[Paper Review] Persistent current and low-field magnetic susceptibility in one-dimensional mesoscopic rings: Effect of long-range hopping

Santanu K. Maiti|ArXiv.org|Sep 7, 2009

Quantum and electron transport phenomena20 references3 citations

TL;DR

This study investigates persistent currents and low-field magnetic susceptibility in one-dimensional mesoscopic rings with long-range hopping (LRH) using a tight-binding model. It shows that LRH delocalizes electron states, restoring persistent current amplitudes in disordered rings to levels comparable to ordered rings. Crucially, it demonstrates that the sign of low-field current—diamagnetic for odd electron counts and paramagnetic for even—can be precisely predicted even with disorder, and identifies a temperature-dependent critical flux φc(T) where current sign flips from paramagnetic to diamagnetic in even-Nₑ rings.

ABSTRACT

Persistent current and low-field magnetic susceptibility in single-channel normal metal rings threaded by a magnetic flux $ϕ$ are investigated within the tight-binding framework considering long-range hopping of electrons in the {\em shortest} path. The higher order hopping integrals try to reduce the effect of disorder by delocalizing the energy eigenstates, and accordingly, current amplitude in disordered rings becomes comparable to that of an ordered ring. Our study of low-field magnetic susceptibility predicts that the sign of persistent currents can be mentioned precisely in mesoscopic rings with fixed number of electrons, even in the presence of impurity in the rings. For perfect rings, low-field current shows only the diamagnetic sign irrespective of the total number of electrons $N_e$. On the other hand, in disordered rings it exhibits the diamagnetic and paramagnetic natures for the rings with odd and even $N_e$ respectively.

Motivation & Objective

• To understand how long-range hopping (LRH) affects persistent current and low-field magnetic susceptibility in disordered one-dimensional mesoscopic rings.
• To resolve the long-standing discrepancy between theoretical predictions and experimental observations of persistent currents, particularly the enhanced current amplitudes in disordered rings.
• To determine whether the sign of low-field magnetic susceptibility can be reliably predicted in the presence of disorder, especially as a function of electron number (Nₑ).
• To analyze the temperature dependence of magnetic response and identify the critical flux φc(T) at which the current sign transitions from paramagnetic to diamagnetic in even-Nₑ rings.

Proposed method

• Employing a tight-binding Hamiltonian with site-dependent disorder (W = 1) and long-range hopping integrals modeled as v_ij ∝ 1/|sin(π|i−j|/N)|^α.
• Using exact numerical diagonalization to compute energy eigenstates and eigenvalues for finite N-site rings (N = 60) under magnetic flux φ.
• Calculating persistent current as I(φ) = −(1/L) ∂E/∂φ, where L is the ring circumference, and evaluating the derivative numerically.
• Computing low-field magnetic susceptibility χ = dI/dφ at φ = 0 to determine the sign of the response (diamagnetic or paramagnetic).
• Analyzing the temperature dependence of current by incorporating thermal population of states via the Fermi-Dirac distribution, with T⋆ defined by the level spacing Δ.
• Identifying the critical flux φc(T) as the point where the low-field current changes sign from paramagnetic (positive χ) to diamagnetic (negative χ) at finite T.

Experimental results

Research questions

• RQ1 How does long-range hopping (LRH) influence the amplitude of persistent current in disordered mesoscopic rings compared to conventional nearest-neighbor hopping (NNH) models?
• RQ2 Can the sign of low-field magnetic susceptibility be reliably predicted in disordered rings, and how does it depend on the total number of electrons (Nₑ)?
• RQ3 What is the role of electron delocalization due to LRH in counteracting the current suppression typically caused by disorder?
• RQ4 How does finite temperature affect the magnetic response, and what is the critical flux φc(T) where the current sign flips from paramagnetic to diamagnetic in even-Nₑ rings?
• RQ5 Does the presence of disorder alter the fundamental magnetic response behavior (diamagnetic vs. paramagnetic) in a way that depends on Nₑ, and can this be predicted reliably?

Key findings

• Long-range hopping (LRH) with α = 1.6 significantly enhances current amplitude in disordered rings, bringing it close to the value observed in perfectly ordered rings due to delocalization of energy eigenstates.
• In disordered rings, low-field magnetic susceptibility exhibits a diamagnetic response for odd Nₑ and a paramagnetic response for even Nₑ, regardless of the specific disorder configuration.
• For perfect rings at T = 0 K, the low-field current is always diamagnetic, irrespective of Nₑ, confirming the absence of paramagnetic contribution in the absence of disorder.
• At finite temperature, the critical flux φc(T) at which the current sign flips from paramagnetic to diamagnetic increases with temperature, with φc(T) rising as T/T⋆ increases from 0.5 to 1.0.
• The critical flux φc(T) is smaller in rings with nearest-neighbor hopping (NNH) than in those with LRH (α = 1.6), indicating that LRH stabilizes the paramagnetic phase to higher flux values.
• The study confirms that the sign of low-field current can be precisely predicted based on Nₑ and disorder presence, even in disordered systems, offering a robust signature for experimental detection.