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Tuning the Metamagnetism of an Antiferromagnetic Metal RAPID COMMUNICATIONS PHYSICAL REVIEW B 87, 060404(R) (2013) Tuning the metamagnetism of an antiferromagnetic metal J. B. Staunton,1 M. dos Santos Dias,1,2 J. Peace,1 Z. Gercsi,3 and K. G. Sandeman3 1Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom 2Peter Grunberg¨ Institut and Institute for Advanced Simulation, Forschungszentrum Julich¨ and JARA, D-52425 Julich,¨ Germany 3Department of Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom (Received 14 June 2012; revised manuscript received 14 September 2012; published 13 February 2013) We describe a “disordered local moment” first-principles electronic structure theory which demonstrates that tricritical metamagnetism can arise in an antiferromagnetic metal due to the dependence of local moment interactions on the magnetization state. Itinerant electrons can therefore play a defining role in metamagnetism in the absence of large magnetic anisotropy. Our model is used to accurately predict the temperature dependence of the metamagnetic critical fields in CoMnSi-based alloys, explaining the sensitivity of metamagnetism to Mn-Mn separations and compositional variations found previously. We thus provide a finite-temperature framework for modeling and predicting different metamagnets of interest in applications such as magnetic cooling. DOI: 10.1103/PhysRevB.87.060404 PACS number(s): 75.30.Kz, 75.10.Lp, 75.50.Ee, 75.30.Sg The application of a magnetic field to an antiferromagnet phase at a second-order transition. However, if anisotropy can cause abrupt changes to its magnetic state.1 While such effects are negligible, the helical order has no favored axis metamagnetic transitions have been known for a long time and, once a magnetic field is applied, the helix plane orients 3 in materials such as FeCl2 (Ref. 2) and MnF2, it is their perpendicular to the field. With increasing field the moments association with technologies such as magnetic cooling4 that cant smoothly into a conical spiral towards the field’s direction has driven recent efforts to control metamagnetism in the with a second-order transition to a high magnetization phase. room temperature range, and in accessible magnetic fields. Thus, in this localized picture, first-order metamagnetism Indeed, a large, or even “giant” magnetocaloric effect can relies on a source of anisotropy. arise at a first-order metamagnetic phase transition such as is Metamagnetism is accounted for differently in an itinerant found in FeRh.5 However, the thermomagnetic hysteresis as- electron system. Seminal work by Wohlfarth and Rhodes,10 sociated with most first-order transitions results in significant Moriya and Usami,11 developed and extended by many others, inefficiency during magnetic cycling. Therefore, (tri)critical e.g., Refs. 12 and 13, derived the coefficients of a Landau- transitions that straddle first- and second-order behavior are of Ginzburg free energy expansion for an AFM exposed to a great interest. It is worth noting that tricritical points have also uniform magnetic field in terms of the order parameter mq, been examined in the context of other technologies, such as with wave-vector modulation q. They considered both Stoner liquid crystal displays.6 particle-hole excitations10 and spin fluctuations11,13 generated Given the interest in finding room temperature magnetic from the collective behavior of the interacting electrons.11,14,15 refrigerants, a reexamination of tricritical antiferromagnetic Field-induced tricriticality occurs when the quadratic and (AFM) metamagnets is warranted. At present there are only quartic terms in mq both equal zero. More recently, by two first-order magnetic refrigerants, La-Fe-Si (Ref. 7) and analyzing a generic mean-field Hamiltonian describing a MnFe(P,As,Si),8 at an advanced stage of deployment in proto- helical state in an applied field, Vareogiannis16 has shown type cooling devices. Both materials are ferromagnets (FMs) how an itinerant electron system can undergo a first-order with field-induced metamagnetic critical points. However, “spin-flip” transition. The AFM polarization is parallel to a tricriticality allows AFMs to emulate the low hysteresis, high weak applied field and flips perpendicularly only if the field entropy change properties of their FM cousins. For such real exceeds a critical value.17 materials the greatest challenge is to understand the mixed In this Rapid Communication we describe an ab initio localized/itinerant electron spin nature of their magnetism, spin density functional theory (SDFT)-based “local moment” and how this influences tricritical properties. Metamagnetic theory for AFM to suit real materials where both itinerant transitions, however, have previously been investigated in and local spin effects are at play. Slowly varying “local terms of either spin effect alone. moments” can be identified from the complexity of the In AFM insulators,6 magnetic field-driven phase transitions electronic behavior. We find that where magnetic anisotropy can be