tric ﬁeld. As a result, the complex magnetic phase diagrams are obtained. Among a variety of emerging states, one can list: Kondo lattice, mixed valency or Kondo insulators. The insight on the electronic structure of these unusual states of matter by means of photoemission spectroscopy is the primary goal of this work. Moreover, the correspondence between the electronic structure and physical properties is studied with application of speciﬁc heat and electrical resistivity measurements. The structure of this thesis is following. This dissertation is divided into three parts. The ﬁrst, introductory part, consists of four chapters. In the ﬁrst chapter general description of properties of systems is provided. The second chapter deals with the theoretical description of these systems. The third chapter is devoted to discussion of crystal structures and physical properties of selected cerium intermetallics. The general ideas lying under the ab initio calculations are brieﬂy sketched in the fourth chapter. The second part is an experimental section. It contains some theoretical aspects of the photoemission process, as well as, details of performed experiments (i.e. descriptions of experimental setups, etc.). The third part is devoted to obtained results. Each chapter of this section has a form of a preprint of article intended for publication in a journal (or already published article). The ﬁrst chapter of this part deals with the description of the Kondo lattice state in CeCu9In2 compound. These results are confronted with those available for isostructural CeNi9In2, which hosts a mixed valency state. The second chapter describes the evolution of the band structure in the CeRhSb1−xSnx as a function of relative content of Sn and Sb. The obtained results are interpreted in the light of quantum phase transition, which is encountered in the system at x = 0.13. The two subsequent chapters deal with the electronic structure of two heavy fermion superconductors from famous CenTmIn3n+2m family (T -transition metal). Namely, we have studied CeCoIn5 and Ce3PdIn11. The ﬁne structures of the spectral function related to low energy excitations originating from mixing between Ce 4f electrons and carriers from conduction band are collated with theoretical calculations based on tight binding approximation or density functional theory. Bibliography Bloch, F. (1929). Z. Phys., 52(7-8):555 – 600. Kasuya, T. (1956). Prog. Theor. Phys., 16(1):45–57. Kondo, J. (1964). Prog. Theor. Phys., 32(1):37–49. Landau, L. D. (1956). J. Exptl. Theoret. Phys. (U.S.S.R.), 30:1058–1064. Landau, L. D. (1957). J. Exptl. Theoret. Phys. (U.S.S.R.), 32:59–60. Ruderman, M. A. and Kittel, C. (1954). Phys. Rev., 96:99–102. Author’s contribution to the published articles The thesis consists of articles already published in scientiﬁc journals and preprints intended for publication. The list of already published papers with bibliographical information and description of contribution of the author of this dissertation is provided. The contribution to the results presented in chapters, which have not been published is also discussed. 1. Kondo lattice behavior observed in the CeCu9In2 compound R. Kurleto, A. Szytuła, J. Goraus, S. Baran, Yu. Tyvanchuk, Ya. M. Kalychak, and P. Starowicz, Journal of Alloys and Compounds 803, 576-584 (2019). doi: 10.1016/j.jallcom.2019.06.140 The author of this thesis (R. K.) is the ﬁrst author of the paper as well as the corresponding author together with his supervisor (P. S.). He performed the photoelectron spectroscopy measurements. His work involved also sample preparation prior to the electrical resistivity and speciﬁc heat measurement as well as the assistance during experiments. He analyzed obtained data and the results of theoretical calculations obtained by theoretician. The ﬁgures were prepared by him. He also wrote the ﬁrst version of the manuscript and discussed the manuscript with coauthors. 2. Studies of Electronic Structure across a Quantum Phase Transition in CeRhSb1−xSnx ´ R. Kurleto, J. Goraus, M. Rosmus, A. Slebarski, and P. Starowicz, The European Physical Journal B 92, 192 (2019). doi: 10.1140/epjb/e2019-100157-3 R. K. performed photoelectron spectroscopy measurements with help of other coauthors. He also analyzed the experimental data and the result of theoretical calculations obtained by J. G. He prepared the ﬁrst version of the manuscript and all ﬁgures. R. K. took part in the discussion of the manuscript. 3. Direct observation of f-electron hybridization effects in CeCoIn5 R. Kurleto, M. Firysiak, L. Nicolaï, M. Rosmus, Ł. Walczak, A. Tejeda, K. Kissner, J. E. Rault, F. Bertran, D. Gnida, D. Kaczorowski, J. Minar, F. Reinert, J. Spałek, and P. Starowicz, intended for publication R. K. performed photoelectron spectroscopy measurements together with P. S., M. R., Ł. W., A. T., K. K., J. E. R., and F. B. He also oriented the samples (Laue x-ray diffraction) obtained by D. G. and D. K. He analyzed the experimental data and the theoretical results obtained by M. F., L. N., J. S. and J. M. He prepared the ﬁgures. The text was written by R. K. and P. S. 4. Electronic Structure of the Ce3PdIn11 Heavy Fermion Compound Studied by Means of Angle-resolved Photoelectron Spectroscopy R. Kurleto, M. Rosmus, L. Nicolaï, Ł. Walczak, A. Tejeda, D. Gnida, J. E. Rault, F. Bertran, D. Kaczorowski, J. Minar, and P. Starowicz, intended for publication R. K. performed photoelectron spectroscopy experiments together with P. S., M. R., Ł. W., A. T., J. E. R., and F. B. R. K. oriented the samples (Laue x-ray diffraction) obtained by D. G. and D. K. He performed data analysis. He also analyzed and discussed theoretical results obtained by L. N. and J. M. He prepared the ﬁgures, wrote the manuscript and consulted it with P. S. Part I Introduction Chapter 1 Ce 4f electrons Among all elements, the f-block of periodic table seems to be the most intriguing from the viewpoint of the electronic properties. The varying number of f electrons as a function of atomic number (Z) makes lanthanides and actinides an ideal playground in studies of many electron phenomena. The cerium, which is in center of interest of this work, is the lightest element from the f-block which has f electron. The conﬁguration of atomic Ce is anomalous: 2 [58Ce]=[54Xe]4 f 15d16s, the Madelung rule is not obeyed in this case (Kramida et al., 2018). It is proven, that in chemical reactions, ﬁrstly 6s and 5d electrons are conveyed into ionic bonding, resulting in pure f1 valence conﬁguration of Ce3+ ion (Kettle, 1996). High values of Z number suggest the signiﬁcance of the relativistic effects for the elements from f-block of the periodic table. Indeed, the spin-orbit interaction has a crucial impact on electronic structure of Ce and its compounds. The 2F spectroscopic (notation: 2S+1LJ) term is split by this interaction into two multiplets: 2F5/2 and 2F7/2. The 2F5/2 is a ground state according to the Hund’s rules. The complex structure of 4f orbital is reﬂected in constant charge density surfaces presented in the Fig. 1.1 together with 4f radial function 1. As we can see the radial density of probability reaches zero exactly at the position of the center of mass (approximately at the position of the atomic nucleus) and has got a peak-like shape at some distance from nucleus. The behavior of f electron in Ce can resemble that encountered in systems with one electron in s valence shell, to some extent. One should have in mind, that orbitals presented in Fig. 1.1, are valid in case of spherical symmetry (e.g. free Ce3+ ion) or in case of octahedral symmetry (e.g. complexes of Ce with ligands in the octahedral bipiramid arrangement). The crystalline electric ﬁeld inﬂuences the electronic structure of Ce3+ (Bauer and Rotter, 2009). Namely, the 2F5/2 is further split. In systems with cubic symmetry of the Ce sites the doublet-quartet conﬁguration is obtained. Any other type of point symmetry of Ce3+ site results in conﬁguration of energy levels which consists of three doublets. The Ce4+ conﬁguration, is obtained by subsequent ionization, which is nonmagnetic in contrast to the former Ce3+ . The usefulness of CEF theory is undeniable in systems with the permanent magnetic moments. However, the value of magnetic moment on Ce atom immersed in a solid is usually reduced. Thus, one should expect the behavior betwixt Ce3+ and Ce4+ . It is noteworthy that the electronic properties of a system with Ce can be related with these in system to Ce changed by Yb (Kettle, 1996). The valence conﬁguration of [Yb3+] is 4 f 13, so instead of one electron occupying the 4f shell we deal with one hole. Hence, one may expect that physics of Yb systems is somehow related by a mirror symmetry with that of with Ce. 1 Plots of orbitals adapted from internet portal: https://chem.libretexts.org The Kondo scattering is usually identiﬁed as a microscopic origin of quenching of local moments in cerium intermetallics (Matar, 2013; Stewart, 1984). A considerable mixing between f states and conduction band together with strong Coulomb repulsion between electrons on the f shell lead to ambiguous electronic conﬁguration of Ce in a solid. Beyond a Kondo effect (Kondo, 1964), in systems with many sites occupied by Ce atom, the RKKY interaction is observed (Kasuya, 1956; Ruderman and Kittel, 1954). This is an indirect exchange interaction between magnetic moments at different vertices of a lattice mediated by carriers from conduction band. It supports a long range magnetic order. The ground state of the cerium intermetallic compound is interpreted as a result of the competition between two forces: Kondo scattering and RKKY interaction. The intuitive description of the behavior of such systems is provided by means of so called Doniach diagram (Fig. 1.2), albeit it is intended only for one dimensional systems and does not represent exact solution of the problem (Doniach, 1977). The diagram provides the representation of the char

acteristic energies in coupling constant (Jc f ) -temperature plane. The mutual ratio of the strength of Kondo and RKKY interaction can be tuned by f-c coupling constant (Jc f ). The strength of RKKY exchange is represented by the characteristic temperature TRKKY (red dotted line), while the strength of the Kondo scattering is given by the Kondo temperature TK (blue dashed line). Beyond them, the Neel temperature as a function of temperature is plotted (black line). The Kondo temperature TK is given by the equation: 1 TK = We D(εF )Jc f , (1.1) where D(εF ) denotes the density of states at the Fermi level and W is a width of conduction band. The formula describing TRKKY is following: TRKKY ∼ J2 (1.2) cf D(εF ) The region of the Doniach diagram between the Neel line and the horizontal axis represents antiferromagnetic state. In the region denoted as ”RKKY” regime magnetic moments prevail at Ce atoms. This is a case of quite weak correlation with Curie-Weiss like behavior observed in a system. The point of the diagram at which the Neel temperature reaches 0 K (the point assigned to the CeNiSn compound) represents the quantum critical point (the instability of an antiferromagnetic phase at 0 K). The value of Jc f depends on the interatomic distances and the overlap between orbitals (Matar, 2013). Hence, several techniques exist which allow for manipulation in c-f coupling strength. The application of external pressure, in general, increases the value of Jc f . The value of Jc f can be reduced by expanding the unit cell, for example by ”isoelectronic” substitution of element with greater atomic radius instead of that present in the structure (e.g. substitution of Pd in place of Ni) or by a hydrogenation. However, the inﬂuence of adsorbed H into a crystal structure of a host compound can be more complex, because H atom donates electron and can participate in a chemical bonding formation (Matar, 2013). The Doniach phase diagram provides a good image of behavior of systems with Kondo-type interaction. However, in many cases it is insufﬁcient to cover complexity of such systems. Thus, the extensions have been developed so far (Doradzi´nski and Spałek, 1998). The idea of similarity between phase diagram of heavy fermions and cuprates or iron pnictides has been explored as well (Weng et al., 2016). Nowadays, it is believed that the proper description of systems with Kondo scattering is obtained by means of periodic Anderson model (Anderson, 1961). Thorough studies of such a model in a parameter space revealed different states. Among them, one can list: Kondo lattice2, mixed valency, Kondo insulator state or heavy fermion superconductivity. The brief description of some of them is provided in subsequent sections. 2Remark The attention should be paid to the used nomenclature, which can be a source of ambiguity. The term ”Kondo lattice” is often used (mainly by experimentalists) in order to describe a system in which the f sites cannot be treated independently. On the other hand, a paramagnetic solution of the Periodic Anderson Model which has Fermi liquid ground state is treated as a true ”Kondo lattice” (mainly by theoreticians). Such a state is sometimes called a heavy fermion state in some works. Bibliography Anderson, P. W. (1961). Phys. Rev., 124:41–53. Bauer, E. and Rotter, M. (2009). Magnetism of Complex Metallic Alloys: Crystalline Elelctric Field Effects, pages 183–248. World Scientiﬁc. Doniach, S. (1977). Physica B & C, 91:231–234. Doradzi´nski, R. and Spałek, J. (1998). Phys. Rev. B, 58:3293–3301. Dorenbos, P. (2012). ECS J. Solid State S. T., 2(2):R3001–R3011. Kasuya, T. (1956). Prog. Theor. Phys., 16(1):45–57. Kettle, S. F. A. (1996). f electron systems: the lanthanides and actinides, pages 238–268. Springer Berlin Heidelberg. Kondo, J. (1964). Prog. Theor. Phys., 32(1):37–49. Kramida, A., Yu. Ralchenko, Reader, J., and NIST ASD Team (2018). NIST Atomic Spectra Database (ver. 5.6.1), [Online]. Available: https://physics.nist.gov/asd [2019, May 4]. National Institute of Standards and Technology, Gaithersburg, MD. Matar, S. F. (2013). Prog. Solid State Ch., 41(3):55 – 85. Ruderman, M. A. and Kittel, C. (1954). Phys. Rev., 96:99–102. Stewart, G. R. (1984). Rev. Mod. Phys., 56:755–787. Weng, Z. F., Smidman, M., Jiao, L., Lu, X., and Yuan, H. Q. (2016). Rep. Prog. Phys., 79(9):094503. Chapter 2 Kondo effect and heavy fermion physics Anomalous properties of systems composed of magnetic atoms immersed in a nonmagnetic crystalline matrix have attracted attention in area of experimental studies as well as in theoretical modeling. Nowadays, it is believed that the broad range of physical phenomena in such systems is well covered within the scope of the Periodic Anderson Model (PAM). It describes the system in terms of the effective Coulomb repulsion between electrons on magnetic impurity shell ( f or d shell, in this thesis we deal exclusively with f electron systems). However, solving of eigenproblem of PAM is a tremendous task (Hewson, 1993). Thus, simpler models, valid for some narrower range of phe

