Spin-Inelastic Friedel Oscillations

• Spin-inelastic Friedel oscillations are spatial patterns resulting from inelastic scattering of conduction electrons by magnetic impurities, altering local electronic densities.
• These oscillations provide insights into spin excitation dynamics, identifiable via spectroscopic features such as peaks in $(d^2I/dV^2)$ through STM and STS techniques.
• Key experimental setups include low temperature conditions and spin-polarized STM tips, enhancing the ability to observe quantum impurity interactions and spin dynamics.

Spin-inelastic Friedel oscillations constitute a spatial oscillatory response in the local electronic structure of conductors, triggered by inelastic scattering events involving localized magnetic impurities and itinerant conduction electrons. When such impurities undergo spin transitions—typically due to anisotropy splitting or coupling to external fields—there is an exchange of discrete energy quanta (ΔE) with conduction electrons. The resultant scattered waves, shifted in energy and wavevector from the incoming states, generate spatial interference patterns in observables such as the local density of states (LDOS) and local magnetization density (LMD). Unlike conventional (elastic) Friedel oscillations, these features can be imaged as distinct resonances in the second derivative of the tunneling current with respect to the applied voltage ($d^2I/dV^2$), enabling direct spectroscopic and spatial resolution of spin-flip dynamics in quantum impurity systems (Fransson et al., 2011, Kozlov et al., 2019).

1. Theoretical Framework and Model Hamiltonians

The canonical framework for spin-inelastic Friedel oscillations models conduction electrons via a surface or two-dimensional (2D) Hamiltonian, commonly given as

$$H_{\text{surf}} = \sum_{\mathbf{k}} \varepsilon_\mathbf{k} c_\mathbf{k}^\dagger c_\mathbf{k},$$

and their coupling to a localized spin $\mathbf{S}$ situated at position $\mathbf{r}_0$ via Kondo exchange:

$$H_K = J_K\, \mathbf{s}(\mathbf{r}_0) \cdot \mathbf{S}, \quad \mathbf{s}(\mathbf{r}) = \sum_{\sigma\sigma'} \psi_\sigma^\dagger(\mathbf{r}) \frac{\boldsymbol{\sigma}_{\sigma\sigma'}}{2} \psi_{\sigma'}(\mathbf{r}).$$

To address spin–orbital entanglement or external magnetic fields, models can incorporate Rashba–Dresselhaus spin–orbit interactions and Zeeman terms:

$$H_0 = \frac{\hbar^2\mathbf{k}^2}{2m}\,\sigma_0 + \alpha(\sigma_x k_y - \sigma_y k_x) + \beta(\sigma_x k_x - \sigma_y k_y) + \mathbf{h} \cdot \boldsymbol{\sigma},$$

where $\alpha$ and $\beta$ are SOI strengths, and $\mathbf{h} = (g^*\mu_B/2)\mathbf{B}$ is the Zeeman energy (Kozlov et al., 2019).

Key quantities—retarded Green's functions, T-matrix, and self-energy corrections—are analyzed perturbatively (up to second order in $J_K$). The self-energy $$H_{\text{surf}} = \sum_{\mathbf{k}} \varepsilon_\mathbf{k} c_\mathbf{k}^\dagger c_\mathbf{k},$$0 encodes the essential physics of inelastic scattering:

$$H_{\text{surf}} = \sum_{\mathbf{k}} \varepsilon_\mathbf{k} c_\mathbf{k}^\dagger c_\mathbf{k},$$1

where $$H_{\text{surf}} = \sum_{\mathbf{k}} \varepsilon_\mathbf{k} c_\mathbf{k}^\dagger c_\mathbf{k},$$2 and $$H_{\text{surf}} = \sum_{\mathbf{k}} \varepsilon_\mathbf{k} c_\mathbf{k}^\dagger c_\mathbf{k},$$3 determines the energy and temperature dependence (Fransson et al., 2011).

2. Mechanism of Spin-Inelastic Friedel Oscillations

Spin-inelastic Friedel oscillations emerge from the quantum interference between incident conduction electron waves at (or near) the Fermi momentum $$H_{\text{surf}} = \sum_{\mathbf{k}} \varepsilon_\mathbf{k} c_\mathbf{k}^\dagger c_\mathbf{k},$$4 and those that have been scattered inelastically (i.e., accompanied by a local spin excitation or relaxation, changing their energy by $$H_{\text{surf}} = \sum_{\mathbf{k}} \varepsilon_\mathbf{k} c_\mathbf{k}^\dagger c_\mathbf{k},$$5). The resultant scattered waves thus acquire a shifted wavevector:

$$H_{\text{surf}} = \sum_{\mathbf{k}} \varepsilon_\mathbf{k} c_\mathbf{k}^\dagger c_\mathbf{k},$$6

