga + E w + ac) (3.35)
The first term in the parenthesis, H;bH,;a = |H;b|2, represents the first Born ap-
proximation, since it involves J 2 as prefactor. The second Born approximation is
represented by the second term, where the sum goes over all possible intermediate
states c, and it involves J3 as prefactor. It is the triple product under the sum which
yields the temperature dependent scattering rates [42].

Kondo assumed that the scattering of the conduction electron by the magnetic
impurity can be divided into two major processes. In the first process, the electron is
scattered into an intermediate state by the magnetic impurity which is also scattered
to its intermediate state. In the second process, the conduction electron is scattered
to its final state while the impurity is scattered back to its original state. The in-
termediate state may or may not involve spin-flip processes. If the electron spin is
flipped in the intermediate state, the impurity spin has to be flipped in order to keep
the total spin conserved. It was found only those processes that involve spin-flip
result in temperature dependent scattering rates.

Kondo considered four possible configurations for the initial and final states of
the conduction electron. The four configurations are (k’ T [H ;d|k T), (k' l |H;d|k l),
(k’ T IHLdIk l), and (k’ l |H;d|k T). In addition, Kondo divided the intermediate
state for each configuration into four groups. For more details about those groups, I
refer the reader to the original Kondo paper [16]. Starting from equation (3.35) and
after a lengthy calculations, Kondo found an expression for the magnetic impurities

contribution to the scattering rate given by [16]

1 37m 2

45

where

3 k k—kp
905k): 27:; I1+2k— log k+kp

 

 

] for T = 0, (3.37)

 

This function is singular at k = kp, because the Fermi function has a discontinuity
at k = kp. At very low temperatures, (It - kp) for thermally excited electrons is on

average (kBT/ep)kp. Then equation (3.37) becomes

kBT

357 ] (3.33)

 

 

3 1
=——1 —1
90%) 25F[ +2 05

where I used k z IQ: and k + kp z 2kp. Substituting this back into equation
(3.36), one can see that there are terms in the scattering rate expression proportional
to J3log(T). Using equation (3.36) Kondo also calculated the magnetic impurities

contribution to the resistivity and found

k T
pimp = c,mp2——e 2’15}: J23 S(S+ 1) [1 +4JN(€F) log < 13)— )] (3.39)

provided J < 0 and D = 25F is the bandwidth, where the DOS was assumed to be
constant. When this magnetic contribution is added to the other contributions to the
resistivity, we recover equations (1.7) and (1.8).

Although Kondo successfully explained the temperature dependence of the resis-
tivity not too far below T = Tmin, his calculations break down as T —-> 0 as discussed
in the first chapter. This is known as the Kondo problem. The search for a solution
for this problem attracted a lot of theoretical work in the late 60s and early 70s. The
main goal of those works was to perform infinite order perturbation in J and sum
the most logarithmically divergent terms. In 1965, Abrikosov carried out a summa-
tion of these leading order terms for the resistivity [18]. Although he solved the zero
temperature divergence in resistivity, he found a more severe divergence, but now, at

finite temperature TK, the Kondo temperature. In summary, the perturbation theory

46

was good enough to describe the Kondo systems for T >> T K, but could not be
extended to the region T << TK. In 1975, Wilson [20] solved the Kondo problem
by devising a non—perturbational approach known as the numerical renormalization
group (NRG). He showed that at temperatures far below the Kondo temperature
(T < TX), the spin of the magnetic impurity is totally screened by the conduction
electrons. For conduction electrons, the screened impurity appears then as a poten-
tial scattering center, hence the resistivity saturates. His approach was successful in

predicting most of the thermodynamic properties of many dilute Kondo systems.

3.5 Adding the decoherence rates

Consider two time-reversed paths (1 , 2) that start and end at the origin, and a
magnetic impurity, with spin S, sitting somewhere along the path. Let Em? be
the position of the magnetic impurity relative to the origin. In principle, there is
a finite probability for the conduction electron to follow either of the two paths. If
the conduction electron follows the first path, it will arrive at Rim? after a time 7'1
and interact with the magnetic impurity there. On the other hand, if the conduction
electron follows the second path (time-reversed path), it will arrive at gimp after a
time 1'2. As a result, the conduction electron may have two spin-flip events at gimp.
The time difference between the two events is |72 - 71L During this time difference
and due to the dynamics of the magnetic impurity, it may interact with other thermal
electrons (with energy of order kBT) from outside the loop, and the spin orientation
of the magnetic impurity may change. This means that the spin configuration of the
magnetic impurity may or may not be the same for both events. The time scale at
which the magnetic impurity re—orients or relaxes is called the Korringa time, TK. It

was found that the Korringa rate, 7K, varies linearly with temperature and can be

47

written as [43]

,K = _1_ = EJ23(S + 1)V%~kBT (3.40)
TX 5

where up is the total DOS at the Fermi level. On the other hand, we found early in
this chapter that the magnetic scattering rate (in the limit T > TX) scales linearly

with the concentration of the magnetic impurities and takes the form

7., = i = %J28(S + 1)1/Fn3 (3.41)

Tm
The ratio of the two rates becomes

h _ ”3
7K VFkBT

 

(3.42)

Below the Kondo temperature TK, where the magnetic impurity is screened, the

Korringa rate saturates and then the ratio reads

7m_ n.

_ __ 3.43
7K VFkBTK ( )

For a Ag host, where up = 1.03 x 1047J’1m‘3 and k3 = 1.38 x 10‘23JK‘1, this reads

7—m -.= 4.15 x 10'2 —"’(ppm)

7x TK(K) (344)

where the concentration is now expressed in ppm unit and as usual TX in Kelvin. In
the next section, we will see that the ratio, 'ym/7K << 1, is the same as the criterion
of the validity of Micklitz’s theory.

The conduction electron can be treated quantum mechanically as if it is composed
of two partial waves traversing the loop in opposite directions. Once the two partial
waves arrive back to the origin, they will interfere constructively or destructively

depending on the final spin states of the two partial waves . If the two partial

48

waves do not undergo any spin rotation on the conjugated paths, then their final Spin
states will be the same and they will interfere constructively. Decoherence results
from non-commutation of the spin rotation Operators the electron experiences as it
traverses the loop in opposite directions. There are two limits to the spin-induced
decoherence depending on how fast the impurity will be randomized. The two limits
are controlled by the ratio 7m/7K. If 7m < 7K, then the impurity spin will randomize
quickly after the first spin-flip event, hence the two partial waves will see the same
average spin orientations of the magnetic impurities. On the other hand, if 7m > 7K,
then the magnetic field created by the magnetic impurities is no longer random, hence
the decoherence rate is doubled. The contribution of magnetic impurity scattering
to the total decoherence rate in the spin-singlet channel of the weak-localization

magnetoresistance can be written as [43]
7’4 = “rm + 7m for 'ym < 7K (3.45)

7c? = ”Yin + 27m for 7m > 7K (346)

where 7m is the dephasing rate due to electron-electron and electron-phonon scat-
tering. In summary, I want to emphasize that it is the ratio 7m/7K that determines

which equation to be used for adding the decoherence rates.

3.6 The theories of Zarand and Micklitz (Numer-
ical Renormalization Group technique)

Although Wilson solved the Kondo problem in 1975, there was no calculation of the
inelastic scattering rate until the paper of Zarand et al. [28] in 2004. In their work,
Zarand et al. used the numerical renormalization group (N RG) method to study the

complete energy dependence of the inelastic scattering cross-section, a,,,el(w, T = 0),

49

of the conduction electrons scattered by a spin 1 / 2 magnetic impurity at T = 0. The
theory of Zarand predicts that the the inelastic scattering cross-section, am; (w, T =
0), has only a very weak (logarithmic) energy dependence above TK, has a broad
maximum around T K, scales approximately linearly with w for 0.1TK < w < TX,
and crosses over to a ~ wz behavior for very small energies, w < 0.1TK. The in-
elastic scattering rate of the conduction electrons scattered by a magnetic impurity,
7m (w, T = 0), is proportional to the inelastic scattering cross-section, aim) (w, T = 0).
Although Zarand et al. computed 0mg; (w, T = 0), they believe that aim.) (w = 0, T)
has a similar form.

In 2006 Micklitz etal. [27] showed that the dephasing rate due to magnetic im-
purities measured in a weak localization experiment, 7m (T), is directly related to
the inelastic cross section, (1,-ml (w), calculated by Zarand et al. [28]. In other words
7m (T) or 0mg; (w). According to this theory, the effect of a small concentrations, n,,
of spin 1 / 2 magnetic impurities on the dephasing rate can be explored from T >> T K
all the way down to T < TX. This theory is valid in the limit of small concentrations,

and this condition can be written as

”s
m << 1 (3.47)

where Up is the DOS at the Fermi level per spin degree of freedom , k3 is the Boltz-
mann constant, and TX is the Kondo temperature of the system under study.
Furthermore, the theory of Micklitz predicts that if the density ns of magnetic im-
purities is sufficiently low, the magnetic scattering rate, 7m(T), is a universal function,
'ym(T) oc n, f (T/TK), and depends only on two parameters, n, and T K. The calcu-
lated magnetic scattering rate, 7m(T), for a weakly disordered 1D-system is shown
in figure (3.3). The figure shows that 7m(T) is almost constant for T > TK. For
T S, 0.3 TK, 7m(T) varies linearly in T and at very low temperatures, one obtains the

expected T2 behavior. The figure also shows that the maximum value of 7m(T) occurs

50

0.25 v I v I ' l ' I ' l '

 

 

 

0.20
cm
N 0.15
“~
u.
>
"E 0.10
E
>.

0.05

 

 

 

 

 

 

 

Figure 3.3: Magnetic scattering rate due to diluted magnetic impurities embedded in
weakly disordered 1D-metal host calculated via N RG [27]. The inset is a blow up of
the low temperature region.

at T = T K. This maximum scales linearly with the concentration of the magnetic

impurities and can be written as

 

2 s
73:“(T = TK) = 0.23 x n (3.48)
7?th
For a Ag host, where up = 0.515 x 1047J‘1m’3, this reads
73‘”(T = TK) z 1.59 X n3(ppm) 703—1 (3.49)

In our work, we compare our results with Micklitz’s theory, which provides us with

the calculated dephasing rate due to magnetic impurities in a weakly disordered metal

51

as measured in a weak localization experiment.

In any Kondo system, there are three major sources of electron dephasing; electron-
electron, electron-phonon, and electron-magnetic impurity interactions. Assuming
that the three rates are independent, then the total scattering rate will be the sum
of them (Matthiessen’s rule). In the context of Micklitz’s theory, this is not strictly
correct. The theory of Micklitz provides us with a new way of adding the dephasing
rate due to e-e interactions with the rate due to all other processes including 7",. At
very low temperatures, where the e-ph interaction is negligible and assuming that
there are only two sources of dephasing; e-e and e-magnetic impurity interactions,

the total dephasing rate, in the context of Micklitz’s theory, can be written as [27]

l 1
705

where the first term represents the e—e contribution to the dephasing rate and the
second term describes the dephasing rate due to the magnetic impuritias, It = A3/2,

where A = h‘1[1rk23R/4uprtRK]l/ 3 (see equation 2.9). Rearranging the last equa-

/ 1
“m = 73 — KT — (3.51)
74

We see that the last equation deals only with two dephasing sources ; e-e and e-

tion yields 7m

magnetic impurity interactions. However, experimentally, the total dephasing rate,
7,3, may contain additional dephasing sources such as scattering of the conduction
electrons by both phonons and low T Kondo impurities. Those additional sources
are non avoidable, and they exist in both the pure and the implanted samples, since
all of our samples have been evaporated from the same source and at the same time.
In our work, we first fit the total dephasing rate of the pure sample, assuming that
it contains three dephasing sources; e-e, e—ph, e— low T Kondo impurity interactions,

then we subtract this total dephasing rate out from the implanted samples to end

52

up with, 7",, the scattering rate due to the implanted magnetic impurities. We have
compared the results of analyzing our data with equation (3.45) and with equation
(3.50), and we found that the difference is at most 15% for the 2 ppm sample at the

lowest temperature.

3.7 Impurity-Impurity interaction

So far, we have been talking about the Kondo effect, which deals with non-interacting
or very dilute magnetic impurities dissolved into a non-magnetic metals. This is called
the Kondo regime, where the magnetic impurity concentration has to be very small to
avoid impurity-impurity interactions. It is also sometimes called the single impurity
limit. Historically, a reasonable approximation for this regime is to work at impurity
concentrations below 100 ppm [44] .

In practice, the single impurity limit is difficult to realize, since the impurity-
impurity interaction is present and falls off rather slowly as (1/1'3). This coupling
between the magnetic impurities is known as Ruderman, Kittel, Kasuya, Yosida
(RKKY) interaction, named after its discoverers [45, 46, 47]. In highly concentrated
Kondo alloys, the RKKY interactions lead to a spin glass transition at a temperature
T39, which is given by [48]

472.3

T8 = (3.52)
g kaupln2(vpfini/3/kBTK)

 

where Up and 01: are the total DOS at the Fermi level and the Fermi velocity of the

host. For Ag host, where Up = 1.03 x 10"‘7J'1m‘3 and ’Up = 1.39 x 106ms‘1, we find

_ 5.285 x 10’2 n3(ppm)
89 1112(413295 ni/3(ppm)/TK(K))

 

(3.53)

where again the concentration is expressed in ppm unit and both TK and T39 are in

53

Kelvin. For Fe impurities with a concentrations of 2 and 10 ppm , and assuming
that TK = 4K, the corresponding ng’s are 4.5 and 18 mK, respectively. To avoid
the RKKY interactions, one has to work at temperatures higher than the spin glass
transition temperature of the system under study, i.e. T > T89. In our experiment, the
working temperatures is always higher than 40 mK, which is the base temperature
of the Oxford dilution refrigerator used in this work, hence our experiment is not

sensitive to the RKKY interactions.

54

Chapter 4

Experimental Methods

4.1 Introduction

In this chapter I discuss the experimental methods used in this work to fabricate,
characterize, ion-implant, and measure our samples. First, I will go over sample
fabrication. This includes wafer processing, electron-beam writing, evaporation and
lift-off, and ion implantation. Second, I will discuss the measurement methods. The
latter includes testing the samples at room temperature, dilution refrigeration, and
weak localization magnetoresistance circuitry. Finally, I discuss the use of weak lo-
calization magnetoresistance to determine the phase coherence time of conduction

electrons in our samples.

