Novel metallic states at low temperatures in strongly correlated systems. Wenlong Wu

Transcription

1 Novel metallic states at low temperatures in strongly correlated systems by Wenlong Wu A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto Copyright c 2010 by Wenlong Wu

2 Abstract Novel metallic states at low temperatures in strongly correlated systems Wenlong Wu Doctor of Philosophy Graduate Department of Physics University of Toronto 2010 This thesis describes experiments carried out on two novel strongly correlated electron systems. The first, FeCrAs, is a new material that has not been studied before, while the second, Sr 3 Ru 2 O 7, has been previously shown to have a very novel so-called nematic phase around the metamagnetic quantum critical end point (QCEP). For these studies, a new variation on an established method for measuring the field dependence of susceptibility in a BeCu clamp cell has been developed, and is described, as is a relaxation heat capacity cell that works from 4 K down to 300 mk. A method of growing stoichiometric crystals of the hexagonal iron-pnictide FeCrAs has been developed, and transport and thermodynamic measurements carried out. The in-plane resistivity shows an unusual non-metallic dependence on temperature T, rising continuously with decreasing T from 800 K to below 100 mk. The c-axis resistivity is similar, except for a sharp drop upon entry into an antiferromagnetic state at T N 125 K. Below 10 K the resistivity follows a non-fermi-liquid power law, ρ(t) = ρ 0 AT x with x < 1, while the specific heat shows Fermi liquid behaviour with a large Sommerfeld coefficient, γ 30 mj/mol K 2. The high temperature properties are reminiscent of those of the parent compounds of the new layered iron-pnictide superconductors, however the T 0 K properties suggest a new class of non-fermi liquid. The metamagnetic critical end point temperature T in Sr 3 Ru 2 O 7 as a function of hydrostatic pressure with H ab has been studied using the ac susceptibility. It is found ii

3 that T falls monotonically with increasing pressure, going to zero at P c = 14 ±0.3 kbar. One sign of the nematic phase observed in the field-angle tuning, i.e. T rises as the novel phase emerges, has not been seen in our study. However, we see a slope change in T vs P at 12.8 kbar, and a shoulder at the upper field side of the peak in χ from 12.8 kbar to 16.7 kbar. These new features indicate that some new physics sets in near the pressure-tuned QCEP. iii

4 Acknowledgements It is a pleasure to acknowledge my deepest appreciation and gratitude to my supervisor, Professor Stephen R. Julian, for his guidance, patience and stimulating discussions throughout the course of my Ph.D study. I am deeply grateful to our postdoc Dr. Alix McCollam for her help in the measurements and the data analyses. I would like to thank Dr. S. A. Grigera for preparing the sample Sr 3 Ru 2 O 7 and his reading and comments on the paper draft of this crystal. I would like to express my thanks to Professor Yong Baek Kim and Professor Young-June Kim for valuable discussions. I also acknowledge Dr. Ian Swainson at Chalk River Laboratories for the collection of neutron data and his analysis of the magnetic structure of FeCrAs. I also thank my fellow graduate student Patrick Rourke for his work on the labview program; to my fellow graduate student F. F. Tafti for his help in the pressure measurements and the discussions we have had; and to graduate student Fan Wang for her assistance in the magnetic hysteresis and DC susceptibility measurements. I would like to thank other graduate students in the Physics Department: Aaron Sutton, Patrick Morales, Cryrus Turel, Igor Fridman, Dan Sun for making my time in the McLennan Physical Laboratories more enjoyable. My appreciation also to Krystyna Biel for her help in the administrative issues during my Ph.D study in the Physics Department; to Robert Henderson for his services in supplying liquid helium and nitrogen; and to James Riffert in the machine shop for his guidance on using the machines. iv

5 Contents 1 Introduction Phenomenological Fermi liquid theory Quantum critical point Metamagnetic quantum critical point Design and development of experimental instruments Nonmagnetic clamp cell Design of the cell Components made by myself Use of the cell Thermal relaxation calorimetry Mathematical model Construction of the heat capacity cell Experiment Data analysis and results Further work Growth of single crystal FeCrAs Procedures and characterizations The first growth The second growth v

6 3.1.3 The third growth Seventh growth Eighth growth Tenth growth Eleventh growth Microprobe analysis Powder X-ray diffraction Powder neutron diffraction A novel non-fermi-liquid state in FeCrAs Material background Experiment Results Discussion Conclusion Pressure induced quantum criticality in Sr 3 Ru 2 O Material background Origin of the metamagnetism Nature of the nematic phase Motivation Experiment Results Discussion Summary Conclusions and suggestions for further work Technical development Crystal growth vi

7 6.3 Novel low temperature state in FeCrAs Pressure induced QCEP in Sr 3 Ru 2 O Further work FeCrAs Sr 3 Ru 2 O Bibliography 139 vii

8 List of Tables 3.1 Summary of crystal growths Surface EPMA analysis results Room temperature phase information viii

9 List of Figures 1.1 Schematic diagram illustrating the Fermi-Dirac distribution A diagram showing the limited k-space Schematic phase diagram in the vicinity of a quantum critical point Schematic diagram showing the density of states A generic phase diagram of a ferromagnetic metal Typical field dependence of the magnetization Schematic phase diagram illustrating the critical point Cross-sectional view of the clamp pressures cell Picture of the clamp cell and the centering mechanism Schematic diagram of a relaxation-time heat capacity cell Typical temperature behavior of the sample Picture of the heat-capacity cell Two irregular-shape FeCrAs samples A rectangular slab of UPt 3 sample Instrumentation of heat capacity measurements Thermometer calibration R vs T curves An example of the fitting results of the UPt C/T vs T around the superconductivity transition of UPt C vs T for UPt 3 from 1 K to 4 K Comparison of C/T vs T for UPt 3 for mks and that for 1 4 K ix

10 2.14 Relaxation time vs temperature for FeCrAs Relaxation time vs temperature for UPt C/T vs T 2 for FeCrAs from 1 K to 4 K A sample of FeCrAs single crystal Laue back scattering pattern of FeCrAs Illustration of sealing the materials in the quartz tube Magnetization vs field at 10 K Specific heat vs temperature for a sample C/T vs T 2 at low temperatures Resistivity as a function of temperature for a sample Magnetization vs field behavior at 2 K and 300 K Resistivity and specific heat as a function of temperature Comparison of the magnetic properties of 7th and 8th growths The magnetic hysteresis behavior and M vs T for the tenth growth Magnetization vs field and M vs T for the eleventh growth Powder x-ray diffraction at room temperature Powder neutron diffraction at 297 K with λ = Å Powder neutron diffraction at 297 K and 2.8 K Magnitude of the magnetic peak vs temperature Crystal structure of FeCrAs Schematic diagram showing the four-point-contact The wired sample for the resistivity measurements Resistivity vs temperature for a-axis Resistivity vs temperature for c-axis The c-axis resistivity vs field at 2 K The a-axis resistivity vs field at 2 K x

11 4.8 The a-axis resistivity vs field at 170 mk The real part of the ac susceptibility vs field The Hall coefficient at 83 mk Temperature dependence of the susceptibility Magnetization vs field at 2 K, 125 K and 300 K C/T vs T 2 at low temperatures and C vs T K The unit cell of Sr 3 Ru 2 O A schematic drawing showing the critical end point Superconducting transition of tin at different temperatures Diagram of the detection coils for the ac susceptibility Picture of the coils for the ac susceptibility measurements The real part of the ac magnetic susceptibility The imaginary part χ of the ac susceptibility The real part of the ac susceptibility for 10.4 kbar and 12.8 kbar The temperature evolution of the ac susceptibility χ for 15.7 kbar, 16.7 kbar and 18.2kbar The critical field vs pressure and phase diagram The magnetization change M vs H and χ max vs P The magnitude of χ at the metamagnetic transition The temperature dependence of the metamagnetic transition xi

12 Chapter 1 Introduction This chapter describes important physical concepts and theories upon which my research projects are based. My first project involves FeCrAs, which is an unusual non-fermiliquid compound. So in the first section I will give a brief description of the phenomenological Landau Fermi-liquid theory the theory that lies at the heart of our understanding of the low temperature metallic states. Our focus is on the predictions of this theory with respect to the electronic specific heat, spin susceptibility and electrical resistivity. The introduction about this theory is quite basic. A fuller description can be found in the book by Pines and Nozieres [1]. My second project is investigating the quantum criticality of the metamagnet Sr 3 Ru 2 O 7. Therefore, in the second section, I will first describe the quantum critical point (QCP) associated with a second-order phase transition: properties of the classical second order phase transition are briefly reviewed, followed by a description of the characteristics of QCP, including the statistics it involves and the profound influence it has on the phase diagram. In the third section, I will start to explain the quantum phase transition most relevant to my research. A new kind of quantum critical point, called quantum critical end point (QCEP), which is associated with a first-order phase transition is introduced. The QCEP in metamagnetism is discussed. An example of the quantum metamagnetic critical point, i.e. the QCEP observed in Sr 3 Ru 2 O 7, is 1

13 Chapter 1. Introduction 2 presented. Again neither section 2 nor section 3 is a comprehensive review of the rich physics of quantum phase transitions. More details can be found in Refs. [2 4]. 1.1 Phenomenological Fermi liquid theory In a metal, some electrons are not bound to any atom, rather they are free to move around in the periodic potential of the crystalline lattice. In the independent electron approximation, in which electron-electron interactions are treated in a mean-field way, the single-particle eigenstates are Bloch waves ψ nk (r) where n, k are the band index and the wave vector, respectively. Note that in this section we omit the spin index, and include it in k. Electrons are fermions, so the Pauli exclusion principle dictates that a maximum of two electrons (one spin up and one spin down) can occupy one energy level (or Bloch wave state). So the ground state of an N-electron system is obtained by filling the N/2 stationary one-electron levels of the lowest energy. If there is at least one band not fully filled, then the solid is a conductor. The highest filled levels form a surface in the 3-d k-space, called the Fermi-surface S F, and the states on the Fermi surface have the same energy ǫ F, called Fermi energy. At finite temperatures, only a limited number of states that lie close enough to the Fermi surface will be excited to the unoccupied levels, making a partially-occupied shell of width k B T about ǫ F. The distribution function of the non-interacting electron system is described by f(ǫ) = exp[(ǫ µ)/k B T], (1.1) where µ is the chemical potential. At absolute zero, f(ǫ) is a step function, taking the value 1 for ǫ < ǫ F and 0 for ǫ > ǫ F ; see the black line in Figure 1.1. As T increases, the step is smeared out; the occupancy numbers for levels immediately below the Fermi surface become less than 1, but levels further down (more than k B T away from ǫ F ) are unchanged; for T T F, where T F is the Fermi temperature (T F = ǫ F /k B ), the distribution function looks like that shown by the red line in Figure 1.1. For temperatures small

14 Chapter 1. Introduction 3 Figure 1.1: Schematic diagram illustrating the Fermi-Dirac distribution for T = 0 (black) and a finite temperature T T F (red), where T F refers to the Fermi temperature. At T = 0, the occupancy number is 1 for levels below ǫ F, and 0 for levels above ǫ F. At finite temperatures, some electrons at levels immediately below the Fermi surface (S F ) are thermally excited to levels above S F, and the step function is smoothed out. compared with T F, the independent electron model predicts a linear-t electronic specific heat c v Tg(ǫ F ), and a temperature independent Pauli paramagnetic susceptibility χ P g(ǫ F ). Here g(ǫ F ) is the density of states at the Fermi surface. The independent electron model has successfully explained a wide range of experimental results on real metals. This is surprising because electrons are charged particles, so strong Coulomb interactions exist between them. Landau s Fermi liquid theory provides the explanation. Landau Fermi liquid theory was initially formulated for the description of the neutral Fermi liquid of 3 He. It is because the Coulomb interactions in metals are screened, and therefore become short-ranged, that Landau theory can also be applied to the interacting electron systems. This theory is based on the concept of the quasiparticle, which in the interacting electron system shares many similarities with the Bloch state.

15 Chapter 1. Introduction 4 Once we acknowledge that we are dealing with quasiparticles, not electrons, the concepts used for describing the independent electron system such as Fermi surface are still valid in the interacting system. The rules for constructing quantities like the electric current from the distribution function are found to be very similar to those of the non-interacting system. Fermi liquid theory assumes that a one-to-one correspondence between the eigenstates of an interacting system and those of the non-interacting system can be established by slowly (adiabatically) switching on the interactions. The ground state of an interacting system can be generated from some eigenstate of the non-interacting system. The excited states of an interacting system can be constructed as follows. Adding a particle to the ground state of the non-interacting system, and slowly turning on the interaction, the added particle will slowly perturb the particles in its vicinity. Once the interaction is completely turned on, the added particle moves together with the distortion of the surrounding particles created by the interaction. The entity of the dressed particle is called a quasiparticle. The state we just created is an excited state with one quasiparticle. With the same procedure, we can define a quasihole. The excited state involving several quasiparticles and quasiholes can be generated in the same manner. The quasiparticle obeys Fermi statistics, and has the same characteristics as a Bloch wave, and likewise is indexed by n and k. We can also define a quasiparticle Fermi surface. Both the interacting and the non-interacting systems have the same distribution function, bounded at T = 0 by a sharp Fermi surface. Quasiparticles do have features that do not appear in the non-interacting systems. The quasiparticle has a finite lifetime, which varies as the inverse square of the energy separation from S F. The quasiparticle concept makes sense in the immediate vicinity of S F because only in this region the quasiparticle has a sufficiently long lifetime for the slow switching-on method to work. This is generally not a problem if our interest is in low temperature physics, because in this limit only the states very close to S F contribute

16 Chapter 1. Introduction 5 to the electronic properties of a metal. Quasiparticles interact with each other. This makes the relation between the energy of the excited state E and the quasiparticle distribution function n k more complicated than in the non-interacting system. E depends on n k in a functional form, E[n k ]. If n k is sufficiently close to the ground state distribution n 0 k, the Taylor expansion can be used to obtain the excitation energy (which is relative to the ground state), E ex = E E 0 = k ǫ k δn k + 1 f 2 kk δn k δn k + O(δn 3 ), (1.2) kk where δn k = n k n 0 k represents the departure from the ground state, ǫ k is the first functional derivative of E, and the coefficient f kk, which is related to the interaction energy of the excited quasiparticles k and k, is the second variational derivative of E with respect to n k. The quasiparticle local energy ǫ k has the form ǫ k = ǫ k + k f kk δn k, (1.3) thus ǫ k also depends on the interactions. Replacing ǫ in equation 1.1 with ǫ k gives the equilibrium distribution function of the quasiparticle. The equations 1.2 and 1.3 play a key role in the development of the Landau theory. For these two equations to be valid in the interacting electron system, the interactions between quasiparticles must be treated as short-ranged, which can usually be justified by considering screening. Due to the existence of the interactions, the construction of the physical quantities from the theory becomes complicated. For instance, in constructing a transport theory, we need to consider the non-equilibrium electronic distribution function. In a non-interacting system, this deviation from the equilibrium distribution has no bearing on the form of the ǫ vs k relation; but for an interacting system, the relation between ǫ and k may well be altered because the quasiparticle energy depends on the configuration of other electrons. Therefore, the transport theory for an interacting system becomes more difficult to build. However, the thermodynamics in a simple nearly isotropic electron system are not

17 Chapter 1. Introduction 6 that complicated. The specific heat is given by c v Tg (ǫ F ) (1.4) where g (ǫ F ) is the density states of the quasiparticles at Fermi surface [1], which has the same form as the density states of Bloch electrons g(ǫ F ), but with the Bloch electron mass m being replaced by the quasiparticle mass m, i.e. g (ǫ F )/g(ǫ F )=m /m. The spin susceptibility is found to be χ P g (ǫ F )/(1 + F0 a ) (1.5) where the parameter F a 0 is a so-called Landau parameter, in this case the spin-antisymmetric quasiparticle-quasiparticle interaction averaged over the Fermi surface. Compared with the Bloch electron system, we see that the temperature dependence of c v and χ P of the quasiparticle system are the same to leading order: c v linearly depends on T and χ is almost temperature independent. In the interacting system, the quasiparticle acquires an effective mass which is usually larger than that of a free electron. In an interacting system, the resistivity due to electron-electron scattering at low temperatures can be analyzed as follows. Due to the exclusion principle, the quasiparticle scattering rate is dramatically reduced. This reduction occurs when the electronic configuration differs only slightly from its thermal equilibrium form as seen in the electrical transport process. The k-space available for the scattering is limited to a thin shell of width k B T around S F, as shown in Figure 1.2. This allows us to estimate the scattering rate in terms of temperature. The process where two quasiparticles in levels ǫ 1 and ǫ 2 scatter into levels ǫ 3 and ǫ 4 must conserve energy, therefore ǫ 1 + ǫ 2 = ǫ 3 + ǫ 4. (1.6) For the quasiparticle ǫ 1, the k-space available for it to choose the scattering partner ǫ 2 has a thickness of k B T; and the k-space for the scattered quasiparticle ǫ 3 also has a magnitude of k B T. Once ǫ 3 is chosen, ǫ 4 is fixed as required by energy conservation.

18 Chapter 1. Introduction 7 Figure 1.2: A diagram showing the limited k-space available for quasiparticlequasiparticle scattering at low temperatures. For illustration, we assume the system is isotropic. At low temperatures, due to the Pauli exclusion principle, only a limited number of electrons at levels close to the Fermi surface are excited to the levels above the Fermi surface, making a partially occupied shell of width k B T. Due to energy conservation, quasiparticle-quasiparticle scattering can only occur in this shell. This dramatically reduces the scattering rate.

19 Chapter 1. Introduction 8 So we obtain a scattering rate proportional to (k B T) 2. Taking the impurity and defect scatterings into account, the resistivity at low temperatures can be expressed as ρ = ρ 0 + AT 2, (1.7) where ρ 0 arises from the impurity and defect scatterings, and the temperature coefficient A contains the average effective mass of the quasiparticle. Note that equation 1.7 is only valid if the electron-phonon scattering can be neglected. The above description of Landau Fermi liquid theory is basic; for more details, see Ref. [1]. Applying Landau s Fermi liquid theory to anisotropic electron systems (in real metals this is certainly the case because the Fermi surface has the symmetry of the crystalline lattice) is much more complicated. In this kind of a system, the electronic specific heat and the Pauli-like susceptibility given above is in principle valid, but their expressions are no longer as simple. Generally speaking, in all the normal metals (i.e. those in which the quasiparticle representation is valid), the Landau theory predicts that c v shows a linear-t behavior, χ is almost temperature independent and ρ quadratically depends on temperature. Particularly, the T-square dependence of the resistivity is regarded as the hallmark of Fermi liquid behavior. These behaviors in c v, χ and ρ have been observed in many metals. Recently, a new class of f-electron heavy fermion materials has been reported to show non-fermi-liquid (NFL) behaviors, in which these power laws of the Landau Fermi-liquid are violated. In most of these cases, the resistivity is metallic in the sense that it falls with decreasing temperature, often in the vicinity of a quantum phase transition [5]. However, there are a few non-metallic non-fermi-liquids that show the following temperature dependencies for T T 0, where T 0 is a characteristic temperature: ρ(t) 1 at/t 0, C(T)/T ( 1/T 0 ) ln(t/bt 0 ), and χ(t) 1 c(t/t 0 ) 1/2. These materials are typically derived by substitution from a parent material, e.g. doping a nonmagnetic element into Ce or U intermetallics, and are therefore crystallographically disordered due to random site occupation or metallurgical imperfections, as seen, for example,

20 Chapter 1. Introduction 9 in M 1 x U x Pd 3 (M = Y, Sc) and UCu 5 x Pd x [6, 7]. However, undoped non-metallic NFL systems have also been discovered, e.g. CeCuAs 2 [8], CeRh 2 Ga [9], UCu 4 Pd [10], URh 2 Ge 2 [11]. Investigations of the undoped materials show that NFL behavior can be an intrinsic property for some metals; its origin is not well understood, although it is often attributed to scattering from quantum fluctuations of disordered magnetic clusters even in the apparently stoichiometric materials. For example, the appearance of Griffiths phase near the quantum critical point (QCP) has been proposed to explain the NFL behaviours, such as ρ(t) = ρ 0 AT, observed in CeRh 2 Ga [9]. In all of these non-metallic NFL systems, ρ, C and χ seem to be closely correlated, in the sense that if one of them deviates from Landau Fermi-liquid theory, then the others generally also show strong NFL signatures. For example, the mildly disordered Ce compound CeRh 2 Ga shows a non-fermi-liquid resistivity which has a linear-t dependence below 5 K. The thermodynamics, the specific heat and magnetic susceptibility, also deviate from the Fermi liquid theory predictions at low temperatures: the specific heat divided by temperature, instead of saturating according to the Fermi liquid theory, rises linearly with decreasing temperature on the log-log scale; the susceptibility is not Pauli-like, as on the log-log scale it also increases linearly as T falls. In comparison to f-electron systems, d-electron non-metallic NFL materials are scarce outside of the cuprate family of superconductors, which famously show this behavior on the underdoped side of their phase diagram [12]. Chapter 4 describes a detailed investigation of the pure, d-electron compound FeCrAs, which I have grown in a singlecrystalline form. I will show that this material has an exotic non-metallic NFL state. In this stoichiometric 3-d electron compound, ρ follows a power law ρ(t) = ρ 0 AT 0.6 at low temperatures, exhibiting non-metallic NFL behavior, however, the specific heat and magnetic susceptibility are FL-like. This combination of NFL transport with FL thermodynamic properties may represent a new class of NFL materials. In chapter 4, I will give more details about this compound.

