First-principles completion of Mott's 1936 s-d scattering theory for the resistivity and thermopower of transition metals, their alloys, and their ferromagnets

The transition metals Ni, Pd, and Pt and their alloys derive their anomalous transport from a nearly full d band into which s-like conduction electrons scatter. N. F. Mott (1936) explained the high resistivity, the large negative thermopower, and the magnetization-dependent transport of these metals with a parametrized one-electron theory: a single fitted ratio alpha of s-s to s-d transition probabilities, estimated band-edge and Debye parameters, the low-temperature (Sommerfeld-limit) thermopower formula, neglect of conduction by d holes, and measured magnetization curves as input. The problem, defined by the approximations the paper itself acknowledges, is to replace this parametrized account by a microscopic, parameter-free theory: scattering matrix elements and their energy, spin, and temperature dependence computed from band wavefunctions (including phonon, disorder, spin-disorder, and magnon channels), transport integrals evaluated exactly over the thermal window at arbitrary temperature and magnetization, d-carrier contributions included, and quantitative agreement demonstrated with the measured resistivity and thermopower of the canonical systems (Pd, Pt, Pd-based alloys, Ni, Cu-Ni/constantan), including the Curie transition of the ferromagnets. No work to date gives a parameter-free account of the thermopower of an elemental ferromagnet through its Curie point, where the 1936 approximations demonstrably fail, or a unified validated account across the canonical systems; that core remains open.

Background

Transition metals - the elements around Ni, Pd, and Pt, whose valence electrons occupy both a broad, nearly-free-electron $s$ band and a narrow, high-density $d$ band derived from atomic $d$ orbitals - were the materials in which the 1930s quantum theory of metals visibly failed. In a normal monovalent metal the resistivity follows the Bloch-Grüneisen law, the prediction of Bloch's theory of electron-phonon scattering (phonons are the quantized vibrations of the crystal lattice, and the Debye temperature $\Theta_D$ sets the temperature scale of the phonon spectrum), up to a constant depending on the atomic mass $M$. Palladium and platinum instead show a large, strongly temperature-dependent resistivity and a large negative thermopower - the Seebeck coefficient $S$, the voltage a material develops per unit applied temperature difference - and ferromagnetic metals such as Ni (ferromagnetic = spontaneously magnetized below the Curie point $T_C$, the magnetic ordering temperature) show abrupt resistivity changes at $T_C$ and unusual thermopower near it.

N. F. Mott resolved these anomalies in a series of papers (Mott 1935; Mott 1936, 'The electrical conductivity of transition metals'; and Mott 1936, 'The resistance and thermoelectric properties of the transition metals'). In these metals the current is carried by $s$-like electrons, while the $d$ band is nearly full: its density of states $N_d(E)$ - the number of one-electron states per unit energy - is large and, near the band top $E_0$, has the square-root form $N_d(E)\propto\sqrt{E_0-E}$ characteristic of a three-dimensional band edge. An $s$ electron can be scattered by lattice vibrations into an empty $d$ state (an $s$-$d$ transition), and because $N_d$ is so large this dominates ordinary $s$-$s$ scattering. The resulting relaxation time $\tau(E)$ - the mean time between collisions for an electron of energy $E$ - takes, near the Fermi level $\zeta_0$ (the energy of the highest occupied state at zero temperature), the form

$$\frac{1}{\tau(E)}=C_0\left[\alpha+\sqrt{\frac{E_0-E}{E_0-\zeta_0}}\right]\frac{T}{M\Theta_D^{2}},$$

where $C_0$ is a constant and $\alpha$ is the ratio of the $s$-$s$ to the $s$-$d$ transition probability at the Fermi level - an empirical parameter, fitted as $\alpha\simeq0.25$ from the drop of the temperature-dependent resistivity of Pd-Au alloys as alloying fills the $d$-band holes. The characteristic temperature $T_0=(E_0-\zeta_0)/k$ (with $k$ Boltzmann's constant) measures how far the $d$-band top lies above the Fermi level; the 1936 fits used estimated values ($T_0\simeq3450$ K for Pd from resistivity, $\simeq4100$ K from thermopower, $\simeq2200$ K for Ni, with estimated Debye temperatures $\Theta_D\simeq270$ K for Pd and $410$ K for Ni). For the thermopower the paper used the Mott formula, the low-temperature (Sommerfeld) expansion $S=\frac{\pi^2}{3}\frac{k^2T}{e}\left(\frac{\partial\log\sigma(E)}{\partial E}\right)_{E=\zeta}$ (where $\sigma(E)$ is the energy-resolved conductivity, $e$ the electron charge, and $\zeta$ the finite-temperature Fermi energy), which assumes every energy-dependent quantity varies slowly across the thermal window of width $\sim kT$ around the Fermi level; with the above $\tau(E)$ this gives $S\approx\frac{\pi^2}{3}\frac{k}{e}\frac{T}{T_0}\frac{1/2}{1+\alpha}$, reproducing the large negative thermopowers of Pd and Pt and the growth of $|S|$ as $d$ holes are filled by alloying Pd with Cu, Ag, or Au, including the behavior of constantan (Cu-Ni). For resistivity, by contrast, the paper evaluated the full thermal-window average exactly, tabulating a reduction factor $\phi(T/T_0)$ (from 1 at $T/T_0=0$ to 0.47 at $T/T_0=0.6$) that reproduces the depressed temperature coefficient of Pd and Pt at high temperature. For the ferromagnets the paper introduced the two-current picture - spin-up and spin-down $s$ electrons scatter independently into the spin-split $d$ bands, so the two spin channels conduct in parallel - derived a resistivity formula in the reduced magnetization $z=I/I_0$ ($I$ the magnetization, $I_0$ its saturation value) predicting $\rho_{z=1}/\rho_{z=0}\simeq0.34$ for $\alpha=0.25$, and, using the measured curve $I/I_0(T)$, matched the calculated $\rho(T)$ of Ni to experiment at several temperatures through the Curie region.