4.2 Sample fabrication

4.2.1 Wafer processing

In this work, I used 4-inch oxidized silicon wafers. The thickness of the oxide layer
was 300 nm. The first thing to do was to cut the wafer into pieces 5 mm x 5 mm
in size using a MicroAutomation 1006 Dicing Saw. But before dicing, we have to

protect the surface of the wafer from scratches, residuals that may come off during

55

 

Material Silicon
Wafer Diameter (mi) 4000
Wafer thickness (mi) 20
Tape thickness (mi) 4

 

 

 

 

 

 

 

 

First index (mm) 5
second index (mm) 5
Hight (mi) 16
depth of cut (mi) 8
Angle (deg) 90

 

 

 

 

Feed speed (rpm) 200

 

Table 4.1: The parameters used to programm the dicing Saw to cut 4 inches Silicon
wafer into 5 mm squares. 1 inch = 1000 mils.

the dicing process, or from any other contamination sources. To do that I first cover
the surface of the wafer by a thick layer of e-beam resist, bake it for half an hour,
and then take it to be cut. All sample fabrication steps were done inside the clean
room except the dicing which was done outside where the Saw is located. The wafer
was then placed, facedown, on a sheet of kapton film with holes punched through to
allow a vacuum chuck to hold the wafer on the saw. The kapton film is important to
reduce scratching of the surface and to help sliding the wafer to the edge while the
vacuum is on, after the dicing is done. This is important, because once the dicing is
done if one turns the vacuum off, the wafer will pop up and break apart. Instead one
has to slide the kapton film to the edge while the vacuum is on, and then take out
the wafer. I usually dice the wafer from the back side, since the wafer will be covered
again by e-beam resist for writing purposes, and I do not dice it deeply. The depth
of cut was usually taken to be one-third of the thickness of the wafer. In this way,
the wafer would not break apart if handled carefully. The Saw can be programmed
to define the characteristics of both the wafer and the size of the chips need to be
cut. Table (4.2.1) lists the parameters used in dicing 4 inch Silicon wafers.

Once the wafer is cut not too deeply from the back side, I take it back to the
clean room to create a bi-layer of resist which will be used in the e-beam writing

process. Having two layers of resist is crucial to form a nice undercut. The two

56

types of resist were chosen in a way that the lower resist has higher sensitivity to
both electron exposure and developer than the top layer. Therefore, the lower layer
will be removed more easily than the top, forming an undercut (see figure 4.1b). In
this work, I used two type of resists to create the lower and the top layers. The
lower layer was created by dropping copolymer of P(MMA(8.5)MAA) 9% in ethyl
lactate on the surface of the wafer (see figure 4.1a). The wafer is then spun at 6000
rpm for 50 seconds. The centrifugal force spreads the resist across the wafer, and a
desired thickness of the resist can be achieved by selecting an appropriate spin speed.
Spinning the wafer this way removes excess material and forms a smooth, uniform
layer across most of the wafer. After spinning the resist, the wafer is placed on top
of a hotplate and baked at 170 degrees Celsius for half an hour. The e-beam resist is
suspended in solvent, therefore by baking the wafer, the solvent will evaporate leaving
a hard layer of resist sitting on the Silicon substrate. The second layer of resist was
created by dropping 495 PMMA 4% in chlorobenzene on top of the first layer. The
wafer is then spun at 6000 rpm, and baked at 170 degrees Celsius for one hour.
Now the wafer is ready to be broken apart into small squares (5 mm x 5 mm).
Since the thickness profile of the resist on the Silicon substrate is not uniform near the
edges of the wafer, I discard the chips which are far away from the center and close
to the edge of the wafer. Then I check each single chip using the optical microscope

to find which chips are clean enough to be used in the writing process.

4.2.2 Electron beam lithography; preparing the electron mi-

croscope for writing
SEM column description

Electron-beam lithography is the technique that enables small patterns to be trans-

ferred to a substrate. Once the resist is baked, the chip is ready to be loaded into

57

 

PMMA(4%)

 

MAA(9%) a) spin on resist

 

Substrate

 

 

 

 

 

PMMA(4%)

 

 

 

 

M ° 0 .
AA(9/ ) b) expose and develop res1st

 

Substrate

 

 

 

 

 

 

 

 

 

 

 

 

 

‘ —
PMMA(4%)
MAA(9%) /- c) deposit metal
Substrate
-
d) lift-off
Substrate

 

 

 

Figure 4.1: Cartoon showing the basic steps in EBL. (a) Covering the substrate by
bi-layer resists. (b) Exposing and developing the resist. (c) Metal deposition. ((1)
lifting-off the resists and excess metal.

58

the electron microscope. In this work, I used a JEOL model JSM-840A scanning
electron microscope with a tungsten filament. The electron microscope column is the
most critical part of any electron beam lithography system. It consists of the electron
source, an alignment system for centering the beam in the column, apertures, a beam
blanker for shuttering the beam, several lenses for focusing the electron beam, and a
stigmator used for correcting astigmatism in the beam [49]. A typical electron beam

column is shown in Figure (4.2).

Saturating the filament and beam current stability

After loading the sample into the microscope, I usually wait from 5—10 minutes for
the column to pump down to ~ 1 x 10’6 Torr, then I start slowly saturating the
filament. The tungsten filament is heated by passing current through it and electrons
are emitted thermionically from it. The current passing through the filament is called
filament current. It may take around 5 minutes to fully saturate the filament. The
filament itself is kept at high negative voltage compared with the anode, which is
at ground or zero potential. Therefore, the anode is positive with respect to the
filament, causing the electrons emitted from the filament to be accelerated toward
the anode. In all samples I fabricated, both the accelerating voltage and the working
distance were fixed to 35 keV and 25 mm, respectively. The anode has a small hole
(aperture), which allows some electrons to pass through and travel down the column
forming what is called the beam current. Some small holes in the sample holder were
designed to measure the beam current. I focus the beam into one of the holes, and
turn on the current meter to watch the beam current. Initially, I set the beam current
to around ~ 7 pA and wait at least half an hour for the current to be stable. For a
new filament, it may take from 1-2 hours for the beam current to get stable. On the
other hand, after a few runs, the filament will be cooked well and then it takes less

time for the beam current to get stable.

59

Tungsten Filament \/
|__ _J

Anode ,Aperture — —
T'lt
Gun alignment coils l E E
Shift E El
Beam blanker [ I
Condenser lens X XI
Aperture — '—
Scan coils (Magnification) E E
Objective lens (Focus) E E
Stigmation E E
Final aperture _ _
Detector

Sample, Stage | |

Figure 4.2: Schematic diagram showing the basic components of the J EOL microscope
column.

60

Gun alignment and beam current adjustment

Once the current becomes stable, I check the gun alignment to maximize the current in
the column. The gun is properly aligned if the direction of the beam current is parallel
to the axis of the column. This is important because I use three current settings in
writing my pattern; low current setting (25 pA) to write the smallest features (wires),
intermediate current setting (0.5 nA) to write the intermediate features (leads), and
high current setting (40 nA) to write the big features (pads). If the gun is not
properly aligned, the high current setting will not be reached. On the other hand, if
the filament has been used many times, it will not provide high currents; in this case
the filament needs to be replaced. While checking the high current setting, I usually
check the performance of the beam blanker. The beam blanker is used to turn off or
blank the beam. This is necessary when the beam needs to be moved from one part

of the wafer to another. Typical ramp times for the beam blanker are 3 - 5 us.

Focusing and stigmating the beam

Next, I start focusing and stigmating the beam. To do this, I use a test sample that
contains circles, horizontal and vertical lines of 120 nm thick Ag. I usually focus
and stigmate at the same current used to write the smallest features in my pattern
(namely at I z 25 pA). I start this process by focusing on Ag lines which are at
right angles with the magnification all the way up to 100,000, then I use the two
stagmator correction knobs (x, y), in combination with the focus knob to get the
sharpest image. I repeat this step several times at higher magnifications, until a
sharp image is obtained at 300,000. Then I go to the real chip to focus on it. For this
purpose, I usually make two scratches at the right and left sides of the chip to be used
for focus purposes. I first focus on the right scratch and record the angular position of
the focus knob, second I focus on the left scratch and record the the angular position

of the focus knob, and finally I set the focus knob in between the two position to

61

correct for tilting the substrate. Now the microscope is ready to start the writing
process. At this point I usually turn off both the detector and the current meter to

reduce the amount of noise inside the chamber.

Pattern design, run file creation, and pattern writing

The JEOL - SEM I used is fed with the Nanometer Pattern Generation System
(N PGS), which controls the pattern generation process. This process can be divided
into three main steps; designing the pattern, creating the parameter run file, and
writing the pattern [50]. My patterns were designed using DesignCAD LT2000. Once
the pattern is designed, N PCS creates a run file which is used to record the exposure
conditions for each layer in the pattern. A typical run file contains information about
the dose, the magnification, and the measured beam current for every single layer in
the pattern. In writing the pattern, I used two types of doses; area dose (MC/m2)
and line dose (726' / cm) The line dose was used to write the smallest features (i.e the
wires). A typical line dose was 6.5 72.0 / cm. The area dose was used to write the small
four-terminal pads attached to the wire, the leads, and the big pads. A typical area
dose was 250 uC/cm2. If the user enters the dose into the run file, the point exposure
time will be calculated automatically.

My pattern consists of four main layers. The first layer contains information
about the parameters used to write the wire. This layer was written at magnification
of 800. The line dose and the measured beam current were 6.5 720/ cm and 25 pA,
respectively. The second layer represents the small pads attached to the wire. This
layer was written also at the same magnification and the measured beam current of
the first layer, but instead with an area dose of 250 pC/cm2. The third layer was
used to write the small leads which are close to the wire. This layer was written at a
magnification of 100. The area dose and the measured beam current of this layer were

250 440/ c7712 and 0.5 nA, respectively. Finally, the last layer was used to write the

62

outer leads, the big pads, and the guards. This layer was written at a magnification of
20. The area dose and the measured beam current of this layer were 250 pC'/cm2 and
40 nA, respectively. The run file also provides an option to correct for the unwanted
offset that occurs when the magnification or the beam current is changed between
different layers. This option allows the origin of a single layer of a pattern to be offset
from the origin of the total pattern, which is always at the center of the field of view
of the microscope. This offset has to be tested experimentally, and then corrected for
in the run file. In my pattern, when the magnification and the beam current change
from 800 and 25 pA to 100 and 0.5 nA, the offset was (-33, -15) pm. To correct for
this shift, I entered (33, 15) pm in the run file. Similarly, when the magnification and
the beam current change from 100 and 0.5 72A to 20 and 40 nA, the offset was (-191,
-82) pm. Again, to correct for this shift, I entered (191, 82) pm in the run file.
Once the run file has been created, the NPGS program starts the writing process.
This program reads the run file and calls the writing program (PG). Typical total
writing time of my pattern was around 8 minutes. Each chip contains two identical
samples (two patterns), so the total writing time for each chip is around 16 minutes.
After writing each layer in the pattern, the program asks the user’s permission to
start writing the second layer. Before hitting enter, one can change quickly the

magnification and the beam current as desired.

4.2.3 pattern deveIOpment, evaporation and lift-off

After the patterns have been written by EBL, the exposed areas have to be removed
from the surface using the appropriate developer. When the electron beam interacts
with the resist, it breaks the bonds inside the resist, which makes it more soluble in
an organic solvent (developer). In this work, the developer used was methyl isobutyl
ketone (MIBK), 99+% pure. This MIBK developer was diluted to 1:3 with isopropyl

alcohol (IPA). To develop the patterns, I use three small beakers. The first beaker is

63

 

Figure 4.3: Optical microscope picture of the pattern after development.

filled with the developer(MIBK) and placed inside an ultrasonic bath. The second and
third beakers are filled with alcohol (IPA) and deionized water (DIW), respectively.
To develop, I swirl the chip for 50 s in the MIBK solution, 10 s in the IPA, and
finally 20 s in the DIW. Then I take it out and immediately dry it with N2 gas. To
check the development performance, I look at the chip using the optical microscope.
If the pattern is not completely developed, I follow the above procedure for 5—10 3
as needed. Figure (4.3) shows the pattern after it had been developed. Once the
patterns have been developed properly, the chips are ready to be transferred to the
thermal evaporator for Ag deposition.

An Edwards 306 thermal evaporator was used for this purpose. In this run, I

loaded 18 chips into the evaporator, to be evaporated at the same time. All sam-

64

ples were evaporated at the same time to make sure that all of them have the same
microstructure and the same amount of disorder (i.e same diffusion constant). Each
chip contains two samples on it, so in total I have more than 30 samples. A Molyb-
denum boat was used on which pellets of 99.9999% pure Ag are placed. The chips
are mounted about 30 cm directly above the source. The chamber is then pumped
down for one hour to a vacuum of about 10’6 Torr. After one hour of pumping down
and keeping the shutter closed, I melt the source by passing a current of about 1.8
A into the boat. At this point the source melts and the chamber pressure goes up,
therefore I wait one more hour for the chamber pressure to come down to 10"6 Torr.
Melting the source first helps cleaning the source and baking the chamber at the same
time. This step is important to get very pure Ag samples. One hour after melting
the source, I add Liquid N2 to the cold trap. This helps lowering the pressure by
freezing the water vapor and other contaminations to the walls of the cold trap. One
hour after adding liquid N2, the chamber pressure goes down to around 10‘7 Torr.
The total pumping time from closing the chamber to starting a real evaporation is
around three hours.

Now the chamber is ready to start the evaporation. I start the evaporation process
by passing the current slowly into the Molybdenum boat. I usually increase the
current by 0.2 A every 15 8. During this heating process and if the chamber pressure
goes up, I wait until the pressure falls again. When the current reaches roughly
1.8 A, the Ag starts to evaporate at a small rate. The evaporation rate is measured
using a crystal thickness monitor, which is placed next to the mounted chips. Once
the evaporation rate approaches 2 A/s, I open the shutter and zero the thickness
monitor. When the desired thickness (45 nm) is deposited, I close the shutter and
decrease the current fast. Then I wait around 20 minutes to let the chamber cool
down before I open it to air.