21 Chapter 1. Introduction Quantum critical point Phase transitions are ubiquitous: from the transformation of water between ice, liquid and vapor to the emergence of superconductivity in a cooled metal to the formation of the universe. Phase transitions can be classified into different categories. Those that involve latent heat are usually called first-order transitions; and those that do not involve latent heat are called second-order phase transitions or continuous transitions. In this section, we are only concerned with continuous transitions. A continuous phase transition can be characterized by an order parameter: a thermodynamic quantity which is zero in the disordered phase and non-zero in the ordered phase. Take the ferromagnetic transition (T c = 770 C) in iron for example: the paramagnetic phase (T > T c ) in which there is no net magnetization is the disordered phase; and the ferromagnetic phase (T < T c ) in which spontaneous magnetization appears is the ordered phase. The order parameter, in this case, is the magnetization. Upon the appearance of the order parameter, a symmetry associated with the disordered phase is broken. Even though the average of the order parameter is zero in the disordered phase, its fluctuations are not. As the critical temperature is approached, the spatial correlation of the order parameter becomes long-ranged. At the critical point, the correlation length ξ diverges. The correlation time τ c, which corresponds to the time scale for decay of the fluctuations, also diverges at the critical point and its relation with the correlation length can be described as τ c = ξ z, where z is called the dynamic exponent. This implies that there is a characteristic frequency ω c associated with the critical fluctuations that vanishes at the transition, a phenomenon known as critical slowing down. Different phase transitions can have different order parameters and origins. However, it was found that these phase transitions often share many fundamental characteristics. The specific heat when the helium becomes superfluid (lambda transition) has exactly the same power-law dependence on temperature as that of a planar ferromagnet near its magnetic ordering transition. This universality of critical phenomena is one of the

22 Chapter 1. Introduction 11 most remarkable features of continuous phase transitions. The critical behavior at a continuous phase transition is completely characterized by a set of critical exponents (e.g. the asymptotic power law exponents for the specific heat (α), the order parameter (β), the susceptibility (γ), etc.) which only depend on the symmetries of the Hamiltonian and the spatial dimensionality. The mechanism behind this universality is the divergence of the correlation length at the phase transition. Phase transitions at finite temperatures are driven by thermal fluctuations, that is, the fluctuations that destroy the long-range order in these phase transitions are thermally excited. If a finite temperature phase transition is suppressed by a non-thermal tuning parameter P (such as pressure, magnetic field, or chemical doping) to absolute zero as shown by the solid line in Figure 1.3, then the transition can no longer be thermal. In this case, the transition is driven solely by the quantum fluctuations. For this reason, the T c = 0 phase transition is called a quantum phase transition, and the point at which it occurs is a quantum critical point (QCP). The Hamiltonian of the quantum system contains two competing parts: one part favors an ordered phase, the other tends to destroy the order. The coefficient of the latter over that of the former gives the coupling constant K, which is the measure of the strength of the quantum fluctuations. At absolute zero, as K is tuned up, the quantum fluctuations grow; at a critical value which corresponds to P c in the phase diagram illustrated in Figure 1.3, the ordered phase is destroyed. Here K plays the role that the physical temperature does in a finite-t transition. At finite temperatures, rising K lowers T c because the system needs less and less thermal energy to destroy the ordered phase; see the solid line in Figure 1.3. Analysis of a quantum phase transition requires the use of quantum statistics. Unlike classical statistics where the dynamic (kinetic) part and static (potential) part of the Hamiltonian are decoupled, in a quantum system the dynamic and static part of the Hamiltonian usually are coupled due to the fact that momentum and coordinates

23 Chapter 1. Introduction 12 Figure 1.3: Schematic phase diagram in the vicinity of a quantum critical point. The solid line is the phase boundary between the ordered and disordered phase. The hatched area marks the region dominated by classical fluctuations. The critical behavior asymptotically close to the transition is entirely classical; however, the quantum fluctuations dominates the physics outside the hatched area at small T c ; measurements along path (a) will reveal the quantum to classical crossover. Right above the quantum critical point (QCP) is the quantum critical region noted for the non-fermi-liquid resistivity in metallic systems. The physics in this region is controlled by the QCP.

24 Chapter 1. Introduction 13 operators no longer commute. The quantum partition function can be calculated via Z = n < n e βh n >. (1.8) The canonical density operator e βh looks exactly the same as a time-evolution operator e ihθ/ h with Θ equal to an imaginary value i hβ where β = 1/k B T. If an imaginary time is introduced as an additional dimension into the system, the partition function of a quantum system can be mapped to that of a classical system. It should be noted that the additional temporal dimension scales as the zth power of a length as indicated in the relation between the correlation length and correlation time τ c = ξ z. Only when space and time enter the theory symmetrically, does z (known as the dynamic exponent) take the value 1. Therefore, a d dimensional quantum system is equivalent to a classical system with d+z spatial dimensions, except that the extra dimension is finite with a thickness hβ. For the quantum phase transition, since it occurs at absolute zero, the temporal dimension goes to infinity and we get a truly effective d+z dimensional classical system. For finite temperature transitions, asymptotic thermodynamics can be captured by doing classical statistics. However, at very low temperatures the influence of quantum fluctuations extends to regions very close to the phase boundary. Therefore, an experiment performed along the path (a) of the phase diagram shown in Figure 1.3 will observe a crossover from d+z dimensional behavior (quantum behavior) to d dimensional behavior (classical behavior) at places close to the phase boundary. The thickness of the temporal dimension hβ sets the condition for the quantum to classical crossover: if the correlation time τ c is smaller than hβ, the system does not realize that the temperature is finite and it is effectively d+z (quantum in nature); once τ c > hβ, the system knows that it is effectively d dimensional (classical). The condition for the system behaving classically, i.e. τ c > hβ, can be rewritten as hω c < k B T, (1.9)

25 Chapter 1. Introduction 14 where ω c is the characteristic frequency of the phase transition, which vanishes as T T c. Equation 1.9 indicates that for all finite temperature phase transitions the critical behaviors asymptotically close to the transition are entirely classical. The hatched area in the phase diagram of Figure 1.3 marks the region where the d dimensional classical treatment is adequate. There is a special area called quantum critical in the phase diagram, in which both types of fluctuations are important. Its boundaries are determined by an equation similar to equation 1.9, i.e. hω c k B T. The physics in the quantum critical region is controlled by the quantum critical point: the system looks critical with respect to P due to the quantum fluctuations but is driven away from criticality by thermal fluctuations. In this region, physical quantities exhibit behavior markedly different from that on either side. For example, both the ordered and the quantum disordered states are typically Fermi liquid in which the resistivity shows a quadratic temperature dependence at sufficiently low temperature; in the critical region, however, a departure from T 2 resistivity, for example a quasi-linear resistivity, is usually observed. Even though matter can never be cooled down to the quantum critical point, drastic effects are felt well before this point is reached. Besides showing the non-fermi-liquid behavior in the quantum critical region, as the QCP is approached by adjusting the tuning parameter, the low temperature limiting behavior is that both the Sommerfeld coefficient of the specific heat and the T coefficient in the the resistivity ρ = ρ o + AT α go toward infinity, indicating that the electron effective mass diverges. What makes quantum phase transitions even more fascinating is that electrons very often appear to re-organize themselves into a new stable phase of matter under the influence of the intense critical fluctuations of a quantum critical point. As such, quantum criticality provides an important new route for the discovery of new phases of matter. The most famous example is exotic superconductors observed around QCPs which inspire the hope of one day achieving room-temperature superconductivity, because the fluctuations associated with

26 Chapter 1. Introduction 15 the quantum critical points are believed to have significant influence on the properties of the strongly correlated systems even at room temperature and above. Quantum criticality still is a field of intensive research. For more details about quantum criticality, refer to Refs. [3, 4, 13]. 1.3 Metamagnetic quantum critical point In the last section, we showed that the key step in realizing a quantum critical point is to produce a phase transition with diverging correlation length at T = 0. At first sight, a standard first-order phase transition would not be a good choice because the susceptibility does not diverge. However, the critical point at which the phase boundary of a first-order transition terminates, for example the end-point at 647 K and 218 atm of the liquid-gas transition line in the P-T phase diagram of water, exhibits properties of a continuous phase transition that could lead to quantum criticality. If this critical point is suppressed to absolute zero by some non-thermal tuning parameter, a new type quantum critical point, called a quantum critical end point (QCEP), could be produced. A QCEP is thought to be observed in a few metamagnetic systems such as Sr 3 Ru 2 O 7. Metamagnetism is empirically defined as a sudden non-linear rise in magnetization at some finite applied field. This can occur in several circumstances. Under a large magnetic field, the spins in an antiferromagnetic insulator may flop and produce a sudden change in the magnetization. Metamagnetism first came into use to describe this situation. A form of metamagnetism more relevant to itinerant systems was first proposed by Wohlfarth and Rhodes [14]. They pointed out that a strongly enhanced paramagnet on the verge of ferromagnetism might undergo a phase transition involving a magnetization jump due to the exchange splitting of the Fermi surface at some finite applied field. A mean field explanation is given below. The molecular field theory says that all spins feel an average exchange field λm

27 Chapter 1. Introduction 16 produced by all their neighbors. In a metal, the molecular field can polarize the electrons, and the resulting magnetization of the electrons M would in turn be responsible for the molecular field. This is a positive feedback, and the process saves some potential energy. However, polarizing the electron gas costs kinetic energy because some electrons are moved to higher energy levels. If the energy reduction due to the interaction between the molecular field and the magnetization of the polarized electrons outweighs the kinetic energy cost, this positive feedback will lead to ferromagnetism. The condition for this spontaneous ferromagnetism to occur is Ug(ǫ F ) 1 (which is known as the Stoner criterion) where U is the Coulomb energy or exchange energy and g(ǫ F ) is the density of states at the Fermi energy per site for one spin direction. If the Stoner criterion is not satisfied, spontaneous ferromagnetism will not occur. But the susceptibility may be enhanced; in this case, the susceptibility is given by χ = χ P 1 Ug(ǫ F ), (1.10) where χ P is the Pauli susceptibility. We see that χ P is enhanced by a factor (1 Ug(ǫ F )) 1. If a system has a large enough Ug(ǫ F ) parameter but not large enough to cause spontaneous ferromagnetism, the system is said to be on the verge of ferromagnetism. For a paramagnet on the verge of ferromagnetism, if its Fermi level sits at a minimum of the density of states, then the magnetic field will raise the density of states at the Fermi level due to Zeeman splitting, see Figure 1.4. The susceptibility, according to equation 1.10, may diverge at a critical field, and give rise to a sudden change of the magnetization, i.e. the metamagnetic phase transition. Note that the density of states presented in Figure 1.4 is for illustration only. In a real system, the density of states usually is complicated, and the evolution of the density of states with field is not as simple as shown here. Paramagnetic metals in which metamagnetism is seen usually have the following characteristics: an enhanced susceptibility and a maximum in the zero-field χ(t). These characteristics are inherent in the strongly exchanged-enhanced

28 Chapter 1. Introduction 17 Figure 1.4: Schematic diagram showing the density of states change as the external magnetic field varies. At H = 0, the density of states of the spin-up and spin-down bands overlap each other, as shown by the solid black line; the thin dashed black line represents the Fermi energy. As H increases, the two bands splits with the spin-up (parallel to H) moving downward (dashed red line) and the spin-down (blue dashed line) moving upward. The density of states at the Fermi level, which is the sum of the density of states of the two bands, increases. paramagnets, and have been seen in many itinerant metamagnets such as UCoAl, UPt 3, Y(Co 1 x Al x ) 2 and Co(S 1 x Se x ) 2 [15 17]. Figure 1.5 shows the suggested generic phase diagram of a ferromagnetic metal which shows metamagnetism when its ferromagnetism is made unstable by some tuning parameter [18, 19]. From this phase diagram, we can see how the quantum critical end point is realized. This phase diagram has been applied, for example to CoS 2 [17] and MnSi [18]. At H = 0, a second order phase transition to a spontaneously ordered ferromagnetic state occurs at T c. T c is depressed by the tuning parameter P which can be pressure or chemical doping, and the paramagnetic to ferromagnetic transition becomes first-order as P > P. If the magnetic field is present, it will induce magnetization in the system and

29 Chapter 1. Introduction 18 Figure 1.5: A generic phase diagram of a ferromagnetic metal that can be tuned away from ferromagnetism and show metamagnetism. At H = 0, the ferromagnetic phase transition is second-order for P < P and first-order for P > P, where P is the tuning parameter. When H 0, the surface of the first-order metamagnetic transition appears and extrudes in the H direction for P > P. The top of the surface is delimited by a line of critical points T (P, H). Below T, e.g. along b (an isotherm with T < T ), the magnetization jumps discontinuously as the sweeping field passes through the surface; if T > T, as in line a, there is no discontinuity, only a cross-over.

30 Chapter 1. Introduction 19 the symmetry associated with the paramagnetic phase is broken. Therefore, a secondorder phase transition can no longer take place in this system. However, since a first-order phase transition does not have the symmetry constraint, the first-order transition line extrudes into a surface in (P, H, T) of first-order transitions, and produces a tricritical point. The emergent surface is represented by a green sheet in the phase diagram. At the green sheet, the magnetization shows a first-order metamagnetic jump as a function of applied magnetic field H. The top of the sheet is delimited by a line of critical points T (P, H), which separates the first-order jump from a continuous super-linear cross-over behaviour in the M vs H curve. This is illustrated in Figure 1.6: as H is increased along the red line labelled b, which shows an isotherm with T < T, the magnetization jumps discontinuously when the line passes through the surface; on the other hand, if T > T, as in line a, there is no discontinuity, only a cross-over. At T the slope, dm/dh, should diverge. The line of critical points terminates at a particular point (P q, H q ) at absolute zero. This point is the quantum critical end-point [20]. So keeping the tuning parameter at P q, it is possible to use the magnetic field to tune in and out of quantum criticality. As mentioned above, CoS 2 [17] and MnSi [18] are thought to follow this scenario. In these systems, the relation between ferromagnetism and metamagnetism can be clearly seen: as T c is pushed down by pressure, the second-order ferromagnetic transition changes into first-order and the ferromagnetism becomes unstable, then metamagnetism appears in a certain temperature range above T c. Some paramagnets sit on the verge of ferromagnetism under ambient conditions, corresponding to the region of P > P c in the phase diagram of Figure 1.5. In these systems, metamagnetism may be seen by adjusting only one parameter, i.e. the magnetic field. Sr 3 Ru 2 O 7 is regarded to be one of them; at ambient pressure, for magnetic fields applied in the ab-plane, Sr 3 Ru 2 O 7 is believed to lie on the generic phase diagram, in the (H,T) plane roughly where the two red lines a and b lie in Figure 1.5. For H ab, in high purity samples (ρ 2.4 µω cm) a first-order metamagnetic phase transition is observed

31 Chapter 1. Introduction 20 Figure 1.6: Typical field dependence of the magnetization in a metamagnetic system. Below the critical temperature T (see Figure 1.7), M shows a jump at the critical field H m (red line). Above T, M shows a superlinear rise at H m (black line). Figure 1.7: Schematic phase diagram illustrating the critical point of the first-order metamagnetic phase transition in Sr 3 Ru 2 O 7 for H ab. The critical temperature T is 1.25 K for a sample with a purity ρ 2.4 µω cm. at 4.9 T below 1.25 K. This transition has a close analogy to the liquid-gas transition in water, with the difference in the magnetizations of the low and high field states play a role analogous to the difference in density of vapor and liquid water. The critical point for this metamagnetic transition is at (H M,T ) with H M = 4.9 T, T = 1.25 K. The schematic T-H phase diagram of the metamagnetic transition is presented in Figure 1.7. Below the critical temperature T, the metamagnetism is first-order; above T, the metamagnetism becomes a crossover. Interestingly, unlike the usual metamagnetic transitions, the metamagnetic transition field H mmt in Sr 3 Ru 2 O 7 moves to lower field as the temperature increases. The significance of this unusual temperature dependence of H mmt (T) will be discussed in chapter 5. The low critical point temperature makes it a good candidate for realizing the metamagnetic QCEP. Grigera et al. [21] showed that as the field is rotated from the ab-plane toward c-axis, the critical point falls toward zero. The QCEP is realized for H within 10 degrees of c-axis. In this case the tuning parameter

32 Chapter 1. Introduction 21 is the field angle. For the metamagnetic phase transition, the high magnetization state has the same symmetry as the low magnetization state. So there is no symmetry breaking in the metamagnetic transition. This characteristic distinguishes the QCEP from the QCP for which the symmetry breaking is a necessary ingredient. Signatures of a QCP, such as non-fermi-liquid resistivity, logarithmically divergent specific heat divided by temperature C/T, have been observed near the QCEP in Sr 3 Ru 2 O 7 [20, 22]. In ultra-pure Sr 3 Ru 2 O 7 (ρ < 0.5 µω cm), a novel phase characterized by an enhanced anisotropic resistivity is observed in the vicinity of the QCEP. Significant interest has been focused on exploring the nature of this new phase [20, 21, 23 33]. However, so far neither the microscopic mechanism of the novel phase nor the properties of the metamagnetic QCEP are well understood. Metamagnetism is also sensitive to pressure. So it would be interesting to explore the metamagnetic quantum criticality of Sr 3 Ru 2 O 7 under hydrostatic pressure. The details of our study of the pressure dependence of the metamagnetic transition of Sr 3 Ru 2 O 7 will be given in chapter 5.

33 Chapter 2 Design and development of experimental instruments My supervisor, Prof. Stephen Julian, joined the Physics Department of the University of Toronto one month before I began my Ph.D program, we (Prof. Julian, our post-doc Alix McCollam, one other graduate student Patrick Rourke and myself) spent quite a lot of time building the lab. In this chapter, I will describe my two major works in the lab: designing the clamp pressure cell and developing the relaxation-time calorimeter. For the former, I will give a brief description about the features of the cell and how to use it; for the latter, I will present the principle of the relaxation-time calorimetric technique, the data analysis method and the test results. 2.1 Nonmagnetic clamp cell Pressure as a clean and controllable means of altering the interatomic spacing of a given material has been used increasingly in modern condensed matter physics research. For instance, in the area of quantum criticality, pressure has become one of the main tuning parameters used in bringing a phase transition at finite temperature to a quantum critical point. These studies are often undertaken at low temperatures, and frequently 22

34 Chapter 2. Design and development of experimental instruments 23 involve the use of magnetic fields. So a multipurpose pressure cell is usually made of nonmagnetic materials which have good mechanical properties at low temperatures. Due to the limitations of the strength of these materials, it is not easy to attain high pressures. A simple piston-cylinder clamp cell can only reach a few 10 s kbars. To realize a pressure greater than 50 kbar, a sophisticated pressure cell such as anvil cell is required. However, the limited sample space in such devices is often a great inconvenience. For low pressures, it is more convenient to employ a piston-cylinder clamp cell. One of my projects is studying the metamagnetic quantum criticality in Sr 3 Ru 2 O 7 under hydrostatic pressure. For this study, relatively low pressures of <20 kbar are needed, so we decided to use a clamp cell. The main body of the cell was designed by myself and made by the machine shop at the Physics Department of the University of Toronto. Other components of the cell, except for the tungsten carbide piston, were made by myself. In the next few sections, I will describe the features and use of this cell Design of the cell A diagram of the pressure cell is presented in Fig When using it, the teflon cap will be filled full of a fluid which serves as pressure transmitting medium, the sample will be connected to the electrical leads and immersed in the fluid. Applying a force on the Pusher (this is usually done by a hydraulic ram), the piston will press the teflon cap, and pressure will build up in the compressed teflon cap assuming it is sealed. The locknut is used to lock the pressure inside the cap. At the desired pressure, the cell can then be removed from the ram and carried to some experimental apparatus. 1) selection of materials Since we are going to use the pressure cell in high fields, the cell cannot have a magnetic part. For the body of the cell which is subject to large tensile stresses, we chose to use Be-Cu. This material has found many applications in cryogenic pressure cells, particularly, in piston-cylinder cells. The advantage of using this material is that the material

35 Chapter 2. Design and development of experimental instruments 24 Figure 2.1: Cross-sectional view of the clamp pressures cell. The components are made of Be-Cu, unless otherwise labelled. The diameter of the bore is 4 mm.

