The Application of Two Fluid Model to Ir Spectra of Heavy

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THE APPLICATION OF TWO FLUID MODEL TO IR SPECTRA OF HEAVY

FERMIONS

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulﬁllment

of the Requirements for the Degree

Master of Science

Prabuddha Madusanka Hathurusinghe Deawage

November, 2018 THE APPLICATION OF TWO FLUID MODEL TO IR SPECTRA OF HEAVY

FERMIONS

Prabuddha Madusanka Hathurusinghe Deawage

Thesis

Approved: Accepted:

Advisor Dean of the College Dr. Sasa V. Dordevic Dr. Linda M. Subich

Faculty Reader Dean of the Graduate School Dr. Ben Yu-Kuang Hu Dr. Chand Midha

Faculty Reader Date Dr. Sergei F. Lyuksyutov

Department Chair Dr. Christopher J. Ziegler

ii ABSTRACT

This work studies conductivity data experimentally determined using IR spectroscopy for three heavy fermion materials: YbFe4Sb12, CeRu4Sb12 and CeCoIn5. We describe them using the two ﬂuid model of electrons and mathematically analyze the materials using the Drude-Lorentz model. In particular, using the program RefFit, the optical data in the frequency range of 4-25,000 cm−1 for temperatures ranging from 8-300

K was considered. We found the plasma frequency and scattering rate and observed how temperature eﬀected their magnitudes and the distributions of electrons in our system.

Through analysis of these Drude-Lorentz model parameters, the characteristic temperature - the point below which large conductivity exists - of all three materials was found to be in the range of 70-80 K. Further, using these characteristic temperatures, we showed that the optical data was described by a modiﬁed form of the hybridization order parameter equation. This demonstrates that the IR spectra of these materials can be described using the two ﬂuid model.

iii ACKNOWLEDGEMENTS

I would like to express my sincere thanks to my advisor Dr. Sasa V. Dordevic for his continuous support of my research and his patience, motivation, and immense knowledge. His guidance helped me in all throughout my research and writing of this thesis.

I would also like to extend my thanks to Dr. Ben Yu-Kuang Hu and Dr. Sergei

F. Lyuksyutov for being on my thesis committee and for their insightful comments and encouragement. I would also like to thank the rest Physics Department’s academic staﬀ for their support.

I warmly thank my friends for sharing their knowledge and for their invaluable help. Finally, last but by no means least, my parents for their unconditional love and support in a number of ways throughout my time here.

iv TABLE OF CONTENTS

Page

LISTOFTABLES...... vii

LISTOFFIGURES ...... viii

CHAPTER

I. INTRODUCTION...... 1

II. LITERATUREREVIEW...... 3

2.1 Electrodynamic Solids ...... 3

2.2 Maxwell’s Equations ...... 4

2.3 Conductivity ...... 5

2.4 DrudeandLorentzmodel ...... 6

III.HEAVYFERMIONS ...... 12

3.1 YbFe4Sb12, CeRu4Sb12, CeCoIn5 ...... 13

3.2 Two-ﬂuid Model and Hybridization Order Parameter ...... 23

IV.REFFIT ...... 25

4.1 UsingRefFIT...... 26

V. RESULTANDDISCUSSION ...... 30

5.1 YbFe4Sb12 ...... 34

5.2 CeRu4Sb12 ...... 47

v 5.3 CeCoIn5 ...... 59

5.4 HybridizationOrderParameter ...... 71

VI.CONCLUSION ...... 76

BIBLIOGRAPHY ...... 78

APPENDICES ...... 81

APPENDIX A. APPENDIX TITLE GOES HERE ...... 82

APPENDIX B. SECOND APPENDIX: THE TWO DIMENSIONAL WAVEEQUATION ...... 83 APPENDIX C. EXAMPLE OF A TABLE AND A FIGURE ...... 84

vi LIST OF TABLES

Table Page

3.1 Properties of YbFe4Sb12...... 13

5.1 Temperaturesforthreematerials...... 33

5.2 Fitting parameters for real and imaginary component of YbFe4Sb12 at10K...... 36

5.3 Drude model ﬁtting parameters for YbFe4Sb12 at various temperatures. 38

5.4 Lorentz models ﬁtting parameters for YbFe4Sb12 at various temperatures. 40

5.5 γ ﬁtting parameters of ﬁrst three modes for YbFe4Sb12 with temperature. 45

5.6 Fitting parameters for real and imaginary componant of CeRu4Sb12 at10K...... 49

5.7 Drude models ﬁtting parameters for CeRu4Sb12 at various temperatures. 51

5.8 Lorentz models ﬁtting parameters for CeRu4Sb12 at various temperatures. 53

5.9 γ ﬁtting parameters of ﬁrst three modes for CeRu4Sb12 with temperature. 57

5.10 Fitting parameters for real and imaginary componant of CeCoIn5 at8K...... 61

5.11 Drude models ﬁtting parameters for CeCoIn5 at various temperatures. . 63

5.12 Lorentz models ﬁtting parameters for CeCoIn5 at various temperatures. 65

5.13 γ ﬁtting parameters of ﬁrst three modes for CeCoIn5 with temperature. 69

5.14 Parameter values for YbFe4Sb12, CeRu4Sb12 and CeCoIn5...... 72

vii LIST OF FIGURES

Figure Page

3.1 YbFe4Sb12 has a cubic structure with the space group Im3. The color representaion is, Blue - Sb, Red - Fe, Yellow - Yb (https: www.researchgate.net/ publication/ 255251265 Lattice dynamics and anomalous softening in the YbFe4Sb12 skutterudite)...... 14

3.2 This graph from Dordevic S.V. et.al. [1] shows the resistivity of YbFe4Sb12 asafunctionoftemperature...... 15

3.3 CeRu4Sb12 crystal has body centered cubic (BCC) structure (https : //www.sciencedirect.com/science/article/pii/S0022459616302079). . . 16

3.4 This graph from Dordevic S.V. et.al. [1] shows the resistivity of YbFe4Sb12 dependenceoftemperature...... 18

3.5 This graph from Yuji Matsuda [2] shows CeCoIn5 crystal structure. . . 19

3.6 This graph from Singley E. J. et.al. [3] shows the conductivity of CeCoIn5 dependenceoftemperature...... 21 3.7 This graph from Dordevic S.V. et.al. [1] shows band structure of conduction electrons (ǫk) and localized f-electrons (ǫf ) at high temperature (dashed lines) and low temperature (solid line)...... 22 4.1 Firstinterface...... 26

4.2 DatasetManagerwindow...... 26

4.3 GraphwindowandContentswindow...... 27

4.4 Model window and Parameter control window with the Graph con- tentwindow...... 28 4.5 Changing the parameters of the Drude and Lorentz model...... 29

4.6 Changing the parameters of Drude and Lorentz model...... 29

viii 5.1 (a) Simple Drude mode, (b) Three Drude modes with diﬀerent parameters for conduction and heavy electrons, (c) Two Drude modes for conduction electrons and heavy electrons, (d) Two Drude modes for conduction and heavy electrons with a Lorentz mode for inter- bandtransition...... 32

5.2 This graph from Dordevic S.V. et.al. [1] shows the real (top) and imaginary (bottom) parts of the conductivity of YbFe4Sb12 as a function of wavenumber for diﬀerent temperatures...... 35

5.3 Graph showing experimentaly determined conductivity (black) with the real part of the ﬁtted conductivity curve (red) for YbFe4Sb12 as a function of frequency at 10 K. The modes represent the individual components(blue)oftheoverallcurveﬁt...... 37

5.4 Graphs showing ω of mode 1 (red), mode 2 (blue) and total ω p − − p (black) for YbFe4Sb12 asafunctionoftemperature...... 39 5.5 Graphs showing ω of mode 3 (red), mode 4 (blue) and total ω p − − p (black) for YbFe4Sb12 asafunctionoftemperature...... 41

5.6 Graphs showing normalized total ωp for both Drude (black) and Lorentz (red) models of YbFe4Sb12 as a function of temperature. . . . . 43

5.7 Graph showing total ωp for both Drude and Lorenz models of YbFe4Sb12 asafunctionoftemperature...... 44 5.8 Graphs showing γ of mode 1 (Drude), mode 2 (Drude) and mode 3 (Lorentz) for YbFe−Sb as a function of− temperature. . . . . 46 − 4 12 5.9 This graph from Dordevic S.V. et.al. [1] shows the real (top) and imaginary (bottom) parts of the conductivity of CeRu4Sb12 as a function of wavenumber for diﬀerent temperatures...... 48