where $$H_{\text{surf}} = \sum_{\mathbf{k}} \varepsilon_\mathbf{k} c_\mathbf{k}^\dagger c_\mathbf{k},$$7 is the Fermi velocity. This shift in $$H_{\text{surf}} = \sum_{\mathbf{k}} \varepsilon_\mathbf{k} c_\mathbf{k}^\dagger c_\mathbf{k},$$8 causes the interference pattern to form standing waves at the inelastic momentum transfer, producing a spatial pattern described by $$H_{\text{surf}} = \sum_{\mathbf{k}} \varepsilon_\mathbf{k} c_\mathbf{k}^\dagger c_\mathbf{k},$$9, where $\mathbf{S}$0 is the zeroth-order Bessel function, $\mathbf{S}$1 (Fransson et al., 2011).

The spatial LDOS correction near the impurity thus exhibits characteristic rings, with wavelength

$\mathbf{S}$2

and the envelope decaying as $\mathbf{S}$3 in two dimensions, modified by thermal and impurity-induced broadening factors.

3. Experimental Detection and Spectroscopic Features

Scanning tunneling microscopy (STM) and, in particular, scanning tunneling spectroscopy (STS) with the capability to record inelastic electron tunneling spectroscopy (IETS, $\mathbf{S}$4) are the primary tools for direct observation. The Tersoff–Hamann framework relates the tunneling conductance to the surface LDOS:

$\mathbf{S}$5

so that

$\mathbf{S}$6

At temperatures $\mathbf{S}$7, sharp peaks in $\mathbf{S}$8 occur when the bias $\mathbf{S}$9 crosses the spin excitation energy difference, and spatial mapping reveals concentric rings with period set by $\mathbf{r}_0$0. For $\mathbf{r}_0$1, typical ring spacings are $\mathbf{r}_0$2 Å, shifting by a few percent for $\mathbf{r}_0$3 meV (Fransson et al., 2011).

STM measurement protocols to optimize contrast include temperatures below 4 K, sub-millivolt bias modulation, use of spin-polarized tips (e.g., Cr-coated, MnNi), and intentional placement of impurity atoms in engineered nanostructures (quantum corrals) to enhance signal by up to a factor of five.

4. Role of Spin-Orbit Coupling and Magnetic Fields

In systems with significant Rashba–Dresselhaus SOI and nonzero in-plane magnetic field, the LDOS and LMD acquire pronounced anisotropic features. The Fermi surface splits into two non-centrosymmetric sheets, yielding multiple characteristic wavevectors $\mathbf{r}_0$4 dependent on real-space angle $\mathbf{r}_0$5 and the direction of the applied field $\mathbf{r}_0$6:

$\mathbf{r}_0$7

The interference of scattered waves from different branches produces LDOS oscillations:

$\mathbf{r}_0$8

where $\mathbf{r}_0$9 is the potential scattering strength, $$H_K = J_K\, \mathbf{s}(\mathbf{r}_0) \cdot \mathbf{S}, \quad \mathbf{s}(\mathbf{r}) = \sum_{\sigma\sigma'} \psi_\sigma^\dagger(\mathbf{r}) \frac{\boldsymbol{\sigma}_{\sigma\sigma'}}{2} \psi_{\sigma'}(\mathbf{r}).$$0 the impurity magnetic moment, $$H_K = J_K\, \mathbf{s}(\mathbf{r}_0) \cdot \mathbf{S}, \quad \mathbf{s}(\mathbf{r}) = \sum_{\sigma\sigma'} \psi_\sigma^\dagger(\mathbf{r}) \frac{\boldsymbol{\sigma}_{\sigma\sigma'}}{2} \psi_{\sigma'}(\mathbf{r}).$$1 curvature factors, and the $$H_K = J_K\, \mathbf{s}(\mathbf{r}_0) \cdot \mathbf{S}, \quad \mathbf{s}(\mathbf{r}) = \sum_{\sigma\sigma'} \psi_\sigma^\dagger(\mathbf{r}) \frac{\boldsymbol{\sigma}_{\sigma\sigma'}}{2} \psi_{\sigma'}(\mathbf{r}).$$2 term represents the spin-flip (inelastic) scattering (Kozlov et al., 2019).

Rotation of the in-plane field $$H_K = J_K\, \mathbf{s}(\mathbf{r}_0) \cdot \mathbf{S}, \quad \mathbf{s}(\mathbf{r}) = \sum_{\sigma\sigma'} \psi_\sigma^\dagger(\mathbf{r}) \frac{\boldsymbol{\sigma}_{\sigma\sigma'}}{2} \psi_{\sigma'}(\mathbf{r}).$$3 modulates both the amplitude and wavelength of the oscillations, enabling extraction of SOI constants and $$H_K = J_K\, \mathbf{s}(\mathbf{r}_0) \cdot \mathbf{S}, \quad \mathbf{s}(\mathbf{r}) = \sum_{\sigma\sigma'} \psi_\sigma^\dagger(\mathbf{r}) \frac{\boldsymbol{\sigma}_{\sigma\sigma'}}{2} \psi_{\sigma'}(\mathbf{r}).$$4-factors from the angular Fourier analysis of STM conductance maps.

