The Electromagnetic Response of a Relativistic Fermi Gas at Finite Temperatures: Applications to Condensed-Matter Systems

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The electromagnetic response of a relativistic Fermi gas at finite temperatures: Applications to condensed-matter systems

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The electromagnetic response of a relativistic Fermi gas at ﬁnite temperatures: Applications to condensed-matter systems

E. Reyes-Gomez´ 1,L.E.Oliveira2 and C. A. A. de Carvalho3,4

1Instituto de F´ısica, Universidad de Antioquia UdeA - Calle 70 No. 52-21, Medell´ın, Colombia 2Instituto de F´ısica, Universidade Estadual de Campinas-Unicamp - Campinas-SP, 13083-859, Brazil 3Instituto de F´ısica, Universidade Federal do Rio de Janeiro-UFRJ - Rio de Janeiro-RJ, 21945-972, Brazil 4Inmetro, Campus de Xer´em - Duque de Caxias-RJ, 25250-020, Brazil

received on 12 April 2016; accepted by B. A. van Tiggelen on 24 April 2016 published online 3 May 2016

PACS 71.10.Ca – Electron gas, Fermi gas PACS 71.45.Gm – Exchange, correlation, dielectric and magnetic response functions, plasmons PACS 78.20.Ci – Optical constants (including refractive index, complex dielectric constant, absorption, reﬂection and transmission coeﬃcients, emissivity)

Abstract – We investigate the electromagnetic response of a relativistic Fermi gas at finite temperatures. Our theoretical results are first-order in the fine-structure constant. The electromagnetic permittivity and permeability are introduced via general constitutive relations in reciprocal space, and computed for different values of the gas density and temperature. As expected, the electric permittivity of the relativistic Fermi gas is found in good agreement with the Lindhard dielectric function in the low-temperature limit. Applications to condensed-matter physics are briefly dis- cussed. In particular, theoretical results are in good agreement with experimental measurements of the plasmon energy in graphite and tin oxide, as functions of both the temperature and wave vector. We stress that the present electromagnetic response of a relativistic Fermi gas at finite temperatures could be of potential interest in future plasmonic and photonic investigations.

editor’s choice Copyright c EPLA, 2016

The response of material media to applied external elec- systems satisfying two requirements: i) they should be rel- tromagnetic fields is central to a host of scientific and ativistic, so as to restore the symmetry between electric technological applications. Indeed, knowledge of how ma- and magnetic effects; ii) they should exhibit negative ε for terials respond to electromagnetic perturbations, coupled some range of frequencies, so that this behavior should to engineering at the nanoscale, has led to the construc- naturally extend to μ for systems satisfying i). The rel- tion of customized devices tailored to exhibit very specific ativistic electron gas was then chosen as a model system properties. Materials obtained in this way are presently that could meet both requirements, as its nonrelativistic called metamaterials. Their origin goes back to a spec- limit was known to comply with ii) in the long-wavelength ulation by Veselago [1], who investigated the then hypo- limit. In fact, the Lindhard formula, which corresponds thetical case of media that could have negative values for to the order α approximation to the permittivity, reduces both the electrical permittivity (ε) and the magnetic per- to the Drude formula which acquires negative values for meability (μ), a behavior believed not to occur in Nature. small frequencies. This behavior persists in the relativis- Such characteristics were, years later, obtained in artifi- tic case, where the permeability may also acquire negative cially constructed systems made up of tiny LC-circuits, values for low frequencies [2]. Nevertheless, the fact that the so-called split ring resonators, responsible for enhanc- the model exhibits negative values for both ε and μ in ing magnetic responses. That was the starting point of the long-wavelength limit at low frequencies is not enough a whole new field of research, oriented towards custom for us to assert that this behavior will occur in Nature. made devices, which were used to obtain perfect lenses, For that, we have to verify that the model, within the ap- invisibility cloaks, and special antennae. proximation used, is a reliable description of experimen- Recently [2], however, it was suggested that systems tal results. Clearly, one needs measurements involving with negative values for both ε and μ might occur in the relativistic electron gases to be found, for example, Nature. That belief relied on the existence of candidate in synchrotron beams or in astrophysical systems for such