Once the samples are taken out of the evaporator, they are ready for lift-off of the

65

excess metal. First, I fill a small beaker with Acetone, and then set it on a hotplate.
Once the Acetone starts boiling (at 65°C), I take the beaker out of the hotplate and
immediately place one sample face up at the bottom of the beaker. One may need
to agitate the solution by rotating the beaker and using an eyedropper to speed up
the process by flushing the surface with hot Acetone. After a few minutes, the excess
metal and the resist will be removed, then I take the sample out and immediately
wash it with IPA and dry it with N2 gas. Finally, I use the optical microscope to check
the sample quality. One can use the dark image option in the optical microscope to
check the continuity of the wire and the other leads. It is not recommended to use the
SEM to check the sample, since the electron beam may harm the wire. I follow the
same procedure to lift-off the other samples. Figure (4.4) shows one of the evaporated
wires. Once done, I keep my samples in a safe holder and then take them outside the

clean room to start preparing for the next step; ion-implantation of the samples.

4.2.4 Ion implantation

Ion implantation is a process by which ions of a material are accelerated to a high
energy and speed to be injected into a solid. A typical ion implantation experiment
consists of an ion source, where ions of the desired element are produced, an acceler-
ator, where the ions are accelerated to a high energi, and a target, where the dopant
ions come to rest. When the ions collide with the surface, they lose their energy due
to collisions with target electrons, and finally come to rest at some depth in the tar-
get. The penetration depth depends on the ion type, ion energy, and the composition
of the target.

The amount of ions that is delivered to the target is known as the dose and this
can be accurately measured by monitoring the ion beam current. The total number
of ions incorporated into the solid is determined by the ion flux and the duration of

implantation. In this work, the target was Ag and the implanted ions were Fe. The

66

 

Figure 4.4: SEM picture of silver wire after evaporation and lift-off. The width of the
wire is around 130 nm.

67

 

n,(ppm) n,(cm‘3) Area Dose(cm'2)
2 11.8 x 1016 0.53 x 1012
6 35.4 x 1016 1.59 x 1012
10 59 x 1016 2.66 x 1012

 

 

 

 

 

 

 

 

Table 4.2: The calculated area doses for three different concentrations of the im-
planted ions into 45 nm Ag host, calculated from equation (4.1).

area dose is defined as the concentration of the ions incorporated into the solid times

the thickness of the film. This can be written as

AreaDose(cm'2) = n,(cm‘3) x t(cm) (4.1)

where n,(cm"3) = n,(ppm) x 10‘6 x n(cm“3), n is the particle density of the target
(for Ag n = 5.9 x 1022crrf3), n,(ppm) is the concentration of the implanted ions
expressed in units of ppm, and t is the thickness of our wires (45 nm). Table 4.2.4
lists the calculated area doses for three different concentrations of the implanted ions
into 45 nm Ag host.

Before performing the ion implantation, we have to know what is the ion energy
needed to have most of the implanted Ions stopped in the thickness of the wire. If the
ions energy is small, the ions will not be able to penetrate deeply inside the wire. On
the other hand, if the ions energy is high, most of the ions will escape to the substrate
under the wires. To simulate the ion penetration through the thicknesses of both the
wire and the substrate, I used SRlM-2003 software. This program is fed with the
type and thickness of each layer involved in the system, and the type of the ion and
its energy used in the implantation. In this work, the 45 nm Ag wire represents the
first layer, and the silicon oxide (300 nm), which sits underneath the wire, represents
the second layer. The results of these simulations indicate that at an ion energy of 70
keV, about 90% of the implanted ions stay in the Ag wires, with the rest going into
the silicon oxide layer (see figure 4.5).

All the evaporated samples were divided into four batches. One of them was kept

68

 

 

 

 

 

 

 

  

 

 

Ion Range = 264 A Skewness = 1.2051

Stragr e = 167 A Kurtosis = 5.0388
2: 74* 7*? A v _.m,‘. 24x104
5
3 20m“
2
O
5 m VA 1514104
\
1; ,7, 4.7-«.7 if -v#—# a; 12x10“1
5 Ion type : Fe
5 8110‘
3 Ion energy = 70 keV
: 4x104

0 A 450 A - Target Depth - 34 0 A

(A)

Figure 4.5: The distribution of the implanted Fe Ions in the target. The energy of the
Fe Ions is 70keV. Notice that most of the implanted Ions come to rest at the middle
of the wire.

pure, and the rest were sent to be ion implanted with 2, 6, and 10 ppm of Fe impurities,
respectively. The samples which needed to be ion implanted, were mounted on silicon
wafers. Each wafer holds three samples. The samples then were sent to IMPLANT
SCIENCE corporation for ion implantation. Unfortunately, the 6 ppm batch did not
survive, and here we report data taken on a pure sample and samples with 2 and 10

ppm Fe impurities.

4.3 Measurement methods

4.3.1 Mounting and testing samples at room temperature

Once done with sample fabrication, the sample is then mounted on top of a sample

holder as shown in figure (4.6a). The sample holder has a sheet of copper-kapton with

69

(a)

 

Gold wire

 

 

 

Sample 2

 

(b)

891:!) V1~0.1v
100k!) V=Read out of the DMM
9V __ V1
_100(VN1)
”‘9 R” _ l-(V/Vl)

 

 

Figure 4.6: Schematic diagram showing a) the sample holder and b) The circuit used
to test samples at room temperature.

copper etched away forming 9 distinct pads. The sample is glued to the central pad
by silver paint. The remaining 8 pads are used for electrical connections. Gold wires
are used to connect the sample pads with the copper pads by silver paint. The copper
pads have Cu wires soldered to them. The other ends of the Cu wires are connected
to 6—pin Microtech connectors. Each chip has two samples on it, where each sample
is connected to 6-pin Microtech connector. While mounting the two samples, I keep
them shorted together by shorting the two 6-pin Microtech connectors together and to

ground through an external cable. The grounding shorts are important to ensure that

static electricity does not flow through the sample and destroy it, but rather through
the ground: the easy path. When I start working on the sample, I wear a wrist strap
to short my body to ground. The working area is also shorted to ground through
a grounding pad. To attach the gold wires to sample pads, I use the dental pick.
Each time I need to use a dental pick, I touch it to the grounding pad to discharge it.
Once all the gold wires are connected to both samples, I carefully transfer the sample
holder to the probe, which is also grounded to the wall through the switch box.

The probe has a total of 18 wires, 4 of which are used for thermometry, 10 of which
for measurements, and the remaining 4 were previously for thermometry, but recently
have been modified for other purposes. The 10 measurement leads are divided into
4 sets. The first two sets of twisted pairs are used for voltage measurement, and the
remaining two sets of twisted triples are typically used for currents. To reduce the
noise in the leads, we use two types of filters; LC-7r filters and RC low-pass filters.
Each lead has these two filters. The LC-7r filter is a commercial filter with 1.75 nF
capacitance and a 50 dB roll-off at 100 MHz. This filter sits at the top of the cryostat.
The RC low-pass filters are located at the lower part of the probe, with a resistance
of 2.2 k9 and a capacitance of 1 nF.

Once the sample is mounted and wired properly to the probe, it is ready to be
tested for electrical continuity at room temperature. To check the electrical continuity,
we perform two—terminal measurements with the current passing through the sample
less than 1 pA. To get such a small current, we use the circuit shown in figure (4.6b).
All samples are made in a way that the outer pads are shorted together by metal
guards. So before testing the sample, the guards have to be removed by scratching
them carefully with a dental pick, while the sample is grounded. If the sample is
electrically continuous and has a reasonable value of resistance, then it is ready to be

lowered into the cryostat.

71

4.3.2 H e3 — H e4 Dilution refrigeration

The dilution refrigerator is a powerful tool to reach very low temperatures where
many interesting effects in mesoscopic physics take place. To cool the fridge down
from room temperature to 77 K, we use liquid nitrogen. Once the core of the fridge
reaches 77 K, the liquid nitrogen is replaced by liquid helium which cools the fridge
down to 4.2 K. To achieve lower temperatures, one can pump on the helium. The
lower the pressure above the liquid helium, the lower the boiling point of the liquid
helium. By pumping on the helium, it is possible to reduce the temperature to
around 1.2 K. The liquid helium undergoes a superfluid transition at 2.2 K. To reach
temperatures below 1.2 K, a different isotope of helium (H 63) is used. This isotope
boils at atmospheric pressure at 3.2 K. By pumping on liquid H e3 only, temperatures
of 300 mK can be reached. Even lower temperatures can be reached by mixing the
H e4 and H e3 in a dilution refrigerator. For the principle of operation of our dilution
refrigerator, I refer the reader to reference [51].

Most of my samples were measured in a top-loading H e3 — H e4 dilution refrigera-
tor. The base temperature of this refrigerator is around 35 mK. The probe, which is
around 0.6 inch in diameter, has a ruthenium thermometer from Oxford Instruments
calibrated from 4.2 K down to the base temperature. The thermometer is placed very
close to the substrate. Both sample and thermometer are immersed in the H e3 — H e4
bath. Conductance bridges were used to record the conductance of the thermometer
which is converted to temperature using a Labview program (TemperatureRecord.vi)
and calibration table. The mixing chamber is surrounded by a superconducting mag-
net provided by American Magnetics. The maximum field obtained by this magnet
is around 9 Tesla. This magnet provides vertical fields (up and down). Once the
current passing through the magnet is known, the magnetic field can be calculated
using the relation B(Tesla)=0.11884 I (Amps), where I is the current in Amperes. A

mixing chamber heater was used to control the temperature between 35 mK and 4 K.

72

Measurements above 4.2 K were performed using a pumped H e4 cryostat (1K-
Cryostat) which has a base temperature of around 1.3 K. This cryostat has a Ger-
manium thermometer calibrated in the temperature range of 1.4 to 100 K. The
conductance of the Germanium resistor was recorded using a Labview program (
TemperatureRecord-Ce.vi ). This Cryostat is equipped with an American Magnetics
magnet with a maximum field of around 9.0 Tesla. The magnetic field generated by
this magnet can be calculated from the relation B(Tesla)=0.12066 I (Amps), where I

is the current in Amperes.

4.3.3 Sample characterization

To calculate the phase coherence time, 71),, of a sample, we need to know the di-
mensions of the sample (L, w, t) and its diffusion constant, D. Table (4.3) lists the
Geometrical and electrical characteristics of the measured samples. The resistivity
was calculated using Ohm’s law, p = th / L, where R is the sample resistance mea-
sured at 1.2 K .

The diffusion constant, D, was determined from the resistivity and the Einstein
relation p"1 = e2D1/F, with the density of states in Ag Up = 1.03 x 1047J’1m'3.
The uncertainty in D mostly comes from the uncertainty of measuring the width of

the wire, w, where (SD/D = \/((5R/R)2 + (ciL/L)2 + ((Sw/w)2 + (6t/t)2. The width

 

was measured using SEM with an accuracy of around 10 %. The thickness was
measured using AFM with an accuracy of less than 3 %. The mean free path, lc,
was then calculated using the relation D = Uple/3, with the Fermi velocity in Ag

up =1.39 x106 m/s.

4.3.4 Low field magnetoresistance

Weak localization is a powerful technique used to measure the phase coherence time,

71,, in metallic wires. The change in the resistance AR(B) as a function of applied

73

 

 

Sample L t w R p D le cimp
(um) (nm) (nm) (9) (uflcm) (cmz/S) (nm) (ppm)
Ag 780 47 130 3307 2.59 146 32 -
AgFel 780 47 1 10 3890 2.59 146 32 2
AgFe2 780 47 185 2330 2.59 146 32 10

 

 

 

Table 4.3: Geometrical and electrical characteristics of the measured samples at 1.2 K.
L, t, and w are the sample dimensions obtained from the SEM and AFM measure-
ments. R and p are the resistance and resistivity of the measured samples. D is the
diffusion constant and c,-mp is the implanted Fe concentration.

magnetic field, B, was measured using the circuit shown in figure (4.7). The applied
magnetic field, B, was perpendicular to the film. The size of the typical resistance
correction due to weak localization is around 10‘3 R or less. Since we are interested
in measuring the change in the resistance, AR(B), rather than the resistance itself,
R, a ratio transformer was used to enhance the lock-in sensitivity. The change in the
resistance was measured using a standard ac four-terminal technique with a lock-in
amplifier.

The drive signal at the output of the lock-in was taken to two different paths. In
the first path the drive signal is fed into the AC input of the sum box, which breaks
the ground of the input signal and divides it by 100. The DC input of the sum box
was shorted. The output of the sum box goes into a current limiting resistor (Ballast
resistor ) of 1 M9, passes through the sample, and finally goes back to the ground
of the lock-in. An NF (LI-75A) preamplifier with a gain of 100 was used to measure
the voltage drop across the sample. The input voltage noise of the preamplifier is
1.5 nV/x/I-E. The output of the preamplifier was then taken to the A-input of the
lock-in. In the second path (see figure 4.7), the drive signal was passed through an
isolation transformer to break its ground and then passed through a ratio transformer,
whose gain could be adjusted to balance the sample resistance. The output of the
ratio transformer was then fed into the B-input of the lock-in. The ratio transformer

enables us to subtract the B—input from the A-input of the lock in. By adjusting the

74

 

 

—[ Isolation Tmnsfonn;[—[;tio Transformer

G=l

  
  

“NH 939:1 MOI OH

  

AC (G=l/100)

 

Sin out

Lock-In

 

 

4r

 

Shielded Room

 

)

Figure 4.7: Schematic diagram showing the circuit used to measure the change in the
resistance due to weak localization. The ground breaking box (GBB) is a differential
amplifier which breaks the ground between the lock-in output signal and the shielded
room.

ratio transformer gain and setting the lock—in to be in (A-B) mode, the X-channel
of the lock-in will read essentially zero. In this case we can View the change of the
resistance at much higher sensitivity. A low pass filter was introduced at the output
of the ratio transformer to adjust for the phase shift of the signal of interest. The
capacitance of this filter was fixed to 0.02 pF and the resistance was varied between 1
and 10 k9, to minimize the Y-channel of the lock-in. At C = 0.02 MP and R = 1 k9,
the low pass filter has a roll-off of about 8 kHz, which is well above the drive frequency

(f o = 228 Hz). Therefore, the filter does not attenuate the signal much at the output

75

of the ratio transformer, it mainly shifts its phase. This phase shifter compensates for
the phase shift in the measured voltage signal across the sample. The phase shift in
the measured signal originates from the capacitance in the circuit which mainly comes
from the capacitance of the low-pass filters. Using this set up, the lock-in measures
a voltage signal corresponding to the resistance difference between the sample and a
reference value set by the ratio transformer. The output of the lock-in can be written

as

Vlock—in : VA _ VB

sum m X GNF — TGLIO) (4.2)

where Gsum and GNF are the gains of the sum box and the NF preamplifier, R, and
R3 are the resistances of the sample and the Ballast resistor, and ratio is the ratio
transformer setting. A typical value of the ratio transformer setting depends on the
value of R,. For example, if R, = 2330 {2, the ratio transformer setting would be
ratio = 0.00233.

