The evolution of the ferromagnetic ordering of Co doped URhGe

MSc Physics

Advanced Matter and Energy Physics

Master Thesis

The evolution of the ferromagnetic ordering

of Co doped URhGe

by Roos Jehee 6064914 August 2015 54 ECTS 2 September 2014 - 20 August 2015 Supervisor: Dr. A. de Visser Second corrector: Dr. R. Spreeuw

URhGe by means of specific heat measurements. The Curie temperature in-creases from 9.5 K for URhGe to 20 K for a Co concentration of 60% and then decreases to 3 K for UCoGe. A theoretical model (by Silva Neto et al. [1])

proposes that the change in TC is governed by the changes in hybridization

be-tween the d-orbitals of the Co and Rh atoms and the f- states of the U-atoms. The evolution of the d-f-hybridization can be probed with specific heat measure-ments. The measured electron contribution of the specific heat and the magnetic entropy both show a decrease with increasing Co concentration. We conclude that the measured trend in the electronic contribution of the specific heat is in contradiction with the expected behaviour determined by the changing d-f-hybridization. This thesis also includes an extensive description of the usage of the PPMS heat capacity option.

1 Introduction 3

2 Theoretical aspects 7

2.1 Specific heat . . . 7

2.1.1 Thermodynamics . . . 7

2.1.2 Lattice specific heat . . . 9

2.1.3 Electronic specific heat . . . 11

2.2 The d-f hybridization in URh1−xCoxGe . . . 14

3 Experiment 19 3.1 Specific heat measurement methods . . . 19

3.2 Thermal models . . . 20

3.2.1 Simple model . . . 20

3.2.2 The 2τ model . . . . 22

3.3 Design of the PPMS Dynacool . . . 23

3.3.1 Calibration of the puck . . . 25

3.4 Sample preparation and mounting . . . 26

3.5 Heat capacity measurement procedure . . . 27

3.6 The addenda measurement . . . 29

3.7 Difference between Cp and Cv . . . 32

4 Results & Discussion 34 4.1 Specific heat of URh_1−xCo_xGe . . . 34

4.1.1 Curie temperature . . . 35

4.2 Fitting to the Debye & Einstein function over the entire temper-ature range . . . 35

4.3 Fitting to the low temperature Debye approximation . . . 38

4.4 The magnetic specific heat & entropy . . . 42

4.5 d-f hybridization . . . 45

5 Summary 49

A Derivation of heating and cooling response curves 51

1

Introduction

A new class of uranium compounds has attracted a lot of attention, due the coex-istence of ferromagnetism (FM) and superconductivity (SC) at ambient pressure. The materials found so far that have this property are URhGe [2] and UCoGe [3].

This property was already encountered in other materials, for example UGe2,

but here superconductivity is induced by applying an external pressure [4]. The coexistence of ferromagnetism and superconductivity is remarkable. Su-perconductivity and ferromagnetism are carried by the same electrons. In con-ventional superconductors, the superconducting state is a s-wave singlet state, which means that the spin components of the electrons in a Cooper pair are aligned anti-parallel. The pairing is mediated by lattice vibrations. It has been predicted that the pairing state in itinerant ferromagnetic superconductors is a p-wave triplet state, where the spins are aligned parallel. Moreover, the pairing is mediated by ferromagnetic spin fluctuations [5, 6]. Also, due to the alignment of the magnetic moments in ferromagnets, it was presumed that the internal mag-netic field would destroy superconductivity and therefore the two could never coexist.

Both URhGe and UCoGe belong to the class of correlated metals. These type of compounds show some interesting properties, due to the f-states of the Ce-, Yb-, or U-atoms. The main characteristic is the enhanced electron mass and therefore, an enhanced low temperature specific heat. Under special circum-stances the correlated U-compounds show a non-Fermi liquid (nFL) behaviour. This is often an indication that the material is near to a magnetic instability, i.e. a quantum phase transition (QPT). In the compounds under consideration, a QPT is a ferro- to paramagnetic transition at T = 0 K. At absolute zero, there are no thermal fluctuations which can control the phase transition. The

● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆

0.0

0.2

0.4

0.6

0.8

1.0

0

5

10

15

20

x

T

C

(K

)

URh1-xCoxGe URhGe1-xSix URh1-xRuxGe

Figure 1.1: The phase diagram of URh1−xRuxGe (squares), URhGe1−xSix(circles) and URh1−xCoxGe (diamonds). Ru doped URhGe shows a small increase in TC and the QCP is at x = 0.38. Doping the URhGe with Si on the Ge site shows no drastic changes in TC. The TC of URh1−xCoxGe shows a maximum at x = 0.6.

quantum phase transitions is mediated by quantum fluctuations. It is already possible to probe for signs of a QPT in a small region above T = 0. In order to tune to the magnetic stability one may apply a magnetic field, chemical or mechanical pressure.

The work presented here is an extension of the PhD thesis of N.T. Huy con-ducted at the University of Amsterdam: Ferromagnetism, Superconductivity and Quantum Criticality in Uranium Intermetallics [7]. This work was a search for ferromagnetic instabilities in ternary uranium intermetallics. The starting point of the research of Huy is the ferromagnetic superconductor URhGe. To examine the magnetic spin fluctuations, which possibly mediate p-wave super-conductivity, URhGe is tuned to a quantum critical point. This is done by alloying URhGe. An element is substituted on the Rh or Ge site in order to tune the material to a paramagnetic state, without inducing major changes in the crystal structure. The investigated end compounds are: URuGe, UCoGe and

URhSi. Huy indeed found that ferromagnetism is suppressed in URh1−xRuxGe

at a concentration of x = 0.38 [8]. The Curie temperature (T_C) increases from

9.5 K for URhGe to 10.5 K for x = 0.05 and reduces until the critical concen-tration at x = 0.38 is reached, see figure 1.1.

In previously conducted experiments [9, 10] UCoGe appeared to be a paramag-net. This and the fact the UCoGe crystallizes in the same structure as URhGe (orthorhombic TiNiSi structure), was the motivating force to alloy URhGe with Co in order to find a quantum critical point. However, UCoGe was found to exhibit ferromagnetic ordering below 3 K and is superconductivity below 0.8

Figure 1.2: Dependence of ordering temperature on the distance between U atoms for several UTGe compounds. When one moves up in the periodic table in the same group (from Pt to Pd to Ni or from Ir to Rh to Co) the distance between the U-atoms decreases. Based on this figure, it is expected that alloying URhGe with Co will decrease the Curie temperature monotonically. Picture taken from [12].

