Transport, Response, and Optics

A measured voltage is not yet a material conductivity, a bright optical feature is not yet a pole, and a successful fit is not yet a microscopic mechanism. Transport and spectroscopy connect a prepared sample to a drive, a current or field, a detector, and an inference model. Changing any one of those objects can change which response function is meaningful.

This gateway owns that selection problem. Given a material state, source, geometry, scale hierarchy, and requested observable, it routes to the narrowest valid material framework and its measurement owner. It also states when a bulk coefficient must give way to terminal conductance, when a kinetic theory must give way to a correlation function, and when a response claim must stop at the level of a calibrated spectrum.

Helpful background. Use Band Theory and Electronic Structure for fillings, velocities, masses, and band versus quasiparticle energies. Use Correlation Functions and Linear Response for operators, retarded correlators, causality, spectral representations, and limit order. Use Conventions for Quantum Matter for signed charge, Fourier, tensor, unit, and field conventions. Use How Quantum Matter Is Measured for the chain from recorded data to a bounded physical claim. None is a universal hard prerequisite: repair only the capability needed by the chosen branch. Use Superfluidity and Superconductivity when zero resistance, condensate spectral weight, Meissner screening, or pair transport becomes a phase claim; this gateway retains response-object and measurement routing. Vortex Matter, Pinning, and Flux Flow owns the force–velocity, electric-field, pinning, creep, and driven mixed-state interpretation when mobile vortices generate the response; this gateway still owns selection of the response object, limit order, and transport or optical handoff.

Enter This Chapter

Three material pages are currently substantive.

• Drude Theory owns the one-relaxation-time bulk benchmark, dc and ac tensors, optical weight, parameter identifiability, and failure tests.
• Boltzmann Transport owns band-resolved semiclassical distributions, collision operators, conservation tests, and electrical and thermal moments.
• Hall Effect owns ordinary, multiband, anomalous, and quantized transverse-response routing with explicit sign and tensor discipline.

Eight narrower material treatments remain planned: Kubo implementation in periodic materials, magnetoresistance, optical conductivity, dielectric response with screening and loss, thermoelectric response, coherent Landauer–Büttiker transport, current noise, and nonlinear transport and optical response. Until each page is substantive, the routes below use live canonical owners and name the remaining coverage boundary. Sidebar order is a catalog, not a prerequisite or accuracy ladder.

Readiness Check

Before choosing a formula, answer five questions.

What is driven? Name the material phase, filling, temperature, field, disorder, dimensionality, boundaries, and preparation. A clean three-dimensional crystal, a disordered film, and an open mesoscopic device do not share one transport limit.

What is the source? Distinguish an electric field, vector or scalar potential, chemical-potential bias, temperature gradient, optical pulse, magnetic field, and nonlinear waveform. A terminal voltage is not automatically the same object as a uniform internal electric field.

What is detected? Name charge current, heat current, polarization, density, terminal current, voltage, transmitted field, emitted field, or a noise spectrum. State the contacts, axes, thickness, and detector bandwidth.

What scale hierarchy applies? Compare sample dimensions with mean free paths, coherence and thermal lengths; compare frequency with scattering and interband scales; and compare drive amplitude with the linear-response window.

What output is requested? Resistance, conductance, resistivity, conductivity, thermopower, dielectric response, loss function, noise, and a nonlinear tensor are different objects. Choose one before choosing a method.

If one answer is missing, repair that input. Do not select Kubo because it sounds more quantum, Boltzmann because the sample is a solid, or Landauer because the device is small.

Write the Response-Claim Ledger

Record ten fields before interpreting transport or optical data.

