Magnon drag effect in Fe-Co alloys

Thermoelectric materials are capable of relating temperature gradients and electrical transport through solid-state processes. This property has garnered increasing interest in improving the efficiency of thermal engines via waste heat recovery from their exhaust gas. Conversely, the same effect has applications in solid-state refrigeration.¹ Low efficiency of commercially viable thermoelectric materials limits applications of thermoelectric refrigerators and power generators. The efficiency is characterized by the dimensionless figure of merit,

where $α$⁠, $σ$⁠, T, and $κ$ are thermopower or Seebeck coefficient, electrical conductivity, absolute temperature, and thermal conductivity, respectively. Thermoelectric devices are also used for active cooling of electronics; it was recently pointed out^2,3 that active cooling processes are very different from those used in refrigeration. In active cooling, the goal is to cool a load to ambient temperature by moving heat from the hot device to ambient. In refrigeration, the goal is to cool a load to below ambient temperature by moving heat from the cold device to ambient temperature. Thus, it is reasonable that a high thermal conductivity is favorable in active cooling applications, where the heat flux from the load to the ambient must be maximized, but not in refrigeration, where the backflow of heat from the ambient to the load must be prevented. Therefore, whereas zT is the right criterion for refrigeration, it not for cooling. It has been shown² that the materials' criterion for cooling is the power factor P,

Furthermore, thermoelectric metals with a high electrical and thermal conductivity are ideally suited for cooling, provided they also have a high thermopower.

Unfortunately, in the classical nonmagnetic electron transport theory, σ and α are inversely related due to the direct connection between σ and the density of charge carriers n_e, and the Pisarenko relation⁴ 

Thus, until now, thermoelectric materials development has concentrated on semiconductors.

The thermopower characterized by the Pisarenko relation is known as diffusion thermopower; however, in magnetically ordered materials, magnon-drag thermopower, which is absent in spin-degenerate materials, has an additive effect on the total thermopower, and in the case of the 3d transition metals (Fe, Co, and Ni) dominates thermoelectric transport.⁵ For magnetically ordered metals, this provides a valuable parameter to tune and optimize. Moreover, metals, which are typically mechanically strong and ductile, relatively easy to manufacture, to shape, and to join, provide greater functionality and easier integration into industrial processing procedures. As discussed above, in the case of cooling of electronics, the high thermal conductivity of metals is beneficial. Even in power generation and refrigeration applications, only the phonon thermal conductivity acts as a loss mechanism that lowers the thermal efficiency of thermoelectric processes.⁵ Designing metal alloys with enhanced thermopower, therefore, provides a great opportunity for the widespread adoption of the thermoelectric technology and allows for the seamless integration of thermoelectric devices into existing industrial and commercial products.

In magnetically ordered metals, the collective excitation of the localized magnetic moments forms magnons⁶ that have associated specific heat, thermal conductivity, and chemical potential. The magnon chemical potential, a recently introduced theoretical quantity that describes magnon accumulation under thermodynamic driving forces,⁵ interacts with temperature gradients resulting in nonequilibrium magnon generation, and, therefore, a magnonic thermopower. The thermally driven magnon diffusion produces a drag effect on the surrounding electron fluid resulting in a contribution to the total thermoelectric power, called the magnon-drag thermopower α_md. A hydrodynamic theory was developed⁷ to quantify it: α_md is proportional to the magnon specific heat C_m over the total number of free electrons n_e,

where e is the effective charge of the dominant charge carrier, and τ_em and τ_m are the scattering time for magnon-electron collisions and magnon collisions, respectively. In metals at sufficiently low temperatures, magnons are scattered dominantly by electrons so that the scattering time portion of α_md approaches unity (this statement does not hold for magnetic semiconductors). The sign of the effective charge e and the magnon-drag thermopower in Eq. (4) is governed only by the polarity of the charge carrier: positive for holes and negative for electron bands. The hydrodynamic theory models magnons and electrons as two types of fluids and neglects Umklapp and magnon nonconserving processes. Further, assuming that the electrons undergo mostly normal electron-phonon and magnetic scattering, the diffusion thermopower in metals is given by⁸ 

where E_F is the Fermi level of that material. A recent study by Watzman et al.⁷ investigated the magnon-electron drag effect in the elemental Fe, Co, and Ni, for which the magnon structure and carrier concentrations are well known. The sum of α_d and α_md calculated by the hydrodynamic theory explains the data on Fe and Co very well up to about 25% of the Curie temperature without adjustable parameters. The result further shows that the magnon-drag thermopower can be one order of magnitude larger than the diffusion thermopower at low temperatures. However, the origin of the changes in the sign observed⁹ in the thermopower of elemental iron remains a mystery to date.