K [3]. This made it the second known compound where superconductivity and ferromagnetism coexist at ambient pressure. UCoGe is closer to a magnetic in-stability and therefore more convenient to use in the search for a QCP. This is done by substituting Si on the Ge site in UCoGe [11]. Another point of discussion

is the evolution of the Curie temperature of the URh1−xCoxGe series, which was

measured by transport and magnetisation experiments. The Curie temperature as function of x shows an unusual behaviour despite the fact that the alloying of

URhGe with Co is an isoelectronic substitution. TC first increases from 9.5 K

for URhGe to 20 K for 60% Co concentration and afterwards decreases to 3 K

for UCoGe, see figure 1.1. This non-monotonic variation of T_C(x) is the main

topic of the work presented here. In order to elucidate the mechanism behind

the non-monotonic behaviour of the T_C(x) curve, we will perform specific heat

measurements (with the Physical Property Measurement System, PPMS) In order to illustrate the problem, we will first explain what is expected, from

basic principles, for the T_C(x) curve of URh_1−xCo_xGe. Rh and Co are

isoelec-tronic elements, which reside above each other in the same group of the periodic table. Both have the same number of electrons in the valence band. Therefore,

it is expected that TC changes in a simple manner, which can be explained by

the distance between the U-atoms. Hill was the first to point out that the di-rect f-f hybridization is important when the distance between the U-atoms is smaller than 3.5 Å when determining the ground state of U compounds [7]. This

critical distance of 3.5 Å is called the Hill-limit. If the distance between the U-atoms is larger than 3.5 Å there is no overlap between the f-orbitals of the U-atoms and there is local magnetism. If this distance is smaller than the Hill limit there is no long-range ordering. This is depicted in the Hill plot for UTGe compounds, figure 1.2. On the left-hand side of the Hill limit are the paramag-netic materials, such as UFeGe and URuGe and on the right-hand side are the (anti-)ferromagnetic UTGe compounds.

Later, it was shown that it is not only the overlap between the f-orbitals of U-atoms which determines the ground state, but also the overlap between the f-band and non-f orbitals, e.g. 3d, 4d or 5d orbitals. The group of Silva Neto et al. developed a model for the ground state of ternary uranium intermetallics, which calculates the hybridization between the 5f-band and the 4d- or 3d-orbital. In order to elucidate the problem of the non-monotonic behaviour we are going to perform specific heat measurements. This was the first time that the heat capacity option of the PPMS was used in the lab. A major part of this thesis is devoted to a description of the heat capacity option of the PPMS. In order to test the specific heat option, we needed a test-case with the correct temperature

range. The problem of the TC(x) curve of U(Rh,Co)Ge is exactly above the

minimum temperature (2 K) of the PPMS. With specific heat measurements we can determine the density of states at the Fermi level and the entropy. There are theoretical and experimental values for the density of states of URhGe and UCoGe, but there are no numbers available of the intermediate concentrations. Another important aspect is that the specific heat is a quantity which is very often predicted by theoretical calculations and therefore ideal to falsify existing theories.

2

Theoretical aspects

2.1

Specific heat

The molar specific heat is a thermal property of a material. It is defined as the amount of heat required to raise the temperature of one mole of a material by one degree. The specific heat is directly linked to the number of available energy levels within a system. Specific heat in an excellent technique to probe phase transitions Here, we will investigate para- to ferromagnetic transitions. The ferromagnetic state can be achieved by decreasing the temperature. The

temperature where the phase transition occurs is the Curie temperature, TC.

Above TC, the magnetic moments are randomly orientated resulting in a net

magnetisation of zero in the absence of an external magnetic field. Below T_C,

it is energetically more favourable for the magnetic moments to align parallel resulting in ferromagnetic domains with a non-zero magnetisation of the mate-rial, also in the absence of an external magnetic field. This sudden change in the magnetic configuration of the system results in a step in the specific heat. In order to present the basic elements of the specific heat here below we follow the chapters one to four of the book ’The Specific Heat of Matter at Low Tem-peratures’ by A. Tari [13].

2.1.1 Thermodynamics

In this section we will give a general derivation of the specific heat from basic laws of thermodynamics.

The specific heat of a substance is defined as the ratio of the amount of heat

(dQ) transferred to the sample over the resulting temperature change (dT ). The specific heat depends on temperature and therefore the change in T is assumed to be very small. This can be expressed in the following limit:

Cx,y... ≡ lim dT →0 _dQ dT  x,y... (2.1) where the subscript x,y.. indicates which quantities are kept constant, often pressure or volume. The convention is to use ’c’ for the specific heat per unit mass and to use ’C’ for the specific heat per mole of the substance measured in J/mol K. The term heat capacity is also encountered often. This is the specific heat of a material of a certain mass and has units J/K.

From thermodynamic relations the specific heat can be derived. The first law of thermodynamics states that the change in energy of a closed system is a combination of the amount of heat added to the system and the amount of work done by the system:

dE = dQ + dW (2.2)

The second law of thermodynamics tells that heat will flow from a warm reser-voir to a cold reserreser-voir in a closed system, i.e. the entropy of a system always increases. The change in entropy is defined as the amount of heat (dQ) flowing in or out a reservoir at a temperature T:

dS = dQ

T (2.3)

For completeness: the third law of thermodynamics states that the entropy in a perfect crystal will be zero at T = 0.

Combining equations (2.1) and (2.3):

C = TdS

dT (2.4)

For a reversible process, the work performed by a system is governed by dW = −P dV . This can be combined with the first and second law of thermodynamics giving:

dE = T dS − P dV (2.5)

It is required to relate the specific heat to the free energy of a system. The change in the Helmholtz free energy of a thermodynamic process is:

dF = d(E − T S) (2.6)

which can be evaluated with equation (2.5) at constant volume (dV = 0) to

Next step is to evaluate equation (2.5) at constant V and to use equations (2.7) and (2.4) to obtain the specific heat:

Cv = dE dT    _v = −T d 2_F dT2     _v (2.8) This relation is very useful, because one can formulate an expression for the energy of a system and calculate the specific heat by taking the derivative. Or, one may, for second order phase transitions, use Landau theory to determine the free energy and calculate the specific heat by taking the second derivative. Consequently, one can derive a theoretical expression for the specific heat. This is the reason why specific heat measurements are such a powerful tool: there is a direct link between experimental data and theoretical models.

2.1.2 Lattice specific heat

The law of Dulong & Petit

In the 19thcentury Dulong and Petit discovered that the specific heat near room

temperature is the same for several elements. The law that has been formulated by Dulong and Petit is a completely classical description. The reasoning is as follows: Classically, the atoms are treated as solid balls which are connected by springs. At room temperature, the vibrations of the atoms can be described by linear harmonic oscillators. The equipartition law states that a linear harmonic

oscillator in thermal equilibrium will have a mean internal energy of k_bT . In

three dimensions there are three degrees of freedom, which gives an average energy of 3kbT per oscillator. The average energy of one mole oscillators is then

Eavg = 3N0kbT = 3RT , where N0 is Avogrado’s number and R the gas constant.