5.10 Graph showing experimentaly determined conductivity (black) with the real part of the ﬁtted conductivity curve (red) for CeRu4Sb12 as a function of frequency at 10 K. The modes represent the individual components(blue)oftheoverallcurveﬁt...... 50

5.11 Graphs showing ω of mode 1 (red), mode 2 (blue) and total ω p − − p (black) for CeRu4Sb12 asafunctionoftemperature...... 52 5.12 Graphs showing ω of mode 3 (red), mode 6 (blue) and total ω p − − p (black) for CeRu4Sb12 asafunctionoftemperature...... 54

ix 5.13 Graphs showing normalized total ωp for both Drude (black) and Lorentz (red) models of CeRu4Sb12 as a function of temperature. . . . . 55

5.14 Graph showing total ωp for both Drude and Lorenz models of CeRu4Sb12 asafunctionoftemperature...... 56 5.15 Graphs showing γ of mode 1 (Drude), mode 2 (Drude), mode 3 − − − (Lorentz) and mode 6 (Lorentz) for CeRu4Sb12 as a function of temperature...... − 58

5.16 This graph from Singley E. J. et.al. [3] shows the real part of the conductivity of CeCoIn5 as a function of wavenumber for diﬀerent temperatures...... 60

5.17 Graph showing experimentaly determined conductivity (black) with the ﬁtted conductivity curve (red) for CeCoIn5 as a function of frequency at 8 K. The modes represent the individual components (blue)oftheoverallcurveﬁt...... 62

5.18 Graphs showing ω of mode 1 (red), mode 2 (blue) and total ω p − − p (black) for CeCoIn5 asafunctionoftemperature...... 64 5.19 Graphs showing ω of mode 3 (red), mode 4 (blue) and total ω p − − p (black) for CeCoIn5 asafunctionoftemperature...... 66

5.20 Graphs showing normalized total ωp for both Drude (black) and Lorentz (red) models of CeCoIn5 as afunction oftemperature...... 67

5.21 Graph showing total ωp for both Drude and Lorenz models of CeCoIn5 asafunctionoftemperature...... 68 5.22 Graphs showing γ of mode 1 (Drude), mode 2 (Drude), mode 3 (Lorentz) and mode 4 (Lorentz)− for CeCoIn −as a function of temperature.− 70 − 5

5.23 Graphs showing hybridization order parameter (black) and ωp of mode 3 (red) for YbFe4Sb12 as a function of temperature. The − ∗ values for f0 = 333, T = 60 and C =5418...... 73

5.24 Graphs showing hybridization order parameter (black) and ωp of mode 6 (red) for CeRu4Sb12 as a function of temperature. The − ∗ values for f0 = 3148.5, T = 80 and C =9400...... 74 5.25 Graphs showing hybridization order parameter (black) and the combination of ωp of mode 3 and mode 4 (red) for CeRu4Sb12 as a − − ∗ function of temperature. The values for f0 = 11220, T = 80 and C =29418...... 75

x CHAPTER I

INTRODUCTION

The discovery of heavy electron materials in the late seventies opened new chapters in condensed matter physics [4]. In this work, the conductivity of three skutterudite, heavy fermion materials of this variety - YbFe4Sb12, CeRu4Sb12 and CeCoIn5 - was analyzed at various temperatures and frequencies ranging from infrared to UV. The goal was to study the eﬀect of temperature on the transformation from localized electrons to itinerant heavy electrons. The chosen materials are particularly useful in this application due to their high (above 50 K) characteristic temperature.

Heavy electron materials are composed of interacting local moments of 4f or

5f electrons coupled antiferromagnetically to conduction electrons. This causes, at sufficiently low temperatures, hybridization (spin entanglement) between the f and conduction electrons. Materials of this variety are described using the two-fluid model of heavy electrons which groups electron types into two fluids: a first composed of the hybridized, heavy electrons, and the second comprised of the remaining conduction electrons [5].

The spectral weight, or effective mass, of the composite state (quasi-particle) particles in the heavy electron fluid of hybridized conduction electron is temperature dependent. When the effective mass changes, the magnitude of the charge gap (or

1 pseudogap) is also altered, resulting in changes to the conductivity and the resitivity of the material. Thus, the conductivity and resistivity of the system are temperature dependent [5, 1].

The conductivites of these materials can be described with the Drude-Lorentz model, corresponding to the free and bound electron models, respectively. RefFIT, a program designed to ﬁt and analyze the optical spectra of solids, was used with the

Drude-Lorentz model to ﬁt the experimentally determined IR spectra of the materials.

Some features of RefFIT are described in Chapter IV. Three parameters - the plasma frequency, transverse frequency, and the scattering rate - are used in the program to

ﬁt individual Drude and Lorentz models to the data. The resulting ﬁt parameters were studied to determine the critical temperature and further explore the electron transformation behavior of our heavy fermion materials.

2 CHAPTER II

LITERATURE REVIEW

2.1 Electrodynamic Solids

Even though the individual laws of classical electrodynamics were discovered by

Coulomb, Ampere, Faraday and others, James Clerk Maxwell connected and pack- aged the previously unrelated laws together. Coulomb’s law deals with the forces due to interacting charge distributions. Ampere postulated that magnetism is a result of charges in motion. Faraday observed that magnetic fields influence the behavior of light and that, as a result, light must be electrical in nature. Maxwell’s theory helped to justify these hypotheses, combining electricity, magnetism and optics into the broader field of electromagnetism [6].

Electrical charge is conserved and quantized and the space around it is perme- ated by electromagnetic fields. Electrodynamics deals with moving charges in space under the influence of electromagnetic fields and is classically described by Maxwell’s equations [6, 7].

3 2.2 Maxwell’s Equations

James Clerk Maxwell (1831-1879), was a Scottish mathematician and physicist who is attributed with formulating classical electromagnetic theory. Maxwell’s equations demonstrated that electricity, magnetism and even light are all manifestations of the same phenomenon, namely the electromagnetic ﬁeld. Although his name is borne, like much of physics, his equations come from the work of other physicists. Maxwell, however, brought them together for the ﬁrst time in his paper, Physical Lines of

Force, published between 1861 and 1862 [8]. Further, he added a correction term to

Ampere’s law which accounts for the presence of nonsteady currents. The following equations represent the ﬁnal forms of Maxwell’s equations [6]

1 E = ρ Gauss’s Law (2.1) ▽· ǫ0

B = 0 (2.2) ▽·

∂B E = Faraday’s Law (2.3) ▽ × − ∂t

∂E B = µ J + µ ǫ Ampere’s Law with Maxwell correction (2.4) ▽ × 0 0 0 ∂t

Maxwell’s equations tell us how charges produce ﬁelds, so these equations emphasize that electric ﬁelds can be produced either by charges (ρ) or by changing

4 ∂B magnetic ﬁelds ( ∂t ), and magnetic ﬁeld can be produced either by current or by

∂E changing electric ﬁelds ( ∂t ).

2.3 Conductivity

In general, there are two types of materials: insulators, also known as dielectrics, and conductors. The electrons in an insulator are attached to the atoms, whereas a conductor has free electrons. Despite there being no perfect conductors - a conductor with unlimited free electrons - some materials are close to this description in practice.

For ideal conductors, no electric ﬁelds or net charge is inside the conductor.

The speed of current normal to a unit area (current density) is proportional to the electric ﬁeld. This proportionality factor is deﬁned as electrical conductivity and it depends on the composition of the material [6]. The electrical conductivity is a property that measures a material’s ability to conduct electric current. Normally it is represented by the Greek letter σ (sigma)

J = σE (2.5)

The resistivity is deﬁned as the reciprocal of the conductivity

1 ρ = (2.6) σ

5 2.4 Drude and Lorentz model

2.4.1 Lorentz model

The electron is bound to the nucleus of the atom by a force that behaves according to Hooke’s Law. An applied electric ﬁeld interacts with the charge of the electron, causing “stretching” or “compression” of the spring, which then sets the electron into an oscillating motion. This is the so-called Lorentz oscillator model.

