Tuning the metamagnetism of an antiferromagnetic metal

We describe a `disordered local moment' (DLM) 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 magnetisation 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 modelling and predicting new metamagnets of interest in applications such as magnetic cooling.

others, e.g. [12,13], derived the coefficients of a Landau-Ginzburg expansion of the free energy of an antiferromagnet exposed to a uniform magnetic field in terms of the antiferromagnetic order parameter, ∆m q , with wave-vector modulation q. The coefficients of the expansion are determined from a consideration of both Stoner particle-hole excitations [10] and spin fluctuations [11,13] generated from the collective behavior of the interacting electrons [11,14,15]. Field-induced tricriticality occurs when the quadratic and quartic terms in ∆m q both equal zero. More recently, from an analysis of a generic mean-field Hamiltonian describing a helical antiferromagnetic state in an applied field Vareogiannis [16] has shown how an itinerant electron system can undergo a first-order 'spin-flip' transition. For a weak magnetic field the AFM polarization is parallel to the applied field and flips perpendicular only when the field exceeds a critical value [17].
In this Letter we describe an ab-initio spin density functional theory (SDFT)-based 'local moment' theory for antiferromagnetic metals that blends these two complementary accounts of metamagnetism to suit materials where slowly varying 'local moments' can be identified from the complexity of the electronic behavior. A principal result is that, in the limit where magnetic anisotropy effects are small or neglected entirely, the local moments' antiferromagnetic order can still undergo a first order transition to a fan or ferromagnetic state owing to the feedback between the local moment and itinerant aspects of the electronic structure.
We test our theory against a detailed experimental case study of CoMnSi-based tricritical metamagnets and show how it provides quantitative materials-specific guidance for tuning metamagnetic and associated technological properties.
We start from a generalisation of SDFT [18] that describes the 'local moment' picture of metallic magnets at finite temperature [19][20][21][22]. Its basic premise is a timescale separation between fast and slow electronic degrees of freedom so that 'local moments' are set up with slowly varying orientations, {ê i }. The existence and behavior of these 'disordered local moments' (DLM) is established by the fast electronic motions and likewise their presence affects these motions. This mutual feedback can then lead to a dependence of the local moments' interactions on the type and extent of the long range magnetic order through the associated itinerant electronic structure. The DLM picture of the paramagnetic state can be mapped to an Ising picture with, on average, one half of the moments oriented one way and the rest antiparallel. However once the symmetry is broken so that there is a finite order parameter { m i } profile, e.g in a FM or AM state and/or when an external magnetic field is applied, this simplicity is lost. Ensemble averages over the full range of non-collinear local moment orientational configurations, {ê i }, are needed to determine the system's magnetic properties realistically [22]. We now develop this idea and extend the DLM theory to enable the effects of an external magnetic field to be modelled along with temperature and field dependence of the magnetic state.
We consider a magnetic material in an external magnetic field B at a temperature T . The probability that the system's local moments are configured according to where the probability of a moment pointing alongê i on a site i is via the generalised SDFT.
The Weiss field at a site l is given by To capture the itinerant electronic component of the problem coming from the overall spinpolarisation of the electronic structure, we approximate the h l as the first two terms of an whereS (2) i,j ( m) is the direct correlation function tensor and describes effective interactions between the local moments that depend on the magnitude of long range magnetic order, m.
A solution of Eqs.1 to 3 at a fixed T and applied field B minimises the free energy, Eq. 1.
l },· · · may be found and the one which produces the lowest free energy, Eq. 1, describes the equilibrium state of the system, { m l } Equil. . Hence metamagnetic transitions can be tracked as functions of T and B -for a given T the solutions for increasing values of B can show a transition from, say, an antiferromagnetic to ferromagnetic state at a critical field, B c (e.g. see Fig. 1(b)). The material's spin-polarised electronic structure also depends on B and T and the state of magnetic order.
We now follow through the general framework laid out in Eqs.1 to 3 for a putative helical antiferromagnetic metal. In order to elucidate the main aspects, we shall assume temporarily that magnetic anisotropy effects are small and can be neglected. In the absence of an applied magnetic field, the solution { m l } which produces the lowest free energy has the form m i = ∆m q (cos( q · R i )x + sin( q · R i )ŷ). The helical axis,ẑ, has no preferred direction in the crystal lattice. The order parameter ∆m q increases from 0 to 1 as T drops from T N to 0 K. On application of a magnetic field, B, defining, say, the x-axis of our coordinate frame, we can find numerical solutions of the mean field equations 1 to 3 of (i) distorted helical form: m i = (m + ∆m q ) cos( q · R i )x + ∆m q sin( q · R i )ŷ, (ii) a conical helix: and (iv) a ferromagnetic state with ∆m q = 0. The free energies are F dh (m, ∆m q ; B, T ), The local moment interactionsS (2) i,j ( m) of Eq. 3 set up the Weiss field and, along with the uniform component, h( m), are the key elements of our theory. We determine these quantities ab-initio using relativistic, spin-polarised, multiple scattering (Korringa-Kohn-Rostoker, KKR) theory and the coherent potential approximation (CPA) to handle the electrons moving in fields set by the orientationally disordered local moments [22,24,25].