nomena have been developed. For example in case of small amount of magnetic atoms which are well separated from each other in diamagnetic matrix, one can use a single impurity Anderson model. Subsequently this model can be mapped for some subset of parameters on the s-d exchange model (a Kondo model). Here we would like to sketch some theoretical aspects of physical phenomena encountered in f -electron systems. Some intuitive description has been provided in the preceding section. In case of one impurity with two internal degrees of freedom (σ = ±21) the single impurity Anderson model (SIAM) is used (Anderson, 1961). The Hamiltonian has got a following form: HSIAM = ∑ εknkσ + ∑ εfnf σ +Unf ↑nf ↓ + ∑ {Vk f c † kσ fσ +Vk f jfσ †ckσ }, (2.1) k,σσ k,σ where the ﬁrst part denotes the energy of conduction band electrons, the second one is the energy of f shell, the third gives the Coulomb repulsive interaction with effective strength U. The last part describes the mixing between f level and conduction band with the Vk f strength. The standard second quantization notation is used: fσ and fσ † are creation and annihilation operators for f electron with † spin σ. Similarly: ckσ and ckσ are creation and annihilation operators for a conduction band carrier with momentum k and spin σ. The electron number operators are: nkσ and nf σ , for conduction band and f shell, respectively. It turns out that the single impurity Anderson model (SIAM) in the limit of small mixing between conduction band with dispersion εk and the impurity level with energy εf is equivalent to the Kondo model. The effective Hamiltonian Hef f in such a regime is obtained by elimination of ﬁrst order terms in Vk f in original hamiltonian (Schrieffer and Wolff, 1966). This is realized by application of a canonical transformation: Hef f = eSHe−S (2.2) together with a condition: [H0,S]= H1, (2.3) where H0 is a hybridization independent part of H, H1 represents the ﬁrst order therm in Vk f and S is a generator of transformation. The generator S has got a form: ( 11 ) † S = ∑Vkf nf ,σ¯+(1 − nf σ¯) ckσ fσ − h.c. (2.4) εk − εf −U εk − εf k,σ The spin dependent part of effective hamiltonian obtained by Schrieffer and Wolff, in a subspace with forbidden double occupancies of f shell is as follows: Hsd = − ∑Jk,k' (Ψ† k' SΨk) · (Ψ† f SΨf ), (2.5) k,k' where the ﬁeld operators (Ψf and Ψk) have been introduced: (ck,↑)(f↑) Ψk = , Ψf = . ck,↓ f↓ σ The symbol S in equation (2.5) refers to spin operator S =2( σ is a column matrix composed of Pauli matrices). Hsd has got the similar form as in the model investigated by J. Kondo (Kondo, 1964). In general the s-d exchange coupling strength (Jkk' ) is energy dependent what is visible in formula: ( 1 111 ) Jkk' = Vk' fVk f j+ −− . (2.6)εk − εf −U εk'− εf −U εk − εf εk'− εf However, assuming that the dominating contribution comes from the excitations from vicinity of the Fermi surface (i.e. k, k'≈ kF ; kF -Fermi wave vector) the strength of the s − d exchange can be represented by the antiferromagnetic coupling constant: U Jkk'≈ J0 = 2|VkF f |2 (2.7) εf (εf +U) The s-d exchange model in slightly modiﬁed form than that presented in equation (2.5) has been introduced by J. Kondo in order to describe the anomalous behavior of low temperature electrical resistivity of a simple diamagnetic metal (eg. Au or Ag) which contains small amount of magnetic impurities (eg. Fe)(Kondo, 1964). His approach relied on perturbation theory (second order Born approximation). He showed that the part of electrical resistivity related to the scattering on the spin degrees of freedom is proportional to the logarithm of temperature. Hence, the total resistivity ρ(T ) can be described by the formula: ρ(T )= ρ0 + aT 5 + cρ1 lnT, (2.8) where the term ∼ T 5 represents the lattice contribution at low T and ρ0 stands for the temperature independent part. The magnetic contribution is proportional to concentration of magnetic atoms (c), ρ1 is some constant which is proportional to exchange integral J0. One can check, that for antiferromagnetic coupling (viz. J0,ρ1 < 0) the function ρ(T ) displays a minimum at temperature Tmin: 5 (−cρ1 )1 Tmin = . (2.9) 5a Such a minimum is a ﬁngerprint of presence of Kondo effect in a metallic system. The remark on the behavior of ρ(T ) at very low temperature should be made. Namely, the formula (2.8) predicts the divergence of resistivity in the limit T → 0 K, instead of experimentally observed saturation. Such a divergence is an artifact related to the breakdown of perturbation theory. Indeed, below some temperature (i.e. Kondo temperature TK ) the effective screening of magnetic moment by conduction electrons is large and cannot be described perturbatively. The magnetic moment can be fully quenched. The antiferromagnetic interaction between conduction electrons and impurity spin results in a singlet and triplet many-body states. The ﬁrst one is a ground state, while the second one has got energy by about U higher (U is a measure of strength of effective Coulomb repulsion on impurity shell). The singlet state is reﬂected in the narrow resonance peak at the Fermi energy in the spectral function (so called Kondo peak). Exact solution of the Kondo problem can be obtained with the aid of numerical methods. However, some compact analytic formulas were proposed in order to provide approximate description of physical properties (Frota, 1992). It turns out that the Kondo resonance encountered in the spectral function as a result of formation of a singlet state can be modeled well by the simple formula for a density of states D(ε) (ε -energy, here the Fermi energy is put to zero): 2 ( iΓK )21 D(ε)= Re, (2.10) πΓA ε + iΓK where ΓK is proportional to the Kondo temperature TK and ΓK parameter is related to the Friedel sum rule. The magnetic susceptibility χ(T ) at low temperatures can be expressed as: k2 (π2 BT 2 ) χ(T )= χ(0)1 − , (2.11) 8 Γ2 K with the susceptibility at 0 K equal to: g2µ2 χ(0)= B , (2.12) πΓA (g is the Lande magnetic factor, µB -the Bohr magneton). Similar dependency as for χ(T ) is followed by electrical resistivity ρ(T ) at low temperature (here the reduced quantity is presented): ρ(T ) − ρ0 π2 kB2 T 2 = 1 − . (2.13) ρ(0) − ρ08 ΓK 2 The coefﬁcient ρ(0) in equation (2.13) is a value of resistivity at 0 K, while ρ0 is given by the formula: m ρ0 = , (2.14) n0e2τ0 where n0 is density of conduction electrons, τ0 is a relaxation time related to non-resonant scattering, m and e describe mass and charge of electron, respectively. The speciﬁc heat (c) calculated within this approach is linear at low T: π kBT c = (2.15) 3 ΓK and the entropy (ΔS) related to the Kondo resonance is equal to: ΔS = kB ln2, (2.16) which agree with the result of exact calculations. It is noteworthy, that expressions: (2.11), (2.13) and (2.15) are universal functions of T /ΓK variable. One can calculate the so called Wilson ratio using the magnetic susceptibility and linear coefﬁcient of speciﬁc heat at low T , i.e.: 3g2µB 2 C ΓA RW == . (2.17) kBπ2 T χ(0) ΓK One obtains RW = 2 after taking ΓA = 2ΓK. This result is consistent with that of Wilson (Wilson, 1975) renormalization group theory for a Kondo problem. In case of free electron gas one gets RW = 1. Of course, the presented description of physical properties is not exact, because the temperature dependency of a Kondo resonance is not taken into account. However, good agreement with the exact results is expected at low temperature (for T << TK). The crucial experimental aspects of single ion Kondo effect, i.e. its scaling properties together with the resonance peak in spectral function, were described above within the simple theory provided by H. O. Frota. However, they are not preserved in case of dense Kondo system, i.e. this means that magnetic atoms cannot be treated as isolated impurities. The minimal variant of the Anderson model, which is able to capture main aspects of a Kondo lattice, can be expressed as: †† HKL = ∑εkckσ ckσ + J0 ∑Sj · (cjασαβ cjβ ), (2.18) k,σ j where the ﬁrst part is given in momentum (k) space, while the second part is represented in real space, in distinction to equation (2.5), consists of summation over whole lattice of magnetic ions with spin Sj. Indeed, the neighboring magnetic atoms can interact by the exchange mediated by conduction band electrons what is reﬂected in RKKY interaction, which can be derived from the model (2.18). Oscillating character of RKKY interaction is visible in its static spin polarization function (Π(r)), which can be presented in the form: sin(2kFr) − 2kFr · cos(2kFr) Π(r) ∼ , (2.19) kF 4 r4 where kF denotes the Fermi wave vector and r is a distance from the arbitrary chosen lattice point. The presence of such oscillating interaction opens a possibility for magnetic order at low temperature. However, when the RKKY interaction does not dominate over Kondo demagnetization, the heavy fermion behavior is observed in the simplest case. In the diluted regime (Kondo regime), the Kondo scattering between f electrons and conduction band carriers occurs with random phases (the phase of the wave function of scattered electron was random). However, for a periodic arrangement of f orbitals the Bloch theorem must be fulﬁlled. As a result the Fermi liquid ground state emerges in the system. The resistivity at low temperatures drops rapidly as a consequence of coherence (Bloch state formation) and beyond the minimum, the maximum can be also observed. In such a case the system is paramagnetic and magnetic susceptibility obeys the simple Curie-Weiss behavior at high temperature, while at low temperature saturation is observed to the value of χ(0): mj 22 D(εF ) χ(0)= gµB · , (2.20) 1 + Fa m 0 where mj is an effective mass of charge carrier, while Fa is a coefﬁcient related to the strength of o interaction between quasiparticles (Landau parameter). The speciﬁc heat is linear with Sommerfeld coefﬁcient γ equal to: π2kB 2 mj γ = · (2.21) 3 m The Wilson ration deﬁned in equation (2.17) is equal to: RW = 1 + Fa (2.22) o and points directly to the strength of interaction between quasiparticles. Presence of heavy quasiparticle band with mostly f orbital character is another aspect of the heavy fermion state. Such a band has got a width of order of TK and is developed at low temperature, namely below the coherence temperature (Tcoh), which is another characteristic of the heavy fermion system. Thus, the f moments participate in the Fermi surface at low T , while at above Tcoh they are expelled from it. Hence, one should observe a signiﬁcant change of the volume of the Fermi surface as a function of temperature, what is sometimes called as a Fermi surface collapse. The sketched scenario is the simplest one. Other more complex states can be encountered, as for example Kondo insulator state (eg. as in CeRhSb or Ce3Bi4Pt3). In such systems the formation of the Kondo singlet is accompanied by the emergence of narrow gap in a band structure. Such a gap opens at low temperature due to mixing between f states and conduction band. The temperature evolution of the width of a gap is different that those encountered in the Bloch or Mott insulators. The competition between RKKY interaction and Kondo scattering can be studied by means of alloying in a physical system (e.g. by substitution of magnetic elements in place of non-magnetic ones). It allows to observe the transition from the single impurity Kondo regime to the coherent Fermi liquid state (e.g. in CexLa1−xCu6). On the other hand it opens a possibility for studies of the magnetic instability at 0 K and the associated breakdown of Fermi liquid theory. There are several theoretical scenarios of such a quantum phase transition and the subject is still controversial up to know. In the vicinity of either, ferromagnetic or antiferromagnetic zero temperature instability, the ground state of many electron system is described in terms of the non Fermi liquid behavior (Coleman et al., 2001). The simple description of physical properties provided above for the Fermi liquid is not obeyed. The crucial observation is the fact, that in the quantum critical point, the mass renormalization factor diverges: mj → ∞. This is reﬂected in the behavior of the speciﬁc heat at low temperature. Namely, m in case of many systems with quantum critical point the speciﬁc heat evolves with the temperature according to the logarithmic law: (T0 ) c(T ) ∼ T ln, (2.23) T where T0 is some constant in units of temperature. The electrical resistivity follows the power law: ρ ∼ T 1+ε (2.24) with 0 < ε < 0.6, instead of characteristic quadratic dependency observed for the Fermi liquid. The Curie-Weiss law describing the magnetic susceptibility is replaced by the formula: 11 =+ cT a , (2.25) χ(t) χ0 where the relation a < 1 was observed in experiments. The quantum phase transition is also visible in a dynamic susceptibility measured by inelastic neutron scattering measurements or in a hall coefﬁcient RH . The measurement of RH as a function of a parameter which drives the quantum critical transition can be used in assessment of the realized scenario of the transition in a system. Bibliography Anderson, P. W. (1961). Phys. Rev., 124:41–53. Coleman, P., Pépin, C., Si, Q., and Ramazashvili, R. (2001). J. Phys. Condens. Matter, 13(35):R723– R738. Frota, H. O. (1992). Phys. Rev. B, 45:1096–1099. Hewson, A. C. (1993). The Kondo Problem to Heavy Fermions. Cambridge Studies in Magnetism. Cambridge University Press. Kondo, J. (1964). Prog. Theor. Phys., 32(1):37–49. Schrieffer, J. R. and Wolff, P. A. (1966). Phys. Rev., 149:491–492. Wilson, K. G. (1975). Rev. Mod. Phys., 47:773–840. Chapter 3 Cerium intermetallics The number of intermetallic compounds of cerium is huge and they display a rich variety of crystal structures. In the propounded thesis we do not even try to face up the problem of systematic description of these systems. However, in subsequent sections, we provide a brief description of selected representatives. The choice is dictated by the reference to the results presented in Part 3 and due to historical reasons. The brief description of the phase diagram of elemental Ce is provided at the beginning. 3.1 Metallic Ce -phase diagram The pressure-temperature phase diagram for elemental Ce is quite complex (Fig. 3.1) (Schiwek et al., 2002). Usually the discussion of phase diagram is limited to ﬁve solid phases under a melting curve ' denoted as: α, α, β, γ, δ . The hexagonal β phase is the most stable at room temperature at ambient pressure conditions. However, both β and γ type structures can exist at room temperature at standard pressure. It should be mentioned that transition pressures and temperatures depend on the mechanical treatment and signiﬁcant effects of hysteresis were observed. The bcc structure is encountered at high temperature in δ phase. The phase diagram displays two fcc phases: γ and α, in neighborhood of β phase. The high pressure phase is an α phase. The isostructural transition from γ to α Ce, encountered at about 8 kbar at room temperature, is accompanied by the volume collapse of order of 15% (Allen and Martin, 1982). It is believed that the main driving force of the transition is a '' delocalization of 4f electrons due to the presence of Kondo effect. The monoclinic α(C2/m space ' group) and orthorombic α(Cmcm space group) phases are also present. Beyond them, the bct-type (body centered tetragonal) ε phase exists under high pressure (>7 GPa). The γ phase displays the Curie-Weiss-like magnetism characteristic of a presence of local moments (Hamlin, 2015). The cerium in the collapsed α phase is a Pauli-paramagnetic metal, characterized by large value of magnetic susceptibility. A superconducting state with Tc lower than 0.05 K at ' pressure of 2 GPa emerges in this phase. Above 5 GPa the superconductivity in the αphase below 1.7 K was also observed (Wittig, 1968). The decrease of the pressure from 5 GPa to 3.6 GPa leads to increase in Tc to 1.9 K (Loa et al., 2012). However, there is no conclusive information which phase is ' '' characterized by that number, both αand αshould be considered. 3.2 RT9In2 compounds Ternary indidies from R-T-In (R -rare earth element, T -transition metal) system display many interesting properties from the viewpoint of crystal chemistry and physics. Beyond the possibility of studies of crucial issues of condensed matter physics (e.g. heavy fermion physics, unconventional superconductivity, etc.), they can be considered as promising materials for hydrogen storage (Kalychak et al., 2004). Among the whole family of R-T-In compounds, intermetallics crystallizing in tetragonal structure of YNi9In2-type (space group: P4/mbm) draw particular attention (Kalychak et al., 2004). The YNi9In2-type structure is an ordered superstructure of CeMn6Ni5-type structure (Fig. 3.2). The high coordination numbers are typical of this structure. Each rare earth atom has got 22 nearest neighbors, while 15 and 12 atoms surround each In and Ni (transition metal site) atom, respectively (Bigun et al., 2013). The arrangement of Ni and In atoms can be considered as the 3D Ni9In2-net with channels along c-axis which are ﬁlled by rare earth atoms. A strong bond between indium atoms is testiﬁed by very low distance between them in all known representatives. So far, magnetic properties of RCu9In2 and RNi9In2 systems have been studied thoroughly (Baran et al., 2016; Bigun et al., 2014). LaCu9In2, LaNi9In2 and YNi9In2 are Pauli-paramagnetic metals without f electrons. Magnetism of PrNi9In2 and NdNi9In2 as well as of their Cu counterparts is dominated by Curie-Weiss-like behavior between 20 and 300 K. The obtained values of paramagnetic Curie-Weiss temperature (θp) are negative. This suggests the presence of antiferromagnetism in the system. For PrNi9In2, PrCu9In2 and NdNi9In2 the phase transition was not encountered down to 2 K. In case of NdCu9In2, the metamagnetic transition was observed at temperature equal to 4.3 K. In case of EuNi9In2 there is no conclusive information, if either ferromagnetic order or intermediate valency state is realized. The interpretation of the data for EuCu9In2 is also not clear, although the antiferromagnetic ground state is anticipated. It is noteworthy, that RT9In2 compounds with T=Cu are more prone to exhibit magnetic order at low temperature than these with Ni. Known representatives of CeT9In2 display ﬁngerprints of heavy fermion physics. CeNi9In2 is considered as a realization of mixed valence state (Kurleto et al., 2015; Moze et al., 1995; Szytuła et al., 2014; Tran et al., 2020), while the presence of the coherent Kondo lattice phase in CeCu9In2 is discussed further in the text of this thesis. The substitution of Cu in the place of Ni in CeNi9In2 leads to signiﬁcant change in electronic structure, what can be explained as follows. The electronic conﬁguration of Cu atom is [Ar] 3d10 4s1, while for Ni we have [Ar] 3d8 4s2. Thus, 3d electrons in Cu form a closed shell. The maximum of the Cu 3d density of states is expected to locate well below the Fermi level. Hence, one can expect that mixing between f level and 3d shell is suppressed and coupling between 4s and 4f levels is dominating. On the contrary, in case of Ni the 3d-4f mixing is considered as dominating, because of suspected large density of states at the Fermi level. Additionally, the lattice parameters of CeNi9In2 are smaller than in case of its counterpart with Cu. This is in line with the smaller atomic radius of Ni than that of Cu. Therefore, the higher inter-atomic distances in CeCu9In2 result in weaker overlap between Ce 4f states with the states originating from other atoms. As a consequence valence ﬂuctuations are suppressed, while the strength of exchange interaction is not affected signiﬁcantly (the strength of RKKY interaction is proportional to J2 cf , while the strength of Kondo scattering is given by exponential function of Jc f ). This reasoning should also apply in the case of the rest of RT9In2 compounds with light lanthanides. 3.3 CeTX compounds The ternary equiatomic CeTX (T -transition metal, X -element from p-block of the periodic table) compounds are considered as the largest family of cerium intermetallics. The great interest in this systems can be explained by the rich variety of phenomena related to presence of f electrons e.g.: valence ﬂuctuations, Kondo insulating state or quantum phase transitions. Representatives of CeTX family crystallize in orthorombic or hexagonal structures which are superstructures of a simple AlB2type structure (Matar, 2013). Here, we would like to focus mainly on ternary stannides and ternary antimonides. The magnetic and transport compounds of representatives of this subfamilies are very interesting. The compounds: CeCuSn, CeAgSn, CePdSn (´ Slebarski et al., 2001), CePtSn (Matar, 2013) exhibits antiferromagnetism at low temperature. The ferromagnetic state was observed in: CePdSb, CePtSb and CeAuSn. CeRhSb and CeNiSn are Kondo insulators. Partial substitution of Sn in a position of Sb in CeRhSb results in solid solution CeRhSb1−xSnx (0

tional can be rewritten in a form: f ρ(r)V (r)d3 r, (4.1) E[ρ(r)] = F[ρ(r)] + R3 where V (r) represents the static part of a potential for the electron gas in solid and F[ρ(r)] is some functional, which can be expressed as: F[ρ(r)] = T [ρ(r)] + 1 2 f f ρ(r)ρ(r ' ) |r − r '| ' d3rd3r + Exc[ρ(r)]. (4.2) R3 R3 The T [ρ(r)] in formula (4.2) denotes the kinetic energy, the second part with double integral repre

