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Quantum Phase Transition and Ferromagnetism in Co 1+X Sn

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Quantum Phase Transition and Ferromagnetism in Co 1+X Sn University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications from Nebraska Center for Materials and Nanoscience, Nebraska Center Materials and Nanoscience for (NCMN) 2019 Quantum phase transition and ferromagnetism in Co1+xSn Rabindra Pahari Balamurugan Balasubramanian Rohit Pathak Manh Coung Nguyen Shah R. Valloppilly See next page for additional authors Follow this and additional works at: https://digitalcommons.unl.edu/cmrafacpub Part of the Atomic, Molecular and Optical Physics Commons, Condensed Matter Physics Commons, Engineering Physics Commons, and the Other Physics Commons This Article is brought to you for free and open access by the Materials and Nanoscience, Nebraska Center for (NCMN) at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications from Nebraska Center for Materials and Nanoscience by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Authors Rabindra Pahari, Balamurugan Balasubramanian, Rohit Pathak, Manh Coung Nguyen, Shah R. Valloppilly, Ralph Skomski, Arti Kashyap, Cai-Zhuang Wang, Kai-Ming Ho, George C. Hadjipanayis, and David J. Sellmyer PHYSICAL REVIEW B 99, 184438 (2019) Quantum phase transition and ferromagnetism in Co1+xSn Rabindra Pahari,1,2 Balamurugan Balasubramanian,1,2,* Rohit Pathak,3 Manh Cuong Nguyen,4,5 Shah R. Valloppilly,1 Ralph Skomski,1,2 Arti Kashyap,3 Cai-Zhuang Wang,4,5 Kai-Ming Ho,4,5 George C. Hadjipanayis,6 and David J. Sellmyer1,2,† 1Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588, USA 2Department of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588, USA 3School of Basic Sciences, IIT Mandi, Mandi, Himachal Pradesh 175005, India 4Ames Laboratory, US Department of Energy, Ames, Iowa 50011, USA 5Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 6Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA (Received 22 April 2019; revised manuscript received 9 May 2019; published 28 May 2019) The onset of ferromagnetism in cobalt-tin alloys is investigated experimentally and theoretically. The Co1+xSn alloys were prepared by rapid quenching from the melt and form a modified hexagonal NiAs-type crystal structure for 0.45 x 1. The magnetic behavior is described analytically and by density-functional theory using supercells and the coherent-potential approximation. The excess of Co concentration x, which enters the interstitial 2d sites in the hypothetical NiAs-ordered parent alloy CoSn, yields a Griffiths-like phase and, above a quantum critical point (xc ≈ 0.65), a quantum phase transition to ferromagnetic order. Quantum critical exponents are determined on the paramagnetic and ferromagnetic sides of the transition and related to the nature of the magnetism in itinerant systems with different types of chemical disorder. DOI: 10.1103/PhysRevB.99.184438 I. INTRODUCTION desirable magnetic properties. The Co1+xSn system is com- pelling because the local-environment effects and associated Quantum phase transitions (QPTs), defined as continu- spin-cluster effects are nontrivial, and whether the apparent ous phase transitions at zero temperature, have remained an quantum phase transition is similar to or distinguishable from intriguing research topic [1–6]. There are several types of other examples is intriguing [5]. QPTs, dealing with various classes of materials, such as Thermodynamic phase transitions such as Curie transitions magnets [7–9], superconductors [10], heavy-fermion com- and QPTs have many features in common but also exhibit pounds [11,12], and ferroelectrics [13], and triggered by important differences. A common feature is that both types different control parameters, for example mechanical pres- of transitions are typically described by power laws that relate sure, magnetic or electric fields, and chemical composi- the response of the system to control parameters. Our focus is tion [14–16]. In some solid-solution alloys of type M − T 1 x x on magnetic alloys, where the order parameter (magnetization [2,17,18] magnetic transition-metal elements (T) cause the M) and the susceptibility χ are controlled by the temperature nonmagnetic metal (M) to become a ferromagnet above some T, the magnetic field H, and the chemical composition x. critical concentration x . The chemical disorder in QPT al- c For example, structurally homogeneous ferromagnets near the loys is normally of the substitutional solid-solution type, β γ Curie temperature T obey M ∼ (T –T ) and χ ∼ 1/|T –T | with nearest-neighbor exchange bonds. This paper deals with c c c [24]. The susceptibility diverges near the critical point, and QPTs caused by interstitial modification of an intermetallic this divergence is accompanied by long-range critical fluc- compound. tuations. These fluctuations, observed for example as critical We consider alloys having the composition Co + Sn ν 1 x opalescence in fluids, have correlation length ξ ∼ 1/|T –T | (0 < x 1), where the excess Co (x) enters the 2d interstitial c [24,25]. A major difference is that QPTs reflect quantum- sites in the NiAs structure (Fig. 1)[19,20]. Current research mechanical fluctuations, as contrasted to the thermodynamic on Co-based alloys is partly motivated by the need to discover fluctuations governing the critical behavior of ferromagnets new magnetic materials with high Curie temperature, high (Curie transition) and of fluids (gas-liquid transition). anisotropy, and high magnetization. Such materials, especially The description of critical behavior and the determina- consisting of earth-abundant and inexpensive elements, are tion of critical exponents are nontrivial, especially in low- needed for advanced energy and information-processing ap- dimensional systems [24–27]. For example, it is well known plications [21–23]. New Co- and/or Fe-rich compounds, es- that long-range fluctuations yield T = 0 in one-dimensional pecially those with noncubic and/or metastable structures, are c magnets [25,26,28]. The simplest approach towards critical of particular interest, since they potentially possess the above phenomena is the mean-field approximation (MFA), where the crystalline environment of a given atom is modeled as an effective medium. In this paper, we distinguish between *[email protected] three types of mean-field approximations, namely thermo- †[email protected] dynamic MFA (Landau theory), quantum-mechanical MFA 2469-9950/2019/99(18)/184438(10) 184438-1 ©2019 American Physical Society RABINDRA PAHARI et al. PHYSICAL REVIEW B 99, 184438 (2019) itinerant systems, such as Co-Sn. The corresponding wave- vector-dependent susceptibility is [29,30] χ χ(k) = 0 . (5) 2 1 − ID(EF ) + wk 2 This equation, where w ≈ 2ID(EF )/kF , describes the low- temperature susceptibility. Due to spin fluctuations, Eq. (5)is difficult to generalize to finite temperatures, but for reasons FIG. 1. Unit cell of NiAs-ordered Co1+xSn: (a) CoSn and discussed elsewhere [30–32], high-temperature susceptibili- (b) Co1.5Sn. The excess Co (x) occupies the interstitial 2d sites, ties of itinerant magnets are often of the Curie-Weiss type, which exhibit a trigonal-prismatic coordination by the Co atoms of Eq. (3). the CoSn host lattice. A third type of mean-field theory is related to the per- colation aspect of disordered alloys. Randomly distributing (Stoner theory), and structural MFA (Bethe-lattice percolation Co atoms over the interstitial sites creates Co-rich clusters theory). and Co-poor regions, and ferromagnetism develops from Co- In its simplest form, the mean-field theory of thermody- rich clusters. In fact, even below the onset of long-range = namic phase transitions considers the order parameter M ferromagnetic order, some clusters are very big, which causes Mez and assumes a Landau free energy the susceptibility to exhibit a quasiferromagnetic singularity 1 2 1 4 known as the Griffiths singularity; the corresponding region is F = a0(T − Tc)M + a4M − μ0HM, (1) 2 4 referred to as Griffith phase [2,33–36]. With increasing x,the where a and a are materials constants and M =M(r). 0 4 Co clusters grow, and at some initial percolation threshold xc, Putting ∂F/∂M = 0 in this equation yields the familiar mean- an infinite backbone develops. Below xc, the average cluster field exponents β = 1/2 and γ = 1. An alternative way of size has the character of a correlation length, obeying ξ ∼ deriving these exponents is to replace the field H acting on an ν 1/(xc − x) [37,38]. atomic spin by H + λM, where λ is referred to as molecular- The mean-field description of percolation is that of the or mean-field coefficient. Bethe lattice [38] and yields the mean-field exponent ν = The assumption of an effective medium of average mag- 1/2. The same mean-field exponent is obtained from Eq. (3), netization M does not necessarily mean that the local 1/2 where ξ ∼ 1/(1 − T/Tc ) , and from Eq. (5), where ξ ∼ magnetization M(r) of homogeneous solids is equal to the 1/2 1/[1 − ID(EF )] . These correlation lengths all diverge at average magnetization M. The corresponding fluctuations the critical point. One question considered in this paper is how are described by the correlation function the different correlation lengths interact with each other in the C(|r − r|) = [M(r) −M][M(r) −M]. (2) cobalt-tin system. Our emphasis is on length scales of a few interatomic The zero-field correlations decay as C(|r − r |) ∼ distances, partially going beyond mean-field theory. In the exp(−ξ/(|r − r|)). experimental part, Sec. II, the Co-Sn alloys are investigated Completely ignoring C(|r − r |) corresponds to the Landau using structural and magnetic measurements, whereas Sec. III theory of Eq. (1), but the Ornstein-Zernike extension of the is devoted to analytical and density-functional theoretical Landau theory includes fluctuations on a mean-field level calculations, as well as an analysis of the data in terms of a [24,27]. The extension is obtained by adding a Ginzburg- quantum phase transition.
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