# The Congress pays Awdhesh Singh

## pdf document

Thermodynamics of multiband superconductors

Thermodynamics of multiband superconductors

DISSERTATION

for obtaining the academic degree

Doctor rerum naturalium

(Dr. rer. Nat.)

submitted

of the Faculty of Mathematics and Natural Sciences

of

Dipl.-Phys. Andreas Wälte

from Emsdetten

made at the Institute for Metallic Materials of

Leibniz Institute for Solid State and Materials Research Dresden

Dresden 2006

1st reviewer: Prof. Dr. Ludwig Schultz

2nd reviewer: Prof. Dr. Michael Loewenhaupt

3rd reviewer: Univ.-Prof. Dr. Dr. Harald Weber

Submitted on: October 11, 2006

Defense Day: February 16, 2007

For my parents, without whose support and understanding this work would not have been possible

Sentence with L A T E X

Bibliography according to DIN 1505 citation convention

Analyzes and numerical calculations with ROOT [BR97]

short version

In the present work the microscopic properties of the superconducting state are examined

of MgCNi 3, MgB 2 and some rare earth transition metal boron carbides based on measurements

the specific heat investigated. The COOPER pair state that causes superconductivity

Electrons are generated by the interaction of the electrons with lattice vibrations. Therefore

becomes in addition to the specific heat of the superconducting state also that of the normally conducting state

State examined. From the latter, taking theoretical results into account

the electron-phonon interaction strength can be determined for the electronic density of states.

With the help of a self-developed computer program, the frequency spectrum of the

Lattice vibrations estimated and compared with results from neutron scattering experiments.

The energy gap of the superconducting state can be determined from the specific heat of the superconducting

State can be determined, which as well as the upper critical magnetic field H c2 (0) point to

provides the electron-phonon coupling. From analyzing these results and comparing them with results

from transport measurements such as tunnel or point contact spectroscopy can be deduced

the extent to which the BCS model of superconductivity needs to be modified to include the superconducting

To be able to describe the state of the examined connections. There are both well-known

Extensions to take into account increased electron-phonon coupling as well as within the framework

two-band analytical formulations developed for this work are available.

Studies on MgCNi 3, which is close to magnetic instability, show that

occurring magnetic fluctuations halve the superconducting transition temperature T c

have as a consequence. The value of T c = 7 K, which is relatively high under this aspect, is a consequence stronger

Electron-phonon coupling, which is essentially due to nickel vibrations stabilized by carbon

will be carried. Multiband effects are one of the dominant ones in this system

Ribbons on the FERMI edge only for the consistent comparison of different experiments from

Importance. Transport experiments primarily measure the properties of the fast load carriers

(Band with the low partial density of states), while the specific heat over the

Averages the band proportions and therefore the properties of the slow charge carriers (band with the high

partial density of states).

A pronounced anomaly observed for the first time in the specific heat of the classic multi-band superconductor

MgB 2 (here with pure boron-10) at about T c / 4 = 10 K can be determined using a two-band model

in accordance with recently made theoretical predictions for the case

particularly weak coupling between the two bands can be understood. The strength of the

Interband coupling is also of practical interest because of the introduction of scattering centers

H c2 (0) is increased, but at the same time the interband coupling generally increases,

which results in a lowering of the common T c's of both bands.

The analysis of the specific heat of the superconducting phase of the non-magnetic rare earth

Nickel-boron carbides YNi 2 B 2 C and LuNi 2 B 2 C lead to the conclusion that there are visible effects of the multiband electron system

both from the earth in the place of the rare earth, and the transition metal

[investigated on Lu (Ni 1 − x Pt x) 2 B 2 C] are dependent.

The signal of the superconducting visible in the specific heat of the antiferromagnetic HoNi 2 B 2 C

Phase transition is smaller than expected. The discrepancy corresponds to about a third

the electronic density of states and roughly corresponds to the results for the likewise magnetic ones

Systems DyNi 2 B 2 C and ErNi 2 B 2 C. In the context of the multi-band model, this can be considered natural

Consequence of the different strong influence of magnetism on the different

Ribbons are interpreted.

Table of Contents

List of figures

List of tables

iii

v

Introduction 1

1 sample preparation 5

2 Specific heat 9

2.1 Measurement method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Basics of superconductivity 15

3.1 Analytical T c formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Renormalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17th

3.3 Specific heat in the superconducting state. . . . . . . . . . . . . . . . . . . . . . . . . . 18th

3.3.1 The α-model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19th

3.4 The isotropic single-band model of superconductivity. . . . . . . . . . . . . . . . . . . . . . . . . . 20th

3.4.1 The R-test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22nd

3.5 Corrections in the two-band model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22nd

3.6 Influence of magnetic disturbances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25th

4 Superconductivity and ferromagnetic fluctuations in MgCNi 3 29

4.1 Peculiarities of the electronic structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Resistance and magnetic properties. . . . . . . . . . . . . . . . . . . . . . . . . . . 30th