36 Chapter 2. Design and development of experimental instruments 25 has fcc crystal structure. It remains strong in tension and tough when cooled to low temperatures. On top of that, Be-Cu has relatively low specific heat at low temperatures, making it easier to cool the device down to very low temperatures [34]. Two parts of this cell are subject to large compressive stress instead, therefore they require large hardness. One is the pushing rod; the other is the piston over the teflon cap. The former is made of drill rod; the later is made of tungsten carbide. The tungsten carbide has nickel binder, rather than cobalt, because nickel WC is less magnetic. 2) features of the cell We take a two-layer design for this pressure cell. Both inner and outer layers are made of Be-Cu. The reason is that the boundary between the two layers can inhibit the propagation of any cracks from the core to the outer surface. Another feature is that the feedthrough has threads on the end close to the Locknut. The purpose of the threads is for taking out the feedthrough without damaging the sample. Once used under high pressure, the feedthrough will get stuck in the cell. If pushing it out from the other end of the cell, the sample would be destroyed. The threads allow one to screw a hollow cylinder on the feedthrough and pull it out Components made by myself The teflon cap was made on the lathe machine by myself. Starting from a teflon rod, a hole about 100 µm under the size of the tip of the feedthrough was drilled first. Then the rod was machined down until it became too thin to stand the push from the tool. After that, the cap was cut off from the rod and pushed on a steel rod, and was finished with the steel rod clamped on the chuck of the machine. This finishing step demands great care, each time removing from the cap a tiny amount of material, otherwise the cap may stop turning with the steel rod and your work is ruined at the last moment. The finished cap should have a force fit on the tip of the feedthrough, and a light press fit to the bore. The top sealing pad and lower sealing ring were also made on the student machines

37 Chapter 2. Design and development of experimental instruments 26 by myself. Ideally, the top sealing pad should be made of Be-Cu. Since we could not get that material in our lab, both these sealing parts were made of copper. They turn out to work fine, at least to a pressure of 18 kbar. Since the upper sealing pad will expand to accommodate a large bore under pressure, it does not have to be a tight fit initially. The lower sealing ring, on the other hand, should have a press fit onto the feedthrough. The tungsten carbide piston was ordered directly from a manufacturer. All parts made of Be-Cu need to be hardened before they can be used. The heat treatment was applied using an electrical furnace. The process was as follows: 26 C 1 hour 321 C (dwelling 2.8 hours) 7 hours 30 C. After this treatment, the hardness of Be-Cu components increased from C 36 to C 42. The Pusher is made of drill rod which also needs to be hardened. I made this piston on the lathe machine in the student machine shop, and then handed it over to a technician for hardening. The electrical wires were sealed into the bore of the feedthrough using Stycast Vacuum degasing of the Stycast, in order to remove trapped air, is very important in achieving high strength. If there are some bubbles of air left in the Stycast, the wires and Stycast can blow out when the cell is pressurized. The loom of the electrical wires should be protected by a rubber tube where they pass through the lower lock-nut, in our case we use heat-shrink to isolate it from the metal parts of the cell. When cooling down the cell to very low temperatures, it is necessary for the cell to be isolated from the inner vacuum can (IVC) of the fridge. For this purpose, I made the centering mechanism. It was made of brass, and composed of two parts, see Figure 2.5. On the left hand side of the cell is the bottom part with its central pin isolated from the outer ring by fishing line. This part will be placed at the bottom of the tail of the inner vacuum can (IVC), and once the pressure cell is lowered into the IVC, the cell will sit on the central pin of the centering mechanism. The top part of the centering mechanism has a similar structure to the bottom one. It is fixed on top of the pressure cell to make sure that the cell will not cant and touch the wall of the IVC. This centering mechanism

38 Chapter 2. Design and development of experimental instruments 27 Figure 2.2: Picture of the clamp cell and the centering mechanism. The device sitting on the left of the cell is the bottom centering mechanism, the one on top of the cell is the upper centering mechanism. worked well, it enabled us to cool the cell down to 70 mk Use of the cell When wiring the sample to the feedthrough, one should leave enough space for the distortion of the teflon cap under pressure. The total length of the cap is 2 cm, one should leave a room of at least 1/3 of that length. After the feedthrough being inserted into the bore of the cell, the lower Locknut (referring to Figure 2.1) should be tightened firmly. Then other parts go into the cell in this order: the top sealing pad, the tungsten carbide piston and the top Locknut. Under the hydraulic ram, pressure should be raised slowly, and once in a while the top Locknut needs to be tightened; to reduce the pressure, just reverse the procedure. When taking apart the cell, the feedthrough should be pulled out using the pre-built stainless steel pipe which has threads on its inner wall and can be screwed onto the feedthrough. The sample will come out intact with the feedthrough.

39 Chapter 2. Design and development of experimental instruments 28 the teflon cap may be left in the bore of the cell. In this case, one can use a rod to push the cap out. The teflon cap may be damaged by the pushing, even if it is not, it is not recommended to use the teflon cap a second time, because it will not seal as safely as a new cap. 2.2 Thermal relaxation calorimetry Heat capacity provides a lot of information about the properties of a material. It allows us to access theoretical quantities such as entropy, internal energy (via S = S 0 + T 0 C v(t) dt T and U = U 0 + T 0 C v(t)dt ). It is also an important channel for obtaining information about lattice, electronic and even magnetic properties of a material [35]. At low temperatures, the specific heat at constant pressure in a paramagnetic material can be written as C P = γt + βt 3. The linear-t term is the specific heat of the electrons, the coefficient γ is related to the electronic density of states at the Fermi energy g(ǫ F ) by g(ǫ F )(1 + λ) = γ, where λ is the coupling strength of the electrons both to lattice phonons and to other electrons; the cubic term is associated with the lattice phonons, its coefficient β can be related to the Debye temperature through θ D = (1944/β) 1/3 10. Specific heat has also been used to study phase transitions. For instance, in a magnetic material, the entropy at the magnetic ordering transition temperature, S = T mag+ T T mag T C/TdT, reveals the spin state s of the ordered moment via S = Rln(2s+1), where R = J/mol K. From the practical point view, materials used in construction of refrigeration systems and crystats must be characterized thermally, so knowledge of specific heat is crucial for any successful thermal design. Prior to 1968, heat capacity had been measured according to the classical definition: C P = lim T 0 ( Q/ T) P (2.1) where Q is the amount of heat applied to the sample and T is the temperature rise. At a given temperature, a pulse of heat was sent to the sample and the temperature change

40 Chapter 2. Design and development of experimental instruments 29 was recorded. This technique is called the adiabatic method. This method depends on good thermal isolation of the sample, which implies that sample of large enough size is required in order to minimize the effects of stray heat leaks. Low temperature modern adiabatic calorimeters were introduced about a century ago [36], since then the adiabatic calorimeters had been employed to measure the heat capacities of large samples ( 10 g). With improved instrumentations and refined experimental methods, physicists had been able to measure smaller and smaller samples. In 1963, Morin and Maita published the low temperature specific heat data measured on samples as small as 100 mg [37]. The operation temperature range and the resolution of the adiabatic calorimeter had also been significantly improved. By 1965, adiabatic calorimeters were already able to measure the heat capacity of a material to an accuracy of 0.5% and many of them could be operated over a wide temperature range ( K) [36]. In spite of this progress, the adiabatic method encounters difficulties when the measurement involves a very small sample or very low temperatures, under these circumstances, the amount of heat required to measure the heat capacity would be relatively small, the heat leaking through the electrical leads of either the heater or the thermometer becomes significant, and adiabatic calorimetry is no longer an appropriate choice. To overcome these difficulties, two relatively new methods have been developed, the ac method and the relaxation time method. Instead of minimizing the heat loss, these techniques exploit heat leaks to advantage. In 1968, Sullivan et al. published a landmark paper describing how to measure small samples using the ac method [38]. They showed that if an ac current of frequency ω is sent to the sample heater, the thermometer on the sample will display an ac temperature, with its magnitude described by T ac = P 0 2ωC (1 + 1 ω 2 τ ω 2 τ const) 1/2 (2.2) where P 0 is the magnitude of the ac power to the heater, τ 1 is the sample-to-bath relaxation time which describes the speed for the sample to reach an equilibrium state with the bath, τ 2 is the combination of the relaxation times of heater-to-sample, thermometer-

41 Chapter 2. Design and development of experimental instruments 30 to-sample and the sample internal thermal equilibrium. The above equation is only valid under the condition that the period of the ac power is much smaller than τ 1, but much larger than τ 2. The major strength of the ac heat capacity technique lies in its ability to detect a small change in heat capacity (with an attainable sensitivity of 0.04% [38]). Concerning the absolute accuracy, it is no better than the traditional methods. The ac method has frequency limitation, i.e. 1 τ 1 < f < 1 τ 2, that makes it inconvenient to use at extremely low temperatures because of the high thermal resistances (between the heater, thermometer and the sample) encountered there. These high resistances lead to large τ 2 that would require operating the ac calorimeter at undesirable frequencies. Above all, the τ 2 factor is more difficult to determine and correct for using ac method than using the relaxation time method. The relaxation time method (RM) was introduced by Bachmann et al. in 1972 [39]. The schematic diagram of a relaxation-time calorimeter is shown in Figure 2.3 (a). This design involves a constant temperature bath connected to the sample with a weak thermal link. If a constant current is sent to the heater, the temperature of the sample will increase until the power into the sample (from the heater) is equal to the power out of the sample (through the thermal link between the platform and the bath). When the heater is turned off, the temperature of the sample decreases toward the bath temperature. A typical temperature curve of this up and down process is presented in Figure 2.4. The temperature evolution process, not the steady state, is used to extract the heat capacity. The thermal link in the diagram represents the electrical wires to the heater and thermometer and the heat sinking wire which is sometimes added between the platform and the bath to shorten the time for the sample to reach equilibrium with the bath. This technique has the advantage that it is easy to signal average, at a given T 0, numerous up and down cycles to improve the signal-to-noise ratio. In addition, it is relatively easier to handle a large τ 2 effect with the RM method than the ac method, so at temperatures below 1 K where τ 2 is large, RM is superior to the ac method.

42 Chapter 2. Design and development of experimental instruments 31 Figure 2.3: (a): schematic diagram of a relaxation-time heat capacity cell. (b): sketch showing the τ 2 effect between the sample and the platform. The thermometer and heater are included into the platform. τ 1 and τ 2 are the platform-to-bath and sample-to-platform relaxation times, respectively. Due to the large τ 2, T is no longer equal to T.

43 Chapter 2. Design and development of experimental instruments Temperature (K) T Time (sec) Figure 2.4: Typical temperature behavior of the sample on the relaxation-time calorimeter: the first 500 seconds is for heater on, the second 500 seconds is for heater off. In one of my Ph.D projects, which will be described in detail in chapter 4, we observed that FeCrAs crystal showed a non-fermi-liquid resistivity, but a conventional specific heat C/T vs T 2 gives a straight line at low temperatures. The specific heat measurements were done using the PPMS, which can only go down to 2 K. We intended to extend the specific heat measurement to millikelvin temperatures in our dilution fridge. For this, we needed to build a heat capacity cell. We chose to build a relaxation time calorimeter under the consideration of the sample size and temperature range. In the next few sections, I will describe the theory, the construction, the data analysis and results of the heat capacity cell made by myself Mathematical model In most cases, the data of a RM calorimeter can be fitted directly using a simple mathematical model to extract the heat capacity value. The condition for this model to work

44 Chapter 2. Design and development of experimental instruments 33 is τ 1 τ 2. One can always modify the design of the RM heat capacity cell to make the simple mathematical model valid. In case that this condition is violated, one can use a technique called integration to analyze the data. Next, we will give some details about these two data-analysis methods. 1. Direct fitting If τ 1 is much larger than τ 2, the heat balance equation for the heater-on process can be written as power into sample = / t(heat in sample) + power out, (2.3) power into sample is equal to the power of the heater P, power out (the power loss to the bath) is described by κ(t T 0 ) where κ is the thermal conductance of the thermal link and T 0 is the bath temperature, the heat change in the sample is represented by C dt dt where C is the heat capacity of the sample. Note that here we ignore the addenda by which we mean the specific heat of the thermometer, heater and platform. If the specific heat of the addenda cannot be neglected, C includes C addenda as well. Inserting these expressions in the equation 2.3, we have P = κ(t T 0 ) + C dt dt. (2.4) Assuming that P, κ and C are all constants while the heater is on, the integration with respect to time t brings us the time dependence of the sample temperature T T 0 = P κt (1 e C ). (2.5) κ From the equation 2.5, we see that the sample temperature will saturate if the time is long enough, and the temperature rise will end up as T = P/κ. Therefore, we can rewrite equation 2.5 as T T 0 = T(1 e κt C ). (2.6) Similarly, we can obtain the sample temperature T and time t relation for the heater-off process. Setting P = 0 for equation 2.4, the integration with respect to t yields T T 0 = Te κt C. (2.7)

45 Chapter 2. Design and development of experimental instruments 34 During the measurement, usually only the heater power P, the sample temperature T and the time t are recorded. To find out C, we need to know κ. Since the heater power P and the temperature rise in the saturated-state T are known, κ can be calculated using κ = P T. (2.8) With these known parameters, one can fit the (T T 0 ) vs t curves and extract the heat capacity C. 2. Integration At very low temperatures it is likely that the condition for direct fitting, namely τ 1 τ 2, is violated. There is then an advantage to using a simple integrating technique. At very low temperatures, the thermal contact between the sample and the platform deteriorates more quickly. Let s assume that the platform-to-thermometer and the platform-to-heater thermal resistances can still be neglected, then during the up or down process the temperature measured by the thermometer is not the sample s, but the platform s, see Figure 2.3 (b). For the up process, the heat loss through the thermal link is given by t 0 κ(t(t) T 0); by the time that the temperature gets saturated and the sample reaches equilibrium with the platform, the total heat absorbed is C T where C = C sample + C platform + C heater + C thermometer ; the heat generated by the heater is Pt. From the law of conservation of energy, we have or Pt = TC + C = 1 T [Pt t 0 t 0 κ(t(t) T 0 ) (2.9) κ(t(t) T 0 )]. (2.10) Note that equation 2.10 can also be used in case the internal conductivity of the sample is poor. So when direct fitting fails, one can try the integration technique. It should be pointed out that integration is not the only technique one can use to solve a large τ 2 problem. Hwang et al. published a paper describing how to analyze

46 Chapter 2. Design and development of experimental instruments 35 the data when the large τ 2 arises from the large thermal contact resistance between the sample and the platform [40]. The theory used there is complicated compared to that presented above. However, this model encounters difficulties when the sample internal thermal conductivity gets poor. We do not intend to use this analysis technique, so it will not be described here Construction of the heat capacity cell To measure a small sample at very low temperatures, the calorimeter should have: 1) small addenda the background heat capacity including that of the platform, the heater and the thermometer; 2) good platform-to-heater and platform-to-thermometer thermal contacts. If the addenda is large such that the ratio of the sample heat capacity to the addenda heat capacity becomes small, it makes subtraction of the addenda difficult. If the thermal contact between the platform and thermometer is poor, it will be hard for us to know the temperature of the platform or the sample. This makes writing the heat balance equations difficult. Poor thermal contact between the platform and the heater also complicates the mathematical model. To make the addenda small, 50µm Cu foil was used for the 3 3 mm 2 platform of the heat capacity cell. Copper has the advantage of small specific heat and good thermal conductivity at low temperatures. A small RuO 2 chip resistor of size mm 3 and mass 0.5 mg was used as thermometer. The resistance of RuO 2 is very sensitive to temperature below 4 K, so it is among the best for making a low temperature thermometer. A flattened 2.5 mm manganin wire (12.9 Ω/ft) was used as the heater, its mass was 0.4 mg. (Note that I made two heat capacity cells, the heater sizes are different for these two cells. The size of the heater quoted here is for the cell used for measuring the UPt 3 sample. The one used for measuring FeCrAs has a heater twice as long, but 1/3 as wide.) The total mass of the platform, the thermometer and the heater is 5 mg. To get good thermal contact, it would be ideal to use an integrated circuit as a sample

47 Chapter 2. Design and development of experimental instruments 36 holder, and two differently doped resistive elements on it as heater and thermometer. As we do not have the facility for making this kind of platform, we chose to use copper foil. The thermometer and the heater were initially glued onto the the platform with GE varnish, isolated from the Cu foil by 12 µm mylar. Later we found that mylar did not provide a thermal contact as good as cigarette paper. We switched to use cigarette paper as the insulation, and found that this mounting method gives good thermal contacts down to 300 mk. Figure 2.5 shows a picture of the heat capacity cell. For a relaxation time calorimeter, it is important to select a proper relaxation time τ 1. If the τ 1 is too small, fast electronics is required and, even so, the measurement is still subject to errors because of the internal (within the system of the sample and the platform) temperature gradient caused by the rapidly changing temperature of the platform. If τ 1 is too large, the measurements will take an extremely long time. We decided to add a manganin wire in between the platform and the bath to adjust τ 1. This material has an electrical resistance that weakly depends on temperature. According to the Wiedemann-Franz law, its thermal conductance should linearly depend on T at low temperatures. The specific heat of the sample also has an approximately linear-t dependence at low temperatures, i.e. C γt, so the relaxation time τ 1 = C/κ would be more or less temperature independent. The 12.9 Ω/ft manganin wire was used for the heat sinking. Their lengths are 0.6 cm and 0.8 cm for the UPt 3 -sample cell and the FeCrAs-sample cell, respectively. A twisted pair of 50 µm Nb wires was used as the leads for both the thermometer and the heater (the lengths of the Nb wires are 0.8 cm 0.6 cm for the UPt 3 -sample cell and the FeCrAs-sample cell, respectively). Nb becomes superconducting at 9.2 K. The reason of using superconducting leads is two fold: first, a superconductor has poor thermal conductivity, so the thermal link will be dominated by the manganin wire, and τ 1 will have a weak temperature dependence; second, we want to know the input heat to the sample precisely, so it would be better if the leads of the heater do not generate

48 Chapter 2. Design and development of experimental instruments 37 Figure 2.5: Picture of the heat-capacity cell. The platform is made of 50 µm copper foil. Its size is 3 mm 3 mm.

49 Chapter 2. Design and development of experimental instruments 38 heat. The Nb wires extend from the thermometer and the heater to the body of the cell, where they were spot-welded to 80 µm diameter superconducting wires made of NbTi filaments in Cu/Ni matrix. This cell is designed for use at temperatures below 4 K. Above this temperature, the accuracy of the RuO 2 thermometer deteriorates with increasing temperature. The high temperature limit is actually set by the Nb wires; when these wires are in normal state, i.e. above 9.2 K, the relaxation time τ 1 will become so short that the cell can no longer be used Experiment I made two heat-capacity cells, one was mounted with two irregular-shape FeCrAs samples (see Figure 2.6), the other was mounted with a rectangular slab of UPt 1 3 sample (see Figure 2.7) for testing the design (the specific heat of UPt 3 is known, it is a heavy fermion material with an electronic specific heat hundreds of times larger than that of the copper, so the addenda of the cell can be neglected). The two cells were tightly screwed on the mixture chamber (M/C) so that the bath temperature of the cell would be equal to that of the M/C. The normal heat capacity measurement procedure consists of measuring the addenda first, then putting the sample on the platform and measuring their total heat capacity, the latter minus the former (addenda) will give us the heat capacity of the sample. However, what we want to know is if the C/T vs T 2 curve of FeCrAs still gives a straight line below 2 K, not the absolute value. Since the addenda is small compared to the heat capacity of the sample and the copper plate of the addenda has a linear in T temperature dependence at low temperatures, the addenda should not make the total heat capacity non-linear if the part from FeCrAs does not. To save time, we decided to skip the first step The instrumentation of the measurements is shown in Figure 2.8. In the heater circuit, a square wave voltage was generated by the labview program of the computer 1 I am grateful to Andrew Huxley at the University of Edinburgh for providing the UPt 3 sample.