The natural resonant frequency ω0 of the atomic dipole is given by

Ks ω0 = (2.7) s µ where Ks is the spring constant and µ the reduced mass. In this case, it is approxi- mated by µ m since the mass of the nuclei is very large compared to the mass of ≈ 0 an electron. The equation of motion of the electron under an electric ﬁeld E is

Driving force Damping Restoring mass Acceleration = + + × due to electric ﬁeld force force

d2x dx m + m γ + m ω2x = eE (2.8) 0 dt2 0 dt 0 0 −

iωt iωt By substituting E(t) = E0e and x(t) = x0e into equation 2.8, the solution can be derived for its equilibrium position

eE /m x = − 0 0 (2.9) 0 ω2 ω2 iγω 0 − − 6 The relative dielectric constant is determined using the electric displacement D =

ǫ ǫ E and polarization P = Nex [9] 0 r − 0

D = ǫ0E + P (2.10)

Ne2 1 ǫ ǫ E = ǫ E + E (2.11) 0 r 0 m (ω2 ω2 iγω) 0 0 − −

Ne2 1 ǫ (ω)=1+ (2.12) r ǫ m (ω2 ω2 iγω) 0 0 0 − − 2.4.2 Drude model

The Drude model is a result of applying kinetic theory to electrons in a solid and is used to model free electrons and electrical conduction. Metals are considered as a classical gas of electrons where the electrons are not bound to the nuclei. In the Drude model, the density of atoms is equal to the density of free electrons, resulting in the high electrical conductivity observed in metals. The potential energy distribution of a metallic lattice makes it as energetically favorable for electrons to jump from one nucleus to another as staying bound to a single nucleus. These delocalized electrons constitute what is known as a “sea of electrons” where the particles are able to move freely around the lattice of a nuclei. It is this which allows metals to conduct electricity. Consider a particular change in our Lorentz oscillator model: in this metallic bond, if the electrons are not bound, then there is no restoring “spring”

7 force. That is, Fspring = 0, hence the equivalent spring constant associated with it is equal to 0.

If an electric ﬁeld is applied, the electrons accelerate away from a nucleus and collide with other electrons. The characteristic scattering time, or relaxation time, τ of the system describes the time needed for the system to return to equilibrium. The relationship between τ, momentum P , and the electric ﬁeld is

dP P = eE (2.13) dt − τ −

dP A DC electric ﬁeld gives dt = 0 and, using equation 2.13 with the current density

J = NeP , we get the DC conductivity − m

Ne2τ σ = (2.14) DC m

dx For the case of an AC input, P = m0 dt . Substituting this into equation 2.13 gives the displacement x of an electron with mass m0 under the AC electric ﬁeld

−iωt E(t)= E0e

d2x dx m + m γ = eE e−iωt (2.15) 0 dt2 0 dt − 0

1 where ω is the frequency of the electric ﬁeld and γ = τ is the scattering rate. The solution to the above equation is

eE(t) x(t)= 2 (2.16) m0(ω + iγω)

8 Using the deﬁnition of the electric displacement D and the polarization P =

Nex , the expression can be written for the relative dielectric constant ǫ − 0 r

D = ǫ0E + P (2.17)

Ne2E ǫrǫ0E = ǫ0E 2 (2.18) − m0(ω + iγω)

ω2 ǫ (ω)=1 p (2.19) r − (ω2 + iγω)

2 Ne 1/2 where ωp =( ǫ0m0 ) is plasma frequency. For lightly damped systems, γ = 0, giving

ω2 ǫ (ω)=1 p (2.20) r − ω2

The reﬂectivity R from the complex refractive index n, n = √ǫr, is

n 1 2 R = − (2.21) n +1

dx Writing equation 2.15 in terms of the velocity v = dt

dv m + m γv = eE (2.22) 0 dt 0 −

−iωt Letting v = v0e gives the solution

eτ 1 v(t)= E(t) (2.23) −m 1 iωτ 0 −

9 The equation for conductivity can be derived by using the current density j =

Nev = σE −

Ne2τ/m σ(ω)= 0 (2.24) 1 iωτ −

1 σ(ω)= ω2ǫ (2.25) p 0 1 iω τ − with real and imaginary components σ1 and σ2, respectively

1 σ (ω)= ω2ǫ τ (2.26) 1 p 0 1+ ω2τ 2

ωτ σ (ω)= ω2ǫ τ (2.27) 2 p 0 1+ ω2τ 2

Using equation 2.25 with equation 2.20, the dielectric constant becomes

iσ(ω) ǫr(ω)=1+ (2.28) ǫ0ω

ω2τ 2 iω2τ ǫ (ω)=1 p + p (2.29) r − 1+ ω2τ 2 ω(1 + ω2τ 2)

The free electron model of metals is attributed to Paul Drude, and so the application of the dipole oscillator model to free electron systems is generally called the Drude-Lorentz model [9, 10]

ǫ˜ ω2 =1 p (2.30) ǫ − (ω2 iγω) 0 − 10 This is particularly useful for developing accurate predictions regarding the optical behavior of metals [11]

There are three parameters - ω0, ωp and γ - which describe the Drude and

Lorentz models. The transient frequency ω0 is the location of peak conductivity for the Lorentz models; however, this value is non-zero only for bound electrons and, as such, is zero for the Drude free electron model. The area under the curve of the Drude or Lorentz models is related to the plasma frequency ωp and describes the number of bound electrons. The width of the peak is related to the scattering rate γ and the relaxation time is given by the reciprocal of γ

1 τ = (2.31) γ

11 CHAPTER III

HEAVY FERMIONS

Heavy fermion materials are materials composed of strongly correlated and localized

4f or 5f electrons with delocalized s or d conduction electrons (Fermi electrons).

If the correlation between f-electrons and Fermi electrons is strong, then collective hybridization, a kind of spin entanglement where the f and Fermi electrons undergo a spin-ﬂip transition, occurs. It is also called a Kondo spin ﬂuctuation; therefore, the fermion system is called a Kondo lattice.

The new state of hybridization between localized f-electrons and conduction electrons is called a quasiparticle, and it has a very large eﬀective mass compared to that of free electrons. Therefore, these new coupled electrons are referred to as heavy fermions.

Comparing the properties of a heavy electron system to normal metals, the resistivity shows very small variation at high temperatures; however, the resistivity highly depends on temperature below the characteristic temperature where the resistivity decreases as temperature decreases. [4, 5]

12 3.1 YbFe4Sb12, CeRu4Sb12, CeCoIn5

3.1.1 YbFe4Sb12

YbFe4Sb12 is an interesting model system due to its unique magnetic and electrical properties. YbFe4Sb12 is in the skutterudites family. Skutterudite materials have good electrical conductivity and poor thermal conductivity. The ﬁlled skutterudites can be represented by the formula LnA4B12, where Ln is a rare earth element, A can be iron or ruthenium, and B is phosphorus, arsenic or antimony. This compound has a cubic type crystal structure with a unit cell containing eight AB3 groups. There are also two large cages in the unit cell which are ﬁlled with rare earth atoms. Due to the weak bonding of the rare earth atoms with its 12 nearest pnictogen neighbours,

ﬁlling atoms are able to interact with a wide spectrum of low frequency phonons.

This results in the substantially lower lattice thermal conductivity value of ﬁlled skutterudites when compared to that of unﬁlled binary skutterudites.

Table 3.1: Properties of YbFe4Sb12.

Property Value

Lattice parameter 0.9156 nm

Liquidus temperature 1151 K

Hole concentration at room temperature 3.6 1021 cm−3 × Electrical resistivity at room temperature 0.62 mΩ cm

13 Figure 3.1: YbFe4Sb12 has a cubic structure with the space group Im3. The color representaion is, Blue - Sb, Red - Fe, Yellow - Yb (https: www.researchgate.net/ publication/ 255251265 Lattice dynamics and anomalous softening in the YbFe4Sb12 skutterudite).

Figure 3.2 shows the resistivity of YbFe4Sb12 as a function of temperature.

The electrical resistivity increases with increasing temperature and YbFe4Sb12 exhibits a maximum between 760 K and 800 K. The estimated total thermal conduc-

−1 −1 tivity of YbFe4Sb12 ranges from 33.6 mW cm K at room temperature to 28.7 mW cm−1 K−1 at 670 K.

In the structure of YbFe4Sb12, FeSb3 is the parent compound with Fe4Sb12 polyanion ﬁller atom Yb. The valence state of Yb in the system is intermediate between Yb2+ and Yb3+. Since Yb exhibits the oxidation state of 2+ in some com-

14 pounds, their large ion radii stabilizes the skutterudite structure. A structural phase transition occurs due to atom position change; further, a magnetic phase transition from ferromagnetism to paramagnetism might take place due to temperature change.

