Transport, Response, and Optics

Transport and optical response describe how a material reacts to an electric field, a temperature gradient or another applied disturbance. The measured response depends on both the material and the experiment: a bulk conductivity, a two-terminal conductance and a transmitted optical field are different quantities. This chapter connects the appropriate microscopic description to each of them.

Helpful background. Band theory supplies occupied states, velocities and effective masses. Correlation functions and linear response introduce causal response and spectral functions. Conventions fixes Fourier, charge and tensor signs. The measurement guide explains how sample geometry and detector response enter a comparison with theory.

Enter this chapter

Choose the article that addresses the quantity you want to understand. The methods have overlapping domains, but they do not form a ladder from less to more accurate physics.

Question	Article	Main idea
How does a relaxing carrier current respond to a field?	Drude theory	A one-time-scale benchmark for dc, ac and magnetic response.
How do band occupations and collisions produce current?	Boltzmann transport	A distribution evolves along semiclassical trajectories and through a collision operator.
How is conductivity calculated from a Hamiltonian?	Kubo formula in materials	Current vertices, contact terms and correlation functions must describe the same Hamiltonian.
What sets the transverse voltage?	Hall effect	Carrier dynamics, multiple bands, Berry curvature and quantized phases give different limits.
Why does longitudinal resistance change in a field?	Magnetoresistance	Tensor inversion, scattering, multiple carriers and current geometry must be distinguished.
What does an optical spectrum measure?	Optical conductivity	Dissipation, intraband and interband weight, and the role of spectral sums.
How do charges screen a perturbation?	Dielectric response	Polarization, screened interactions and loss depend on dimensionality and boundaries.
How are charge and heat flow coupled?	Thermoelectric response	Open-circuit conditions, Seebeck and Peltier effects, and electronic and lattice heat currents.
How do reservoirs drive a coherent conductor?	Landauer–Büttiker formalism	Transmission probabilities connect incoming reservoir occupations to terminal currents.
What information is in current fluctuations?	Current noise	Thermal noise, partition noise and detector ordering reveal information beyond the mean current.
What changes beyond linear response?	Nonlinear transport and optics	Frequency mixing and symmetry constrain nonlinear currents, with heating and contacts as possible alternatives.

From device resistance to material conductivity

A small dc voltage and current define a terminal resistance $R=V/I$ and conductance $G=1/R$. A local material description instead relates fields and current densities:

$$\mathbf j=\boldsymbol\sigma\mathbf E, \qquad \mathbf E=\boldsymbol\rho\mathbf j, \qquad \boldsymbol\rho=\boldsymbol\sigma^{-1}.$$

The inversion is a matrix operation. For example, an isotropic planar response with a Hall component obeys

$$\begin{aligned} \boldsymbol\rho&= \begin{pmatrix}\rho_{xx}&\rho_{xy}\\-\rho_{xy}&\rho_{xx}\end{pmatrix},\\ \sigma_{xx}&=\frac{\rho_{xx}}{\rho_{xx}^2+\rho_{xy}^2}. \end{aligned}$$

Thus a change in longitudinal conductivity need not mean the inverse change in longitudinal resistivity. The measured transverse voltage also needs a declared current direction and contact polarity; Hall measurements explains those signs.

For a homogeneous rectangular bar at zero magnetic field, uniform local current gives $R=\rho L/A$, with voltage-probe separation $L$ and cross-sectional area $A$. This conversion fails when contacts, spreading currents or spatial inhomogeneity dominate. In a thin film, the intrinsic sheet current has units of amperes per metre and is related to the in-plane field by a sheet conductance. Converting it to a three-dimensional conductivity requires an appropriate conducting thickness; the sheet quantity itself does not.

A short coherent constriction is described naturally by terminal transmissions. Even a perfectly transmitting channel has a finite two-terminal resistance because the voltage is imposed between reservoirs. It need not have a meaningful local bulk resistivity. Büttiker’s multi-terminal formulation makes the role of contacts explicit. Mesoscopic physics compares the sample size with mean free path and coherence length, while transport measurements discusses practical geometry.

Distributions, correlations and relaxation

Drude theory assumes that one current-carrying degree of freedom relaxes exponentially. Boltzmann theory resolves occupations throughout a band and the collisions that redistribute them. The latter requires well-defined quasiparticles and a suitable semiclassical regime; it does not follow merely from having a crystal lattice. Scattering through a small angle can broaden a single-particle spectrum while relaxing current weakly, so a spectral lifetime is not automatically a transport lifetime.

Kubo’s response formula relates a small perturbation to equilibrium correlations of the appropriate operators. That structure does not require a quasiparticle distribution. Calculating its correlation function may nevertheless require approximations. In a periodic model, the current operator, its field-dependent contact term and the Hamiltonian must be constructed consistently; dropping one of them can violate gauge invariance or a spectral sum rule. The general derivation is in Kubo formula.

Relaxation also depends on conservation laws. Electron collisions that conserve total momentum cannot alone relax every momentum-carrying current. On a lattice, current need not be proportional to momentum, and Umklapp, disorder and boundaries change the conclusion. Which component survives is a question about the actual model, not about the number of collisions.