Exploiting equation 2.8 the specific heat of one mole of oscillators is obtained:

Cv =

∂E

∂T = 3R ' 24.94 J/mol K (2.9)

This is the Dulong-Petit law: All mono-atomic compounds have a specific heat of approximately 25 J/mol K. Due to the classical argument, this law is only valid at high temperatures. Note that the specific heat does not depend on the mass of the compound, but on the number of atoms. This implies that for a

binary compound AxBy : C(AxBy) = xC(A) + yC(B).

However, after the discovery of cryogenic liquids it was possible to measure the specific heat at lower temperatures. These experiments revealed that the specific heat decreases, rather than staying constant, for temperatures below approxi-mately 150 K. The classical approach is no longer valid in this regime. This problem could only be solved using quantum mechanics. An attempt to

formu-late an expression for the specific heat over a large temperature range was made by Einstein and later by Debye.

Einstein specific heat

With the introduction of quantum mechanics, Einstein made an attempt to describe the temperature dependence of the specific heat. The first main as-sumption is that all atoms vibrate independently of each other. Secondly, all

atoms vibrate at the same angular frequency, ωE. Einstein utilizes Planck’s idea

of quantization, i.e. energy levels are quantized and evenly spaced by ħhω. With this approach Einstein was able to explain the decreasing trend in the experi-mentally observed specific heat. The energy for a collection of N oscillators with

Einstein frequency ω_E is calculated by:

E = 3sN Z

(hni + 1/2) ħhω δ(ω − ωE)dω (2.10)

where s is the number of atoms per formula unit, (hni + 1/2)ħhω is the mean energy of n oscillators in thermal equilibrium with hni = (exp(ħhω/kbT ) − 1)−1. All the oscillators have the same frequency, which results in a delta function for the distribution of vibrational modes. By taking the derivative the specific heat of an Einstein solid is obtained:

Cv = 3Rs _θ E T 2 _exp(θ E)/T [exp(θE/T ) − 1]2 (2.11)

where θ_{E = ħhωE}/kB is the Einstein temperature. At high temperatures (T 

ΘD) equation (2.11) approaches to the Dulong-Petit law. In the limit T → 0, the

Einstein specific heat exhibits an exponential behaviour. At the time Einstein obtained this result it was not yet possible to measure to such low temperatures. Later it was shown that the low temperature specific heat is not given by an exponential function. Although, the model is incorrect at low temperatures, the ideas of Einstein led to other proposals for describing the behaviour of the low temperature specific heat.

Debye specific heat

Debye based his ideas on those of Einstein. Rather than considering each atom individually, he treated the lattice as an elastic continuum. Due to this assump-tion, there is no single allowed frequency, but rather there is a distribution of allowed frequencies. The expression for the average energy of a Debye solid is given by:

E = Z ωD

0 (hni + 1/2)ħhω · 3D(ω)dω

(2.12) where hni is the average number of phonons given by the Fermi-Dirac distribu-tion, ħhω is the energy of a single phonon (1₂ħhω is the zero-point energy) and

3D(ω) is the number of modes. The latter is the main difference with the Ein-stein model, where the number of atoms is equal to the number of modes, which is reflected by the delta function in equation 2.10. The maximum value of ω is confined by the lattice. The wavelength cannot be smaller than the inter atomic distance, which results in a maximum frequency. This maximum

fre-quency is the Debye frefre-quency, ωD, and is determined by the number of states

3sN = RωD

0 3D(ω). The corresponding temperature is the Debye temperature,

ΘD = ħhωD/kb. When T  ΘD, the atoms are decoupled from each other and

we enter the classical regime described by Dulong and Petit, where the atoms vibrate independently. Typical values for the Debye temperature are between 100 K and 400 K.

By taking the derivative of equation (2.12), the specific heat for a Debye solid can be obtained: Cv= 9Rs  _T ΘD 3Z x_D 0 x4exdx (ex_{− 1)}2 (2.13)

where x = ħhω/kBT . In the high temperature limit (T ΘD) Dulong-Petit law

is retrieved. At low temperatures (T ΘD), the upper limit of the integral in

equation (2.13) can be expanded to infinity. This results in the Debye T3-law:

Cv = βT3 (2.14) β = 12π 4 5 Rs ΘD (2.15) Equation (2.14) is generally accepted to approximate the low temperature phonon

specific heat. However, at intermediate temperatures (Θ/50 < T < Θ/2 ) the

Debye theory fails. Often, the T3_{-law is used to extract the low temperature}

phonon specific heat from the total specific heat.

2.1.3 Electronic specific heat

The aforementioned theories for the specific heat were all derived before the dis-covery of the electron. After this disdis-covery, successful theories describing the behaviour of electrons were developed. However, they lack to describe the con-tribution of electrons to the specific heat. Following the equipartition law, a free electron will have an internal energy of 1/2kBT . For N free electrons with three degrees of freedom, the electron specific heat would be 3R/2. This results in a total specific heat (electrons + lattice) of 3R/2 + 3R = 9R/2. This is bigger than the 3R obtained by Dulong and Petit for the lattice specific heat. Therefore, it seems that electrons do not participate to the specific heat. This was a puzzling

develop-ment of the free electron model. Electrons obey the Fermi-Dirac distribution, this means that all available states are filled up to the Fermi level at T = 0. When T > 0 electrons can be excited to a higher state by receiving an energy of

kbT . However, not all the electrons can receive this energy, because the states

within a range of kbT are all occupied, due to Pauli exclusion principle. Only

the electron around k_bT can excite to a higher state, a state above the Fermi

level. Normally, the thermal energy, k_bT , received by electrons, is of the order of meV and the Fermi energy for metals is in the order of eV. So, the amount of electrons that can excite to a higher state is fairly small and most electrons will not contribute to the specific heat. The specific heat of the electrons is

dominant at low temperatures. In the case of URh1−xCoxGe the contribution of

the electrons and the lattice to the total specific heat is equal at a temperature of ∼10 K (UCoGe) or ∼20 K (URhGe). These values are already quite large, because they belong to the group of correlated metals, where γ is enhanced due to the enhanced electron mass. Above these temperature the electronic specific

heat is rapidly dominated by the T3 _{contribution from the lattice specific heat.}

With the help of the free electron model the mean energy of N electrons at temperature T can be calculated:

E =

Z ∞

0

f ()η()d (2.16)

where η() is the density of states for both spin directions and f () is the Fermi-Dirac distribution.