sents the Hartree energy, while the Exc is exchange-correlation energy functional. The last quantity is unknown, and several schemes provide construction of its approximation. It is noteworthy, that proposed scheme is an exact theory, provided that the exact form of Exc is known. In order to obtain ground state ρ(r) the minimum of total energy should be found, what is equivalent with ﬁnding a solution of a set of Kohn-Sham equations: (1 Δi +Vef f (r) − εi 2 ) ψi(r)= 0, (4.3) where εi is Kohn-Sham orbital energy and Vef f (r) is an effective functional given by: f ρ(r ' ) ' Vef f = Vext + d3r +Vxc, (4.4) |r − r '| with Vxc related to exchange-correlation energy functional by: δExc Vxc = , (4.5) δρ ( δ denotes the variational derivative with respect to the function ρ). The symbol Vext in equa δρ tion (4.4) denotes the part of potential related to external sources (e.g. magnetic ﬁeld). The electron density is recovered from Kohn-Sham orbitals (ψi) after solving equations (4.3) with application of formula: N ρ(r)= ∑ |ψ(r)|2 . (4.6) i=1 The attention should be paid to the interpretation of obtained ψi functions. Namely, they are not equivalent to the eigenfunctions of original hamiltonian. Bibliography Landau, L. and Lifshitz, E. (2001). Mechanika kwantowa. Wydawnictwo Naukowe PWN, Warszawa. Matar, S. F. (2013). Prog. Solid State Ch,, 41(3):55 – 85. Nalewajski, R. (2001). Podstawy i metody chemii kwantowej. Wydawnictwo Naukowe PWN, Warszawa. Part II Experimental methods Chapter 5 Photoemission Spectroscopy 5.1 Basics of Photoemission Spectroscopy The basis of photoemission spectroscopy, which is nowadays a powerful technique in studies of electronic structure of materials, was established together with the birth of the quantum mechanics. The photoelectric effect was discovered collaterally by Heinrich Hertz and Wilhelm Hallwachs in 1887. Its theoretical interpretation provided in 1905 by Albert Einstein was one of the milestones in modern physics. The arrangement of typical photoemission experiment (Fig. 5.1) has been used. It is similar to the original experiment performed over hundred years ago. The beam of monochromatic light is directed into a sample subjected to the scrutiny. If the energy of incident photons (h¯ω) is greater than the work function (φ) of a studied system, the electrons are freed from the sample into a vacuum. The kinetic energy (Ekin) of the emitted photoelectron can be used in order to calculate the binding energy (EB) of charge carrier in the sample. Namely, the famous Einstein-Millikan formula can be used: Ekin = h¯ω − φ −|EB|. (5.1) Nowadays, the kinetic energy of photoelectrons is usually measured with application of hemispheric analyzers. The common sources of quasi monochromatic radiation used in small in-house laboratory setups are helium lamps and X-ray tubes. The ﬁrst ones generate radiation with the energy 21.2 eV (He I spectral line) and 40.8 eV (He II spectral line). X-ray tubes for photoemission purposes usually contain Mg or Al anode, what corresponds to the radiation with energy equal to 1253.6 eV (Mg Kα ) and 1486.6 eV (Al Kα ) respectively. Another important sources of radiation are synchrotrons and free electrons lasers. Despite the photoemission can be successfully performed for gases and liquids, this thesis deals only with the description of this process for crystalline solids. Here, we discuss the photoemission within the scope of simpliﬁed three step model with the so called sudden approximation. The model allows to split the act of photoemission into three steps: absorption of the photon, transport of excited electron to the surface and escape of the electron into vacuum. The sudden approximation assumes that the time of the photoemission act is many times shorter than characteristic time scale of the system, so the interaction between photoelectron and created hole can be neglected (this assumption is not justiﬁed in case of photoemission excited with extremely low photon energies). The direction of the momentum of excited electron leaving the crystalline sample, as well as its energy, can be referred to the components of its wave vector inside a solid. The component of wave vector which is parallel (k||) to the sample surface is conserved during the act of photoemission due to the translational symmetry on the surface of perfectly periodical system, provided the momentum of the exciting photon can be neglected (which is well justiﬁed for UV radiation). The formula describing k|| is following: √ 2mEkin k|| = sinθ, (5.2) h¯where the notation introduced in (Fig. 5.2) has been used. Figure 5.2: Schematic plot allowing to derive basic formulas obeyed during photoemission process. The part of the electron’s wave vector which is perpendicular (k⊥) to the sample surface is not conserved, because of the broken translation symmetry in this direction. Additionally, one should know the dispersion relation for the ﬁnal states. Assuming the free electron like (i.e. parabolic) dispersion relation, one can derive the following formula for k⊥: k⊥ = f2m h2 · vEkin cos2 θ +V0, (5.3)¯where V0 denotes the so called inner potential, which describes the height of the potential barrier between sample and surface (V0 is usually of order of 10 eV). One should mention, that not all excited electrons leave the sample. Firstly, the inelastic scattering between excited electrons implies ﬁnite mean free path of a charge carrier. As a result, the electrons freed into a vacuum come mainly from a few ﬁrst atomic layers, especially in photoemission excited by the ultraviolet radiation. However, in the soft X-ray regime the contribution from the bulk of the sample to the number of emitted photoeletrons is signiﬁcant. The mean free path of the excited electrons is a universal function (Hüfner, 2003; Suga and Sekiyama, 2014) of radiation energy and it has a minimum at some energy corre

sponding to ultraviolet radiation. This last fact points to the usefulness of the method in surface science. Secondly, some number of excited particles is reﬂected back to the interior of the crystalline sample on the barrier between the sample and vacuum. This can be explained by the analogy with total internal reﬂection effect in optics. The directions of momentum of electrons which can be freed into a vacuum must lie into so called Mahan cone, i.e. angle (θ) between the wave vector and the normal to the sample surface must be lesser than some limit angle (θMahan) given by the formula: (f Ekin ) θMahan = arcsin(5.4)Ekin +V0 The variation of the photoemission experiment allowing for concurrent measurement of kinetic energy of photoelectron and components of its wave vector is called angle-resolved photoemission spectroscopy (ARPES). Today, technological development allows for many types of measurements based on photoelectric effect, e.g. spin resolved photoemission spectroscopy or time resolved photoemission. However, that rich family of experimental techniques is not described in this thesis, because here we concentrate on angle-resolved photoelectron spectroscopy in UV or soft X-ray regime. Now, the important question arises: what is the relation between measured number of photoelectrons as a function of a kinetic energy and emission angles? In order to give an answer one should refer to the quantum mechanical description of the system composed of crystalline solid and electromagnetic wave. The perturbation of a system due to the coupling to the photon ﬁeld results in additional term H' in hamiltonian: H ' e = A · p, (5.5) mc where p denotes the momentum operator for electrons conﬁned into a solid. The vector potential A contains the information about the light polarization and is described by a plane wave, but it can be treated as a constant, because typically the wavelength (λ) of the used radiation is many times greater than the size of an atom: ikλ ·r ≈ A0. A = A0e The form of the coupling to the photon ﬁeld presented in equation 5.5 is obtained from the general minimal coupling scheme assuming the conditions: φ = 0,∇ · A = 0 are fulﬁlled and neglecting the two photon term A2. Sometimes this approach is called the dipole approximation. The photocurrent excited by radiation with energy h¯ω, according to the golden Fermi rule, is given by the proportionality: I ∼ ∑ |�φf ,Ek A0 · p φi,k)|2 ∑|cs|2δ (Ekin + Es(N − 1) − E0(N) − h¯ω), (5.6) f ,i,ks where φf ,Ek ) represents the ﬁnal free electron like state of photoelectron with the kinetic energy Ek and φi,k) stands for the k-th orbital from which the photoelectron is emitted. The total energy of ground state of N electrons in absence of a radiation is denoted by E0(N), while the energy of s-th excited state after removal of electron is given by Es(N − 1). The factor |cs|2 is interpreted as the probability of obtaining the s-th excited state by emission of electron from the initial φi,k orbital. If we make a restriction k = s, then the equation 5.6 simpliﬁes to: I ∼ ∑ |Mf ik|2δ (Ek + |εk|− h¯ω), (5.7) f ,i,k where the symbol Mf ik refers to the same matrix element as in 5.6 and εk stands for the energy of φk,i orbital. Such an assumption is equivalent with treating the remaining electrons in the system during the removal of electron from the particular shell as spectators, which are not involved in this process. Such an approach is justiﬁed for systems without strong correlation between electrons. In such a situation the electronic structure does not depend on the ﬁlling, so the presence of the hole does not lead to the reorganization of energy levels in a system. The equation 5.7 informs us that the measured spectrum consists of a sharp peak (described by a Dirac δ ) at the energy which corresponds to the energy of the state in a solid. Hence, for the uncorrelated system, one should expect one-to-one correspondence between the measured photoelectric spectrum and electronic structure. Of course, one should have in mind, that measured spectrum is a smooth function of kinetic energy, because of convolution with the instrumental function and to the non-zero lifetime of a photohole. In case of presence of the correlations in the system, foregoing simpliﬁcation is not justiﬁed. The measured spectra are not a simple blueprint of the electronic structure. Beyond the main lines corresponding to the energy levels of the system, the satellite peaks are observed due to rearrangement of electronic structure in presence of a hole in a system, what is reﬂected by the presence of many excited states labeled by s in 5.6. For interacting system, the equation 5.7 can be rewritten with introduction the notion of spectral function A0: I(ε) ∼ ∑ f ,i,k |Mf ik|2A0(ε;k), (5.8) where the experimental variable is denoted by ε: ε = Ek − ¯hω. The spectral function A is related to the one-particle retarded Green function G(ε;k) via: A0(ε;k) = − 1 π ImG(ε;k). (5.9) One should have in mind that the photoemission probes only occupied states. Indeed, the measured photocurrent: I(ε;k) ∼|Mfik|2A0(ε;k) fFD(ε,T ) is also determined by the Fermi function fFD: 1 fFD = (5.10) 1 + e(ε−εF )/kBT , where T is a temperature, kB is Boltzmann constant, εF denotes the Fermi energy. 5.2 Analysis of spectral function in photoemission The results obtained for non-interacting system are a starting point for description of spectra of a system with electronic correlations. In general, photoemission can probe the spin and momentum resolved spectral function Aσ (ε,k) as a function of energy ε. Similarly as in equation 5.9 we have the relation between the spectral function and one-particle retarded Green function for interacting system: Aσ (ε,k)= − 1 ImGσ (ε,k) (5.11) π The inﬂuence of many-body interactions is described by means of so called ”self-energy” operator Σ: Gσ (ε,k)= 1 . (5.12)ε − ε0(k) − Σσ (ε,k) The bare band dispersion is denoted as ε0(k). The poles of the Gσ (z;k) function in the complex plane of z determine the shape of the spectral function. The real part of z corresponding to the pole describes the peak position in spectral function, while the imaginary part describes its width. In case of relatively small effects of ”self-energy” operator and for a single ﬁrst order pole, one can divide the spectral function into two parts (here we omit the spin dependency): Zk G(ε,k)= +(1 − Zk)Gn(ε,k), (5.13)ε − z1(k) where the ﬁrst term of the sum is a coherent part, while the second one (Gn) is a smooth incoherent part. The so called quasiparticle weight is denoted by Zk, and the self-energy effect is included in self-consistent equation: z1 = ε0(k)+ Σ(z1,k). The equation 5.13 contains the essence of the renormalization concept. Namely, the interaction shifts the bare energy levels according to the Σ function and the mass enhancement factor is reﬂected in Zk factor. G(z,k) is a Green function, so it must obey some rigid mathematical rules (Landau and Lifshitz, 2001). In particular, causality impose that G(z,k) must be an analytic function of z in the upper half of complex plane. As a consequence, one can use the following representation of the Green function: 1 f G(z ' ,k) ' G(z,k)= − P dz , (5.14) π z '− z IR where the P J symbol has a meaning of Cauchy principal value of integral. This equation can be rewritten separately for both, imaginary and real part: ReG(z,k) = 1 π P f ImG(z ' ,k) z ' − z dz ' (5.15) IR ImG(z,k) = − 1 π P f ReG(z ' ,k) z ' − z dz ' . (5.16) IR The obtained formulas are known as Kramers-Kroning relations and are useful in analysis of effects of interaction in photoelectric spectra (i.e. for determination of self-energy operator). Same relations obeyed by the Green function G(z,k) hold on also for the ”self-energy” operator Σ(z,k). Another important constraint is a normalization condition for the spectral function: f A(ε,k)dε = 1, (5.17) IR which is closely related with the deﬁnition of momentum distribution n(k): f n(k)= A(ε,k) · fFD(ε;T )dε . (5.18) IR Similar sum rules can be also written for the coherent and incoherent parts of the spectral function: ff Acoh(ε,k)dε = Zk An(ε,k)dε = 1 − Zk, (5.19) IR IR where Acoh and An are related to coherent and incoherent part of the Green function, respectively. The spectral function of interacting system is qualitatively different than that for non-interacting system. Namely, only coherent part is present in the latter case. In case of system of interacting electrons described by the Fermi liquid theory, in general both coherent and incoherent parts are present and well separated. At the wave vector corresponding to the Fermi vector (kF ), the incoherent part vanishes. The n(k) function in this case has got a jump at kF equal to Zk. In case of absence of interactions the unity jump is observed at kF . The ARPES method gives the intensity of photocurrent as a function of energy and momentum I(ε,k). The analysis of the data can be performed in two possible ways. Namely, one can analyze cuts corresponding to the constant value of momentum. These are energy distribution curves (EDC). On the other hand, one can ﬁx the energy and deal with so called momentum distribution curves (MDC). The peak ﬁtting procedure utilizing MDC or EDC approach can be used in order to obtain the dispersion relation from the ARPES data together and information about the lifetime of excitations (given by the width of the peak). However, in order to obtain the ”self-energy” operator, the bare dispersion ε0(k) should be known. The theoretical calculations can be used in order to provide ε0(k). On the other hand, recently the self-consistent approach based on Kroning-Kramers relations which allow to determine Σ(z,k) directly from ARPES data has been developed (Kordyuk et al., 2005). 5.3 Resonant photoemission The photoemission spectrum of valence band of a solid can be treated, to some extent, as a super-position of contributions related to particular orbitals. Thus, the measured photocurrent I(ε) can be approximately written as a linear combination: I(ε)= ∑In,l(ε)σn,l(h¯ω), (5.20) n,l where Il(ε) represents the contribution from n,l shell and σn,l(h¯ω) denotes the cross section for a photoionization of this shell. The quantity σn,l depends on the energy of incident radiation (h¯ω) in a highly rapid manner. The non-monotonic behavior of σn,l(h¯ω) function, i.e. the presence of resonant peaks and minima between them, allows for estimation of In,l(ε) functions. The resonant enhancement of intensity of photocurrent is realized, if the h¯ω is close to the energy corresponding to the absorption threshold of some core level. This observation is an underlying principle of the so called resonant photoemission technique (Hüfner, 2003; Suga and Sekiyama, 2014). Here, we limit the discussion to the Ce compounds. Two main resonant transitions are encountered in such systems. These are: ”4d → 4 f ” at h¯ω ≈120 eV and ”3d → 4 f ” at h¯ω ≈880 eV. If the resonance condition is fulﬁlled, then the photoemission is a result of interference between two processes: direct photoionization and Auger process. Here, the description for ”3d → 4 f ” transition is provided. The change of the valence conﬁguration of Ce during the direct photoemission can be written as: 3d104 fn hω → 3d104 fn−1 − + ¯+ e. In contrary, two shells are involved during the indirect process. Firstly, the electron from the 3d shell is excited by photon and transferred to the 4f shell. Subsequently, the Auger decay takes place (actually super Koster-Kronig process). Namely, one electron from the 4f shell recombines with the hole in 3d level, while the second one is emitted to the vacuum. Thus the following sequence of valence changes: 3d104 fn hω → 3d94 fn+1 → 3d104 fn−1 − + ¯+ e, takes place. The analogous description can be formulated for the resonant photoemission near the 4d threshold. The increase in population of 4f electrons in resonance implies increased signal from 4f states. Hence, the difference between on-resonance and off-resonance spectra can be identiﬁed with 4f spectral function, providing that cross sections for other orbitals in the system does not rise signiﬁcantly. The interference between two possible channels in a quantum process resulting in the same ﬁnal state was investigated by U. Fano (Fano, 1961). The intensity (N(h¯ω)) of the particular feature in the photoelectron spectrum excited with radiation of energy h¯ω can be expressed as: (ε + q)2 Δ(E0) N(h¯ω) ∼ ε2 + 1 , (5.21) with: ε = , 2(h¯ω − E0) (5.22) where E0 is the energy of a core level (3d in case of ”3d → 4 f ” transition), Δ(E0) is a width of this level and q is some constant (characteristic of core level). The thorough analysis of the Ce 3d5/2 X-ray absorption spectrum (XAS) shows that several resonances exist for h¯ω ≈ 880 eV (Cho et al., 2003). They arise due to presence of splitting of 3d multiplet in the intermediate step of indirect photoemission process. This fact is also reﬂected in photoelectric spectra. Namely, data collected with different h¯ω contains the peak-like structure, which corresponds always to the same kinetic energy, what means that this structure arise due to the Auger effect. It turns out that the contamination of the spectra by such an effect is the smallest when the kinetic energy corresponds to the excitation of the lowest lying state of the core level multiplet, but not to the highest value of the integral photocurrent. Otherwise, the interpretation of the data is not unambiguous. The enhancement of the signal in satellite line due to the Auger effect does not reﬂect an intrinsic property of 4f spectral function, but can be spuriously identiﬁed as a result of the Kondo effect. Similar problem is also encountered in studies of d-electron systems (Hüfner, 2003). There is another drawback of resonant photoelectron spectroscopy. The photoemission from 5d orbital is also enhanced with application of radiation with h¯ω corresponding to ”3d → 4 f ” transition. However, the expected contribution from 5d states is not as large as for 4 f ones (Sekiyama et al., 2001). 5.4 Photoemission spectroscopy on Ce intermetallics The photoelectric spectrum measured on a polycrystalline sample of a simple metal (e.g. Au or Cu) at low temperature reproduces well the shape of the Fermi function (5.10) near the Fermi energy (εF ). The completely different situation is encountered for many Ce intermetallic compounds, which display signiﬁcant anomalies in measured spectrum close to εF due to many body effects. It is believed that a proper theoretical description of many body interactions which are present in Ce systems is provided by Anderson lattice model. The hamiltonian of the periodic Anderson model (PAM) has got a form: †† H = ∑εkckσ ckσ + εf ∑ fi† σ fiσ +U ∑ni↑ni↓ + ∑ (Vikck,σ fiσ + H.C.) (5.23) k,σ i,σ ii,k,σ where the ﬁrst term stands for the bare conduction band, the second part describes the lattice of f moments (U -intrasite Coulomb repulsion), the third part describe the mixing (Vk f -hybridization function) between the f states and the conduction band. This model takes into account the exchange interaction between different f -sites. Hence, it is suitable for description of systems, which consists of dense sublattice of f orbitals in a nonmagnetic matrix. In case of a system with relatively small concentration of Ce, each Ce atom can be treated to some extent as isolated impurity, what can be described by the single impurity Anderson model (SIAM). The hamiltonian of the SIAM can be written as: H = ∑ εkc † kσ ckσ + εf ∑ fσ † fσ +Un↑n↓ + ∑ (Vkc † (5.24) k,σ fσ + H.C.), k,σσ k,σ with application of the same notation as in equation (5.23). However, in this case we have not inter-action between f moments at different sites of crystal lattice. The spectral function for both models (PAM and SIAM) has been calculated within many theoreti