17009-p1 E. Reyes-Gómez et al. verification. A specific experiment where an external ap- and plied field is compared to the field measured inside a syn- α 1 chrotron beam is definitely called for to hopefully settle C = − + 3+γ2 [γ arccot(γ) − 1] , (12) the question in the near future. 3π 3 The present study shows preliminary tests of the accu- where α is the fine-structure constant, racy of the electron gas description by using relativistic expressions at finite temperatures, in their nonrelativistic ∞ y2 2 − q˜2 +˜ω2 I = dy F0 1+ F1 , (13) limit, to describe quasi-free electrons in condensed-matter 0 y2 +1 8yq˜ systems, for which experimental results are already avail- ∞ y2 4(y2 +1)− q˜2 +˜ω2 able. In particular, we have compared the dependence on J = dy F0 1+ F1 2 temperature and wave vector of the experimental values of 0 y +1 8yq˜ the electric plasmon frequencies for graphite and tin oxide ω˜ y2 +1 with the predictions of our formulae. Unfortunately, we − F2 , (14) 2yq˜ cannot yet test the equivalent behavior in the magnetic case, as we are in the nonrelativistic regime. 1 1 F0 y,β,˜ ξ˜ = √ − √ , (15) The constitutive relations for the electromagnetic field β˜ y2+1−ξ˜ β˜ y2+1+ξ˜ e +1 e +1 in a relativistic Fermi gas at finite temperature are given (˜q2 −ω˜2 +2yq˜)2 −4(y2 +1)˜ω2 by [2] F1(y,q,˜ ω˜)=ln , (16) j jk k jk k (˜q2 −ω˜2 − 2yq˜)2 −4(y2 +1)˜ω2 D = ε E + τ cB , (1a) ω˜4 − 4(˜ω y2 +1+yq˜)2 and F2 k (y,q,˜ ω˜)=ln 4 2 2 , (17) j −1 jk k jk E ω˜ − 4(˜ω y +1− yq˜) H =(μ ) B + σ . (1b) c In the above relations, which are given in reciprocal and 4 (Fourier) space (q,ω), one has γ = − 1. (18) q˜2 − ω˜2 jk jk j k ε = εδ + ε qˆ qˆ , (2) The electromagnetic response of a relativistic Fermi gas −1 (μ−1)jk = μ−1 δjk + μ qˆjqˆk, (3) at finite temperature may be straightforwardly evaluated τ jk = τ jkl qˆl, (4) through eqs. (6)–(9) by computing the integrals given by eqs. (13) and (14). To do that, it is first necessary to and obtain the ξ chemical potential of the relativistic Fermi σjk = σ jkl qˆl, (5) gas by solving [2] the transcendental equation jk jkl where δ is the Kronecker delta, is the Levi-Civita − + ΔN = N − N = gf0(p, β, ξ), (19) symbol,q ˆj = qj/q, q = |q|, p ω˜2 ω˜2 ε =1+A + 1 − B + 2 − C, (6) where N − and N + are the number of particles and an- q˜2 q˜2 − ω˜2 tiparticles in the Fermi gas, respectively, ω˜2 q˜2 μ−1 =1+A−2 B + 2+ C, (7) q˜2 q˜2 − ω˜2 1 − 1 f0(p, β, ξ)= β(Ω −ξ) β(Ω +ξ) (20) 2 e p +1 e p +1 −1 q˜ ε = −μ = −A + C, (8) q˜2 − ω˜2 is the distribution function accounting for the presence of 2 2 2 4 both particles and antiparticles, Ωp = p c + m c is and ω˜ q˜2 the relativistic energy of a carrier with momentum p,and τ = σ = −B + C . (9) g = 2 is the spin degeneracy factor of the Fermi gas. By q˜ q˜2 − ω˜2 defining the effective carrier density η =ΔN/V , eq. (19) Here we have defined the dimensionless variablesq ˜ = reduces to 2 2 q/qc,˜ω = ω/ωc, β˜ = mc β,andξ˜ = ξ/(mc ), where 2 +∞ qc = mc/¯h is the Compton wave vector, ωc = mc /¯h is 2 η = η0 dyy F0 y,β,˜ ξ˜ , (21) the Compton frequency, β =1/(kBT ), T is the absolute 0 temperature, and ξ is the chemical potential of the Fermi 3 3 2 3 30 −3 gas. The dimensionless scalar functions in eqs. (6)–(9) are where η0 = gm c /(2π ¯h ) ≈ 1.76 × 10 cm . given by We denote εi and εi as the real (i = 1) and imaginary (i = 2) parts of the electromagnetic-response functions 2 − 2 A 4α 1 I − 3 q˜ ω˜ J given by eqs. (6) and (7), respectively. First we have fo- = 2 2 + 1 2 , (10) π q˜ − ω˜ 2 q˜ cused on the dielectric tensor defined by eq. (2). It is pos- jk 4α J sible to see that ε may be diagonalized diag(ε, ε, ε+ε ). B = , (11) π q˜2 − ω˜2 We display in fig. 1 the real and imaginary parts of ε and