4.3 Specific heat in the normal state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.1 Electron-phonon and -paramagnon interaction. . . . . . . . . . . . . . . . . . . 33

4.4 Thermodynamics of the superconducting phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.5 The upper critical magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.6 Corrections in the two-band model of superconductivity. . . . . . . . . . . . . . . . . . . . . . . . 41

4.6.1 The energy gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.6.2 The jump in the specific heat at T c. . . . . . . . . . . . . . . . . . . . . . 42

4.7 Influence of the carbon concentration on T c. . . . . . . . . . . . . . . . . . . . . . . . . . . 42

i

ii

TABLE OF CONTENTS

5 Two-band superconductivity in MgB 2 49

5.1 Specific heat in the normal state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2 Electron-Phonon Interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.3 Thermodynamics of the superconducting phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4 The upper critical magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Multi-band superconductivity in rare earth nickel boron carbides: LuNi 2 B 2 C and YNi 2 B 2 C 65

6.1 Electronic specific heat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 Characterization of the superconducting phase. . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 transition metal substitutions: Lu (Ni 1 − x Pt x) 2 B 2 C and Y (Ni 1 − x Pt x) 2 B 2 C 75

7.1 Determination of the electron contribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.2 Determination of the phonon contribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.3 Thermodynamics of the superconducting phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8 Superconductivity in magnetic systems: HoNi 2 B 2 C 91

8.1 Magnetic properties, resistance and specific heat. . . . . . . . . . . . . . . . 91

8.2 Specific heat in the normal state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.3 Electron-Phonon Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

8.4 Superconducting state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

9 Summary and Outlook 99

A programs 105

A.1 HEAT - a specific Heat Excitation Analysis Tool. . . . . . . . . . . . . . . . . . . . . . . . . 105

A.2 α model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Bibliography 121

Own publications 133

Thanks 135

List of figures

1 DE GENNES scaling of the order temperatures of RNi 2 B 2 C. . . . . . . . . . . . . . . . 2

1.1 X-ray diffractograms and crystal structures of Mg 1.2 C 1.6 Ni 3 and MgB 2. . . . . . . . . . 6th

1.2 X-ray diffraction pattern and crystal structure of LuNi 2 B 2 C. . . . . . . . . . . . . . . . . . . . 7th

2.1 Temperature dependence of the specific heat in the EINSTEIN and DEBYE models. . . . . . 12th

2.2 Temperature spectrum of the EINSTEIN or DEBYE model. . . . . . . . . . . . . . . . . . . . 13th

2.3 Temperature resolution of the computer algorithm. . . . . . . . . . . . . . . . . . . . . . . . . 14th

3.1 Temperature dependence of the energy gap in the BCS model. . . . . . . . . . . . . . . . . . . . 20th

3.2 Accuracy of the single-band model approximation formula for the penetration depth λ L. . . . . . . . . . . . 21

4.1 FERMI surfaces and electronic density of states N (E) of MgCNi 3. . . . . . . . . . . . . . 30th

4.2 Specific resistance of MgCNi 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Magnetization and susceptibility of MgCNi 3. . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4 Specific heat of MgCNi 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.5 Dispersion and density of states F ph (ω) of the phonons of MgCNi 3. . . . . . . . . . . . . . . 34

4.6 Electron-phonon spectral density α 2 (ω) F (ω) of MgCNi 3. . . . . . . . . . . . . . . . . . . . 35

4.7 Electronic specific heat γ N (T) of MgCNi 3 in the normal state. . . . . . . . . . . . 35

4.8 Specific heat, entropy and µ 0 H (T) of the superconducting phase of MgCNi 3. . . . . . . 36

4.9 Normalized electronic specific heat and D (t) of MgCNi 3. . . . . . . . . . . . . . . . 38

4.10 Upper critical magnetic field µ 0 H c2 (T) of MgCNi 3. . . . . . . . . . . . . . . . . . . . . . . 39

4.11 Jump height of the specific heat of MgCNi 3 at T c in the two-band model. . . . . . . . . . 41

4.12 Dependency between carbon content, lattice constant and T c of MgC x Ni 3. . . . . . . . . 43

4.13 Specific heat c p of MgC xn Ni 3 up to 300 K and relative error of the model adaptation. . . 44

4.14 c p for T < 10="" k="" von="" mgc="" xn="" ni="" 3="" und="" entropie="" ∆s="" des="" supraleitenden="" zustands="" für="" x="" n="1,00." .="" .="">

