Weak Conductance in Quantum Systems

• Weak conductance is a descriptor for transport regimes where small quantum corrections, such as weak antilocalization and conductance fluctuations, modify classical conductance.
• It encompasses diverse scenarios including weak tunnel junctions with Nyquist noise, weakly interacting Luttinger liquids, and weakly coupled quantum dots, each with distinct temperature and scaling behaviors.
• Understanding weak conductance informs the design of nanoelectronic devices by elucidating phase-coherent transport, quantum interference imaging, and the impact of weak perturbations on device performance.

Searching arXiv for the cited papers to ground the article in published sources. arXiv search query: (Matsuo et al., 2012) weak antilocalization conductance fluctuation Bi2Se3 arxiv_search({"query":"(Matsuo et al., 2012)"}) In the literature represented here, conductance is analyzed in regimes characterized by weak antilocalization and conductance fluctuations, a weak tunnel junction, weakly interacting Luttinger liquids, and a quantum dot weakly coupled to a conducting channel. These regimes are not unified by a single microscopic mechanism; rather, they are linked by the fact that conductance is controlled by small quantum corrections, weak tunneling, weak coupling, or weak perturbations superposed on a larger transport background. The resulting observables include low-field magnetoconductance corrections, zero-bias conductance suppression, temperature-dependent transmission through impurities, and scanning-gate conductance maps (Matsuo et al., 2012, Mošková et al., 2016, Das et al., 2018, Kolasiński et al., 2014).

1. Regimes in which conductance is controlled by weak effects

The sources describe four distinct settings.

Setting	Weak element	Conductance quantity
Epitaxial Bi$_2$Se$_3$ wire	Weak antilocalization and universal conductance fluctuation	$\Delta \sigma(B)$, $B_c(T)$, $\delta G_{\rm rms}(T)$
Tunnel junction in resistor environment	Weak tunnel junction with Nyquist noise	$G(T)$, $G/G_t$
Inhomogeneous Luttinger liquid	Weakly interacting limit and weak barrier	$G(T)=G_0\,\tau(T)$
Side-coupled quantum dot	Weak coupling to channel and weak tip perturbation	$G(x_t,y_t)$, $\delta T$

In the Bi$_3$0Se$_3$1 system, weak conductance phenomena appear as “small, but experimentally resolvable, quantum corrections to the classical Drude conductance.” In the tunnel-junction problem, the defining assumption is a conductance much smaller than $_3$2. In the Luttinger-liquid problem, the weakly interacting limit is the regime in which the Matveev–Yue–Glazman result is recovered, while the full non-chiral bosonization technique extends beyond that limit. In the conductance-microscopy problem, weak coupling means that the connection of the channel to the dot transmits a single transport mode only, and the tip is treated as a weak, delta-like perturbation (Matsuo et al., 2012, Mošková et al., 2016, Das et al., 2018, Kolasiński et al., 2014).

A plausible implication is that “weak conductance” is best understood as a transport descriptor tied to perturbative or quasi-perturbative control parameters, rather than as a synonym for simply small absolute conductance.

2. Phase-coherent weak conductance corrections in a Bi$_3$3Se$_3$4 wire

A sub-micrometer-sized Hall bar made of epitaxial Bi$_3$5Se$_3$6 thin film was used to probe phase coherent transport below 22 K. The geometry was a wire of width $_3$7 nm, thickness $_3$8 nm, and length $_3$9. Two signatures were examined: weak antilocalization (WAL) and universal conductance fluctuations (UCF) (Matsuo et al., 2012).

For WAL, the magnetoconductance correction is described by the Hikami–Larkin–Nagaoka formula

$\Delta \sigma(B)$0

with

$\Delta \sigma(B)$1

Here, $\Delta \sigma(B)$2 is a dimensionless prefactor and $\Delta \sigma(B)$3 is the phase coherence length. In experiment, $\Delta \sigma(B)$4 is obtained by converting the measured resistance $\Delta \sigma(B)$5 to conductance and subtracting its zero-field value, and the fit is performed over the low-field range $\Delta \sigma(B)$6, where $\Delta \sigma(B)$7 and $\Delta \sigma(B)$8 is the elastic mean free path. For the Bi$\Delta \sigma(B)$9Se$B_c(T)$0 wire, fitting at $B_c(T)$1 K yields $B_c(T)$2. The extracted $B_c(T)$3 grows from $B_c(T)$4 at 22 K to $B_c(T)$5 at 2 K (Matsuo et al., 2012).

The temperature scaling of $B_c(T)$6 is used to infer transport dimensionality. Theory predicts $B_c(T)$7 with $B_c(T)$8 for two-dimensional Nyquist dephasing and $B_c(T)$9 for one-dimensional dephasing. In this wire, a log-log plot of $\delta G_{\rm rms}(T)$0 versus $\delta G_{\rm rms}(T)$1 yields $\delta G_{\rm rms}(T)$2 above $\delta G_{\rm rms}(T)$3 K, close to the one-dimensional expectation $\delta G_{\rm rms}(T)$4. Physically, once $\delta G_{\rm rms}(T)$5 exceeds the wire width $\delta G_{\rm rms}(T)$6 at $\delta G_{\rm rms}(T)$7 K, the system crosses over to quasi-1D and further coherent transport occurs predominantly along the wire axis (Matsuo et al., 2012).