17009-p2 The electromagnetic response of a relativistic Fermi gas at ﬁnite temperatures

60 100 80 80 (a) (b) (a) (b) 40 q = 0.25 qs 80 60 60 20 60 q = 0.25 qs q = 0.5 qs 1 2

ε 0 ε 40 40 40 (meV) (meV) p p

−20 20 ω ω h h q = 0.5 q − − → s 20 20 −40 0 → −60 −20 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 100 200 300 400 0 0.05 0.1 0.15 0.2 0.002 0.001 −1 q = 0.25 q (c) (d) T (K) q|| (Å ) s → 0.001 0 q = 0.5 qs Fig. 2: (Color online) (a) Temperature dependence of the plas- q = 0.5 qs 0 −0.001 ε′ ε′ mon energy of graphite. Dots and squares correspond to the experimental values by Jensen et al. [5] and Portail et al. [6], −0.001 −0.002 respectively, for plasmons excited by an electromagnetic field q = 0.25 qs c → with electric-field component parallel to the unitary vector −0.002 −0.003 0 5 10 15 20 25 30 0 5 10 15 20 25 30 normal to the basal plane. Solid and dashed lines correspond h− ω (eV) h− ω (eV) to theoretical results obtained from eq. (6), in the limit q → 0, by replacing the free-electron mass by the effective masses ∗ ∗ Fig. 1: (Color online) Real and imaginary parts of ε and ε mc =3.7 m and mc =6m, respectively. The η carrier den- as functions of ¯hω. Calculations were performed for various sity was assumed as a temperature-dependent function (see values of q in units of qs (see text) and for η corresponding to text). (b) Plasmon energy of graphite as a function of the q the electron density in a primitive cell of a silicon crystal. Solid wave vector parallel to the c direction. Solid squares corre- and dashed lines correspond to the present theoretical results spond to the experimental measurements by Portail et al. [6] (cf. eqs. (6) and (7)) at T = 5 K and to the Lindhard dielectric at T = 300 K, whereas the solid line corresponds to present function [3,4], respectively. calculations from eq. (6), at the same value of T , computed by ∗ taking [8] mc =6m. The dashed line corresponds to the lower zero of eq. (6). The η carrier density was taken as in panel (a). ε as functions ofhω ¯ . Results were obtained for η corresponding to the electron density in a primitive cell of a silicon crystal and for various values of the wave vector q where we have replaced the free-electron mass by the s ∗ ∗ expressed in units of q =2π/a. Solid lines correspond estimated effective masses mc =3.7 m and mc =6m, to the present numerical results obtained at T =5K, respectively, in the c direction normal to the basal plane. whereas dashed lines correspond to the Lindhard dielectric Calculations were performed in the limit q → 0. Here, function [3,4]. As expected, results computed from eq. (6) we have considered the η carrier density as a function coincide with the Lindhard dielectric function in the non- of the temperature and performed a parabolic