AFM regime

Below the ordering temperature of FMs and AFMs, the magnon-drag thermopower is shown (5) to be proportional to the magnon specific heat C m and inversely proportional to the number of free electrons or holes n, following the equation α md = ± 2 3 C m ne ⋅ 1 1 + τ em / τ m (1)

The second factor in the right-hand side of Eq. 1 contains magnon relaxation times. A summary of the different relaxation times and lifetimes used in this paper follows:

1) τ m is the total momentum relaxation time for magnons. It accounts for all magnon interactions, e.g., 4-magnon Umklapp processes, magnon-phonon, or magnon-electron interactions.

2) τ em is the magnon momentum relaxation time limited by magnon-electron interactions, which enters Eq. 1. Thus, the second factor in Eq. 1 represents the fraction of magnon scattering events that impel momentum to electrons.

3) τ L is the magnon lifetime in the AFM (τ L,AFM ) and PM (τ L,PM ) regimes, as measured from neutron scattering. It does not a priori concern momentum exchange, because neutron scattering measures the spin pair correlations. The linewidth is a measure of how the spin-spin pair correlation function decays with time at a specific value of magnon momentum q. However, because τ m is a measure of the perturbations in q and the relaxation mechanisms that limit τ m are those that limit τ L , it is reasonable to assume that τ L ≈ τ m , just as the phonon lifetime measured by neutron spectroscopy is a measure of phonon scattering times.

4) τ e quantifies the electron momentum relaxation by all scattering mechanism; it is the relaxation time that enters the electron mobility.

5) τ me quantifies the electron momentum relaxation in electron-magnon interactions; because the resistivity of magnetic materials is often limited mostly by magnetic scattering, this is likely the dominant term in τ e .

The sign of α md is determined by the polarity of the majority carriers (5, 6). The theory is confirmed at low temperature (<½ T Curie ) in FM Fe, Co, and Ni (5), where the second factor, with the relaxation times, can be assumed to be unity. This is not an assumption that can be made in a semiconductor, where the concentration of electrons is smaller than in metals.

The carrier concentration n is determined experimentally from the Hall effect measurements (Fig. 3A) in the AFM regime. The Hall coefficient shows an anomaly at T N and, in some samples, can give a different value in the PM regime from that in the AFM regime. This is discussed in (15) and attributed to magnetic scattering; the value in the AFM regime, reported here, is accepted as representative of the true carrier concentration (15). Because the carrier concentration is determined by the Li-doping level, which does not depend on T, it is reasonable to assume that it is essentially temperature independent for n > 6 × 1019 cm−3.

Fig. 3 Hole concentration, mobility and relaxation time, and specific heat of Li-doped MnTe. (A) Carrier concentration of all samples from Hall measurements. (B) Specific heat analysis of 6% Li-doped sample. The black dots are measured specific heat; the dashed line at low temperature is electron specific heat. Assuming that the high-temperature plateau is the Dulong-Petit high-temperature limit, a Debye model is fitted to the data to calculate the phonon contribution. The difference between measured specific heat and Debye model plus electronic specific heat is then the magnetic contribution. (C) Sample mobility μ. (D) Electron relaxation time τ e from mobility. The legend for all frames is shown in frame (C).

The magnon specific heat C m is determined experimentally from the total specific heat measurements. The specific heat (C) of all six samples has an identical temperature-dependence curve and shows no field dependence up to 7 T. The specific heat of 6% Li-doped sample is shown in Fig. 3B (black dots); it consists of a Debye contribution, an electron contribution at T < 6 K, and a magnetic contribution. The electronic part at low temperature follows a T1 law, the phonon part follows a Debye function, and the magnetic part follows a T3 law, except very close to T N . At low temperature, both phonon and magnon specific heat is proportional to T3, while electron specific heat is proportional to T. Therefore, at low temperature, the specific heat can be expressed as C p = γ ⋅ T + A ⋅ T3 or C p T = γ + A ⋅ T 2 (2)

Equation 2 is solved graphically by plotting C p /T as a function of T2 to yield γ = 2.07 mJ/mol-K2. After subtracting the linear term, a Debye model is fitted to the difference of measured specific heat and the electron specific heat (Fig. 3B, solid line). The plateau above T N is set to be the Dulong-Petit high-temperature limit; the fit gives a Debye temperature of Θ = 223 K. The magnetic contribution C m is then the measured specific heat subtracted by the Debye model and electron specific heat (Fig. 3B, cross marks). At T < T N , it follows a T3 law, as expected from AFM magnons, which have a linear dispersion (16). This is the value of C m that we will combine with the experimental data for n to calculate α md .

It is informative to compare the order of magnitude of different relaxation times τ m , τ em , and τ me . However, we have not found a way to estimate the second factor in Eq. 1 quantitatively, especially as a function of temperature and doping level. An order-of-magnitude estimate comes from the following considerations. We can estimate τ m assuming τ L ≈ τ m . The neutron spectroscopy data on polycrystalline samples permit only a coarse estimate of the magnon lifetime in the dispersion-less region of the spectrum, where the bandwidth ~3 to 5 meV is an upper bound (T = 250 K). Thus, τ L, AFM is of the order of 1 × 10−13 to 2 × 10−13 s. τ me is related to τ em via mutual scattering of the charge carriers and magnons. The electron scattering time τ e can be estimated from the Hall mobility according to τ e = m*μ/e, where m* and μ are the hole effective mass and mobility, respectively. The mobility is shown in Fig. 3C, and τ e is calculated using a value of m* = 0.53 m e (the free electron mass) (11) in Fig. 3D. If magnetic scattering dominates the mobility, τ me is expected to be of the same order of magnitude as τ e . This relation is valid only if we can ignore the second-order effect, i.e., the momentum transferred from the carriers to magnons is randomized before it can be transferred back to another carrier (17). Furthermore, magnons and charge carriers must have similar drift velocity provided that the effects of other scattering mechanisms are small, like the case of phonon-electron drag (18). On the basis of this argument, one can show that in the degenerate regime, τ em = τ me k B T m * c 2 , where c is the magnon group velocity (11, 15). The original calculation of this relation by Zanmarchi and Haas (15) is smaller by a factor of 2, because they assumed a single magnon mode. However, the magnon modes in AFMs are usually doubly degenerate, resulting in τ em two times larger. From these considerations and the value of τ L , we derive that the range for the ratio τ em /τ m falls in the bracket ~2 to 200 in the AFM regime; we will treat this ratio as the sole adjustable parameter in the thermopower.

The total thermopower is given by the sum of the magnon-drag and diffusion thermopower (5) α = α md + α d (3)with, for metals (assuming acoustic phonon scattering) (5) α d = 2 3 ( π 3 ) 2 / 3 k B e m * ℏ 2 k B T n 2 / 3 (4)

Except for the ratio 1 + τ em /τ m in Eq. 1, all the quantities are known, and the experimental thermopower can be fit to Eq. 3. We set 1 + τ em /τ m = 100 for all temperatures and all doping levels. The resulting values for the calculated thermopower are compared to the data in Fig. 4 for x = 0.01, 0.03, and 0.06. The agreement in the AFM regime is within a factor of 2. Thus, the thermopower in the AFM regime fits quite well to a model that includes a diffusion term (T1 law) plus a magnon-drag term (T3 law).