4.15 F (ω) and electron-phonon coupling of MgC xn Ni 3. . . . . . . . . . . . . . . . . . . . . . . 46

5.1 Specific heat and deviations of the lattice model adaptation for Mg 10 B 2. . . . . . . . 51

5.2 Comparison of the adapted lattice model with data for T 40 K of Mg 10 B 2. . . . . . . . . 52

5.3 Dispersion and density of states F ph (ω) of the phonons for Mg 10 B 2. . . . . . . . . . . . . . . . 54

iii

iv

LIST OF FIGURES

5.4 Comparison between F (ω) and point contact spectroscopy results for MgB 2. . . . . . . . . . 55

5.5 Electronic specific heat of Mg 10 B 2 in the superconducting state. . . . . . . . . . . . 56

5.6 Entropy difference and thermodynamically critical field of Mg 10 B 2. . . . . . . . . . . . . . . 57

5.7 Normalized electronic specific heat of Mg 10 B 2 for T < t="" c="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="">

5.8 D (t) of Mg 10 B 2 in comparison with the BCS and analytical two-band model. . . . . . . . . . . 59

5.9 Comparison between experiment and ELIASHBERG two-band model for Mg 10 B 2. . . . . . . . . 60

5.10 Field dependence of the specific heat for T < 15="" k="" von="" mg="" 10="" b="" 2="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="">

5.11 Upper critical magnetic field µ 0 H c2 (T) of Mg 10 B 2. . . . . . . . . . . . . . . . . . . . . . . 62

5.12 Energy gap dependence of the charge carrier density within the two bands of MgB 2. 62

6.1 Electronic density of states of HoNi 2 B 2 C. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2 Specific heat and model adaptation of (Lu, Y) Ni 2 B 2 C. . . . . . . . . . . . . . . . . . 67

6.3 Specific heat of the superconducting state of (Lu, Y) Ni 2 B 2 C. . . . . . . . . . . . . . . 68

6.4 Normalized entropy and thermodynamically critical field of (Lu, Y) Ni 2 B 2 C. . . . . . . . . . 69

6.5 D (t) of (Lu, Y) Ni 2 B 2 C in comparison with the BCS and analytical two-band model. . . . . . . . 70

6.6 Electronic specific heat from LuNi 2 B 2 C for T < t="" c="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="">

6.7 Electronic specific heat of YNi 2 B 2 C for T < t="" c="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="">

7.1 Upper critical magnetic field µ 0 H c2 (T) of (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . . . . . . . 75

7.2 Transition from the “clean limit” to the “quasi-dirty limit” from (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . 76

7.3 Specific heat and lattice model adaptation of (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . . 77

7.4 Determination of the SOMMERFELD parameter for (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . . . . 79

7.5 Accuracy of the lattice model fit for (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . . . . . . . 79

7.6 Phononic density of states F (ω) of (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . . . . . . . . . 80

7.7 Interpolated phononic density of states F (ω) of (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C for x = 0 ... 0.5. . . 81

7.8 Specific heat of the superconducting state of (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . 81

7.9 Normalized entropy difference for (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . . . . . . . . . . . . . 82

7.10 Thermodynamically critical field µ 0 H c (T) of (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . . . . . 83

7.11 D (t) of (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C in comparison with the analytical two-band model. . . . . . . 84

7.12 Normalized electronic specific heat of (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . . . . 85

7.13 "R-Test" for (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.1 Susceptibility χ (T) of HoNi 2 B 2 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8.2 Specific heat and resistance of HoNi 2 B 2 C. . . . . . . . . . . . . . . . . . . . . . . 92

8.3 c p (T) from HoNi 2 B 2 C to T = 300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8.4 Dispersion of the phonon branches of HoNi 2 B 2 C. . . . . . . . . . . . . . . . . . . . . . . . . 95

8.5 F ph (ω) and spectral density αtr 2F (ω) of HoNi 2B 2 C. . . . . . . . . . . . . . . . . . . . . . . . 96

8.6 Experimental and theoretical jump in the specific heat of HoNi 2 B 2 C at T c. . . 97

List of tables

4.1 Characteristic thermodynamic conditions for MgCNi 3. . . . . . . . . . . . . . . . . . 37

4.2 Properties of superconductivity of MgC xn Ni 3 depending on the carbon content. . . . . . . . 43

5.1 Parameters of the lattice model adaptation for Mg 10 B 2. . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 The sizes of the bands of Mg 10 B 2 that characterize superconductivity. . . . . . . . . . . . . . 58

7.1 Characteristic quantities of superconductivity of (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C. . . . . . . . . . . . . 87

7.2 Results of the "R-Test" for (Lu, Y) (Ni 1 − x Pt x) 2 C. . . . . . . . . . . . . . . . . . . . . . . . . 88

8.1 References to the crystal electrical field of HoNi 2 B 2 C. . . . . . . . . . . . . . . . . . 93

8.2 Results of the grid model adaptation for HoNi 2 B 2 C. . . . . . . . . . . . . . . . . . . . . . 94

v

vi

LIST OF TABLES

introduction

The properties of superconducting materials have been intensifying since the discovery of superconductivity [Onn11]

researched in order to steadily improve the understanding of the superconducting state. The focus is on

the goal of the special magnetic and electronic properties of this exotic state

use. The search for superconductors with a high transition temperature T c, which are naturally particularly attractive

for applications poses a particular challenge. The discoveries are prominent examples

the cuprate high-temperature superconductor with transition temperatures up to ∼ 130 K [BM86, SCGO93] and

of superconductivity in MgB 2 with T c ≈ 40 K [NNM + 01]. To do a systematic search for such materials