fitting in relativistic limit. It is apparent from fig. 1 that the electric T of the experimental data [7,8] η =3× 1018 cm−3 at permittivity is essentially isotropic, in the non-relativistic × 18 −3 jk T =4.2K, η =4.2 10 cm at T =77.5K, and limit, due to the negligible contribution of ε to the ε η =1.13 × 1019 cm−3 at T = 300 K. The use of the ∗ tensor (cf. figs. 1(c) and (d)). effective mass mc =3.7 m leads to a good agreement Present theoretical results may be used to compute the between present theoretical calculations and experimen- plasma frequency (or equivalently, the plasmon energy) tal results by Jensen et al. [5]. It should be noted that as a function of the system temperature. It is well known this value of the effective mass is smaller that the effec- ∗ that the plasma frequency ωp corresponds to the upper fre- tive mass mc =6m reported by Chung [8], which leads quency zero of the real part of the electric permittivity [4]. to a good agreement between present results and the ex- In the nonrelativistic limit, the zeroes of the eigenval- perimental measurements performed by Portail et al. [6]. ues of the dielectric tensor essentially coincide due to its Of course, an appropriate study of the plasmon energy almost-isotropic behavior. Therefore, one may compute as a function of the temperature should include the band the plasma frequency of the Fermi gas from one of the structure information of the specific material. Obtaining eigenvalues of εjk . Here, we have compared the present the plasmon energy from the dielectric function of an elec- theoretical results for the plasmon energy ¯hωp with some tron gas necessarily implies the use of fitting parameters, experimental measurements. In this respect, the exper- such as the effective mass and the carrier density, in or- imental dependences of the plasmon energy of graphite der to describe the experimental results. Moreover, even as a function of the T temperature, reported by Jensen in a quantum-mechanical calculation, the specific condi- et al. [5] and Portail et al. [6], are depicted in fig. 2(a). tions of each experiment should be considered in detail to In all cases, the plasmon modes were excited by an elec- account for the strong dispersion observed in the exper- tromagnetic field with electric-field component parallel to imental results (see, for instance, the remarkable differ- the unitary vector c. Solid and dashed lines correspond ences between the experimental results by Jensen et al. [5] to the present theoretical results obtained from eq. (6), and Portail et al. [6] depicted in fig. 2(a)). The behavior

17009-p3 E. Reyes-G´omez et al.

21 10 ∗ (a) (b) have replaced the free-electron mass by the mc =0.31 m conduction-eﬀective mass of tin oxide [10]. In this case, 18 8 )