1. Material background. State phase, dimension, boundaries, disorder, interactions, lattice, and dielectric or magnetic environment.
2. State and preparation. State filling or density, chemical potential, temperature, field, magnetic state, ensemble, and nonequilibrium preparation.
3. Source. Name the electric field, scalar or vector potential, thermal gradient, terminal bias, optical field, strain, or other drive, including amplitude, momentum, frequency, and switch-on convention.
4. Detector and output. Name charge, particle, energy, heat, or spin current; polarization or density; terminal voltage or current; emitted field; or the other measured quantity.
5. Geometry and conventions. Fix bulk, sheet, wire, or terminal geometry; axes, contacts, thickness, signed charge, Fourier sign, tensor order, units, and open- or short-circuit conditions. Distinguish resistance, conductance, resistivity, conductivity, impedance, thermopower, thermal conductivity, dielectric response, loss, noise, and nonlinear tensors.
6. Regime and scale hierarchy. Declare linear or nonlinear drive, momentum, frequency, temperature, magnetic field, disorder, coherence, sample size, resolution, and ratios such as $\omega\tau$, $\omega_c\tau$, $L/\ell$, and $L/L_\phi$.
7. Framework and closure. Name the Drude closure, Boltzmann collision operator, Kubo kernel, dielectric or Maxwell problem, scattering matrix, master equation, or nonlinear expansion, and say why it is controlled.
8. Relaxation and conservation. State which current, momentum, particle, and energy modes relax or remain conserved; identify collision, vertex, contact, lead, reservoir, and magnetization-current treatments.
9. Limits and forward model. Declare the order of $L\to\infty$, $\omega\to0$, $\mathbf q\to0$, and regulator removal, then connect the intrinsic response to terminal voltages, transmitted or reflected fields, detector loading, backgrounds, and resolution.
10. Audit. Record parameter provenance, units, continuity, gauge invariance, causality and Kramers–Kronig consistency, passivity, Onsager–Casimir reciprocity under field and magnetization reversal, positivity, sum rules, heating, uncertainty, credible alternatives, and a falsification or escalation test.

A defensible claim has the form: “for this sample, geometry, source, scale window, and forward model, this response framework accounts for the declared observable to the stated accuracy.” It does not silently identify a unique microscopic mechanism.

Distinguish the Response Objects

For a two-terminal device in linear dc response, $V=RI$ and $I=GV$ define resistance and conductance. A nonlinear record instead distinguishes chord quantities such as $V/I$ from differential quantities such as $dV/dI$; a finite-frequency terminal record uses complex impedance or admittance. For a spatially resolved bulk description in local linear response, $\mathbf j=\boldsymbol\sigma\mathbf E$ and $\mathbf E=\boldsymbol\rho\mathbf j$ define conductivity and resistivity tensors. Geometry relates these pairs only in a justified local, homogeneous regime. In a Hall geometry, tensor inversion is essential: generally $\rho_{xx}\ne1/\sigma_{xx}$.

Intrinsic two-dimensional sheet conductance and sheet resistance require no thickness. A thickness is required only when converting between a sheet quantity and an effective three-dimensional coefficient. Contact resistance can dominate a two-terminal record while being absent from an ideal bulk resistivity. A coherent finite conductor can therefore have a well-defined terminal conductance even when a local conductivity is not the useful description.

At finite frequency, complex conductivity, dielectric response, reflectance, transmittance, absorbance, and a specified longitudinal loss quantity—such as a scalar inverse dielectric response, an inverse-matrix element, or a probe-weighted contraction—are related through an electrodynamic boundary problem but are not interchangeable. The probe owner must perform that inversion.

Read the Dependency Graph

The central formalisms are parallel regime choices.

1. Effective-mass and carrier information can feed the one-fluid Drude benchmark.
2. A controlled quasiparticle distribution and collision closure lead to Boltzmann transport. Drude is one reduction, not a mandatory first course.
3. A declared Hamiltonian, density and current operators, source, and state lead to the Kubo Formula. Material implementation must then preserve contact terms, gauge identities, limit order, and sum rules.
4. Ordinary and anomalous Hall problems branch according to carrier dynamics, Berry geometry, scattering, magnetization, and field regime. They do not all descend from one Boltzmann calculation.
5. An irreducible polarization or density response together with the bare Coulomb kernel defines the dielectric matrix. Its inverse then produces the screened interaction and longitudinal loss used in collective-mode tests.
6. A coherent open conductor connected through ideal leads to reservoirs and described by a scattering matrix leads to Landauer–Büttiker terminal transport. It is not a more accurate Boltzmann theory of the same object.
7. Equilibrium noise branches from fluctuation–dissipation; partition noise branches from transmission statistics; sequential-tunneling noise branches from open-system counting dynamics.
8. Nonlinear response begins only after a drive-amplitude window and symmetry data are declared, and after heating and contacts are treated as competing explanations.

