[Paper Review] Power laws in surface state LDOS oscillations near a step edge

Rudro R. Biswas, Alexander V. Balatsky|arXiv (Cornell University)|May 26, 2010

Quantum and electron transport phenomena3 citations

TL;DR

This paper develops a general power-counting method to predict the spatial decay power law of surface state local density of states (LDOS) oscillations near a step edge, based on the geometry of the constant-energy Fermi surface and wavefunction characteristics. It shows that the decay follows a power law ∝ 1/x^(α+β+γ+1)/η, with the exponent determined by scaling behaviors of the density of states, reflection amplitude, spin overlap, and momentum-space dispersion near characteristic scattering wave vectors.

ABSTRACT

In this paper we indicate a general method to calculate the power law that governs how electronic LDOS oscillations decay far away from a surface step edge (or any local linear barrier), in the energy range when only 2D surface states are relevant. We identify the critical aspects of the 2D surface state band structure that contribute to these decaying oscillations and illustrate our derived formula with actual examples.

Motivation & Objective

• To determine the spatial decay power law of LDOS oscillations induced by a step edge on a 2D surface state.
• To identify how the geometry of the constant-energy Fermi surface and wavefunction properties govern the decay exponent.
• To provide a general method applicable to various surface states, including those with non-circular Fermi surfaces like Bi2Te3.
• To enable prediction of oscillation decay from qualitative knowledge of band structure and scattering wavevector behavior.

Proposed method

• Uses power-counting analysis of momentum-space integrals to derive the asymptotic decay of LDOS oscillations far from a step edge.
• Identifies 'characteristic' scattering wave vectors where the scattering wavevector varies most slowly with ky, corresponding to coherent contributions.
• Expresses the x-dependent LDOS as a sum over contributions from these characteristic regions, each governed by scaling laws: ρ₀ ∝ δky^α, r ∝ δky^β, χ†·χ ∝ δky^γ, and Δkx ∝ Δ₀ + Δ₁ δky^η.
• Performs a change of variables to μ = δky^η x, transforming the integral into a form revealing the power-law decay ∝ 1/x^(α+β+γ+1)/η.
• Applies the method to specific systems: 2DEG (circular Fermi surface), and Bi2Te3 with hexagonal warping, using known band parameters.
• Suggests using 1D Fourier transform of LDOS data to isolate contributions from different scattering processes via power-law fitting.

Experimental results

Research questions

• RQ1 What determines the power-law decay exponent of LDOS oscillations far from a surface step edge in 2D electronic systems?
• RQ2 How do the scaling behaviors of the density of states, reflection amplitude, and spin overlap near characteristic wave vectors influence the decay rate?
• RQ3 Can the decay power law be predicted from qualitative knowledge of the Fermi surface geometry and band structure?
• RQ4 How do different scattering processes (e.g., equator vs. corner scattering) contribute to the observed LDOS oscillations in systems like Bi2Te3?
• RQ5 Can the 1D Fourier transform of LDOS data be used to isolate and quantify contributions from distinct scattering mechanisms?

Key findings

• The decay of LDOS oscillations far from a step edge follows a power law ∝ 1/x^(α+β+γ+1)/η, where α, β, γ, and η are scaling exponents of the DOS, reflection amplitude, spin overlap, and momentum-space dispersion, respectively.
• For a 2DEG with a circular Fermi surface, the decay is ∝ 1/x, with α=0, β=1, γ=1, η=2, yielding exponent (0+1+1+1)/2 = 1.5, but the paper notes a correction to ∝ 1/x due to symmetry.
• In Bi2Te3 with hexagonal warping, corner-to-corner scattering leads to α=0, β=0, γ=0, η=1, resulting in a 1/x decay, consistent with experimental observations.
• For step edges perpendicular to the ΓM direction in Bi2Te3, the dominant decay is ∝ 1/x due to linear band dispersion and finite spin overlap at corners.
• For step edges perpendicular to the ΓK direction, two contributions are possible: ∝ 1/x (corner scattering) and ∝ 1/x^(3/2) (equator scattering), with the latter arising from α=0, β=1, γ=1, η=2.
• The 1D Fourier transform of LDOS data reveals power-law features ∝ |k±K|^(n−1) near characteristic wave vectors, enabling experimental isolation of distinct scattering contributions.