The quantities e, E_F, and n_e in Eqs. (4) and (5) originate from the band structure and density of states (DOS) of the material, allowing for the qualitative calculation of the thermopower of metals from ground state ab initio density functional theory (DFT) calculations. The number of electrons is proportional to the DOS at the Fermi level, which can be orbital-resolved to estimate the number of s-, p-, and d-orbital electrons. As d-orbital electrons generally are localized, the number of free electrons can be considered proportional to the partial DOS of the s- and p-electrons. Thus, as tuning the carrier concentration in metals is not practical via alloying, a method of optimizing the redistribution of electrons into localized d-orbitals can be used to optimize the thermopower of magnetic metal alloys.

The sign of the effective charge e is that of the energy derivative of the DOS at the Fermi level, ΔDOS. A positive ΔDOS results in electrons being the dominant charge carrier and corresponds to a negative value of e; a negative value of ΔDOS results in hole-dominated behavior and a positive e. In the hydrodynamic theory, the sign of α_md depends solely on this quantity.⁷ However, a more precise microscopic theory was developed subsequently,¹⁰ which is equivalent to the hydrodynamic theory at low temperatures but suggests the possibility of a change in the sign of α_md when itinerant magnetism becomes an important part of the total magnetization of a ferromagnetic sample. Therefore, we discuss a possible role of itinerant magnetism in our high-temperature data further in the paper. Nevertheless, in the light of the success of the hydrodynamic theory on Fe below 200 K, we will not consider this point in the analysis of our data on Fe-rich Fe-Co alloys below 200 K.

In this work, thermopower of body-centered-cubic (BCC) Fe-Co Fe-rich solid solutions has been investigated and compared with DFT calculation of DOS. Although good agreement is achieved, the goal of enhancing thermopower is obstructed by the compensating effect of the carriers in this alloy system.

Seven Fe-Co alloy samples were prepared in total, with nominal Co content of 1%, 5%, 8%, 10%, 12%, 15%, and 28%. According to the binary phase diagram,¹¹ this covers the concentration region where solid solutions of Co in Fe crystallize in a single body-centered-cubic solid solution phase (the models do not cover multiphase alloys). The starting materials are 99.9% pure Co pieces and 99.98% pure Fe granules, both from Alfa Aesar. The starting materials were arc-melted in flowing argon gas. The melting was repeated 5 times to increase the homogeneity of the resulting ingots. These alloy ingots then were sealed in ampoules and annealed at 1150 °C for 7 days for homogeneity. The structure of the samples was confirmed by X-ray diffraction.

The low temperature (80 K to 400 K) thermopower, electrical conductivity, Hall effect, and Nernst effect were measured in a ST-300 T Janis flow-cryostat system up to 1.4 T controlled by a LabVIEW program, using a static heater-and-sink method with copper-constantan thermocouple thermometry. The high-temperature (400 K to 1000 K) thermopower and electrical conductivity were measured in the commercial thermoelectric property measurement system Linseis LSR-3 in a helium atmosphere. The experimental uncertainties are the same as in previous similar studies on other materials and are discussed elsewhere.¹² 

The computational portion of this study was performed with the Vienna ab initio simulation package (VASP), using projector augmented wave (PAW) pseudopotentials, and the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional.^13–17 Spin-polarized calculations were performed on 64 atom supercells using special quasirandom structures (SQS) to approximate solid-solution alloys.^18,19 The Brillouin zone was sampled using a 5 × 5 × 5 Monkhorst-Pack k-point mesh, using Methfessel-Paxton smearing for structural relaxation and the tetrahedron method with Blöchl corrections for DOS calculations.^20,21 An energy cutoff of 400 eV was used throughout all calculations. To facilitate numerical differentiation of the DOS around the Fermi level, a Savitzky-Golay filter was applied using a 3rd order polynomial with a window length of 7.^22,23

The measured thermopower for all candidate alloys between 80 K and 1000 K is shown in Fig. 1(a). With Co atom percentage increasing from 1% to 28%, the thermopower becomes generally more negative. The data on pure Fe from Ref. 7 are added as a reference. Figure 1(b) shows thermopower vs Co concentration at 200 K, and the behavior of the same curves at lower temperatures is similar, and the zero crossing point remains within the experimental inaccuracy on sample composition, at 11 ± 1 at. % Co from 77 to 200 K