Choose the Shortest Live Route

Low-frequency one-fluid benchmark

Use Drude Theory when one effective carrier fluid and a momentum-relaxation time are a controlled or deliberately phenomenological reduction. It separates $n/m^*$ spectral weight from $\tau$, treats tensor and Hall variants, and states where multiband, incoherent, nonlocal, or coherent-device physics invalidates the model.

Band-resolved occupations and collisions

Use Boltzmann Transport when quasiparticle occupations, semiclassical trajectories, and a collision operator are meaningful. It owns scattering-in terms, conservation-aware linearization, electrical and thermal moments, magnetic histories, and the limits of a constant relaxation time. A finite one-particle lifetime is not automatically the transport lifetime.

Transverse response

Use Hall Effect for ordinary, multiband, anomalous, and quantized transverse-response taxonomy. Pair it with Hall Measurements for signed terminal acquisition, parity reduction, contact misalignment, hysteresis, and uncertainty. Hall sign is mobility weighted in multiband systems and is not a direct pocket census.

General quantum linear response

Use Correlation Functions and Linear Response to select the operator pair, retarded object, spectral representation, and equilibrium assumptions. Then use the canonical Kubo Formula and Density and Current Operators. The future material bridge will own periodic-Hamiltonian current vertices, contact terms, intraband and interband separation, finite-size regularization, and conservation audits; it will not rederive Kubo theory.

Complex optical response

For an intrinsic free-carrier baseline, begin with Drude. For the exact linear-response structure, add Kubo and Sum Rules. Terahertz and Infrared Probes owns transmission and reflection inversion, substrate and thickness effects, calibration, and resolution. The planned Optical Conductivity page will own material intraband/interband weight and bounded spectral-weight transfer claims, not the measured-intensity forward model.

Screening, dielectric response, and loss

Use Susceptibilities for response conventions and Random-Phase Approximation for the standard screening closure. Given an irreducible polarization $\boldsymbol\Pi$ and bare Coulomb kernel $\mathbf v$, one common convention is $\boldsymbol\epsilon=\mathbf 1-\mathbf v\boldsymbol\Pi$ and $\mathbf W=\boldsymbol\epsilon^{-1}\mathbf v$; matrix ordering and signs must follow the declared convention. A longitudinal loss function is the appropriate scalar, inverse-matrix element, or probe-weighted contraction of $-\operatorname{Im}\boldsymbol\epsilon^{-1}$, not an unspecified scalar in every geometry. Plasmons Preview owns collective-mode existence, poles, continua, and damping. A plasmon claim requires a dielectric eigenvalue that vanishes at the generally complex mode frequency—equivalently $\det\boldsymbol\epsilon=0$ in a finite or truncated representation—and a compatible pole of inverse dielectric or density response with nonzero weight, plus damping, continuum, boundary, and probe checks. A zero of $\operatorname{Re}\epsilon$ or a loss peak alone is not enough. The future material page will unite this structure with macroscopic extraction, local-field effects, static and dynamic screening, conductivity relations, and loss inference.

Magnetoresistance mechanism audit

Begin with Drude, Boltzmann, and Hall for tensor and multiband baselines. Add Weak Localization for interference. For a magnetic contribution, enter Magnetism and Spin Systems before using Spintronics for spin-dependent transport. Use Weyl and Dirac Semimetals for anomaly claims. Transport Measurements owns contacts, geometry, current homogeneity, field parity, and raw-data reduction. Negative longitudinal magnetoresistance alone proves none of those mechanisms.

Thermoelectric response

Use Boltzmann for semiclassical electronic moments, Transport Coefficients Preview for general coupled-current and limit conventions, and Phonons for the lattice heat channel. The future material page will own open-circuit Seebeck, Peltier and Kelvin relations, Nernst conventions, drag and bipolar effects, open-circuit thermal conductivity, and the full figure-of-merit heat budget.