50 Chapter 2. Design and development of experimental instruments 39 Figure 2.6: Two irregular-shape FeCrAs samples of total mass 14.0 mg were mounted on the platform with Apiezon N grease. The thermometer and the heater are on the backside of the platform. Figure 2.7: A rectangular slab of UPt 3 sample of mass 35.2 mg was mounted on the platform with Apiezon N grease. The thermometer and the heater are on the backside of the platform. and output by the PXI Box 2. This voltage was applied to the shunt resistor ( 1 kω) and the heater of ( 0.1 Ω for the UPt 3 cell, 0.3 Ω for FeCrAs cell). When the voltage is high, electrical current runs through the heater and the temperature of the sample rises; when the voltage is low (actually U = 0), the temperature of the sample falls exponentially. The temperature of the sample is monitored by the thermometer circuit using the lock-in SR830, the shunt resistor in this circuit is 1.04 MΩ. The RuO 2 chips on the two cells had not been calibrated; we calibrated them with temperature sweeping rates 5 mk/min and 2 mk/min for temperatures between 1 4 K and temperatures below 800 mk, respectively, using the calibrated RuO 2 thermometer on the mixing chamber; the time constant of the lock-in SR830 was set at 1 second. The R vs T data are shown in Figure 2.9. The millikelvin R vs T curves show that the thermal contacts between the RuO 2 chips and the platforms become poor below 2 I am very grateful to Patrick Rourke, who coded the labview program for my specific heat measurements.

51 Chapter 2. Design and development of experimental instruments 40 Figure 2.8: (a): the square wave voltage over the shunt resistor and the heater turns the heater on and off. (b): the temperature of the thermometer is monitored by the lock-in amplifier SR830.

52 Chapter 2. Design and development of experimental instruments (a) 100 (b) 20 UPt 3 cell thermometer 80 UPt 3 cell thermometer R (kω) 15 R (kω) T (K) T (mk) (c) FeCrAs cell thermometer (d) FeCrAs cell thermometer R (kω) 15 R (kω) T (K) T (mk) Figure 2.9: (a), (b): RuO 2 thermometer calibration curves R vs T for the UPt 3 cell; (c), (d): the R vs T curves for the RuO 2 chip on the FeCrAs cell. (b), (d) show that the thermal contacts between the RuO 2 chips and the platforms deteriorate below 300 mk and 200 mk for the FeCrAs cell and the UPt 3 cell, respectively. 300 mk and 200 mk for the FeCrAs-sample cell and UPt 3 -sample cell, respectively; these two temperatures set the low temperature limits for the two cells. The model used to describe these calibration curves is ln( 1 3 T ) = a i [ln(r R 0 )], (2.11) i=0 where R 0 is the room temperature resistance of the RuO 2 chip; the rest of the parameters are determined by fitting the calibration data. Once these parameters are known, equation 2.11 can be used in determining the temperature via the resistance of the RuO 2 chip.

53 Chapter 2. Design and development of experimental instruments 42 The electronic setup for this measurement is simple. In order to measure the the fast-changing temperature in the vicinities of the heater-on and heater-off moments, we lowered the time constant of the lock-in amplifier to 30 millisecond so that it could respond to the temperature change more quickly. The sine-wave frequency used in the lock-in amplifier was Hz. There is a trade-off between fast response and averaging. So the temperature data were only lightly averaged by the lock-in and consequently quite noisy. To compensate, at each temperature we either repeated the measurements a few times or let the labview program take multi-cycle data (one cycle means one up and down process) in order to average out the noise. This measure is effective, but imposes some limitation on our data analysis Data analysis and results The simple instrumentation restricted our choice of the data analysis techniques. Given the noisy data, it is impossible for us to apply the integration technique, or the τ 2 model proposed by [40]. Thus, for the whole temperature range ( 300 mk 4 K) we used direct fitting to analyze the data. To make this model valid at millikelvin temperatures, we used a thin manganin wire for heat sinking in order to increase the relaxation time between the platform and the bath, i.e. τ 1. For the UPt 3 cell, τ 1 was found to be 190 s at 150 mk, that means we had to allow 35 minutes for one cycle so that the platform would have enough time to fall back down to the bath temperature (τ 1 for the FeCrAs cell is one order of magnitude shorter). This considerably slowed down our measurements, but makes the data relatively easy to analyze. Even with a large τ 1, the warming-up and cooling-down temperature curves can still sightly deviate from the exponential form. They tend to go in the opposite directions. So averaging the up and down data can restore the exponential form and improve the signal to noise ratio. However, the up and down processes were measured continuously. Separating them and flipping one to match the other may introduce human error.

54 Chapter 2. Design and development of experimental instruments (a) data sample temperature fit base temperature fit Temperature (K) Time (sec) (b) Temperature (K) Time (sec) Figure 2.10: (a): an example of the fitting results of the UPt 3 data. This data set was taken at 1.1 K. (b): The fitting result for the 0.57 K data of FeCrAs. The data is noisy, several cycles were measured in order to average out some noise. For both (a) and (b), the fitting function chooses the heater-on equation or the heater-off equation according to the least squares principle to fit different segments of the data. The blue line is a fit to the average temperature with a linear function which is added to compensate the base temperature drifting.

55 Chapter 2. Design and development of experimental instruments 44 So we decided to combine the up and down equations (see equations 2.6, 2.7) into one fitting function and use this function to fit the whole data set. The up and down equations have the same parameters, and were used to fit different segments of the data according to the least squares principle. This technique reduced the human error and made averaging data easy. Examples of such kind of the fitting, one for each sample, are given in Figure The data for FeCrAs sample is noisy, so we measured several cycles for averaging at each temperature. A linear function, represented by the blue line, was added to the fitting function to fit the average temperature in order to compensate the drifting base temperature. At 0.5 K, UPt 3 undergoes a double superconducting transition. In the superconducting state, it has a poor thermal conductivity. The large relaxation time between the platform and the bath guarantees the internal temperature gradient (within the platform, the sample, the heater and the thermometer) is negligibly small during the up and down processes, making the direct fitting applicable. The measured UPt 3 specific heat around the superconducting transition is shown in Figure The double superconducting transitions can be clearly seen. Our results compare well with that given by [41]. Figure 2.12 shows the specific heat of the UPt 3 sample for temperature 1 4 K. Our result for this temperature range also agrees well with the published data. The solid line in Figure 2.12 is obtained from the formula given in [42]. This formula fits the data there very well. Note that for the UPt 3 cell the addenda is negligibly small in comparison with the heat capacity of the UPt 3 sample. The specific heat data around the superconductivity transition and that in the normal state are compared in Figure We see that the measurements in these two different temperature ranges are consistent with each other. The results for FeCrAs sample is not as good as that for the UPt 3 sample. The reasons might be: 1) the sample is thick and its contact with the platform is poor due to

56 Chapter 2. Design and development of experimental instruments C/T (mj/mol.k 2 ) UPt3 C ( 10 3 mj/mol.k) UPt T (K) T (K) Figure 2.11: C/T vs T around the superconductivity transition of UPt 3. Addenda included. Figure 2.12: C vs T for UPt 3 from 1 K to 4 K. Addenda included. The solid line is drawn based on the formula derived from the published data [42]. the rugged bottom. 2) the relaxation time τ 1 is much shorter than that for the UPt 3 cell; τ 1 vs T for the FeCrAs-sample cell is shown in Figure 2.14, which demonstrates that τ 1 for this cell does not change much in the temperature range mk, and is much smaller than the relaxation time of the UPt 3 -sample cell, see Figure 2.15 (note that the rising of τ 1 in Figure 2.15 around 0.5 K is due to the increase of the heat capacity of UPt 3 in the vicinity of the double superconductivity transitions). These two reasons 1) and 2) would result in the internal temperature gradient during the up and down processes, and make the direct fitting invalid. In addition, the FeCrAs cell was used without a radiation shield, this will increase the temperature fluctuation of the platform. The result of C/T vs T 2 above 1 K is presented in Figure 2.16 (a), which exhibits a roughly linear behavior. The result for T < 1 K, see Figure 2.16 (b), shows that C/T starts to rise below 0.6 K. Note that this behavior is not reliable for the two reasons given above. However, this result rules out a dramatic rise or fall in C/T at millikelvin temperatures. Our results for T > 1 K, though noisy, are not off much from the PPMS data, see Figure 3.9 in chapter 3.

57 Chapter 2. Design and development of experimental instruments Around the superconducting transition Normal state C/T (mj/mol.k 2 ) UPt T (K) Figure 2.13: Comparison of C/T vs T for UPt 3 at mks and that at temperatures between 1 4 K Further work Our data for the UPt 3 sample show that this design works well at millikelvin temperatures, and that if τ 1 is large enough the analysis of data is straightforward direct fitting will work. However, there is still some room for improvement. The measurements for FeCrAs is not sufficiently accurate at millikelvin temperatures. So it would be worthwhile to redo it. Measures that can be taken in the future measurement are: 1) polish the sample into a thin slab Depending on the thermal conductivity of the sample, the slab can be made thinner or

58 Chapter 2. Design and development of experimental instruments FeCrAs cell τ 1 (s) τ 1 (s) UPt 3 cell T (K) T (K) Figure 2.14: Relaxation time τ 1 vs T in the temperature range mk for the FeCrAs-sample cell. Figure 2.15: Relaxation time τ 1 vs T between mk for the UPt 3 -sample cell. thicker than the UPt 3 sample used this project. The thinner the slab is, the easier for it to reach equilibrium with the platform, but if the slab is too thin, subtraction of the addenda would be difficult. 2) increase the thermal resistance of the heat sinking wire A large relaxation time between the platform and the bath can not only reduce the temperature gradient in the sample, but also keep the platform, the thermometer and the heater at approximately the same temperature during the measurement. Reducing the temperature difference between the platform and thermometer is especially helpful, otherwise the data are very difficult to analyze. 3) improve the thermal contact between the sample and the platform One can try to use electronic coupling between the sample and platform, e.g. spot-welding gold wire in between them. 4) use a heater with large resistance The resistance of the heater used this project is too small ( 0.1 Ω and 0.3 Ω for the UPt 3 and FeCrAs cells, respectively). It was hard to measure the resistance of the heater and calculate the output heat accurately. With a large resistance heater, one can not only know the heat more accurately, but also avoid heating up the fridge by reducing the

59 Chapter 2. Design and development of experimental instruments (a) C/T (mj/mol.k 2 ) FeCrAs T 2 (K 2 ) (b) C/T (mj/mol.k 2 ) T 2 (K 2 ) Figure 2.16: (a): C/T vs T 2 for FeCrAs from 1 K to 4 K. Addenda included. The solid line is a guide to eye. (b): C/T vs T 2 for FeCrAs at millikelvin temperatures. Note that the rise of C below 0.6 K may not be real, because validity of the direct fitting may be compromised due to the short relaxation time τ 1.

60 Chapter 2. Design and development of experimental instruments 49 current to the heater. But overall, considering how simple the setup is, it works very well, and seems to be a promising approach for the future.

61 Chapter 3 Growth of single crystal FeCrAs The binary pnictides, Fe 2 As and Cr 2 As, have been studied in some detail during the last few decades. Besides having the isomorphic crystal structure of tetragonal Cu 2 Sb-type, both compounds are known to be antiferromagnets with Neel temperatures of 353 and 393 K, respectively [43]. The ternary pnictide FeCrAs, however, has been ignored for some reason. Only the crystallography of this crystal has been worked out, see [44], which shows that FeCrAs has a hexagonal crystal structure at room temperature. The magnetic structure of this compound was unknown. The magnetic properties have only been investigated by Mossbauer spectroscopy which shows that there is no magnetic ordering on the Fe sites down to 4.2 K [45]. FeCrAs seems to be a good candidate for investigating geometric frustration in a metallic system. This was our motivation for growing and investigating this crystal. Indeed, as we will see in next chapter, FeCrAs is a unique compound which exhibits a novel non-fermi-liquid state at low temperatures; at high temperatures it shows properties reminiscent of those of the parent compounds of the new layered iron-pnictide superconductors. FeCrAs was initially grown following the recipe for growing Fe 2 As [46]. It turned out that this recipe is not wholly suitable for growing FeCrAs. That recipe always gives a batch containing a minority phase of pure Fe. To find the right growth conditions, 50

62 Chapter 3. Growth of single crystal FeCrAs 51 Figure 3.1: A sample of FeCrAs single crystal. Figure 3.2: Laue back scattering pattern of FeCrAs single crystal ( x-ray beam parallel to [120] direction). procedures including slow-cooling, remelting and annealing have been applied. After many tests, eventually, we mastered the technique of growing high quality single crystals of FeCrAs. Two batches that contain pieces of pure single crystal FeCrAs of size as big as 3 5 mm have been obtained (see Figure 3.1). The crystals look shiny and metallic, but are brittle. Samples from different growths have been characterized by electrical resistivity, magnetic susceptibility, magnetic hysteresis and heat capacity measurements. Two batches were examined by x-ray powder diffraction. The x-ray spectra showed that the samples crystallized in the correct structure, i.e. space group P 62m. One sample was examined by powder neutron diffraction which showed that there is a very low disorder on the Fe and Cr sites. Both x-ray and neutron results have been analyzed using GSAS, a Rietveld fitting program. 3.1 Procedures and characterizations When growing a crystal, it is very helpful to have a phase diagram showing the relation between the desired solid phase and other phases. For the Fe-Cr-As system, however, this kind of phase diagram is still unavailable, which means we had to find out its crys-

63 Chapter 3. Growth of single crystal FeCrAs 52 Table 3.1: The date means the starting date of the growth. If the sample can be picked up by a magnet, it is called magnetic ; the symbol / in this column means not checked. The symbol / in the quality column means the quartz tube had not been open, so the quality of the sample is unknown. growth date magnetic quality No. 1 14/02/2005 yes powdery; contains pure iron. 2 10/03/2005 yes contains impurities. 3 2/06/2005 no contains impurities. 4 6/02/2006 / quartz tube cracked. 5 28/02/2006 yes powdery; contains pure iron. 6 20/03/2006 / quartz tube cracked. 7 4/04/2006 tiny nonmagnetic contains impurities and pieces pure iron. remelt: 17/04/2007 / quartz tube cracked. 8 20/04/2006 yes contains pure iron remelt: 29/05/2006 small nonmagnetic contains pure iron pieces anneal: 9/03/2007 no not very pure. 9 4/04/2007 / quartz tube cracked /04/2007 / / anneal: 10/05/2007 yes contains pure iron. remelt: 4/07/2007 yes contains pure iron. remelt: 22/02/2008 / / anneal: 7/03/2008 nonmagnetic lumps good 11 20/01/2008 yes contains pure iron. remelt: 1/02/2008 / / anneal: 11/02/2008 nonmagnetic lumps best (see section for the growth conditions).

64 Chapter 3. Growth of single crystal FeCrAs 53 tallization conditions on our own. When we started the growth of this compound, all the relevant information we had was: 1) the melting points of the three elements Fe, Cr and As are 1538 C, 1907 C and 817 C, respectively; 2) iron and arsenic can form Fe 2 As crystal, which has a melting point 919 C [46]; 3) chromium and arsenic can crystallize into Cr 2 As at temperature 900 C [47]; 4) iron and chromium can form an alloy, and for the composition of Fe between percent, the melting temperature of the alloy is 1260 C 1650 C. Growth of FeCrAs has to be carried out carefully to avoid the contamination of undesired phases such as binary arsenides and/or pure elements. Crystals of FeCrAs were obtained by melting stoichiometric amounts of Fe, Cr and As in vacuum. First, stoichiometric amounts of Fe, Cr and As were put into an alumina crucible, then the crucible was sealed in an evacuated quartz tube. After that, the materials were melted in an electric furnace. An issue to emphasize is that when melting the materials, care has to be taken to prevent the quartz tube from cracking and releasing arsenic vapor into the air, because both arsenic and its compounds are poisonous. But it should be noted that when the quartz tube cracked, only minimal mass was lost, indicating minimal release of arsenic. Probably it had reacted with iron and chromium before the cracking happened, and thus was much less volatile than pure arsenic. Nevertheless, it is important to carry out these growths in a fume hood, so that in the event of a release of arsenic it will not go into the laboratory. Next, we give a brief description about the sample growth procedures, 1) clean the crucible: After the initial cleaning, i.e. boiled in 10% nitric acid for 10 minutes and then rinsed with distilled water, the crucible was baked in a vacuum furnace at 1300 C for 1.5 hours to get rid of the contamination that would evaporate at high temperatures. Important issues are: 1) the crucible has to be heated up to a temperature above the highest crystal growth temperature. 2) the crucible should be properly protected, e.g. being placed in a large crucible, so that the vapor of the metallic elements of the furnace will not deposit

65 Chapter 3. Growth of single crystal FeCrAs 54 on it. If this happens, the coating can be etched away with 10% nitric acid. 2) make the quartz tube: The quartz tube was made from a long, new, thick-wall quartz pipe of 1 inch diameter. First, the pipe was sealed at one end. Note that a flat bottom is safer than a round one, because the round bottom is subject to intense stress exerted by the crucible due to the different thermal expansions, and occasionally it was found broken after the thermal cycle. Then, the crucible was put in the quartz tube, and a neck was made to facilitate the later sealing. This is illustrated in Figure 3.3 (a). 3) clean the materials: Chemical etching was applied in order to get rid of the oxides on the surfaces of the materials. Here are the formulae of the etching solutions for the three elements: Chromium: H 2 O : H 2 SO 4 = 9 : 1, Iron: EtOH(ethanol) : HNO 3 = 9 : 1, Arsenic: H 2 O : HNO 3 : HCl = 1 : 1 : 1. After the etching, the materials were rinsed with distilled water three times, with the second rinse being done in an ultrasonic cleaner (10 minutes). 4) seal the quartz tube: The materials were put in the crucible through a Pyrex glass tube, then the quartz tube was pumped with a roughing pump for 1/2 1 hour (due to the oil vapor problem, it is not recommended to pump the quartz tube with a roughing pump loner than this suggested time). After that, the bottom of the quartz tube was dipped in liquid nitrogen, and the neck was sealed (see figure 3.3 (b)). The nitrogen is used to keep the crucible cool so that As will not evaporate during sealing the neck of the quartz tube. The quartz tube has to be pumped well before being dipped in nitrogen, otherwise the moisture in the air will condense out and stay in the quartz tube. 5) melt the materials: The temperature control programs vary from growth to growth. One should look up the

66 Chapter 3. Growth of single crystal FeCrAs 55 Figure 3.3: Illustration of sealing the materials in the quartz tube for the growth of FeCrAs. (a): put the materials in the crucible; (b): seal the quartz tube. section related to the growth in which one is interested for the appropriate temperature control. Here we only quote the thermal cycle for the first growth, which is based on the methods for growing Fe 2 As [46]. The quartz tube was heated up in an electric furnace to 600 C in one day. As the vapor pressure of arsenic exceeds atmospheric pressure at this temperature, the furnace was kept at 600 C for 24 hours, to allow the As vapor to react with the iron and chromium, before being raised to 1100 C over 2 hours. Dwelling at 1100 C for half hour, the furnace was then brought down to room temperature at a rate of 1 C. The most difficult part of growing FeCrAs lies in the thermal treatment. Firstly, the cooling rate has to be slow enough, otherwise it may precipitate the formation of other compounds. For instance, if FeCrAs is an incongruently melting compound, i.e. during melting, the compound undergoes a process of solid 0 (parent) solid 1 (descendant)+liquid 1 liquid 0 (which has the same composition as the parent solid), fast cooling will hasten the intermediate stage and produce a mixture containing solid 1 and solid state of liquid 1.