Here, for the first time, we account for the variation ofS (2) i,j ( m) with increasing extent of longrange magnetic order m driven by spin-polarisation of the itinerant electrons which mediate the interaction between the local moments [26,27]. Since spin-orbit coupling effects are also included [25] we can assess whether the neglect of magnetic anisotropic effects is reasonable.
Details on howS (2) i,j ( m) is calculated can be found in references [18,25,28,29]. Our previous experimental investigations have revealed a class of magnetic materials based on the orthorhombic CoMnSi metallic antiferromagnet to be an ideal testing ground for the theory [30][31][32]. Those studies considered both the composition-dependent metamagnetism and pronounced magneto-elasticity in CoMnSi by extensive magnetic and structural measurements, including a characterisation of the anomalous temperature variation of structural parameters in zero magnetic field. CoMnSi orders into an non-collinear, helical antiferromagnetic state in zero magnetic field at T N ≈380 K. In an applied field this transition becomes a metamagnetic one to a high magnetisation/ferromagnetic state at a temperature T t . As the applied field is increased T t decreases before going through a tricritical point at around 2 Tesla where an enhanced magnetocaloric effect is observed along with marked changes to the interatomic spacings. All three subsystems, Co, Mn and Si can be doped and substituted which also affects the magnetic properties [32]. This material is typical of many useful magnetic metals in that its magnetism possesses both localised and itinerant electron spin attributes [17]. Localised magnetic moments can be identified with the Mn sites whereas the magnetism associated with the Co sites is reflected in the long range spin polarisation of the electronic structure. Our measurements have also found that the magnetism of CoMnSi is dominated by the behavior of the Mn moments. Their interactions, however, are delicately poised and depend on the spacing between them, the compositional environment in which they sit and the overall long range magnetisation.
We previously modelled a commensurate approximation to the helical AFM state [31] found in experiment and other calculations have examined an assumed FM state [33]. This Letter makes the first ab-initio exploration of the non-collinear helical magnetism of CoMnSi and its magnetic field and temperature variation. We start with self-consistent KKR-CPA calculations [34][35][36] for the paramagnetic (m = 0) DLM state of CoMnSi. Local moments of magnitude µ ≈ 3.0µ B are established on the Mn sites which are very close to the magnetisation per Mn site we find in calculations of CoMnSi in a FM state (m = 1) and we can assume that the sizes of the Mn local moments are rather insensitive to the orientations of moments surrounding them. We thus use the potentials within the 'frozen potential' approximation [24,37] to study CoMnSi for probability distributions {P i (ê i )} with a set of values of the magnetic order parameter m i , each ranging between 0 and 1. We use an angular momentum cut-off of l = 3, a muffin-tin approximation for the potentials and local spin density approximation for exchange and correlation. For our DLM calculations for a finite long range order parameter, m = 0, we consider a probability distribution on a fine grid of local moment orientations [22,24]. No local moment forms on the cobalt sites in the DLM paramagnetic state whereas for finite m, a small magnetisation associated with each Co site is induced by the lining up of the Mn moments and the consequent overall spin polarisation of the electronic structure. For the FM state, m = 1, this is ≈ 0.6µ B per Co site [31,33].
The onset of magnetic order in CoMnSi and the temperature at which this transition takes place is obtained from an examination ofS (2) i,j ( m) for the paramagnetic state, m = 0. By calculating the lattice Fourier transform of this interaction,S (2) ( q, 0) and finding the wave-vector, q max , where the largest eigenvalue, s (2) ( q, 0), maximises, we find the magnetically ordered state that the system forms below our mean-field theory estimate of T N = s (2) ( q max , 0)/3k B . If q max = 0 a FM state is indicated whereas q max = 0 points to an AF state. Fig.1(a) shows s (2) ( q, 0) using structural data obtained by neutron diffraction [30].
