Volume plasmons

Colliex, C.

doi:10.1107/97809553602060000593

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 397-399

Section 4.3.4.3.1. Volume plasmons

C. Colliex^a

4.3.4.3.1. Volume plasmons

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The basic concept introduced by the many-body theory in the interacting free electron gas is the volume plasmon. In a condensed material, the assembly of loosely bound electrons behaves as a plasma in which collective oscillations can be induced by a fast external charged particle. These eigenmodes, known as volume plasmons, are longitudinal charge-density fluctuations around the average bulk density in the plasma n $[\simeq]$ 10²⁸ e⁻/m³). Their eigen frequency is given, in the free electron gas, as $[\omega_p=\left({n\,e^2\over m\varepsilon_0}\right)^{1/2}. \eqno (4.3.4.8)]$ The corresponding $[\hbar\omega_p]$ energy, measured in an energy-loss spectrum (see the famous example of the plasmon in aluminium in Fig. 4.3.4.3), is the plasmon energy, for which typical values in a selection of pure solid elements are gathered in Table 4.3.4.2. The accuracies of the measured values depend on several instrumental parameters. Moreover, they are sensitive to the specimen crystalline state and to its degree of purity. Consequently, there exist slight discrepancies between published values. Numbers listed in Table 4.3.4.2 must therefore be accepted with a 0.1 eV confidence. Some specific cases require comments: amorphous boron, when prepared by vacuum evaporation, is not a well defined specimen. Carbon exists in several allotropic varieties. The selection of the diamond type in the table is made for direct comparison with the other tetravalent specimens (Si, Ge, Sn). The results for lead (Pb) are still subject to confirmation. The volumic mass density is an important factor (through n) in governing the value of the plasmon energy. It varies with temperature and may be different in the crystal, in the amorphous, and in the liquid phases. In simple metals, the amorphous state is generally less dense than the crystalline one, so that its plasmon energy shifts to lower energies.

Table 4.3.4.2| top | pdf |
Plasmon energies measured (and calculated) for a few simple metals; most data have been extracted from Raether (1980)

Monovalent			Divalent			Trivalent			Tetravalent
$[\hbar\omega_p]$ (eV)			$[\hbar\omega_p]$ (eV)			$[\hbar\omega_p]$ (eV)			$[\hbar\omega_p]$ (eV)
	Meas.	Calc.		Meas.	Calc.		Meas.	Calc.		Meas.	Calc.
Li	7.1	(8.0)	Be	18.7	(18.4)	B	22.7	(?)	C	34.0	(31)
Na	5.7	(5.9)	Mg	10.4	(10.9)	Al	14.95	(15.8)	Si	16.5	(16.6)
K	3.7	(4.3)	Ca	8.8	(8.0)	Ga	13.8	(14.5)	Ge	16.0	(15.6)
Rb	3.4	(3.9)	Sr	8.0	(7.0)	In	11.4	(12.5)	Sn	13.7	(14.3)
Cs	2.9	(3.4)	Ba	7.2	(6.7)	Sc	14.0	(12.9)	Pb	(13)	(13.5)

The above description applies only to very small scattering vectors q. In fact, the plasmon energy increases with scattering angle (and with momentum transfer $[\hbar{\bf q}]$ ). This dependence is known as the dispersion relation, in which two distinct behaviours can be described:

(a) For small momentum transfers $[(q\,\lesssim\, q_c)]$ , the dispersion curve is parabolic: $[\hbar\omega_p(q)=\hbar\omega_p(0)+{\alpha\hbar^2 \over m_0}q^2. \eqno (4.3.4.9)]$ The coefficient α has been measured in a number of substances and calculated for the free-electron case in the random phase approximation (Lindhard, 1954); see Table 4.3.4.3 for some data. A simple expression for α is $[\alpha=\textstyle{3\over5} \displaystyle{E_F \over\hbar \omega_p(0)}, \eqno (4.3.4.10)]$ where [E_F] is the Fermi energy of the electron gas. More detailed observations indicated that it is not possible to describe the dispersion curve over a large momentum range with a single [q^2] law. In fact, one has to fit the experiment data with different linear or quadratic slopes as a function of q [see values indicated for Al and In in Table 4.3.4.3, and Höhberger, Otto & Petri (1975)]. Moreover, anisotropy has been found along different q directions in monocrystals (Manzke, 1980). In parallel, refinements have been brought into the calculations by including band-structure effects to deal with the anisotropy of the dispersion relation and with the bending of the experimental curves. Electron–electron correlations have also been considered, which has slightly improved the agreement between calculated and measured values of α (Bross, 1978a,b).

