Electromagnetic propagation in a relativistic electron gas at finite temperatures
D. M. Reis, E. Reyes-Gómez, L. E. Oliveira, C. A. A. de Carvalho
aa r X i v : . [ c ond - m a t . o t h e r] M a r Electromagnetic propagation in a relativistic electron gas at finite temperatures
D. M. Reis , E. Reyes-G´omez , L. E. Oliveira , and C. A. A. de Carvalho , Centro Brasileiro de Pesquisas F´ısicas - CBPF,Rua Dr. Xavier Sigaud 150, Rio de Janeiro - RJ, 22290-180, Brazil Instituto de F´ısica, Universidad de Antioquia - UdeA, Calle 70 No. 52-21, Medell´ın, Colombia Instituto de F´ısica, Universidade Estadual de Campinas - Unicamp, Campinas - SP, 13083-859, Brazil Diretoria Geral de Desenvolvimento Nuclear e Tecnol´ogico da Marinha - DGDNTM,Rua da Ponte s/n, Ed. 23 do AMRJ, Rio de Janeiro - RJ, 20091-000, Brazil Instituto de F´ısica, Universidade Federal do Rio de Janeiro - UFRJ,Caixa Postal 68528, Rio de Janeiro - RJ, 21945-972, Brazil (Dated: September 6, 2018)We describe electromagnetic propagation in a relativistic electron gas at finite temperatures andcarrier densities. Using quantum electrodynamics at finite temperatures, we obtain electric andmagnetic responses and general constitutive relations. Rewriting the propagator for the electromag-netic field in terms of the electric and magnetic responses, we identify the modes that propagate inthe gas. As expected, we obtain the usual collective excitations, i.e., a longitudinal electric and twotransverse magnetic plasmonic modes. In addition, we find a purely photonic mode that satisfiesthe wave equation in vacuum, for which the electron gas is transparent. We present dispersion rela-tions for the plasmon modes at zero and finite temperatures and identify the intervals of frequencyand wavelength where both electric and magnetic responses are simultaneously negative, a behaviorpreviously thought not to occur in natural systems.
PACS numbers: 71.10.Ca; 71.45.Gm; 78.20.Ci.
I. INTRODUCTION
The knowledge of the responses of a relativistic elec-tron gas (REG) at finite temperatures and densities toelectromagnetic (EM) radiation is a useful tool for un-derstanding the physics of several systems, as we willoutline below. The present article uses a quantum fieldtheory treatment to describe the interaction of the EMfield with the REG. It focuses on the calculation of theresponses and on the description of the modes of propa-gation within the gas.Investigations of the response of a REG to the actionof an external electromagnetic (EM) perturbation havemany similarities with studies in the fields of photonicsand plasmonics [1, 2], where it is crucial to understandthe propagation of EM and plasmonic modes that arealso present in the REG.In photonics and plasmonics one normally uses phe-nomenological expressions for the responses of the mediaof interest. Here, however, one may actually computesuch responses from first-principles, so that we envisageapplying our techniques in the future to the actual com-putation of the responses of artificially constructed ma-terials.In the context of plasma physics, the REG has beenthe subject of many articles [3]. The treatment presentedhere uses a field theory approach that can be quite natu-rally generalized to encompass the study of several otherplasmas, whose physical properties may be then com-pared to those of the REG.In astrophysical scenarios, it has been shown thatelectron-positron pairs may be obtained in systems suchas compact stars [4, 5]. There again, the study of theREG is of importance, especially with respect to the propagation of EM and collective modes through the gas,which may even show up in the observations.Related situations of interest whose analysis can profitfrom a good understanding of the REG may be ob-tained in the laboratory by using high intensity lasersin a plasma gas [6], with the laser photons acquiring aneffective mass in the medium, thus characterizing a col-lective excitation [7, 8].Finally, heavy-ion collison experiments at RHIC andLHC [9] produce collective excitations in the hot QCDliquid matter, known as the quark-gluon plasma (QGP).Its transverse and longitudinal collective plasma excita-tions, also known as plasmons, have similar behaviour tothe plasmons in a REG [10].Recently, it has