From electronic states to friction
Choose the electronic states, count their transitions, and calculate a simple friction spectrum. Then see how Gaussian broadening changes the result.
Which electronic states are available?
The density of states (DOS) tells us how many electronic states lie near each energy. We set the chemical potential μ to zero. At zero temperature, states below zero are filled and states above zero are empty. At higher temperature, the occupation f gives the probability that a state contains an electron.
Try a flat DOS, a peak, a gap, or discrete levels. These controls update the transition and friction plots below.
The band extends from −W to +W. The shaded region marks energies below μ = 0.
How temperature changes occupation
The Fermi–Dirac function and the chemical potential
T is the electronic temperature in kelvin; kBT sets the energy range over which occupations change. Here μ stays at zero. With an asymmetric DOS, the total electron number can then change with temperature. Keeping that number fixed would require adjusting μ.
What the DOS controls mean
The peak adds a Gaussian with twice the background height. The gap is centred on zero. W sets the band half-width. Discrete levels come in symmetric pairs, each with weight w = 2Wρ₀/N. Increasing N gives finer sampling of the same total weight. The spacing control moves the levels within the band.
How many excitations are possible?
An electron can move from an occupied state into an empty state. Moving up costs energy; moving down releases energy. The net transition weight is the upward weight minus the downward weight. This occupation-weighted JDOS is what we use to calculate friction.
At zero temperature, a transition of energy E starts between −E and 0. At higher temperature, both directions contribute. Subtracting their occupation factors gives f(ε) − f(ε + E).
Here ρ is the DOS, ε is the starting electron energy, and E is the transition energy. The integral adds the contributions from all starting energies.
Occupation factors
The upward probability is f(ε)[1−f(ε+E)]. The downward probability is f(ε+E)[1−f(ε)]. Multiplying by the DOS at both energies and integrating gives the two curves. At equilibrium, J↓(E) = exp[−E/(kBT)] J↑(E).
At zero energy, the upward and downward weights are equal, so their difference is zero. The slope of this difference near zero determines the continuum friction at zero frequency.
Model assumptions
This model pairs states using their energies and occupations. A calculation for a crystal also needs the electronic wavefunctions, coupling strengths, and crystal-momentum selection rules.
From transition weight to friction
Friction measures how strongly electronic transitions damp atomic motion. Here we multiply the net transition weight by a constant coupling g² and divide by the transition energy E.
Each pair i < j gives a line at Δ = εⱼ − εᵢ. Its net transition weight is wᵢwⱼ(fᵢ − fⱼ). Dividing by Δ and multiplying by g² gives the friction line weight:
Each stick shows the integrated weight of a delta peak. We include pairs of distinct levels, so every line has a positive transition energy.
Section 4 broadens each line into a Gaussian. The partner at −Δ has the same friction weight: reversing the pair changes the signs of both the occupation difference and the energy denominator.
For E > 0, the tool divides the net transition weight by E. At E = 0, it uses the limit as E approaches zero:
As E approaches zero, [f(ε) − f(ε + E)]/E approaches −df/dε. This gives the integral above, which the tool evaluates numerically at finite temperature. At zero temperature, it becomes Λ(0) = g²ρ(0)² for a DOS continuous at μ.
This limit applies to the continuum model, where transition energies can approach zero continuously.
For a wide, flat DOS, the net transition weight grows approximately in proportion to E. Dividing by E gives nearly constant friction. A peak or gap changes this energy dependence. Values are in arbitrary units, with common physical prefactors left out.
The next section broadens this friction spectrum. Use the button above to return to this input after trying another spectrum.
Limits, Γ-point calculations, and negative frequencies
We assume equilibrium occupations at fixed μ and the same coupling for every state pair. In a real material, coupling strengths and momentum selection help determine the low-energy friction.
For a fixed set of distinct levels, every transition costs a finite energy. Friction then tends to zero as both frequency and Gaussian width approach zero. A continuum with states at μ supports transitions of arbitrarily small energy and can have a finite limit. This distinction matters when interpreting a finite interband sum at Γ.
The real scalar friction response has equal weight at positive and negative frequencies. Reversing a state pair changes the signs of both the occupation difference and the energy denominator, leaving their ratio unchanged. The two frequency branches describe the response to real atomic motion.
The friction formula follows the energy-difference denominator in Box, Stark & Maurer (2023), Eq. (14). Diagonal terms, degenerate states, and broadening of phonon peaks in α²F each require their own treatment.
What changes when we smooth the spectrum?
Gaussian broadening spreads each sharp contribution into a bell curve. The width σ sets how far its weight spreads. Compare three ways of doing this near zero frequency.
“Positive only” uses peaks at positive energies. “Rescaled peaks” increases each peak so its area above zero equals its original weight. “Both partners” adds an equal-weight peak at the corresponding negative energy.
Positive excitations only
The thicker curve shows your selected treatment. The dashed reference shows the input before broadening.
Values at zero: the tool evaluates the broadened spectrum at E = 0 with the selected width σ. It integrates over a continuum input or sums the Gaussian tails from discrete lines.
The ideal flat-spectrum test
The ideal flat spectrum extends to arbitrarily high energies. For a starting value of 1, broadening gives these values at zero:
Λ(0) is the friction at zero frequency. Changing σ changes the width of the dip; the three values above stay fixed.
This example isolates the effect of the boundary. For a particular material, the starting spectrum determines the result. Negative-partner tails become small when the observation energy is much larger than σ.
One peak and its partner
The orange Gaussian is centred at +Δ. Its green partner is centred at −Δ. Move the peak towards zero to see how much of each curve lies on the positive side.
Area measures total weight. Height measures the contribution at one energy. Rescaling raises the orange curve everywhere, adding most weight near +Δ. The green tail adds most of its positive-side weight near zero. At zero, the orange and green heights are equal.
Gaussian weights
G is a Gaussian with total area 1. A is the fraction of the positive peak above zero. The two treatments are:
Both have area 1 above zero. Rescaling multiplies the positive peak by 1/A. Including the partner adds its tail. These give different shapes. Including both partners preserves the height of the ideal flat spectrum.
The error function erf gives the Gaussian area. This writes the positive-domain normalization of Maurer et al. (2016), Eq. (23) with Δ > 0 and an explicit width σ. The integral definition gives the expression used here and in the scalar code.
FHI-aims settings and numerical details
friction_delta_normalisation .true. rescales positive peaks in the scalar tensor routine. Its scope is that routine. To include both partners, use friction_negative_energies .true. with delta normalization off. This tool compares these choices separately.
Here we broaden the friction spectrum directly. Broadening the dissipative susceptibility before dividing by frequency gives another finite-width approximation. The double-delta expression evaluates states at the Fermi surface.
The test peak has width 0.07 eV. The test gap has a smooth edge of width 0.025 eV. The rising spectrum is background × (1 + Δ / 1 eV). These presets extend beyond the plot; the DOS model keeps its finite band edges.
Discrete broadening sums the Gaussian contributions from every distinct state pair. Continuum broadening uses midpoint integration, with spacing at most σ/45 and an upper bound eight Gaussian widths beyond 1 eV. The DOS-derived friction uses a 0.002 eV grid. Material calculations also need convergence checks for electronic states and k-points.