Resonant-level electronic friction

Compare a known continuum result with different Gaussian broadening expressions.

1 · Model

One orbital coupled to a metal

Moving x shifts the orbital energy. Electrons are noninteracting; the metal has a wide flat band and constant coupling.

ε_{d} (x) = ε_{0} + g x, ρ_{d} (ε) = \frac{1}{π} \frac{\frac{Γ}{2}}{{(ε - ε_{d})}^{2} + {(\frac{Γ}{2})}^{2}}

ρ_d integrates to one over all energies. Only part of its tails is shown.

2 · Published equilibrium result

Friction at zero frequency

η (0) = π ℏ g^{2} \int d ε ρ_{d}^{2} (ε) [- \frac{\partial f}{\partial ε}]

At T = 0, only the density of states at μ enters:

η (0) = π ℏ g^{2} ρ_{d}^{2} (μ) = \frac{ℏ g^{2}}{π} \frac{{(\frac{Γ}{2})}^{2}}{{[{(μ - μ_{d})}^{2} + {(\frac{Γ}{2})}^{2}]}^{2}}

Selected temperatureAnalytic T = 0

The dashed curve evaluates the published constant-Γ, zero-temperature limit; the solid curve evaluates its thermal integral. At T = 0 they coincide. These are recalculated results, not digitised data. This equilibrium integral is also the zero-width limit of the double-delta expression below.

3 · Finite-frequency result

Compare broadening expressions

η (E) = π ℏ g^{2} \int d ε ρ_{d} (ε) ρ_{d} (ε + E) \frac{f (ε) - f (ε + E)}{E}

At T = 0 and E > 0, integrate from −E to 0. At E = 0, use the equilibrium limit above.

Zero-frequency values
Expression	η(0)/ℏ (Å⁻²)	Error vs reference

The double-delta line is a zero-frequency value, shown horizontally for comparison. It is not a finite-frequency spectrum.

Change σ to test the numerical approximation. Change Γ to change the physical model. The reference does not depend on σ.

Expressions used in the comparison

For a positive transition Δ, including its reverse gives the kernels below. Both multiply the same positive-transition occupation difference and squared coupling.

K_{1A} (E, Δ) = \frac{G_{σ} (E - Δ) - G_{σ} (E + Δ)}{E}

K_{1B} (E, Δ) = \frac{G_{σ} (E - Δ) + G_{σ} (E + Δ)}{Δ}

1A broadens the dissipative response before dividing by E. 1B broadens friction directly. At E = 0, 1A uses its analytic limit. Both approach the continuum result as σ tends to zero.

Positive-only 1B retains G_σ(E − Δ)/Δ. Normalised 1B divides that kernel by A(Δ) = [1 + erf(Δ/(√2σ))]/2. This restores each peak’s positive-energy area, not necessarily the correct spectrum.

Double delta

At T = 0, the two delta functions select electronic states at μ. Replacing each by a Gaussian gives

η_{DD}^{σ} = π ℏ g^{2} {[\int d ε ρ_{d} (ε) G_{σ} (ε - μ)]}^{2}

At finite temperature, we average the squared Gaussian-smoothed DOS over the thermal Fermi window:

η_{DD}^{σ} = π ℏ g^{2} \int d u [- f^{'} (u)] {[\int d ε ρ_{d} (ε) G_{σ} (ε - u)]}^{2}

Here σ broadens each electronic energy separately; in 1A and 1B it broadens the transition energy. Equal σ values therefore represent different approximations. As σ tends to zero, double delta approaches η(0). No positive-transition boundary is imposed.

These plots broaden the continuum transition distribution. They do not model a finite bath or a discrete electronic mesh.

Reference and conventions

Dou and Subotnik (2016), Electronic friction near metal surfaces: spectral function and equilibrium friction, specialised here to constant Γ. Their spectral function A = 2πρ_d. The finite-frequency reference is the equilibrium linear response of this same model.

η multiplies nuclear velocity in the friction force. Divide by nuclear mass to obtain a damping rate. Two identical independent spin channels double all friction values. Numerical integrals resolve Γ and kBT; Gaussian convolution extends eight σ beyond the plotted energy range.