Table 4.3.4.3| top | pdf |
Experimental and theoretical values for the coefficient α in the plasmon dispersion curve together with estimates of the cut-off wavevector (from Raether, 1980)

	Measured α	Calculated α	(Å⁻¹)
Li	0.24	0.35	0.9
Na	0.24	0.32	0.8
K	0.14	0.29	0.8
Mg	0.35	0.39	1.0
Al	0.2 (< 0.5 Å⁻¹)
Al	0.45 (> 0.5 Å⁻¹)	0.43	1.3
In	0.40 (< 0.5 Å⁻¹)
In	0.66 (> 0.5 Å⁻¹)
Si	0.41
Si	0.3	0.45	1.1

(b) For large momentum transfers, there exists a critical wavevector [q_c] , which corresponds to a strong decay of the plasmon mode into single electron–hole pair excitations. This can be calculated using conservation rules in energy and momentum, giving $[\hbar\omega_p(0)+\alpha{\hbar^2\over m_0}\,q^2_c = {\hbar^2\over 2m_0}\,(q^2_c+2q_c q_F), \eqno (4.3.4.11)]$ where [q_F] is the Fermi wavevector. A simple approximation is $[q_c\simeq\omega_p/v_F]$ , [v_F] being the Fermi velocity. Single pair excitations can be created by fast incoming electrons in the domain of scattering conditions contained between the two curves: $[\left.\matrix{\Delta E_{\max}=\displaystyle{\hbar^2\over 2m_0} (q^2+2qq_F) \cr \Delta E_{\min}=\displaystyle{\hbar^2 \over 2m_0}\,(q^2 - 2qq_F)} \right\}\eqno(4.3.4.12)]$ shown in Fig. 4.3.4.13 . They bracket the curve $[\Delta E=\hbar^2q^2/2m_0]$ corresponding to the transfer of energy and momentum to an isolated free electron. For momentum transfers such as $[q\gt q_c]$ , the plasmon mode is heavily damped and it is difficult to distinguish its own specific behaviour from the electron–hole continuum. A few studies, e.g. Batson & Silcox (1983), indicate that the plasmon dispersion curve flattens as it enters the quasiparticle domain and approaches the centre of the continuum close to the free-electron curve. However, not only is the scatter between measurements fairly high, but a satisfactory theory is not yet available [see Schattschneider (1989) for a compilation of data on the subject].

Figure 4.3.4.13| top | pdf |

The dispersion curve for the excitation of a plasmon (curve 1) merges into the continuum of individual electron–hole excitations (between curves 2 and 4) for a critical wavevector q_c. The intermediate curve (3) corresponds to Compton scattering on a free electron.

Plasmon lifetime is inversely proportional to the energy width of the plasmon peak $[\Delta E_{1/2}]$ . Even for Al, with one of the smallest plasmon energy widths ( $[\simeq0.5]$ eV), the lifetime is very short: after about five oscillations, their amplitude is reduced to 1/e. Such a damping demonstrates the strength of the coupling of the collective modes with other processes. Several mechanisms compete for plasmon decay:

(a) For small momentum transfer, it is generally attributed to vertical interband transitions. Table 4.3.4.4, extracted from Raether (1980), compares a few measured values of $[\Delta E_{1/2}(0)]$ , with values calculated using band-structure descriptions.

