Fermi edge singularities in the mesoscopic regime: Photoabsorption spectra

Martina Hentschel, Denis Ullmo, and Harold U. Baranger

Phys. Rev. B 76, 245419 – Published 18 December 2007

Abstract

We study Fermi edge singularities in photoabsorption spectra of generic mesoscopic systems such as quantum dots or nanoparticles. We predict deviations from macroscopic-metallic behavior and propose experimental setups for the observation of these effects. The theory is based on the model of a localized, or rank one, perturbation caused by the (core) hole left behind after the photoexcitation of an electron into the conduction band. The photoabsorption spectra result from the competition between two many-body responses, Anderson’s orthogonality catastrophe and the Mahan-Nozières-DeDominicis contribution. Both mechanisms depend on the system size through the number of particles and, more importantly, fluctuations produced by the coherence characteristic of mesoscopic samples. The latter lead to a modification of the dipole matrix element and trigger one of our key results: a rounded $K$ -edge typically found in metals will turn into a (slightly) peaked edge on average in the mesoscopic regime. We consider in detail the effect of the “bound state” produced by the core hole.

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Received 19 June 2007

DOI:https://doi.org/10.1103/PhysRevB.76.245419

Authors & Affiliations

Martina Hentschel^1,2, Denis Ullmo^2,3, and Harold U. Baranger²

¹Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Straße 38, Dresden, Germany
²Department of Physics, Duke University, Box 90305, Durham, North Carolina 27708-0305, USA
³CNRS, Université Paris-Sud, LPTMS UMR 8626, 91405 Orsay Cedex, France

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Vol. 76, Iss. 24 — 15 December 2007