To be able to operate efficiently, a constant refinement of theoretical models is indispensable. Conventional

In the context of a theory developed by Bardeen, Cooper and Schrieffer (BCS 1 model), superconductors can

[BCS57a, BCS57b] or their form extended by Eliashberg [Eli60]. The exchange bosons

the interaction leading to superconductivity are usually vibrational excitations of the

Atomic lattice (phonons). To check the validity of this theory for a superconducting connection,

three microscopic characteristics of the material must be known:

1. the spectrum of the lattice vibrations (phonon spectrum),

2. the electronic density of states at the FERMI edge and

3. the interaction strength of the electrons with the lattice vibrations.

The phonon spectrum is determined with the help of neutron scattering experiments, the electronic density of states

can be determined from band structure calculations. The interaction between electrons and

Phonons are determined using tunnel or point contact spectroscopy. In reality, this proves to be true

Despite the already high experimental challenge, the concept is usually too simple because it is in solids

The multitude of excitation states that can influence the superconducting state are not taken into account

remains. Magnetic excitations can weaken or completely weaken the superconducting phase

suppress [AG61]. Special features of the band structure can also play a role, for example

The properties of the superconducting state of MgB 2 are quantitative only taking into account two

to understand separate load carrier groups. Other influences such as the interaction between the

superconducting electrons and crystal field excitations 2 a crystal electric field are largely still

misunderstood.

For the present work three related systems with different complexity regarding the

Electronic structure and non-phononic excitations for a detailed analysis of both the multiband and

also selected the magnetic influences on superconductivity. These are the multi-band superconductor MgB 2,

as well as the newly discovered superconductor MgCNi 3, which is close to magnetic instability, and finally

the class of both superconductivity and magnetism exhibiting rare earth transition metal

Boron carbides. Specific heat is used as the preferred measured variable. Calorimetric measurements are

volume-specific, so that single crystals are not required. The specific heat forms the at the same time

1 Bardeen Cooper Schrieffer

2 Excitations of f -electrons (the rare earth atoms) which occur in rare earth compounds and which are not directly connected to the metallic

Conductivity or the COOPER pair formation in the superconducting state are involved.

1

2 INTRODUCTION

Figure 1: Order temperature of

RNi 2 B 2 C applied against the DE GEN-

NES factor of the respective rare earth ion R 3+.

The circles (squares) indicate the superconducting

(magnetic) transition temperature T c (T N)

Order temperature [K]

0

0 2 4 6 8 10 12 14

dG = (g an. The lines are for orientation. J

- 1) J (J + 1)

20

15

10

5

Lu Tm Er Ho Dy Tb Gd

Y

sl

T c

afm + sl

T N

2

afm

Properties of the electron system and the atomic lattice and also allows statements about the

the interactions taking place. To analyze the superconducting state of a material using the

specific heat must be the proportion of specific heat caused by the superconductivity of the

be separated by the normal state certain contribution. Such a separation can be achieved by a reference measurement

in magnetic fields in which the superconducting state is destroyed (µ 0 H> µ 0 H c2)

become. However, this procedure does not always lead to the desired result, because it is external

Magnetic field other excitations occurring in the solid, such as the fine and hyperfine structure,

are amplified so that the measurement in the magnetic field no longer represents the normal state. Around

To minimize these effects, a previously rarely used approach was chosen in the course of the present work,

in which the specific heat is modeled in an extended temperature range. 3 measurements

in magnetic fields are only used for orientation. This was based on simple

Based on the assumptions of the model, a computer algorithm has been developed that reliably and quickly eliminates the effects of lattice vibrations

caused proportion of the specific heat is determined. The phononic and also electronic ones

Properties of superconducting materials can thus be at least approximately determined with relatively little effort

can be determined from the specific heat.

The new PEROVSKIT superconductor MgCNi 3 is close to a ferromagnetic one due to its high nickel content

Phase transition, 4 which, however, is not achieved. In fact, MgCNi 3 shows superconductivity with

a transition temperature of T c ≈ 7 K. From band structure calculations it is known that two of each other

almost decoupled strips cut the FERMI edge, one of them with a contribution of around 85%

to the total density of states. In particular, the influence of ferromagnetic instability on superconductivity should be considered here

to be examined. Both phenomena are dominated by the Ni-3d states [RWJ + 02].