All samples were measured at the same drive frequency (f0 = 228 Hz). The voltage
drop across the sample was limited to eV 5 kBT, to avoid heating the electrons in the
wires. Measurements of weak localization magnetoresistance below 500 mK, required
using longer time constants (r = 3 sec) to have better signal to noise ratio. Above
500 mK, one can drive harder, and hence use shorter time constants (r = 1 sec, or
0.3 sec). The output of the lock-in goes through ground breaking boxes, and then
is fed into a Keithley 199 model digital multimeter. The output of the Keithley was
recorded by Labview program (XYchartrecorder.vi).

The magnetic field was applied perpendicular to the surface of the film (z-direction).
Figure (4.8a) shows the circuit used to generate magnetic fields in the z-direction. By

connecting the two magnet power supplies (1, 2) in parallel, one can change the cur-

76

rent direction in the solenoid, hence changing the field direction. To reduce the sweep
rate, four shunt resistors were connected in parallel to the power supplies with several
configurations. The slowest sweep rate (0.0015 A / s) was achieved by connecting the
four shunt resistors in parallel (see figure 4.8b). The current through the supercon-
ducting magnet was determined by measuring the voltage across the 0.1 Q resistor
connected in series with the magnet. This 0.1 Q resistor can handle currents up to
15 A. The voltage across the 0.1 Q resistor was recorded by Labview program (XY-
chartrecorder.vi), where the X-channel measures the voltage across the 0.1 Q resistor
and the Y-channel measures the voltage across the sample. Figure (4.9) shows typical

magnetoresistance data for the pure sample taken at 1.6 K.

77

(a)

power supply (1) 11

 

I2

 

V

_©+

power supply (2)

+@-

 

‘7

 

 

 

 

go...

Keithley [

VI

 

 

ww

Shunt Resistors

.l
Persistaht swich
heater

 

 

(

 

Labview

 

 

 

 

WW

 

PC

 

 

k

‘—___l

Superconductor wire

[.___

 

Solenoid

 

 

 

Cryostat

 

(b)

fast ramping configaration

slow ramping configaration

 

 

Figure 4.8: (a) Schematic of the circuit used to generate magnetic fields in the z-
direction. (b) Two different configurations for connecting the shunt resistors to control
the sweep rate. Typical value of the shunt resistor (R) is around 1 m9.

78

 

AR/R

 

 

 

L l I. l A I I I l 1 l

-150 -100 -50 0 50 100 150

B(mT)

 

Figure 4.9: Typical magnetoresistance trace of the pure sample taken at 1.6 K. the
solid line is the fit to the lD-weak localization expression

79

Chapter 5

Results and Discussions I: Phase

Coherence

5.1 Introduction

In this Chapter, we first discus how L¢, and hence T4,, can be determined from mag-
netoresistance measurements on our quasi-1D Ag wires. Then, we show how the
magnetic scattering rate 7m, due to the implanted Fe impurities, can be extracted
from the measured total dephasing rate, 74,. And finally, we compare our experimen-
tal 7", data with a recent theory [27] of scattering of conduction electrons by dilute

magnetic impurities in quasi-1D wires.

5.2 Weak localization and phase coherence

Our wires are quasi-1D in the sense that w,t << L4, << L. In our wires, the electron
motion is diffusive due to scattering off static disorder. The phase coherence length
of the conduction electrons, L¢, depends on both disorder and temperature. All our
samples, presumably, have the same amount of disorder, since they were evaporated

simultaneously. The disorder dependence shows in the value of the diffusion constant,

80

D. Here, we investigate the temperature dependence of the phase coherence length,
L», and hence the phase coherence time, 71,, (L), = M), in two type of samples;
pure and implanted samples. Typically, L4, can be several microns in pure metallic
wires at liquid helium temperature [5]. The most reliable way to estimate L, is to
use the method of weak localization magnetoresistance (WLM). The WLM signal is
proportional to L4,. Measuring L, at low T provides information about the scattering
mechanisms of the conduction electrons.

In the presence of magnetic field perpendicular to the wires, the weak localization
contribution to the resistance will be suppressed, as discussed in Chapter 2. The
magnetoresistance is the change in the resistance of the wire due to an applied mag-
netic field. In pure weakly disordered wires, the field scale of this effect is around 0.03
Tesla, therefore it is called low-field magnetoresistance. However, in samples contain-
ing small amount of magnetic impurities, the field scale depends on L4,. We have
mentioned in Chapter 4 that the size of the resistance correction due to this effect
is around AR(B) / R = 10'3 or less. In the low-field limit, the classical magnetore-
sistance due to the Lorentz force, AR(B) / R as (war)2 [13], where we = eB/m is the
cyclotron frequency, is negligibly small. In our wires where 1,, z 32 nm, in a magnetic
field of 1 T this classical magnetoresistance is of the order of 10‘5 R, which is typically
two order of magnitudes smaller than the weak-localization magnetoresistance. By
scanning the magnetic field between -0.03 Tesla to 0.03 Tesla, one can measure the

resistance change due to electron weak localization, hence indirectly measure L¢.

5.2.1 Total dephasing rate : pure samples

In this work, two pure samples were measured in two different cryostats. The first
pure sample was measured in an Oxford dilution refrigerator between 40 mK and 4.0
K. I will call this sample (puresamplel). This sample was measured directly after the

evaporation. The diffusion constant of this sample was 146 cmz/s. Unfortunately,

81

 

puresample1

 

 

AR/R

 

 

 

 

B(mT)

Figure 5.1: Magnetoresistance data of puresample1 and fits to equation (2.8). This
sample was measured in the dilution refrigerator. The Ag wires were made from
source materials of nominal purity 6N (99.9999%). The geometrical and electrical
characteristics of this sample is the same as the Ag sample given in table (4.3). The
spin-orbit length extracted from the MR fits was 0.76 pm. The B offset is due to
trapped flux in the superconducting magnet.

this sample did not survive to be measured again in the high temperature cryostat
(1K-cryostat). Therefore, we had to measure another pure sample to acquire data
above 4.0 K. The second pure sample was measured using a pumped H e4 cryostat
(1K-cryostat) between 1.3 K and 18.0 K. I will call the second sample (puresample2).
This sample was measured five months after the evaporation. The diffusion constant
of this sample was 120 cmz/s. The lower value of the diffusion constant of this sample
could be due to some oxidation of the silver wire.

Figures (5.1) and (5.2) show magnetoristance data and fits to equation (2.8).

Magnetoristance data for each sample were fit using the following procedure [52]. We

82

 

        

l..’

3‘
' "~._’ 0
5

kg (5‘
I I e (\\\

'f
f ' \‘\\
' V. I .l‘l‘r’" v (6., \
“01.4“ 11.0!“ I“ .l (\ - \‘\'L\\ ,
‘ -‘ x - \

 

'- '2
o-‘.
‘~‘

l-n' '
\\ "”(r\\‘u‘"‘"'(‘_’ -.,,- luf"
‘ . "\\\\ \ I
\ ‘ “

(C.

AR/R

 

 

 

. ‘.
' "
( \ke ‘. " ‘1' ('0
-3 5 K 1' \ fi' " \ .\
" - , we” “w ,
r f mm ' 'rs“"~ .. ‘. (““5fv} it (
7.0.41” .r» , -“"'.‘".3“ "‘ "W n 0 “We" (.4 (""ef": ,, , , m
\‘ \N 609° 0 '8‘ '\\"\'«\\ u
0
0‘”
_4 0.0
1 O
I
q"
1'»
‘I
n I 1 l 1 I 1
-200 -1 00 0 1 00 200

B(mT)

Figure 5.2: Magnetoresistance data of puresample2 and fits to equation (2.8). This
sample was measured in the 1K-cryostat. The geometrical and electrical character-
istics of this sample are: L = 780 pm, 10 = 137 nm, t = 47 nm, D = 120 cmz/s,
L80 = 0.72 pm, and R = 3824 k9. The Ag wires were made from source materials of
nominal purity 6N (99.9999%).

have mentioned in Chapter 2 that what determines the MR lineshape (peak or dip
around the zero field) is the strength of the spin-orbit coupling. Silver is considered
to have moderate spin-orbit scattering [53] (r80 z 40 ps for all of our samples). At
low T, where 1",, >> 7‘”, the MR is positive (dip). At higher temperatures, where
r¢ < 130, the MR starts out positive but then turns around at a field scale B as
20 mT (see figure 5.2). Data at several temperatures in this higher temperature
range (which was different for each sample) were first fit with three free parameters:
L), = \/Dr_¢, where D is the diffusion constant, L80 = m, and the sample width,
w. Furthermore, the MR data are known only up to an additive constant which was

adjusted to fit each MR curve. For each sample, these fits gave consistent values of

83

L50 and w over a broad temperature range. We then fixed those values of L30 and w,
and fit the MR curves for all temperatures with L, as the only free parameter. The
difference between the width measured from the scanning electron microscope (SEM)
and the best fit value of w, was found to be always less than 20%. To calculate the
diffusion constant, D, we used the width value measured from the SEM. Using the
value of D obtained from the resistance and sample dimensions, we finally obtain 71,,
as a function of temperature for each sample. Figures (5.1) and (5.2) also show that
the size of the weak-localization signal increases with decreasing temperature due to
the progressive freezing of the inelastic scattering events, specifically e-ph scattering
events.

Figure (5.3) shows the total dephasing rate, 7,5, as a function of temperature for
both puresample1 and puresample2. Horn now on, I will refer to this figure shortly as
puresample or Ag sample, which will be used in the comparison with the implanted
samples. Furthermore, the parameters of puresample1 given in table (4.3) will be
used in future calculations. In pure metallic wires, as discussed in chapter 2, the total
dephasing rate takes the form, 7¢(T) = AT2/3 + ET3 [1, 11], where the first term
describes the electron-electron scattering rate, dominating at lower T, and the second
term describes the electron-phonon scattering rate, dominating at higher T. The e-ph
scattering rate (BT3) corresponds to the clean limit (que >> 1), where qT is the wave
vector of a typical thermal phonon at temperature T, and la is the elastic mean free
path of conduction electrons. The clean limit approximation is usually valid at high
temperatures. The expected values of the prefactors Athy and Bthy are 0.32 ns‘lK‘W3
and 0.002 ns‘lK‘3, respectively. The solid line in the figure is the fit of 7¢(T) data to
this functional form. It can be seen from the figure that the e-e interactions dominate
the scattering below 1 K, and the e-ph interactions dominate at temperatures above
1 K. The inset shows the output of the prefactors Am, and Bap extracted from the

least-square fit. While the prefactor Awp a: 0.36 ns'lK‘f‘V3 is in good agreement with

84

the expected value, Amy z 0.32 ns'lK‘r‘V3 [52], the prefactor Ben, m 0.026 ns‘lK‘3
is higher than the predicted value, Bthy m 0.002 ns'lK“3 by roughly one order of
magnitude. As discussed in Chapter 2, the relatively big difference between Bthy and
Bap, indicates that the e-ph scattering is not limited to one regime (clean limit) over
the temperature range of 40 mK - 18 K, instead it crosses over between them (see
the discussion of e-ph interactions given in Chapter 2). The 7,), data, as seen from the
figure, follow the model above 200 mK, with a modest amount of saturation at lower
temperature. This, however, may possibly be attributed to the presence of extremely
dilute magnetic impurities with low Kondo temperatures, T1,», in the source material,
Ag(6N) [5]-

At low temperatures, the scattering of conduction electrons by magnetic impu-
rities plays a major rule in electron dephasing. In the presence of extremely dilute
magnetic impurities (qmp << 1 ppm) with low T1,», the total dephasing rate takes the
form, 7¢(T) = ATI‘V3 + BT3 + 78f(T) [5], where 73f is the spin-flip scattering rate of
conduction electrons. Above TK, the spin-flip scattering rate can be described well
by the Suhl-Nagaoka (SN) expression [17, 21] (see equation 1.15). Looking carefully
at this expression (SN expression), one can see that the spin-flip scattering rate, 73;,
increases when T approaches TK from above. On the other hand, the e-e scatter-
ing rate, 766(T), decreases with decreasing temperature. The competition between
these two rates creates a nearly constant dephasing rate above TX. This might be
responsible for the observed saturation seen in the pure samples below 200 mK.

To test this hypothesis and to have an estimate of the concentration of the pro-
posed magnetic impurities in our pure wires, we refitted the total dephasing rate,
7¢(T), taking into account the additional contribution of spin-flip scattering interac-
tion due to extremely dilute low TK magnetic impurities, as well as the two major
sources of electron dephasing at low T, e-e and e-ph interactions. Assuming the ex-

istence of Mn magnetic impurities, for example, with s=5/ 2 and TK = 40 mK [5],

85

 

I UUII‘Tr I I U'UUWT r I IUI'I'I 1 U

_ puresample

 

100

Data: sample1 0ppm_J
Model: 1/Tou-phi

I 'T'UUU'
I j LlIIIl

Chi‘2/DOF = 70.00201
R"2 = 0.99216

10

A 0.35657 i0.00666
8 0.0261 1 $000092

 

I IU'U'I'
L I I Illll

-1
r, ("S )

I I IIIIII

V
I

0.1

 

1 U U "I"
I IIIIIII

 

 

I IIIIJII I I IIIIIII I I IIIIIII I I

0.1 1 10
T (K)

Figure 5.3: The total dephasing rate, '74,, as a function of temperature for both
puresample1 and puresample2. The solid line is the fit of 7,, to the form 'y¢(T) =-
ATS”3 + BT3, assuming that our wires are free of magnetic impurities. The inset is
the output of the least square fit of the two prefactors, A and B.