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Figure 1
(Color online) Schematic illustrating processes contributing to the photoabsorption cross section in a Fermi golden rule approach. Conduction electrons (mean level spacing $d$ ) in a generic chaotic system initially occupy levels $ϵ_{0} \dots ϵ_{M - 1}$ (filled) and $ϵ_{M} \dots ϵ_{N - 1}$ (empty), that lower to $λ_{0} \dots λ_{N - 1}$ when the core electron $c$ is excited. Optically active electrons contribute via the coherent superposition of (a) direct and (b) replacement processes. In addition, one (or more) electron-hole pairs can be generated in shakeup processes, (c) for optically active channel and (d) for spectator channel, which are especially important away from threshold. Their presence reflects the suddenness of the perturbation. The vertical arrow in (a) is the bare process which represents the photoabsorption cross section in the naive picture without many-body effects. It depends only on the dipole matrix element $w_{c j}$ between the core electron $c$ and the single particle state $j$ , $A^{b} (ω) \propto ∣ w_{c j} ∣^{2}$ . The effect of AOC is accounted for in the direct process by the additional factor $∣ Δ ∣^{2} < 1$ , $A^{d} (ω) \propto ∣ w_{c γ} ∣^{2} ∣ Δ ∣^{2}$ .Reuse & Permissions
Figure 2
(Color online) Chaotic single-particle wave functions subject to a rank one perturbation. (a) Unperturbed and (b), (c) perturbed wave-function probabilities for $i = 0, 1, 2$ and $M$ , i.e., for the lowest three eigenstates and the state at the Fermi energy $E_{F}$ . We assume a two-dimensional chaotic system ( $N = 100$ , $M = 50$ ) with unperturbed energies at their mean (bulklike) values and model the unperturbed wave functions as random superpositions of $100$ plane waves. The intensities along a line that contains the perturbation, located at ${\vec{r}}_{c} = 0$ , are shown. The normalization volume of the wave function is $Ω = N$ . In (b), a weak perturbation causes only slight changes in the intensities. In contrast, a strong perturbation in (c) causes the wave-function intensity $∣ ψ_{0} ∣^{2}$ corresponding to the bound state to pile up at the position of the perturbation. Screening of the core hole is done by the bound state on a length scale of the order of the Fermi wavelength, $λ_{F}$ , indicated by the black bar.Reuse & Permissions
Figure 3
(Color online) Spectral weight of the unperturbed many-body ground state $∣ Φ_{0} ⟩$ in terms of perturbed many-body states $∣ Ψ_{f} (ω) ⟩$ , classified by their (average) energy from threshold measured in units of mean level spacings $d$ . ( $v_{c} ∕ d = - 10$ , $M = 50$ , $N = 100$ , COE statistics.) The threshold for the secondary band (right panel), when the bound state is empty in the final state, is $M d + λ_{0}$ greater than the threshold with bound state filled (left). The lower (thick) curve is the energy-resolved width of $∣ Φ_{0} ⟩$ in the perturbed basis, equivalent to the contribution of the spectator channel to the absorption cross section. The upper (thinner) curves show the cumulative spectral weight taking into account terms with one, up to two, and up to three shakeup pairs (dashed, dotted with symbols, and full lines, respectively). Remarkably, less than 0.1% of the weight is missed when including only up to two shakeup pairs. The slow saturation of the total weight (taking place on the scale of the band width) is a characteristic of AOC.Reuse & Permissions
Figure 4
(Color online) One-pair shakeup and replacement overlaps for the bulklike case. (a) Intermediate perturbation strength, $v_{c} ∕ d = - 0.5$ . (b) Strong perturbation, $v_{c} ∕ d = - 10$ . For almost all $(μ, γ)$ the replacement overlap $∣ Δ_{\bar{μ} γ}^{b} ∣^{2}$ is zero. Nonzero values arise for (1) replacement through the bound state with the excited electron close to $E_{F}$ ( $μ = 0$ and $γ ≳ M$ , black arrow), and (2) shakeup pairs formed in the vicinity of the Fermi edge ( $μ, γ$ both close to $M$ , red arrow). For a strong perturbation, replacement through the bound state becomes the dominant process.Reuse & Permissions
Figure 5
(Color online) Average mesoscopic absorption spectra at the $K$ edge ( $N = 100$ , $M = 50$ , $v_{c} ∕ d = - 10$ , CUE). (a) The total absorption cross section in the active channel (full line) is the sum of the direct/replacement (triangles) and shakeup (diamonds) contributions. For comparison, the bare contribution (dashed-dotted line) and the direct process alone (squares) are shown. (b) Active (full circles) and spectator (open circles) channel spectra separately, as well as the full spin photoabsorption cross section obtained after convolution (down triangles). The edge is slightly peaked.Reuse & Permissions
Figure 6
(Color online) Average mesoscopic (triangles) and bulklike (quadrangles) spectra as a function of energy from threshold at both a $K$ and $L$ edge ( $N = 100$ , $M = 50$ , $v_{c} ∕ d = - 10$ , CUE). Whereas the bulklike and mesoscopic-chaotic results coincide for the $L$ edge, there is a clear difference in the $K$ -edge spectra: The bulklike edge is rounded whereas a generic mesoscopic system yields a slightly peaked edge on average. The dashed curves are the mesoscopic spectra in a COE situation; they are nearly indistinguishable from the CUE case.Reuse & Permissions
Figure 7