These data show the Mn-Mn distances, d 1 , to vary by more than 2% over the temperature range 100-400K. Our calculations show that CoMnSi should order into an incommensurate helical AFM state along the c-axis, set by the orthorhombic crystal structure, at T N ≈ 400K in good agreement with experiment [30,31]. They confirm the expectation that spin-orbit coupling effects are small in CoMnSi, with only a very weak anisotropy favoring the Mn moments to lie in the a-b plane. This agrees with the very small level of magnetic anisotropy seen in measurement [38].
CoMnSi is evidently very close to a ferromagnetic instability as shown by the value of s (2) ( q = 0, m = 0) being close to that of s (2) ( q max , m = 0), a convenient signature for a potentially useful metamagnet. Its marked magnetoelastic properties are confirmed by the tendency to order ferromagnetically taking over from the incommensurate ordering propensity as d 1 is increased, which experiment also shows to happen with increase in temperature.
When the bath of electrons in which the Mn local moments sit develops a long range spin polarisation as a magnetic field is applied and m increases, our calculations show that the magnetic interactions,S (2) ( q, m), weaken significantly only strengthening again in the spinwave limit of m ≈ 1. The inset to Fig. 1(a) illustrates this. The magnetic anisotropy effects remain very small, ( e.g we find a uniaxial anisotropy K < 0.01mRy. per unit cell for the completely ferromagnetically ordered state with a small orbital moment of < 0.03µ B on the Mn sites). The ferromagnetically-induced weakening of the magnetic interactions acts to promote the stability of the distorted helical magnetic state over other states when the magnetic field is applied [33]. There can therefore be a first order transition to a fan, conical helix or ferromagnetic state at critical field B c despite the absence of magnetic anisotropy. Fig. 1(b) illustrates this effect, and shows our solutions of the mean field theory equations 1 to 3 where we have used our calculated ab-initio uniform Weiss field h( m) [22] andS (2) ( q max , m) values. We point out that there are no adjustable parameters for these curves which show a transition from a distorted helical AF state (small m, B) to a high magnetisation state above a critical field B c . The transition is first order up to a tricritical temperature of 372K and second order thereafter. Fig. 2 shows effect on the temperature dependence of the critical field, B c of either changing Mn nearest-neighbor separations, d 1 , (Fig. 2(a)) in CoMnSi or varying composition slightly away from CoMnSi ( Fig. 2(b)). If the itinerant electron effect which givesS (2) ( q, m) its m-dependence is neglected and only the effects of a weak on-site magnetocrystalline anisotropy are included, we find the values of B c to be 2 orders of magnitude smaller. B c increases sharply as d 1 decreases. This is in line with measured pressure dependence of B c of CoMnSi [39]. B c is also very sensitive to compositional doping -the AFM tendency strengthens as electrons are removed so that B c increases, as shown by an application to  Fig. 3(a). The lower panel, Fig. 3(b) shows our theoretical estimates of the critical fields for these same three systems, where we have used the zero field experimental structural data. Overall the magnitudes and trends are given very well by the theory. The variation in temperature of the relative spacings between the Mn atoms is responsible for concave appearance of B c (T ) in CoMnSi. The Ni-doped structure is shown to be extremely sensitive to structure, as observed experimentally [32]. When B c > 2 Tesla, dBc dT appears to be overestimated by the theory. This discrepancy could be partly mitigated by allowing for the magnetic-field induced increase in d 1 which can inferred from our calculations of h( m) and S (2) ( q max , m) which show that ferromagnetic tendency strengthens with increase in d 1 .
Detailed theoretical models are needed to use in concert with experimental studies if the behavior of highly sensitive magnetic materials is to be analysed and tuned. We have discussed and demonstrated one such model here and showed how the temperature and field dependence of the metamagnetic transition of an antiferromagnetic metal can be affected by varying composition and atomic spacing. Useful magnetic metals typically have large magnetic moments which are established locally. The Disordered Local Moment model works well for these [22,28,29] and we have set out a theory for metamagnetic transitions based on this picture. An important aspect for magnetic material design is to distinguish first and second order magnetic transitions and to get some control of the location of precious tricritical points. Our CoMnSi case study reveals how reducing the Mn-Mn separation, d 1 , increases B c . Small degrees of compositional doping which reduce the number of electrons have a similar effect. One notable feature in our modelling is the dependence of the interac-tion between the local moments upon the extent of long range magnetic order. This comes from the change in the behavior of the itinerant electrons that mediate these interactions and produces a mechanism for first-order 'spin-flip' transitions even for cases where there is little magnetic anisotropy. This aspect has an important role in the analysis and design of adaptive, magnetic metals.