By differentiating equation (2.16) and using the expression of the Fermi-Dirac distribution the electron specific heat is obtained:

Ce= γT (2.17)

γ = 1

3π 2_k2

bη(F) (2.18)

Ceis the heat capacity in units of J/ K, if η(F) is expressed in units of states/eV · atom. The electron specific heat scales linearly with T . One only needs to know the density of states at the Fermi level, not the full bands, in order to calculate the electronic specific heat. We stress that this equation is only valid at low temperatures, where the density of states not varies as a function of  in the vicinity of η(F). γ is also directly linked to the electron mass by:

γ = k

2 b

3ħh2kFme (2.19)

where k_F is the momentum at the Fermi level and m_e the electron mass. In the

case of heavy fermion systems, due to the enhanced electron mass, the electron contribution to the specific heat will also be enhanced. The parameter γ is called

the linear coefficient of the electron specific heat or the Sommerfeld parameter and is an important parameter.

In general, γ can directly be obtained from specific heat measurements by taking the limit:

γ = lim T →0

C

T (2.20)

Magnetic specific heat

Magnetic materials have an extra contribution to the specific heat. Ordering of the magnetic moments will lead to a reduction of the entropy, which is released as thermal energy when cooling a material from the para- to a ferromagnetic state. This can be seen by a step in the specific heat. When the exchange inter-action between the electrons dominates over thermal fluctuations, the magnetic moments of the electrons will align parallel to each other. This is a ferromagnetic state. At T = 0 all the moments are aligned parallel in domains. When T > 0, spin-waves or magnons might be excited. These collective excitation of electron spins is called a magnon. These give an extra term to the specific heat which scales with T3/2_.

The low temperature specific heat of a ferromagnet below TC can be described,

in its most simple form, by the following equation:

Cv = γT + βT3+ δT3/2 (2.21)

The first term is only present in a metal and describes the specific heat of the conduction electrons. The second term comes from the lattice vibrations. The last term describes the magnon specific heat in ferromagnets.

2.2

The d-f hybridization in URh

1−x

Co

x

Ge

In this section an overview will be given of the hybridization model proposed by Silva Neto et al. [1]. With this model, the authors explain the unusual

be-haviour of the Curie temperature of URh_1−xCo_xGe. The model was developed

for ternary uranium intermetallics with the general formula UTX, where T =

Rh, Co and X = Ge, Si. According to Ref[1], the variation in TC depends solely

on the transition metal bandwidth of Rh or Co and the distance between the centres of the f-band of the U-atoms and the d-band of the T-atoms. With this

argument some elementary properties of URh_1−xCoxGe can be explained, such

as the density of states, η(_F), the ground state magnetisation, M₀, the resistiv-ity coefficient, A, and the specific heat coefficient γ.

In figure 2.1 the total density of states (DOS) of URhGe and UCoGe is de-picted. As URhGe is doped with Co on the Rh site, the 4d-band is converted to a 3d-band. During this process, the d-band will become narrower and lower in energy with respect to the 5f-band. The 5f-band is a narrow band around the Fermi level. The p-band of Ge is located far below the Fermi level, ∼ -9 eV, and therefore not taken into consideration [14].

4d-band

3d-band

5f

5f

Figure 2.1: The total DOS of URhGe and UCoGe. The blue points are calculations using the model of Silva Neto et al. and the black lines come from density functional theory (DFT) calculations by Divis [15, 14]. The red line at zero eV is the Fermi level. In both URhGe and UCoGe there is a broad d-band stretching from approximately -6 eV to 2 eV. Around 1 eV there is a pronounced peak from the 5f-band.

The hybridization of the f-band and d-band can be expressed as: Vdf =

WdWf

∆Cdf

(2.22)

W_f and W_dare the bandwidths of the f-band and the d-band, respectively. The

difference between the centre of the d-band of the T-atoms and the centre of the

are made. Firstly, it is assumed that the width of the f-band, W_f, does not change. Secondly, the width of the d-band changes linearly with doping:

Wd(x) = WdRh(1 − x) + WdCox (2.23)

WRh_d and WCo_d are the widths of the 4d-band in URhGe and the 3d-band in

UCoGe, respectively. Thirdly, the difference between the centres of the f-band and the d-band changes via a non-linear function:

∆Cdf(x) =∆CdfRh(1 − x) +∆CdfCox + δ 0

x2(1 − x) + δ00x(1 − x)2 (2.24)

where ∆C_dfRh and ∆C_dfCo are the differences between the centres of the d- and

f-band of URhGe and UCoGe, respectively. WRh_d , WCo_d , ∆C_dfRh and ∆C_dfCo are extracted from DFT calculations. See the black lines in figure 2.1 and Ref. [15, 14]. The adjustable fit parameters, δ0 and δ00, control the minimum of Vdf as a function of x.

Under influence of doping the d-band becomes more narrow with respect to the 5f-state. This will reduce the hybridization and increase the f-DOS at the Fermi level. The enhanced DOS is a general feature of correlated metals. The spin of the 5f-moments are screened (Kondo-effect) by the surrounding spins. This give rise to a narrow f-DOS peak near the Fermi level (Kondo resonance). Upon lowering the temperature, the f-peak becomes more narrow (fig. 2.1). This will reduce the hybridization by equation 2.22.

Secondly, the self-consistent renormalization (SCR) theory deals with the dual (itinerant or localized) magnetic character of hybridized f-d-electrons [17]. SCR

theory provides a relation between T_C and the DOS at the Fermi level for weak

Figure 2.2: The f-DOS for different temperatures, indicated by the different dashed lines, showing the increase of the f-DOS at the Fermi level as a function of temperature,

itinerant ferromagnets:

TC ∝ (Iη(F) − 1)3/4 (2.25)

where I is the Stoner exchange interaction. From equation (2.25) one can see

that an increase in the DOS at the Fermi level causes an increase in T_C. From

this relation we can deduce that a system is ferromagnetic when the condition Iη(F) > 1 is fulfilled. This is called Stoner’s criterion.

Shortly stated: When Rh is substituted by Co in URhGe, the d-band (W_d)

be-comes narrower, reducing the hybridization (Vdf) and increasing the DOS at the

Fermi level. This leads to an increase in TC. However, for x values exceeding a

certain doping concentration, the centres of the d-band and f-band will move

to-wards each other, making∆Cdf smaller. This process increases Vdf and reduces

the DOS at the Fermi level. This leads to a decrease in T_C. A quantitative

analysis is made in section 4.5. The hybridization as a function of doping con-centration x is investigated in section 4.5. The aforementioned arguments show how the energy bands change under the influence of Co doping, which is the

reason for the non-monotonic behaviour of T_C. Next, the hybridization needs

to be related to measurable quantities. Therefore, we give a short description of the periodic Anderson model.