cal schemes (Czycholl, 1985; Glossop and Logan, 2002; Horvati´c et al., 1987). The crucial parameter is the ﬁlling of the f

shell (nf

). This quantity is a measure of the degree of localization of the f elec

trons and plays the same role as the Jc

f

coupling constant (cf. Doniach diagram). In case of nf

≈

1, i.e. in so called Kondo regime, the spectral function reveals the sharp peak in a vicinity of εF . This is a universal feature, obtained within many computational schemes, for both PAM and SIAM. The sharp resonant peak (so called Kondo peak) is interpreted as a symptom of a Kondo singlet (Frota, 1992). The spectrum consists also of some side peaks, albeit their shape and location depend on the model parameters such as onsite Coulomb repulsion (U) and the dispersion of bare conduction band (εk). The mixing between f shell and conduction band, described by Vc f , results in delocalization of f electrons. In case of mixed valence state (i.e. for 0.5 < nf < 0.8) the peak is still present in vicinity of εF , but it becomes broadened. Signiﬁcant spectral weight is transferred above the Fermi level. The enhancement of the spectral function in vicinity of εF is visible in large values of Sommerfeld γ coefﬁcient reported for many Ce compounds. However, direct measurement of the Kondo peak was a serious challenge for earlier photoemission spectroscopy experiments. It should be noted that the Kondo peak is located above εF at energy: δ = kBTK sin(πnf ) (5.25) in order to satisfy the Friedel sum rule, whereas the photoemission probes the occupied states. Hence, it provides mainly the information about spectral features with binding energy below εF . On the other hand, from the spectrum collected at temperature T , one can recover the spectral function up to 5kBT (kB -Boltzmann constant) dividing it by the Fermi function. Another issue, which will be discussed further in the text, is a suppression of the Kondo peak at elevated temperature. Therefore, actually a “tail” of the Kondo peak is recorded in experiment. To sum it up, these points together with exigency of very high resolution (the resolution should be of order of kBTK , in order to observe the Kondo peak (Garnier et al., 1997)) make that the direct measurement of a Kondo resonance is a very demanding task. Several fundamental questions arise in studies of cerium intermetallic compounds. The problems of key importance for the modern physics are encountered in such systems. One can consider for example such ideas as asymptotic freedom or quantum entanglement, which emerge in theoretical studies of the Kondo problem. The direct scientiﬁc aim which underlies under the research in the ﬁeld of cerium compounds is to provide a realistic description of them. There is a long standing debate on the interpretation of valence band photoelecton spectra. According to some studies, it seems that a single impurity Anderson model, treating f electrons as localized, captured well the physical situation. On the other hand, there are reports which contain results which can be explained exclusively by considering itinerant character of f electrons, what is in a scope of periodic Anderson model. The degree of delocalization of f electrons, presence of the spectroscopic signatures (i.e. presence of a Kondo peak in spectra) as well as their thermal evolution (i.e. large to small Fermi surface crossover) are a subject of controversy in the literature (Aynajian et al., 2012; Hackl and Vojta, 2008). The technical development of experimental methods such as angle-resolved photoelectron spectroscopy, allows to probe the dispersion relation in a close vicinity of the Fermi energy. This opens a route to the comparison of microscopic parameters derived from the model ﬁt to the directly measured band structure with results of transport and thermal measurements. Here, a brief overview of the most important results of studies of cerium intermetallics with photoelectron spectroscopy will be given, The analysis of the bulk 4f spectral function of CeRu2Si2 and CeRu2 compounds obtained by means of photoelectron spectroscopy at Ce 3d-4f resonance (Sekiyama et al., 2000) provides a good illustration of applicability of SIAM. The strength of a mixing between f electrons and conduction band, represented by the Kondo temperature TK, is different in these compounds . CeRu2Si2 is characterized by TK ≈22 K, while for CeRu2 TK is of order of 1000 K. In the ﬁrst compound the effects related to Coulomb repulsion on 4f shell have crucial meaning, what is reﬂected in a valency of Ce which is close to 3. On the other hand, the physics in CeRu2 is dominated by a strong mixing between f level and conduction band. One can treat that compounds as two different limits of PAM. Namely, the CeRu2Si2 compound corresponds to SIAM which treats f electrons as localized, while CeRu2 can be described in terms of a quasiparticle band. According to SIAM, the ground state of an f electron system can be described as a linear combination of f 1 5 and f 0 conﬁguration due to non-zero 2 hybridization Vc f . However, the photoelectron spectroscopy probes also excited states. Indeed, the spectral function of a typical Ce compound characterized by low TK , consists of three peaks which correspond to different number of f electrons in the ﬁnal state. These are: f 0, f 51 and f 71 ﬁnal states. 22 The ﬁrst one, f 0 forms usually a broad hump-like structure with a maximum at binding energy equal to -0.2 eV. The f 51 spectral feature is located at the Fermi level and it is interpreted as a tail of the 2 Kondo resonance peak, which itself has the maximum above εF . The f 71 ﬁnal state is a spin-orbit 2 partner peak, and it is observed at about -0.3 eV. Beyond such structures, the crystal ﬁeld side bands give a contribution to the spectrum. However, there are usually not resolved because of not sufﬁcient resolution. The analysis of the spectra of low TK systems within the framework of the SIAM with the non-crossing approximation (NCA) has shown that, the effects of the crystal ﬁeld should be taken into account, in order to obtain realistic value of TK from spectra ﬁtting (Ehm et al., 2007). On the other hand, the spectra of CeRu2 compound cannot be described properly with reference to different ﬁnal states of f shell. The angle-integrated spectrum (collected with the same resolution as that for CeRu2Si2) of such a compound displays a broad peak below the Fermi energy. This is interpreted as a ﬁngerprint of the itinerant character of f electrons. Such a state should be well modeled with application of band structure calculations for example the tight binding method. Numerous experimental studies testiﬁes that SIAM provides sufﬁcient framework for interpretation of the (angle-integrated) photoemission spectra on polycrystalline samples of Ce compounds characterized by low value of TK (see for example (Garnier et al., 1997)). However, the high resolution angle resolved spectra of display features beyond the simple SIAM picture. The main disadvantage of SIAM is the lack of the coherence scale Tcoh. Some experiments point to the thermal evolution of the spectra which is inconsistent with SIAM (Joyce et al., 1992). Such an inconsistency can be explained by reference to the solution of PAM. The exact numerical solution of such a model (Tahvildar-Zadeh et al., 1998) obtained within quantum Monte Carlo approach shows that the thermal evolution of a spectral function is not as rapid as in the case of SIAM. Moreover, it shows that the Kondo peak persists in the spectral function up to 10TK, what explains very weak dependency of the spectra on temperature. Another predictions of the PAM is the absence of universal scaling of the Kondo peak intensity with the hybridization. In case of impurity models the Kondo peak height is proportional to the 1/Vc f 2 factor. Among discrepancies between experimental data and the description of spectra in terms of SIAM, the problem of dispersion of f states was reported many times (Andrews et al., 1996, 1995; Vyalikh et al., 2006). However, the ﬁrst systematic studies of dispersion of spectral features related to the Kondo peak is presented in (Im et al., 2008). The authors have studied the electronic structure of quasi two dimensional heavy fermion compound CeCoGe1.2Si0.8. Such a system is characterized by the following energy scales: TK ≈ 300 K, Tcoh ≈180 K. The ARPES spectra collected below Tcoh with application of resonance spectroscopy near 4d edge clearly shows the itinerant character of f electrons. The dispersion of a quasiparticle E±(k) band was ﬁtted with the simpliﬁed formula based on PAM: ε˜ f + εk ± (ε˜ f − εk)2 + 4|V˜k|2 E±(k)= , (5.26) 2 where the bare dispersion of conduction band εk± was assumed as a linear function of k vector. The constant strength of the hybridization between f electrons and conduction band carriers was assumed: V˜k ≈const. The renormalized f level position was used to estimate the Kondo temperature: ε˜ f TK = . (5.27) kB The coherence temperature is equal to the energy corresponding to the maximum of hole quasiparticle band divided by kB. Obtained values of Tcoh, TK and mass enhancement mj turned out to be in good m agreement with these obtained from thermodynamic and transport measurements. On the other hand, one should have in mind that the hybridization strength Vc f should reﬂect the symmetry of both, the crystal lattice and f orbital character. The variation of quasiparticle weight at the Fermi level, observed in Ce2Co0.8Si3.2 by means of ARPES was interpreted in a light of this effect (Starowicz et al., 2014). The cornerstone of such an analysis is the theoretical analysis of hybridization effects in PAM by means of slave boson approach (Ghaemi et al., 2008). According to such a theory, there are some directions in the Brillouin zone which correspond to nodes in the hybridization function. This opens a way to possibility of extraction of the variation of Vc f function from the ARPES data. 5.5 Matrix elements in ARPES The measured signal related to photoemission from states characterized by wave vector k and energy ε is given by the proportionality: I(ε,k) ∼|Mf ik|2A(ε,k) fFD(ε,T ) (5.28) The spectral function A(ε,k) and the Fermi function have been discussed in preceding chapters. Here, we would like to concentrate on the role of the matrix element Mf ik. It turns out, that the measured intensity of photocurrent can be signiﬁcantly modiﬁed, even totally suppressed, due to details of geometric arrangement of experimental setup (Damascelli et al., 2003). Such a behavior is explained by the inﬂuence of Mfik quantity. Indeed, matrix elements depend on the geometry of the measurement, i.e. on the energy and polarization of incident radiation as well as on the geometrical relation between the propagation vector of radiation and detection direction with respect to the sample position. Mf ik identiﬁed from equation 5.6 can be rewritten with application of commutation rules as: f Mf ik = φf ,Ek A · p φi,k) ∼ φf ,Ek ξ · r φi,k) = φj(r)ξ · rφi,k(r)d3r , (5.29) f ,Ek R with ξ being a unit vector describing the polarization of the incoming photons. The formula 5.29 shows, that Mfik is proportional to the matrix element of dipole moment operator between an initial and a ﬁnal state. The initial state φi,k) usually can be treated as the linear combination of atomic orbitals, which can be separated into orbital and radial part. The ﬁnal state can be approximated by a plane wave function: iκ·r r φf ,Ek ) ∼ e, where κ is a vector connecting a sample with the detection point. The dependency of Mf ik on polarization of incident radiation can be useful in determination of dominating orbital character of measured spectral features. Imagine that the photoemission experiment in geometry depicted in (Fig. 5.3) is performed. We can distinguish some mirror plane, i.e. the plane that is determined by the direction of detection κ and the propagation vector (q) of incoming light. At the beginning we can assume that the polarization of light ξ is perpendicular to the mirror plane (ξ = Es). Let’s apply the reﬂection with respect to the mirror plane to the expression under integral giving matrix element 5.29. This operation changes the sign of the dipole operator, while the function describing the ﬁnal state remains unchanged. Thus, if the function describing the initial state is invariant under performed reﬂection (e.g. s atomic orbital), the overall expression under integral changes sign and the integral gives zero. On the other hand, if the initial state has got an odd symmetry, then integration results in some non-zero value. Hence, photoemission in mirror plane with a polarization of the radiation perpendicular to this plane probes only states which have got odd symmetry with respect to the mirror plane. The same reasoning can be repeated for the light with polarization lying in mirror plane. In such a case, the situation is reversed. Namely, the contribution from even parity orbitals with respect to the mirror plane is observed, while that connected with the odd ones vanishes. The table 5.1 displays the symmetry of the atomic like orbitals (here we deal with real atomic orbitals, which have well deﬁned parity under space reﬂections, in distinctions to the complex orbitals given by spherical harmonics) with respect to mirror plane reﬂection in described experimental geometry, as well as assess visibility in photoemission induced by radiation with particular polarization. Table 5.1: The atomic orbitals together with their parity with respect to mirror symmetry (cf. (Fig. 5.3)). The suppression of contribution related to particular orbital character in a spectrum collected with radiation with polarization in-plane Ep or out off plane Es is denoted by ×. These results are valid only for the photoemission in mirror plane. orbital parity ξip ξop s even × px even × py odd × pz even × dz2 even × dxz even × dyz odd × dxy odd × dx2−y2 even × fz3 even × fxz2 even × fyz2 odd × fxyz odd × fz(x2−y2) even × fx(x2−3y2) even × fy(3x2−y2) even × 5.6 Experimental setups At this point, the detailed description of used experimental setups will be given. The measurement of speciﬁc heat (two-tau method) and electrical resistivity (four-probe technique) have been performed with application of standard Physical Property Measurement System (PPMS, Quantum Design). The description of these methods would be mainly technical, so it is omitted. 5.6.1 ARPES setup available in Department of Solid State Physics, Institute of Physics, Jagiellonian University The signiﬁcant part of the data presented in this thesis was collected with application of equipment available in the Laboratory of Photoelectron Spectroscopy in Department of Solid State Physics (Insititute of Physics, Jagiellonian University in Krakow). The equipment was bought thanks to the ﬁnancial support from ATOMIN project. The single crystals were oriented along high symmetry directions using Laue method, before photoemission studies. Laue patterns were collected using the setup (Laue camera) allowing for experiments in backscattered radiation geometry. The intensity of the reﬂected X-rays were collected by CCD detector. X-ray radiation was generated by standard X-ray tube with Mo anode. The band structure of investigated systems was measured with conventional in-house ARPES setup. The ultra high vacuum (uhv) conditions were realized in the system of vacuum chambers. Typical base pressure in the analysis chamber was equal to 5 · 10−11 mbar. Measurements were performed at ﬁxed temperature, selected between 10.5 K and 340 K. The sample cooling was provided by He refrigerator working in a closed cycle. During the measurements sample was mounted on the manipulator with ﬁve degrees of freedom, what enabled for Fermi surface mapping. Hemispherical photoelectron energy analyzer Scienta R4000 present in the system allowed for measurements with the minimal resolution of 1.8 meV and 0.1◦, for energy and angles respectively. The photoelectrons are excited by the radiation generated by helium lamp (He I and He II spectral lines with energies 20.1 eV and 40.8 eV respectively) or X-ray tube (MgKα and AlKα radiation with energies 1253.6 eV and 1486.6 eV respectively). The analysis chamber is also equipped with instruments allowing for performing low energy electron diffraction (LEED). Additionally, several vacuum cham-bers allowing for sample preparation are present within the same uhv system. The heating stage enables heating the samples up to 2000◦C. The samples can be also cleaved and sputtered by Ar+ ions generated by ion gun. 5.6.2 Cassiopee beamline, synchrotron Soleil High intensity and broad energy range makes synchrotron radiation a powerful tool in edge-cutting research in material science. The opportunity of performing a resonant photoemission is of particular interest for Ce intermetallics. Thorough studies of ﬁne spectral feature in band structure close to the Fermi level need good resolution and considerable statistics. The enhancement of cross section for photoionization of Ce 4f electrons is encountered at h¯ω ≈ 120 eV, 880 eV. Such energies are inaccessible in conventional in-house experimental setups. Thus, synchrotrons allow to explore subtle effects, e.g. related to Kondo physics, which are hardly visible with conventional sources. One of these types of sources is the synchrotron center SOLEIL, which is located near Paris, in France. Particularly good conditions to study compounds of Ce are provided in the Cassiopee ARPES endstation 3. Such a beamline allows for experiments with application of exciting radiation between 8 eV and 1500 eV. Actually, there are two independent ARPES setups in this endstation: the ﬁrst one used for conventional ARPES measurements and the second one for spin resolved photoemission. However, they cannot be used simultaneously because the beam coming from the ring is directed by the reﬂecting mirror to the one selected part. There is also a setup allowing for preparation of samples with application of molecular beam epitaxy (MBE) technique. These parts are connected with each 3 Detailed description and technical information of the Cassiopee beamline is available in the web page: https://www.synchrotron-soleil.fr/en/beamlines/cassiopee other within a single ultra-high vacuum system enabling a transfer of samples between chambers. Two undulators are used in order to obtain electromagnetic radiation. One generates photons with energies between 8 eV and 155 eV, while the second one works between 100 eV and 1500 eV. The spin resolved spectra are collected with application of Mott detector and energy electron analyzer SES2002. It is possible to collect a spectrum corresponding to a single angle. The ARPES part of setup is equipped with Scienta R4000 hemispheric analyzer. It allows to measure with many angular channels simultaneously. The optimal energy resolution obtained at h¯ω ≈120 eV was dominated by the spectral width of radiation and it was equal to about 15 meV (it was possible to set the better resolution, but it lead to worse quality of the spectra). The sample is mounted on the four-axis manipulator. The lowest possible temperature near the sample is equal to 6 K. The manipulator is cooled by liquid He from a Dewar ﬂask (open system). There is a possibility to change a polarization of incoming radiation between linear vertical and linear horizontal. Bibliography Andrews, A. B., Joyce, J. J., Arko, A. J., Fisk, Z., and Riseborough, P. S. (1996). Phys. Rev. B, 53:3317–3326. Andrews, A. B., Joyce, J. J., Arko, A. J., Thompson, J. D., Tang, J., Lawrence, J. M., and Hemminger, J. C. (1995). Phys. Rev. B, 51:3277–3280. Aynajian, P., Neto, E., Gyenis, A., Baumbach, R., Thompson, J., Fisk, Z., Bauer, E., and Yazdani, A. (2012). Nature, 486:201–6. Cho, E.-J., Jung, R.-J., Choi, B.-H., Oh, S.-J., Iwasaki, T., Sekiyama, A., Imada, S., Suga, S., Muro, T., Park, J.-G., and Kwon, Y. S. (2003). Phys. Rev. B, 67:155107. Czycholl, G. (1985). Phys. Rev. B, 31:2867–2880. Damascelli, A., Hussain, Z., and Shen, Z.-X. (2003). Rev. Mod. Phys., 75:473–541. Ehm, D., Hüfner, S., Reinert, F., Kroha, J., Wölﬂe, P., Stockert, O., Geibel, C., and Löhneysen, H. v. (2007). Phys. Rev. B, 76:045117. Fano, U. (1961). Phys. Rev., 124:1866–1878. Frota, H. O. (1992). Phys. Rev. B, 45:1096–1099. Garnier, M., Breuer, K., Purdie, D., Hengsberger, M., Baer, Y., and Delley, B. (1997). Phys. Rev. Lett., 78:4127–4130. Ghaemi, P., Senthil, T., and Coleman, P. (2008). Phys. Rev. B, 77:245108. Glossop, M. T. and Logan, D. E. (2002). J. Phys. Condens. Matter, 14(26):6737–6760. Hackl, A. and Vojta, M. (2008). Phys. Rev. B, 77:134439. Horvati´c, D., and Zlati´ c, B., Sokcevi´c, V. (1987). Phys. Rev. B, 36:675–683. Hüfner, S. (2003). Photoelectron Spectroscopy: Principles and Applications, pages 133–148. Springer Berlin Heidelberg. Im, H. J., Ito, T., Kim, H.-D., Kimura, S., Lee, K. E., Hong, J. B., Kwon, Y. S., Yasui, A., and Yamagami, H. (2008). Phys. Rev. Lett., 100:176402. Joyce, J. J., Arko, A. J., Lawrence, J., Canﬁeld, P. C., Fisk, Z., Bartlett, R. J., and Thompson, J. D. (1992). Phys. Rev. Lett., 68:236–239. Kordyuk, A. A., Borisenko, S. V., Koitzsch, A., Fink, J., Knupfer, M., and Berger, H. (2005). Phys. Rev. B, 71:214513. Landau, L. and Lifshitz, E. (2001). Mechanika kwantowa. Wydawnictwo Naukowe PWN, Warszawa. Sekiyama, A., Iwasaki, T., Matsuda, K., Saitoh, Y., Onuki, Y., and Suga, S. (2000). Nature, 403:369– 398. Sekiyama, A., Suga, S., Iwasaki, T., Ueda, S., Imada, S., Saitoh, Y., Yoshino, T., Adroja, D., and Takabatake, T. (2001). J. Electron Spectrosc., 114-116:699 – 703. Starowicz, P., Kurleto, R., Goraus, J., Schwab, H., Szlawska, M., Forster, F., Szytuła, A., Vobornik, I., Kaczorowski, D., and Reinert, F. (2014). Phys. Rev. B, 89:115122. Suga, S. and Sekiyama, A. (2014). Photoelectron Spectroscopy: Bulk and Surface Electronic Struc tures, pages 7–31, 155–217. Springer Berlin Heidelberg. Tahvildar-Zadeh, A. N., Jarrell, M., and Freericks, J. K. (1998). Phys. Rev. Lett., 80:5168–5171. Vyalikh, D. V., Kucherenko, Y., Danzenbächer, S., Dedkov, Y. S., Laubschat, C., and Molodtsov, S. L. (2006). Phys. Rev. Lett., 96:026404. Part III Results Chapter 6 Kondo Lattice Behavior Observed in CeCu9In2 Compound R. Kurleto, A. Szytuła, J. Goraus, S. Baran, Yu. Tyvanchuk, Ya. M. Kalychak, and P. Starowicz, Journal of Alloys and Compounds 803, 576-584 (2019). doi: 10.1016/j.jallcom.2019.06.140 Highlights • CeCu9In2 is a new system which hosts Kondo lattice state with coherence temperature of 45 K • electrical resistivity varies with a temperature in a way typical of Kondo lattice systems • Ce 4f contribution to the speciﬁc heat displays pronounced anomaly at 1.8 K • at higher temperatures (≈30 K) contribution from crystal ﬁeld effect is visible in speciﬁc heat • extracted Ce 4f spectral function near the Fermi energy with application of ultraviolet photoelectron spectroscopy shows features related to f1 and f1 ﬁnal states 7/25/2 • Gunnarsson-Schönhammer analysis has been applied to XPS spectra of Ce 3d core levels conﬁrming presence of mixing between 4f orbitals and conduction band • ab initio calculations suggest that one may expect a Fermi surface nesting in CeCu9In2 compound Chapter 7 Studies of Electronic Structure across a Quantum Phase Transition in CeRhSb1−xSnx ´ R. Kurleto, J. Goraus, M. Rosmus, A. Slebarski, and P. Starowicz, The European Physical Journal B 92, 192 (2019). doi: 10.1140/epjb/e2019-100157-3 Highlights • CeRhSb1−xSnx is a Kondo insulator for x < 0.13 and a weakly magnetic non-Fermi liquid for x > 0.13, x = 0.13 corresponds to a quantum critical transition • sharp peak at the Fermi level (4f15/2 ﬁnal state) is not visible in photoemission spectra, while the structure corresponding to 4f15/2 ﬁnal state is clearly visible in measurements for each studied composition • observed shifts of the band structure as a function of x agree with the hole doping scenario, both in experiment and ab initio calculations • however, there are some deviations from a simple rigid band shift scenario • ab initio calculations predict that the hole doping induces a series of modiﬁcations of the Fermi surface topology (Lifshitz transitions) Chapter 8 Direct observation on f-electron hybridization effects in CeCoIn5 Highlights • the electronic structure of heavy fermion superconductor CeCoIn5 has been mapped at 6 K using ARPES method • a band characterized by large effective masses and high f orbital contribution has been observed in spectra near the Fermi level • the signiﬁcant effect of matrix elements on the ARPES spectra has been observed • a symmetrization of experimentally obtained Fermi surface allowed to extract variation of 4f electron spectral intensity, which is related to the effect of hybridization, what was conﬁrmed by tight binding calculations Abstract A heavy fermion superconductor, CeCoIn5 was studied by angle-resolved photoemission spectroscopy (ARPES) with excitation energy corresponding to Ce 4d-4f resonance. Fermi surface (FS) maps with pronounced Ce 4f electron spectral weight have been obtained at temperature of T=6 K in wide angular range. Various effects of a hybridization between valence band and f-electrons (Vc f ) have been observed. These are e.g. heavy f-electron dispersing bands and variable f-electron contribution along FS. Due to a strong variation of hybridization between FS branches certain bands are strongly correlated with high f-electron contribution while others are weakly correlated. The hot spots at EF found around the Γ point in ARPES spectra, which represent highly correlated states, coincide with high contribution from In atoms (5p electrons) located in Ce-In layers, which have been identiﬁed as principal surface termination with a help of one-step model calculations. Spectral weight of f-electrons at EF is determined by both matrix element effects and the form of wave vector dependent Vc f (k). FS scans covering a few Brillouin zones allowed to observe large matrix element effects. In contrast, the inﬂuence of Vc f on spectral intensity obeys the reciprocal lattice symmetry. A symmetrization of experimental FS, which reduced matrix element effects to some extent, yielded a speciﬁc variation of 4f electron spectral intensity at EF around Γ and M points, which was attributed to a pure effect of Vc f hybridization. Tight binding approximation calculations for Ce-In plane yielded the same, quite universal distribution of 4f electron density for a variety of parameters used in TBA. This proves that we were able to extract the effect of Vc f on f-electron intensity in ARPES spectra. 8.1 Introduction Hybridization between 4f electrons and conduction band (Vc f ) leads to various fascinating phenomena such as Kondo effect, heavy fermion state or mixed valence (Stewart, 1984). It can also play a role in quantum phase transitions and superconductivity. In case of many Ce systems photoemission spectroscopy reveals a high intensity peak near the Fermi energy (EF ) with strong contribution from 4f electrons and this spectral feature is called a Kondo resonance in some cases (Klein et al., 2011; Sekiyama et al., 2000; W. Allen, 2005). Already early ARPES studies of the Kondo resonance for Ce systems pointed that its intensity depends on wave vector, which is a ﬁngerprint of anisotropic hybridization (Garnier et al., 1997). Later it was observed that this f-electron related peak is intensive at Fermi vectors (Danzenbächer et al., 2005). Finally, it is known that bands, which are strongly correlated with 4f electrons, may coexist with those weakly correlated (Koitzsch et al., 2013). In fact, anisotropic structure of Vc f is a source of f-electron peak intensity variation along the Fermi surface (FS) (Ghaemi et al., 2008). However, matrix element effects play an important role here and also have a serious inﬂuence in this intensity. Although, it is believed that anisotropic Vc f yields a speciﬁc distribution of f-electron peak along FS, its structure described consistently by ARPES and theory was not presented so far. The effects described above can be conveniently studied on the example of intermetallics from CemTnIn3m+2n family (T -transition metal) (Thompson et al., 2003). We choose CeCoIn5, which de-spite of relatively simple, layered crystal structure hosts many interesting physical states (Shimozawa et al., 2016; Steglich and Wirth, 2016). Namely, the superconductivity with Tc of 2.3 K has been observed in this compound (Petrovic et al., 2001). The heavy fermion state, which appears due to hybridization between 4f electrons and conduction band, can be characterized by Sommerfeld coefﬁcient γ which amounts to 0.35 J/(mole·K2) (Thompson et al., 2003) and coherence temperature equal to 45 K (Jia et al., 2011). Such a system provides an opportunity to study exotic phenomena like quantum criticality (Paglione et al., 2003) or Fulde-Ferrel-Larkin-Ovchnnikov (FFLO) type superconductivity at high magnetic ﬁelds (Kumagai et al., 2006). In this paper we report a systematic ARPES study performed for a CeCoIn5 heavy fermion compound. Observation of different hybridization effects, which depend on band and Fermi vector, gives an image of Vc f effects in ARPES spectra. A principal role in a formation of the observed heavy fermion bands is played by Ce-In planes. Distribution of f-electron spectral weight at εF obtained with tight binding approximation (TBA) calculations is found in the ARPES FS mapping, what allows to propose an image of Vc f hybridization for the investigated system. 8.2 Material and methods Single crystalline samples of CeCoIn5 have been obtained using the ﬂux method. Their synthesis and characterization is described elsewhere (Jia et al., 2011). They have been oriented along important crystallographic directions using X-ray diffraction (Laue method). The ARPES measurements have been realized in Cassiopee beamline of Synchrotron Soleil (Paris, France). All measurements have been performed at temperature equal to 6 K. The Scienta R4000 photoelectron energy analyzer has been used. The samples have been cleaved in ultra high vacuum conditions prior to the measurements (base pressure: 10−10 mbar). The (001) plane was exposed after cleaving. Spectra have been collected using radiation with energy equal to 122 eV (Ce 4d-4f resonant transition) and 87 eV. The electronic structure of CeCoIn5 has been calculated using Korringa-Kohn-Rostoker (KKR) method in single particle Green function approach with application of multiple scattering theory (Ebert et al., 2011). The implementation of the method in a spin polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) code for Calculating Solid State Properties (SPR-KKR) has been used. The tight binding approximation (TBA) model (Maehira et al., 2003) based on the generalized Slater-Koster approach (Slater 83