67 Chapter 3. Growth of single crystal FeCrAs 56 If it happens to be solid 1 = CrAs and liquid 1 = Fe, we will get a batch with trace phase of pure Fe. It very likely is the reason why the as-grown ingots often contain pure Fe. Secondly, the temperature gradient in the electric furnace makes it hard to grow this crystal. If no precaution has been taken, unknown amounts of Fe, Cr, and As vapors will deposit on the cold area of the quartz tube, and leave the materials in the crucible nonstoichiometric. To make the crystal extremely pure is also difficult. Firstly, oxygen is difficult to completely eliminate. It can get into the sample through the oxides on the surfaces of materials, or through the tiny amount of air left in the quartz tube. We tried to etch away the oxides using different acids. We found that the formula given above works best. To reduce the air left in the quartz tube, the quartz tube should be pumped for at least one hour with the roughing pump. Secondly, impurities can be introduced during transferring the materials into the crucible. Everything that can touch the materials needs to be cleaned carefully. In particular, one should wait until the Pyrex glass tube is dry before using it, otherwise fine particles of the materials may stick on the wall of tube. The last thing worthy of note is that in order to get a stoichiometric compound, the mass of each element has to be measured precisely, in our case the mass was measured down to 0.1 mg to ensure the deviation from the stoichiometric ratio is less than 0.005% of the total mass. We came up with the right procedures for growing FeCrAs after eleven trials. The results of these trials are summarized and presented in table 3.1. We have given a general description of the crystal growth procedures above. In the next few subsections, we will present some sample specific information, including detailed growth procedures and characterization using a PPMS system for specific heat and resistivity measurements, and an MPMS system for magnetization and susceptibility measurements.

68 Chapter 3. Growth of single crystal FeCrAs The first growth In the first growth of FeCrAs, the material used was a mixture of Fe, Cr and As powders at atomic ratio 1:1:1. The mixture was supplied by Professor D.G. Rancourt, of the University of Ottawa. This was a part of the batch that had been used for the Mossbauer study [45]. The powder had been in air for more than 20 years. Because it is powder, it is impossible to apply chemical etching, the material must contain some oxides. About 0.6 g mixed powder was put into the crucible, the quartz tube was then evacuated and sealed by the glass blowers. The thermal cycle of the first growth has already been described in section 3.1. This batch is not a firmly consolidated ingot, but a blob made up of grains. The problems with this batch are: 1) there are green color impurities on the surface of the blob; 2) it contains a considerable amount of pure iron and, thus, is magnetic (by that we mean they can be picked up by a bar magnet). Figure 3.4 (b) shows the temperature dependence of the resistivity of a sample from the first growth. Though the resistivity rises as T falls at high temperatures, it becomes metallic, i.e. ρ drops with decreasing T, at low temperatures. We will later see that this resistivity behavior is not consistent with that observed in samples that are free from pure iron inclusions. Even though it is magnetic, no hysteresis was observed from room temperature down to 10 K. Figure 3.4 (a) shows an example of the field dependence of the magnetization at 10 K. However, it is noted that M vs H curve is s-shaped. This observation, combined with the metallic resistivity and the fact of being magnetizable, suggests that this batch contains a significant amount of a ferromagnetic material, which our later powder neutron diffraction measurements showed to be pure iron. The specific heat, as shown in Figure 3.5 looks fairly normal. Figure 3.6 shows C/T vs T 2, that gives a γ value 30 mj/mol K 2 which is quite large for a d-electron system (γ is the coefficient of electron term in C = γt + βt 3 ). This γ value agrees well with that of the more pure samples, suggesting that the bulk part of this batch is FeCrAs. Note that the noisy data

69 Chapter 3. Growth of single crystal FeCrAs (a) M (emu) decreasing field (10K) increasing field (10K) H (T) (b) ρ (µω cm) T (K) Figure 3.4: (a): Magnetization vs field at 10 K, the s-shape hysteresis curve is an indication of the presence of pure Fe. (b): Resistivity vs temperature, the metallic resistivity at low temperatures is in contrast with the non-metallic resistivity of the pure FeCrAs. The samples came from the first growth.

70 Chapter 3. Growth of single crystal FeCrAs 59 C (10 4 mj/mol K) T (K) C/T (mj/mol K 2 ) T 2 (K 2 ) Figure 3.5: Specific heat vs temperature for a sample from the first growth. It looks fairly normal in the whole temperature range. Figure 3.6: C/T vs T 2 at low temperatures. It gives a quite large γ value (γ 29.4 mj/mol.k 2 ). The sample is from the first growth. below helium temperature is due to the poor temperature control of PPMS. No sign of a phase transition can be seen from these transport and thermodynamic measurements The second growth In the second growth of FeCrAs, the materials used was the same as the first growth. As the first batch was not well crystallized, in the second growth we heated the materials up to a higher temperature and let the sample dwell at this temperature longer so that the materials could thoroughly melt. The thermal cycle became 30 C 2 hours 600 C (dwelling 24 hours) 1 hour hours C (dwelling 1.5 hours) 30 C. The quality of the second batch seems to be better than the first one. The Laue back scattering picture showed a clear, regular pattern. The temperature behavior of the resistivity, see Figure 3.7, further shows that the properties of this batch is quite different from the first one. This resistivity has a stronger temperature dependence at higher temperature, and apart from a drop at about 120 K it is non-metallic in the whole temperature range observed. We will later see that this drop is due to an antiferromagnetic transition. However, the second batch is still magnetizable, indicating

71 Chapter 3. Growth of single crystal FeCrAs ρ (µω cm) T (K) Figure 3.7: Resistivity as a function of temperature for a sample from the second batch. the existence of trace amounts of pure iron The third growth The powder material was used up in the first two growths. From the third one, we started growing FeCrAs using iron foil and lumps of chromium and arsenic with shipping purity %, % and %, respectively. The materials were etched by diluted acids in order to get rid of the oxides on their surfaces. The etching solutions were made of H 2 O : H 2 SO 4 = 9 : 1, EtOH(ethanol) : HNO 3 = 9 : 1 and H 2 O : HNO 3 = 11 : 4 for chromium, iron and arsenic, respectively. The chromium looked clean after etching; the iron and arsenic did not. They are no longer shiny and the surfaces of Fe and As became brown and black, respectively. They were put in the crucible through the Pyrex tube with all pieces of arsenic wrapped in the iron foils. The temperature control program was hours C 600 C (dwelling 48 hours) 2 hours 1150 C (dwelling 2.5 hours) power off 30 C.

72 Chapter 3. Growth of single crystal FeCrAs 61 M (emu/mol) (a) c-axis(2 K) a-axis(2k) a & c-axis(300k) H (T) (b) FC M (emu/mol) 3 2 H // c-axis H // a-axis T (K) Figure 3.8: (a): Magnetization vs field behavior at 2 K and 300 K for field along the a-axis and the c-axis. (b): magnetization vs T with a dc field of 1000 Oe applied along the a-axis and the c-axis, recorded in both zero-field-cooled (ZFC) and field-cooled (FC) modes. The samples are from the third growth.

73 Chapter 3. Growth of single crystal FeCrAs (a) 1200 ρ (µω cm) T (K) C ( 10 4 mj/mol.k) (b) C/T (mj/mol.k 2 ) T 2 (K 2 ) T (K) Figure 3.9: (a): resistivity as a function of temperature. (b): the temperature dependence of the specific heat from 2 K to 300 K. Inset, a plot of C/T vs T 2 for low temperatures (γ 25.0 mj/mol.k 2 ), the dashed line is a guide to eye. The samples came from the third growth.

74 Chapter 3. Growth of single crystal FeCrAs 63 This batch of crystal is nonmagnetic, though there are still signatures of disorder. The magnetization curve shown in Figure 3.8 (a) is a straight line unlike the s-shape curve seen in the magnetic samples. This indicates that no detectable pure iron exists. However, this batch may contain dislocations or possibly other impurities that disrupt the crystalline order. Figure 3.8 (b) shows the temperature dependencies of the magnetizations, with a DC field of 1000 Oe applied along a-axis and c-axis, in both zero-fieldcooled (a procedure that involves cooling in zero field and measuring the magnetization upon warming with the field on) and field-cooled (which involves cooling the sample in a field and measuring the magnetization with the field on) modes. The zero-field-cooled (ZFC) and field-cooled (FC) magnetizations start to deviate from each other at a quite high temperature ( 40 K). Spin freezing at such a high temperature maybe due to the breakdown of geometric frustration due to dislocations or impurities. The suppressed antiferromagnetic transition is another indication of the existence of disorder. In this sample, the transition represented by the bump occurs at 100 K, while in pure samples the transition takes place at 125 K. The in-plane resistivity is non-metallic, growing continuously as temperature falls in the range of K, see Figure 3.9 (a). The hump at 100 K originates from the antiferromagnetic transition. Figure 3.9 (b) shows the temperature dependence of the specific heat measured in the PPMS. In the whole temperature range, the specific heat behaves fairly normally. The wiggles in the range of 2 4 K are caused by the unstable temperature which the PPMS failed to control, probably because the needle valve was partially blocked. It should be noted that no anomaly is visible at 100 K, the temperature at which both the magnetization and resistivity show bumps. At low temperatures, C/T vs T 2 gives a straight line (see the inset of Figure 3.9 (b)), with a large γ value which is comparable to that observed in the first growth sample. The Laue diffraction photos show clear patterns with the right symmetry, indicating that we have got single crystals which have the correct crystal structure.

75 Chapter 3. Growth of single crystal FeCrAs Seventh growth To improve the purity of the sample, we changed the method of cleaning Fe and As 1. The Fe foils were cleaned with ethanol only; the arsenic was etched with the diluted nitric acid (H 2 O : HNO 3 = 11 : 4) the same as used in the third growth, but this time we heated the solution up to 80 C to accelerate the reaction. The effect was weak, the arsenic looks slightly less dark compared to the third growth. The temperature program for melting the materials was changed to be: hours C 600 C (dwelling 50 hours) 10 hours hours C (dwelling 6 hours) 30 C. This batch is inhomogeneous, most of it is magnetic, only two small pieces are nonmagnetic. One of the nonmagnetic pieces has been characterized by measuring its magnetization as a function of field and temperature. Figure 3.10 (a) shows the magnetization data at 2 K (green curve) and 300 K (blue curve). Even though there is no hysteresis, the M vs H curves are not straight for either temperature, indicating that this piece still contains traces of pure Fe. This fact demonstrates that M vs H measurement is more sensitive in determining the presence of pure Fe than a bar magnet. The temperature dependence of both FC and ZFC magnetizations is given in Figure 3.10 (b). Both the temperature at which M shows a hump ( 102 K) and the temperature at which the field-cooled and zero-field-cooled magnetizations start to deviate each other ( 35 K) show that this batch is only slightly purer than the batch of the third growth (see Figure 3.8). One of the non-magnetic piece was examined by Laue diffraction. The Laue photos show that it is a single cystal with the correct symmetry. The spots on the photos are sharper in comparison with the Laue patterns of the crystals of the third growths. 1 Batches from the fourth, fifth and sixth growths are so bad that none of them is included.

76 Chapter 3. Growth of single crystal FeCrAs 65 M (emu/mol) (a) 2 K 300 K H // c_axis Seventh growth H (T) M (emu/mol) (b) Seventh growth Field-cool Zero-field-cool T (K) 100 (c) 21.5 (d) M (emu/mol) a_axis c_axis Eighth growth H (T) 60 (e) M (emu/mol) Eighth growth Zero-field-cool Field-cool T (K) 3 (f) M (emu/mol) a_axis c_axis M (emu/mol) ZFC ZFC a_axis c_axis Eighth growth annealed H (T) 1.5 Eighth growth (annealed) T (K) Figure 3.10: Comparison of the magnetic properties of the seventh growth, the eighth and eighth-annealed samples. (a): magnetization (M) vs field (H) for H along the c-axis at 2 K (green) and 300 K (blue); (c), (e): M vs H at 2 K for H along the a-axis (black) and along the c-axis (green). (b), (d): M vs T in both zero-field-cooled (ZFC) (green) and field-cooled (FC) (black curve) modes with a 1000 Oe dc field applied along the c-axis; (f): M vs T with a dc field of 1000 Oe applied along the a-axis (black) and the c-axis (green), recorded in both ZFC and FC modes.

77 Chapter 3. Growth of single crystal FeCrAs Eighth growth In the eighth growth, the three elements were carefully etched. After etching, they all looked clean. The etching solutions are: H 2 O : H 2 SO 4 = 9 : 1, EtOH(ethanol) : HNO 3 = 9 : 1 and H 2 O : HNO 3 : HCl = 1 : 1 : 1 for Cr, Fe, As, respectively. During etching it is important to keep stirring the solution. Otherwise, some dirty substances would deposit on the surfaces of the pure materials. The crystal obtained after a heating process of 25 C 35 hours 600 C (dwelling 60 hours) 10 hours 1150 C (dwelling 10 hours) 40 hours 30 C was magnetic. Then this batch was sealed in quartz tube again and remelted. The quartz tube was sealed in our lab by myself. Since then most of the glass blowers work was done in our lab. The temperature control program for remelting is 24 C 6 hours hours C (dwelling 48 hours) 800 power off C room temperature. After remelt, we obtained a well-crystallized batch, the Laue back scattering photos show very clear, regular patterns. But it is still magnetic. The sample purity was improved, but not much. Some impurities must have been introduced into the sample during the remelting processes. A sample of the remelt batch has been characterized through M vs H measurements and M vs T measurements. Figure 3.10 (c) shows the field dependence of the magnetization at 2 K for fields along a-axis (red) and c-axis (green). Compared to the non-magnetic sample of the seventh growth (see Figure 3.10 (a)), the fast rise of M of this sample extends to even higher field, indicating that the sample contains more pure iron. On the other hand, Figure 3.10 (d) shows that the magnetization splits at a relative low temperature ( 15 K) into zero-field-cooled and field-cooled branches, suggesting that the disorder level is lower than that of the seventh growth. The antiferromagnetic transition correspondingly moves to a slightly higher temperature ( 107 K). This fact implies that the two features the hump and the the spin freezing are good for looking at the disorder level, but are insensitive to the presence of trace amount of pure Fe. A magnetic piece of the remelted sample was sealed in an evacuated quartz tube, and

78 Chapter 3. Growth of single crystal FeCrAs 67 annealed following the procedure of 30 C 3 hours 900 C (dwelling 150 hours) 5 hours 30 C. It becomes nonmagnetic after this process. Its magnetization linearly depends on field, see Figure 3.10 (e), indicating that no detectable pure Fe present. Due to the absence of pure Fe, the magnitude of M, as shown in Figure 3.10 (f), becomes one order of magnitude smaller than the unannealed sample, though the antiferromagnetic transition temperature and the spin-freezing temperature are unchanged Tenth growth In the eighth growth, we learnt that annealing can significantly reduce the amount of pure Fe in the sample. We also noticed that transferring a sample to another quartz tube to anneal would introduce some impurities. We decided to grow a new batch of FeCrAs with melting and annealing in the same quartz tube. The melting process is 30 C 10 hours 570 C 30 hours 600 C (dwelling 60 hours) 60 hours 1160 C (dwelling 40 hours 10 hours) 530 power off C room temperature. Without breaking the quartz tube, the following anneal process was used: hours C 900 C (dwelling 240 hours) 20 hours 500 power off C room temperature. After the first anneal the sample is still magnetic. Then the sample was remelted at 1200 C and cooled down to room temperature by powering off the furnace (note that the cooling rate of the furnace is not that fast, so this is not a quench process). After that, the sample was annealed at 900 C for 150 hours. Despite all these treatments, the sample is magnetic. All this points towards the fast-cooling being at fault. so we remelted the sample and cooled it down very slowly (1150 C 150 hours 800 C), then annealed it at 900 C for 130 hours. Finally, we got large pieces of nonmagnetic FeCrAs crystal in this batch. But some pieces are still magnetic. What we learnt from this growth is that anneal itself will not completely eliminate pure Fe from the sample; for the anneal to work, the to-be-annealed sample must come out from a slow cooling batch. Figure 3.11 (a) shows the magnetization curve of a non-magnetic sample at 2 K for

79 Chapter 3. Growth of single crystal FeCrAs (a) M (emu/mol) a_axis c_axis (b) H (T) M (emu/mol) c-axis a-axis T (K) Figure 3.11: (a): magnetization vs field at 2 K for field along a-axis (black) and along c- axis (green). (b): magnetization as a function of temperature with a dc field of 1000 Oe applied along a-axis (black) and c-axis (green), recorded in both zero-field-cooled and field-cooled modes. The samples came from the tenth growth.

80 Chapter 3. Growth of single crystal FeCrAs 69 field along a-axis (black) and c-axis (green). The a-axis magnetization curve is a straight line; the c-axis magnetization curve has a barely visible kink at fields close to zero. So this sample contains nearly no pure Fe. It is quite pure, as can be seen from Figure 3.11 (b) which shows that the antiferromagnetic transition moves to higher temperature (121 K) while the spin freezing temperature goes down to 10 K Eleventh growth In this growth, we tried to grow a more pure sample. The iron foil was for the first time cut into small pieces so that it could mix with other materials better. The three materials Fe, Cr and As were etched in the same way as in the eighth growth. The procedures for cleaning the crucible and sealing the quartz tube were the same as that given in section 3.1. The initial melting process is hours C 570 C (dwelling 0.1 hour) 30 hours hours C (dwelling 60 hours) 1150 power off C (dwelling 10 hours) room temperature. The ingot was found to be magnetic after taking out of the quartz tube. So it was remelted and annealed as follows: the ingot was put into a cleaned crucible, then the crucible was placed into a new quartz tube. After being pumped for 1 hour, the quartz tube was sealed. The ingot was melted as 22 C (dwelling 70 hours) 120 hours 800 C 24 hours 300 C 20 hours 1150 C power off room temperature; and without breaking the quartz tube, the sample was then annealed as 30 C 900 C (dwelling 150 hours) 5 hours 30 C. 10 hours We obtained large nonmagnetic pieces from this growth, which turned out to be the best sample we have. A piece of crystal from this batch is shown in Figure 3.1. No detectable pure Fe is present as indicated by the perfect linear field dependence of the magnetization, see Figure 3.12 (a). This batch is also very pure, as shown in Figure 3.12 (b), the spin freezing temperature is low ( 7 K) and the antiferromagnetic transition moves up to 125 K.

81 Chapter 3. Growth of single crystal FeCrAs (a) M (emu/mol) a_axis c_axis H (T) M (emu/mol) (b) a_axis c_axis T (K) Figure 3.12: (a): magnetization vs field at 2 K for field along the a-axis (black) and along the c-axis (green). (b) magnetization as a function of temperature with a dc field of 1000 Oe applied along the a-axis (black) and the c-axis (green), recorded in both zero-field-cooled and field-cooled modes. The samples came from the eleventh growth.

82 Chapter 3. Growth of single crystal FeCrAs Microprobe analysis In some growths before the ninth trial, green spots were often seen on the surfaces of the ingots. We decided to find out what the impurities on the surface are using Electron Probe Micro-analyzer (EPMA). Samples from two different growths were examined. The results are summarized in table 3.2. Table 3.2: Surface EPMA analysis results. Normalized atomic ratio (in %) are quoted here. Note that elements less than 1% cannot be definitely identified. The two samples come from the third growth and eighth growth. Element 3rd growth Atomic (%) 8th growth O Al Cr Fe As Si / 0.69 The high content of Cr and O elements indicates that the green impurity on the surface probably is Chromium oxide. Please note that the impurities were only found on the surfaces, the inner part of the sample is shiny and looks very clean; and that in our last two growths (the tenth and the eleventh) the surfaces of the ingots also look clean.