 

and using the Suhl-Nagaoka expression for 7,;(T) and leaving the concentration c as
a free parameter, the fit is improved below 200 mK, as shown in figure (5.4). The
parameter c z 0.01 ppm in the inset is the concentration of the Mn magnetic impu-
rities, extracted form the fit, expressed in units of ppm. This concentration is about
100 times smaller than the nominal total impurity concentrations of any silver source
available in the market. Such a small concentration is undetectable by any means
other than measuring the phase coherence time in films [5] A similar fit was obtained

by assuming the presence of Cr magnetic impurities with s = 2 and TK = 10 mK. The

86

 

I- IrIIII' r I II—IIII' I I IIIIII' I I

puresample

I I

 

100

Data: sample10ppm_J
Model: 1/T-phi-mag

I I fIIII'
I J I IIIII

Chi‘2/DOF = 48.84105

 

- R"2 = 0.99472 .

10 ,- A 0.32702 10.00976 1

A 5 B 0.02747 40.00035 3

‘7 - c 0.0122 40.00331 .

U) . .

C _ ‘

v

Q 1

I IIIIII'
I IIIIIII

I
I

0.1

 

I I I IIII‘I’
I IIIIIII

 

 

I I IIIIII I 4 I LLIIII I I I IIIIII I I

0.1 1 10
T (K)

Figure 5.4: The total dephasing rate, 7,», as a function of temperature for both
puresample1 and puresample2. The lower solid line is the fit of 74, to the form
7¢(T) = AT2/3 + BT3, assuming that our wires are free of magnetic impurities.
The upper solid line is the fit of 74, to the form 7¢(T) = AT2/3 + BT3 + 7,f(T), where
7,, is the spin-flip scattering rate of conduction electrons due to extremely dilute
magnetic impurities, (c,-,,,p << 1 ppm). The parameter c z 0.01 ppm in the inset is
the concentration of the Mn magnetic impurities, extracted form the fit, expressed
in units of ppm. Similar fit was obtained by assuming the presence of Cr magnetic
impurities with s = 2 and TK = 10 mK. Notice how the fit is improved with the
addition of a very small number of magnetic impurities.

 

87

issue of electron dephasing by extremely dilute magnetic impurities was discussed in
detail in reference [5], in the context of the debate over zero-temperature dephasing
in disordered metals [4].

In summary, the total dephasing rate of our pure wires was measured in the
temperature range of 40 mK - 18 K. In the next sections, this total dephasing rate
of the pure wires will be subtracted from the measured total dephasing rate of the
implanted samples, to extract the magnetic scattering rate (spin—flip scattering rate)

due to the implanted Fe magnetic impurities.

5.2.2 Total dephasing rate : 2 ppm sample

In the implanted samples, the phase coherence length, L», must be shorter due to
the scattering of conduction electrons by Fe magnetic impurities. How short L,» is
depends on the concentration of the magnetic impurities. As discussed in Chapter 4,
2 ppm of Fe magnetic impurities were introduced into the thickness of the wires by
ion-implantation. The area dose equivalent to this concentration was 0.53 x 1012 cm”.
This sample was measured only in the dilution refrigerator from 37 mK - 4 K. Figure
(5.5) shows the MR curves of this sample taken at different temperatures.

Figure (5.6) shows the total dephasing rate, W» as a function of temperature for the
2 ppm sample. Notice that ’ch decreases with decreasing temperature. According to
theory, the total dephasing rate must vanish at zero temperature, where all inelastic
scattering events freeze out. At the lowest accessible temperature (35 mK in this
work), 7,, z 0.2 ns‘1 for this sample (see figure 5.6). As will be discussed later in this
Chapter, the Kondo temperature of our system (AgFe) is around 5 K [52], therefore
the 7,, data of this sample (2 ppm) falls below TK. Far below TK, Fermi liquid theory
predicts a T2 dependence of 7,, [25]. This, however, has never observed experimentally.
The phase coherence length, L45, of this sample at the base temperature was around

9.1 pm, which is shorter than that of the pure sample (L, = 14.6 pm). This decrease

88

 

AR/R

 

 

 

 

-60 .40 -20 0 20 40 60
B(mT)

Figure 5.5: Magnetoresistance data of the 2 ppm sample and fits to equation (2.8).
This sample was measured in the dilution refrigerator between 37 mK and 4 K. The
geometrical and electrical characteristics of this sample are listed in table (4.3) under
the name ”AgFel”. The spin-orbit length extracted from the fit was 0.86 pm. The
B offset is due to trapped flux in the superconducting magnet.

in L), is due to the addition of 2 ppm of Fe magnetic impurities.

5.2.3 Total dephasing rate : 10 ppm sample

This sample was ion-implanted with 10 ppm Fe magnetic impurities. The area dose
equivalent to this concentration was 2.66 x 1012 cm”. The sample was first measured
in the dilution refrigerator between 35 mK and 4 K. The magnetoresistance data of
this sample are shown in figure (5.7). The geometrical and electrical characteristics of
this sample are given in table (4.3), under the name ”AgFe2”. The phase coherence

length, L», of this sample at the base temperature was around 4.1 pm, which is

89

 

Ifif‘ I I I IIIII'

10- 2 -.
f FWD .....0 5
O
’4'
. .
7w 1? . .
5 : ,.- '
>3 4 .0
. -1
O

I jIJIII

 

 

J I IIIIII I I J IIIIII I I JJ

0.1 1
T(K)

Figure 5.6: The total dephasing rate, 74,, as a function of temperature for the 2 ppm
sample in the temperature range of 37 mK - 4 K. Notice that 74, continue to decrease
and vary lineally with temperature over almost on decade. The fit of data will be
discussed later in this chapter.

 

shorter than both the pure and the 2 ppm samples. This decrease in L4, again is due
to the addition of 10 ppm of Fe magnetic impurities.

To acquire data above 4 K, the sample was then measured again in the high tem-
perature cryostat (1K-cryostat). The magnetoresistance data of this sample, above
4 K, are shown in figure (58). Unfortunately, the sample resistance went up from
2330 k0 to 3879 kl). This may have happened during the mounting process. Even
one single bad spot created on the wire can take the resistance up by this magnitude.
The effect of this jump in the resistance of the wire reduced the measured diffusion
constant of the sample from 146 cm2/s to 87 cmZ/s. Nevertheless, after fitting the

MR data carefully in this high temperature regime, above 4 K, we found that the 74,

90

 

10 ppm (Fridge)

'—

- -gllr (if.lullll'_.~

2018 K --.e“:v ' \\‘ \ \ ‘-.f.“
v— -

. . -- > _ :v;
'...V VOU'O'. -;;

(p I. 1'”, p , r0-
T=860 mK - - ., . -
' ll”""'...'“°... . \ _ k \\\~-\ \|"quu'(\‘o;“‘ e
‘. “

1‘
r'rrur- .‘ ‘\~\'-'. '."'- .00 o.

:\‘,'" -\v- 1‘ “"\\W us-

T=305 mK

AR/R

T=36 mK

   

1 10"

 

 

I I I I I I I I M I I I I I I

-80 -60 .40 -20 0 20 40 60 80
B(mT)

Figure 5.7: Magnetoresistance data of 10 ppm sample and fits to equation (2.8).
This sample was measured in the dilution refrigerator. The geometrical and electrical
characteristics of this sample are listed in table (4.3) under the name ”AgFe2”. The
spin-orbit length extracted from the MR fits was 0.72 pm. The MR data at 36 mK
are noisy due to the fact that the voltage across the wire was limited to eV S, kBT,
to avoid heating the electrons in the wire. The B ofiset is due to trapped flux in the
superconducting magnet.

 

data, extracted from fitting the MR data taken in the 1K-cryostat match well with
the 74,, extracted from fitting the MR data taken in the dilution refrigerator, below
4 K, (see figure 5.9).

The total dephasing rate of this sample, as shown in figure (5.9), decreases as T
approaches TK from above, has a plateau around the Kondo temperature (TK as 5 K),
and decreases linearly with T below TK. Again the T2 behavior of 74,, predicted by
Fermi liquid theory [25], was not reached down to the lowest accessible temperature

(35 mK).

91

 

AR/R

   
      

 

 

 

T=7.1 K .,

[2.10‘5
T=4.9K
400 I -2b0 I 0 I 200 . 400

B(mT)

Figure 5.8: Magnetoresistance data of the 10 ppm sample and fits to equation (2.8).
This sample was measured in the 1K-cryostat. The geometrical characteristics of
this sample are the same as the AgFe2 sample given in table (4.3). The electrical
characteristics of this sample are: R = 3879 k0, D = 87 cmz/s. The spin-orbit length
extracted from the MR fits was 0.49 pm. The B offset is due to trapped flux in the
superconducting magnet.

5.3 Extracting the magnetic scattering rate from
the total dephasing rate

In the presence of a small amount of magnetic impurities, there is an additional con-
tribution to 74,(T) from the scattering of the conduction electrons by the magnetic
impurities, 7m(T). As discussed in Chapter 1, the temperature dependence of the
magnetic scattering rate, 7m(T), is usually obtained by subtraction of the total de-

phasing rate of the pure sample from the total dephasing rate of the implanted sample

92

 

I I IrITI' I I I IIIII' I I I IIIII' I I

IIII

10 ppm -

I

100 '

IIIII

I 'IIIIII'

-1
7,018 )
I
'l

 

I IIIIIIT
I IJIIILI

I
I
IIILI

I I IIIIII I I I ILIIII I I I IIIII. I L

0.1 1 10
T00

Figure 5.9: The total dephasing rate, 74, as a function of temperature for the 10 ppm
sample in the temperature range of 36 mK - 18 K. Notice that 74 decreases with
decreasing temperature, has a plaeau around TK, and continues to decrease linearly
with temperature over almost one decade below TK. The fit of data will be discussed
later in this chapter.

 

 

 

(see figure 1.4). The method of addition (subtraction) of the total dephasing rates
was discussed in detail in Chapter 3, where we showed that there are two ways of
adding them (see equations 3.45 and 3.46), depending on the ratio 7m/7K, where 7g
is the Korringa scattering rate [43]. If 7m < 7K, equation (3.45) should be used for
adding the dephasing rates. On the other hand, if 7m > 7K, then equation (3.46)
should be used instead.

Below the Kondo temperature, T K, the Korringa rate saturates and the ratio
reads 7m/7K = ns/quBTK, where n, is the concentration of magnetic impurities

and up is the total DOS at the Fermi level. For the 2 ppm sample, and assuming

93

 

IIIII' I I I IIIII' I fTIIIII' r I

100 '

 

 

E f
V
>9 1 . A"
’3‘
A
0.1
n a 111m 1 a n 11“.!
1 10
100 .
w...-
10 r"
A '- all"
'm f
5 1 O.
o
:5

0.1

II... A A ALLA“. A L

1 10

T00

 

Figure 5.10: a) Total dephasing rate vs. temperature for a pure Ag sample (A), and for
samples with implanted Fe concentrations of 2 ppm (0) and 10 ppm (I). b) Magnetic
scattering rate vs. temperature for samples with implanted Fe concentrations of 2 ppm
(0) and 10 ppm (I), after subtracting the total dephasing rate of the pure sample.

94

TK z 5 K (to be shown later), 7m/7K = 0.02 < 1; hence we should use equation
(3.45) to extract 74,(T) from 74(T). Similarly for the 10 ppm sample, the ratio reads
7m/7K = 0.2 < 1; hence equation (3.45) should be used again to extract 74,(T) from
74(T). Figure (5.10b) shows the magnetic scattering rate for both' 2 ppm and 10 ppm

samples after subtracting the total dephasing rate of the pure sample.

5.4 Fitting the magnetic scattering rate to theory

of Micklitz et al.

In this section, we fit our magnetic scattering data to the theory of Micklitz et al.
[27]. This theory, as mentioned earlier, uses the numerical renormalization group
(N RG) method to calculate the complete temperature dependence of the magnetic
scattering rate of the conduction electrons from dilute magnetic impurities, in weakly
disordered metals. This theory was originally developed to describe the effect of a
small concentration, 72,, of spin 1 / 2 magnetic impurities on the dephasing rate from
T > TX all the way down to T << TK. However, in our wires, n, is small but 3 = 2
instead. As predicted by this theory, the magnetic scattering rate, 74,(T), due to
very dilute magnetic impurities is a universal function, 74,(T) or n, f (T/TK), and
depends only on two parameters, n, and TX. This theory has maximum predictive
power if both the concentration , n,” and the Kondo temperature, TK, are known.
In this work, only the concentration, as, is known from the ion-implantation process.
Therefore, TK, will be a free parameter in the following fit. The following chapter will
discuss our attempt to obtain an independent estimate of TX.

The criterion 7m/7K << 1 is a necessary condition for the theoretical approach
of Micklitz- et al. [27] in their NRG calculations of 7",. In this work, our 7m data
are within this limit, therefore we used equation (3.45) to add the dephasing rates

due to all mechanisms (Matthiessen’s rule). Figure (5.11) shows 7m for the 2 ppm

95

and 10 ppm samples obtained from the 74 data using the equation (7m = 74 — 7m),
with 74,, obtained from the data on the nominally pure sample (figure 5.3). The
2 ppm data is multiplied by 5 to scale it with the 10 ppm sample. The two data
sets are consistent with each other, indicating that our measurements of 74 for both
samples were accurate. This figure summarizes our work for the phase coherence in
the implanted samples [52].

The dotted line in the figure is the Suhl-Nagaoka (SN) approximation for s=1/2
magnetic impurities. As seen from the figure, the SN approximation describes well
our 7m data for T > TK. On the other hand, for T < TK, the SN approximation does
not come close to reproducing the temperature dependence of 7",, consistent what
was observed in reference [54].

The solid line in figure (5.11) shows a fit of the NRC calculation from [27] to our
data at temperatures above 0.4 K only. Our fit was restricted to T > 0.4 K, because
including data below 0.4 K makes the overall fit distorted. Fitting to the 2 ppm
(10 ppm) data alone gives TK = 4.8 K (5.4 K). The Kondo temperature here represents
the temperature at which the magnetic scattering peaks. This peak is broad around
the Kondo temperature, as predicted by theory. The NRC theory of Micklitz et al.
[27] fits the data reasonably well over the temperature range T /TK = 0.1 — 2 [52].
This fit is much better than that obtained using the SN approximation for the decade
below TK. For T/TK < 0.1, the NRC theory of Micklitz deviates strongly from the
7m data. Above TK, 7m data is almost constant. Below TK, 7m varies linearly in T.
Below 0.1 TK, the temperature dependence of 7m is much weaker than that given by
theory, and it certainly doesn’t approach the T2 behavior predicted by Fermi liquid
theory [25].