The UCF analysis gives an independent coherence-length estimate. Small changes in magnetic field scramble interference patterns and produce reproducible conductance fluctuations $\delta G_{\rm rms}(T)$8 of order $\delta G_{\rm rms}(T)$9. The correlation field $G(T)$0 is defined through the autocorrelation $G(T)$1, and for a 1D diffusive wire one finds

$G(T)$2

For a 1D wire at finite temperature, if $G(T)$3 and $G(T)$4, theory gives

$G(T)$5

with $G(T)$6. Experimentally, the autocorrelation yields $G(T)$7 and hence $G(T)$8 consistent with WAL, while $G(T)$9 scales as $G/G_t$0, in excellent agreement with the $G/G_t$1 law for 1D UCF (Matsuo et al., 2012).

These results establish quasi-one-dimensional phase-coherent transport below approximately 22 K. The same work states that quasi-1D topological-insulator wires open routes to Aharonov–Bohm oscillations of Dirac surface modes, one-dimensional Majorana bound states when proximitized by a superconductor, and other interference-based spintronic functionalities relying on long $G/G_t$2 (Matsuo et al., 2012).

3. Zero-bias conductance of a weak tunnel junction with Nyquist noise

A different weak-conductance problem concerns the Coulomb blockade of a tunnel junction with conductance much smaller than $G/G_t$3, capacitance $G/G_t$4, and a series resistance $G/G_t$5 producing Nyquist noise. The single-electron charging energy is

$G/G_t$6

In the semiclassical regime

$G/G_t$7

the resistor’s phase fluctuations charge the junction with a random offset charge $G/G_t$8 whose steady-state distribution is Gaussian (Mošková et al., 2016).

The zero-bias conductance is written in orthodox theory as

$G/G_t$9

In the semiclassical high-$G(T)=G_0\,\tau(T)$0 limit,

$G(T)=G_0\,\tau(T)$1

With

$G(T)=G_0\,\tau(T)$2

the conductance becomes

$G(T)=G_0\,\tau(T)$3

where

$G(T)=G_0\,\tau(T)$4

Since $G(T)=G_0\,\tau(T)$5 as $G(T)=G_0\,\tau(T)$6 and remains $G(T)=G_0\,\tau(T)$7 for all $G(T)=G_0\,\tau(T)$8, the leading temperature dependence is

$G(T)=G_0\,\tau(T)$9

The formula is stated to be valid for

$G(x_t,y_t)$0

and also reproduces earlier asymptotic forms (Mošková et al., 2016).

The comparison with earlier results is explicit. For $G(x_t,y_t)$1, Joyez–Esteve give

$G(x_t,y_t)$2

For $G(x_t,y_t)$3 but still $G(x_t,y_t)$4, Averin and Odintsov give

$G(x_t,y_t)$5

The present result is stated to hold for all $G(x_t,y_t)$6 subject to the Nyquist-noise condition above (Mošková et al., 2016).

The factor of $G(x_t,y_t)$7 in the activation energy is a central physical point. If one ignored environmental fluctuations and took $G(x_t,y_t)$8, the activation would be $G(x_t,y_t)$9. Including Nyquist noise broadens $\delta T$0 to a Gaussian and shifts the most probable energy cost to $\delta T$1; physically, the resistor’s charge fluctuations “help” the tunneling. Numerical data for $\delta T$2 and 10 are reported to agree with the analytic result down to $\delta T$3 within 30%, and experimental measurements on two samples with $\delta T$4 show an Arrhenius-like slope close to $\delta T$5 over a broad temperature range (Mošková et al., 2016).

4. Weak interactions, weak barriers, and conductance in inhomogeneous Luttinger liquids

For a one-dimensional spinful fermion with forward-scattering interactions and an arbitrary impurity cluster $\delta T$6 near the origin, the Hamiltonian is

$\delta T$7

The interaction Fourier component satisfies $\delta T$8 for $\delta T$9, zero otherwise. In the RPA limit one introduces the holon velocity

$_3$00

and the dimensionless interaction parameter

$_3$01

The finite-bandwidth conductance of this Luttinger liquid with a cluster of impurities is studied as a function of temperature (Das et al., 2018).

In the weakly interacting limit, Matveev–Yue–Glazman obtain

$_3$02

where $_3$03, $_3$04, and

$_3$05

For a very weak barrier with $_3$06,

$_3$07

The non-chiral bosonization technique gives instead a transcendental equation for the effective transmission:

$_3$08

with

$_3$09

Since $_3$10, this single equation covers all interaction strengths, and by expanding around $_3$11 it reduces exactly to the Matveev–Yue–Glazman formula when $_3$12 (Das et al., 2018).

Two nonstandard regimes are highlighted. First, if $_3$13, then

$_3$14

and the $_3$15-dependence in the ansatz disappears. In this special case,

$_3$16

The conductance is therefore nearly temperature-independent. Second, if $_3$17 and $_3$18, then one finds $_3$19 and approximately

$_3$20

so that

$_3$21

as $_3$22 (Das et al., 2018).