83 Chapter 3. Growth of single crystal FeCrAs Powder X-ray diffraction Some of our samples are magnetic, others are not. These two classes of samples must have different contents. Based on the hysteresis measurements, we deduced that the magnetic samples may contain trace amounts of pure iron. To obtain direct evidence, powder X-ray diffraction were carried out at Geology Department of the University of Toronto on two samples: a non-magnetic one (from the third growth) and a magnetic one (from the eighth growth). Monochromatic Cu X-ray with dual wavelength Å and Å of ratio 0.5 was used. The scan time is 3 hours for each sample. The results have been fit with a Rietveld program GSAS (General Structure Analysis System) [48] in order to identify the peak produced by the impurity phase; however, we found that for both samples all the observed peaks have been matched by the calculated patterns which are based on the phase of FeCrAs; see Figure This finding proves that the crystal we obtained has the correct structure, i.e. space group P 62m. However, the diffraction spectra failed to reveal the impurity phase. The minority phase which we think should exist in the magnetic sample must be so rare that its signal is below the noise level. To identify it, a longer scan time (>10 hours) is probably needed, however as described below the powder neutron diffraction suggested that the impurity phase is pure iron. 3.4 Powder neutron diffraction In a typical metallic compound, the resistivity falls as temperature decreases due to the reduction of the thermally excited disorder. We find that FeCrAs does not belong to this category. Apart from the antiferromagnetic transition region, ρ rises as temperature falls. This unusual resistivity might be related to the disorder on the Fe and Cr sites. To find out if the non-metallic resistivity is intrinsic, we did powder neutron diffraction

84 Chapter 3. Growth of single crystal FeCrAs 73 Intensity (a) Obs Calc bckgr diff Nonmagnetic sample Theta 600 (b) Intensity Obs Calc bckgr diff Magnetic sample Theta Figure 3.13: Powder x-ray diffraction at room temperature for (a) a non-magnetic sample from the third growth and (b) a magnetic sample from the eighth growth. Obs, Calc, bckgr, diff stand for raw data, calculated pattern, background and difference between data and calculation, respectively. Except for the intensity, the diffraction patterns of these two samples look the same. Note that the calculated patterns do not match the intensity of the measured peaks very well, but this should not prevent us from identifying any peak coming from the impurity phase.

85 Chapter 3. Growth of single crystal FeCrAs 74 Intensity ( 10 3 ) K Obs Calc bckgr diff Theta Figure 3.14: Powder neutron diffraction at 297 K. The neutron wavelength λ is Å. Obs, Calc, bckgr, diff represent raw data, calculated pattern, background and difference between data and calculation, respectively. measurements at Chalk River 2. The sample comes from a magnetic batch which was obtained in the eighth growth. The data are analyzed using the Rievteld refinement program GSAS. The phase information extracted from the room temperature data is presented in table 3.3. The occupancy rates on both the Fe and Cr sites are very high, indicating both sublattices are strongly ordered. Note that the occupancies of the Arsenic sites are quoted as 1 based on the assumption that cations are unlikely to occupy the sites of anions. Figure 3.14 shows an example of the Rievteld fitting result. The data was collected at 297 K with neutron wavelength Å. Our previous resistivity and susceptibility data show an anomaly at a temperature in 2 Powder neutron diffractions measurements were carried out in collaboration with Dr. Ian Swainson of Chalk River Laboratories. The analysis of the magnetic structure were carried out by Dr. Swainson.

86 Chapter 3. Growth of single crystal FeCrAs (a) Intensity ( 10 3 ) K Obs Calc bckgr diff Theta (b) Intensity ( 10 3 ) K Obs Calc bckgr diff Theta Figure 3.15: Powder neutron diffraction at (a) 297 K and (b) 2.8 K. Obs, Calc, bckgr, diff represent raw data, calculated pattern, background and difference between data and calculation, respectively. New peaks at low angles shown in (b) are due to long-range magnetic ordering. The peaks marked by the blue arrows are related to the pure Fe.

87 Chapter 3. Growth of single crystal FeCrAs 76 Table 3.3: Room temperature phase information obtained from the Rievteld analysis of the powder neutron data. The multiplicity denotes the number of the equivalent sites in a unit cell. Space group: P 62m; Lattice parameters (Å): a=b=6.0954, c= Sites Occupancy Fractional coordinates Multiplicity Fe 0.97(3) (1) Cr 0.96(2) (0) As As between K. We claimed that this anomaly is due to an antiferromagnetic transition. Powder neutron diffraction data provide the direct evidence. These measurements were carried out at 2.8 K, 5.6 K, 11 K, 52.8 K, 61.5 K, 77.5 K, 88 K, 98.2 K, K, K, K and K with the neutron wavelength Å. We see new peaks in all the diffraction patterns below 100 K at lower angles, but these peaks disappear in the data sets collected at temperatures above 100 K. An example of the low temperature diffraction pattern is shown in Figure 3.15 (b). At low angles, a few new peaks show up that do not match the calculated pattern (red lines). These peaks fall off quickly at high angles, showing that they originate from magnetic ordering. The magnitude of the peak at 15 degree in 2θ as a function of temperature is shown in Figure We see that the peak grows gradually with decreasing temperature from 100 K to 53 K, then gets saturated. The magnetic-ordering occurs at the same temperature as the anomaly in susceptibility see Figure 3.10 (d). So the cause of the susceptibility anomaly must be the magnetic ordering. Our collaborator Ian Swainson at Chalker River has worked out the magnetic structure of this system. It is found that the magnetic moments on the Cr sites order antiferromagnetically in the ab-plane with the propagation vector k=( 1 3,1,0). 3

88 Chapter 3. Growth of single crystal FeCrAs Peak Height (Intensity) T (K) Figure 3.16: Magnitude of the magnetic peak at angle 15 degree, see 3.15 (b), as a function of temperature. The crystal used in the powder neutron diffraction is not the best sample we have. It is magnetic, so we think it must contain some ferromagnetic impurities. The neutron diffraction data confirm our speculation. The peak at 70, see the blue arrows in Figure 3.15, is consistent with pure Fe. For this particular sample, the fraction of this impurity phase is about 2%. After eleven trials, I determined that the best procedure for growing pure single cystal FeCrAs is according to the recipe described for the eleventh growth. The key steps are: 1) the materials should be cleaned properly. To this end, chemical etching is good enough. The best etching solutions for these three elements are as follows, Chromium: H 2 O : H 2 SO 4 = 9 : 1, Iron: EtOH(ethanol) : HNO 3 = 9 : 1, Arsenic: H 2 O : HNO 3 : HCl = 1 : 1 : 1. One has to keep stirring the solution during etching the materials, otherwise some im-

89 Chapter 3. Growth of single crystal FeCrAs 78 purities will deposit on the surface of the materials. 2) the sample obtained from the initial melt needs to be remelted and the cooling rate for this process has to be slow enough; and after remelt, the sample needs to be annealed. The thermal cycle for remelting is recommended to be 22 C 800 C 30 C 24 hours 300 C 20 hours hours C (dwelling 70 hours) power off room temperature; and the annealing treatment 10 hours 900 C (dwelling 150 hours) 5 hours 30 C will be good. Through the charaterization measurements, I also found the charateristics of the best samples, which can be used to check the quality of samples for future growths of FeCrAs. Good samples have a low spin freezing temperature, but a high antiferromagetic transtion temperature; they also have a linear in H magnetization. To find the spin freezing temperature and the antiferromagnetic transition temperature, DC magnetic susceptibilities in FL and ZFL modes are desirable.

90 Chapter 4 A novel non-fermi-liquid state in FeCrAs In this chapter, we report the transport and thermodynamic properties of stoichiometric single crystals of the hexagonal iron-pnictide FeCrAs. This compound exhibits interesting properties: below 10 K the resistivity shows non-metallic, non-fermi-liquid behavior, while the specific heat shows Fermi liquid behaviour with a large Sommerfeld coefficient, γ 30 mj/mol K 2 ; at high temperatures, the transport and thermodynamic properties are reminiscent of those of the parent compounds of the new layered iron-pnictide superconductors. We will first present the background information related to this compound, including a brief description of the relevant properties of other arsenides and the known properties of FeCrAs before our study; the emphasis is on the the crystallography of FeCrAs. Then, we will describe the experimental details for this study, and the four terminal resistivity measurement setup will be introduced. Finally, we will show the experimental results, followed by the discussion and conclusion of our investigations. This work has been published on Europhysics Letters [49]. 79

91 Chapter 4. A novel non-fermi-liquid state in FeCrAs Material background The intermetallic compounds Mn 2 As, Cr 2 As and Fe 2 As have the tetragonal Cu 2 Sb-type structure, and they all are antiferromagnetic below the Néel temperature of 573, 393 and 350 K, respectively [47, 50]. The ternary-metal arsenic system of the type FeMAs (M=3d transition metal) shows a greater variety of properties. Of them, FeCoAs and FeMnAs have been relatively well studied [51 53]. FeCoAs has the hexagonal Fe 2 P-type structure at room temperature. It is ferromagnetic below T c = 293 K. FeMnAs takes one of the two structures, the T (tetragonal Cu 2 Sb-type) or the H (hexagonal Fe 2 P-type), according to the synthesis conditions. Under normal conditions FeMnAs crystallizes into the T structure; if the synthesis is performed under high pressure and temperature (P = 3.5 GPa, T = 800 C) FeMnAs will have the H structure. The T FeMnAs was found to be an antiferromagnet below T N = 463 K, while H FeMnAs is a c-axis ferromagnet with T c = 190 K [53]. The properties of other FeMAs compounds have not been well studied. The ternary-metal arsenide FeCrAs has a crystal structure of hexagonal Fe 2 P type. Before the work we will present in this chapter, this compound had only been studied by x-ray diffraction [44, 54] and Mossbauer spectroscopy [45]. The former gives the crystallography of this compound, the latter shows that the iron moments do not display long range magnetic order down to 4.2 K. However, from previous studies [45, 55, 56] we know that for the ternary-metal arsenide with hexagonal Fe 2 P type crystal structure, there are two crystallographically nonequivalent metal positions, a tetrahedral site and a pyramidal site. The usual rule is that the more electropositive transition element occupies the pyramidal site [57]. In the case of FeCrAs, the pyramidal sites and tetrahedral sites are occupied by Cr and Fe, respectively [55, 57, 58]. The Fe 2 P crystal structure can be viewed in a number of ways [44]. Figure 4.1 (c) emphasizes the distinct chromium (blue atoms) and iron (red atoms) sublattices. The Cr sublattice can be viewed as a distorted Kagome lattice, while the iron atoms form

92 Chapter 4. A novel non-fermi-liquid state in FeCrAs 81 Figure 4.1: Crystal structure of FeCrAs. Iron atoms are shown as small red balls, Cr atoms are blue and As atoms are green. (a) and (b): the primitive unit cell from the side and the top respectively. The blue tetrahedron shows the tetrahedral coordination of iron by arsenic. (c): looking down the c-axis at a three by three slice. The network of pink triangles illustrates the distorted Kagome lattice on which the Cr atoms sit, while the gray triangles illustrate the triangular lattice of Fe trimers in the front layer.

93 Chapter 4. A novel non-fermi-liquid state in FeCrAs 82 trimers, the gray triangles of Fig. 4.1, that lie on a triangular lattice. These planes of iron trimers and the chromium Kagome planes are alternately stacked along the c- axis, and the triangular network of trimers in particular offers interesting possibilities for frustration. However, the interaction between the iron atoms within a trimer is not known. The relevant information we know are the interatomic distances, which are given below: Fe-Fe within trimer Fe-Fe between trimers Å Å Fe-Fe to atom in layer below 3.666Å Fe- nn Cr Fe- nnn Cr Cr-Cr nearest neighbor Cr-Fe nearest neighbor Cr-Fe nnn Fe-As nearest neighbor (nn) Fe-As nnn Å Å Å Å Å Å Å This is the chromium that is along the bond joining the Fe atom with the center of the next trimer. There are two such nearest neighbors, one in the plane above, and one in the plane below. There are four such neighbors, two in the next plane up, two in the next plane down, located in the bonds that are 120 o to the bond on which the Fe atom sits. The chromium atom sits near the centre of a triangular prism of Fe atoms, displaced slightly towards the Fe atom along the same bond direction as the chromium, so these distances are slightly different. There are thus two Fe atoms at the nearer distance (one in the next plane up and one below), and four Fe atoms at the farther distance, two in the next plane up and two in the next plane below. Four As atoms tetrahedrally coordinate the Fe atom. The two in the same plane as

94 Chapter 4. A novel non-fermi-liquid state in FeCrAs 83 the Fe trimer lies are slightly nearer to the Fe atom, and therefore denoted as nearest neighbor ; the two out of the plane are called nnn, means next nearest neighbor. FeCrAs seems to be a good candidate for a frustrated metallic spin system. This prospect initially motivated us to grow and investigate this compound. Our preliminary studies unveiled a number of interesting properties. In particular, we found that it exhibited a rarely-seen non-metallic, non-fermi-liquid resistivity. Some of the properties of FeCrAs are commonly seen in geometrically frustrated systems; but others are not generic properties of a frustrated system. This makes FeCrAs even more interesting. My project of studying FeCrAs started in In early 2008, the first iron-pnictide superconductor La[O 1 x F x ]FeAs was discovered [59]. We noticed that FeCrAs shares a number of characteristics with the new type of superconductor. However, significant differences exist between FeCrAs and the iron-pnictide superconductor. Metallic states that violate Landau s Fermi liquid paradigm appear in many strongly correlated electron systems such as doped cuprates, quantum critical metals, and disordered Kondo lattices [60 62]. The new layered iron-pnictide high temperature superconductors also show non-fermi-liquid behaviour, most notably in the undoped parent compounds which have a high, roughly constant resistivity at temperatures above a magnetic spin-density-wave transition that typically occurs around T SDW 150 K (see, for example, [59, 63 65]). For T < T SDW, however, this incoherent charge transport gives way to a rapidly falling (i.e. metallic ) resistivity. The basic structural units of the new iron-pnictide superconductors are layers in which iron atoms are tetrahedrally coordinated by arsenic. Some theoretical approaches to the incoherent T > T SDW state in these systems suggest that the physics is different from that of the cuprates, not least because this tetrahedral coordination produces only a small crystal-field splitting of the Fe 3d-shell. Sawatzky et al. [66] argue that the electronic structure is dominated by narrow iron 3d bands coupled to a highly polarizable arsenic environment. Dynamical mean field theory (DMFT) treatments [67 69] find an

95 Chapter 4. A novel non-fermi-liquid state in FeCrAs 84 incoherent metallic state that arises from strong on-site inter-orbital interactions in the 3d-shell that perhaps puts some [67] or all [68] of the weakly crystal field split 3d-orbitals close to a Mott transition. The local physics of this so-called bad semiconducting state may represent a qualitatively new kind of incoherent metal. In this chapter I present magnetic, thermodynamic and transport properties of Fe- CrAs, and point out that the behaviour shows intriguing similarities to the parent compounds of the layered iron-pnictide superconductors, including an antiferromagnetic transition in a similar temperature range (T N 125 K). However there are key differences: firstly, the crystal structure is hexagonal as opposed to tetragonal; secondly, although the iron atoms are tetrahedrally coordinated by arsenic, FeCrAs is three-dimensional (it does not have insulating layers); thirdly, as shown below, in FeCrAs the magnetism for T < T N resides primarily on the Cr sites, whereas in the layered iron-pnictides the magnetism is on the iron sites; and finally, the behaviour of the electrical resistivity of FeCrAs is much more extreme in that it is non-metallic over a huge temperature range the in-plane resistivity rises with decreasing temperature from 800 K to below 100 mk without any sign of saturation or a gap at low temperature. Moreover, there is a profound and novel incompatibility at low temperature between the resistivity, which is non-fermi-liquid with a sub-linear dependence on temperature, and the specific heat, which has a Fermi-liquid linear-in-t dependence as T 0 K. 4.2 Experiment Single crystals of FeCrAs were grown by melting stoichiometric quantities of high purity Fe, Cr and As following the recipe given in chapter 3. Several batches of crystals were grown and annealed at 900 C in vacuum for five to ten days. The two batches obtained from the tenth and eleventh growths (see chapter 3) are our best samples, which for clarity will be labelled with the subscript pure. The reasons will be given in the

96 Chapter 4. A novel non-fermi-liquid state in FeCrAs 85 Figure 4.2: Schematic diagram showing the four-point-contact configuration of the resistivity measurements. The current leads are soldered on the sample, and the voltage leads (A,C) and the Hall-effect leads (C,D) are spot-welded on the sample. Figure 4.3: The wired sample for the resistivity measurements in our dilution fridge. The silver wire at the lower end of the sample (which is in the middle of the rotation bobbin) is the current lead to which the heat sinking wire is connected. The coil on the left is for determining the orientation of the sample in the magnetic field. next section. All the data shown in this chapter came from the pure samples, unless otherwise indicated. At some places, we also show the data from the less pure samples, the purpose is to demonstrate if the physical quantity has a strong sample dependence. Transport measurements used a conventional four terminal technique, either in a dilution refrigerator with a superconducting magnet for measurements below room temperature, or in a tube furnace under vacuum for high temperatures. The wiring of the sample for the resistivity measurements in the dilution fridge is illustrated in Figure 4.2 and Figure 4.3. To hold the sample in place, a thick silver wire (1 mm) is used as one of the current leads; a thinner copper wire (70 µm) is used for another current lead so that stress will not build up (due to the thermal expansion) in the sample as temperature changes. The voltage leads (A,C) and the Hall effect leads (C,D) are made of 25 µm

97 Chapter 4. A novel non-fermi-liquid state in FeCrAs 86 gold wires. For the measurements in the tube furnace, the maximum temperature was limited by evaporation of the gold leads. Magnetic susceptibility measurements were done in a Quantum Design MPMS system, while specific heat measurements were done in a Quantum Design PPMS system. 4.3 Results Figure 4.4 and Figure 4.5 show the temperature dependence of resistivity for current along the a-axis and c-axis, respectively. The absolute resistivity ρ along the a and c axes ranges from 200 to 500 µω cm, which at high temperature at least is typical for a strongly correlated metal. There is only weak anisotropy in the resistivity, reflecting the absence of insulating layers in this material. As shown in Figure 4.4, the a-axis resistivity ρ a (T) shows a remarkable non-metallic behaviour, growing with decreasing temperature from 800 K down to at least 80 mk. An expanded plot of the a-axis resistivity at low temperature is shown in the Figure 4.4 (a), where it can be seen that there is no sign of saturation or a gap that would give rise to exponentially increasing resistivity as T falls. Instead, ρ a (T) has an exotic non-metallic and sub-linear in T behaviour as T 0 K: ρ a (T) = ρ a,0 AT 0.60±0.05. The c-axis resistivity shows similar behaviour except that there is a very pronounced maximum, or cusp, at a certain temperature that is sample dependent. Figure 4.5 (b) shows the resistivities from three different samples. The cusps in resistivity for S1 pure and S2 pure appear at 125 K, while that for S3 shows up at 100 K. The two samples that exhibit this resistivity maximum at higher temperatures also show spin freezing (the deviation of χ vs T for field-cooled (FC) and zero-field-cooled (ZFC) modes) at lower temperatures. Based on this, we believe that these two batches of samples are less disordered, and thus we label them pure. Referring to Figure 4.4, examined closely, the in-plane resistivity, which was measured on one of the pure samples, also shows a

98 Chapter 4. A novel non-fermi-liquid state in FeCrAs (a) ρ a (µω.cm) T (K) 500 (b) ρ a (µω cm) a-axis T (K) Figure 4.4: (a): resistivity vs. temperature for a-axis between 80 mk and 4 K, demonstrating that it does not saturate or show a gap-like structure down to the lowest temperature measured. (b): the a-axis resistivity between 2 K and 800 K.

99 Chapter 4. A novel non-fermi-liquid state in FeCrAs (a) 388 S2 pure (eleventh growth) S3 (eighth growth) ρ c (µω cm) T (K) (b) S1 pure (tenth growth) S2 pure (eleventh growth) S3 (eighth growth) ρ c (µω.cm) T (K) Figure 4.5: Resistivity vs. temperature for c-axis. Data from different samples are presented to show the sample dependence of the c-resistivity. (a): an expand plot for low temperatures (80 mk 5 K); the resistivities in two different samples behave similarly in this temperature range. (b): ρ vs T in the range 2 K< T < 900 K; for medium temperatures ( 25 K< T < 150 K), the resistivity is sample-dependent. Note that data for S3 has been scaled (by multiplying a number 1.2) to match the T >150 K curves of S1 pure and S2 pure.