and Koster, 1954) has been ﬁtted to the experimental band structure. The ﬁve orbital TBA model for Ce-In planes comprises three doublets from Ce 4f5 manifold ( fa, fb, fc) together with two doublets 2 (pa, pb; jz = ±1) from In 5p manifold. The 5p state with jz = 0 does not contribute to the band structure of Ce-In layer near the Fermi level. Average occupancy of the f shell was ﬁxed during ﬁts as nf = 1. 8.3 Results and Discussion Fermi surface (FS) of CeCoIn5 was studied by ARPES at low temperature (T=6 K) in a wide angular range for two sample orientations (Fig. 8.1 a-c). Such experiment allows to visualize not only FS topography but enables a closer look into a contribution from f-electrons and matrix element effects. These last depend on experimental geometry, photon energy and on a probed Brillouin zone. Fig. 8.1 a-c presents spectral intensity at a Fermi energy (EF ) along semi-planar cuts of the reciprocal lattice. The spectra recorded with hν=122 eV photons, namely at Ce 4d-4 f resonance, assure an increased contribution from Ce 4f electrons (Fig. 8.1 a, b) as compared to off-resonance data obtained with hν=87 eV (Fig. 8.1 c). It is known that ARPES mapping with photon energy of 122 eV probes the region close to Γ point in three-dimensional Brillouin zone (Fig. 8.1 d) for normal emission (Chen et al., 2017). The assumption of inner potential V0 = 16 eV (Chen et al., 2017) allowed to ﬁnd the off-resonance energy of 87 eV, for which the scanned surface in k-space crosses the Γ point at kx=0 and ky=0 as well. Even a superﬁcial analysis of spectral intensity dominated by Ce 4f electrons (Fig. 8.1 a, b) leads to a conclusion that it breaks the reciprocal lattice symmetry and such a result is independent of the applied method of spectra normalization. Hence, the reason for such asymmetry must be in matrix element effects. Moreover, the matrix element effects must be responsible for different distribution of spectral intensity in Figs. 8.1 a and b as these differ just by sample orientation. It also is known that a variation of f-electron density and related spectral intensity may also depend on a speciﬁc form of hybridization between f-electrons and the valence band (Vc f ) (Ghaemi et al., 2008; Starowicz et al., 2014) but such a variation must ﬁt into the reciprocal lattice symmetry in contrary to matrix element effects, which depend i.a. on experimental geometry (Fig. 8.1 e). Therefore, one can distinguish the areas of higher 4f-electron signal, while equivalent k-space regions exhibit a depletion of intensity. Such high intensity areas are indicated with black dashed line rectangles in Fig. 8.1 a-c. It is note

worthy that the off-resonance FS (Fig. 8.1 c) regains more reciprocal lattice symmetry, what signiﬁes that quite prominent matrix element effects on intensity are related to f-electrons. The ﬁrst approach to understand the spectral intensity are one-step model calculations realized with SPR-KKR code (Ebert et al., 2011), which treat the system as a semi-inﬁnite crystal with a spe