The frequency of the ﬁller phonon mode and the valence state may also be altered.

YbFe4Sb12 is considered to be a heavy fermion system because of the high eﬀective mass of its conduction electrons at low temperature. [12, 13]

Resistivity of YbFe Sb as a function of temperature

4 12

400 cm)

300 ( DC

200 Resistivity Resistivity

100

0 100 200 300

Temperature (K)

Figure 3.2: This graph from Dordevic S.V. et.al. [1] shows the resistivity of YbFe4Sb12 as a function of temperature.

15 3.1.2 CeRu4Sb12

Ce-based ﬁlled skutterudites have been subjected to experimental work due to their unique structural, electronic and optical properties. CeRu4Sb12 has the ﬁlled skutterudite crystal structure and it shows metallic characteristics and a temperature dependent electric resistivity. These compounds have high carrier concentration, high electrical conductivity and narrow band gap semiconducting behavior.

Figure 3.3: CeRu4Sb12 crystal has body centered cubic (BCC) structure (https :

//www.sciencedirect.com/science/article/pii/S0022459616302079).

CeRu4Sb12 crystals have a body centered cubic (BCC) structure with 17 atoms per unit cell. The experimental value of the equilibrium lattice constant is about 9.26 A˚. The Ce atoms are weakly bound to the Ru-Sb cages; therefore, the compound has low lattice thermal conductivity at room temperature. Due to the

16 intermediate valence of Ce, it has an electronic speciﬁc heat coeﬃcient which increases with temperature.

Materials with hybridized local f-electrons and conduction electrons at low temperature - also known as strongly corrleated electron systems - have a variety of interesting phenomena including superconductivity, ferromagnetism, antiferromag- netism and metal insulator transition. Figure 3.4 shows the resistivity of CeRu4Sb12 dependence of temperature. In particular, the electrical resistivity of CeRu4Sb12 decreases with decreasing temperature below 100 K. Below 80 K, this compound behaves like a metal due to the pseuedogap present in Kondo insulators. The 4f state of

Ce strongly hybridizes with the 4d state of Ru at very low temperatures and, due to it then being in a heavy electron state, its optical conductivity is strongly suppressed.

[14, 15, 16]

17 Resistivity of CeRu Sb as a function of temperature

4 12

150 cm) (

100 DC

Resistivity Resistivity 50

0 100 200 300

Temperature (K)

Figure 3.4: This graph from Dordevic S.V. et.al. [1] shows the resistivity of YbFe4Sb12 dependence of temperature.

18 3.1.3 CeCoIn5

The heavy fermion material CeCoIn5 has a diﬀerent two dimensional layered crystal structure with a = 4.62 A˚ and c = 7.56 A˚. The antiferromagnet CeIn3 layers are separated by CoIn2 layers. The layers form a cubic lattice with the In atoms located between the Ce and Co. f-electrons only exist in the Ce-layer and conduction electrons

(spd-electrons) are only present in the Co layer [17, 18, 19].

Figure 3.5: This graph from Yuji Matsuda [2] shows CeCoIn5 crystal structure.

CeCoIn5 is considered as a Kondo lattice and it shows superconductivity due to the formation of a strongly correlated resonance at Fermi-energy. The f-electrons in CeCoIn5 strongly inﬂuence the material properties by increasing the eﬀective mass of bare electrons by 10-50 times [19].

19 Heavy fermion materials exhibit superconductivity due to strong electronic correlation. This occurs in some celenium based materials since the Ce atom is a magnetic ion. Even though heavy fermion compounds CeCu2Si2, CeIrIn5 (this shows zero resistivity near 1 K, but it does not show superconductor behavior until

0.4 K) and other Ce-based materials show superconductivity at ambient pressure,

CeCoIn5 has the highest transition temperature at ambient pressure. Figure 3.6 shows the conductivity of CeCoIn5 as a function of temperature. The conductivity of

CeCoIn5 weakly depends on temperature above 30 K, after which it rapidly increases and becomes nearly linear with low temperature. CeCoIn5 achieves superconductor behavior at 2.3 K in ambient-pressure environments [20]. Further, the heat capacity is high at its characteristic temperature. A dip of the chemical potential on the Ce layer suggests that there is a hybridization gap between the f and conduction electrons.

[17, 18, 19]

Around 70 K, the material starts to form heavy quasiparticles as f-electrons hybridize with the spd-electrons. CeCoIn5 transitions to a superconductor at 2.3 K due to qusiparticle excitations changing the magnitude of the pseudogap [21]. Figure

3.7 shows band structure and at low temperatures, a direct gap opens.

20 Figure 3.6: This graph from Singley E. J. et.al. [3] shows the conductivity of CeCoIn5 dependence of temperature.

21 Figure 3.7: This graph from Dordevic S.V. et.al. [1] shows band structure of conduction electrons (ǫk) and localized f-electrons (ǫf ) at high temperature (dashed lines) and low temperature (solid line).

22 3.2 Two-ﬂuid Model and Hybridization Order Parameter

The two ﬂuid model has been demonstrated to successfully model a number of unusual phenomena in heavy electron materials. These materials contain a lattice of localized electrons surrounded by mobile conduction electrons. Some of these local 4f or

5f electrons, at suﬃciently low temperatures, become coupled antiferromagnetically to conduction electrons. This coupling causes hybridization (spin entanglement) between the f and conduction electrons, resulting in heavy electrons with substantially altered properties.

This model posits that these coexisting states of electrons can be described as two individual fluids: the first composed of hybridized, heavy electrons, and the second comprised of the remaining unhybridized conduction electrons. As temperature decreases, the former fluid begins to emerge, resulting in a transfer of the f-electron spectral weight from the localized electrons to the newly formed heavy electrons. The amount of the spectral weight transferred to heavy electron states is described by the hybridization order parameter

T f(T )= f (1 )3/2 (3.1) 0 − T ∗ where T is temperature; f0 is the hybridization eﬀectiveness, a parameter which determines the plasma frequency ratio of heavy to unhybridized electrons at low temperature; and T ∗ is the coherence temperature, the point at which the coupling process begins. The parameters f0 and T are dependent upon both internal and

23 external sources - including the possible material doping and external ﬁelds - and can be determined either through experiment or theoretical analysis.

Superconductors have normal conduction electrons and cooper pairs. For superconductors, the order parameter f(T ) is related to the energy gap. Heavy fermion materials also have normal conduction electrons and cooper pairs. Therefore, we suppose that the band gap of heavy fermion materials follows the same order parameter f(T ) [5].

24 CHAPTER IV

REFFIT

RefFit was inented by scientist Alexy Kuzmenko at the University of Geneva, Switzer- land. It is a fast, ﬂexible program designed to analyze and model optical spectra.

The program accomplishes this by using reﬂectivity or conductivity data to determine a material’s frequency dependent dielectric function. Once solved for, RefFIT allows a user to analytically model the initial data using a set of adjustable parameters discussed later; in our case, we used RefFIT to ﬁt a Drude-Lorentz model to the conductivity data. Though other methods may be used to accomplish this same task, RefFIT allows a user to see results in real time [22].

In particular, this software mainly uses the Fresnel formula to analyze the data and determine the dielectric function

2 1 ǫ(ω) R = − Fresnel formula (4.1) 1+ ǫ(ω) p

p Here, R is the normal-incident reﬂectivity and ǫ is the complex dielectric function dependent upon the frequency ω

ǫ(ω)= ǫ1(ω)+ iǫ2(ω) (4.2)

25 Figure 4.1: First interface. Figure 4.2: Dataset Manager window.

To create a fitting curve which perfectly matches the original data set, more than one Drude or Lorentz mode is required. Using a large number of modes provides a better fit than fewer modes, but it substantially increases analysis difficulty. The goal is to choose the smallest amount of modes needed to faithfully fit the data.

4.1 Using RefFIT

Once open, the RefFIT software window appears as in Figure 4.1. To load the data set, the “Dataset Manager” icon in the above toolbar has to be clicked. Figure 4.2 shows the window of “Dataset Manager” window where data sets can be loaded or unloaded via the load/unload buttons on the left hand side. The proper data set can be selected using the “Quantity” column’s drop down boxes. For instance, S1 represents the conductivity data.

26 Figure 4.3: Graph window and Contents window.

To see the graph, the “Graph” icon in the top tool bar has to be clicked.