The paper is explicit that this account rests on approximations: conduction by the $d$ holes themselves is neglected, their effective mass being taken so large that they carry negligible current; the algebraic thermopower expressions use the low-temperature expansion of the transport integrals, which fails when $\tau(E)$ varies on the scale $kT$ - at high temperature and near the Curie point, where the spin-split $d$-band edge moves rapidly with magnetization; the single constant $\alpha$ lumps together all microscopic matrix elements, ignoring their energy, spin, and temperature dependence; and the numerical comparisons use estimated $T_0$ and Debye temperatures, the square-root band-edge form valid only near the band head, and (for Ni) measured rather than computed magnetization curves. The question derived from these acknowledged approximations is whether the theory can be placed on a microscopic, parameter-free basis and still validated at the same level of quantitative detail. Modern tools make this concrete: density functional theory (DFT, the standard first-principles ground-state electronic-structure method), the Korringa-Kohn-Rostoker coherent-potential approximation (KKR-CPA, the standard first-principles treatment of substitutional chemical disorder), the Kubo-Greenwood linear-response formulation of transport, disordered-local-moment averaging for finite-temperature spin disorder, and many-body (GW/GT-type) self-energies for electron-phonon and electron-magnon lifetimes (a magnon is a quantized spin wave; in magnon drag nonequilibrium magnons transfer momentum to conduction electrons, adding to the thermopower).

Problem Statement

Replace every fitted or estimated ingredient of Mott's 1936 $s$-$d$ scattering theory of transition-metal transport by quantities computed from the microscopic band structure, and validate the result quantitatively against the same classes of measurements the 1936 paper compared to. The problem has four linked components serving this single objective:

(1) Microscopic scattering: compute the scattering matrix elements - the $s$-$d$ and $s$-$s$ transition probabilities and their dependence on electron energy, spin, and temperature, i.e. the microscopic content of Mott's fitted ratio $\alpha$ (the ratio of $s$-$s$ to $s$-$d$ transition probability at the Fermi level, fitted as $\alpha\simeq0.25$ in 1936) - from band wavefunctions, including the phonon, chemical-disorder, and, in ferromagnets, spin-disorder and magnon scattering channels, rather than fitting them to resistivity data. (2) Exact transport integrals: evaluate the transport integrals for both the resistivity $\rho(T)$ and the thermopower (Seebeck coefficient) $S(T)$ exactly over the thermal window of width $\sim kT$ around the Fermi level ($k$ being Boltzmann's constant), at arbitrary temperature and magnetization, removing the Sommerfeld-limit approximation (the low-temperature expansion of the transport integrals, valid only when all energy-dependent quantities vary slowly across the thermal window) that the 1936 paper applies to the thermopower and that fails at high $T$ and near small demagnetizations. (3) $d$-carrier conduction: include the direct contribution of $d$-band carriers to the current wherever their effective mass is not negligible, rather than neglecting it outright. (4) Quantitative scope: demonstrate the account across the canonical systems of the 1936 paper - Pd, Pt, Pd-based alloys (Pd-Ag, Pd-Au, Pd-Cu), Ni, and Cu-Ni (constantan) - with accurate band-structure inputs and, for the ferromagnets, magnetization curves that are predicted or independently measured rather than adjusted to the transport data.

The account sought is a single microscopic theory that, using no transport parameter fitted to the data being explained, reproduces the measured $\rho(T)$ and $S(T)$ of these systems over their measured temperature ranges - including the magnetization-dependent behavior of an elemental ferromagnet through its Curie transition (at the Curie temperature $T_C$), the regime where the 1936 approximations demonstrably break down - and identifies which microscopic mechanisms (spin-dependent $s$-$d$ scattering, $d$-carrier conduction, spin disorder, magnon drag, phonon drag) account for the observed magnitudes, signs, and anomalies. A demonstration that one mechanism of Mott's framework fails microscopically settles only that mechanism, not the joint account formed by components (1)-(4).

References

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