A finite isolated quantum system has discrete absorption lines. Replacing each by a Lorentzian of width $\eta$ helps display the spectrum, but does not supply a microscopic scattering process. A bulk dc calculation must specify how system size, frequency and the regulator approach their limits, and separate any zero-frequency delta weight from the regular response. Zero resistance also does not by itself establish superconductivity: superfluidity and superconductivity adds phase stiffness and magnetic screening, while vortices explains dissipation from moving flux in the mixed state.

Combining dc, Hall and optical measurements

Suppose a film is approximately a single isotropic parabolic carrier system with a constant transport time. With signed charge $q$, the Drude model predicts

$$\sigma(\omega)=\frac{nq^2\tau/m^*}{1-i\omega\tau}, \qquad R_H=\frac{1}{nq}.$$

Here $R_H=E_y/(j_xB_z)$ with transverse current zero, and fields vary as $e^{-i\omega t}$. Dc data alone constrain the combination $n\tau/m^*$. A resolved complex optical response can separate its weight $n/m^*$ from the time scale $\tau$. A Hall measurement then estimates $n$ only if the single-carrier model and its unit Hall factor apply. An energy-dependent scattering time or multiple bands can change that inference.

For example, take an electron-like Hall coefficient $R_H=-6.242\times10^{-9}\,\mathrm{m^3/C}$, dc conductivity $10^6\,\mathrm{S/m}$, and optical time $10\,\mathrm{fs}$. Within the stated model,

$$\begin{aligned} n&\simeq10^{27}\,\mathrm{m^{-3}},\\ m^*&=\frac{ne^2\tau}{\sigma_0}\simeq0.282m_0. \end{aligned}$$

The agreement is a test of a joint model. It does not establish a microscopic scattering mechanism. If the frequency range lies entirely below $1/\tau$, the optical width itself may be poorly determined; a good fit can hide that parameter uncertainty.

Measured reflectance and transmission are not conductivity directly. They follow from an electromagnetic boundary problem involving the sample, substrate and surrounding media. A density-loss spectrum similarly weights inverse screening rather than the same operator as optical current absorption. Terahertz and infrared probes and dielectric response explain these relations. A shifted peak can result from changing an interface or a probe matrix element as well as from changing an intrinsic excitation.

Exercises

1. A film and a short constriction

A rectangular film at zero magnetic field has voltage-probe separation $L=200\,\mu\mathrm m$, width $W=20\,\mu\mathrm m$, conducting thickness $t=50\,\mathrm{nm}$ and four-terminal resistance $R=100\,\Omega$. Assuming uniform local current flow, find its sheet resistance, bulk resistivity and conductivity. Would the same conversion be justified for a short coherent two-terminal constriction with the same $V/I$?

Solution

The geometry gives

$$\begin{aligned} R_\square&=R\frac{W}{L}=10\,\Omega,\\ \rho&=R_\square t=5\times10^{-7}\,\Omega\,\mathrm m,\\ \sigma&=2\times10^6\,\mathrm{S/m}. \end{aligned}$$

For the constriction, the measured quantity includes the connection to reservoirs and its transmission eigenvalues. Equal values of $V/I$ need not imply equal material resistivity. The local conversion would require separate justification of current homogeneity and a meaningful interior electric field.

2. Equal conductance, different current noise

Consider coherent elastic conductors at zero temperature and small nonzero bias, with energy-independent transmissions across that bias window. Every channel below includes its spin label. Compare channel sets $(1,1)$ and $(1/2,1/2,1/2,1/2)$ using

$$G=\frac{e^2}{h}\sum_nT_n, \qquad F=\frac{\sum_nT_n(1-T_n)}{\sum_nT_n}.$$

What does the noise add to the conductance measurement? Does it determine all transmissions in an arbitrary conductor?

Solution

Both sets give $G=2e^2/h$. The first has $F=0$: fully transmitted incoming particles produce no partition noise. The second has $F=1/2$. For the conventional low-frequency noise normalization $S_I=2e|I|F$, its noise is $e|I|$. These are finite-bias partition fluctuations; at zero temperature and exactly zero bias this excess noise vanishes.

Conductance fixes one sum of transmission probabilities, while noise supplies a second moment. Two moments do not determine arbitrarily many channel transmissions. Temperature, energy dependence, interactions and detector normalization introduce further requirements, developed in current noise and the review by Blanter and Büttiker.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976). Band transport, collisions and the Drude–Sommerfeld approximation.
• Y. M. Blanter and M. Büttiker, “Shot noise in mesoscopic conductors,” Physics Reports 336, 1–166 (2000), doi:10.1016/S0370-1573(99)00123-4. Transmission statistics and current fluctuations.
• M. Büttiker, “Four-terminal phase-coherent conductance,” Physical Review Letters 57, 1761–1764 (1986), doi:10.1103/PhysRevLett.57.1761. Terminal response and reciprocity in coherent conductors.
• M. Dressel and G. Grüner, Electrodynamics of Solids: Optical Properties of Electrons in Matter (Cambridge University Press, 2002). Complex optical response and its relation to measured fields.
• R. Kubo, “Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems,” Journal of the Physical Society of Japan 12, 570–586 (1957), doi:10.1143/JPSJ.12.570. The equilibrium-correlation formulation of linear response.