According to theory of Micklitz, the maximum value of 7m, occurring at T = TK,
is equal to 0.23 x 4n8/(7rpr). For the 10 ppm sample, the predicted maximum of 7m

is 7Q“ = 16 ns‘l, whereas the data in figure (5.11) show a maximum value about

96

 

firl‘VVII W I IVI’II'" I l Ill'lIl

 

 

 

 

 

 

 

Figure 5.11: Inelastic scattering rate due to magnetic impurities for the 2 ppm (0)
and 10 ppm (I) samples. Data for the 2 ppm sample are multiplied by 5. The solid
line is the theoretical calculation of Micklitz et al. [27] fit to the data for T > 0.4 K.
The dotted line is the Suhl-Nagaoka approximation for s=1 / 2. Inset: Same plot with
a linear vertical scale. Uncertainties are larger for the 10 ppm sample due to smaller
signal-to-noise in the measurements.

twice as large (m 34 ns'l). This factor 2 difference between theory and experiment
could be due to the inadequacy of the spin-1/2 theory of Micklitz et al. [27] to
account for the large spin (s=2) of Fe impurities in Ag. On the other hand, previous
measurements of the magnetic scattering rate, 7",, in Ag samples implanted with Mn
impurities [5], with s=5/ 2, were consistent with the theoretical estimate. Due to the
low Kondo temperature of Mn in Ag (TX z 40 mK), those data were analyzed with
the SN approximation, which is in close agreement with the theory of [27] for T > TK.

We conclude that the inelastic scattering cross section of Fe in Ag is roughly twice

that of Mn in Ag.

97

The most serious discrepancy between theory and experiment is the flattening of
74,(T) for T / T K < 0.1, shown in figure (5.11). The N RG calculations of Micklitz were
based on the assumption that for spin 1 / 2 magnetic impurities and at temperatures
far below TK, the magnetic impurities are totally screened by conduction electrons.
This state is called compensated state, with (n = 25) [18], where n is the number of
channels and S is the spin of magnetic impurity; hence n = 1 in this model (single—
channel Kondo model). For Fe, which has 6 electrons in the 3d orbital, the ground
state according to Hund’s rule has a spin S=2, so there will only be perfect spin
compensation (n = 25) if the number of channels is 4 (n-channel Kondo model)
[18]. Sacramento and Schlottmann showed that a multi-channel Kondo model with
S=2 and n=4 gave a good fit to data for several equilibrium quantities in the Ange
and CuzFe systems [29]. A calculation of the phase decoherence rate in that model
might resolve the discrepancy between theory and experiment shown in figure (5.11).
The issue of screening for Fe in Ag is unclear. The Fe spin (S=2) is coupled to the
conduction electrons through five channels corresponding to the d-orbitals of Fe, so
the S=2 Fe spin should be overscreened. In reality, however, the different channels
have different coupling constants, and the impurity spin and orbital degeneracies are
broken by crystal fields and spin-orbit scattering. A recent calculation of the inelastic
scattering cross section for the the case where screening is incomplete (with 8:1
and n=1) shows that such a model can not fit our data for 0.1 < T/TK < 1 [55].
The multi-channel Kondo model of Fe magnetic impurities in Ag has been recently
investigated in [56] for both underscreened (n < 25') and overscreened (n > 23).
Those authors conclude that neither underscreened nor overscreened models can fit
the low-temperature inelastic scattering rate. Pedro Schlottmann believes that the
flatting of 7m in our samples may be a finite—size effect [57]. As the temperature is
decreased, the effective dimensionality of the Ag wire seen by the Kondo impurity

may cross over from 3D to 2D or even 1D.

98

Chapter 6

Results and Discussions II: High

Field Magnetoresistance

6.1 Introduction

In this chapter we first show our data of the resistivity as a function of temperature
for both pure and implanted samples . Then we discuss the high field magnetore-
sistance of our samples. Specifically, we show how the Kondo temperature, TK, can
be determined independently from analyzing the high field magnetoresistance data
in the context of Numerical Renormalization Group (NRG) calculations of reference

[58].

6.2 Resistivity vs temperature

The electrical resistivity of a pure metal is dominated at high temperatures by colli-
sions of the conduction electrons with phonons. On the other hand, at lower temper-
atures, the electrical resistivity is dominated by collisions of the conduction electrons
by static disorder (static impurities, film boundaries, ....) Figure (6.1) shows the re-

sistivity as a function of temperature of puresample2 measured between 1.3 K-32 K.

99

 

' I I I l I l T I I l I I I

     
 

90 pure (1 K-cryostat) -
8° _ B=0.1 T l
70 - ‘
60 - 4

01
O
l

static disorder

Ap(n£2.cm)
0) «h

—t N
O O
l l

 

 

O
L

 

0 5 10 15 20 25 30 35
T(K)

Figure 6.1: Resistivity vs temperature of puresample2 measured between 1.3 K-32 K.
The resistivity was measured in a finite magnetic field of 0.1 T to suppress the weak
localization magnetoristance. The data do not have the right offset, so the value of
po shown in the figure is not the correct one. To calculate p0, one has to use Ohm’s
law, R = poL/wt, where R is the measured resistance at low T, and L,w, and t are
the sample dimensions.

A magnetic field of 0.1 T was applied to suppress the weak localization contribution
to the resistance. As seen from the figure, the resistivity decreases monotonically with
decreasing temperature and reaches a constant residual value at low temperatures p0.

In the presence of a small amount of magnetic impurities, the electrical resistiv-
ity shows anomalous behavior, whereby it increases when the temperature is lowered
[16]. The temperature at which the resistivity starts increasing is called the resistivity
minimum, Tm,” (see figure 6.2). This minimum is due to the competition between
scattering of conduction electrons by phonons dominating at high T and by mag-

netic impurities dominating at low T. Below the minimum, the resistivity was found

100

 

10‘T'r'lrl'r'l'r'l'

- 10 ppm (1K-cryostat) ]
9 - 13:0.1 r O -

Ap(nQ.cm)

 

 

 

 

Figure 6.2: Resistivity vs temperature for 10 ppm Fe magnetic impurities in Ag mea-
sured at a finite magnetic field of 0.1T to suppress the weak localization magnetore-
sistance. This sample was measured in the 1K-cryostat between 1.3 K-18 K. Notice
that the resistivity minimum occurs at around 8 K. The increase in the resistivity
below the minimum is due to the presence of Fe magnetic impurities.

to increase logarithmically (— In T) when the temperature is lowered. Figure (6.2)
shows the electrical resistivity as a function of temperature for 10 ppm Fe magnetic
impurities in Ag measured at a finite magnetic field of 0.1 T between 1.3 K-18 K. The
resistivity minimum of this sample occurs around 8 K. To extract the Kondo contri-
bution to resistivity, one has to subtract the e—ph scattering contribution to resistivity
from figure (6.2). Ideally, one has to measure the same sample before and after ion-
implantation to be able to subtract the e-ph contribution to the resistivity properly.
However, for this sample, the e-ph contribution to resistivity was not measured before

the ion-implantation. Furthermore, it was hard to compare the resistivity data of the

101

 

7 ...-.., ........, ....
10 ppm (Fridge) ‘
B=O.1T

    

Ap(nQ.cm)

 

 

0 ......l n I IIIIIII I a n-

0.1 1
WM

Figure 6.3: The e—e and the magnetic impurities contributions to the resistivity for
the 10 ppm sample measured between 40 K - 4 K. This sample was measured in the
dilution refrigerator at a finite magnetic field of 0.1 T. This magnetic field (0.1 T) is
big enough to kill the weak localization (antilocalization) effects, but small enough
to not polarize the magnetic impurities. The resistivity data below ~ 1 K are noisy
due to the fact that the voltage across the wire was limited to 6V S kBT, to avoid
heating the electrons in the wire.

 

pure sample with this sample; since the two samples are totally different. Due to
these reasons, we could not extract the Kondo resistivity and fit it to the empirical
formulas given in reference [17] to extract the Kondo temperature,TK.

At low temperatures (T 5 4 K), the e-ph contribution to the resistivity is small.
On the other hand, the e-e contribution to resistivity increases with decreasing tem-
perature, and it dominates below 1 K (see equation 2.14). Figure (6.3) shows the e—e
and the magnetic impurities contributions to the resistivity for the 10 ppm sample

below 4 K. The temperature dependence of the magnetic impurities contribution to

102

 

6 ......, ......., f...

  
 

L 10 ppm (Fridge) 4
5 - B=O.1 T -
A 4 - J
E . .
0
d 3 - _
C _ .
V
2]- 2 _ Unitary limit _
1 — .1
0 A A A A A A L A A A A I A A A l A A A A

 

 

 

0.1 1
WM

Figure 6.4: The magnetic impurities contribution to resistivity for the 10 ppm sample
after subtraction of the e—e contribution to resistivity using equation (2.14). Notice
that the resistivity saturates below ~ 0.2 K, approaching its unitary limit (its value
at T = 0). The resistivity data below ~ 1 K are noisy due to the fact that the voltage
across the wire was limited to eV S, kBT, to avoid heating the electrons in the wire.
the resistivity for the same sample (10 ppm) is shown in figure (6.4) after subtraction
of the e-e contribution to resistivity using equation (2.14). This sample was measured
in the dilution refrigerator at a finite magnetic field of 0.1 T between 40 mK - 4 K.
This magnetic field (0.1 T) was big enough to kill the weak localization (antilocaliza-
tion) effects, but small enough not to polarize the magnetic impurities. As seen from
the figure, the resistivity increases with decreasing temperature down to 0.2 K and
then levels off, approaching its unitary limit (its value at T = 0). The unitary limit
observed here is a sign of the formation of the singlet state proposed by Wilson [20],

where he showed that at temperatures far below the Kondo temperature (T < TX),

103

the spin of the magnetic impurity is totally screened by the conduction electrons;
hence the impurity becomes non-magnetic. For conduction electrons, the screened
impurity appears then as a potential scattering center. The Kondo temperature is
the temperature at which the Kondo resistivity drops to half its value at T = 0 (uni—
tary limit [17]). The data shown in figure (6.4) have an unknown vertical offset, so
we do not know where the zero is. Guessing where to assign the value Ap = 0 is not
reliable for our samples, since the e-ph scattering contribution to resistivity starts to
grow significantly when T > 8 K, before the Kondo contribution has completely died
out [52].

Although figures (6.2) and (6.3) are for the same sample (10 ppm) measured in
two different cryostats, they do not match in the overlap temperature range. This,
however, is due to the fact that the sample was initially measured in the dilution
refrigerator and a few months later when the sample was measured again in the high
temperature cryostat (1K-cryostat), it was found that its resistance had increased
by around 50%. Therefore, our resistivity data are not reliable enough to be used
to extract the Kondo temperature. Nevertheless, the resistivity data serve to show
some experimental signatures of the Kondo effect such as the resistivity minimum,
unitary limit, and the logarithmic rise of the resistivity below the minimum. In the
following sections, we will discuss our efforts for obtaining an independent estimate

of the Kondo temperature from the high field magnetoresistance.

104

6.3 High field magnetoresistance and NRG calcu-

lations

6.3.1 High field MR data

At high temperatures (T >> TK), the magnetic moment of the Fe impurity behaves like
a free magnetic moment. Below the Kondo temperature, TK, the magnetic moment
of the Fe impurity is progressively screened by the surrounding conduction electrons
as the temperature is lowered. Far below the Kondo temperature (T < TK), the
Fe impurity is completely screened; hence it behaves like a static scatterer. In this
limit, the scattering of the conduction electrons by the Fe impurity is pure elastic,
and the resistivity saturates towards its unitary limit. This transition from magnetic
to nonmagnetic state is not a sharp transition but happens gradually. It was found
that this transition results in the anomaly in the resistivity of the Kondo alloys. In
the previous section, we found that for 10 ppm Fe in Ag, the resistivity saturates
below 0.2 K, indicating that the unitary limit was reached below this temperature
(0.2 K). In this section, we discuss the anomalies (negative MR) observed at high
magnetic fields for 10 ppm Fe in Ag. Then we show how the Kondo temperature can
be obtained from fitting the high field MR data to the theory of reference [58].

In a pure metallic film, the classical magnetoresistance due to the Lorentz force
takes the form AR(B) / R z (4047')2 [13], where we = eB / m is the cyclotron frequency
and 7' is the elastic mean free time. At low magnetic fields, this classical MR is
negligible. The classical 32 dependence of one of the pure samples measured in the
dilution refrigerator below 4 K is shown in figure (6.5). This is a positive magne-
toresistance. The big spikes seen at [B I < 0.03 T are due to the weak localization
magnetoresistance. The inset of the figure shows how the resistance changes with
decreasing temperature at zero magnetic field. This increase in the resistance with

decreasing temperature is mainly due to the contribution from e-e scattering as T is

105

 

 

 

 

 

 

 

11.3 - 2-0 . ' . ' Ag(6N)
“PC, 1.5-‘\ -
10.1 1' :- 10 O,
E ' i '\
3,3 - E 0.5» A O
V 0.0 . . . . from top
A 7.6 5 0 1 2 3 4 54mK
9; W) 292mK
m 6.3 - 555mK
q ' ; 1580mk
5.0 - ‘ 3305mK
3.8 -
2.5 L
1.3 —
A 1 A L A l A l A l A l A I A l A l

 

 

 

-3-e-4-20 2 4 6 810
I3(T)

Figure 6.5: High field magnetoresistance of a pure sample measured in the dilution
refrigerator between 54 mK - 3.3 K. The geometrical and electrical characteristics of
this sample are: L = 780 pm, t = 47 nm, and R = 3147 k9. This wire was very
rough, where its width was between 50 nm - 200 run when measured using the SEM.
Therefore, this sample will not be used in future calculations. However, it is still
good to show the behavior of the classical MR in pure films. The inset shows the
increase in the overall resistance with decreasing temperature at B = 0. This increase
is mainly due to e-e interactions (see text).