(Color online) Average mesoscopic absorption spectra at the $L$ edge ( $N = 100$ , $M = 50$ , $v_{c} ∕ d = - 10$ , CUE). (a) The spectrum of the optically active channel is the sum of the direct/replacement (triangles) and shakeup (diamonds) contributions. (b) Active (full circles) and spectator (open circles) channel spectra separately, as well as the full spin photoabsorption cross section obtained after convolution (down triangles). The contribution of the spectator channel is identical to that in Fig. 5b. The peak at the $L$ edge is much more pronounced than that at the $K$ edge (Fig. 5) and extends over several mean level spacings in photon energy.Reuse & Permissions
Figure 8
(Color online) Mesoscopic averaged photoabsorption cross section as a function of the number $2 M$ of particles in a half-filled band for (a), (b) $K$ edge and (c), (d) $L$ edge, and both weak coupling [(a),(c) $v_{c} ∕ d = - 0.3$ ] and strong coupling [(b),(d) $v_{c} ∕ d = - 10$ ]. Results are for the COE with full spin up to excitation energies a quarter of the bandwidth above threshold, and are normalized by the bare photoabsorption value. The $K$ edge appears, apart from the behavior directly at threshold, rounded, and the rounding increases for more particles in the system. In this sense AOC wins the competition with MND as the thermodynamic limit is approached. However, the slight peak at the edge persists as the signature characteristic of a mesoscopic-coherent system. The $L$ edge clearly is peaked, and this peak sharpens with increasing particle number. For strong coupling, the $L$ edge is completely dominated by replacement through the bound state, making the magnitude at the $L$ edge much larger. Comparison is made to the bulk power law behavior in each case (light gray, yellow online).Reuse & Permissions
Figure 9
(Color online) Individual $K$ -edge absorption spectra of four mesoscopic samples, illustrating the outcome expected in real single-sample measurements ( $N = 100$ , $M = 50$ , $v_{c} ∕ d = - 10$ , COE). Triangles (green online): Direct and replacement processes. Crosses (red online): Shakeup processes. Solid line: Total absorption cross section assuming an (experimental) resolution of $d ∕ 6$ . The spectra have been shifted such that their threshold energies coincide. Fluctuations occur in both the energy and the cross section.Reuse & Permissions
Figure 10
(Color online) Fluctuations of the photoabsorption cross section at the $K$ edge for (a) weak and (b) strong perturbation ( $N = 100$ , $M = 50$ , CUE, optically active spin). The average photoabsorption cross section as a function of energy, $⟨ A (ω) ⟩$ , is shown for processes with different average excitation (marked $ω_{th} + d$ , $ω_{th} + 2 d, \dots$ ). Note that energies are measured with respect to the threshold energy $ω_{th}$ (vertical line) as would be the case in experiments. The behavior of the threshold energy and cross section are shown in the inset: Both are approximately Gaussian distributed. The peak next to threshold is clearly asymmetric with the maximum photoabsorption cross section found at energies distinctly below the average value. The curves broaden and symmetrize away from threshold. The area under the curves is the total photoabsorption.Reuse & Permissions
Figure 11
(Color online) Fluctuations of the photoabsorption cross section at the $L$ edge for (a) weak and (b) strong perturbation ( $N = 100$ , $M = 50$ , CUE, optically active spin). Explanations are the same as for Fig. 10. The strongly peaked FES is evident.Reuse & Permissions
Figure 12
(Color online) Distribution of the mesoscopic photoabsorption fluctuations ( $N = 100$ , $M = 50$ , CUE): for a strong perturbation $v_{c} ∕ d = - 10$ , (a) the $K$ edge and (b) the $L$ edge, and for a weak perturbation $v_{c} ∕ d = - 0.3$ at (c) the $L$ edge with the $K$ edge in the inset. The probability distribution of the photoabsorption $A (ω)$ normalized by its mean value $⟨ A (ω) ⟩$ is shown for different mean excitation energies $⟨ ω ⟩$ , both near threshold (curves with symbols) and further away from threshold (full line) where a Gaussian shape emerges. For comparison, a Porter-Thomas distribution is indicated by the dashed line. Near threshold, the distributions are Porter-Thomas in almost all cases because the fluctuations in the dipole matrix elements dominate and cancel all correlations from the overlap. An exception occurs at threshold for the $L$ edge where the excited electron sits in the first level above the Fermi energy. In this case, the distribution resembles that of a ground state overlap.Reuse & Permissions
Figure 13
(Color online) Contribution of bound state processes to the average photoabsorption cross section as a function of photon energy above threshold ( $N = 100$ , $M = 50$ , CUE, optically active spin). All replacement and shakeup processes involving the bound state are included. Upper (lower) panels: Weak (strong) perturbation. Left panels: $K$ edge. Right panels: $L$ edge. For weak perturbation, the contribution is, as expected, small. For strong perturbation, processes through the bound state make a significant but decaying contribution at the $K$ edge; at the $L$ edge, they dominate.Reuse & Permissions
Figure 14
(Color online) Experimental arrangement to test our prediction of a transition from a rounded $K$ edge to a slightly peaked edge as the system size is diminished to the mesoscopic coherent regime. We propose a quantum dot array in a semiconductor heterostructure where the plane of the 2DEG also contains impurities that provide a localized state in the $GaAs$ band gap.Reuse & Permissions

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