Periodic Anderson model

The periodic Anderson model (PAM) describes localized magnetic impurities in a metal. The PAM is often used to describe heavy fermion systems. The Hamiltonian of the PAM is given by:

H = −tX hi,ji (d†_iσdjσ+ h.c.) + 0d X i nd_iσ+ 0_fX i nf_iσ (2.26) + Vdf X i

(f_iσ†diσ+ d†iσfiσ) + U X

i

nf_i↑nf_i↓

where t is the nearest neighbour hopping integral for the d-band, d†_iσ(d_iσ) creates (removes) a f-electron at site i with spin σ, f_iσ† (fiσ) creates (removes) a d-electron at site i with spin σ, 0 is the energy dispersion relation, nf_iσ = f_iσ†fiσ and nd

iσ = d

†

iσdiσ are number operators and U is the Coulomb repulsion on the

localized f-electron site.

This Hamiltonian describes a dispersive band of conduction d-electrons, which are hybridized with the localized f-states due to magnetic impurities. The first term of the Hamiltonian describes the hopping of electrons in the d-band. In a more general expression for the Hamiltonian there is also a hopping integral for the f-band. However, the f-electrons are localized and the hopping integral is assumed zero for the f-electrons. The second and third term give the energy

(a) (b)

Figure 2.3: a) Magnetisation of the ground state as function of the hybridization. Two regimes are visible with the boundary at Vdf = 1.45. The green lozenge indicates the location of URhGe in the weak hybridization regime and the yellow lozenge indicates the location of UCoGe in the strong hybridization regime. At Vdf = 1.9 the magneti-sation becomes zero at a QCP. b) The resistivity coefficient A as function of doping concentration calculated from the hybridization model (blue points). The red squares give the experimental values from Huy [7]. Picture taken from [1].

dispersion of the d-band and the f-band, respectively. The fourth term shows

the coupling between the local site and the conduction band by Vdf. The last

term reflects the on-site Coulomb repulsion between the f-electrons.

The magnetisation of the ground state, M0, can be obtained from equation

(2.26). The Hamiltonian can be diagonalized in order to obtain the eigenvalues of H in terms of V_df. After that, M₀ can be calculated as a function of V_df. In figure 2.3a the magnetisation of the ground state as function of the hybridization is shown.

According to Ref.[1], URhGe corresponds to V_df = 1.42 eV and UCoGe with

Vdf = 1.73 eV, which is indicated in figure 2.3a by the green and yellow lozenges, respectively. There are two different kinds of behaviour in the magnetisation

associated with the hybridization. First, the regime with V_df < 1.45 where M0

is quadratic in Vdf. This is the weak hybridization regime. Second, the regime

where V_df > 1.45, which results in a linear dependence of M0 on Vdf. This is the strong hybridization regime. URhGe belongs to the weak hybridization regime and UCoGe belongs to the strong hybridization regime. According to Silva Neto

et al. the existence of these to regimes in URh_1−xCo_xGe is the cause of the

unusual behaviour in TC.

Experimental values for the resistivity coefficient, A (where ρ = ρ₀+ AT2), have been determined by transport measurements conducted by Huy [7]. A can also be derived from the PAM by utilizing the SCR theory, which provides a relation between the DOS at the Fermi level and A for weakly-itinerant ferromagnets:

A ∝ (Iη(F) − 1)1/2 (2.27) In figure 2.3b A calculated from the PAM by Silva Neto et al. is shown. The

curve displays a behaviour opposite to TC and matches well the experimental

data.

The d-f-hybridization model and the calculations are also applied to URhGe1−ySiy.

Experimentally, the URhGe_1−ySi_y series show no change in the Curie

temper-ature [7]. This is in agreement with the calculation of the d-f-hybridization performed by Silva Neto et al., which shows that there is no change in the hy-bridization between the f-state and the d-band when Ge is substituted by Si.

Therefore, no change in the Curie temperature of URhGe1−ySiy is expected.

Quantum critical point

If Vdfcontrols TC, it controls the proximity to a QCP. This can also be calculated from the PAM. In figure 2.3a the magnetisation as function of the hybridization is calculated in the limit T = 0, which shows a ferro- to paramagnetic transition at Vdf=1.9. This is a prediction for a QCP in the URh1−xCoxGe series. Experi-mentally, this can de accomplished by doping UCoGe with a different element in order to reduce the hybridization even further. This was also done by Huy [7], where UCoGe is tuned to a QCP by substituting Ge by Si. The magnetisation decreases when the Si concentration increases, till it vanishes at a critical doping concentration x ∼ 0.12.

3

Experiment

All the specific heat measurements have been performed with the Physical Prop-erty Measurement System (PPMS) Dynacool developed by Quantum Design. This is a fully automated system, which can measure several physical properties such as resistivity, magnetisation and heat capacity. The Dynacool is equipped with a 9 T superconducting magnet and the temperature can be sweeped from 400 K to 2 K. This is the first time that the heat capacity option of the PPMS was used in the lab. Therefore, a more extensive treatment on the hardware, measurement and fitting procedure is given here. Parts of this section are based on the heat capacity option manual [18] and seminar slides provided by Quantum Design (https://www.qdusa.com/pharos/).

3.1

Specific heat measurement methods

There are different methods for measuring the specific heat, where a considera-tion needs to be made between accuracy and resoluconsidera-tion.

The adiabatic method is a classical method which uses the definition of the heat

capacity: Cp= dQ/dT .

The specific heat is determined experimentally by applying a heat pulse (dQ) to the sample and recording the change in temperature (dT). This method has some drawbacks such as the large sample size (∼ 10 g) in order to minimize the amount of stray heat leaks and the need of a mechanical or superconducting switch to establish thermal contact between the sample and the environment (often the liquid He bath) [19]. Moreover, the sensitivity is not good enough to detect small changes in the heat capacity [13]. In order to circumvent these problems a different method was developed by Sullivan and Siedel, the ac-method, which

enabled to measure the specific heat of small samples (1-100 mg)[20]. In the ac-method no heat switch is needed. The sample is thermally coupled to a heat reservoir and a heat pulse with a known frequency is applied to the sample. The measured ac temperature response is then inversely proportional to the specific heat of the sample. Although this method is better than the adiabatic method

in measuring small changes in the specific heat (∆C ∼ 0.04%), yet it is not more

accurate for measuring the absolute specific heat (∆C ∼ 8 %) compared to the

adiabatic method [19].