The stated physical interpretation is that the standard “cutting” versus “healing” dichotomy is incomplete once finite bandwidth and arbitrary interaction strength are retained. In particular, a weak barrier under very strong repulsion need not simply suppress transport. The abstract additionally states that inclusion of backward scattering leads to non-monotonic temperature dependence of conductance when dealing with fermions with spin (Das et al., 2018).

5. Weak coupling and conductance microscopy of an open quantum dot

A further meaning of weak conductance arises in scanning-gate conductance microscopy of an open quantum dot connected to a conducting channel. The system is modeled in two dimensions with an effective-mass Hamiltonian

$_3$23

where $_3$24 is the GaAs effective mass, $_3$25 is zero inside the 70 nm-wide channel and the $_3$26 nm$_3$27 dot and tends to infinity elsewhere, and $_3$28 is the scanning-tip perturbation. The wave function is solved with a finite-difference approach and asymptotic subband boundary conditions in the leads (Kolasiński et al., 2014).

At zero temperature, the two-terminal conductance is given by the Landauer–Büttiker formula

$_3$29

with

$_3$30

Weak coupling of the dot to the channel is realized when the narrow 130 nm-long opening of width $_3$31 supports exactly one transverse mode at the Fermi energy. In that case the $_3$32 bypass subbands in the main channel transmit with $_3$33 each, unaffected by the dot, while the single channel that enters the dot has $_3$34. Hence

$_3$35

The paper identifies this interval as the numerical signature of the weak-coupling one-mode regime (Kolasiński et al., 2014).

The imaging result is formulated through Lippmann–Schwinger perturbation theory. The exact scattering states satisfy

$_3$36

and the first-order change in transmission is

$_3$37

For a delta-like tip,

$_3$38

so that

$_3$39

When the dot-channel coupling supports a single mode, the left and right solutions inside the dot differ only by a global phase, $_3$40, and therefore

$_3$41

Because this quantity equals the local density of states, the conductance variation satisfies $_3$42 to first order in $_3$43 (Kolasiński et al., 2014).

The same source contrasts this with strong coupling. If the opening is wide enough that two or more transverse modes feed the dot simultaneously, $_3$44 and $_3$45 are different superpositions, the inter-mode interference terms no longer track $_3$46, and the conductance map typically bears little resemblance to the LDOS except at isolated Fano-resonance energies where one quasi-bound state dominates the transport. In the weak-coupling perturbative limit, by contrast, the single-mode condition is stated to be both necessary and sufficient for the normalized conductance map to reproduce the unperturbed LDOS (Kolasiński et al., 2014).

6. Comparative interpretation and recurrent misconceptions

Across these studies, weak conductance does not denote a single universal conductance law. The weak element is system-specific: a weak-localization-type correction in a topological-insulator wire, a weak tunnel junction in a noisy electromagnetic environment, a weakly interacting or weak-barrier sector of a Luttinger liquid, or a weakly coupled and weakly perturbed quantum dot geometry (Matsuo et al., 2012, Mošková et al., 2016, Das et al., 2018, Kolasiński et al., 2014).

The observables likewise differ. In the Bi$_3$47Se$_3$48 wire, the relevant quantities are low-field $_3$49, the coherence length $_3$50, the correlation field $_3$51, and $_3$52. In the tunnel-junction problem, the central quantity is the zero-bias $_3$53 normalized to the bare conductance $_3$54. In the Luttinger-liquid problem, the conductance is determined through an effective transmission $_3$55. In the microscopy problem, conductance is spatially resolved as $_3$56 or through the first-order $_3$57 induced by the tip (Matsuo et al., 2012, Mošková et al., 2016, Das et al., 2018, Kolasiński et al., 2014).

Several misconceptions are explicitly corrected by the sources. Weak conductance effects are not necessarily synonymous with weak absolute transmission: in the Bi$_3$58Se$_3$59 wire they are small corrections to the classical Drude conductance, while in the quantum-dot problem weak coupling still permits $_3$60 to lie between $_3$61 and $_3$62. Nor does Coulomb blockade in a noisy environment generically imply activation by $_3$63; in the semiclassical Nyquist-noise regime the leading activation is $_3$64. Likewise, a weak barrier in a Luttinger liquid does not always simply cut the chain: for $_3$65 and $_3$66, the formal low-temperature trend is toward $_3$67 (Matsuo et al., 2012, Mošková et al., 2016, Das et al., 2018, Kolasiński et al., 2014).

A plausible synthesis is that the weak-conductance label is most useful when it identifies the controlling approximation. In the four cases summarized here, those approximations are respectively the low-field phase-coherent interference correction, the Gaussian Nyquist-noise form of $_3$68, the effective-transmission equation of non-chiral bosonization, and first-order Lippmann–Schwinger perturbation theory for a delta-like tip. The conductance response is therefore weak not in a universal numerical sense, but in the sense that it is generated, renormalized, or imaged by a weak sector of the transport problem.