The influence of the structure of the electron system on superconductivity is particularly evident in the case of MgB 2.

The situation at the FERMI edge can be described by two effective, weakly coupling bands.

These differ particularly in their coupling to the lattice vibrations, which is when it disappears

Interband coupling leads to marked differences in their respective intrinsic phase transition temperatures

T c would result. The very different electron-phonon interaction in the respective sub-systems

and the relatively weak interband scatter is reflected in significantly different energy gaps

both bands reflected (with a factor of 3 - 5 at T = 0 K), with the bands having a common effective

Transition temperature must be assigned [SMW59, SW67]. From Choi et al. was based on

3 Comparable approaches in [MHDH95, WFD + 01, WPJ01]

4 The electronic density of states of MgCNi 3 shows a VAN HOVE singularity.

INTRODUCTION 3

Calculations within the framework of the two-band ELIASHBERG model show a distribution of the energy gap within the

both bands predicted [CRS + 02a, CRS + 02b], the experimental detectability of which is controversial

becomes [MAJ + 04, CRS + 04]. Previous investigations into the energy gaps of MgB 2 were limited to

essential to spectroscopic investigations [SZGH03]. Determining the energy gap from the specific

Heat is more complicated and depends on the model, which is why it is mainly used for estimations

becomes. Transport measurements such as tunnel contact spectroscopy are based on the FERMI speeds of the

Charge carriers are influenced and are therefore generally influenced by the properties of the fastest quasiparticles

certainly. Therefore a detailed analysis of the specific heat is of particular interest because

this is independent of the FERMI speeds. In addition there were polycrystalline Mg 10 B 2 - and Mg 11 B 2 -

Samples available.

In contrast to MgCNi 3 and MgB 2, the rare earth transition metal boron carbides are RT 2 B 2 C (R = rare

Earth, T = transition metal) has been the subject of research for over 10 years [CTZ + 94, NMH + 94]. The

Group of rare earth metals is formed by the elements yttrium, scandium 5 and the lanthanoids 6.

All of them and especially the 4f metals (cerium

to ytterbium) have very similar chemical properties. Systematic dependencies can be found, for example

in the ionic radii (lanthanide contraction) and the magnetic moment. Depending on

rare earth and transition metal, the rare earth transition metal boron carbides show a complex

Behavior in which superconducting and magnetic phases can even be observed at the same time

[CTZ + 94, NMH + 94]. Figure 1 shows the respective phase transition temperatures for RNi 2 B 2 C as a function

from the so-called DE GENNES factor, which is part of the

as well as later by Kasuya and Yosida the RKKY 7 theory expanded the strength of the magnetism of the respective

Rare earth ions indicates [RK54, Kas56, Yos57]. The RKKY interaction is the more localized interaction

magnetic moments via conduction electrons. It is relevant when there is no overlap of neighboring atomic

4f wave function occurs or it is only very weak. In the case of the rare earth nickel boron carbides

this indirect interaction is mediated, among other things, via the 5d states of the rare earths. Around

To facilitate identification, the corresponding rare earth is given at the top of Figure 1.

Due to the small distance between the magnetic order temperature T N and the superconducting temperature

Order temperature T c are the rare earth nickel boron carbides RNi 2 B 2 C (with magnetic rare earth R)

better suited for investigating the influence of magnetism on superconductivity than, for example

the CHEVREL phases RMo 6 S 8 and RMo 6 Se 8 or the rhodium borides RRh 4 B 4 [FM82, MF83]. The electronic

In the case of boron carbides, the density of states at the FERMI edge is essentially determined by two effective bands

certainly. Band structure calculations have shown that the coupling between the bands is comparative

is strong. Correspondingly, measured variables averaging over the sample volume such as the specific heat can be approximated

can already be understood in an effective binding model [MHDH95, EHMH00]. The top

critical magnetic field of LuNi 2 B 2 C and YNi 2 B 2 C can be due to the important different FERMI-

On the other hand, speeds can only be meaningfully described with the help of the multiband character [SDF + 98].

Despite extensive research in recent years, there are still numerous unanswered questions about the

Superconductivity as well as to the magnetic phases [MN01, MFDN02]. The symmetry of the order parameter

is still controversially discussed [MMEHH01, MTW02, YT03, MWH04]. Only recently was through

the analysis of the temperature dependence of the energy gap in LuNi 2 B 2 C by means of point contact spectroscopy

Two-band character confirmed and a pronounced anisotropy of the order parameter was found [BBT + 05].