5. Spin Polarization and Selective Channel Probing

If the STM tip and/or substrate are spin-polarized, the spin-dependent density of states $$H_K = J_K\, \mathbf{s}(\mathbf{r}_0) \cdot \mathbf{S}, \quad \mathbf{s}(\mathbf{r}) = \sum_{\sigma\sigma'} \psi_\sigma^\dagger(\mathbf{r}) \frac{\boldsymbol{\sigma}_{\sigma\sigma'}}{2} \psi_{\sigma'}(\mathbf{r}).$$5 enters the scattering matrices and self-energies, causing majority and minority channels to oscillate with different amplitudes and possibly phases. Complete tip polarization selectively enhances a given spin channel, increasing the contrast of spin-inelastic Friedel oscillations and enabling differentiation between magnetic and nonmagnetic scattering contributions by comparison (Fransson et al., 2011).

In these settings, even nonmagnetic impurities can induce oscillatory LMD signals if both spin–orbit coupling and Zeeman splitting are present, while pure spin-flip contributions (i.e., inelastic effects in LDOS) require finite local magnetic moments and Zeeman fields (Kozlov et al., 2019).

6. Spatial Structure, Symmetry, and Decay

The spatial structure of spin-inelastic Friedel oscillations is governed by the symmetry of the impurity and the substrate. For isolated impurities, the oscillations are circularly symmetric, while placement in engineered structures such as quantum corrals can impose specific mode symmetries or focus standing-wave patterns. The envelope of the oscillations in two dimensions decays as $$H_K = J_K\, \mathbf{s}(\mathbf{r}_0) \cdot \mathbf{S}, \quad \mathbf{s}(\mathbf{r}) = \sum_{\sigma\sigma'} \psi_\sigma^\dagger(\mathbf{r}) \frac{\boldsymbol{\sigma}_{\sigma\sigma'}}{2} \psi_{\sigma'}(\mathbf{r}).$$6, modulated by possible exponential damping due to finite lifetime or temperature effects (i.e., $$H_K = J_K\, \mathbf{s}(\mathbf{r}_0) \cdot \mathbf{S}, \quad \mathbf{s}(\mathbf{r}) = \sum_{\sigma\sigma'} \psi_\sigma^\dagger(\mathbf{r}) \frac{\boldsymbol{\sigma}_{\sigma\sigma'}}{2} \psi_{\sigma'}(\mathbf{r}).$$7). The number and periodicities of observed oscillation rings directly reflect underlying impurity excitation spectra and Fermi surface geometry (Fransson et al., 2011).

7. Applications, Significance, and Experimental Conditions

Imaging spin-inelastic Friedel oscillations provides direct, spatially resolved access to the quantum dynamics of localized spin excitations and their coupling to itinerant electrons. These observations yield quantitative spectroscopic information on anisotropy splittings, Kondo coupling, spin–orbit strength, g-factors, and Fermi surface features, and enable disentangling of elastic/inelastic, spin-conserving/spin-flip, and orbital/spin-orbit scattering channels.

Essential conditions for their experimental detection are: temperatures $$H_K = J_K\, \mathbf{s}(\mathbf{r}_0) \cdot \mathbf{S}, \quad \mathbf{s}(\mathbf{r}) = \sum_{\sigma\sigma'} \psi_\sigma^\dagger(\mathbf{r}) \frac{\boldsymbol{\sigma}_{\sigma\sigma'}}{2} \psi_{\sigma'}(\mathbf{r}).$$8 K, bias sweeps within $$H_K = J_K\, \mathbf{s}(\mathbf{r}_0) \cdot \mathbf{S}, \quad \mathbf{s}(\mathbf{r}) = \sum_{\sigma\sigma'} \psi_\sigma^\dagger(\mathbf{r}) \frac{\boldsymbol{\sigma}_{\sigma\sigma'}}{2} \psi_{\sigma'}(\mathbf{r}).$$9 mV, lock-in detection for $$H_0 = \frac{\hbar^2\mathbf{k}^2}{2m}\,\sigma_0 + \alpha(\sigma_x k_y - \sigma_y k_x) + \beta(\sigma_x k_x - \sigma_y k_y) + \mathbf{h} \cdot \boldsymbol{\sigma},$$0, tunable tip polarization (up to 60%), and surface engineering capacities (e.g., corral construction). The characteristic spatial and spectroscopic patterns provide direct probes for the interplay of spin–orbit and Kondo physics, and the influence of external fields in nanostructured quantum materials (Fransson et al., 2011, Kozlov et al., 2019).