The DFT calculations, shown in Fig. 2, indicate that both s- and p-orbits have a minimum at around 12% Co content. Since the exchange interaction of Fe-Co alloys is larger than that of the pure elements,²⁴ and the Curie temperature of iron is above 1000 K, below 200 K all the alloys can be assumed to be in the regime where Eq. (4) holds, as it does for iron. If all the carriers on s- and p-orbits had the same sign of the ΔDOS, then the hydrodynamic theory would predict that the magnon-drag thermopower had a maximum when the total number of s- and p-carriers was at its minimum. In fact, the ΔDOS, and thus the polarity of carriers on both s- and p-orbits, changes from holes to electrons when Co content increases. At 12% Co content, the effective mass of the s-electrons changes from negative to positive, but p-electrons maintain a negative effective mass, and, therefore, have p-type behavior. The different polarities of s- and p-carriers make the 12% Co-Fe alloy nearly a compensated metal. As a result, both the diffusion thermopower and magnon drag thermopower approach zero for this composition. This conclusion is supported by the experimental results. As shown in Fig. 1(b), in the low temperature range, the thermopower decreases with increasing Co atomic percent and changes the sign from positive to negative. At around 10%, the thermopower is zero. Therefore, the low temperature behavior of thermopower is well explained and understood.

The temperature dependence of the thermopower, i.e., the direct effect of the temperature gradient on the electrons, at the higher temperature range is abnormal for all the alloys, as is that of the elemental iron.⁹ The diffusion thermopower is linear with temperature for metals,⁸ and since the band structure has little temperature dependence, cannot give rise to a sign change. It is, therefore, likely that the peculiar features of α(T), such as sign change in the samples with 1%, 5%, and 8% Co and the valleys in α(T) shared by all the curves at high temperatures, are caused by changes in the magnon drag thermopower. The alternative magnon drag theory by Flebus et al.¹⁰ mentioned in Sec. I suggests a possible explanation. In this theory, α_md has a prefactor $(β−αG)$⁠, where α_G is the Gilbert damping parameter and β is the spin transfer torque parameter.⁹ If less than 1/3, the ratio (α_G/β) can change the sign of α_md. This ratio is equal to the ratio of itinerant spin-angular momentum to the total spin-angular momentum.²⁵ Rosengaard and Johansson²⁶ used DFT to calculate the ground state properties of Fe and used the Monte Carlo method to calculate the magnetic properties of Fe at finite temperature. Because of the quenching of the orbital angular momentum of 3d elements, the local moment of Fe ion is proportional to the number of localized d electrons. The local moment of Fe is shown to increase slowly with temperature, which means that more electrons become localized as temperature increases. In other words, the ratio of itinerant electrons to the total number of electrons decreases with temperature. Therefore, this calculation roughly supports our explanation; work is in progress to develop a more rigorous and quantitative explanation for the thermopower of the elemental iron up to 1000 K.

For completeness, the resistivity measured through the whole temperature range is shown in Fig. 3(a). Power factor (PF) then is calculated following Eq. (2). The results are shown in Fig. 3(b). The 28% Co sample shows a PF comparable with commercially available Bi₂Te₃, whose peak is around 40 at 200 K;²⁷ we point out that the PF of the elemental cobalt in its FCC phase is much higher yet, reaching 160 μW K⁻¹ cm⁻¹at 300–400 K.⁵ 

Using the hydrodynamic theory of magnon-drag thermopower, we use a DFT directed search for thermoelectric alloys in the Fe-Co system. Thermopower, resistivity, and power factor of Fe-Co alloys from 80 K to 1000 K are explored experimentally. The sign of the low temperature thermopower is described well by DFT calculations. The high-temperature features of thermopower deviated from predictions derived from ab initio calculations, and are, in fact, unexplained to date, even on the elemental iron. Using recent theoretical work published by Flebus et al.,¹⁰ as well as calculations performed by Rosengaard and Johansson,²⁵ we suggest that this deviation is due to the increasing population of localized d electrons.

Although we obtain good agreement between experiment and DFT calculation, the thermoelectric properties of Fe-Co alloys are optimized fully by tuning the population of electron orbitals at the Fermi level. As predicted by DFT calculations, the s- and p-orbitals have different polarities at the same Co concentration, where the number total free carriers is at minimum. The number of the two types of carriers is comparable, making the Fe-Co alloy a compensated metal. Since magnon drag is a mechanism that increases the thermopower of magnetic metals by an order of magnitude,⁷ the results reported here offer a way to discover and prepare high-power-factor metal thermoelectrics suitable, in particular, in active cooling applications.² 

This work was supported by the Center for Emergent Materials, a National Science Foundation (NSF) MRSEC under Grant No. DMR-1420451. Y.Z. and J.P.H. are also supported by the Army Research Office (ARO) MURI under Grant No. W911NF-14-1-0016. Computational work was performed using resources allocated by the Ohio Supercomputing Center under Grant No. PAS0072.