Electronic structure of copper studied by electron momentum spectroscopy

We present electron momentum spectroscopy measurements of the electronic structure of copper single crystals. Generally, good agreement was found with the band dispersion as measured by photoemission. The energy-resolved momentum densities are quite anisotropic. Observed diffraction effects can be disentangled in first order, and the experiment compares well to calculated momentum density. Deviations of the Fermi surface from spherical symmetry are resolved by this scattering experiment. Many-body effects cause lifetime broadening of the quasiparticle peak and a smooth tail extending to higher binding energies, but no clear satellite structures were found.

Figure 1

(Color online) The dispersion as calculated by the full-potential linear tin-muffin orbital method along the three major symmetry directions (upper panel). The momentum densities, split up for the different bands, is shown in the lower three panels; the linestyle of each momentum density curve is the same as that of the dispersion of the corresponding band.

Figure 2

In (a) we show a schematic representation of the measurement. Incoming electrons (momentum ${\mathit{k}}_0$) impinge on a thin film and two analyzers, measuring simultaneously a series of azimuthal angles ${\varphi }_{1,2}$ and energies $E_{1,2}$, select coincident pairs of emerging electrons with momenta ${\mathit{k}}_1$ and ${\mathit{k}}_2$. The sample is indicated as a block with a surface normal along the $⟨0,\overline{1},1⟩$ direction, the edges oriented along the ⟨0,1,1⟩ and ⟨1,0,0⟩ crystal directions. In (b) we measure the SMD along the ⟨1,0,0⟩ direction, and the incoming beam is directed along $⟨0,\overline{1},1⟩$. The outgoing electrons are moving close to ⟨0,1,0⟩ and ⟨0,0,1⟩ directions. In (c) we measure the SMD along the same direction, but the crystal has been rotated along the $y$ axis by 10° moving the incoming and outgoing electrons away from high-symmetry directions. In (d) the crystal was rotated along the $⟨0,\overline{1},1⟩$ axis by 54.4° and the SMD along a ⟨1, 1, 1⟩ direction is measured. A rotation over 90° results in a measurement along ⟨0, 1, 1⟩.

Figure 3

A cut through reciprocal space (with $p_{⟨0,\overline{1},1⟩}=0$) showing the first three Brillouin zones. The free electron Fermi sphere for one electron per unit cell is drawn as a dashed circle. The three directions along which the spectral momentum density was determined are indicated by dashed lines. Some of the special points customarily used to describe high-symmetry points of the Brillouin zone are indicated as well.

Figure 4

Spectra of crystals with different orientations for selected momentum intervals. The three measurements for different orientations were normalized for the spectra near zero momentum.

Figure 5

A comparison of the measured dispersion with that obtained by a FP-LMTO calculation. The open circles are the band energies corrected for self-energy effects, as calculated by Marini et al.[11] Without diffraction, significant intensity is expected only for the part of the band structure that is indicated by a thick line.

Figure 6

The measured spectra near several special points. Energies as calculated by Marini et al.[11] are indicated by bars. Without diffraction, intensity is only expected at the thick bars.

Figure 7

The momentum density integrated over a $1\text{−}\mathrm{eV}$-wide window near the Fermi level for the three main symmetry directions. The theory is integrated over a $2\text{−}\mathrm{eV}$-wide window.

Figure 8

(Color-online) In the top-left panel we show an energy loss spectrum, taken with $25\text{−}\mathrm{keV}$ incoming electrons. The full line represents the experimental data, the dotted line an empirical fit used in the deconvolution procedure. In the bottom-left panel we compare the trajectories in this EELS experiment with those of the EMS experiment. The measured energy loss distribution is used to deconvolute the EMS data for inelastic scattering. The result of this procedure is shown in the right panel for the measurement along the ⟨110⟩ direction. The raw data (thin lines) are compared after deconvolution (dots) with GW calculation (blue, dotted lines) and cumulant expansion calculations (thick, red line)

Figure 9

The effect of alignment on diffraction and peak positions for measurements near the ⟨1,0,0⟩ alignment. Diffraction effects are largest for the best aligned crystal (open squares). The position of the outer diffracted peaks move slightly inward with increasing misalignment. This is explained in the top inset. Here the cut through the Fermi surface is approximated by a sphere, and the diffracted spheres (spheres centered on the neighboring reciprocal lattice points) are shown as well. Peaks are observed when the measurement line intersects the spheres. The outermost intersection moves slightly to lower momenta for the misaligned (dashed) measurement.

Figure 10

The momentum profiles on the ⟨100⟩ binding energies as measured (thin lines), and after removal of the diffracted contribution (thick lines).

Figure 11

The momentum density as measured (filled circles), after subtraction of the diffracted peak (open squares). The measurement is compared to the calculated momentum density without any broadening (full line) and with broadening of $0.1\mathrm{a.u.}$ (dashed line)

Figure 12

Energy spectra at the indicated momentum values along the ⟨100⟩ direction. The raw measured intensity (dots) and after corrections for the inelastic and elastic energy loss processes (thin line) compared to the LMTO theory with additional life-time broadening based on jelium calculations (thick line).