Then, double tapping on the graph window, the contents window will appear. The data set loaded to the software can be seen under “Available curves.” To place that data on the graph, the data set should be put under the graph curves space using the arrow icons. All data points are made visible as in Figure 4.3 by double tapping on the axes.

To create a ﬁtting curve, the ﬁrst icon in the top tool bar that is called ”New model” has to be clicked. Then two windows, the model window and Parameter control window, will appear simultaneously. As in Figure 4.4, any number of modes can be added or removed by using the Add and Remove buttons on the bottom of the model window. The values of the parameters ω0, ωp and γ, can be changed by selecting and typing or by moving the slider on the Parameter control window. As

27 Figure 4.4: Model window and Parameter control window with the Graph content window.

shown in Figure 4.5, to make visible the ﬁtting curve on the graph window, the graph content window has to be opened. Once opened, select the correct data model from under the Available curves and put it under the graph curves.

The red line on the graph window represents the fitting curve and the black line with the data points represents the actual data. Figure 4.6 shows a fitting curve that is very close to the actual data curve using four modes, the first two Drude modes and the last two Lorentz modes.

28 Figure 4.5: Changing the parameters of the Drude and Lorentz model.

Figure 4.6: Changing the parameters of Drude and Lorentz model.

29 CHAPTER V

RESULT AND DISCUSSION

This section includes the conductivity results determined using the Kramers-Kroning analysis of reflectivity [1] as a function of frequency for materials, YbFe4Sb12, CeRu4Sb12 and CeCoIn5 obtained from UV-visible spectroscopy. The Drude-Lorenze fits of the conductivity for different temperatures were generated and analyzed using RefFIT.

The plotting program Origin was used to generate the graphs.

Normal conduction electrons at high temperature can be represented by a simple Drude model. Figure 5.1 (a) shows a simple Drude model with the conductivity from normal electrons as a function of frequency. The area under the curve gives the total number of conduction electrons. Figure 5.1 (b) shows three Drude modes with diﬀerent parameter values. When the temperature goes down, some normal electrons become heavy electrons. The black dashed line represents the conductivity from normal conduction electrons at high temperature. The black solid line represents the conductivity from normal conduction electrons after temperature goes down. The area under the black dashed line approaches the area under the black solid line as temperature decreases. That means the number of conduction electrons at high temperature decreases as the temperature decreases. The area diﬀerence between black dashed line and the solid black line is related to the total number of

30 transformed electrons (normal conduction electrons to heavy electrons). The blue solid line represents the conductivity due to heavy electrons at low temperature, and the area under this line is related to the number of heavy electrons; further, it is equal to the area diﬀerence between the black dashed line and the black solid line due to the charge conservation. At very low temperature, all normal conduction electrons become heavy electrons. Figure 5.1 (c) shows the conductivity from normal electrons

(black line) and heavy electrons (blue line) at very low temperatures. The area under the black line is near zero since there are no normal electrons at very low temperatures whereas the area under the blue line is a maximum because almost all the conduction electrons are heavy electrons. Figure 5.1 (d) shows the contribution of normal conduction electrons (black line) and heavy electrons (blue line) with the inter-band transition (red line) at low temperature. The eﬀect of the inter-band transition on the conductivity can be studied using the Lorentz model.

31 Figure 5.1: (a) Simple Drude mode, (b) Three Drude modes with diﬀerent parameters for conduction and heavy electrons, (c) Two Drude modes for conduction electrons and heavy electrons, (d) Two Drude modes for conduction and heavy electrons with a Lorentz mode for inter-band transition.

32 The conductivity data of three materials (YbFe4Sb12, CeRu4Sb12 and CeCoIn5) were analyzed for different temperatures (Table 5.1 shows the temperature values in the range of 8 K - 300 K) to figure out the critical temperatures for each material and to study the behavior of different electrons.

Table 5.1: Temperatures for three materials.

YbFe4Sb12 CeRu4Sb12 CeCoIn5

10 K 10 K 8 K

20 K

30 K 30 K 25 K

40 K

50 K 50 K 50 K

60 K

80 K 80 K 77 K

100 K

200 K

300 K 300 K 300 K

33 5.1 YbFe4Sb12

The real and imaginary parts of the conductivity of YbFe4Sb12 as a function of frequency for diﬀerent temperatures are shown in Figure 5.2. The conductivity is on the y-axis and frequency is on the x-axis. The range of frequency is 4 to 25,000 cm−1 and is in log scale. The solid line describes the variation of conductivity at 10 K, the lowest temperature, while the dashed and dot lines are for higher temperatures.

There are two sharp peaks at 114 and 267 cm−1 due to infrared active phonons while the conductivity is suppressed at frequencies below 90 cm−1[1]. The little bump between frequencies 100 cm−1 and 300 cm−1 is related to the hybridization gap shrinking as temperature increases. At low temperatures, the localized f-electrons interact with conduction electrons, making quasi-particles which lead to the opening of the charge gap [1].

According to the real part of Figure 5.2, the ﬁrst minimum values of conductivity are at temperature 10 K (solid line) and lie in the frequency range of 60 to 70 cm−1. The conductivity at that point is not zero which means there is a contribution to conductivity from the few normal conduction electrons which still exist very low temperatures. So, at this temperature, both normal and heavy conduction electrons contribute to the conductivity of this two ﬂuids material.

Table 5.2 includes the parameter values of two Drude models and four Lorentz models for YbFe4Sb12. These six modes create both the real and imaginary components which describes conductivity as a function of frequency at the temperature 10

34 10 K

YbFe Sb - Real 20 K

4 12

30 K

40 K 10000

50 K

60 K

80 K

100 K

300 K

5000 ) -1 cm (

YbFe Sb - Imaginary

4 12

5000 Conductivity Conductivity

10 100 1000 10000

-1

Frequency (cm )

Figure 5.2: This graph from Dordevic S.V. et.al. [1] shows the real (top) and imaginary (bottom) parts of the conductivity of YbFe4Sb12 as a function of wavenumber for diﬀerent temperatures.

35 Table 5.2: Fitting parameters for real and imaginary component of YbFe4Sb12 at 10

Mode No Model ω0 ωp γ

1 Drude 0 3104 18

2 Drude 0 1530 2.3

3 Lorenz 153 5670 222

4 Lorenz 704 12408 2300

5 Lorenz 11370 68816 14950

6 Lorenz 39861 33406 14813

At very low temperature (10 K), the conductivity in the frequency range of 4 cm−1 to 30 cm−1 is very high. This frequency region is described by the combination of the two Drude models related to the normal conduction electrons and heavy electrons.

36 Figure 5.3: Graph showing experimentaly determined conductivity (black) with the real part of the ﬁtted conductivity curve (red) for YbFe4Sb12 as a function of frequency at 10 K. The modes represent the individual components (blue) of the overall curve

ﬁt.

37 The parameter values of ωp for the two Drude models, total ωp and normalized values of the total ωp at diﬀerent temperatures are shown in Table 5.3. The total ωp values are related to the total number of the normal and heavy electrons and were calculated using the following equation

2 2 ωp(total) = ωp(mode1) + ωp(mode2) (5.1) q

Table 5.3: Drude model ﬁtting parameters for YbFe4Sb12 at various temperatures.

Temperature ωp of mode ωp of mode Total ωp Normalized

1 2 ωp

10 3104 1530 3460.6 0.23

20 3150 1468 3475.3 0.23

30 3199 1275 3443.7 0.22

40 3610 868 3712.9 0.24

50 3764 654 3820.4 0.25

60 3850 623 3900.1 0.25

80 3560 14386 14820 0.97

100 3630 14530 14976.6 0.98

300 3148 15025 15351.2 1.00

38 Figure 5.4 shows the ωp variation of the two Drude models (mode-1 and mode-2) and the total ωp at various temperatures. Since the variation in mode-1 is not large, it does not eﬀect signiﬁcantly the total ωp. According to the variation of mode-2, when the temperature decreases there is a steep drop in ωp for temperatures below 80 K. This indicates that the number of normal conduction electrons is getting smaller for temperatures below 80 K.

for Drude modes as functions of temperature

15000

mode-2

mode-1

10000

Total p

5000

0 100 200 300

Temperature (K)

Figure 5.4: Graphs showing ω of mode 1 (red), mode 2 (blue) and total ω (black) p − − p for YbFe4Sb12 as a function of temperature.