The scaling of the magnetic order temperature with the DE GENNES factor corresponds - taking into account

the magnetic anisotropy caused by the crystal electric field - expectations

the RKKY theory [SC05]. It is less well understood why the transition temperature T c of the superconducting

Phase roughly scaled with the DE GENNES factor (Figure 1). This behavior became qualitative

treated so far within the framework of the ABRIKOSOV-GOR’KOV theory [AG61, SBMW64]. The applicability of the

ABRIKOSOV-GOR’KOV theory, which is explicit only for the case of small magnetic impurity concentrations

was derived, but is limited. For example, Cho et al. the collapse of the DE GEN

5 Due to its position in the periodic table and the three valence of Sc 3+

6 The 15 elements of the periodic table following barium, also referred to as lanthanides

7 ruderman smock Kasuya Yosida

4 INTRODUCTION

NES scaling demonstrated in a (Ho, Dy) Ni 2 B 2 C mixed series (small black squares in Figure 1)

[CHJC96]. On the basis of measurements of the specific heat of a HoNi 2 B 2 C single crystal it should be checked,

to what extent the ABRIKOSOV-GOR’KOV theory is sufficient for the description and whether through an extension

this model on correlated systems [MY79a, MY79b, YM79, Mac79, MNM80, ZF81, FZ82] a better one

Description can be achieved.

But changes in the lattice vibrations can also influence T c (in the BCS theory, T c is direct

proportional to the frequency of the lattice vibrations 8). These are determined both by the distance between the atoms, as

also influenced by the mass of the vibrating atoms. If one considers the non-magnetic rare earth

Nickel-boron carbides, there is an almost constant increase in T c when the mass on the square is doubled

the rare earth with ScNi 2 B 2 C ∆T c≈1.5 K

−−−−−−− → YNi 2 B 2 C ∆T c≈1.1 K

−−−−−−− → LuNi 2 B 2 C. The ionic radii of the rare earths,

r (Sc 3+) = 0.0732 nm < r="" (="" lu="" 3+)="0,0848" nm="">< r="" (="" y="" 3+)="0,0905" nm="" skalieren="" andererseits="" nicht="" mit="" t="" c="">

Accordingly, the mass of the rare earth plays the dominant role for T c, a fact that has so far almost been

is not taken into account.

With the help of the computer program "HEAT" developed within the scope of this work, the specific

Heat from LuNi 2 B 2 C and YNi 2 B 2 C can be studied as the superconductivity by these two different

Effects is influenced. To improve the role of the transition metal site for superconductivity

understand, a polycrystalline Lu (Ni 1 − x Pt x) 2 B 2 C- and a single crystal Y (Ni 1 − x Pt x) 2 B 2 C-

Mixed series examined. 9

8 For elements this is the EINSTEIN frequency, for simple connections mostly the DEBYE frequency.

9 The designation (Lu, Y) (Ni 1 − x Pt x) 2 B 2 C used in the following means R (Ni 1 − x Pt x) 2 B 2 C with rare earth R = Lu or Y.

Chapter 1

Sample preparation

For the measurements presented in this work, polycrystalline materials [MgB 2, MgC x Ni 3 and

Lu (Ni 1 − x Pt x) 2 B 2 C] as well as high quality single crystals [Y (Ni 1 − x Pt x) 2 B 2 C and HoNi 2 B 2 C] are available.

The latter were made within the framework of the Collaborative Research Center 463 in the Institute for Solid State and Materials Research

(IFW).

The preparation of the polycrystalline samples is similar for the three systems. First of all, the

Raw materials mixed in pure form in a stoichiometric ratio. The powder is means

a pressing tool with a punch diameter of ∅ = 8 mm at a pressure of p = 5 × 10 8 Pa

cold compacted. The rare earth transition metal boron carbide compounds are initially cold-pressed

several times in an arc furnace under an argon atmosphere on a copper plate cooled with water

melted and turned in between to achieve optimal homogeneity. Due to

Due to the high vapor pressure of magnesium, this step does not make sense for MgB 2 and MgC x Ni 3. Samples

are then packed in tantalum foil, sealed in quartz glass under a protective atmosphere (180 mbar argon) and

homogenized in a tube furnace in several annealing processes depending on the target system. The tantalum foil

should be possible reactions in particular between carbon, magnesium and boron with the quartz glass

minimize. After the annealing process, the samples are quickly solidified in cold water.

MgC x Ni 3

Due to the relatively large differences between the boiling points of the elements of MgC x Ni 3 were

20 mol% of magnesium are added to the stoichiometric composition [HHR + 01b, AHL + 02]. It was found,

that the transition temperature of the superconducting state slightly varies with the carbon content used

varies. This can be explained by the tendency for graphite to form during manufacture. So lie in the stoichiometric

Composition basically at least three phases: the superconducting MgCNi 3 phase, the

non-superconducting MgC 1 − x Ni 3 phase and the unreacted carbon in the form of graphite. The MgC 1 − x Ni 3 -