A different technique, also designed to measure the heat capacity of small sam-ples, is the relaxation technique developed by Bachmann et al. [21]. By applying a heat pulse to the sample and tracking the temperature response, the heat flow equations with the corresponding boundary conditions can be determined. By solving the equations C is found to be proportional to the relaxation constant of the temperature response curve. More details of the method of Bachmann et al. are given in section 3.2. The PPMS utilizes the relaxation technique developed by Hwang et al. [22] which is an extension to the method of Bachmann et al.. Huang et al. also take into account the heat flow between the sample and the

platform, the tau two effect (τ₂). The experimental set-up for the relaxation

technique is shown in figure 3.3.

3.2

Thermal models

The heat capacity is measured by applying a heat pulse and tracking the tem-perature reponse of the sample. Here we will elaborate on the fitting procedure of the temperature response curves. To fit the temperature response curve, the software uses two thermal models, the ’simple’ model and the two tau model. The simple model treats the sample and the platform (on which the sample is mounted) as one system, whereas in the two tau model also a heat flow between the sample and the platform is considered, see figure 3.1.

3.2.1 Simple model

The simple model assumes perfect contact between the sample and the platform, i.e. the heat flow between the between sample and the platform is not modelled. The temperature T of the whole system can be found by solving the heat flow

equation 1

Ctot dT

dt = P (t) − κw(T − T0) (3.1)

where C_tot is the specific heat of the sample, platform, thermometer and heater.

T(t) is the temperature at time t, P(t) is the power of the heater, κw is the

1

Note, Ctotis the heat capacity of the total system and Cplthe heat capacity of the platform,

Sample Platform Ctot T κW Thermal bath T0 (a) Platform Cpl Tpl κW κG Thermal bath T0 Sample Cs Ts (b)

Figure 3.1: a) Heat flow diagram for the simple model. Ctot is the heat capacity of the whole system and κW is the thermal conductivity of the wires. The thermal bath has a temperature of T0. b) Heat flow diagram in the two tau model. κG is the thermal conductance of the grease. Cpl, Tpl and Cs, Ts are the heat capacity and temperature of the platform and sample respectively.

thermal conductivity of the wires and T₀ is the temperature of the bath. There

is a heat flow directly to the sample/platform from the heater. The power of the heater has a constant magnitude P during heating and is zero during the cooling part. The second flow is from the sample/platform to the environment. This is determined by the thermal conductance of the wires and the temperature difference between the sample/platform and the bath formulated in the second term of equation 3.1.

The differential equation can be solved for T(t) for the heating part and for the cooling part of the temperature response curve. An example of a temperature response curve is given in figure 3.2. The solution is given by equation (3.2) of which a full derivation is given in Appendix A.

T (t) =    P τ1 − e−t/τ/Ctot+ T0 0 < t < t0 P τ1 − e−t0/τ_e−(t−t0)/τ_/C tot+ T0 t > t0 (3.2)

where τ = C_tot/κ is the relaxation constant. The measured T(t) curve is fitted

to equation 3.2. The three unknown variables (κw, Ctot, T0) are varied in such a way that the sum of the differences S between the modelled T at the measured

time t_i and the measured temperature T_i are minimal:

S(Ctot, κw, T0) =

X

i

(T (t_i) − T_i)2 (3.3)

The set of parameters C_tot, κ_w and T₀ for which S is minimal yields the value

0 t₀ T_0+2%

T₀

ON OFF

Figure 3.2: An example of a temperature response curve with base temperature T0. The heater will stay on until the temperature T0+2% is reached at time t0. The whole curve is measured in 2τ seconds.

3.2.2 The 2τ model

The two-tau model is a more sophisticated model, which also models a flow between the sample and the platform. The heat flow diagram is given in figure 3.1b. The equations that govern this model are given by:

Cpl dTpl dt = P (t) + κg(Ts(t) − Tpl(t)) − κw(Tpl(t) − T0) (3.4a) Cs dTs dt = −κg(Ts(t) − Tpl(t)) (3.4b)

Equation 3.4a shows the flow from and to the platform. The two incoming heat flows to the platform are: (i) from the heater, with magnitude P when the heater is on, and (ii) a flow from the sample to the platform, where the magnitude is

determined by κ_g. The last term shows the flow from the platform to the bath,

which is governed by κw.

Equation 3.4b shows the heat flow from and to the sample. When the heater is on there will be a flow from the platform to the sample and when the heater is off there will be a flow from the sample to the platform. The simple model (eq. 3.1)

is a simplified version of the 2τ model and assumes that κ_g  κ_w. How much

the 2τ model deviates from the simple model is reflected in the sample coupling, which is defined as κg/ (κg+ κw). The sample coupling is 1 in the ideal case. Equation (3.4) cannot be solved directly as in the simple model. A clear deriva-tion is given in a paper of Shang Hwang et al. [22], where it is demonstrated that the two differential equations can be converted into three equations with four

unknown parameters: κw, κg, Cs, Cpl, where Cpl is known from the addenda

measurement (the heat capacity of the platform, heater and thermometer. See section 3.6). By fitting the three parameters to the three solutions of the cou-pled differential equations one can determine the heat capacity of the sample.

) (t T b T b T Thermometer Grease Platform Sample Heater

Wires _{Thermal bath}

Thermal bath _К

w

Figure 3.3: Setup for the specific heat measurement. The sample is placed with grease on a platform. Under the platform are the heater and thermometer connected. The wires from the platform to the sample holder provide a link to the thermal bath. The thermal conductance of the wires is κwThe temperature of the sample is T(t) and the temperature of the thermal bath is Tb.

Note that in practise we measure the temperature of the platform and not of the

sample, but in the derivation of Ref. [22] Ts is eliminated and throughout the

calculation T_pl is used.

The software first fits the measured T_pl(t) to the simple model and after that to the two tau model. The result of the model with the smallest fit deviation will be used for further calculations. In our experiments almost all data above 200 K is obtained from the simple model. Note, that in the model we neglect the thermal resistance between the thermometer and the platform and between the heater and the platform.

3.3

Design of the PPMS Dynacool

All the measurements are conducted in the Dynacool, which is schematically depicted in figure 3.4. It is not necessary to transfer liquid cryogens, which makes it very efficient to use. The system makes use of a two stage pulse tube cooler. The pulse tube cooler provides a small amount of liquid helium in the bucket and cools the superconducting magnet to 4 K. A mixture of liquid helium and gaseous helium flows from the bucket through the cooling annulus surrounding the sample chamber. This, combined with a block heater below the sample chamber, controls the temperature of the insert between 2 K and 400 K. The Dynacool works with ’pucks’. A puck is a sample holder which provides thermal and electrical contact when it is placed in the bottom of the sample chamber. In the case of heat capacity measurements it is important to stress that the bottom of the sample chamber is an isothermal region. The isothermal region is made of copper in order to provide a uniform thermal region. The block thermometer is connected to the bottom of the isothermal region below the puck interface and measures the temperature of the puck.

