advanced36 min studyLesson 519 in Physics

Track phase-coherent return paths in a diffusive conductor

Explain weak localization, estimate its coherence length and magnetic scale, and distinguish dephasing from elastic scattering.

Before you begin

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The question for today

What you’ll understand

Explain the interference of time-reversed diffusive loops.
Relate phase-coherence time to length and magnetic scale.
Identify the symmetry assumptions behind a weak-localization magnetoconductance sign.

A conductor can be diffusive and still preserve quantum phase. Elastic disorder changes electron momentum repeatedly without necessarily recording which path the electron took. Amplitudes traveling around a closed sequence of scatterers in opposite orders can therefore interfere. In an ordinary time-reversal-invariant conductor with weak spin-orbit effects, their enhanced return probability slightly suppresses conductance relative to the classical result. This weak-localization correction offers an experimental probe of coherence rather than requiring a perfectly ballistic sample.

Pair a loop with its time reverse

In weak disorder, the elastic mean free path $\ell$ is long compared with the Fermi wavelength but short compared with the distances over which diffusion is observed. Equivalently, $k_F\ell\gg1$ supports a metallic perturbative starting point. A path and its time reverse encounter the same static scatterers in opposite order. With time-reversal symmetry and negligible spin complications, their phases agree at zero field, so the amplitudes add constructively for return to the starting region.

Enhanced return probability means a small reduction of transport away from that region, producing a negative quantum correction to conductivity. Random phases of unrelated long paths usually cancel in the disorder average, but the time-reversed pairing survives. This is not the claim that all electrons stop moving or become permanently bound to individual impurities. Weak localization is a perturbative interference correction; strong Anderson localization involves a different regime where treating that correction as small is no longer justified.

L_\phi=\sqrt{D\tau_\phi},\qquad \tau_{\rm el}\ll\tau_\phi,\qquad \ell\ll L_\phi,\qquad \delta\sigma_{2D}\ \propto\ -\ln\frac{L_\phi}{\ell}

D is the diffusion coefficient in m²/s, τ_phi is phase-coherence time, τ_el is elastic scattering time, and L_phi and ℓ are lengths. The logarithmic dependence is the ordinary two-dimensional weak-localization form over its diffusive scaling range; its coefficient depends on spin, channels, and conventions. Finite sample size can replace L_phi as the large-distance cutoff.↔ Scroll wide equations horizontally.

Distinguish momentum relaxation from dephasing

Elastic collisions with static disorder randomize momentum and establish diffusion, yet need not destroy the relative phase of alternative amplitudes. Inelastic processes and fluctuating environments can make the alternatives distinguishable and limit the phase-coherence time. Since motion between many collisions is diffusive, coherence distance scales as the square root of diffusion coefficient times coherence time, rather than as Fermi velocity times coherence time. That latter ballistic distance would overestimate the size of the coherent diffusive region.

An applied perpendicular magnetic field adds a relative phase between the two loop orientations. The relevant phase is twice the single-path magnetic phase because the paths traverse the enclosed flux in opposite directions. Different loop areas then acquire different phase shifts, suppressing their collective zero-field enhancement. A conventional two-dimensional field scale uses $B_\phi=\hbar/(4eD\tau_\phi)$ . It is a defined crossover parameter of the diffusive interference theory, not a sharp field at which every path suddenly loses coherence.

|\Delta\varphi|=\frac{2e|B\mathcal A|}{\hbar},\qquad B_\phi=\frac{\hbar}{4eD\tau_\phi}=\frac{\hbar}{4eL_\phi^2}

The first expression is the relative phase of a time-reversed pair enclosing signed area 𝒜 in uniform perpendicular field B; its sign depends on orientation and electron charge convention. B_phi is the conventional two-dimensional dephasing field. e is positive elementary charge, and B is in tesla when SI units are used.↔ Scroll wide equations horizontally.

Use magnetoconductance with the correct symmetry class

For the ordinary weak-localization case described here, suppressing a negative zero-field correction raises conductance, giving positive low-field magnetoconductance and negative magnetoresistance. Strong spin-orbit coupling can instead produce weak antilocalization, with the opposite low-field tendency. Magnetic impurities, spin splitting, multiple valleys, and interactions can change the fit. Consequently, a sign claim must specify the symmetry and scattering assumptions rather than treating every low-field feature as the same universal interference effect.