100 Chapter 4. A novel non-fermi-liquid state in FeCrAs K S pure from tenth growth K S pure from eleventh growth ρ c (µω cm) 418 ρ a (µω cm) H (T) H (T) Figure 4.6: The c-axis resistivity vs field at 2 K for the pure sample, S pure, which was obtained from the tenth growth. Figure 4.7: The a-axis resistivity vs field at 2 K for the pure sample, S pure, which was obtained from the eleventh growth. slope change at approximately 125 K. The low temperature resistivities (T < 5 K) for a pure and less pure samples are presented in Figure 4.5 (a). Though their magnitudes are slightly different, the power law dependence on T is the same, i.e. ρ c (T) = ρ c,0 AT 0.70±0.05, which is again sub-linear in T. At high temperatures, we followed the negative dρ/dt to nearly 900 K without finding any sign of saturation. It should be noted that both the temperature at which the cusp appears and the magnitude of the drop in ρ c below this cusp have large sample to sample variations, however the low temperature power law, and the resistivity above 150 K, are sample independent. The Figures 4.6, 4.7, 4.8 show the field dependence of the resistivities from three different samples. These results show that the field dependence of the resistivity is very weak for both ρ a and ρ c ; ρ a decreases by 0.08% from 0 to 15 Tesla at 170 mk for the sample obtained from the third growth. Note that the magnetoresistivity is nearly sample independent. The field dependence of the ac susceptibility is also very weak, see Figure 4.9. We have also measured the Hall effect at room temperature, 120 K and between mk, but find that the Hall signal is too small to be resolved. Here we only present the data at 83 mk and 120 K for two different samples, see Figure

101 Chapter 4. A novel non-fermi-liquid state in FeCrAs 90 4 ρ a (µω cm) mk Sample from the third growth χ ( 10-6 emu) K Sample from the third growth H (T) H (T) Figure 4.8: The a-axis resistivity vs field at 170 mk for the sample obtained from the third growth. Figure 4.9: The real part of the ac susceptibility vs field at 2 K for the sample obtained from the third growth Based on the resolution of our devices, the low-temperature and the intermediate temperature ( 120 K) Hall coefficients are estimated to be less than m 3 /C and m 3 /C, respectively. This suggests that FeCrAs is a compensated metal with a scattering rate that is uniform over the Fermi surface. Fig (a) shows the temperature dependence of the magnetic susceptibility χ of one of the pure samples for fields along the a- and c-axes, under both field cooled (FC) and zero field cooled (ZFC) conditions. For both orientations, a clear peak is observed at 125 K, below which χ a and χ c deviate from one another. In addition, at lower temperatures a weak difference appears between the FC and ZFC susceptibilities, which is indicative of magnetic freezing as is observed in spin-glasses. In Fig (a) this magnetic freezing sets in below 7 K and 15 K for H c and H a respectively; however, as noted above we have found that this behaviour is very sample dependent, starting as high as 45 K in some crystals. However, we emphasize that, aside from the size of the drop in ρ c (T), the physical quantities, ρ(t), C(T) and χ(t), shown in this chapter do not show qualitatively different behaviour between samples. The overall change of the susceptibility is small in the whole temperature range ob-

102 Chapter 4. A novel non-fermi-liquid state in FeCrAs 91 Hall coefficient (m 3 /C) 2e-08 1e mk c-axis 83 mk ab-plane 0 4e H (T) Hall coefficient (m 3 /C) 3e-08 2e-08 1e K ab-plane H (T) Figure 4.10: (a): the Hall coefficient at 83 mk for both field in the ab-plane and along the c-axis; the sample was obtained from the third growth. (b): the Hall coefficient at 120 K for field in the ab-plane; the pure sample from the tenth growth, is used.

103 Chapter 4. A novel non-fermi-liquid state in FeCrAs (a) χ (10 3 emu/mol ) H // a-axis H // c-axis T (K) 15 (b) 1/(χ χ ο ) (10 3 mol /emu) 10 5 H // a-axis H // c-axis T (K) Figure 4.11: (a): Temperature dependence of the susceptibility χ for fields along a- axis (red curve) and c-axis (black curve). A small difference between zero-field-cooled and field-cooled conditions is visible at the lowest temperatures. (b): the temperature dependence of 1/(χ χ 0 ) showing an absence of Curie-Weiss behaviour.

104 Chapter 4. A novel non-fermi-liquid state in FeCrAs 93 served, that is, χ(t) is quite Pauli-like and does not show Curie-Weiss temperature dependence. Figure 4.11 (b) shows that even after a temperature independent term χ 0 has been subtracted to make 1/(χ-χ 0 ) as linear as possible, the inverse of χ still deviates from a straight line. However, it seems that the curvature of the inverse susceptibility decreases as temperature rises towards room temperature. So it is likely that the linearity of the inverse of the susceptibility will appear at T T N, and that the deviation of the linearity shown in Figure 4.11 (b) is due to the geometric frustration. For all our pure samples, no magnetic hysteresis was observed down to 2 K. Figure 4.12 shows the field dependence of the magnetization of the pure sample obtained from the eleventh growth. The data were taken below liquid helium temperature (2 K), around the transition (125 K) and room temperature (300 K) for both a-axis and c-axis. We see that M vs H gives a straight line in these three cases, and no hysteresis behavior can be resolved. Our powder neutron diffraction measurements found magnetic peaks below 100 K, demonstrating that the features in χ(t) and ρ c (T) in the temperature range 100 K 125 K arise from magnetic ordering. The temperature dependence of χ and the hysteresis measurement at 2 K indicate that this ordering is antiferromagnetic. Indeed, the magnetic peaks in the neutron diffraction were indexed to a propagation vector of k=( 1, 1, 0) 3 3 corresponding to a commensurate spin-density wave that triples each in-plane unit cell dimension. Fitting the neutron spectra gives a moment on the Cr sites that varies between a maximum of 2.2 and a minimum of 0.6 µ B. The moment on the iron sites is much weaker, and indeed within the noise it is consistent with an earlier Mössbauer spectroscopy measurement [45] which found very weak magnetic order on the iron sites at 4.2 K, with the magnetic moment estimated to be less than 0.1 ± 0.03 µ B. 1 In striking contrast to the resistivity, the specific heat, shown in Fig (a), exhibits 1 Powder neutron diffraction experiments were performed in collaboration with Ian Swainson, and he carried out the magnetic structure analysis.

105 Chapter 4. A novel non-fermi-liquid state in FeCrAs 94 M (emu/mol) (a) 300 K 125 K 2 K a-axis H (T) 60 (b) M (emu/mol) K 125 K 2 K c-axis H (T) Figure 4.12: (a): magnetization vs field at 2 K, 125 K and 300 K for field along a-axis. (b): magnetization vs field at the three different temperatures quoted in (a), but with field along c-axis.

106 Chapter 4. A novel non-fermi-liquid state in FeCrAs (a) C/T (mj/mol K 2 ) Pure sample from the tenth growth Sample from the first growth Sample from the third growth T 2 (K 2 ) 80 (b) 60 C (J/mol K) Pure sample from the tenth growth Sample from the first growth Sample from the third growth T (K) Figure 4.13: (a): C/T vs T 2 at low temperatures for three different samples. The green line is a linear fit to the data of the pure sample. (b): the temperature dependence of C(T) from room temperature to 2 K for the same three samples. The Néel transition produces a weak anomaly at 125 K for the pure sample, see the black arrow.

107 Chapter 4. A novel non-fermi-liquid state in FeCrAs 96 classic Fermi liquid behaviour at low temperatures: C(T)/T depends linearly on T 2. The Sommerfeld coefficient, γ = C el /T, extracted from the data of the pure sample (which was obtained from the tenth growth) shown in Fig (a) gives a value of 31.6 mj/mol K 2, which is quite high for a d-electron system. We find that γ varies between samples by up to 20%, however the linear relationship of C(T)/T vs T 2 holds in all samples down to at least 1 K. At high temperatures, the specific heat shows a weak anomaly near the Néel temperature for the pure sample (see the black arrow in Figure 4.13 (b)), but nothing for the other two samples. To compare the enhancements of the specific heat coefficient and the spin susceptibility, the Wilson ratio, R W = Kχ/γ, is commonly applied, where K is a scale factor which gives a dimensionless value R W = 1 for a free electron gas. For a Kondo system, a value of R W = 2 is expected. In FeCrAs, however, we find R W 5 from the c-axis susceptibility and R W 4 from the a-axis susceptibility, note that the γ value of the pure sample obtained from the tenth growth is used in the calculation of the Wilson ratios. 4.4 Discussion The unusual features of our results are as follows: 1) both the in-plane and c-axis resistivities show strong non-metallic behaviour with negative dρ/dt to beyond 800 K, the highest temperatures we measured; 2) at low temperatures, both the in-plane and c-axis resistivities have strongly non-metallic, non-fermi-liquid behaviour, while the specific heat is Fermi-liquid like; 3) the linear coefficient of specific heat γ is unusually large for a d-electron material; 4) there is a large, roughly temperature independent (i.e. Pauli-like) susceptibility, giving a Wilson ratio of between 4 and 5; and 5) the ordered moment on the iron sites is unusually small for an iron compound. Some of these features are common to the parent compounds of the new iron-pnictide superconductors, notably the large value of γ, the high resistivity above the magnetic or-

108 Chapter 4. A novel non-fermi-liquid state in FeCrAs 97 dering transition, the weakness of magnetism on the iron sites, the Pauli-like susceptibility and the high Wilson ratio (see e.g. [59, 63 65]). Even the magnetic ordering transition temperature is similar, however the magnetism here is of a very different origin: FeCrAs adopts a ( 1, 1, 0) in-plane antiferromagnetic structure but this ordering is clearly driven 3 3 by the chromium sublattice, while in the layered pnictides the SDW-magnetic order is driven by the Fe sites. This may explain why the incoherent transport in the layered iron-pnictides disappears below T SDW, while in FeCrAs it continues to T 0 K. (It should be noted that in polycrystalline LaFeOAs the resisitivity at low temperature does show an upturn as T 0 K, which may be intrinsic or a grain boundary effect [59].) The similarities suggest that there may be physics that is common to the incoherent states of both FeCrAs and the layered iron-pnictides. For example, the fall in ρ c (T) just below T N 125 K is similar to that for T < T SDW in the layered materials, and may be a signature of a spin-fluctuation contribution to the resistivity. Dai et al. [70] have suggested that the incoherence of the layered iron-pnictides arises from spin-fluctuation scattering enhanced by frustrated exchange interactions. Similar arguments may apply in FeCrAs, and indeed the crystal structure of FeCrAs is more amenable to frustration than that of the layered iron-pnictides. In FeCrAs, the planes of iron trimers and the chromium Kagome planes offers interesting possibilities for frustration. We note that the small ordered Fe magnetic moment, the very weak dependence of ρ(t), χ(t) and C(T)/T on magnetic field, and signatures of magnetic freezing in these crystallographically well-ordered crystals, are all typical of magnetically frustrated systems [71, 72]. However, the antiferromagnetic order is a major complication in this scenario, as is the absence of Curie-Weiss susceptibility. An even more serious difficulty is the energy scale: the related tetragonal materials Fe 2 As and Cr 2 As order antiferromagnetically at 350 K [50] and 393 K [73]. If we take this as the typical scale of the exchange energy J in these systems, it is then difficult to see how the non-metallic resistivity could persist

109 Chapter 4. A novel non-fermi-liquid state in FeCrAs 98 to temperatures that are twice as high. In most spin fluctuation systems the resistivity is a rising function of temperature, and it saturates when T is larger than J. Finally, we note that the weak magnetism on the iron sites is probably due to electronic structure effects, not magnetic frustration. LMTO calculations by Ishida et al. [74] consistently find for the Fe 2 P structure that the tetrahedrally coordinated site (the Fe site in FeCrAs) has a small moment, while the pyramidal site, coordinated by five As atoms (the Cr site in FeCrAs), has a large moment. Indeed for FeCrAs they find that the density of states of the 3d-bands of Fe is below the Stoner criterion, and although they found that a ferromagnetic ground state is energetically favoured, the antiferromagnetic state that they used for comparison was quite different from the one that we have found. A very different explanation of the high temperature resistivity might be the local physics arising on the tetrahedrally coordinated Fe sites [66 69]. The bad semiconductor/bad metal picture of the incoherent metallic state of the layered iron-pnictides that arises in DMFT calculations is appealing because the energy scale is not the inter-site exchange coupling J. Rather it is the on-site inter-orbital Coulomb interaction, estimated to be on the order of a few ev [66]. We therefore believe that FeCrAs may be an interesting system for exploring this new physics. We emphasize, however, that the strong temperature dependence of the non-metallic resistivity of FeCrAs is different from the roughly temperature independent resistivity seen at high temperatures in the parent compounds of the iron-pnictide superconductors. Whether the DMFT picture explains the low temperature limit is not clear. Laad et al. [69], in attempting to explain optical conductivity, find a scattering rate that is sub-linear in ω, which may translate into a resistivity that is sub-linear in T. However, the most striking feature of our low temperature results the incompatibility of the nonmetallic and non-fermi-liquid resistivity with the classic Fermi-liquid specific heat has not been discussed within either the spin-fluctuation or the DMFT pictures. There are a number of materials that show strong non-metallic resistivity over a

110 Chapter 4. A novel non-fermi-liquid state in FeCrAs 99 large temperature range, terminating at low temperature in a non-fermi-liquid state. Underdoped cuprates are an obvious example [12], however in that case the T 0 K limit of ρ(t) has a logarithmic divergence, not power-law behaviour. Some disordered heavy fermion systems [6, 62] have non-metallic resistivity over a large temperature range, with low temperature linear-in-t behaviour, however in all of these cases the specific heat also shows strong non-fermi-liquid behaviour as T 0 K, typically C(T)/T ln(t). For example, in CeRh 2 Ga, C(T)/T is logarithmic in T and rises from 150 mj/mol K 2 to 480 mj/mol K 2 between 6 K and 2 K [9]. In contrast, the specific heat of FeCrAs is only weakly temperature dependent at low temperature and it is linear-in-t as T 0 K. Moreover, the disordered Kondo systems show dramatic rises in χ(t) over the whole temperature range, typically by a factor of around 10 between room temperature and 1 K, with increasing slope as T 0 K. The Pauli-like χ(t) of FeCrAs is completely different. We note additionally that the disordered Kondo systems have a large magnetoresistance at low temperature, while FeCrAs does not. It should be noted that there is one intriguing exception among the heavy fermion materials to the divergent C(T)/T accompanying a non-metallic non-fermi-liquid resistivity, and that is another arsenide, CeCuAs 2, in which C(T)/T exhibits a fall, instead of logarithmic increase, below 2 K [8]. However in this material too the susceptibility rises very strongly at low temperature, unlike FeCrAs. The under-screened Kondo effect has been suggested as an explanation of the non-metallic non-fermi-liquid resistivity in CeCuAs 2 [75], but given the apparent quenching of iron local moments by tetrahedral coordination with As [66, 74], this scenario seems unlikely in FeCrAs. Finally, disorder-induced localization effects, which have been invoked to explain nonmetallic non-fermi-liquid behaviour in the moderately site-disordered U heavy fermion compounds URh 2 Ge 2 [11] and UCu 4 Pb [10] in the range of 1 K to 300 K, seem to be ruled out because those materials exhibit quite a large magnetoresistance, clear signals in the Hall effect, and non-fermi-liquid behaviours in heat capacity and magnetic

111 Chapter 4. A novel non-fermi-liquid state in FeCrAs 100 susceptibility, while FeCrAs has none of these properties. Thus we believe that the behaviour of FeCrAs is qualitatively different from any previous observations of non-metallic, non-fermi-liquid behaviour in the T 0 K limit. It is natural to hypothesize that the linear low-temperature specific heat arises from fermionic excitations that are distinct from the non-metallic non-fermi-liquid charge carriers. Such neutral fermions arise in some theories of fractionalization in insulating, frustrated spin liquids [76 78], and may recently have been observed experimentally [79]. Fractionalization has also been introduced to describe antiferromagnetic quantum criticality in some heavy fermion compounds [80, 81]. A key ingredient of most fractionalization scenarios is a geometrically frustrated spin system (see e.g. [82]). We have noted some difficulties with this picture in FeCrAs, but it may be relevant that the DMFT studies of the layered iron-pnictides [67, 69] find orbitally selective effects that can be tuned by varying the on-site interactions. Although it is not believed that any of the 3d-orbitals of Fe in the layered iron-pnictides are Mott localized, the local environment of the iron sites in FeCrAs may be sufficiently different from the layered systems that some of the 3d-orbitals do localize, producing local degrees of freedom that do not contribute to charge transport. It is far from clear, however, if these degrees of freedom would be fermionic at low temperatures. 4.5 Conclusion In summary, FeCrAs is an unusual d-electron system in which non-fermi-liquid behaviour has been observed. Its resistivity shows non-fermi-liquid behaviour obeying ρ(t) = ρ 0 AT x at low temperatures with x = 0.60±0.05 for ρ a and x = 0.70±0.05 for ρ c, while at high temperatures ρ decreases with increasing temperature up to at least 800 K. The low temperature specific heat, on the other hand, exhibits typical Fermi-liquid behaviour with a linear temperature dependence. This low temperature state is very robust in

112 Chapter 4. A novel non-fermi-liquid state in FeCrAs 101 the sense that it exhibits very weak dependence on field, and the same behaviour is observed in all the samples we have measured. The high temperature behaviour may be an extreme example of the bad semiconductor state found by DMFT calculations for the layered iron-pnictides, and although the low temperature state may have the same origin, it deserves further study as it has some features expected of a fractionalized electron system which have not, to our knowledge, been observed before.

113 Chapter 5 Pressure induced quantum criticality in Sr 3 Ru 2 O 7 In this chapter, we will describe our hydrostatic pressure study of metamagnetic quantum citicality of Sr 3 Ru 2 O 7. Previous studies show that there is a new kind of quantum critical point, called quantum critical end point (QCEP) in this system, and that a novel phase exists in the vicinity of QCEP. In those studies, the QCEP is realized by tuning the field angle. We decided to approach the quantum criticality of Sr 3 Ru 2 O 7 using a different tuning parameter, i.e. the hydrostatic pressure. The goal is to compare the QCEP realized by hydrostatic pressure with that produced by tuning the field angle, especially, to check if a new phase that was discovered at the field-angle-tuned QCEP is also present at the pressure-induced QCEP. This chapter will be organized as following: firstly, the characteristics or properties of the QCEP and the new phase revealed by the previous studies will be summarized; secondly, the experimental details of our measurements, especially, the ac susceptibility probe used in this work will be described; then, our experimental results are presented; finally, the significance of the data is discussed and our findings from this work is summarized. 102

114 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O Material background The strontium ruthenates, represented by Sr n+1 Ru n O 3n+1, have the Ruddlesden-Popper type structure containing layers of RuO 6 octahedra. As the number of layers, n, is varied, the series show ground state behavior covering a wider range than that seen in almost any other transition metal oxide. The bands at the Fermi level involve the hybridization of oxygen 2p and ruthenium 4d, and in contrast to 3d oxides such as most manganites and many cuprates, no explicit chemical doping is required to produce metallic conduction. This gives a unique opportunity to probe a wide range of correlated electron physics in the low disorder limit. These characteristics provide a strong motivation for studying the compounds in this series. The single-layered ruthenate Sr 2 RuO 4 (n=1) is the first layered perovskite oxide superconductor without copper. It has a transition temperature T c 1.5 K, and exhibits a triplet spin pairing, instead of the singlet spin pairing as seen in cuprate high T c superconductors [83]. The n = ruthenate SrRuO 3 is a ferromagnet with T c = 160 K. At medium and high temperatures, it is a bad metal, but for thin films below 10 K the behavior of the resistivity is Fermi-liquid like [84]. The double-layered ruthenate Sr 3 Ru 2 O 7 (n = 2) lies in between the system with n = 1 and n =. The structure of the double-layered ruthenate Sr 3 Ru 2 O 7 is shown in Figure 5.1. The crystal is formed by stacking the bilayers of Ru-centered octahedra, which are sandwiched between layers of Sr, along the c-axis. Neighbouring RuO 6 octahedra are rotated around the c-axis by about 7.8. The crystal symmetry was first considered to be that of tetragonal space group Pban [85], but later single and powder neutron diffraction work [86, 87] showed that the orthorhombic space group Bbcb is more accurate. At ambient pressure, Sr 3 Ru 2 O 7 is a paramagnetic metal on the verge of ferromagnetism down to the lowest temperature. Its resistivity is highly anisotropic (ρ c ρ ab ), with ρ c and ρ ab falling rapidly below 50 K and 20 K, respectively, followed by a

115 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O Figure 5.1: The unit cell of Sr 3 Ru 2 O 7. This diagram is derived from the space group Bbcb, which is claimed to be the most reliable space group [87]. The lattice parameters used are a=5.4979(4) Å, b=5.5008(10) Å and c= (19) Å. Each RuO 6 octahedron rotates around the c-axis with respect to the next one by about 7.85.