Coherent multi-terminal transport

Use What Is Mesoscopic Physics? to audit wavelength, mean-free-path, coherence, thermal, and contact scales. Use Conductance Quantization for plateau physics and transmission eigenchannels, the canonical S-Matrix owner for scattering-operator conventions and unitarity, and Mesoscopic Transport for sequential tunneling, master equations, and counting statistics. The planned Landauer–Büttiker page will own coherent elastic multi-terminal current matrices, ideal leads, thermalizing reservoirs, voltage probes, Onsager–Casimir reciprocity under field and magnetization reversal, and heat currents.

Current noise

Use Fluctuation–Dissipation Theorem for equilibrium constraints, Noise Spectra for ordered and symmetrized spectral conventions, Conductance Quantization for partition noise, and Mesoscopic Transport for counting statistics. A Fano factor or fitted effective charge is not unique evidence for fractionalization.

Nonlinear transport and optics

Start from Hall Effect and Symmetry of Bloch States, then pair the theory with Transport Measurements or Terahertz and Infrared Probes. A nonlinear material claim requires a reproducible amplitude-scaling window, frequency bookkeeping, and separation from Joule heating, contact rectification, state changes, and surface contributions. The local synthesis page remains planned.

Material Specializations and Canonical Owners

The eight planned leaves have bounded roles: periodic-material Kubo implementation; magnetoresistance mechanism comparison; intrinsic optical conductivity; dielectric response, screening, and loss; thermoelectric inference; coherent multi-terminal Landauer–Büttiker transport; current noise; and nonlinear transport and optical response. These entries are marked Planned; use the substantive treatments above for the corresponding physics until these material specializations are written.

General response theory belongs in the Many-Body response chapter. Material conductivity is developed through the controlled Drude, Boltzmann, Kubo, Hall, and optical realizations. Dielectric response brings screening and loss into the same material description. Coherent multi-terminal scattering belongs with Landauer–Büttiker transport; mesoscopic platforms, localization, Kubo response, and open-system transport retain their distinct questions.

Worked Routing Audit: DC, Hall, and THz on One Film

Suppose a conducting film is measured with four-terminal dc transport, a Hall bar, and complex terahertz transmission. A simultaneous Drude fit returns a carrier density, mass, and lifetime. Those three numbers are not automatically three independently measured material properties.

Declare geometry and observables. Record film thickness and uncertainty, contact arrangement, current direction, Hall polarity, substrate stack, temperature, magnetic field, and THz frequency window. Keep sheet conductance separate from three-dimensional conductivity. Reduce raw terminal records with the transport and Hall probe owners, and invert the optical stack with the THz owner.

Identify parameter combinations. In the one-fluid Drude model, dc data measure $nq^2\tau/m^*$, while low-frequency optical weight constrains $n/m^*$. Write the weak-field Hall coefficient as $R_H=r_H/(nq)$. It reduces to $1/(nq)$ only when the Hall factor is known to be unity, for example in the appropriate isotropic constant-$\tau$ Drude limit. Multiband compensation, mobility weighting, anisotropic masses or scattering, inhomogeneity, thickness error, or a frequency-dependent memory function can reproduce different parts of the same data.

Route the mechanism. Use Drude as the falsifiable baseline. Use Boltzmann if band-resolved velocities and collision kernels are controlled. Use Kubo when coherence, interband response, vertex corrections, or strong interactions make the distribution closure inadequate. Check optical sum rules, Hall tensor inversion, field and temperature dependence, and independent band or density information.

The bounded conclusion reports which combinations are identified, which parameters depend on the assumed model, and which alternative survives. A numerically excellent joint fit does not by itself prove a unique $n$, $m^*$, or microscopic scattering mechanism.

Worked Regime Audit: Bulk Resistivity or Terminal Conductance?

Suppose the same material is patterned first as a long four-probe bar and then as a short phase-coherent constriction. Both measurements report a voltage divided by a current, but they need not measure the same response object.

