Parallel conduction channels: multiband transport and identifiability

Scope

Two bands, two Drude components, two scattering processes and two pairing channels are different claims. Independent channels experiencing the same electric field add their conductivities. Scattering events within one channel instead enter a collision problem and can add as rates only under appropriate approximations. A good fit to DC resistivity alone rarely identifies which microscopic description is correct. MgB2 illustrates why band-resolved superconducting gaps must also be distinguished from independently conducting subsystems. R1 R2 R3 R5

This entry concerns transport decomposition and spectral-weight accounting. A fitted component can represent a band, an orbital sector, a spatial layer or an effective relaxation scale; the label must be justified with additional evidence. Identifying carriers that contribute to the condensate does not by itself identify the interaction responsible for pairing. R4 R5

Mathematical assumptions

For independent isotropic channels, use angular frequency $\omega$ and the convention $e^{-i\omega t}$: $\sigma(\omega)=\sum_j S_j/(\Gamma_j-i\omega)$, where $S_j=n_je^2/m_j$ and $\Gamma_j>0$ in the elementary Drude model. At zero field, $\rho_{\rm DC}=1/\sum_j(S_j/\Gamma_j)$, not the sum of channel resistivities. A broad component has $\sigma_j\simeq S_j/\Gamma_j$ only within a window where $\omega\ll\Gamma_j$; its constant approximation cannot be integrated to infinite frequency. R5

Even exact DC data admit infinitely many positive pairs with the same total conductivity. Frequency information helps, but if two linewidths are identical, their sum is exactly a single Drude peak with weight $S_1+S_2$ at every frequency. A finite frequency window can also leave a very broad component's weight and linewidth strongly correlated. Report uncertainty and competing models, not just a residual sum of squares. R4 R5

Independent scattering contributions can approximately give $\Gamma=\Gamma_{\rm imp}+\Gamma_{\rm ph}+\cdots$ within one relaxation model. This Matthiessen approximation is not a prescription for adding the resistivities of parallel bands. Interchannel momentum exchange generally couples their kinetic equations: an illustrative positive relaxation matrix has diagonals $\gamma_1+g,\gamma_2+g$ and off-diagonals $-g$. In normalized equal-density units its conductivity is $\mathbf{1}^{T}C^{-1}\mathbf{1}$, reducing to $1/\gamma_1+1/\gamma_2$ at $g=0$. This toy model exposes the independence assumption; it is not a material fit. R5

Normalization matters in spectral-weight sums. For one Drude term, $\int_0^\infty\sigma_1(\omega)d\omega=\pi S/2$. With condensate response $\sigma_{2,s}=S_s/\omega$, the positive-frequency missing area is $\pi S_s/2$, provided the relevant total optical weight is conserved and backgrounds are treated consistently. Thus $S_s=(2/\pi)\int_0^\infty[\sigma_{1,N}-\sigma_{1,S}^{\rm reg}]d\omega$ in this convention. Cyclic frequency $\nu=\omega/(2\pi)$ requires $\Gamma_\nu=\Gamma/(2\pi)$ and $S_\nu=S/(2\pi)$ in the same Drude form. Delta-function boundary conventions must also be stated. The numerical checks establish these model identities, not the complete historical derivation or experimental sum-rule saturation. R5 R6 R7

Experimental criteria

Romero's FeTe1−xSex study uses complex terahertz conductivity with a DC constraint to distinguish a narrow response from a broad background. The checked author version covers two films, x=0.45 and x=0.35, over 0.2–3 THz. The broad component is not resolved as a second complete peak, so assigning it a unique scattering rate or a second Planckian coefficient would exceed the measurement. Its displayed rate uses cycle frequency and a thermal scale proportional to $k_BT/h$; substituting $\hbar$ without conversion changes the convention. R5

The same study compares a fitted superconducting term, a low-frequency extrapolation of $\nu\sigma_2$, and a missing-area analysis. These are useful consistency comparisons but share measured spectra and modeling assumptions; they are not three independent experimental replications. Near the transition, a finite-width low-frequency response complicates extrapolation to zero frequency. The normal reference temperature, unmeasured low-frequency region and residual background must remain part of any condensate-weight claim. R5

Qazilbash's LaFePO experiment combines in-plane reflectance with ellipsometry and Kramers–Kronig-compatible optical extraction. The final paper analyzes a low-frequency Drude response, higher-energy interband transitions and an extended-Drude scattering rate. Its comparison of integrated optical weight with band theory supports a correlation interpretation under that analysis; it does not establish exactly two independent Drude carriers throughout the iron-based family. The reported kinetic-energy ratio has extraction bounds, rather than being an exact universal one-half. R4

MgB2 supplies a complementary comparison. Liu and Choi calculate band-dependent electron–phonon pairing, while Souma resolves different superconducting gaps on sigma and pi bands by photoemission. Choi's coupled calculation has a common transition. These results do not authorize treating the bands as two unrelated superconductors with separately assigned transition temperatures, or translating gap values into Drude scattering rates. Surface-sensitive spectroscopy and bulk optical transport retain different measurement conditions. R1 R2 R3

Limits and disputes

A spectral decomposition is not unique simply because it is causal or fits both real and imaginary conductivity. Independent Fermi-surface, mass, Hall and magnetoresistance constraints can narrow possibilities, while interband optical transitions and coupled scattering can invalidate a simple channel interpretation. Optical spectral weight, a transport lifetime and a superconducting gap are different quantities; no universal cross-material conversion is justified here. R2 R3 R4 R5

Complete scientific acceptance remains open. The Romero author manuscript's printed missing-area prefactor differs from the normalization implied by its Drude numerator; the final paper and calibration must be compared before importing precise condensate fractions. The LaFePO supplement and complete historical sum-rule derivations have not been checked. Independent model integrations do not resolve those source-version questions or reproduce the experimental data. R4 R5 R6 R7

Discovery implications

First specify what would distinguish two physical channels from an effective two-component fit. Extend the measurement window where possible, constrain the DC limit, document tensor geometry, and test whether fitted weights and rates remain stable when backgrounds or cutoffs change. For superconducting spectral transfer, compare a justified normal-state reference with the regular superconducting response and quantify missing bandwidth. A successful transport decomposition constrains candidate mechanisms; it does not identify a unique pairing interaction. R4 R5

Related topics

Planckian dissipation, Multiband pairing, and MgB2.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.