39 Even though there are four Lorentz models used to ﬁt the conductivity for the region of high frequency, there are only two Lorentz models in region useful for our study. The parameters of the other two Lorentz models in the high frequency region do not depend on the temperature and also do not eﬀect the hybridization phenomena. However, the plasma frequency ωp of the third mode (Lorentz model) is, unlike the other Lorentz modes, temperature dependent.

Table 5.4: Lorentz models ﬁtting parameters for YbFe4Sb12 at various temperatures.

Temperature ωp of mode ωp of mode Total ωp Normalized

3 4 ωp

10 5670 12408 13642.1 1.00

20 5600 12408 13613.2 1.00

30 5539 12408 13588.2 1.00

40 5483 12408 13565.5 0.99

50 5441 12408 13548.5 0.99

60 5418 12408 13539.3 0.99

80 0 0 0 0

100 0 0 0 0

300 0 0 0 0

For high temperatures (above 60 K), the third and fourth modes are not

40 required to ﬁt the conductivity curve. This indicates that these kind of electrons no longer contribute to the conductivity.

Figure 5.5 shows the variation of plasma frequency for mode-3 (red), mode-4

(blue) and the total ωp (black) from the combination of both mode-3 and mode-4 with respect to temperature.

for Lorentz modes as functions of temperature

16000

mode-3

mode-4

12000

Total

8000 p

4000

0 100 200 300

Temperature (K)

Figure 5.5: Graphs showing ω of mode 3 (red), mode 4 (blue) and total ω (black) p − − p for YbFe4Sb12 as a function of temperature.

According to the data of total ωp for Lorentz modes, when temperature de-

41 creases, there is a huge jump in the values of ωp for temperatures below 70 K. That means there is not a contribution to the conductivity from the electrons which are represented by Lorentz modes at high temperature. But, for low temperature, these electrons are hybridized with some conduction electrons and make a contribution to the conductivity.

Taken together, the variation of normalization values of ωp for both the Drude and Lorentz models with temperature clearly shows the transformation of some electrons. At about 70 K, both Drude and Lorentz models show step changes.

For temperatures above 70 K, the Drude electrons dominate and there is no contribution from Lorentz electrons, suggesting only normal conduction electrons are present. But, for temperatures below 70 K, most conduction electrons transform to heavy electrons, increasing the number of Lorentz electrons while decreasing the amount of conduction electrons.

42 Normalized for Drude and Lorentz modes as functions of temperature

1.0

Drude mode

0.8 Lorentz mode p

0.6

0.4 Normalized total total Normalized

0.2

0.0

0 100 200 300

Temperature (K)

Figure 5.6: Graphs showing normalized total ωp for both Drude (black) and Lorentz

(red) models of YbFe4Sb12 as a function of temperature.

43 Total as a function of temperature

80000

70000 p Total Total

60000

50000

0 100 200 300

Temperature (K)

Figure 5.7: Graph showing total ωp for both Drude and Lorenz models of YbFe4Sb12 as a function of temperature.

According to the Figure 5.7, the total ωp (total ωp is the combination of all the ωp values of both Drude and Lorentz models) is constant with temperature. The

Ne2 ωp (plasma frequency ωp = ǫ0m0 ) is related to the number of electrons N. Therefore, q the total ωp is related to the total number of electrons causing it to be constant with temperature due to charge conservation.

44 Table 5.5: γ ﬁtting parameters of ﬁrst three modes for YbFe4Sb12 with temperature.

Temperature γ of mode 1 γ of mode 2 γ of mode 3

10 18 2.3 222

20 20 2.7 226

30 26 3 237

40 35 4 259

50 48 4 297

60 54 4 339

80 154 2054

100 158 2317

300 284 2341

The parameter values of γ for only the ﬁrst three modes (ﬁrst two modes are

Drude while the third one is Lorentz) vary with temperature. It is constant for the last three Lorentz modes. For temperatures below 60 K, the third Lorentz mode was used to get a better ﬁt under the hybridization gap. But, for high temperatures, only the two Drude modes are used to describe that frequency range.

For temperatures above the characteristic temperature, only two Drude modes contribute to the conductivity since the Lorentz mode was not used in the high temperature case. According to Figure 5.8, γ for all three modes decrease as temperature 45 for YbFe Sb as a function of temperature

4 12

2500

2000

mode-1

mode-2

1500

mode-3

1000

500

0 100 200 300

Temperature (K)

Figure 5.8: Graphs showing γ of mode 1 (Drude), mode 2 (Drude) and mode 3 − − −

(Lorentz) for YbFe4Sb12 as a function of temperature.

decreases. For temperatures below the characteristic temperature, the values of γ are small compared to that of the high temperatures. This indicates that the relaxation time (the average time between collisions) of the electrons and conductivity are very large for low temperatures.

46 5.2 CeRu4Sb12

The real and imaginary parts of the conductivity of CeRu4Sb12 as a function of frequency for diﬀerent temperatures are shown in Figure 5.9. The conductivity is on the y-axis and frequency is on the x-axis. The range of frequency is 4 to 25000 cm−1 and in log scale. The solid line for describes the variation of conductivity at 10 K, the lowest temperature, while the dashed and other dashed and dot lines are for high temperatures.

There are two sharp peaks at 116, 221 and 248 cm−1 due to infrared active phonons while the conductivity is suppressed at frequencies below 400 cm−1 [1]. The little bump between frequencies 200 cm−1 and 1000 cm−1 is related to the hybridization gap shrinking as temperature increases.

According to the real part of Figure 5.9, the ﬁrst minimum values of conductivity at the temperature 10 K (solid line) lie in the frequency range of 40 to 50 cm−1. The conductivity at that point is almost zero and that means there is not a contribution to conductivity from the normal conduction electrons at very low temperature. So, at this temperature, only heavy conduction electrons contribute to the conductivity of this two ﬂuids material.

Table 5.6 includes the parameter values of two Drude models and ﬁve Lorentz models for CeRu4Sb12. These seven modes create both the real and imaginary components which describe conductivity as a function of frequency at the temperature

10 K.

47 CeRu Sb - Real 10 K

4 12

30 K

50 K

80 K

300 K

4000 ) -1 cm -1 (

CeRu Sb - Imaginary

4 12

4000 Conductivity Conductivity

10 100 1000 10000

-1

Frequency (cm )

Figure 5.9: This graph from Dordevic S.V. et.al. [1] shows the real (top) and imaginary (bottom) parts of the conductivity of CeRu4Sb12 as a function of wavenumber for diﬀerent temperatures.

48 Table 5.6: Fitting parameters for real and imaginary componant of CeRu4Sb12 at 10

Mode No Model ω0 ωp γ

1 Drude 0 1058 7

2 Drude 0 1036.8 1.3

3 Lorenz 197.8 1615 79

4 Lorenz 221 579 5.9

5 Lorenz 247.5 395 6.3

6 Lorenz 703.5 11977 1099

7 Lorenz 7289.6 21024 9292

At very low temperature (10 K), the conductivity in the frequency range of 4 cm−1 to 60 cm−1 is very high. This frequency region is described by the combination of the two Drude models related to the normal conduction electrons and heavy electrons.

49 CeRu Sb at 10 K

4 12

Original Data

Fitted Data

) 4000

Mode 1 -1

Mode 2 cm

Mode 3

Mode 4 (

Mode 5

Mode 6

Mode 7 2000 Conductivity Conductivity

10 100 1000 10000

-1

Frequency (cm )

Figure 5.10: Graph showing experimentaly determined conductivity (black) with the real part of the ﬁtted conductivity curve (red) for CeRu4Sb12 as a function of frequency at 10 K. The modes represent the individual components (blue) of the overall curve ﬁt.

50 The parameter values of ωp for the two Drude models and total ωp and normalized values of the total ωp at diﬀerent temperatures are shown in Table 5.7.

The total ωp values are from the equation 5.1.

Table 5.7: Drude models ﬁtting parameters for CeRu4Sb12 at various temperatures.

Temperature ωp of mode ωp of mode Total ωp Normalized

1 2 ωp

10 1058 1037 1481 0.267

30 1449 1041 1785 0.322

50 1967 1047 2228 0.402

80 4084 1259 4274 0.772

300 5074 2208 5534 1.00

Figure 5.11 shows the variation of ωp of the two Drude models (mode-1 and mode-2) and the total ωp of them at various temperatures. Since the variation in mode-2 is not large, it does not eﬀect signiﬁcantly the total ωp. According to the variation of mode-1, when the temperature decreases there is a steep drop in ωp for temperatures below 80 K. This indicates that the number of normal conduction electrons is getting smaller for temperatures below 80 K.