Audit the bulk claim. A bulk resistivity interpretation requires a region where local current density and electric field are meaningful, current flow is controlled, and the voltage probes sample the interior rather than the current contacts. Record the cross-section, probe spacing, mean free path $\ell$, phase-coherence length $L_\phi$, inelastic scales, and homogeneity. In a diffusive hierarchy with probe spacing much larger than $\ell$ and, normally, than $L_\phi$, Drude, Boltzmann, or Kubo may supply the intrinsic conductivity. Only with a justified local geometry may one convert the measured four-terminal resistance into a material resistivity.

Audit the coherent-device claim. If the active region is shorter than $L_\phi$ and is connected through specified coherent ideal leads to thermalizing reservoirs, route through Conductance Quantization and the planned Landauer–Büttiker material page. The terminal conductance then depends on transmission eigenvalues, channel degeneracies, contacts, and the reservoir occupations. A contact contribution is part of that open-system observable, not automatically a bulk scattering rate or a resistivity.

Apply the stopping rule. Stop reporting bulk resistivity when local fields, length scaling, cross-section, or contact separation are not controlled. Stop using a purely coherent scattering matrix when dephasing, energy relaxation, or distributed inelastic collisions dominate. Equal numerical values of $V/I$ do not license identifying the two experiments with one material coefficient.

Canonical Boundaries

• Computational Quantum Matter owns the material-facing numerical workflow, convergence, benchmark, uncertainty, and probe-comparison record; this gateway retains response-regime and observable selection.
• This gateway owns material-regime selection, response-object discipline, live fallbacks, and stopping tests. It owns no leaf derivation.
• The Many-Body response chapter owns general correlators, Kubo theory, susceptibilities, fluctuation–dissipation, sum rules, and general transport coefficients.
• Band Theory owns fillings, velocities, masses, and electronic-state inputs. Symmetry and geometry pages own abstract Berry data; topology pages own invariants and quantized phases.
• Probe pages own acquisition, contacts, optical inversion, calibration, backgrounds, detector loading, and measured intensity.
• Mesoscopic pages own platform and plateau physics. Measurement and Open Systems owns generic quantum noise, master equations, and sequential tunneling. Disorder and magnetism pages own their specific mechanisms.
• Plasmon pages own collective-mode existence and damping. Computational volumes own production Kubo, Boltzmann, and scattering-matrix algorithms.

Exit Checkpoint

Before leaving the gateway, you should be able to:

• distinguish terminal, sheet, bulk, longitudinal, transverse, static, dc, optical, and nonlocal response objects;
• name the material state, source, detector, geometry, scale hierarchy, and order of limits;
• choose Drude, Boltzmann, Kubo, dielectric, Landauer–Büttiker, noise, or nonlinear language by regime rather than prestige;
• name the conserved and relaxed quantities and the required contact, collision, vertex, reservoir, or electrodynamic model;
• select the shortest substantive theory and measurement owners;
• apply a causality or Kramers–Kronig, Onsager–Casimir field-reversal, conservation, sum-rule, tensor, heating, or finite-size check;
• identify where the local chapter remains planned; and
• write a bounded response claim with a falsification or escalation condition.

Common Routing Errors

• Conductance is not conductivity, and resistance is not resistivity.
• Intrinsic sheet quantities need a declared two-dimensional convention; thickness is needed only for conversion to an effective three-dimensional coefficient.
• In a magnetic field, $\rho_{xx}$ is generally not $1/\sigma_{xx}$.
• A one-particle lifetime is not automatically a transport lifetime.
• Kubo is exact only as a linear-response statement for the declared Hamiltonian and source; its evaluation can still be approximate.
• A numerical broadening or finite-size line is not an intrinsic optical linewidth or finite dc resistivity.
• Reflectance, absorption, $\sigma_1$, $\epsilon_2$, and a loss function are not synonyms.
• A Hall sign is not a direct multiband carrier census.
• Thermopower sign is not a universal carrier-sign measurement.
• A conductance quantum requires declared channel degeneracy and transmission.
• A Fano factor does not by itself identify fractional charge.
• A nonlinear harmonic does not by itself exclude heating or contact rectification.
• Ballistic, diffusive, hydrodynamic, incoherent, and localized regimes are alternatives, not successive levels of accuracy.