Cryopump First Stage Cryopump Second Stage Sample Chamber High Vacuum valve

Cooler first stage

SC magnet Isothermal region Sample puck

Cooler second stage

Helium gas Bucket

Helium liquid 4K plate

Figure 3.4: A simplified picture of the Dynacool. In green is the bucket with the liquid helium which flows to the pot in the bottom of the annulus. Around the sample chamber flows helium gas. In the bottom of the sample chamber the puck is placed in the isothermal region, under which a block heater is attached for temperature control. To achieve high vacuum the sample chamber is exposed to a two stage cryopump. Picture taken from [18].

transfer from the sample to the environment. Therefore, the PPMS is equipped with a cryopump, which can evacuate the sample chamber to a high vacuum (< 10−6 Torr 2). The cryopump consists of two stages. The first stage is at 70 K and the second stage is at 4 K. The second stage cryopump contains charcoal which at 4 K will absorb any remaining gas from the sample chamber. This high vacuum option is fully integrated in the system. When starting a heat capacity measurement the sample chamber is automatically evacuated to less than 5 Tor with an external scroll chamber pump. To obtain a higher vacuum the chamber isolation valve is closed and the high vacuum valve is opened in order to expose the sample chamber to the cryostat. The pressure will drop within a few seconds to a high vacuum. Extra comments on the pressure regulation are provided in Appendix B.

Design of the heat capacity option

The sample is mounted on a thin sapphire (Al₂O₃) platform of 3×3 mm2 which

is embedded in the puckframe (∅=2.3 cm), see figure 3.5. On the backside of the

platform a RuO2 heater (R = 1 − 2 kΩ) and a Lakeshore Cernox thermometer

(R = 0.1−10 kΩ) are mounted. The resistivity dependence of the heater and the

thermometer on temperature is shown in figure 3.7. The platform is connected 2

Figure 3.5: The puck (left) and the platform (right). 1: The sapphire platform 2: Au-Pd wires 3: Gold contacts 4: Thermometer 5: Heater.

to the puckframe with eight Au-Pd wires (R = 10Ω, ∅ = 75 µm). Four wires

are for the 4-point probe method of the heater and four wires for the 4-point probe method of the thermometer. In each 4-point probe method two wires are for determining the voltage and two for the current. These eight wires act as a thermal link to the bath. The current supplied to the thermometer is ∼1 nA and to the heater is ∼10 nA. The contacts between the heater and thermometer to the wires are provided by a gold layer evaporated on the sapphire platform. Throughout all measurements the heat capacity puck with serial number 1505 is used.

3.3.1 Calibration of the puck

Before measuring the specific heat of the sample it is necessary to calibrate the thermometer and heater of the puck. The calibration consists of two parts, Pass 1 and Pass 2. In Pass 1 the resistance of the platform thermometer is determined in order to relate the resistance to the temperature. The reference thermometer for calibration is the block thermometer in the bottom of the PPMS. The results of the calibration are shown in figure 3.7. In Pass 2 the resistance of the heater is measured, as well as the thermal conductivity of the wires connected to the platform. Between Pass 1 and 2 the chamber needs to be opened in order to place the baffle assembly, see figure 3.6. At the end of the baffle assembly is a charcoal holder with a small amount of charcoal inside. The charcoal will adsorb He gas from the sample chamber, in order to prevent adsorption on the sample

Figure 3.6: The baffle assembly with the baffles and the charcoal holder at the end. The charcoal holder is placed just above the puck and will adsorb He gas.

▼▼▼▼ ▼▼▼▼▼▼▼▼_{▼▼▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼} ▼ ▼ ▼ ▼ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●_●● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● 0 100 200 300 400 0 1 2 3 4 5 6 7 T(K) R (k Ω ) Puck 1505 Heater Thermometer ● ● ● ●●●●●●●●●●●●●●● ●●● ● ●● ● ● ● ● ● 0 100 200 300 400 0.0 0.1 0.2 0.3 0.4 T(K) Κ (mW /K )

Figure 3.7: The results of the calibration (Pass 1 and Pass 2) of puck 1505. The resis-tance of the platform thermometer (orange circles) and heater (blue triangles) measured during the calibration. The inset shows the thermal conductivity (black circles) of the eight Au-Pd wires during Pass 2 of the calibration.

platform. During Pass 2 the heat capacity of the puck is determined, therefore a high vacuum is necessary.

3.4

Sample preparation and mounting

Sample preparation

The experiments were carried out on polycrystalline URh_1−xCo_xGe samples,

with x=0.2 ,0.4, 0.6, 0.8, 0.9, 0.93, 0.95, 0.98 and 1. Specific heat data of

URhGe were taken from the thesis of N.T. Huy [7]. The samples used for this thesis are pieces from bar shaped samples prepared by Huy and are all pre-pared at the Van der Waals-Zeeman Institute of the University of Amsterdam. The bar shaped samples were cut with a spark-erosion machine into one or two

small cubes with a size of approximately 1 × 1 × 1 mm3 and 1 × 1 × 0.5 mm3.

After cutting the surfaces were cleaned with sandpaper. Ideal samples for spe-cific heat measurements are thin and have a flat surface, in order to make sure that the heat is distributed evenly throughout the sample in reasonable amount of time. The samples have a mass between 5 mg and 12 mg, as listed in table 3.1.

x mass (mg) label 0.2 8.1 U1.03Rh0.8Co0.2Ge 2005 (3) 0.4 12.0 U1.03Rh0.6Co0.4Ge 2005 2 #1 0.6 11.8 U1.03Rh0.4Co0.6Ge 2005 1 #1 0.8 6.0 U1.015Rh0.2Co0.8Ge 16102006 4 #2 0.9 6.3 URh_0.1Co_0.9Ge 11aug2005 2 #1 0.93 7.0 U_1.03Rh_0.07Co_0.93Ge 5656 4 #1 0.95 7.4 U_1.03Rh_0.05Co_0.95Ge 5656 3 #1 0.98 10.5 U_1.03Rh_0.02Co_0.98Ge 5656 2 #2 1 5.1 UCoGe polycrystal 11062013 #1

Table 3.1: The masses of the URh1−xCoxGe samples. All samples were prepared by Huy, except the UCoGe sample, which is prepared by A. Nikitin. The error in the mass is ± 0.1mg.