Even though there are four Lorentz models used to ﬁt the conductivity for the region of high frequency, there are only two Lorentz models in the useful region

51 for Drude modes as functions of temperature

6000

4000 mode-1

mode-2

Total p

2000

0 100 200 300

Temperature (K)

Figure 5.11: Graphs showing ω of mode 1 (red), mode 2 (blue) and total ω p − − p

(black) for CeRu4Sb12 as a function of temperature.

for our study. The parameters of the other two Lorentz models in the high frequency region do not depend on the temperature since they are represent the two phonons and also do not eﬀect the hybridization phenomena.

For high temperatures (above 100 K), the sixth mode is not required to ﬁt the conductivity curve. This indicates that this kind of electron no longer contributes to the conductivity.

52 Table 5.8: Lorentz models ﬁtting parameters for CeRu4Sb12 at various temperatures.

Temperature ωp of ωp of ωp of ωp of Total ωp Normalized

mode 3 mode 4 mode 5 mode 6 ωp

10 1615 579 395 11977 12106 1.024

30 1868 579 395 11106 11284 0.954

50 2824 579 395 9944 10361 0.874

80 3300 579 395 9400 9987 0.845

300 10048 579 395 0 10072 0.852

Figure 5.12 shows the variation of plasma frequency of mode-3 (red), mode-6

(blue) and total ωp (black) from the combination of both mode-3 and mode-6 with respect to temperature.

According to the data for Lorentz modes, when temperature decreases, the values of total ωp increases for temperatures below 80 K. That means there is not a contribution to the conductivity from the electrons which are represented by Lorentz modes at high temperature. But, for low temperature, these electrons are hybridized with some conduction electrons and make a contribution to the conductivity.

Taken together, the variation of normalization values of ωp for both the Drude and Lorentz models with temperature clearly shows the transformation of some electrons. At about 100 K, both Drude and Lorentz models show step changes.

53 for Lorentz modes as functions of temperature

12000

10000

8000 p

mode-3

6000

mode-6

Total

4000

2000

0 100 200 300

Temperature (K)

Figure 5.12: Graphs showing ω of mode 3 (red), mode 6 (blue) and total ω p − − p

(black) for CeRu4Sb12 as a function of temperature.

For temperatures above 100 K, the Drude electrons dominate and there is a small contribution from Lorentz electrons, suggesting normal conduction electrons are more prevalent than heavy electrons. But, for temperatures below 100 K, most conduction electrons transform to heavy electrons, increasing the number of Lorentz electrons while decreasing the amount of conduction electrons.

54 Normalized for Drude and Lorentz modes as functions of temperature

1.0

0.8 p

0.6 Normalized Normalized

Drude mode

Lorentz mode 0.4

0.2

0 100 200 300

Temperature (K)

Figure 5.13: Graphs showing normalized total ωp for both Drude (black) and Lorentz

(red) models of CeRu4Sb12 as a function of temperature.

55 Total as a function of temperature

30000

25000

20000 p

15000 Total Total

10000

5000

0 100 200 300

Temperature (K)

Figure 5.14: Graph showing total ωp for both Drude and Lorenz models of CeRu4Sb12 as a function of temperature.

According to the Figure 5.14, the total ωp (total ωp is the combination of all the ωp values of both Drude and Lorentz models) is constant with temperature. The total ωp is related to the total number of electrons causing it to be constant with temperature due to charge conservation.

56 Table 5.9: γ ﬁtting parameters of ﬁrst three modes for CeRu4Sb12 with temperature.

Temperature γ of mode 1 γ of mode 2 γ of mode 3 γ of mode 6

10 7.134 1.256 79 1099

30 11.803 1.904 117 1066

50 21.691 3.676 483 1037

80 69.633 8.516 984 1000

300 192.57 19.483 1006

The parameter values of γ for the first three modes and sixth (first two modes are Drude while the third and sixth are Lorentz) vary with temperature. It is constant for the fourth and fifth Lorentz modes since they represent phonons. For temperatures below 80 K, the sixth Lorentz mode was used to get a better fit under the hybridization gap. But, for high temperatures, only the two Drude modes and one Lorentz mode are used to describe that frequency range.

For the temperature above the characteristic temperature, only the ﬁrst three modes contribute to the conductivity since the sixth Lorentz mode wasn’t used. Ac- cording to Figure 5.15, γ for the ﬁrst three modes decreases as temperature decreased.

For temperatures below the characteristic temperature, even though mode 6 shows a high value for γ, the other three modes have smaller values compared to the high temperatures. However, the contribution from the ﬁrst three modes to the conductivity

57 for CeRu Sb as a function of temperature

4 12

1200

1000

800

mode-1

mode-2

600

mode-3

mode-6

400

200

0 100 200 300

Temperature (K)

Figure 5.15: Graphs showing γ of mode 1 (Drude), mode 2 (Drude), mode 3 − − − (Lorentz) and mode 6 (Lorentz) for CeRu Sb as a function of temperature. − 4 12

is very large due to the small values of γ.

58 5.3 CeCoIn5

The real part of the conductivity of CeCoIn5 as a function of frequency for diﬀerent temperatures are shown in Figure 5.16. The conductivity is on the y-axis and the frequency is on the x-axis. The range of frequency is 4 to 2500 cm−1 and in log scale.

The solid line describes the variation of conductivity at 8 K, the lowest temperature, while the dashed and other dot lines are for high temperatures.

The bump between frequencies 60 cm−1 and 2000 cm−1 is related to the hybridization gap disappearing as temperature increases. At low temperature, the localized f-electrons interact with the conduction electrons by making quasi-particles lead to the opening of a charge gap [1].

According to the Figure 5.16, the ﬁrst minimum values of conductivity at the temperature 10 K (solid line) lie in the frequency range of 60 to 70 cm−1. The conductivity at that point is not zero and that means there is a contribution to conductivity from the few of normal conduction electrons which still exist very low temperature. So, at this temperature, both normal and heavy conduction electrons contribute to the conductivity of this two ﬂuids material.

Table 5.10 includes the parameter values of two Drude models and two

Lorentz models. These four modes describe conductivity as a function of frequency at the temperature 8 K.

At very low temperature (10 K), the conductivity in the frequency range of 4 cm−1 to 30 cm−1 is very high. This frequency region is described by the combination of

59 CeCoIn - Real

80000

8 K

25 K ) 50 K -1

60000

77 K cm

200 K

300 K (

40000 Conductivity Conductivity

20000

10 100 1000

-1

Frequency (cm )

Figure 5.16: This graph from Singley E. J. et.al. [3] shows the real part of the conductivity of CeCoIn5 as a function of wavenumber for diﬀerent temperatures.

the two Drude models related to the normal conduction electrons and heavy electrons.

60 Table 5.10: Fitting parameters for real and imaginary componant of CeCoIn5 at 8 K.

Mode No Model ω0 ωp γ

1 Drude 0 5408 14.8

2 Drude 0 15573 293

3 Lorenz 374 28051 630

4 Lorenz 678 25099 521

61 CeCoIn at 8 K

Original Data

Fitted Data

60000

Mode 1 ) -1

Mode 2

cm Mode 3 -1

Mode 4 (

40000

20000 Conductivity Conductivity

10 100 1000

-1

Frequency (cm )

Figure 5.17: Graph showing experimentaly determined conductivity (black) with the

ﬁtted conductivity curve (red) for CeCoIn5 as a function of frequency at 8 K. The modes represent the individual components (blue) of the overall curve ﬁt.

62 The parameter values of ωp for the two Drude models, total ωp and normalized values of the total ωp at diﬀerent temperatures are shown in Table 5.7. The total ωp values were calculated using the equation 5.1.

Table 5.11: Drude models ﬁtting parameters for CeCoIn5 at various temperatures.

Temperature ωp of mode ωp of mode Total ωp Normalized

1 2 ωp

8 5408 15573 16485 0.404

25 8037 18313 19999 0.490

50 9441 25352 27053 0.663

77 12525 27571 30283 0.742

200 40812 40812 1

300 39902 39902 0.978

Figure 5.18 shows the ωp variation of the two Drude models (mode-1 and mode-2) and the total ωp of them at various temperatures. At temperatures around

100 K there are signiﬁcant changes in both mode-1 and mode-2. However, the total ωp decreases as temperature decreases below 100 K due to the decreasing of the number of normal conduction electrons.