106

lowered. In Chapter 2, we mentioned that the e-e interaction contribution to resis-
tance takes the form AR“ or 1/\/T. From this expression one can see that ARee
dominates the temperature dependent resistivity below 1 K, consistent with what is
seen in the inset of figure (6.5). Unfortunately, when this sample was later observed
under the SEM, the wire was very rough, so the results of this sample will not be
used in future calculations or comparisons. Nevertheless, the high field data of this
sample is good for demonstration and to show the behavior of the classical MR in
pure metallic wires.

Adding a small number of Fe magnetic impurities to Ag results in an anomaly
in the high field data, where the magnetoresistance is now negative at an interme-
diate fields scale. This negative MR is totally due to the scattering of conduction
electrons by Fe magnetic impurities (Kondo scattering). Figure (6.6) shows the high
field magnetoresistivity data for 10 ppm Fe impurities in Ag measured in the dilution
refrigerator between 83 mK - 1.88 K. Again, the spikes seen at low field are due to
the weak localization magnetoresistance (B < 0.03 T). Notice that below 200 mK,
where 74 > 7'”, the spikes are downward, whereas above 550 mK, where 74 < 7,0,
the spikes are upward. At intermediate fields, the magnetoresistivity curves are nega-
tive. At fields bigger than 4 T, the classical magnetoresistivity dominates the Kondo
scattering. This, however, is expected because at very high magnetic fields, the spins
of magnetic impurities are completely polarized to the direction of the applied mag-
netic field; hence the magnetic moments lose their dynamics and behave like static
impurities. Notice here how the peak amplitude increases with decreasing tempera-
ture, indicating that the Kondo scattering is stronger at lower temperatures. This
is consistent with the Kondo effect, where the resistivity increases with decreasing
temperature below Tmin. Looking carefully at the lowest two data sets (83 mK and
188 mK), one can see that the amplitudes of magnetoresistances are almost equal,

indicating that the unitary limit was reached below 200 mK, consistent with the p vs

107

 

l ' l ' l ' l ' l ' I ' I '

  
  
 
  

10 ppm from top ‘
83 mK
' 188 mK 1
550 mK
4 ' 860 mk ‘

1880 mK

Ap(n§2.cm)

 

 

 

B(T)

Figure 6.6: High field magnetoresistivity for 10 ppm Fe impurities in Ag measured at
temperatures of 83, 188, 550, 860, and 1880 mK, from top to bottom. The last four
data sets were shifted downward by 0.75 nQ.cm relative to the 83 mK data set for
clarity.

T data shown in the last section for the same sample (see figure 6.4). Actually the
two data sets are identical, but for clarity all data sets for T 2 188 mK were shifted
downward by 0.75 chm relative to the 83 mK data set.

At high temperatures, the Kondo scattering is less effective and dies out gradu-
ally with increasing temperature; hence the total magnetoresisitvity will be mostly
classical. Figure (6.7) shows the total magnetoresisitvity vs B for the 10 ppm sample
measured at 10 K. The 32 dependence of the magnetoresisitvity is obvious. In the
figure, the low-field data points were masked between -1 T < B < 1 T, and the
remaining points were fit to the functional form Ap(B) = a + b(B — BO)2, where a is

the extrapolated value of the magnetoresisitvity at zero field (Ap(0)) and B0 compen-

108

 

2.0 F —T I V ' I l T fir I l 1 1 t T l '

 

Data: RUNOZJAN20_DeItaROhm

1-6 ’ Model:a+b‘(B-B°)2 "
L Chm/Dos =0.00004
‘ R"2 = 0.99971
1'2 ' , a 0.04765 40.00077
‘A b 0.03364 40.00004
[ BO 0.02099 40.0013

 

Ap(nQ.cm)

 

 

 

 

l3(T)

Figure 6.7: High field magnetoresistance for 10 ppm Fe impurities in Ag measured
in the dilution refrigerator at 10 K. The low-field data points were masked between
—1 T < B < 1 T, and the remaining points were fit to the functional form Ap(B) =
a + b(B — Bo)2, where a is the extrapolated value of the magnetoresisitvity at zero
field (Ap(0)) and Bo compensate for the B offset. The solid line in the figure is the
fit to this functional form. The window appears in the figure shows the output of the
least-square fit with a, b, and Bo being the adjustable parameters.

sates for the small B offset due to trapped flux in the magnet. The solid line in the
figure is the fit to this functional form. The window appearing in the figure shows the
output of the least-square fit with a, b, and Bo being the adjustable parameters. The
value of the coefficient b will be used later to subtract the classical MR (figure 6.7)
from the total MR (figure 6.6) to extract the Kondo contribution to MR. A typical
subtraction example for the 83 mK data set is shown in figure (6.8). Notice that
the two curves in the figure are identical between -2 T and 2 T, indicating that the

classical MR is weak below fields of 2 T. Figure (6.8) shows also that the Kondo effect

109

 

I I t I I ' T j I I '

6 ' 10 ppm , T=83 mK

5- .4
E l
0. 4‘ , '
Cl ' .
C 3- M '. -
‘5’. L w *c
<

2- 4

L
1— _
0 I; l 1 l L L+ + l

 

 

 

B(T)

Figure 6.8: High field magnetoresistance for 10 ppm Fe impurities in Ag measured in
the dilution refrigerator at 83 mK. The upper curve is the total magnetoresisitvity.
The lower curve is the Kondo contribution to magnetoresisitvity, which was obtained
by subtraction of the classical magnetoresisitvity (figure 6.7) from the upper curve.
Notice that one needs to go to higher fields to completely polarize the Fe magnetic
moments, at which the magnetoresisitvity becomes flat (field independent).

still survives in a magnetic field up to 6 T. This means that at this filed (6 T), the
Fe moments are not completely polarized to the direction of the applied field, so one

needs to go to higher fields to totally kill the Kondo effect.

6.3.2 Theory of Costi for high field MR in dilute Kondo al-
loys

The theory of Costi [58] describes the magnetic field dependence of the spectral

density of a S=1/2 magnetic impurity at both zero and finite temperatures. The

110

total spectral density of a magnetic impurity, A (w,T, B), is defined by the sum of
spin up and spin down spectral densities, A (w, T, B) = 20 140(0), T, B). According
to this theory, the magnetic impurity contribution to the resistivity, at very small

concentrations (723 << 1), is given by [58]

—1(TB)—EEZ/+oodwr(wTB)(—a—f) (61)
p a _ 2m 0 _oo 0’ a a aw ‘
where 70.1 (w, T, B) = n,J A, (w, T, B) is the inverse transport time of electrons of

spin a, f is the Fermi function, and m, n, and e are the mass, density, and charge,

 

respectively, of the conduction electrons. At T = 0, 3—5 = —6(w), then equation (6.1)
reads
—1 7282 +00
p (T=0,B) = 577—1 glee dw'ra(w,0,B)5(w) (6.2)
2
me
= — a a aB '
2m 0 7' (0 0 ) (6 3)
— "6221(003) (64)
_ 2m T 7 a '
n62 1
= _ .5
m ln.AT (0.0.B>l (6 )

where I used AT (0,0,B) = A1 (0,0, B). Finally, taking the inverse of the last
equation, we end up with an expression for the resistivity as a function of magnetic

field calculated at T = 0.
m
p (0, B) = ——5 mt (0. o. B) (6.6)
ne

The quantity AT (0,0, B) was calculated in reference [58] using the Numerical
Renormalization Group method (NRG). The results of these calculations are shown
in figure (6.9). The Kondo field, BK, is defined as the magnetic field at which the

spectral function A1 (0, 0, B), shown in figure (6.9), falls to half its value at T = B = O.

111

 

0.18 . . . , . , . , .
0.16 b
0.14 h
0.12 -
0.10 .
0.08 D

0.06

AT(O, 0, B)

0.04

 

0.02

 

 

0.00 . n l . l n l n l

 

Figure 6.9: The spin up spectral density fimction, AT (0,0, B), vs magnetic field B
calculated at T = 0 using NRG method. The x-axis is normalized to the parameter
BK, which is related to T K by TK z 1.34 BK. The data in the figure were provided
by the author of reference [58].

The x-axis of figure (6.9) is normalized to the parameter BK, which is related to T K
by ngBK = kBTK, where p3 is the Bohr magneton. Using 9 = 2 and 113/kg z
0.67 K.T"1, this reads TK(K) a: 1.34 BK(T). Here, I want to emphasize again that
the numerical calculations shown in figure (6.9) are valid only at T = 0. Nevertheless,
our resistivity data for 10 ppm Fe impurities in Ag show that the resistivity saturates
below 0.2 K, approaching its unitary limit (its value at T = 0); hence we can, in
principle, fit our lowest temperature high field MR data set, 83 mK, to this model.

The following subsection will discuss this fit.

112

6.3.3 Fitting the 83 mK high field data set to theory of Costi

In this section, we fit the high field magnetoresistivity data of the 83 mK data set
using the NRC calculations of reference [58]. As mentioned earlier, below 0.2 K the
resistivity of the 10 ppm sample saturates towards its unitary limit. Therefore, below
0.2 K, the high field data of the 83 mK data set are expected to be nearly identical
to the T = 0 analytical solution of reference [58]. The solid line in figure (6.10) shows
the fit of our 83 mK data set to the theory of Costi [58] using the functional form
Ap(B) = ApK f(B/BK), where ApK E p(B = 0,T = 0)—p(B >> BK,T = 0) and BK
is the magnetic field at which the magnetoresisitvity falls to its half value at B = 0
(i.e 1 / 2 Ap(0)). The data set were fit with two adjustable parameters, ApK and BK.
The result of the fit gives the values ApK = 0.57 chm/ ppm and BK = 1.2 T, which
in turn gives TK = 1.6 K [52]. This value of TK is considerably lower than the value
of TK z 5 K obtained from the fit to 7m, discussed in Chapter 5.

The expected value of the unitary limit for s-wave (l = 0) scattering is given by
ApK = 47rhns/n62kp [59], where n, is the concentration of magnetic impurities, and
n and kp are the density and the Fermi wave number of the host, respectively. The
d—wave unitary limit is 5 times the s—wave unitary limit. The factor 5 comes from
(21 + 1), where l = 2 for the d-waves. In a Ag host, where kp = 1.2 x 1010 m‘l,
the s-wave unitary limit is ApK = 0.43 chm/ ppm. Our measured value of ApK =
0.57 nflcm/ppm is larger than the theoretical s-wave value, but smaller than the
measured values in other Fe-noble metal Kondo systems: 1.3 chm/ ppm in Cu:Fe

and 1.0 chm / ppm in AuzFe [60].

113

 

 

6.10ppm . T=83mK ..

5.. _
E '
0. 4’ ‘
G t .
C 3- -
v
Q. .
4

2% -

1- _

0 . l n I 1 l n I n l j l n l L

 

 

-8 -6 -4 -2 0 2 4 6 8
Figure 6.10: Kondo contribution to magnetoresistivity for 10 ppm Fe impurities in
Ag measured in the dilution refrigerator at 83 mK. The data were obtained after

subtraction of the classical magnetoresisitvity (figure 6.7). The solid line is the fit to
reference [58].

6.3.4 Fitting the high temperature MR data sets to theory

of Costi

The high field N RG calculations shown in figure (6.9) were good enough to fit the
lowest MR data set, 83 mK, since it is close to the unitary limit. However, this
approximation is no longer valid at high temperatures (T > 0.2 K); hence more N RG
calculations at higher temperatures were necessary to fit our high field MR data. Upon
our request, the high temperature N RG calculations were provided to us by the author
of reference [58] (not shown in this thesis). We tried to fit our high field MR data

to these calculations, while fixing ApK to its unitary limit (ApK = 0.57 chm/ ppm),

114

 

 

  
  
   

 

 

 

10 ppm 1% s.
6 — d -
from top E 4'
' 63 mK if 2- ‘
4 _ 188 mK ‘q’ n 4
<1 " 1
550 mK T (K)
- 860 mk 1

1880 mK

  

Ap(nQ.cm)

 

 

 

B (T)

Figure 6.11: Kondo contribution to magnetoresistivity for the 10 ppm sample with
fits to the spin-1/2 NRG theory from [58], as discussed in the text. The zero for
the 83 mK data is chosen so that the fit curve approaches Ap —> 0 for [B] > BK.
Subsequent data sets are shifted downward by 0.75 chm for clarity. Top right inset:
Ap(B = 0) vs. temperature [52].

but unfortunately we could not fit them consistently with such a low value of TX =
1.6 K. The problem is that the temperature scale over which the high-field MR
decreases is much larger than the value 1.6 K obtained from the field dependence at
T = 0. Instead, we have performed a global fit to all the p(B) data sets using the
functional form Ap(B, T) = ApK f (T/TK, B / BK), with only three parameters: ApK,
BK, and TX. The parameter BK determines the magnetic field scale over which p(B)
decreases and the parameter TK determines the temperature scale over which p(B =

0) decreases. The solid lines in figure (6.11) are the results of this global fit, with the
values BK = 1.36 T, TX 2 2.96 K, and Apx = 0.58 chm/ppm [52]. The value of

115

BK corresponds to a Kondo temperature of 1.8 K, which is close to the TK = 1.6 K
obtained from fitting the 83 mK data set alone to the T = 0 NRG calculations (see
figure 6.10). The Kondo temperature extracted from the temperature scale (2.96 K)
is almost twice as that extracted from the field scale (1.8 K).

The discrepancy between the values of TX extracted from the field scale (1.8 K)
and from the temperature scale (2.96 K) highlights the fact that we are using a
spin-1/2 theory. In a metal containing small amount of magnetic impurities with-
out Kondo effect, the magnetoresistance is proportional to — < M; >2, where
< Mz >= N pBJ(:r) is the average magnetization of the system, N is the total
number of magnetic impurities, u = 9,113.] with J = L + 5 being the total angular

momentum of the localized moment, and B J(:1:) is the Brillouin function given by [61]

BM) = (1 + %) coth [(1+ 513) x] — 517 coth [$23] (6.7)

where a: = ngJ B / kBT is the Zeeman energy of the local moment in the external
field B in units of kBT. The Brillouin function is normalized and varies between 0 and
1. Figure (6.12) shows the calculated magnetoresistance using the function 1 — 83(3")
for both S = 1 / 2 and S = 2, where the function B J(a:’ ) is given by

BJ(x’) = (1 + %) coth [(J + g) :c'] — 53— coth [éx’] (6.8)

with x’ = ngB/kBT.