Routing Exercises

1. Choose the framework

Route four problems: a broad low-frequency metal with one relaxation scale; a band-resolved semiconductor with known phonon collision rates; a finite interacting lattice whose current correlator is computed exactly; and a short phase-coherent device attached to reservoirs.

Solution

Use Drude for the first as a one-fluid benchmark. Use Boltzmann for the second because a distribution and collision kernel are controlled. Use Kubo for the third, with current and contact terms derived from the same Hamiltonian and with finite-size and limit-order audits. Use Landauer–Büttiker language for the fourth if coherent elastic scattering and reservoir occupations are justified. The four methods address different state descriptions and boundary conditions; they are not an accuracy ranking.

2. Convert a device record

A rectangular film has length $L$, width $W$, thickness $t$, and measured four-terminal resistance $R$. What additional assumptions license a bulk resistivity, and how does the answer change for a coherent two-terminal device?

Solution

For homogeneous local current flow with negligible contact and spreading corrections, the longitudinal bulk resistivity is $\rho=R\,Wt/L$. The sheet resistance is $R_\square=\rho/t=R\,W/L$. State thickness and geometry uncertainties and use the tensor relation in a magnetic field. In a coherent two-terminal device, $G=1/R$ is a terminal quantity containing contacts and transmission. Converting it to a local bulk resistivity can be meaningless; route instead to the mesoscopic regime and scattering framework.

3. Identify Drude parameters

DC conductivity, a weak-field Hall slope, and a THz Drude peak are available for a film. Explain when they determine $n$, $m^*$, and $\tau$, and give two failure modes.

Solution

In a validated one-carrier weak-field model, the Hall slope estimates $R_H=r_H/(nq)$. Only with a known unit Hall factor—for example in the appropriate isotropic constant-$\tau$ Drude limit—does it directly give $1/(nq)$. The optical weight estimates $n/m^*$ and the optical width estimates $1/\tau$; dc conductivity cross-checks their product. The inference fails in a multiband or mobility-weighted Hall response, and it can also fail with anisotropic masses or scattering, inhomogeneous thickness, non-Drude memory, or an optical background that trades spectral weight with the fitted peak. Report the fitted combinations and model dependence rather than three supposedly direct measurements.

4. Take a dc limit

A finite closed lattice calculation replaces every optical delta line by a Lorentzian of width $\eta$ and reports a nonzero $\sigma_{\mathrm{dc}}$. Why is that not yet a bulk resistivity?

Solution

The finite spectrum is discrete, and $\eta$ is a numerical regulator unless a physical bath or scattering process has been defined. One must declare the order of $L\to\infty$, $\omega\to0$, and $\eta\to0$, separate any zero-frequency Drude weight from the regular part, and test size and broadening convergence. A clean momentum-conserving system may retain a delta contribution rather than a finite resistivity. Kubo supplies the response identity; it does not turn an arbitrary broadening into dissipation.

5. Sort four optical objects

Distinguish the real optical conductivity $\sigma_1$, dielectric loss $\epsilon_2$, reflectance, and $-\operatorname{Im}\epsilon^{-1}$.

Solution

The regular positive-frequency part of $\sigma_1$ describes absorption for a passive medium under the declared convention, while a zero-frequency Drude or superfluid delta weight can be nondissipative. $\epsilon_2$ is the imaginary part of dielectric response and is related to conductivity only after units, locality, tensor structure, and frequency conventions are fixed. Reflectance is a boundary-dependent intensity ratio requiring a Maxwell and sample-stack model. A specified scalar, inverse-matrix element, or probe-weighted contraction of $-\operatorname{Im}\boldsymbol\epsilon^{-1}$ weights longitudinal screening channels. None can be substituted for another without the relevant electrodynamic inversion, and a loss maximum alone does not establish a collective pole.

6. Audit negative magnetoresistance

For an in-plane isotropic Hall sample with $\boldsymbol\rho=\begin{pmatrix}\rho_{xx}&\rho_{xy}\\-\rho_{xy}&\rho_{xx}\end{pmatrix}$, invert the tensor. Then explain why a longitudinal magnetoresistance claim must include field parity and a current-jetting test before invoking a Weyl anomaly.