Mounting the sample

Mounting the sample is a fairly straightforward procedure. The puck is placed in a sample mounting station and fixed in position with the interlock arm. A small vacuum pump is connected to the mounting station which stabilizes the platform and ensures that no force is exerted on the fragile wires during mounting. When the puck is fixed correctly we need to apply a small amount of Apiezon N Grease on the platform and the sample can be pressed into the grease. Before measuring the specific heat of a sample we need to measure the specific heat of the sample holder (puck) and grease together, the addenda measurement. Quantum Design recommends Apiezon N grease when measuring below 200 K, due to anomalies in its specific heat above 200 K. Apiezon H Grease is recommended for measure-ments above 200 K, because below 200 K H Grease can pop off the platform. For the whole temperature range of 2 K - 400 K Apiezon N Grease is recommended. When doing so, it is recommended that above 200 K a temperature spacing of ∼5 K between the datapoints is used in order to resolve any anomalies. These were not detected here. See section 3.6 for a more extensive treatment of the addenda measurements.

On the puck a thermal radiation shield is placed, to prevent heating by radia-tion from the rest of the chamber. After the puck is placed in the bottom of the sample chamber, one needs to place the charcoal baffle assembly in the sample chamber.

3.5

Heat capacity measurement procedure

This section will elaborate on which steps need to be taken to start an accurate heat capacity measurement. The PPMS is a fully automated system. Therefore a description of the whole process, which may not be directly obvious for the user, is given here. When the heat capacity is activated, a dialogue box, see

figure 3.8, will open where some parameters can be modified.

3

2

1

4

5

6

7

Figure 3.8: The dialogue box. 1) Temperature range: Temperature startpoint, endpoint and spacing. 2) Temperature rise: Height of temperature pulse. 3) Repetitions at each temperature: At each temperature point the specific heat is measured three times in order to check stability and reproducibility. The results are not averaged. 4) Relaxation measurement: The measurement time is always set to 2τ . During this time data acqui-sition takes place. 5) Determination of initial time constant: In general determined by applying a trial heat pulse, or, if known, it can be fixed. 6) Measurement stability: A heat pulse will only be applied when the temperature is within 1% of the requested tem-perature rise. 7) Extra wait time: When measuring at high temtem-peratures it is necessary to wait longer before the measurement starts.

In principle, the default settings are suitable to start a first measurement, but some settings need to be adjusted to get a more stable and accurate measurement of the heat capacity over the whole temperature range. Firstly, the temperature start- and endpoints and the spacing needs to be adapted. Another important setting to change is ’Extra wait time at new temperature’ under the tab ’Ad-vanced’. At each new temperature point the measurement will start immediately when the temperature has settled within 1% of the requested temperature rise, but at temperatures above ∼ 40 K the temperature can fluctuate causing a scattered heat capacity in the three measurements at each temperature point.

Increasing the measurement time was not necessary in the case of URh_1−xCoxGe,

but can be useful when the measured samples have a large diffusion time in com-parison with the relaxation time. It can be time efficient to assign a value to the initial time constant when the time constant at a specific temperature is already known. In this way the software does not have to make an estimation based on a trial heat pulse.

When all the previous steps have been taken, a measurement can start.

Data acquisition only takes place during the first 2τ after applying a heat pulse. When the measurement time is in units of τ , we first need to apply a trial heat pulse in order to estimate τ . The magnitude of the trial heat pulse is determined by reading the calibration file with the values for the thermal conductivity of

the wires in order to get the requested temperature rise which is the ratio of the power and the thermal conductivity of the wires, as shown in section 3.2. When an initial time constant is determined the temperature will settle again to the base temperature and a heat pulse is applied. The heater will stay on

until the requested temperature is reached at time t0. At this point the heater

is turned off and the sample will cool down. The temperature response curve is shown in figure 3.2. After 2τ seconds the data acquisition stops, this is the end of the curve shown in figure 3.2. The temperature response curve obtained in this way will be fitted to the simple model and the two tau model discussed in section 3.2. After 2τ the T(t) curve stops, but the temperature of the sample still decreases to the base temperature. During this time, every τ seconds the drift rate, dT (t)/dt, is checked and when the temperature is not varying within 1% of the requested temperature rise, the same procedure is repeated by applying a second heat pulse. At each temperature point, the heat pulses is repeated three times and with that three values for the heat capacity are obtained. The base temperature can now be increased to the next temperature point using the block heater below the sample chamber. At the next temperature point, the obtained time constant of the previous measurement is used as the initial time constant.

3.6

The addenda measurement

The addenda measurement gives the heat capacity of the sample holder (plat-form, heater, thermometer and the grease) and is measured prior to the sample heat capacity. After performing the sample heat capacity measurement the ad-denda measurement is subtracted from the total heat capacity in order to obtain the sample heat capacity. The addenda measurement turns out to be a very im-portant measurement, because the samples under consideration are fairly small, with a mass of only a few mg. Therefore, there is a large contribution (88 % at 300 K) to the total specific heat from the platform itself, see figure 3.9.

The specific heat of Apiezon N Grease

The specific heat measurements for all the URh1−xCoxGe compound were

con-ducted with one addenda measurement instead of remeasuring the addenda prior to any new measurement. We tried to reproduce the specific heat of Apiezon N Grease. If we know the specific heat of the grease we can calculate the contribu-tion to the specific heat of one mg extra grease, and with that make an estimate of the error in the determination of the specific heat of each sample. Therefore, we measured the specific heat of the empty puck and the specific heat of the puck with a known mass of grease. The results are shown in figure 3.10. The specific heat of the grease is obtained by subtracting the ’empty puck’ curves from the ’addenda’ curves. We made four different combinations (Addenda # 1- Puck # 1, Addenda # 2-Puck # 1, etc.) in order to obtain an average result

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Figure 3.9: Heat capacity from the addenda and UCoGe sample (black squares) and addenda heat capacity (black circles) as function of temperature. It shows the large contribution of addenda to the total heat capacity. The inset shows the low temperature behaviour. Note that the units are in mJ/K.

æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à ì ìììììììììììììì ìì ì ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò

æ Addenda ð1 with 1 mg grease à Addenda ð2 with 0.3 mg grease

ì Empty puck ð1 ò Empty puck ð2

0

100

200

300

400

0

2

4

6

8

10

T

HKL

C

HmJ

K

L

æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ à à à à à à à à à à à à à à à à à à à à à à à ììììì ì ìì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò 0 5 10 15 0.00 0.01 0.02 0.03 0.04

Figure 3.10: The heat capacity of the puck with 1 mg of grease (blue circles) and of the puck with 0.3 mg grease (yellow squares). The heat capacity of the empty puck is measured twice (red diamonds and green triangles). Throughout all measurements on the U(Rh,Co)Ge samples ’Empty puck #1’ is the calibration curve and ’Addenda #2’ was used as addenda.