Phase content can be reduced by adding carbon beyond the stoichiometric ratio

[RCJ + 02]. An optimal heat treatment consists of two stages. First, the sample is for a period of time

annealed for 0.5 h at 600 ◦ C, with a subsequent one-hour annealing at 950 ◦ C. It has

shown that two to three repetitions of the annealing processes are sufficient to produce homogeneous samples

to obtain high T c values of the superconducting phase. Higher temperatures and / or additional annealing processes

lead to an increased porosity of the samples due to the evaporating magnesium

an additional magnesium admixture must be collected. Lower annealing temperatures lead

to an increased formation of carbon-deficient MgC 1 − x Ni 3, its superconducting transition temperature

is significantly reduced [RCJ + 02]. Attempts to additionally place the samples produced in this way in a hot press system

compacting and thereby minimizing grain boundary influences were unsuccessful and resulted in the

Usually too strong MgNi 2 formation as the elimination phase. Comparisons between samples with very different

5

6 SAMPLE PREPARATION

(a)

(b)

Figure 1.1: (a): adapted X-ray diffraction pattern of a Mg 1.2 C 1.6 Ni 3 sample. (b): adapted X-ray diffraction pattern

a Mg 1.1 10 B 2 sample. Symbols: measuring points. Solid line: RIETVELD adjustment. Lines: reflective layers

of the phases found. Curve in the lower area: Difference between measurement and adjustment. The structural images

show the PEROVSKIT structure of MgCNi 3 and the hexagonal layered structure of MgB 2.

Lichen residual resistances, 1 show that the grain boundaries have no significant influence on the superconducting

Have properties such that the relevant measured physical quantities are intrinsic to MgCNi 3

can be viewed.Figure 1.1 (a) shows the X-ray diffraction pattern of a Mg 1.2 C 1.6 Ni 3 sample. A

RIETVELD analysis [RC90] shows a foreign phase proportion of ≈ 10% by volume of carbon in graphite modification

and about one to two volume percent magnesium oxide. Due to the high carbon content, the MgC 1 − x Ni 3 -

Phase fraction below the detection limit. The lattice constant of the MgCNi 3 phase became in agreement

with literature values [AHL + 02] determined to be a = 0.38107 (1) nm.

MgB 2

It has been shown that the quality of the MgB 2 samples and in particular the value of T c can be improved by adding

10 mol% boron can be optimized. The optimal heat treatment of the MgB 2 samples consists of one

two-hour annealing at 900 ◦ C. This process is repeated twice in order to be as homogeneous as possible

Get sample. In order to counteract the high steam pressure of the magnesium here, too, a piece is needed

Magnesium packed separately in tantalum foil and added to the quartz glass. The relatively porous

MgB 2 samples are compressed in a hot press machine. This subsequent treatment is necessary

in order to obtain a sample that is as dense as possible with a low residual resistance ρ (T c) and the influence of the grain boundaries

to minimize [FLM + 01]. In contrast to MgCNi 3, the greater hardness of MgB 2 results here

no excretion phases detectable by means of X-ray measurements. Figure 1.1 (b) shows the X-ray diffraction pattern

a Mg 1.1 10 B 2 sample. The RIETVELD analysis [RC90] showed a proportion of foreign phases of

≈ 5 vol .-% magnesium and ≈ 4 vol .-% magnesium oxide. The lattice constants of the Mg 10 B 2 phase became

a = 0.30861 (1) nm and c = 0.35227 (1) nm.

Rare earth transition metal boron carbides

The optimal method for preparing polycrystalline rare earth transition metal boron carbide samples depends

on the rare earth used. For rare earths with a relatively high melting point (similar to that of

1 ≈ 30 times between the polycrystalline samples from He et al. and Kumary et al. [HHR + 01b, KJM + 02]

SAMPLE PREPARATION 7

Figure 1.2: Adapted X-ray diffraction pattern

one in the arc furnace

melted LuNi 2 B 2 C sample. Symbols:

Measuring points. Solid

Line: RIETVELD adjustment. Strokes:

Reflective layers of the LuNi 2 B 2 C phases. The

Curve in the lower area corresponds to

the difference between measurement and

Adaptation. The framed structure picture

shows the layer structure of the

Rare earth transition metal boron carbides

using the example of LuNi 2 B 2 C.

Nickel), i.e. the heavier lanthanides from gadolinium (with the exception of ytterbium) can use the compounds

are generated by solidification in an electric arc furnace. Due to the different vapor pressures of the

It is necessary to add 10 mol% boron and 2 mol% carbon to individual reagents. The samples are

Melted several times and turned over in between to achieve optimal homogeneity. The number

the melting process has hardly any influence on the further evaporation of the individual elements if they

are alloyed once or the 1: 2: 2: 1 phase (R: T: B: C with R = rare earth and T = transition metal)

has formed. In order to further improve the homogeneity of the samples, they are subsequently placed in a tube furnace

Heat-treated at 1200 ◦ C for a period of 14 days. The X-ray diagram of a thus obtained

polycrystalline sample is shown in Figure 1.2. A detailed description of the synthesis of the rare earth

Transition metal-boron carbide compounds have been used, for example, by Freudenberger et al. specified [Fre00].