Solution

The inverse is

$$\boldsymbol\sigma = \frac{1}{\rho_{xx}^2+\rho_{xy}^2} \begin{pmatrix} \rho_{xx} & -\rho_{xy}\\ \rho_{xy} & \rho_{xx} \end{pmatrix}.$$

Thus $\sigma_{xx}=\rho_{xx}/(\rho_{xx}^2+\rho_{xy}^2)$ and, for the displayed sign convention, $\sigma_{xy}=-\rho_{xy}/(\rho_{xx}^2+\rho_{xy}^2)$. Tensor inversion separates conductivity changes from Hall mixing, while signed field sweeps expose odd contamination and contact misalignment. Strong field-induced anisotropy can focus current between contacts, producing an apparent negative voltage response without a homogeneous negative bulk magnetoresistance. Vary contacts, geometry, angle, current, and sample aspect ratio, then compare multiband, interference, magnetic, heating, and inhomogeneity alternatives. Only after those tests and independent Weyl-node evidence can an anomaly model become one bounded explanation.

7. Impose an open-circuit condition

Suppose coupled linear response is written schematically as $\mathbf j=L_{11}\mathbf E-L_{12}\boldsymbol\nabla T$ and $\mathbf j_Q=L_{21}\mathbf E-L_{22}\boldsymbol\nabla T$. Derive the open-circuit thermopower and explain why the measured thermal conductivity is not simply $L_{22}$.

Solution

Setting $\mathbf j=0$ gives $\mathbf E=L_{11}^{-1}L_{12}\boldsymbol\nabla T$. For the stated convention $\mathbf E=S\boldsymbol\nabla T$, this means $S=L_{11}^{-1}L_{12}$. Substitution gives $\mathbf j_Q=-\left(L_{22}-L_{21}L_{11}^{-1}L_{12}\right)\boldsymbol\nabla T$, so $\kappa_{\mathbf j=0}=L_{22}-L_{21}L_{11}^{-1}L_{12}$, not $L_{22}$ alone. A material measurement must also separate electronic, lattice, bipolar, drag, contact, and radiation heat channels and state whether magnetization currents have been removed.

8. Combine conductance and noise

At zero temperature, two coherent conductors have the same zero-frequency, infinitesimal-bias conductance $G=2e^2/h$. Does that determine their transmission eigenvalues or shot noise?

Solution

No. At zero temperature, zero frequency, and infinitesimal bias, conductance constrains a sum of Fermi-level transmission eigenvalues, including declared degeneracy, but different sets can share that sum. In the same elastic limit, zero-frequency partition noise adds the moment $\sum_nT_n(1-T_n)$. The same compact moments also describe a finite bias window only when the transmissions are effectively energy independent there; otherwise conductance, current, and noise require energy integrals. Two perfectly transmitted resolved channels have no partition noise, whereas several partial channels can yield the same conductance and finite noise. Detector ordering, temperature, bandwidth, environment, interactions, and circuit loading must be included before inferring an effective charge or Fano factor.

9. Separate nonlinearity from heating

A second-harmonic voltage scales quadratically with drive over one decade. What additional tests are needed before calling it an intrinsic nonlinear Hall or optical response?

Solution

Verify tensor and symmetry selection rules, frequency and phase relations, signed field behavior, contact permutation, thickness and surface dependence, and a reproducible low-amplitude expansion window. Track sample temperature and resistance to exclude Joule heating, and test contact rectification, thermoelectric pickup, state changes, and instrumental mixing. An intrinsic claim also requires a model with the appropriate band, scattering, and Berry or optical matrix elements. Quadratic scaling is necessary for a second-order assignment but is not a unique mechanism identifier.

References

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• R. W. Boyd, Nonlinear Optics, 4th ed. (Academic Press, 2020).
• M. Büttiker, “Four-terminal phase-coherent conductance,” Physical Review Letters 57, 1761–1764 (1986), doi:10.1103/PhysRevLett.57.1761.
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