There are only two Lorentz models used to ﬁt the conductivity for the region of high frequency where the hybridization gap is located.

63 for Drude modes as functions of temperature

40000

30000 p

20000

mode-1

mode-2

Total 10000

0 100 200 300

Temperature (K)

Figure 5.18: Graphs showing ω of mode 1 (red), mode 2 (blue) and total ω p − − p

(black) for CeCoIn5 as a function of temperature.

For high temperatures (above 100 K), the third and fourth modes are not required to ﬁt the conductivity curve. This indicates that these kind of electrons no longer contribute to the conductivity.

Figure 5.19 shows the variation of plasma frequency of mode-3 (red), mode-4

(blue) and total ωp (black) from the combination of both mode-3 and mode-4 with respect to temperature.

64 Table 5.12: Lorentz models ﬁtting parameters for CeCoIn5 at various temperatures.

Temperature ωp of mode ωp of mode Total ωp Normalized

3 4 ωp

8 28051 25099 37640 1

25 27989 23603 36612 0.973

50 25598 19259 32033 0.851

77 23030 17469 28905 0.768

According to the data of total ωp for Lorentz modes, when temperature decreases, there is a huge jump in the values of ωp for temperatures below 100 K. That means there is not a contribution to the conductivity from the electrons which are represented by Lorentz modes at high temperature. But, for low temperature, these electrons are hybridized with some conduction electrons and make a contribution to the conductivity.

Taken together, the variation of normalization values of ωp for both the Drude and Lorentz models with temperature clearly shows the transformation of some electrons. At about 80 K, both Drude and Lorentz models show step changes.

For temperatures above 80 K, the Drude electrons dominate and there is no contribution from Lorentz electrons, suggesting only normal conduction electrons are present. But, for temperatures below 70 K, most conduction electrons transform

65 for Lorentz modes as functions of temperature

40000

mode-3

mode-4

Total

30000 p

20000

10000

0 100 200 300

Temperature (K)

Figure 5.19: Graphs showing ω of mode 3 (red), mode 4 (blue) and total ω p − − p

(black) for CeCoIn5 as a function of temperature.

to heavy electrons, increasing the number of Lorentz electrons while decreasing the amount of conduction electrons.

66 Normalized for Drude and Lorentz modes as functions of temperature

1.0

0.8 p

Drude mode

0.6

Lorentz mode

0.4 Normalized Normalized

0.2

0.0

0 100 200 300

Temperature (K)

Figure 5.20: Graphs showing normalized total ωp for both Drude (black) and Lorentz

(red) models of CeCoIn5 as a function of temperature.

67 Total as a function of temperature

50000

40000

30000 p Total Total

20000

10000

0 100 200 300

Temperature (K)

Figure 5.21: Graph showing total ωp for both Drude and Lorenz models of CeCoIn5 as a function of temperature.

According to the Figure 5.21, the total ωp (total ωp is the combination of all the ωp values of both Drude and Lorentz models) is constant with temperature. The total ωp is related to the total number of electrons causing it to be constant with temperature due to charge conservation.

68 Table 5.13: γ ﬁtting parameters of ﬁrst three modes for CeCoIn5 with temperature.

Temperature γ of mode 1 γ of mode 2 γ of mode 3 γ of mode 4

8 14.86 293.03 630.10 520.96

25 29.65 312.45 649.85 539.19

50 42.46 351.28 684.94 543.36

77 91.70 381.22 699.36 562.84

200 399.61

300 416.16

For temperatures above 80 K, only the ﬁrst Drude mode was needed to get a better ﬁt under the hybridization gap. According to Figure 5.22, γ for all three modes decreases as temperature decreases. This indicates that the relaxation time

(the average time between collisions) of the electrons and the conductivity of the material increases as temperature increases.

69 for CeCoIn as a function of temperature

800

mode-1

mode-2

600 mode-3

mode-4

400

200

0 100 200 300

Temperature (K)

Figure 5.22: Graphs showing γ of mode 1 (Drude), mode 2 (Drude), mode 3 − − − (Lorentz) and mode 4 (Lorentz) for CeCoIn as a function of temperature. − 5

70 5.4 Hybridization Order Parameter

In this experiment, the plasma frequency was used as the hybridization order parameter, f(T ). The original equation of hybridization order parameter, equation 3.1, was modified without affecting the universal temperature dependence of order 3/2. The modified equation is given in equation 5.2 where C is a constant dependent only on the material.

T f(T )= f (1 )3/2 + C (5.2) 0 − T ∗

For all three materials, only the Lorentz models were used to ﬁt the experimental conductivity data under the energy gap. The frequency ranges 100-300 cm−1,

−1 −1 300-1000 cm and 100-1500 cm are relevant to the energy gaps of YbFe4Sb12,

CeRu4Sb12 and CeCoIn5 respectively. Table 5.14 shows the parameter values of f0,

T ∗ and C which were used to calculate the hybridization order parameter for all three materials. Figures 5.23, 5.24 and 5.25 show the hybridization order parameter variation (black solid line) for the modiﬁed equation 5.2 with the plasma frequency

(red square).

71 Table 5.14: Parameter values for YbFe4Sb12, CeRu4Sb12 and CeCoIn5.

∗ Material f0 T C

YbFe4Sb12 333 60 5418

CeRu4Sb12 3148.5 80 9400

CeCoIn5 11220 80 29418

72 f(T)

5700

5600

5500 Hybridization order parameter order Hybridization

5400

10 20 30 40 50 60

Temperature (K)

Figure 5.23: Graphs showing hybridization order parameter (black) and ω of mode 3 p − ∗ (red) for YbFe4Sb12 as a function of temperature. The values for f0 = 333, T = 60 and C = 5418.

73 13000

f(T)

12000

11000

10000 Hybridization order parameter order Hybridization

9000

10 20 30 40 50 60 70 80

Temperature (K)

Figure 5.24: Graphs showing hybridization order parameter (black) and ω of mode 6 p − ∗ (red) for CeRu4Sb12 as a function of temperature. The values for f0 = 3148.5, T = 80 and C = 9400.

74 f(T)

40000

p Hybridization order parameter order Hybridization

30000

0 10 20 30 40 50 60 70 80

Temperature (K)

Figure 5.25: Graphs showing hybridization order parameter (black) and the combination of ω of mode 3 and mode 4 (red) for CeRu Sb as a function of temperature. p − − 4 12 ∗ The values for f0 = 11220, T = 80 and C = 29418.

75 CHAPTER VI

CONCLUSION

We studied and analyzed the IR spectroscopy conductivity data of heavy fermion materials YbFe4Sb12, CeRu4Sb12 and CeCoIn5 for different temperatures in the frequency range of 4 - 25000 cm−1. This was accomplished by studying the plasma frequency ωp and scattering rate γ and the effect they had on the distribution of electrons in our system. When temperature is low, the scattering rate is low, leading to an increased relaxation time and, thus, higher conductivity; the opposite is true when temperature is above the characteristic point. The different scattering rates γ correspond to different electron types.

For all three materials, we found that the characteristic temperature T ∗ was between 60 and 80 K. The plasma frequency from the Drude model dominates for temperatures above the characteristic temperature. Thus, normal conduction electrons contribute to the conductivity for temperatures above this point. However, for temperatures below the characteristic temperature, the plasma frequency from the

Lorentz models dominate, indicating that heavy electrons are the primary contributor to conductivity. Heavy electrons are present because the localized f-electrons interact with normal conduction electrons when temperature is below the characteristic point, building quasi-particles with eﬀective masses larger than that of its constituents. It

76 is these heavy electrons which produce the observed high conductivity levels.

The Drude-Lorentz ﬁt also supports the presence of heavy electrons below the characteristic temperature for another reason: the opening of the energy gap. We found that, for all materials, the strength of the Lorentz mode’s ωp at low temperatures causes this opening and the subsequent generation of quasi-particles.

We also demonstrated that our materials could be described by a modiﬁed form of the order parameter below their characteristic temperatures. This parameter describes superconductors with many heavy electrons, suggesting that CeRu4Sb12,

CeCoIn5, and YbFe4Sb12 can be described with the two ﬂuid model. The last of these materials in particular is strongly compatible with this description.

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