ciﬁc surface termination. A projection of both bulk and surface initial states on time-reversed LEED ﬁnal states yields a simulation of ARPES spectra, which takes into account photon energy, light polarization, experimental geometry, ﬁnal state effects and surface termination. As there are 3 possible surface terminations for cleaved CeCoIn5 (Fig. 8.1 f) such calculations have been performed for Ce-In (Fig. 8.1 g), Co (Fig. 8.1 h) and In2 (Fig. 8.1 i) terminations as marked with arrows in Fig. 8.1 f. They indicate that Ce-In termination prevails as corresponding simulated ARPES spectra for hν=122 eV (Fig. 8.1 g) ﬁt the experimental results, what is also the case for the off resonance (hν=87 eV) spectra (not shown). The simulations also visualize a break of left-right symmetry, which is characteristic of f-electron intensity in the ARPES FS image. To extract the contribution of f-electrons a simulated spectrum for only 4f initial states was added to the spectrum comprising all initial states (Fig. 8.1 j). This ﬁgure presents increased asymmetric contribution of 4f states to photoemission spectra and the lack of symmetry is particularly visible around the Γ point. Thus, we observe that the matrix element effects are strong and they have to be separated, if one wishes to study bare Vc f hybridization effects on 4f electron spectrum. Figure 8.1: (Color on-line) Fermi surface of CeCoIn5. Intensity maps along Fermi surface obtained by ARPES at T=6 K for photon energies of 122 eV (a,b) and 87 eV (c). (d) First Brillouin zone with marked high symmetry points. (e) Crystal structure of CeCoIn5. (f) Geometry of the experiment. Simulation of the ARPES spectra for Ce-In (g), Co (h), In2 (i) surface terminations using one-step photoemission model based on SPR-KKR calculations for photon energy of 122 eV as well as with enhanced contribution from 4f initial states (j) in case of Ce-In termination. Constant energy contours at 0 eV (k-p) and −0.30 eV (q-v) of Bloch spectral functions obtained in SPR-KKR calculations. These represent a topology of the Fermi surface. The total Bloch spectral function (k,l, q, r) is projected onto atomic wave functions of Ce (m,s), Co (n,t), In from In2 layers (o,u) and In from Ce-In planes (p,v). The total Bloch spectral function calculated in the absence of hybridization effects (4f in core) is shown in (k) and (q). Experimental FS topography can be confronted with KKR calculations yielding Bloch spectral functions (BSF), which have been extracted in constant energy contours of EF (Fig. 8.1 k-p) and 0.25 eV below EF (Fig. 8.1 q-v) for wave vector kz = 0 (Fig. 8.1 k-v). A presentation for two binding energies gives already some idea about dispersions, allows to distinguish hole and electron bands and helps to ﬁnd theoretical counterparts of the experimental FS. Total BSF distributions with felectrons frozen in a core are given in Fig. 8.1 k, q and the corresponding plot for the system with itinerant f-electrons is shown in Fig. 8.1 l, r. The difference between these ﬁgures shows an effect of Vc f hybridization with f-electrons on electronic structure. The most prominent changes are located around Γ point and along Γ-X direction, where a large contribution from f-electron states is present. This is visible in particular at EB = 0 deﬁned in calculations (Fig. 8.1 l,m). This f-electron spectral intensity is diminished at lower binding energy (Fig. 8.1 r,s). The calculations show a presence of hole pockets at Γ and X and one large electron pocket at M. In fact, branches of experimental FS correspond to theoretical FS contours located either at EB = 0 or EB = −0.25 eV. Generally, the calculations for EB = −0.25 eV represent well the shape of experimental FS. However, the theoretical contours for EB = 0 show two hole pockets at Γ, which are observed experimentally. BSF for itinerant f-electrons have been projected on atomic wave functions (Fig. 8.1 m-p, s-v) in order to ﬁnd a contribution of particular atoms to FS. Ce atoms donate mainly 4f electrons at EF with high intensity in the region of Γ and M points. Co (mainly 3d) electrons and In atoms from In planes shown by blue arrows in Fig. 8.1 f (high contribution from 5p) contribute rather to the whole FS. However, Co 3d states have a high population in the electron pockets around M (well seen for EB = −0.25 eV), while this population is reduced in some bands around Γ. Interestingly, In 5p electrons from Ce-In planes (green arrows in Fig. 8.1 f) exhibit four prominent maxima around the Γ point visible in particular at at EB = −0.25 eV. These maxima correspond to the location of four hot spots in ARPES maps located around the Γ point along Γ-M direction (Fig. 8.1 a-c). BSF represent states from the whole Brillouin zone, whereas one-step model calculations show that the contribution from Ce-In planes dominates the ARPES image. Even if 4 In atoms per unit cell give high intensity in BSF, one indium atom per unit cell located in the surface Ce-In planes may deliver quite signiﬁcant contribution to ARPES spectra. Therefore, the hot spots observed in ARPES should correspond to increased contribution from In located in Ce-In planes. Moreover, one can see that also Ce 4f electrons yield a high amplitude of BSF at the same points in k-space (Fig. 8.1 s). This will be reﬂected in high Vc f hybridization effects observed by ARPES. The experimental FS obtained in this work reproduces some shapes also present in previous ARPES studies (Chen et al., 2017; Dudy et al., 2013; Jang et al., 2017; Koitzsch et al., 2013). However, FSs obtained before were recorded at higher temperatures and, depending on a report, with other photon energies, geometry and in some cases other surface terminations. Hence, also four hot spots around Γ were not discussed before. Band structure obtained by ARPES is shown along important crystallographic directions as well as along other interesting paths in k-space (Fig. 8.2 a-p). These paths are numbered from 1 to 7 and are drawn on experimental FSs (Fig. 8.1 a,b). The increased f-electron contribution reﬂected in a high signal at EF is well visible in the spectra presented as energy distribution curves (Fig. 8.2 a-l). One can resolve an electron pocket α around M (Fig. 8.2 b, d, e), hole pockets (β , γ) around the Γ point (Fig. 8.2 a-c) and two hole-like dispersions observed at the X point along M-X-M direction (Fig. 8.2 d, e). These last are attributed to β and γ bands. The measured dispersions are in qualitative agreement with calculated BSF (Fig. 8.2 q). On the other hand, an electron like band around the X point detected along X-Γ-X direction (Fig. 8.2 a) is not found in calculations. The labels α, β and γ correspond to bands observed previously in ARPES spectra of CeCoIn5 (Chen et al., 2017; Dudy et al., 2013; Jang et al., 2017) and these are also named 135, 133 and 131, respectively, in other ARPES study (Koitzsch et al., 2013). BSF with frozen f-electrons is also shown to visualize the effect of these electons on band structure (Fig. 8.2 r). Roughly speaking this is an appearance of heavy bands at or above EF . Projections of BSF on particular atoms along high symmetry directions is shown in the Supplementary material. Previous ARPES measurements performed by Chen et al. delivered interesting information on heavy fermion formation in CeCoIn5. These studies showed that along the M-Γ direction the bands β and γ become broad and gain high f-electron related intensity with lowering temperature down to 17 K. We had the opportunity to measure the same region at much lower temperature of 6 K. Therefore, the hybridization effects are quite pronounced; bands β and γ are much broadened and yield high intensity hot spots at FS, while the α band remains sharp and weakly correlated. This is visible in Fig. 8.2 b, c presenting single measurements assuring the same experimental conditions for the band structure along M-Γ-M and the path ”3” located next to M-Γ-M direction (Fig. 8.2 b, c). Panels d and e from the Fig. 8.2 present scans along M-X-M measured in different Brillouin zones and differing by certain value of kz vector. The α and β bands are better separated in the Fig. 8.2 e. Spectral intensity in these panels differs also due to matrix element effects for 4f electrons. Namely, a contribution from Ce 4f electrons is visible below ky ∼−1Å−1 and above ky ∼ 1Å−1 in Fig. 8.2 d, whereas it is enhanced around X in Fig. 8.2 e. This agrees with the general tendency of high and low intensity of 4f electrons related to matrix element effects. Finally, effects of c-f hybridization are also visible in Fig. 8.2 f. High f-electron intensity starts at wave vectors, which would be Fermi vectors in the absence of Vc f hybridization. These images present the spectral shape characteristic of Vc f hybridization described by the periodic Anderson model (Tahvildar-Zadeh et al., 1998). Finally, the spectra allowed to detect dispersion of f-electron bands. The spectra normalized by Fermi Dirac distribution were subjected to MDC ﬁtting (Fig. 8.2 m,n). The found dispersions were approximated by parabolas, which allowed to estimate values of effective masses, which range from 30 to 130 free electron mass. This does not exclude the existence of higher effective masses in other regions of BZ. (g) -(l). Weakly dispersive band is visible near the Fermi level in spectra corresponding to cut ”7” (m) and ”6’ (n) after dividing by Fermi function. The corresponding EDCs are shown in (o) and (p). Panels (q) and (r) present Bloch spectral functions along important crystallographic directions calculated with SPR-KKR package in presence (4f in VB) and absence (4f in core) of hybridization effects, respectively. Anisotropic Vc f hybridization should affect ARPES spectra considerably. The evidence includes not only variation of effective masses and quasiparticle weight (Ghaemi et al., 2008) but also various contribution of 4f electrons to spectra along the Fermi surface. The latter inﬂuences spectral intensity at EF together with matrix element effects. The last should be separated in order to discuss a bare contribution of 4f electrons to the spectra related to Vc f hybridization. In order to remove partially geometrical effect of matrix elements, FS was symmetrized by adding spectra rotated by 90◦, 180◦ and 270◦ to the original Fermi surface (Fig. 8.3 a). We have also performed alternative symmetriza

tion procedure using reﬂections or their combination with rotations (what corresponds to full 4/mmm point symmetry of Γ point), but this resulted in an image of worse quality. The contour of FS around corresponding to electron pocket around M point has been described by the circle which was ﬁtted to points belonging to FS. Then we deﬁned the φ angle, as the angle between horizontal M-X direction and the line crossing the particular point at the contour and M point. The angles are measured in counterclockwise direction. The momentum distribution curves (MDCs) crossing M point and corresponding to different φ angles have been extracted at the Fermi energy (Fig. 8.3 b). The integration over energy window of ±20 meV around EF was applied. A closer inspection shows that higher intensity of f-electron related peak at EF is present along M-X direction whereas it is lower at M-Γ. Energy distribution curves (EDCs) shown for the contour around M point exhibit a variation indicating higher f-electron contribution along same directions as for MDCs (Fig. 8.3 c). Finally, the integrated (over 0.1 x 0.1 Å−2 square surroundings) intensity along the electron pocket roughly reproduces minima along M-Γ and maxima along M-X directions. Hence, the symmetrized experimental FS yields the variation of 4f electron related spectral intensity around the M point with minimized matrix element effects. Moreover, Fig. 8.3 a presents a similar variation of spectral intensity around the Γ point. This shows maxima along Γ-M and minima along Γ-X for 4f electron contribution. The spectral intensity at EF (Fig. 8.3 e) is plotted along the rectangular contour (Fig. 8.3 a) around the Γ point. We did not analyze MDCs and EDCs in this case, as more bands are present around Γ. Here, we propose that such spectral intensity variations observed around M and Γ points are bare Vc f hybridization effects. Figure 8.3: (Color on-line) (a) Symmetrized experimental Fermi surface obtained by a superposition of ARPES data rotated 4 times by 90◦. Red circle presents the pocket around M subjected to f-electron intensity analysis. (b) Momentum distribution curves extracted from symmetrized data along lines crossing M point in different directions given by φ angle as deﬁned in (a). The exemplary red and cyan curves represent paths which are drawn with the same colors in (a). (c) Energy distribution curves from the electron pocket around M for selected φ angles. (d) Intensity at the Fermi energy along the electron pocket indicated by the black circle in (a) as a function of φ. Red arrows in (d) denote minima in intensity, which appear to be located along Γ-M direction, while blue arrows show M-X direction characterized with higher spectral intensity. (e) Intensity at the Fermi energy along the hole pocket around Γ indicated by the black circle in (a) as a function of φ1. Red arrows denote minima in intensity, which are complementary to these in (d). The assumption that the effect of Vc f hybridization on 4f spectral intensity has been found can be veriﬁed by tight-binding model calculations. One should keep in mind that ARPES spectra have a considerable contribution from surface termination, which are Ce-In planes as was proved by a comparison with one-step model calculations (Fig. 8.1). Moreover, the highest contribution to FS from In located in Ce-In planes (5p states) is found at 4 hot spots around the gamma point, where strong hybridization leading to high spectral broadening at low temperature is observed. These arguments support the idea to consider a simple model of Ce-In planes in TBA calculations. The TBA parameters have been obtained by ﬁtting theoretical dispersions to the experimental binding energies corresponding to M and X points, as well as to experimental Fermi wave vectors in the Γ-X and X-M directions. The obtained theoretical bands are compared with measured ARPES spectra along M-X-M and X-Γ-X directions in Fig. 8.4 a-b . The area of the marker is proportional to the 4f electron contribution to the band. One can see that TBA model describes well the electron pocket around M point. Several bands with dominating f electron contribution are visible above the Fermi level. It is noteworthy, that calculations predict the presence of a hole band around Γ point (Fig. 8.4 b), which becomes ﬂat and gains f electron character close to the Fermi level. This is obvious that not every band visible in measured spectra can be identiﬁed in the theory as the TBA minimal theoretical model has to reproduce f-electron intensity variation originating from surface Ce-In layer, which have important contribution to ARPES spectra. Particular attention should be paid to the distribution of the spectral weight around Γ point, which in accordance with our as well as with previous studies, is related to the heavy quasiparticle band developing at low temperature. The idea of minimal TBA model for CeCoIn5 appeared before (Maehira et al., 2003). However, previously the model has been ﬁtted to the DFT calculations. According to our knowledge, this is the ﬁrst attempt of ﬁtting such a model to photoemission spectra of CeCoIn5. The distribution of spectral weight at FS predicted by the model is shown in Fig. 8.4 c. The projec

tions on different base states have been shown in Fig. 8.4 d-h. Almost rectangular hole pocket around Γ point shows speciﬁc variation of f-electron related spectral intensity. It reaches its maximum at the Γ-M direction, while the minima appears at X-Γ direction. The spectral weight visible in the contour around M point (corresponding to the electron pocket) shows the complementary pattern. On the other hand, the distribution corresponding to In states is much more uniform on the Fermi surface. The observed variation of a spectral weight is consistent with the results of experiments. In fact, one can see that this is exactly the same dependency as shown in Fig. 8.3. Moreover, this is not just an accidental coincidence, because the series of ﬁts for different TBA parameters including those from Maehira et al. (Maehira et al., 2003) have been performed. Therefore, our minimal model can provide some universal description of electronic structure for Ce-In planes in CeCoIn5. It is noteworthy that maxima and minima of 4f related experimental intensity are reproduced by TBA calculations both around Gamma and M points. It should be stressed that this result is robust against quite considerable variation of TBA parameters (see Supplementary material). Hence, such a distribution of 4f electrons is related to original orientation of orbitals building the electronic structure of Ce-In planes and determining anisotropic form of Vc f hybridization. Thus, this is the ﬁrst interpretation of ARPES spectral function indicating the variation of Ce 4f electron intensity, which is related to Vc f hybridization after having geometrical factor of matrix element effects eliminated. Although, one might question, if symmetrized experimental data really conﬁrm this variation, as the matrix element effects have not been eliminated completely, the comparison with TBA results with a variety of parameters assures the validity of the conclusions. Finally, TBA calculations with parameters adjusted to ARPES spectra can yield the effective form of Vc f hybridization, which may be used to further modeling the system. (c) shows the distribution of the spectral weight on the Fermi surface. The projections on different base states are shown in (d)-(h). The lorentzian broadening of 15 meV and cuttof by the Fermi function with temperature equal to 20 K were applied in order to plot such maps. 8.4 Conclusions The electronic structure of CeCoIn5 system has been studied with resonant ARPES at low temperature using Ce 4d-4f transition. The determined topography of band structure is in general agreement with that reported in previous studies. The thorough comparison between measured Fermi surface and the results of one-step model calculations assuming different surfaces shows that the Ce-In termination prevails in our experiment. In this work, we have focused on the hybridization effects Vc f between f-electrons and other valence band electrons. These effects include a presence of weakly dispersive states near the Fermi energy and strong broadening of bands close to the Γ point. The spectra reveal important band dependency of the hybridization. A comparison with theoretical calculations shows that the most signiﬁcant hybridization effects present in the spectra, are related to Ce-In planes. We observe a signiﬁcant variation of f-electron spectral intensity in momentum space, originating from both, matrix element effects and momentum dependent Vc f hybridization. A symmetrization of FS obtained experimentally in wide angular range allows to separate these effects and extract the momentum dependence of f-electron intensity related to a form of Vc f . 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Supplementary material Band structure of CeCoIn5 calculated along high symmetry directions using DFT method Calculated Bloch spectral function for CeCoIn5 along high symmetry directions is shown in (Fig. 8.5 c