116 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O crossover to Fermi-liquid behavior, i.e. ρ AT 2, for T < 10 K. The dc magnetic susceptibility is nearly isotropic in the temperature range 2 K T 300 K. It exhibits an enhanced Pauli paramagnetic susceptibility, with a pronounced peak around T max =16 K. The peak in χ(t) seems to be an inherent feature of exchange-enhanced paramagnets. Though the resistivity is highly anisotropic, the magnetic susceptibility becomes completely isotropic below 5 K. The Sommerfeld coefficient of the specific heat γ 110 mj/(k 2 mol Ru) is among the highest in any oxide [22, 88]. The crystal field scheme of the Ru 4+ d orbitals has been found by optical studies [89, 90]. In the RuO 6 octahedra, the 2p levels of oxygen and the 4d levels of Ru 4+ are hybridized. In the octahedral crystal field, these orbitals are split by an energy 10Dq, into a triplet t 2g and a doublet e g. In Sr 3 Ru 2 O 7, the crystal field is strong enough that 10Dq > J, where J is the Hund energy. Therefore, in the ground state the four electrons are all in the t 2g state, and the total spin S is 1. The degeneracy of the t 2g state is lifted by the Jahn-Teller distortion. The Fermi surface of Sr 3 Ru 2 O 7 is complicated due to bilayer splitting. There are six bands crossing E F. So it has many Fermi surfaces derived from both the in plane Ru 4d xy orbital and out-of-plane Ru 4d yz,zx orbitals, in a striking contrast to the cuprates which have a Fermi surface derived from only the in-plane Cu 3d x 2 y 2 orbital [91]. Naively, one might expect the electronic structure of Sr 3Ru 2 O 7 to be a simple doubling of that of the single layer ruthenate Sr 2 RuO 4, i.e. four electron-like and two hole-like Fermi sheets. However, unlike in Sr 2 RuO 4 whose Hall coefficient has two temperature dependent sign changes and is negative at low temperatures [92], the Hall coefficient in Sr 3 Ru 2 O 7 is positive over the temperature range 30 mk 300 K, indicating that the dominant carriers are holes in the whole temperature range. The double-layered ruthenate attracts a lot of interest because it undergoes a firstorder metamagnetic transition (MMT) at low temperatures, whose critical end point temperature T can be tuned to zero, thus realizing a metamagnetic quantum critical end point (QCEP) [21, 22]. Metamagnetism is empirically defined as a superlinear change

117 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O of magnetization (indeed a discontinuous jump in the case of a first-order MMT) in a narrow magnetic field range. For a sample with purity ρ res 2.4 µωcm, ac susceptibility shows that for H ab the first-order metamagnetic phase transition (MMT) occurs at 4.9 T, which terminates in a critical point T = 1.25 K; for the same field orientation, two weaker metamagnetic anomalies are seen at higher fields H 5.8 T and H 6.3 T. As the field is rotated from the ab-plane toward the c-axis, the MMT gradually moves to higher field, and the critical end point temperature T of the MMT falls from 1.25 K toward zero. By H c, the metamagnetic transition is shifted to 7.9 T and T falls to below 50 mk; a metamagnetic QCEP is believed to exist for this field orientation. Here the tuning parameter is the field angle. Signatures of a quantum critical point, such as non-fermi-liquid resistivity, logarithmically divergent specific heat divided by temperature C/T, have been observed near the QCEP [20, 22]. What makes Sr 3 Ru 2 O 7 even more interesting is that a new kind of ordered electron state has been observed around the QCEP. In extremely clean samples, i.e. having ρ o < 0.5 µω cm and denoted as ultra-pure in this thesis, T does not go to zero, rather it has a minimum around θ 60, and then it rises again accompanied by another, nearby, first-order jump at slightly higher field. The T s for the two first-order metamagnetic transitions grows as the putative QCEP is approached, and each of these MMTs terminates in a finite temperature critical end point with T < 1 K. The region enclosed by the two MMTs exhibits transport anisotropies which are incompatible with the point group symmetry of the host lattice, implying the existence of a nematic-like phase. The phase diagram for the ultra-pure Sr 3 Ru 2 O 7 is presented in Figure 5.2. The green sheets in Figure 5.2 (c) represent the first-order MMT surfaces which are delimited by a line of the critical points on top. The nematic phase region is bounded by the green sheets at its left and right sides and delineated by a roof as indicated by the blue dome. At the boundary represented by the dome, the transition is second-order; at the boundaries represented by the green sheets below the dome, the phase transition is first-order. At

118 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O T (K) (b) H (T) Figure 5.2: (a): a schematic drawing showing the critical end point temperature T of the first-order metamagnetic transition (MMT) in ultra-pure Sr 3 Ru 2 O 7 as a function of field angle θ (relative to the ab plane); at 60 degree another first-order MMT appears at higher field, then T s of both MMTs rise with increasing field angle. A nematic phase is thought to exist between these two MMT transitions. The inset shows the firstorder transition lines at field angles θ 1 and θ 2, with θ 1 < θ 2 (θ 1, θ 2 < 60 degree). (b): a schematic diagram [23] to illustrate the nematic phase for θ = 90 degree. The red arrows mark the T s of the MMTs at the phase boundaries. (c): a schematic phase diagram illustrating the nematic phase (the blue region) of ultra-pure Sr 3 Ru 2 O 7. The two green sheets, which are derived from the ac susceptibility measurements [25], represent the surfaces of the MMTs. Their upper boundaries correspond to T (θ). Between these two green sheets, in the blue region, highly enhanced anisotropic electrical resistivity is observed. At low field angles, considerably enhanced anisotropic resistivity appears in between two metamagnetic anomalies at upper fields. This region is marked by the blue dome around 6 T.

119 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O lower field angles, as indicated by the small blue dome, a less pronounced resistivity enhancement is observed in between the two upper metamagnetic anomalies; though weaker in magnitude, the resistivity has the same characteristic anisotropy as that of the higher field angles [24, 32]. Apart from the anomalous resistivity, other evidence for the existence of the emergent phase has been revealed through a series of experiments. For H c, a number of physical quantities exhibit well-defined features at the boundaries of the nematic region. Both the real part of the ac susceptibility and the c-axis magnetostriction coefficient show three peaks around the metamagnetic critical field H M 7.8 T below 1 K, with the upper two peaks matching the left and right sides of the phase boundaries. The imaginary part of the ac susceptibility also shows two consecutive peaks at the phase boundaries, but the peaks disappear above the critical temperatures of the two metamagnetic transitions. The entropy change S has also been observed as the two first-order phase boundaries are crossed. The indications of a roof bounding the nematic region were provided by a kink in the magnetization, a jump in C/T, and a qualitative change in the temperature dependence of the resistivity upon entering this region from higher temperatures [23, 30, 93] Origin of the metamagnetism The metamagnetism in Sr 3 Ru 2 O 7 has generally been interpreted as a field-tuned Stoner transition into a highly polarized magnetic state. Theoretical work suggests that the phase diagram of Sr 3 Ru 2 O 7 may be understood from the band structure which results in either a local minimum in the density of states at the Fermi level or a sharp increase in DOS over the minute energy scale of Zeeman splitting ( 1 mev for B=10 T) [14, 26, 94]. Density-functional calculations show a number of sharp features in the DOS [95] and have been invoked to support a band-structure-based model of metamagnetism. A number of experiments have been carried out in order to unveil the microscopic origin of the metamagnetism. STM measurements [31] reveal peaks in the tunnelling conductivity

120 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O around ±4 mev of the Fermi energy, which might be understood as a set of van Hove singularities in the DOS. These singularities can produce the metamagnetic transitions in the Photoemission studies reported by [91, 96], on the other hand, failed to detect unusual features on the relevant energy scale. However, the most recent ARPES study [97] analyzed the d xy orbital in the vicinity of the chemical potential, and revealed a complex density of states with Van Hove singularities near the Fermi level. Two peaks at -4 and -1 mev were identified. Therefore, the metamagnetism in Sr 3 Ru 2 O 7 is most likely to be driven by the proximity of the Fermi level to van Hove singularities in bands based on the quasi-two-dimensional Ru d xy orbitals. Based on the metamagnetic features, we might expect that the relevant magnetic correlations in Sr 3 Ru 2 O 7 are predominantly ferromagnetic in nature. This is only true for T > 20 K. At low temperatures, it seems that the ferromagnetic and antiferromagnetic correlations are competing with each other. Single crystal inelastic neutron scattering shows that at H = 0 there are 2-D spin fluctuations which become predominantly ferromagnetic T > 20 K; at low temperatures the fluctuations cross over to 2-D incommensurate antiferromagnetic fluctuations, with wavevectors close to those expected for nesting vectors of the Fermi surface [98]. In the field, for H c, O-NMR [99] gives similar results. On the other hand, single crystal inelastic neutron scattering measurements reported by Ramos et al. show that at T = 1.5 K incommensurate antiferromagnetic fluctuations are present in a wide field range, and remarkably at the metamagnetic field a ferromagnetic peak has also been observed at small energy transfer [100]. Ferromagnetic fluctuations are also observed in the thermal conductivity measurements. Ronning et al. [101] argued that it is the ferromagnetic fluctuations that cause the non-fermi-liquid behaviors in the vicinity of the QCEP down to 40 mk.

121 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O Nature of the nematic phase Usually the Fermi surface has the point-group symmetry of the underlying lattice structure. However, in theory, the Fermi surface can undergo a d-wave type deformation in which the Fermi surface expands along the k x direction and shrinks along the k y direction or vice versa, and thus break the symmetry of the host lattice. This type of Fermi surface deformation is often called a d-wave Pomeranchuk instability, referring to Pomeranchuk s stability criterion for isotropic Fermi liquids. However, a d-wave Fermi surface deformation can also occur without violating Pomeranchuk s stability criterion, it can even occur in strongly correlated systems [27]. It is thought that the new ordered phase, i.e. the namatic phase, near the QCEP of Sr 3 Ru 2 O 7 is an example of the Pomeranchuk Fermi surface deformation. As the magnetic field increases, the spin-up Fermi volume increases, and when the spin-up Fermi surface gets close to a van Hove singularity, the nematic distortion of the spin-up Fermi surface occurs via a first-order transition and the symmetry of the underlying lattice is broken. When the magnetic field increases further, another first-order transition occurs; and the discrete rotational symmetry is restored in the high field phase. The two consecutive first-order transitions are accompanied by jumps in magnetization. In this scenario, it is assumed that only one branch of the Fermi surface, the spin-up or spin-down, spontaneously breaks the discrete rotational symmetry of the lattice [23, 28]. The nature of the emergent phase has been a recent subject of intensive theoretical and experimental research [20, 21, 23 33]. Most of these efforts have been focused on the nematic phase for H c. Some intriguing properties of the nematic phase have been revealed. The most remarkable phenomenon observed for the field close to c-axis is the unusual anisotropic resistivity which depends on the relative orientation of current, lattice and the in-plane magnetic field. Borzi et al. [24] reported that when current flows in the crystallographic direction most parallel to the in-plane component of the field, the resistivity peak persists as the field is moved away from the c-axis; when it is nearly

122 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O perpendicular to the in-plane field, the peak decreases rapidly. This indicates that the in-plane field does not simply remove the anomalous resistive peak, rather it exposes an intrinsic asymmetry of the underlying phase, defining easy and hard directions for magnetotransport. The microscopic origin of this phenomenon has not been uncovered yet. In addition, quantum oscillations were reported to be observed both in and outside the nematic phase through de Haas-van Alphen (dhva) Effect measurements by Mercure et al. [33]. This measurement also reveals that at least some of the mobile charges in the nematic phase are Landau quasiparticles. Prior to Mercure s work, the quantum oscillations had been investigated by Borzi et al. via dhva and Perry et al. via the Shubnikov-de Haas oscillations measurements [30, 102]. Though they failed to observe the quantum oscillations in the nematic phase, both measurements resolved the quantum oscillations above and below the metamagnetic transition. The former also observed that the quasiparticle effective masses are significantly enhanced as the metamagnetic transition is approached, but the Fermi surface topography change is small across the metamagnetic transition. Electrons may break up at a quantum critical point [4]. This phenomenon is thought to be observed in YbRh 2 Si 2 [103]. In Sr 3 Ru 2 O 7, even though non-fermi liquid behaviors have been observed near the QCEP, thermal conductivity shows that the Wiedemann-Franz law is valid at all fields indicating that there is no electron breakdown at quantum criticality [101]. Since the discovery that the QCEP in Sr 3 Ru 2 O 7 can be realized by applying the field along c-axis [21], most attention has been attracted to the properties for H c. In fact, the system shows more features when the field is applied along the ab-plane. Both the ac susceptibility [21] and DC magnetization [32] measurements show that for H ab there are three metamagnetic anomalies. The first is related to the first-order metamagnetic phase transition; the two at higher fields are different from the first. Perry et al. [32] show that these two anomalies exhibit a double peak feature in dm/dh vs H,

123 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O but no DC magnetization hysteresis can be resolved. Interesting phenomena are observed in between these two anomalies: the electronic scattering is significantly enhanced a resistivity anomaly, i.e. a mesa, is observed in this region; moreover, a logarithmic increase of C/T with decreasing T is seen at 6 T, i.e. in between the two anomalies. Borzi et al. reported that the resistivity anomaly with H ab, though less pronounced, has the same anisotropy as that with H c [24]. This region has been marked with the blue dome in Figure 5.2 (c), little is known about the properties of this region except for the anisotropic resistivity. Despite tremendous efforts, the microscopic details of the nematic phase in Sr 3 Ru 2 O 7 are not well understood. For instance, though a spin-dependent Pomeranchuk deformation of the Fermi surface has been proposed to explain the nematic phase, thermal expansion studies [104] revealed an enhanced entropy within the new phase, in contrast to the expectation for a symmetry-breaking phase transition. Different explanations for the phenomena observed around the metamagnetic transition for H c exist: Honerkamp put forward the scenario of phase separation leading to charge inhomogeneities in the nematic phase [94]; Yamase et al. explained the Pomeranchuk instability and the unconventional, non-fermi liquid properties as arising from tuning of the van Hove singularity of either spin band close to the Fermi energy by magnetic field without invoking the putative quantum critical point [27, 105]. Both the quantum criticality and the emergent phase in the double-layered ruthenate require further study. 5.2 Motivation Prior to Sr 3 Ru 2 O 7, metamagnetic transitions had been reported in several other d- or f-electron metals such as UPt 3 [16] and URu 2 Si 2 [106]. However, only in Sr 3 Ru 2 O 7 has it been possible to study the quantum critical end point, and these studies have been limited to field-angle tuning as described above. This field-angle tuning is proposed to play a role

124 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O analogous to pressure in the sense that rotating the field to the c-axis moves the system away from ferromagnetism, as does applying pressure. The analogy may be more direct, if the mechanism of the dependence of T on field angle is angle-dependent magnetostriction [21]. In this sense, the phase diagram with field angle as tuning parameter would have a close relation to the pressure-induced phase diagram obtained from Ginzburg-Landau treatments [18, 107]. However, in changing the field angle the symmetry changes, and nematic signatures appear only when the symmetry is high, i.e. when the field is close to either the c-axis or the ab plane [24]. With pressure we can tune towards/away from the QCEP without changing the symmetry. This provides a strong motivation for exploring the metamagnetic quantum criticality of Sr 3 Ru 2 O 7 under hydrostatic pressure. To the best of our knowledge, metamagnetism has not yet been tuned to a QCEP by hydrostatic pressure in any metamagnetic system [21]. However, previous pressure studies show that metamagnetic critical points can be tuned to lower temperature by hydrostatic pressure, e.g. from 13 K to 11 K by a pressure of 1.2 GPa in UCoAl [15]. In a strongly 2-D system, changing field angle may have more effects than simple hydrostatic pressure can achieve, so finding out the difference between these two tuning parameters would be especially interesting. An intriguing question is whether the new nematic phase appears with pressure tuning. It is important for our later discussion to note that the region of nematic phase extends in a dome above the critical end point of the first order transition lines that define the low temperature boundaries of the phase, see Figure 5.2 (c). For field in the ab-plane, only the metamagnetic anomaly corresponding to the lower-field boundary of the nematic phase may show first-order behavior (Grigera et al., private communication): it is as though T = 0 K cuts across the dome of a nematic phase. As described below, such a scenario may also apply to our results for the lower, stronger ab-plane MMT near its QCEP. In this chapter, we report a hydrostatic pressure investigation of the quantum critical-

125 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O ity in ultra-pure crystals of Sr 3 Ru 2 O 7 for fields in the ab-plane. This approach enabled us to follow the critical end point down as far as 70 mk, and it allowed us to go beyond the QCEP with P > P c. We also searched for signs of a new phase around the MMT QCEP via ac susceptibility a sensitive probe for detecting very weak first-order transitions. 5.3 Experiment Hydrostatic pressure was applied using a BeCu clamp cell. To achieve a highly homogeneous pressure, Daphne oil 7373 was used as the transmitting medium. The effective hydrostatic pressure at low temperatures was calculated from the known temperature dependence of the superconducting transition of tin. This transition should be measured in zero field; however, quite often some magnetic flux is trapped in the magnet, and the transition temperature is suppressed by the residual field to a low temperature T s. So we decided to measure the tin resistance as a function of field, which is applied in the opposite direction to the residual field, at different temperatures. Figure 5.3 shows the voltage over the tin wire vs H for pressure P 13.4 kbar. If T s < T < T c (T c refers to the zero-field transition temperature), as the applied field increases, the voltage over the tin wire will show a pronounced drop at certain field. This allows us to find out the zero-field T c. As mentioned before, ac susceptibility is the best probe for determining the critical end-point temperature T of a first-order phase transition line. In our study, the ac susceptibility was measured using a set of detection coils and a drive coil. The detection coil set is comprised of three coils, with the central coil connected antiparallel to the two end coils. The drive coil is concentrically wound around the three pick-up coils, see Figure 5.4 and Figure 5.5. The detection coils were wound using 25 µm copper wire, with the number of turns satisfying C 1 = C 1 = 1C 2 2 = 227 turns; the drive coil

126 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O V tin ( 10-5 mv) K 2.9 K 3.12 K 3.14 K 3.15 K 3.16 K K 3.17 K H (T) Figure 5.3: Superconducting transition of tin at different temperatures for P 13.4 kbar. The current to the tin wire is 0.5 ma. The suppressed superconducting transitions are recovered as the effect of the residual field is reduced by the applied field, and produce a fall of the voltage over the tin wire. The temperature at which the drop is half of the maximum is taken as the zero-field transition temperature T c, in this case, T c = 3160 ± 5 mk. consists of 102 turns of 70 µm copper wire. This configuration significantly reduces background pick-up from the feedthrough that carries the wires into the high pressure region, allowing us to see the metamagnetic peak more clearly. For all measurements presented in this paper, a low frequency excitation field of 14 Hz generated by the ac current in the drive coil was employed in order to reduce finite-frequency effects [108]. A sample with approximate dimensions mm 3 was placed in the central pickup coil and thermally grounded to the mixing chamber by electrical coupling through a silver and copper wires. The response of the sample was recorded by an SR830 lock-in

127 Chapter 5. Pressure induced quantum criticality in Sr 3 Ru 2 O Figure 5.4: The detection coils for the ac susceptibility measurements. Coil C 1 and C 1 are connected in parallel, then they are antiparallel connected to coil C 2. The sample sits in the bore of C 2. A variable resistor is usually connected in between P and A (or Figure 5.5: The coils for the ac susceptibility measurements of Sr 3 Ru 2 O 7. What we see is the drive coil; the detection coils lie inside of the drive coil, their connections are shown in Figure 5.4. The tin wire in front of the coils is for determining the pressure. B) in order to cancel the unbalanced signal. The signal from the sample is measured from A and B. amplifier preceded by a low temperature transformer, with a turns ratio of 100, and a 1000 low-noise pre-amplifier. The sample used here was cut from a single crystal of Sr 3 Ru 2 O 7 grown at St. Andrews University 1. The residual resistivity was measured to be ρ res < 0.5 µω cm. The ac susceptibility χ ac is obtained by measuring the ac voltage u ac produced by the alternately changing magnetization of the sample; their relation satisfies χ ac u ac. The principle is as follows. The ac current in the drive coil generates a longitudinal ac magnetic field inside of the detection coils. This ac field induces a varying magnetization in the sample, and an electromotive force (emf) in the detection coils. The varying 1 I am grateful to R. S. Perry, who grew this ultrapure crystal Sr 3 Ru 2 O 7.