−3 present theoretical calculations slightly overestimate the

15 cm 6 plasmon energy as compared with the experimental re- (meV) 16 p ω h −

(10 sults. Nevertheless, the tolerance intervals of experimental 12 η 4 and theoretical data overlap, which indicates good agreement between both results. 9 2 0 100 200 300 0 100 200 300 Summing up, in the present work we have investigated T (K) T (K) the electromagnetic response of a relativistic Fermi gas at finite temperatures, in the nonrelativistic regime. Gen- Fig. 3: (Color online) (a) Plasmon energy of tin oxide nanowire eralized permittivities and permeabilities, which result films as a function of the T temperature. Solid circles corre- from a first-order correction on the fine-structure con- spond to experimental measurements by Zou et al. [10]. Open circles correspond to numerical results obtained from eq. (6) stant, were introduced through general constitutive rela- by using the carrier density, at each value of T ,reportedby tions in reciprocal space. Results obtained in the limits Zou et al. [10] (see panel (b)). Calculations were performed of low temperatures and low carrier densities were used in the limit q → 0 by replacing the free-electron mass by to study the behavior of the electric plasmon energy, as ∗ the mc =0.31 m conduction-effective mass of tin oxide [10]. a function of the temperature and wave vector, in some The solid line connecting open circles is a guide to the eye. The condensed-matter systems such as graphite and tin oxide. dark area corresponds to the uncertainty interval of the calcu- The plasmon energy was calculated from the electric per- lated plasmon energy at each value of T and was computed by mittivity and found in good agreement with previous ex- propagating the error of the carrier density estimated by the perimental measurements in such systems. We do hope error bars in panel (b). that present theoretical results will be of importance in condensed-matter applications involving plasmonics and of the plasmon energy of graphite, as a function of the photonics. q wave vector parallel to the c direction, is depicted in fig. 2(b). Solid symbols correspond to the experimental ∗∗∗ measurements reported by Portail et al. [6] at T = 300 K, whereas the solid line corresponds to present theoretical The authors would like to thank the Scientific Colom- results obtained from eq. (6) at this temperature value. In bian Agency CODI - University of Antioquia, and Brazil- addition, we have shown in fig. 2(b) the behavior of the ian Agencies CNPq, FAPESP (Procs. 2012/51691-0 lowest-frequency zero of ε1 (cf. dashed line in fig. 2(b)), and 2013/21320-3), and FAEPEX-UNICAMP for partial given by eq. (6), as a function of T . The free-electron mass financial support. ∗ was replaced by mc =6m for computing purposes [8]. Moreover, the η carrier density was taken as in fig. 2(a). REFERENCES Once again, the present theoretical results are consistent with the experimental measurements by Portail et al. [6]. [1] Veselago V. G., Sov. Phys. Usp., 10 (1968) 509. The overall behavior of the dielectric function as a func- [2] de Carvalho C. A. A., to be published in Phys. Rev. tion of the wave vector is quite similar to that obtained by D; arXiv:1510.00360v2 [quant-ph]. Yi and Kim, in wurtzite GaN, from nonrelativistic RPA [3] Lindhard J., K. Dan. Vidensk. Selsk., Mat. Fys. Medd., theoretical calculations [9]. 28 (1954) 8. Finally, the temperature dependence of the plasmon en- [4] Walter J. P. and Cohen M. L., Phys. Rev. B, 5 (1972) ergy of tin oxide nanowire films is displayed in fig. 3(a). 3101. Solid and open circles correspond to measurements re- [5] Jensen E. T., Palmer R. E., Allison W. and Annett ported by Zou et al. [10] and theoretical calculations from J. F., Phys. Rev. Lett., 66 (1991) 492. Portail M., Carrere M. Layet J. M. Surf. Sci. eq. (6) in the limit q → 0, respectively. Present theoret- [6] and , , 433-435 ical values of the plasmon energy were obtained by using (1999) 863. [7] Spain I. L., Ubbelohde A. R. and Young D. A., Philos. the carrier densities reported by Zou et al. [10] at spe- Trans. R. Soc. London, Ser. A, 262 (1967) 345. cific values of T , as depicted in fig. 3(b). The dark area [8] Chung D. D. L., J. Mater. Sci., 37 (2002) 1475. in fig. 3(a) corresponds to the uncertainty interval cor- [9] Yi K.-S. and Kim H.-J., Physica B, 457 (2015) 149. responding to the calculated plasmon energy, which was [10] Zou X., Luo J., Lee D., Cheng C., Springer D., Nair computed by propagating the error of the carrier den- S.K.,CheongS.A.,FanH.J.and Chia E. E. M., sity estimated by the error bars in fig. 3(b). Here, we J. Phys. D: Appl. Phys., 45 (2012) 465101.

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