Table 4.3.4.4| top | pdf |
Comparison of measured and calculated values for the halfwidth ΔE_1/2(0) of the plasmon line (from Raether, 1980)

	Experimental (eV)	Theory (eV)
Li	2.2	2.55
Na	0.3	0.12
K	0.25	0.15
Rb	0.6	0.64
Cs	1.2	0.96
Al	0.53	0.43
Mg	0.7	0.7
Si	3.2	5.4
Ge	3.1	3.9

(b) For moderate momentum transfer q, a variation law such as $[\Delta E_{1/2} (q) = \Delta E_{1/2}(0)+Bq^2+O(q^4) \eqno (4.3.4.13)]$ has been measured. The q dependence of $[\Delta E_{1/2}]$ is mainly accounted for by non-vertical transitions compatible with the band structure, the number of these transitions increasing with q (Sturm, 1982). Other mechanisms have also been suggested, such as phonons, umklapp processes, scattering on surfaces, etc.
(c) For large momentum transfer (i.e. of the order of the critical wavevector ), the collective modes decay into the strong electron–hole-pair channels already described giving rise to a clear increase of the damping for values of $[q\gt q_c]$ .

Within this free-electron-gas description, the differential cross section for the excitation of bulk plasmons by incident electrons of velocity v is given by $[{{\rm d} \sigma_p \over{\rm d} \Omega} (\theta) = {\Delta E_p \over2\pi Na_0 m_0 v^2}\ {1\over \theta^2+\theta^2_E}, \eqno (4.3.4.14)]$ where N is the density of atoms per volume unit and $[\theta_E]$ is the characteristic inelastic angle defined as $[\Delta E_p/2E_0]$ in the non-relativistic description and as $[\Delta E_p/\gamma m_0v^2]$ {with $[\gamma=[1-(v^{2}/c^{2})]^{-1/2}]$ } in the relativistic case. The angular dependence of the differential cross section for plasmon scattering is shown in Fig. 4.3.4.14 . The integral cross section up to an angle $[\beta_0]$ is $[\sigma_p(\beta_0)=\int\limits^{\beta_0}_0\,\left(\displaystyle{{\rm d}\sigma_p \over {\rm d}\Omega}\right){\rm d}\Omega= \,\displaystyle{\Delta E_p\log(\beta_0/\theta_E) \over Na_0 m_0 v^2}. \eqno (4.3.4.15)]$ The total plasmon cross section is calculated for $[\beta_0=\theta_c=q_c/k_0]$ . Converted into mean free path, this becomes $[\Lambda_p={1\over N\sigma_p} = {a_0\over \theta_E} \, \left(\log{\theta_c\over \theta_E}\right)^{-1} \quad \hbox {(non-relativistic formula)}\semi \eqno (4.3.4.16)]$ and $[\Lambda_p= \,{a_0\gamma m_0v^2 \over \Delta E_p}\, \left(\log{\hbar q_c v \over 1.132\,\hbar\omega_p}\right)^{-1} \quad \hbox{(relativistic formula)}. \eqno (4.3.4.17)]$

Figure 4.3.4.14| top | pdf |

Measured angular dependence of the differential cross section dσ/dΩ for the 15 eV plasmon loss in Al (dots) compared with a calculated curve by Ferrell (solid curve) and with a sharp cut-off approximation at θ_c (dashed curved). Also shown along the scattering angle axis, θ_E = characteristic inelastic angle defined as ΔE/2E₀, $[\tilde\theta]$ = median inelastic angle defined by $[\int^{\tilde\theta}_{0}({\rm d}\sigma/{\rm d}\Omega)\,{\rm d}\Omega=1/2\int^{\theta_{c}}_{0}({\rm d}\sigma/{\rm d}\Omega)\,{\rm d}\Omega]$ , and $[\bar\theta]$ = average inelastic angle defined by $[\bar\theta = \int {\theta}({\rm d}\sigma/{\rm d}\Omega)\,{\rm d}\Omega/\int({\rm d}\sigma/{\rm d}\Omega)\,{\rm d}\Omega]$ [courtesy of Egerton (1986)].

The behaviour of $[\Lambda_p]$ as a function of the primary electron energy is shown in Fig. 4.3.4.15 .

Figure 4.3.4.15| top | pdf |

Variation of plasmon excitation mean free path Λ_p as a function of accelerating voltage V in the case of carbon and aluminium [courtesy of Sevely (1985)].

References

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International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 397-399