Temperature-dependent narrowing or broadening of the low-field feature can reveal changes in coherence time, but the interpretation requires a suitable dimensionality and field range. Classical orbital magnetoresistance and interaction corrections may coexist. The logarithmic two-dimensional expression also assumes a broad diffusive scale interval and a correction small enough for perturbation theory. A visually sharp conductance cusp is evidence to model, not a standalone measurement of the elastic mean free path or a proof that the sample is ballistic.

Work it through

A micrometre coherence length from picoseconds

A two-dimensional diffusive conductor has $D=0.020\,\mathrm{m^2/s}$ , $\tau_\phi=50\,\mathrm{ps}$ , and $\ell=50\,\mathrm{nm}$ . Estimate its coherence length, the ratio to mean free path, and conventional dephasing field. Use $\hbar=1.05\times10^{-34}\,\mathrm{J\,s}$ and $e=1.6\times10^{-19}\,\mathrm C$ .

The diffusion product is $D\tau_\phi=(0.020)(50\times10^{-12})=10^{-12}\,\mathrm{m^2}$ . Taking its square root gives $L_\phi=10^{-6}\,\mathrm m=1.0\,\mu\mathrm m$ ; the picosecond time alone does not imply a ballistic distance.
The scale ratio is $L_\phi/\ell=1000/50=20$ . Thus many elastic-scattering lengths fit within the coherence length, consistent with diffusive phase-coherent motion. A full weak-disorder check would additionally require the Fermi wavevector and $k_F\ell\gg1$ .
The conventional field is $B_\phi=(1.05\times10^{-34})/[4(1.6\times10^{-19})(10^{-12})]=1.64\times10^{-4}\,\mathrm T=0.164\,\mathrm{mT}$ . This is a crossover scale, not a universal hard dephasing threshold.

The result

The coherence length is 1.0 μm, twenty times the elastic mean free path, and the conventional dephasing field is 0.164 mT. These numbers support a coherent diffusive regime if the remaining weak-disorder and sample-size conditions hold.

The living notebook

Explore the model

Interactive lab

Interactive lab · waves

Expose sensitivity to a relative phase

Reset and set both controls to phase difference 0° and amplitude ratio 1; predict resultant amplitude 2. Reset and set both controls to phase difference 180° and amplitude ratio 1; predict cancellation with resultant amplitude 0. Play or Step to compare the sums. Both component waves have equal fixed frequency and propagate in the same direction. This is a phase-interference analogy, not literal time-reversed electron paths, a magnetic-loop calculation, or a conductivity model.

SumFirst componentSecond component

Sum at x = 0: 0.866
Resultant amplitude: 1.732
Phase difference: 60°

Phase difference60°Amplitude ratio1

t = 0 s

Two waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forRelative phase can change an amplitude sum dramatically; mapping that effect to conductance requires a transport theory and its symmetry assumptions.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Static elastic scattering can preserve phase relationships. Multiple scattering creates the time-reversed loops responsible for weak localization, while dephasing is controlled by additional processes and occurs on a longer scale in the stated regime.

Your turn

Try it yourself

Keep D and the elastic mean free path from the example fixed, but reduce the coherence time by a factor four. Find the new coherence length and dephasing field. In the stated two-dimensional weak-localization regime, does the magnitude of the logarithmic correction fall by a factor four?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Length scales as the square root of time, field as its inverse, and the conductivity correction depends logarithmically on length.

Compare with the solution

The new time is 12.5 ps, so $L_\phi$ halves to $0.50\,\mu\mathrm m$ and $B_\phi$ grows fourfold to $0.656\,\mathrm{mT}$ . The length ratio to ℓ falls from 20 to 10. If $\delta\sigma=-C\ln(L_\phi/\ell)$ with the same positive coefficient C, the correction changes from $-C\ln20$ to $-C\ln10$ , becoming less negative by $C\ln2$ . Its magnitude is not reduced by four; neither is the total conductance, which includes the classical background.

Keep the idea

Take it with you

Diffusion and quantum phase coherence can coexist.
Time-reversed return paths yield a symmetry-dependent interference correction.
Coherence length, magnetic crossover, and logarithmic conductivity change scale differently.

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