In this figure, it is assumed that the total angular moment of the localized moment
is due to its spin only, L = 0. As seen from the figure, the full width at half maximum
of the MR curve with S = 1 / 2 is 1.85 times larger than that with S' = 2. If a similar
relationship holds for the p(B) curves in the Kondo regime, then the value B K = 1.2T
we found from the s=1/2 fit to the 83 mK data would become BK 5:: 2.2 T for S=2,

which corresponds to a Kondo temperature TK = 3.0 K [52]. That is consistent with

116

 

1-Bf(x')

 

 

 

 

Figure 6.12: Magnetoresistance of a system of magnetic impurities calculated using
equation (6.8). In this figure, we assume, for simplicity, that N = p3 = g = 1,
and L = 0. Notice that the full width at half maximum of the MR curve with
S = 1/2 is 1.85 times larger than that with S = 2. Notice also that at high fields,
the magnetoresistance saturates with the magnetic moments completely aligned with
the applied field.

the value of T K obtained from the temperature dependence of p(B = 0), but still
lower than the value TX 2 5 K obtained from the fit to 7",.

Recently, Mallet et al. [56] reported measurements of resistivity and phase coher-
ence in three Ag wires implanted with Fe impurities. The implanted ion concentra-
tions were 1.3, 13, and 33 ppm, respectively. From fitting the resistivity data to the
NRG theory for spin 1 / 2 magnetic impurities, they also obtained a Kondo temper-
ature of TK = 3 K. However, the unitary limit observed in their implanted samples
was an order of magnitude less than the expected value of the s-wave unitary limit for

this system (ApK = 0.43 nQ.cm/ ppm). Their phase coherence results were similar

117

to ours, where they found a Kondo temperature of T K z 4.3 K and observed a low T
deviation from theory in the 7",.

In summary, the Kondo temperature obtained from the resistivity, TK a: 3.0 K,
is somewhat lower than the Kondo temperature obtained from the dephasing data,

TK z 4.3 K by Mallet et al. [56], or TK z 4.8 — 5.4 K by us [52].

118

Chapter 7

Conclusions

7.1 Overview

The topic of electron phase coherence in metals at low temperature has been contro-
versial in the past decade. It is important to understand the role of dilute magnetic
impurities in this context. In this work, we have studied the inelastic scattering
mechanisms of conduction electrons in AgFe Kondo alloys. The concentration of the
Fe impurities was very small such that these alloys lie in the regime of the Kondo
effect (single impurity limit). The goal of this work was to study the temperature
dependence of the magnetic scattering rate, 7",, not too far below the Kondo tem-
perature, TK. As discussed in Chapter 1, at the time we started this work, there was
neither experimental data nor theory to describe the intermediate regime for T not
too far below TK (see figure 1.5). To study the temperature dependence of 7",, it was
important to have a system with T K below 8 K, otherwise the Kondo effect would be
masked by the e—ph interactions. On the other hand, it was necessary to keep TK as
high as possible to be able to acquire data below TK. The best Kondo system for this
study, then, should have a Kondo temperature between 2 - 5 K. The AgFe system was
a good candidate to study the temperature dependence of 7",; since it has a relatively

high TK a: 5 K. In this work, two types of samples were studied, pure and implanted

119

samples. It was necessary to perform measurements on both types of samples to be
able to extract the effect of adding Fe impurities on the transport properties of our

Kondo system.

7 .2 Summary of Results

7 .2.1 Phase Coherence Results

In this study, we have used the method of weak localization magnetoresistance to
measure the total dephasing rate in AgFe Kondo wires. By measuring the weak
localization magnetoresistance between 40 mK — 18 K , we were able to indirectly
measure the phase coherence length L¢, and hence 7,». The magnetic scattering rate
of the implanted samples, 7",, was obtained by subtraction of the total dephasing rate
of the pure sample from the total dephasing rate of the the implanted samples. The
7m data were fitted using the Numerical Renormalization Group (NRG) calculations
of reference [27]. From the fit, we found that the theory of reference [27] fits the 7m
data resonably well over the temperature range T/TK = 0.1 — 2 [52] (see figure
5.11). Above the Kondo temperature, TK z 5 K, the 7m data were almost constant,
consistent with the prediction of reference [27]. The maximum value of 7m, occurring
at T = TK, was almost twice as large as that predicted by theory. Around TK, the 7m
data has a broad maximum. Below TK, the 7m data were found to vary linearly in
T. For T/TK < 0.1, we found that the NRC theory of reference [27] deviates strongly
from the 7m data, where the temperature dependence of 7m was found to be much
weaker than that predicted by theory. The T2 dependence of 7m was not reached
down to the lowest accessible temperatures.

The factor 2 discrepancy in the overall magnetic scattering rate was attributed
to the inadequacy of the spin 1 / 2 theory of reference [27]. As mentioned earlier, the

theory of reference [27] was developed for spin 1 / 2 magnetic impurities in the context

120

 

of the single- channel Kondo model (n = 1). The strong deviation between theory
and experiment for For T/ T K < 0.1, could be due to the finite cross-section size of our
wires [57]. Further calculations of the magnetic scattering rate due to dilute magnetic
impurities with S > 1 / 2 and n > 1 might resolve the observed discrepancy between

theory and experiment.

7 .2.2 High Field Results

To obtain an independent estimate of T K, we have measured the high field magne-
toresistance of the 10 ppm sample in a magnetic fields up to 6 Tesla. The high field
MR data were taken between 83 mK - 1.8 K. At an intermediate field scale, the MR
data of the 10 ppm sample was found to be negative (see figure 6.6), whereas in the
pure films it was positive (see figure 6.5). The negative MR observed in the 10 ppm
sample was attributed to the presence of Fe magnetic impurities. From the high field
MR data, we found that the unitary limit was reached below 200 mK, indicating that
the Fe spin is completely screened by the surrounding conduction electrons; hence
the Fe impurity behaves like a static impurity and does not contribute to dephasing
processes.

We have fitted our high field MR data of the 10 ppm sample using the the
Numerical Renormalization Group (N RG) calculations of reference [58] (see figure
6.10). The theory of reference [58] was also developed for spin 1/2 magnetic im-
purities. From the fit, we found that the measured value of the unitary limit was
ApK = 0.58 nflcm/ ppm [52], which is close to the theoretical value for s-wave scat-
tering, ApK = 0.43 chm/ ppm. The value of the Kondo temperature, TK, extracted
from the fit to the T = 0 theory (TK z 1.6 K), was found to be smaller than that
obtained from the fit to 7", data (TK z 5 K). Again, we attribute the discrepancy
between the value of TK obtained from fitting the 7”, data and that TK obtained from

fitting 7m data to the inadequacy of the spin 1 / 2 theories. Estimating the effect of

121

S=2 by comparing the Brillouin function for S=1/2 and S=2 led us to an estimate
TK z 3.0 K, which is consistent with the temperature dependence of the high-field
MR data. Future calculations of the Kondo contribution to MR with S > 1 / 2 would

be very helpful.

7 .3 Future work

The two theories used in this work to analyze our experimental data for both phase
coherence and high field MR were mainly developed for spin 1 / 2 magnetic impurities
in the context of single channel (n = 1) Kondo model. In reality, the Fe has bigger
spin, S = 2, which is quite different from S = 1 / 2. New calculations of the inelastic
scattering cross-section, due to magnetic impurities with S > 1 / 2 and in the context of
multi-channel Kondo model (n > 1), might resolve the observed discrepancies between
experiment and theory. However, for n > 1 one faces a multi-channel Kondo problem
which is much more difficult to track numerically [62]. As discussed earlier, Pedro
Schlottmann thinks that the weak temperature dependence of our 7m data observed
at very low T might be due to the finite cross-section size of our wires [57]. To test
this hypothesis, one has to measure 7",, for films with bigger dimensionality. However,
measuring 7}), in thicker films using the method of weak localization magnetoresistance
is hard to achieve because the weak localization signal becomes smaller as the sample

dimensionality increases.

122

References

[1] B. Altshuler, A. Aronov, and D. Khmelnitsky, J.Phys. C 15, 7367 (1982).

[2] S.Wind, M. Rooks, V. Chandrasekhar, and D. Prober, Phys.Rev. Lett 57, 633
(1986).

[3] P. Echternach, M. Gershenson, H. Bozler, A. Bogdanov, and B. Nilsson, Phys.
Rev. B 48, 11516 (1993).

[4] P. Mohanty, E. Jariwala, and R. A. Webb, Phys.Rev. Lett 78, 3366 (1997).

[5] F. Pierre, A. B. Gougam, A. Anthore, H. Pothier, D. Esteve, and N. O. Birge,
Phys. Rev. B 68, 085413 (2003).

[6] D. Golubev and A. Zaikin, Phys.Rev. Lett 81, 1074 (1998).

[7] D. Golubev, A. Zaikin, and G. Schon, J. Low Temp. Phys 126, 1355 (2002).

[8] D. Golubev and A. Zaikin, J. Low Temp. Phys 132, 11 (2003).

[9] A. Zawadowski, J. von Delft, and D. Ralph, Phys.Rev. Lett 83, 2632 (1999).
[10] Y. Imry, H. Fukuyama, and P. Schwab, Europhys. Lett 47, 608 (1999).
[11] I. Aleiner, B. Altshuler, and M. Gershenson, Waves Random Media 9, 201 (1999).

[12] F. Marquardt, J. von Delft, R. A. Smith, and V. Ambegaokar, cond-
matt/0510556v1 (2005).

[13] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Harcourt Brace College
Publishers, Fort Worth, 1976).

[14] W. de Haas, J. de Boer, and G. van den Berg, Physica 1, 1115 (1934).

[15] J. Franck, F. Manchester, and D. Martin, Proc. Roy. Soc. (London) A263, 494
(1961).

[16] J. Kondo, Prog. Theor. Phys 32, 37 (1964).

[17] H. Suhl, Magnetism, 1201.5 (Academic Press, New York and London, 1973).

[18] A. Hewson, The Kondo Problem to Heavy Fermz'ons (Cambridge University
Press, Cambridge, 1997).

123

[19] P. Anderson, J. Phys. Czsolid st.phys 3, 2436 (1970).
[20] K. Wilson, Rev. Mod. Phys 47, 773 (1975).

[21] C. V. Haesendonck, J. Vranken, and Y. Bruynseraede, Phys.Rev. Lett 58, 1968
(1987)

[22] R. Peters, G. Bergmann, , and R. Mueller, Phys.Rev. Lett 58, 1964 (1987).
[23] P. Mohanty and R. Webb, Phys.Rev. Lett 84, 4481 (2000).

[24] F. Schopfer, C. Bauerle, W. Rabaud, and L. Saminadayar, Phys.Rev. Lett 90,
056801 (2003).

[25] P. Nozieres, J. Low Temp. Phys 17, 31 (1974).
[26] M. Vavilov and L. Glazman, Phys. Rev. B 67, 115310 (2003).

[27] T. Micklitz, A. Altland, T. A. Costi, and A. Rosch, Phys.Rev. Lett 96, 226601
(2006).

[28] G. Zarand, L. Borda, J. von Delft, and N. Andrei, Phys.Rev. Lett 93, 107204
(2004).

[29] P. Sacramento and P. Schlottmann, Physica B 171, 122 (1991).

[30] Y. Imry, Introduction to Mesosc0pic Physics (Oxford University Press, Inc, New
York, 1997).

[31] S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University
Press, Cambridge, 1995).

[32] G. Bergmann, Solid State Communication 42, 815 (1982).

[33] S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor. Phys 63, 707 (1980).
[34] S. Chakravarty and A. Schmid, Phys. Rep 140, 193 (1986).

[35] B. Huard, Limits and effects of coherence in conductors, PhD thesis (2005).
[36] R. Meservey and P. M. Tedrow, Phys. Rev. Lett 41, 805 (1978).

[37] B. Kramer, Quantum Coherence in Mesosc0pic Systems (Plenum Press, New
York, 1990).

[38] J. J. Lin and J. P. Bird, J. Phys.: Condens. Matter 14, R501 (2002).
[39] A. Sergeev and V. Mitin, Phys. Rev. B 61, 6041 (2000).
[40] S.Gasiorowicz, Quantum Physics (John Wiley and Sons, INC, New York, 1974).

124

 

 

 

[41] E.Merzbacher, Quantum Mechanics (John Wiley and Sons, INC, New York,
1998).

[42] E. Bauer, A script for a lecture given at the Vienna University of Technology,
summer semester (2006).

[43] v. Fal’ko, JETP Lett 53, 340 (1991).

[44] J .A.Mydosh, Spin glasses (Taylor and Francis, London.Washington.DC, 1993).
[45] M. Ruderman and C. Kittel, Phys. Rev 96, 99 (1954).

[46] T. Kasuya, Prog. Theor. Phys 16, 45 (1956).

[47] K. Yosida, Phys. Rev 106, 893 (1957).

[48] M. Vavilov, L. Glazman, , and A. Larkin, Phys. Rev. B 68, 075119 (2003).

[49] P.R.Choudhury, Microlithography, Micromachining, and Microfabrication (SPIE
press IEE, Washington, 1997).

[50] N anometer Pattern Generation System version 7.5 User Manual (1996).

[51] O.V.Lounasmaa, Experimental principles and methods below 1 K (Academic
Press, New York, 1974).

[52] G. Alzoubi and N. Birge, Phys. Rev. Lett 97, 226803 (2006).
[53] G. Bergmann, Physics Reports 107, 1 (1984).

[54] C. Bauerle, F. M. andl F. Schopfer, D. Mailly, G. Eska, and L. Saminadayar,
Phys.Rev. Lett 95, 266805 (2005).

[55] W. Koller, A. Hewson, and D. Meyer, Phys. Rev. B 72, 045117 (2005).

[56] F. Mallet, J. Ericsson, D. Mailly, S. Unlubayir, D. Reuter, A. Melnikov, A.
Wieck, T. Micklitz, A. Rosch, T. Costi, L. Saminadayar, and C. Bauerle, Phys.
Rev. Lett 97, 226804 (2006).

[57] P. Schlottmann, Private Communication.

[58] T. Costi, Phys.Rev. Lett 85, 1504 (2000).

[59] D. R. Hamann, Phys. Rev 158, 570 (1967).

[60] J. Loram, T. Whall, and P. Ford, Phys. Rev. B 2, 857 (1970).

[61] R. Pathria, Statistical Mechanics (Pergamon Press Ltd, England, 1972).

[62] A. Rosch, Private Communication.

125

 

 

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