understood qualitatively using a localized (classical) effects are small or neglected entirely, the local moments’ spin Hamiltonian with pairwise isotropic exchange interac- AFM order can still undergo a first-order transition to a tions, a source of magnetic anisotropy, and a Zeeman external fan or FM state owing to the feedback between the local magnetic field term. For example, Nagamiya9 showed that, if moment and itinerant aspects of the electronic structure. We the localized spins of a helical AFM are pinned by anisotropy test our theory against a detailed experimental case study and crystal field effects to spiral around a particular direction, of CoMnSi-based tricritical metamagnets and show how it the effect of a magnetic field brings about a first-order provides quantitative materials-specific guidance for tuning transition to a fan structure where the moments now oscillate metamagnetic and associated technological properties. about the field direction. At higher fields, the fan angle A generalization of SDFT18 describes the “local moment” smoothly reduces to zero to establish a high magnetization picture of metallic magnets at finite temperature.19–22 Its basic 1098-0121/2013/87(6)/060404(5)060404-1 ©2013 American Physical Society RAPID COMMUNICATIONS J. B. STAUNTON et al. PHYSICAL REVIEW B 87, 060404(R) (2013) premise is a time-scale separation between fast and slow structure, we approximate the hl via an expansion about a electronic degrees of freedom so that local moments are set up uniform m = 1 m , i.e., { } N i i with slowly varying orientations eˆi . The existence of these “disordered local moments” (DLMs) is established by the fast 2 ∂ ∂ electronic motions and likewise their presence affects these hl ≈− − · (m j −m) ∂m ∂m ∂m l m j l j m motions. Moreover, the local moments’ interactions depend on the type and extent of the long range magnetic order through (2) = h(m) + S˜ (m) · (m −m). (3) the associated itinerant electronic structure. The DLM picture l,j j j of the paramagnetic state maps to an Ising picture18 with, on average, one half of the moments oriented one way and the S˜(2)(m) is the direct correlation function and crucially de- rest antiparallel. However, once the symmetry is broken so i,j that there is a finite order parameter {m } profile, e.g., in a FM scribes effective interactions between the local moments that i depend on the magnitude of m. A solution of Eqs. (1)–(3) or AFM state and/or when an external magnetic field is applied, this simplicity is lost. Ensemble averages over the full range at a fixed T and applied field B minimizes the free energy F , {(1)} {(2)} of noncollinear local moment orientational configurations {eˆi } Eq. (1). Several solutions, ml , ml ,..., may be found and are needed to determine the system’s magnetic properties the one with the lowest F describes the system’s equilibrium { } realistically.22 state ml Equil.. Hence metamagnetic transitions can be tracked We now develop the DLM theory for a magnetic material as functions of T and B—for a given T the solutions for in an external magnetic field B at a temperature T .The increasing values of B can show a transition from, say, an AFM probability that the system’s local moments are configured to a high magnetization state at a critical field Bc [e.g., see { } { } = − { } Fig. 1(b)]. The material’s spin-polarized electronic structure according to eˆi is Pi ( eˆi ) exp[ β( eˆi ,B)]/Z, where = − { } = also depends on B and T and the state of magnetic order. the partition function Z j deˆj exp[ β( eˆi ,B)], β −1 In order to elucidate the main aspects of the general (kB T ) , and the free energy F =−kB T ln Z. A “generalized” framework laid out in Eqs. (1)–(3) for a putative helical electronic grand potential ({eˆi },B) is in principle available 18 metal, we assume temporarily that magnetic anisotropy effects from SDFT, where the spin density is constrained to be are small and can be neglected. Without a magnetic field orientated according to the local moment configuration {eˆi }. B, the solution {ml} which produces the lowest free energy It thus plays the role of a local moment Hamiltonian but = · + · its genesis can give it a complicated form. Nonetheless, is mi mq[cos(q Ri )xˆ sin(q Ri )yˆ]. The helical axis zˆ has no preferred direction in the crystal lattice. The order by expanding about a suitable reference “spin” Hamiltonian { }= · 23 parameter mq increases from 0 to 1 as T drops from TN 0 eˆi i hi eˆi and, using the Feynman inequality, we find a mean-field theoretical estimate of the free energy,18 to 0 K. On applying B, defining, say, the X axis of our coordinate frame, we find numerical solutions of the mean- F ({m },B,T) =({eˆ },B){ } field equations (1)–(3) of (i) distorted helical form, m i = (m + i i mi mq) cos(q · Ri )xˆ + mq sin(q · Ri )yˆ, (ii) a conical helix, + kB T Pi (eˆi )lnPi (eˆi )deˆi m i = mxˆ + mq[cos(q · Ri )yˆ + sin(q · Ri )zˆ], (iii) a fan state, i m i = mxˆ + mq cos(q · Ri )yˆ, and (iv) a high magnetization = − B · μi m i , (1) state with mq 0.
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