l). Projection on different atoms in unit cell were calculated. Two types of calculations have been performed: with 4f electrons in valence band ("4f in VB") and with 4f electrons excluded from the valence band ("4f in core"). The results obtained with the second approach can be interpreted as the electronic structure in absence of hybridization effects. Therefore, the differences between results obtained in these two ways are directly related to the Vc f hybridization effects. Figure 8.5: (Color on-line) Bloch spectral function calculated along high symmetry directions in the BZ. (a) The crystal structure of CeCoIn5 with characteristic structural features. (b) The ﬁrst Brillouin zone with marked high symmetry points. Two types of calculations are provided: with included hybridization between 4f states and rest of orbitals (c), as well as the results without the hybridization (4f in core) (d). The projection on Ce (e,f), Co (g,h), In from Ce-In planes (i,j) and In form In planes (k,l) are also provided. Description of TBA model ﬁtted to ARPES data In this paper we use previously proposed idea (Maehira et al., 2003) of description of essential features of electronic structure of CeCoIn5 by a tight binding model (within generalized Slater Koster approach) for Ce-In layer. In and Ce atoms in such layers are arranged into NaCl-type lattice in 2D. In such an approach the missing Co and In atoms can be regarded as a charge reservoir, which possible leads to renormalization of band structure. Spin orbit interaction, which is rather strong in case of heavy elements, reduces the degeneracy of 4f orbitals in Ce. In consequence two manifolds characterized by total momentum j equal to 5/2 and 7/2 appear. According to Hund’s rules the ground state in case of Ce with one f electron is a manifold with j = 5/2. In case of In atoms electrons from partially ﬁlled 5p shell are important. It turns out that in case of Ce-In layer p orbitals jz = 0 do not contribute to dispersive bands near the Fermi level. Therefore, we take into account only two doublets with jz = ±1. Finally, our basis states consists of Ce 4f5/2 sextet and two doublets from In 5p1 manifold: |pa ↑>= −|m = 1,s =↑> |pa ↓>= |m = −1,s =↓> |pb ↑>= −|m = −1,s =↑> |pb ↓>= |m = 1,s =↓> | fa ↑>= −| j = 5/2, jz = −5/2 > | fa ↓>= | j = 5/2, jz = 5/2 > | fb ↑>= −| j = 5/2, jz = −1/2 > | fb ↓>= | j = 5/2, jz = 1/2 > | fc ↑>= −| j = 5/2, jz = 3/2 > | fc ↓>= | j = 5/2, jz = −3/2 > The interactions between orbitals at neighboring lattice sites can be described in terms of electron hopping through the respective bond. The ﬁnal form of the Hamiltonian is following: ⎛⎞ † Tˆ (ppξ ) Hˆ =⎜(ppξ ) · pˆpˆi,σ + H.c.⎟+ ⎜∑ i+δ i,σδ i ⎟ ⎝ξ =σ,π ⎠ i,δ i,σ ⎛⎞ Tˆ ( ff ξ ) ⎜( ff ξ ) · fˆ† ⎟+ ⎜∑ i+δ i,σδ i fˆi,σ + H.c.⎟ ⎝ξ =σ,π,δ ,φ ⎠ i,δ i,σ ⎛⎞ † V (pf ξ ) (pf ξ ) · pˆˆfˆi,σ + H.c. + ⎜ ∑ i+δ i,σδ i,σ ⎟ ⎝ξ =σ,π ⎠ i,δ i,σ † ∑Δp · pˆi,σ pˆi,σ , (8.1) i,σ where ξ denotes the bond type (ξ = σ,π,δ,φ), T ppξ and Tff ξ are hopping matrices (δi is a hopping δ i δ i range, in our case only nearest neighbor hopping was taken into account). Vδ p f i,σξ is a p-f hybridization , Tff ξ and Vpf ξ matrix (hopping between p and f orbitals). The explicit form of T ppξδ i,σ can be written δ i δ i down using Clebsch-Gordan coefﬁcients. Δp deﬁnes the location of an atomic p level with respect to f level. The symbols: (ppξ), ( ff ξ ), (pf ξ ) are the Slater Koster integrals, which describe the strength Table 8.1: Fit parameters and orbital occupancies for a model with non-zero (ppπ) integral. Parameter Value [eV] (ppσ) 0.327 (ppπ) -0.038 ( ff σ) 0.048 (pf σ) 0.114 µ -0.035 Orbital Occupancy pa 0.891 pb 0.834 fa 0.492 fb 0.243 fc 0.264 of the hopping between orbitals through ξ bond. In our analysis we perform ﬁts taking into account exclusively σ bonds or σ bonds together with some p-p hopping through π bond. While the second approach results in better agreement between theoretical bands and experimental ARPES spectra near the Fermi level (Fig. 8.6), both give essentially similar distribution of spectral weight on the Fermi surface, which mimics well the experimental one. Following assumptions have been made for a ﬁtting procedure: • quantity Δp has been ﬁxed as 0.1 eV • Slater Koster integrals: (ppσ), (ppπ), ( ff σ), (pf σ), together with chemical potential µ have been treated as free ﬁt parameters • the binding energy at X point determined has been taken from experiment: EB(X)= −0.586 eV • the binding energy at M point determined has been taken from experiment: EB(M)= −0.469 eV • the Fermi wave vector along Γ-X directions has been determined from experiment: kF (Γ-X)= 0.235 Å−1 • the Fermi wave vector along M-X directions has been determined from experiment: kF (M-X)= 0.32 Å−1 • mean occupancy of f orbital has been set as nf = 1 Additionally, in case of a model with (ppπ)= 0 it was possible to use a quantity Δp as a free ﬁt parameter. The obtained ﬁt parameters together with orbital occupancies are shown in Table 1. and Table 2. The results of TBA calculations corresponding to a model with (ppπ)= 0 are shown in Fig. 8.6. Table 8.2: Fit parameters and orbital occupancies for a model with (ppπ) integral set to 0. Parameter Value [eV] (ppσ) 0.3255 ( ff σ) 0.102 (pf σ) 0.262 µ -0.112 Δp 0.239 Orbital Occupancy pa 0.744 pb 0.658 fa 0.574 fb 0.231 fc 0.195 Figure 8.6: (Color on-line) Theoretical band structure for Ce-In planes (solid lines) obtained by tightbinding (TB) calculations. TB parameters have been adjusted by ﬁtting theoretical bands to ARPES data with assumption (ppπ)= 0. This is the alternative version of TBA ﬁt model presented in Fig. 8.4. The comparison of ﬁtted model with the spectra collected along M-X-M (a) and X-Γ-X direction is shown. The marker size is proportional to the contribution of f orbital to the band. Panel (c) shows the distribution of the spectral weight on the Fermi surface. The projections on different base states are shown in (d)-(h). The lorentzian broadening of 15 meV and cuttof by the Fermi function with temperature equal to 20 K were applied in order to plot such maps. Chapter 9 Electronic Structure of the Ce3PdIn11 Heavy Fermion Compound Studied by Means of Angle-resolved Photoelectron Spectroscopy Highlights • the band structure of the Ce3PdIn11 heavy fermion superconductor is studied by means of ARPES for the ﬁrst time • the ﬂat band close to the Fermi level has been observed in measurements performed at Ce 4d-4f resonant transition and temperature equal to 6 K • ab initio calculations allow to distinguish contributions related to Ce 2g and Ce 1a atoms in the measured spectra • it turns out that hybridization effects related to Ce 2g atoms are stronger that these related to Ce 1a atoms Abstract The electronic structure of heavy fermion compound Ce3PdIn11, which has got two inequivalent Ce atoms in the unit cell (Wyckoff positions: Ce 1a and Ce 2g, P4/mmm space group) has been studied by means of angle-resolved photoelectron spectroscopy. The ﬂat band near the Fermi level has been observed in data collected at 6 K using incident radiation with energy corresponding to Ce 4d-4f resonant transition (hν =122 eV). This band emerges as a result of mixing between 4f electrons and conduction carriers. The comparison with ab initio calculations suggests that the hybridization related to 4f orbitals at Ce 2g site is more effective than that related to orbitals at Ce 1a site. 9.1 Introduction Intermetallic compounds of cerium are a testing ground for the studies of many intriguing physical phenomena due to the presence of weakly localized 4f electrons (Stewart, 1984). The strong Coulomb repulsion between electrons on the f orbital together with mixing between f orbitals and conduction band states lead to complex behavior of physical properties as a function of temperature. It is commonly believed that well above the coherence temperature the physics of such systems is dominated by local interactions (i.a. Kondo screening), which are described by single-impurity Anderson model (SIAM). On the other hand, below the coherence temperature the local description is not sufﬁcient, because heavy quasiparticles emerge, and one has to use periodic Anderson model (PAM). The photoelectron spectroscopy provides a direct insight into the spectral function of a system with strongly correlated f electrons (Klein et al., 2011; Sekiyama et al., 2000; W. Allen, 2005). It can be also used in order to track a crossover from the coherent regime to a local (Kondo) regime (Klein et al., 2011). The hallmark of the Kondo coherent state is a ﬂat quasiparticle band, which is visible in the spectral function in the vicinity of the Fermi energy (εF ) and can be identiﬁed with the aid of dynamical mean-ﬁeld theory (DMFT) and Anderson model. Ce3PdIn11 crystallizes in tetragonal structure (space group P4/mmm) (Latturner, 2018; Tursina et al., 2013). It belongs to CemTnIn3m+2n family (T -transition metal), which displays a rich variety of ground states and allows for studies close to the quantum critical point (Matusiak et al., 2011; Paglione et al., 2016; Tokiwa et al., 2013; Urbano et al., 2007). Unit cell of each representative of this family is built of m layers of CeIn3 and n layers of T In2 stacked along c-axis. Ce atoms occupy two inequivalent positions in the unit cell of Ce3PdIn11: Ce 1a and Ce 2g (cf. Fig. 9.1 a). Ce atoms at 1a positions have got the same chemical surrounding and symmetry (C4v point symmetry) as Ce atoms in antiferromagnetic CeIn3. On the other hand, Ce at 2g site shares the same chemical ordering (D4h point symmetry) with Ce in the Ce2PdIn8 heavy fermion superconductor. Consequently, they are characterized by different sets of crystal ﬁeld parameters and different Kondo couplings with conduction band. The presence of more than one inequivalent Ce atom per unit cell places Ce3PdIn11 among heavy fermion compounds such as: Ce5CuSb3 (Tran, 2004b), Ce5CuSn3 (Tran, 2004a), Ce7Ni3 (Sereni et al., 1994), Ce5Ni6In11 (Tang et al., 1995) and Ce3Pd20Si6 (Custers et al., 2012; Strydom et al., 2006). The complex magnetic phase diagram of Ce3PdIn11 was a subject of thorough studies. It hosts heavy fermion superconductivity and antiferromagnetism (Das et al., 2019; Kratochvilova et al., 2015). The coherent heavy fermion state is observed below 20 K. There are two antiferromagnetic phase transitions at 1.68 K and 1.56 K. The superconducting state appears below 0.58 K. Interestingly, the realization of the ﬁeld induced Lifshitz transition in Ce3PdIn11 has been proposed (Das et al., 2019). It is supported by the behavior of speciﬁc heat as a function of magnetic ﬁeld and temperature. According to previous studies, anomalous properties of Ce3PdIn11 at low temperature are directly related to the presence of two inequivalent Ce atoms in the unit cell (Kratochvilova et al., 2015). Such sublattices of Ce can be characterized by different Kondo scales. Consequently, different type of ordering can be realized in each sublattice at the same time. Such a proposition is in line with theoretical results (Benlagra et al., 2011) obtained for a system with two inequivalent Kondo sublat

tices. The non-trivial interplay between the Kondo effects in two sublattices together with complex phase diagram is predicted by this model. The standard Doniach diagram (Doniach, 1977), which describes the competition between RKKY interaction and the Kondo screening, is complemented i.a. by the partial screening phase. Such a phase is characterized by the presence of antiferromagnetic type ordering together with the heavy quasiparticles. On the another hand, one must have in mind, that the coexistence of antiferromagnetism and Kondo effect is not an exclusive feature of systems with inequivalent Ce sites. Such a situation is also realized in uranium compounds (e.g. USb2) as a consequence of orbital selective Kondo effect (Giannakis et al., 2019). In this paper we discuss the electronic structure of Ce3PdIn11 measured with angle-resolved photoelectron spectroscopy (ARPES) method. The experimental results are compared with ab initio calculations. Some attempts on quantiﬁcation of the inﬂuence of the hybridization between 4f electrons and conduction band are made. We also outline possible consequences of presence of two independent sublattices of Ce. We would like to provide some step in better understanding the relations between the crystal structure and the physical properties of Ce3PdIn11 by means of an electronic structure measured with ARPES method. In this article we check: I) to what extent both independent Ce atoms in unit cell give rise to heavy fermion properties; II) if it is possible to quantify hybridization effects using a band structure measured with ARPES. 9.2 Material and methods Single crystals of Ce3PdIn11 have been grown using ﬂux method. Their synthesis and characterization were described elsewhere (Das et al., 2019; Kratochvilova et al., 2014). ARPES measurements have been performed at the Cassiopee beamline of Soleil synchrotron. Samples were oriented by means of x-ray diffraction (Laue method). They were cleaved in ultra-high vacuum conditions (base pressure: 5·10−11 mbar) prior to the measurements, exposing (001) crystallographic plane. The data have been collected at temperature equal to 6 K, using Scienta R4000 photoelectron energy analyzer. The incident radiation with linear polarization and energy corresponding to Ce 4d-4f resonant transition (hν=122 eV) has been applied. Overall resolution was equal to 15 meV. The electronic structure of Ce3PdIn11 has been calculated using Korringa-Kohn-Rostoker (KKR) method in single particle Green function approach (Ebert et al., 2011) with application of multiple scattering theory. The spin polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) code for Calculating Solid State Properties has been used. 9.3 Results and Discussion We have measured electronic structure of Ce3PdIn11 using resonant ARPES near Ce 4d-4f absorption threshold, in order to study the relation of 4f orbitals to enigmatic properties of this system. The sample was oriented in such a way that the analyzer slit direction was parallel to the a-axis. The experimental geometry is shown in Fig. 9.1 b. The Fermi surface map in Γ-X-Z plane (see Brillouin zone in Fig. 9.1 c) has been obtained by performing measurements using incident radiation with different energies (Fig. 9.1 d). Several weakly dispersive features are visible. The intensities of these features are noncontinuous as a function of photon energy, probably due to signiﬁcant variation of the cross section for the photoionization close to the absorption threshold. This fact together with almost twodimensional character of observed bands makes that the reliable determination of the inner potential was impossible with the obtained data. On the other hand, one can expect that the three-dimensional bands, related to interlayer coupling, are also present in the electronic structure of Ce3PdIn11, what is supported by the results of theoretical calculations (Azam et al., 2014). However, they have not been observed in our particular experimental geometry. The spectra measured at different photon energies were compared with calculated bands along Γ-X and Z-R directions. They suggest that the photon energy equal to 98 eV corresponds to measurement in Γ-X-M plane. The Fermi surface map measured at Ce 4d-4f resonance is presented in (Fig. 9.1 e). The bands visible in the photon energy dependent scan are also identiﬁed. The measured Fermi surface is affected by the matrix elements, but not as much as that of CeCoIn5 previously measured by us in the same geometry (more details about matrix elements effects are provided in Supplementary information). Figure 9.1: (Color on-line) (a) Crystal structure of Ce3PdIn11. The Ce 1a and Ce 2g coordination polyhedra are shown. (b) Schematic plot showing the experimental geometry for ARPES measurements. (c) The ﬁrst Brillouin zone of Ce3PdIn11. (d) The map showing the measurement of the electronic structure of Ce3PdIn11 as a function of photon energy. Values of ky corresponding to high symmetry points are marked by black arrows. (e) The Fermi surface of Ce3PdIn11 collected at wide region in k-space at temperature equal to 6 K using photon energy of 122 eV (Ce 4d-4f resonance). The description of the electronic structure of Ce3PdIn11 using measured ARPES spectra is a de

manding task. The signiﬁcant number of electron and hole bands is crossing a Fermi level. Measured FS map is similar with the Fermi surface topography obtained for other CemTnIn3m+2n

compounds. In particular, the structure resembling the ﬂower with four petals centered in Γ

point has been ob

served in the cross section of FS (in Z-R-A plane) calculated for Ce2PdIn8. Additionally, the FS of Ce2PdIn8 consists of many disjoint parts (Werwi´nski et al., 2015). The crystal structure of Ce3PdIn11 shares some similarities with that of Ce2PdIn8, so one can expect some similarities between their electronic structures. Further analogies with the electronic structures of compounds from the same family are also possible. The attention should be paid to the structures centered in the M corner of BZ (kx