8 SAMPLE PREPARATION

Chapter 2

Specific heat

The heat capacity of a solid is determined by its temperature change generated by the supply of heat.

In the case of quasi-static heating, the following applies (equilibrium heat capacity):

C = δ Q

δ T.

The specific heat capacity c is based on the unit of mass (1 kg) or amount of substance 1 (1 mol). At

When measuring the specific heat, the external conditions must be taken into account. Experimental

the specific heat at constant pressure, c p is determined. The theoretical formalism takes place

which uses the specific heat at constant volume, c V. c p differs from c V due to

the additional heat change caused by thermal expansion:

c p - c V = 9α 2 BAT,

with the linear thermal expansion coefficient α, volume V and the compression modulus B [Kit02]. The

The difference between c p and c V is only small and can be neglected, especially in the range of low temperatures

become.

The heat supplied is distributed over the degrees of freedom available in the solid. Hence the

specific heat mainly dominated by lattice vibrations. Since the lattice vibrations for the

superconducting state are of particular importance, a detailed treatment follows in Section 2.2.

As a first approximation, the electron system makes a linear contribution to the specific heat:

c V = γ N T + c lattice, (2.1)

where c lattice is the lattice contribution and γ N is the SOMMERFELD parameter which determines the electron contribution, the

also plays an essential role for superconductivity (see Section 3.2). Tried to determine γ N

it is customary to estimate the residual contribution c V / T for T → 0, since here the thermal excitation of the

Lattice (c lattice) can be neglected. To do this, one makes DEBYE’s low-temperature approximation for

use acoustic oscillation branches [Kit02]:

c grid = 12π4 rk B N A

5

(T

Θ D

) 3

. (2.2)

The calculation of a DEBYE temperature using this approximation usually does not correspond to reality, since

Differentiate transversal and longitudinal acoustic oscillation branches in real crystals often up to 30% and more.

Also, the T 3 dependency is usually only well fulfilled for T Θ D / 50. To determine the electronic

However, this approximation is generally sufficient for T 4 K.

1 Is then also referred to as “molar heat capacity” or “molar heat”

9

10 SPECIFIC HEAT

Further contributions to specific heat result from the splitting and shifting of degenerate energy levels

in the crystal electric field of a solid. The influence of the charge distribution of the

Ions and electrons generated electrostatic field on the non-spherically symmetrical electronic

Orbital levels lead to a partial or complete cancellation, depending on the symmetry of the environment

the 2J + 1-fold degeneration. Compounds whose atoms have a magnetic moment can

also make contributions to the specific heat. In this case the energy levels become degenerate

split by paramagnetic centers in the solid or by an external magnetic field.

The work of Fulde and Loewenhaupt provides a comprehensive overview of crystal field effects in 4f systems

[FL86]. The spin-orbit interaction also contributes to the specific one via the fine structure splitting

Warmth at. The multiplicity of the respective split is calculated according to the PAULI principle and

the HUND’s rules. Another contribution results from the interaction between the spin of the

Atomic nucleus and the total angular momentum of the electron shell. The relative distance between the energy levels depends

here only from the core and shell spin, the absolute value of the value of the magnetic moment µ. This

Due to the mass difference between electron and nucleon, the effect is much smaller than that caused by the

Caused spin-orbit interaction. 2 The resulting split is therefore called a hyperfine structure split

designated. Both interactions make a contribution to the specific heat and are as

SCHOTTKY anomalies (“electronic” or “core” SCHOTTKY anomalies) known. These influences

including the crystal field splitting can be described using a generalized SCHOTTKY model

become [BCW80, GLD + 91]:

〈 〉

E.

2

c s (T) = i - 〈Ei〉 2

k B T 2. (2.3)

The terms in brackets denote thermal mean values of the shape

( )

∑ i f i x i exp - E i

k B T

〈X i〉 = (),

∑ i f i exp - E i

k B T

with the degrees of degeneracy f i and the eigenvalues E i of the energy eigenstates.

There are many other forms of stimulation, which are essentially specific, especially at low temperatures

Can contribute warmth. The most important of these are the paramagnons and magnons, which are located above or

occur below magnetic phase transitions. They are for the analyzes presented in this thesis

and results of MgCNi 3 and HoNi 2 B 2 C are significant. Phase transitions themselves express themselves due to the

changed energy states through characteristic jumps in the specific heat. They are either

Transitions of the first type that generate latent heat (for example magnetic transitions) or entropy

sustaining transitions of the second kind (for example with conventional superconductivity).

2.1 Measurement method

The specific heat was measured by means of a PPMS ("Physical Property Measurement System") from Quantum

Design determined. In the mode for determining the specific heat, the system enables measurements

in a temperature range of 2 K ≤ T ≤ 300 K and in external magnetic fields up to µ 0 H = 9 T. At

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