=

±π

, ky

=

±π

). In our case, two sheets of FS are clearly visible close to the M point. Previously, aa such structures were observed e.g. for: CeCoIn5, Ce2PtIn8 and Ce2RhIn8 and were assigned to the cylindrical, quasi two-dimensional parts of FS. On the other hand, highly corrugated, three dimensional parts of FS are present around Γ point. Interestingly, some studies suggests that heavy fermion properties are essentially related to 3D bands, while the quasi 2D bands almost do not hybridize with f states (Koitzsch et al., 2013). Our measurements show also signiﬁcant enhancement of the signal around X point. Indeed, one can see some oval-like structure in this region. Similar shapes around X point were observed in the case of Ce2IrIn8 heavy fermion compound (Liu et al., 2019). The electronic structure of Ce3PdIn11 can be understood with the aid of the ab initio theoretical calculations. The calculated bands are in good agreement with the experimental ones as well as with those previously calculated (Azam et al., 2014) (see Supplementary information for more details). The total Bloch spectral function as well as projections on Pd and both inequivalent Ce atoms have been calculated (Fig. 9.2 a-e) along high symmetry directions in the ﬁrst Brillouin zone. A rough look at the band structures gives the impression that both Ce atoms give completely different contributions to spectral function. In order to better visualize the difference between Ce 1a and Ce 2g contributions, we classiﬁed (Fig. 9.2 f) the electronic structure according to prevailing contribution of Ce 1a, Ce 2g and Pd states to the spectral function. (The In states were not taken into account to make an analysis simpler, however they can have important contribution in the region of interest -corresponding to the charts in Fig. 9.2 a-e. On the other hand one can expect large but rather uniform contribution of In atoms in this energy region.) One can see clearly that in principle the Pd states are important only be-low −1.5 eV. Above −0.5 eV the Ce contribution is signiﬁcant. The close region of the Fermi level is dominated by the contribution from the Ce 2g states, what points to their anticipated role in heavy fermion properties of Ce3PdIn11. However, there are some features crossing the Fermi level, with dominating Ce 1a character. The differences in contribution from both Ce sublattices are well visible in the distributions of BSF at the Fermi level (Fig. 9.2 g-k). Despite the complicated topography of the Fermi surface, one can see that there are some regions in the BZ where the contributions to BSF from both, Ce 1a and Ce 2g, can be clearly separated. Such an observation opens a way to describe a Ce3PdIn11 in terms of non-trivial interplay between two Kondo sublattices (Benlagra et al., 2011). The performed classiﬁcation of the FS according to Ce 2g, Ce 1a and Pd (with In contribution excluded from the analysis) states is shown in Fig. 9.2 l. One can see the oval like structure around X point, which is elongated along X-Γ direction. It originates mainly from Ce 1a states. Such a structure is related to the shallow electron pocket. On the other hand, when one looks at the X-M direction in the FS some petals are visible around aforementioned oval. These are characterized by the dominating contribution coming from Ce 2g. Interestingly, around M point one can observe nearly circular shape. In fact there are two electron pockets with bottoms roughly located at binding energy of about −0.7 eV. The outer one is dominated by contribution from Ce 1a sublattice, while the inner pocket is built mainly from Ce 2g states. It is also noteworthy, that that according to Fig. 9.2 f the signiﬁcant contribution to spectral function from Ce 1a states is visible in Γ-Z direction. These states are giving a rise to ﬂat, non-dipersive feature which is located directly at the Fermi energy. (f) Band structure classiﬁed according to the contribution from Ce 1a, Ce 2g and Pd atoms. Red color corresponds to states characterized by contribution to BSF from Ce 1a greater than that from Ce 2g and Pd. Similarly, blue and green colors are related to Ce 2g and Pd states, respectively. (g)-(k) the distribution of BSF at the Fermi surface. (l) Fermi surface classiﬁed using the same method as for picture (f). In ﬁgure Fig. 9.3 a we show the experimental band structure of Ce3PdIn11. The FS map measured with high resolution in a small purview of k-space is shown Fig. 9.3 b. The weakly visible sheets around M point have been marked by lines (red and blue dashed lines).We have thoroughly inspected the spectra corresponding to X-M and Γ-X directions (Fig. 9.3 c and d). Two hole pockets (α, β ) centered at X are clearly visible. The outer (α) one follows parabolic relation dispersion for εB between −0.25 eV and −0.01 eV. Close to εF , it seems to bend more rapidly toward the X point (i.e. it is more ﬂat), but it is rather clear that it crosses the Fermi level according to our analysis of momentum distribution curves. On the other hand, it is difﬁcult to describe the inner pocket (β) by the simple quadratic dispersion. Below εF the dispersion β band seems to be almost linear, while very close to the Fermi level it rapidly changes its shape and becomes almost ﬂat. Such a behavior is a result of a hybridization with f states and has been observed for many Ce systems and interpreted successfully using Anderson model (Im et al., 2008). It is difﬁcult to asses if this band crosses the Fermi level using our data. It is most probably that the top of β pocket touches the Fermi level or locates several milielectronovolts above, still giving a hot spot on FS map. Such a conﬁguration of β and α bands recalls the idea of renormalization of point-like Fermi surface proposed for some uranium compound (Durakiewicz et al., 2008). However, we reckon that our data are not sufﬁcient to push forward full analogy with this scenario, which also incorporates the presence of the kink in the dispersion of the outer band. In Fig. 9.3 c and d we marked the points in the k-space at which the band should cross the Fermi level (the red and blue dots stand for Ce 1a and Ce 2g, respectively; the area of the dot means the contribution to the total BSF, the offset of -0.2 eV and -0.18 eV is used for clarity). It turns out that β band close to εF is dominated by Ce 1a contribution, while the Ce 2g contribution plays a crucial role in the case of α pocket. In order to visualize better the dispersion of the hole bands around X point we have calculated the curvature in MDC direction (Zhang et al., 2011) (Fig. 9.1 d and f). Such an approach indeed makes the α and β band clearly visible. However, one has to have in mind that the disadvantage of this procedure is that it removes any non-dispersive feature (i.e. k-independent) from the spectra. We have also tried to calculate full two-dimensional curvature of ARPES spectra, but we did not obtain results with sufﬁcient quality for further analysis. The calculated curvature in the X-M direction was used to determine the band dispersion for α and β bands. Subsequently we have ﬁtted the band using quadratic function in the widest range of energy still providing reasonable good quality of ﬁts. The ﬁts are marked by solid black lines in Fig. 9.3 c. The black dashed lines indicates the extrapolation of ﬁtted function. The ﬁts yield the effective masses equal to 0.8me and 0.23me (me -free electron mass) for α and β band, respectively. However, these values are valid well below εF (εB<−35 meV), because the experimental dispersion strongly deviates from that predicted by parabolic ﬁts. Indeed, the ﬁtted functions suggest that both β and α band crosses the Fermi level, while actually one cannot make such a statement about β band because it becomes very ﬂat close to εF . However, one may notice, that the higher effective mass has been determined for the band which is related mainly to Ce 2g sublattice. Therefore, we may expect that the states from this sublattice are crucial to fermion properties (i.e. the enhanced value of Sommerfeld coefﬁcient) and participates in a formation of a ﬂat band near the Fermi level at low temperature. The deviation from parabolic dispersion relation and the mass enhancement due to c-f hybridization is further analyzed in Supplementary information using recently proposed method (Rosmus et al., 2019). Figure 9.3: (Color on-line) The electronic structure of Ce3PdIn11 obtained by ARPES at 6 K using photon energy of 122 eV. (a) 3D plot showing the band structure of Ce3PdIn11 from different points of view in a wide region in k-space. (b) The Fermi surface of Ce3PdIn11 measured with high resolution in a narrow region of the k-space. The weakly visible sheets of FS around M point are marked by red and blue dashed lines in order to guide the eyes. (c)-(d) The band structure near the Fermi level is shown along high symmetry directions (X-M and Γ-X). The contributions of different Ce sublattices to spectral function have been marked by dots (Ce 1a -red dots, Ce 2g -blue dots, the radius of the dot is proportional to the contribution to spectral function, the offset of −0.2 eV and −0.18 eV is used for better clarity) (e)-(f) MDC curvature calculated for spectra from (c) and (d). (g)-(h) Energy distribution curves showing the spectra collected along X-M and Γ-X directions. 9.4 Conclusions We have provided the electronic structure of Ce3PdIn11 in the paramagnetic state by means of ARPES. This is the ﬁrst study of an electronic of Ce intermetallic with two inequivalent Ce sublattices employing photoelectron spectroscopy, according to our knowledge. Our results show that Ce3PdIn11 can be to some extent regarded as a quasi two-dimensional system. Electronic structure calculations (KKR method) reproduced well observed bands of Ce3PdIn11 and allow to distinguish the contributions from Ce 1a and Ce 2g inequivalent atoms in the unit cell. We have assessed that the enhancement of Sommerfeld coefﬁcient of this system is mainly an effect of band hybridization with f states from Ce 2g site. On the other hand, we reckon that further studies concerning Ce3PdIn11 are still necessary. 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Supplementary information Comparison of different theoretical calculations and experimental results. In Fig. 9.4 we show the comparison between our theoretical calculations and these published previ

ously (Azam et al., 2014). The previous calculations were obtained using the full potential linearized augmented plane wave (FP-LAPW) method based on density functional theory (DFT) implemented in WIEN2K code. In case of Γ-M and X-M directions the agreement is very good. For Z-R direction some signiﬁcant differences between both versions of calculations are visible. In Fig. 9.4 d, we su

perimposed previous calculations on the measured band structure with ARPES along X-M direction. One can see that the outer hole pocket (α band) around X point is quite well reproduced. On the other hand, it is quite difﬁcult to describe the inner pocket (β band) using these calculations. Maybe after some rescaling in k-vector and shift in energy scale one can achieve the agreement between theory and experiment. Figure 9.4: (Color on-line) (a)-(c) Comparison of theoretical band structures of Ce3PdIn11 obtained with different methods. The previous results (Azam et al., 2014) were digitalized (red lines) and superimposed on the Bloch spectral function (gray scale, color maps) calculated using KKR approach for several directions in the ﬁrst Brillouin zone. (d) Comparison between measured band structure (gray scale, color map) of Ce3PdIn11 along M-X direction and theoretical calculations (red and blue dashed lines) available in the literature (Azam et al., 2014) (the curves are shown in different colors only for better readability). The α symbol denotes the outer hole band, which is discussed in the text. The experimental data have been collected at 6 K using photon energy of 122 eV (Ce 4d4f resonance). Mass enhancement in bands around X point In Fig. 9.3 c we have presented the spectrum along X-M together with ﬁtted relation dispersion to both α and β bands in some energy region below εB. One can notice that close to the Fermi level the band dispersion deviates from that predicted by extrapolation of ﬁts (dashed lines in Fig. 9.3 c). In order to quantify this deviation we have analyzed the shape of the experimental dispersion thoroughly. Namely, we have deﬁned the quantity µ in a following manner: µ = h¯2k dk . (9.1) dE Such a quantity can be interpreted as an effective mass in case of parabolic relation dispersion. In other cases it can be used to quantify the wave vector depended mass enhancement effects. Moreover such deﬁnition of µ is particularly useful when the dispersion is obtained from MDC proﬁles, because it allows to get rid of problems with singularities during calculation of derivative. We have calculated µ for both α and β band using the branch of dispersion corresponding to negative values of wave vector. We also made some attempts on calculation µ using the second branch of dispersion or both of them, but signiﬁcant noise in data make results in poor quality of ﬁnal output. Fig. 9.5 shows µ obtained in such a way in the units of free electron mass. Blue and red dots denote the values obtained for β and α bands, respectively. We have also marked the values of effective masses (mα , mβ ) obtained from parabolic ﬁts by blue (β band) and red (α band) dashed lines. One can see that determined values of µ for β band are close to the mβ value below −30 meV. Above such a value of binding energy one can see the rapid change in a shape of µ as a function of k. The absolute value of µ rapidly rises with wave vector approaching zero, becoming about three and half times greater than the mass of free electron in vacuum. Such a behavior is directly related to the mass enhancement close to the Fermi level, which has a source in strong hybridization between s, p, d states and 4f orbitals. Similar behavior is also visible in the case of α band. However, the values of µ corresponding to this band seem to deviate signiﬁcantly from mα . Probably taking into account both branches of dispersion relation would improve the agreement. On the other hand, similarly as for β band the rapid enhancement of effective mass close the Fermi energy (above −40 meV) can be deduced from obtained µ values. It is noteworthy that for both, α and β band, µ is lower than the mass of free electron well below the εF , while close to the Fermi energy it is several times greater. This fact is reﬂected in a very rapid change of the slope of the dispersion relation. Matrix element effects The obtained ARPES spectra of Ce3PdIn11 are slightly affected by matrix elements effects. In order to visualize this effect we analyze some cuts of the band structure corresponding to high symmetry directions in the k-space. After a glance at Fig. 9.6 b, one can notice that the hole bands at X point corresponding to ky=0 are clearly visible, while they are almost difﬁcult to resolve around second X point, at ky = π . The opposite effect is visible in Fig. 9.6 d and f, which show the cut along a X-M directions corresponding to different values of kx. Similar effects are visible along Γ-X direction (Fig. 9.6 c and e). We have checked that this effect is independent of applied normalization procedure, therefore it is not an artifact produced during data analysis. 0.03 eV and 0.018 eV have been used for better readiness. In Fig. 9.2 f and l we have shown the result of classiﬁcation of the band structure according to dominating contribution of Ce 1a, Ce 2g or Pd states to the full spectral function. Such an approach is useful if one wants to separate the regions of k-space where the contribution from particular atom is dominating. However, such a procedure has got some drawbacks. Namely, it cannot give conclusive answer in the regions of BZ where the contributions of Ce 1a and Ce 2g are similar. In order to be sure, that we did not make wrong interpretation of calculated BSF, here we deﬁne a different measure allowing to separate Ce 1a and Ce 2g contributions. Namely, we have calculated a simple difference Δ between BSF projected on both sublattices: Δ = BSF(Ce 2g) − BSF(Ce 1a). The result is shown in Fig. 9.7. After the analysis of this ﬁgure one can come to the conclusions, which are essentially the same as these obtained using classiﬁed band structure. However, the simple difference gives result which allows for more detailed analysis of some subtle differences in Ce 1a and Ce 2g contributions, which are present near the Γ point. Appendix List of published articles coauthored by the author of the thesis 1. Evidence of momentum-dependent hybridization in Ce2Co0.8Si3.2 P. Starowicz, R. Kurleto, J. Goraus, H. Schwab, M. Szlawska, F. Forster, A. Szytuła, I. Vobornik, D. Kaczorowski, and F. Reinert, Physical Review B 89, 115122 (2015). doi: 10.1103/PhysRevB.89.115122 2. Valence Band of Ce2Co0.8Si3.2 and Ce2RhSi3 Studied by Resonant Photoemission Spectroscopy and FPLO Calculations P. Starowicz, R. Kurleto, J. Goraus, Ł. Walczak, B. Penc, J. Adell, M. Szlawska, D. Kaczorowski, and A. Szytuła, Acta Physica Polonica A 126, A144-A147 (2014). 3. Electronic structure and transport properties of CeNi9In2 R. Kurleto, P. Starowicz, J. Goraus, S. Baran, Yu. Tyvanchuk, Ya. M. Kalychak, and A. Szytuła, Solid State Communications 206, 46-50 (2015). doi: 10.1016/j.ssc.2015.01.014 4. Electronic Structure of TmPdIn B. Penc, R. Kurleto, J. Goraus, P. Starowicz, and A. Szytuła, Acta Physica Polonica A 129, 1184-1186 (2016). doi: 10.12693/APhysPolA.129.1184 5. Kondo lattice behavior observed in the CeCu9In2 compound R. Kurleto, A. Szytuła, J. Goraus, S. Baran, Yu. Tyvanchuk, Ya. M. Kalychak, and P. Starowicz, Journal of Alloys and Compounds 803, 576-584 (2019). doi: 10.1016/j.jallcom.2019.06.140 6. Studies of Electronic Structure across a Quantum Phase Transition in CeRhSb1−xSnx R. Kurleto, J. Goraus, M. Rosmus, A. ´ Slebarski, and P. Starowicz, The European Physical Journal B 92, 192 (2019). doi: 10.1140/epjb/e2019-100157-3 7. Effect of electron doping in FeTe1−ySey realized by Co and Ni substitution M. Rosmus, R. Kurleto, D. J. Gawryluk, J. Goraus, M. Z. Cieplak, and P. Starowicz, Superconductor Science and Technology 32, 105009 (2019). doi: 10.1088/1361-6668/ab324f Contents I Introduction

9 1 Ce 4f electrons

11 2 Kondo effect and heavy fermion physics

15 3 Cerium intermetallics

21 3.1 MetallicCe-phasediagram .............................. 21 3.2 RT9In2compounds ................................... 22 3.3 CeTXcompounds.................................... 23 3.4 CenTmIn3n+2m

family .................................. 24 4 Ab initio band structure calculations

29 II Experimental methods

31 5 Photoemission Spectroscopy

33 5.1 BasicsofPhotoemissionSpectroscopy ......................... 33 5.2 Analysisofspectralfunctioninphotoemission. . . . . . . . . . . . . . . . . . . . . 36 5.3 Resonantphotoemission ................................ 38 5.4 Photoemission spectroscopy on Ce intermetallics . . . . . . . . . . . . . . . . . . . 39 5.5 MatrixelementsinARPES ............................... 42 5.6 Experimentalsetups................................... 44 5.6.1

ARPES setup available in Department of Solid State Physics, Institute of Physics,JagiellonianUniversity ........................ 44 5.6.2

Cassiopeebeamline,synchrotronSoleil . . . . . . . . . . . . . . . . . . . . 45 III Results

49 6 Kondo Lattice Behavior Observed in CeCu9In2 Compound

51 7 Studies of Electronic Structure across a Quantum Phase Transition in CeRhSb1−xSnx

63 8 Direct observation on f-electron hybridization effects in CeCoIn5

79 8.1 Introduction....................................... 83 8.2 Materialandmethods .................................. 83 8.3 ResultsandDiscussion ................................. 84 8.4 Conclusions....................................... 91 9

Electronic Structure of the Ce3PdIn11 Heavy Fermion Compound Studied by Means of Angle-resolved Photoelectron Spectroscopy 99 9.1 Introduction....................................... 103 9.2 Materialandmethods .................................. 104 9.3 ResultsandDiscussion ................................. 104 9.4 Conclusions....................................... 109