been theorized [11] that the REGmight be a candidate for the realization of a natu-ral metamaterial, understood, as proposed originally byVeselago in 1968 [12], as a material that can have simul-taneously negative values for its electric permittivity andmagnetic permeability.Although the existence of natural metamaterials hasnot been reported so far, such systems have already beenartificially constructed. A successful design of an artifi-cial metamaterial was reported by Smith et al [13] andconsisted of a periodic array of metallic split-ring res-onators and wires, which exhibited simultaneously nega-tive values of the electric permittivity ( ǫ ) and magneticpermeability ( µ ) in a microwave-frequency band of theEM spectrum. Subsequently, metamaterials were real-ized up to the visible range [14] and all-dielectric meta-materials based on Si/SiO heterostructures have beenreported [15].The possibility that the REG may behave as a natu-ral metamaterial [11] deserves further investigation. Toinvestigate the EM responses of the REG it is naturalto adopt the quantum-electrodynamics (QED) formal-ism for physical systems at finite temperatures [16, 17],within linear response and in the random-phase approx-imation (RPA).In that formalism, it has been shown that, in the limitof temperature T →
0, theoretical results for the electricpermittivity agree with previous non-relativistic calcu-lations performed by Lindhard [18]. Moreover, in thelong-wavelength limit both ǫ and µ − exhibit Drude-likeEM responses and may be simultaneously negative [11].In addition, the validity of the model has been testedin the non-relativistic limit by successfully [19] describingthe experimental behavior of the plasmon energy, as afunction of both temperature and wave vector, in low-energy condensed-matter systems such as graphite [20,21] and tin oxide [22]. The study of the EM response ofa REG in the regime of high temperatures and densitiesis, therefore, in order.The aim of the present work is to investigate the behav-ior of the EM response functions of the REG, as functionsof the temperature, density, frequency, and wave vector,as well as the EM propagation modes within the REG.This study is organized as follows: In section II we de-scribe the theoretical procedure used for obtaining theEM responses and propagation in the REG. Results arepresented in section III and conclusions are in section IV. II. ELECTROMAGNETIC PROPAGATION
In [11] and [27], we have used a semiclassical expan-sion, where the electromagnetic part is treated as a clas-sical external field plus quantum fluctuations, but theelectrons are subject to a full quantum treatment, justas it is done in the nonrelativistic case that leads to thecelebrated Lindhard expression [18].In fact, integrating over the electron field yields afermionic determinant which is expanded in the classi-cal gauge field, assumed to be weak. Only the lowest-order term in the expansion is kept, which is equiva-lent to the linear response approximation (terms of order α ( αE /m ) or α ( αB /m ) are neglected), and electron-electron interactions are also neglected, as they give riseto higher-order terms in α ≡ e / (4 π ) = 1 / A µ ( β, ~x ) = A µ (0 , ~x ) [11, 16] Z [ A ] = I [ dA µ ] det[ − ∂ ] exp( − XZ ˜ A µ ˜Γ µν ˜ A ν ) , (1)˜Γ µν = q δ µν − (1 − λ ) q µ q ν − ˜Π µν . (2) q = q + | ~q | and ˜ A µ is the Fourier transform of the gaugefield A µ . The determinant comes from the Lorentz gaugecondition and λ is a gauge parameter. We have used thesimplified notation XZ ≡ β + ∞ X n = −∞ Z d q (2 π ) , (3)and the polarization tensor of QED [17]˜Π µν = − e β + ∞ X n = −∞ Z d p (2 π ) Sp[ γ µ G ( p ) γ ν G ( p − q )] , (4)where G is the free electron propagator at finite density.The summation over n in (4) is performed over Matsub-ara frequencies p = (2 n + 1) πT .One may write ˜Π νσ = ˜Π ( v ) νσ + ˜Π ( m ) νσ to separate vacuumand medium contributions. The vacuum contribution is − ˜Π ( v ) νσ q = (cid:18) δ νσ − q ν q σ q (cid:19) C ( q ) , (5)whereas the medium contribution is given by − ˜Π ( m ) ij q = (cid:18) δ ij − q i q j | ~q | (cid:19) A + δ ij q | ~q | B , (6) − ˜Π ( m )44 q = B , − ˜Π ( m )4 i q = − q q i | ~q | B , (7)where the three scalar functions A ( q , | ~q | ), B ( q , | ~q | ), and C ( q = q + | ~q | ) are determined from the Feynman graphin Eq. (4) [11].Following [16], we introduce the projector P µν = δ µν − q µ q ν q , (8)and the transverse P Tµν projector P Tij = δ ij − ˆ q i ˆ q j , (9) P T = P T i = 0 , (10)with ˆ q i = q i / | ~q | . The longitudinal projector is then P Lµν ≡ P µν − P Tµν so that the polarization tensor is givenby ˜Π µν = ˜Π ( v ) µν + ˜Π ( m ) µν = FP Lµν + GP Tµν , (11)where F = − q (cid:18) C + B + q | ~q | B (cid:19) , (12) G = − q (cid:18) C + A + q | ~q | B (cid:19) . (13)The quadratic kernel is then˜Γ µν = ( q − F ) P Lµν + ( q − G ) P Tµν + 1 λ q µ q ν , (14)and its inverse, the photon propagator, reads˜Γ − µν = P Lµν q − F + P Tµν q − G + λq q µ q ν q . (15)On the other hand, one may also obtain EM responsesfrom the polarization tensor. Indeed, we have recently[11] shown that the (Fourier transformed) constitutiveequations of the REG are D j = ǫ jk E k + τ jk B k , H j = ν jk B k + τ jk E k , where we have used the notation ν ≡ µ − .The linear-response RPA tensors are ǫ jk = ǫδ jk + ǫ ′ ˆ q j ˆ q k , ν jk = νδ jk + ν ′ ˆ q j ˆ q k , and τ jk = τ ǫ jkl ˆ q l . The eigenvalues of ǫ jk are ǫ + ǫ ′ and ǫ . The eigenvector associated to ǫ + ǫ ′ isalong ˆ q k , thus longitudinal, whereas the two eigenvectorscorresponding to the eigenvalues ǫ are in directions trans-verse to ˆ q k . The same occurs for ν jk , with eigenvalues ν + ν ′ and ν , whereas τ jk is clearly transverse.The permittivities and inverse permeabilities are de-termined by the three scalar functions A ∗ , B ∗ and C ∗ ,with the asterisk denoting the continuation to Minkowskispace, q → iω − + , of the Euclidean scalar functions A ( q , | ~q | ), B ( q , | ~q | ) and C ( q + | ~q | ) obtained from thepolarization tensor ˜Π µν [11, 23, 24], i.e., ǫ = 1 + A ∗ + (cid:18) − ˜ ω ˜ q (cid:19) B ∗ + (cid:18) ω ˜ q − ˜ ω (cid:19) C ∗ , (16) ν = 1 + A ∗ − ω ˜ q B ∗ + (cid:18) − ˜ q ˜ q − ˜ ω (cid:19) C ∗ , (17) ǫ ′ = − ν ′ = − (cid:20) A ∗ + ˜ q ˜ q − ˜ ω C ∗ (cid:21) , (18)and τ = − ˜ ω ˜ q (cid:20) ˜ q ˜ q − ˜ ω C ∗ + B ∗ (cid:21) . (19)For the longitudinal responses, one then obtains ǫ L = ǫ + ǫ ′ = 1 + C ∗ + (cid:18) − ˜ ω ˜ q (cid:19) B ∗ , (20) ν L = ν + ν ′ = 1 + 2 C ∗ + 2 A ∗ − ω ˜ q B ∗ . (21)Our formulae make use of the dimensionless variables˜ q = | ~q | /q c , ˜ ω = ω/ω c , ˜ β = mc β , and ˜ ξ = ξ/ ( mc ), where q c = mc/ ~ is the Compton wave vector, ω c = mc / ~ isthe Compton frequency, β = 1 / ( k B T ), T is the abso-lute temperature, and ξ is the chemical potential of theelectron gas.We now use Eqs. (12), (13) and write the propagatorsin Minkowski space by letting q → iω − + , q = q + | ~q | → − q = | ~q | − ω and A , B , C → A ∗ , B ∗ , C ∗ . Then,one obtains 1 q − F → − q ǫ L , (22)1 q − G → − q [ ν L + 1] . (23) Eq. (22) leads to a pole in the P Lµν longitudinal propa-gator whenever ǫ L ( ω, | ~q | ) = 0 . (24)Note that the pole at q = 0 is not realized in this caseas it corresponds to a transverse mode, as we shall ex-plicitly show. We remark that (24) corresponds to theusual condensed matter dispersion relation of longitudi-nal plasmon collective excitations.The P Tµν transverse propagator [cf. Eq. (23)] has poleswhenever ν L ( ω, | ~q | ) = − , (25) q = ω − | ~q | = 0 . (26)Analogously to the longitudinal case, Eq. (25) yieldsthe dispersion relation of transverse plasmon collectiveexcitations whereas Eq. (26) yields a photonic mode thatpropagates with the speed of light c (= 1) in vacuum.In order to have more explicit expressions for the plas-mon modes, it is useful to write the projectors as P µν = n (1) µ n (1) ν + n (2) µ n (2) ν + n (3) µ n (3) ν , (27) P Tµν = n (1) µ n (1) ν + n (2) µ n (2) ν , (28)where n ( i ) µ = (0 , ˆ n ( i ) ), ˆ q. ˆ n ( i ) = 0, | ˆ n ( i ) | = 1, for i = 1 , n (1) i ˆ n (1) j + ˆ n (2) i ˆ n (2) j + ˆ q i ˆ q j = δ ij . For n (3) µ , wefind n (3) µ = −| ~q | p q , q ˆ q p q ! , (29)if we demand that it must be normalized and orthogonalto q µ and n ( i ) µ , i = 1 ,
2, thus satisfying n (1) µ n (1) ν + n (2) µ n (2) ν + n (3) µ n (3) ν + ( q µ q ν /q ) = δ µν . Then P Lµν = n (3) µ n (3) ν , (30) P Tµν = n (1) µ n (1) ν + n (2) µ n (2) ν . (31)A few observations are in order:(i) in Minkowski space, we have n (3) µ = i | ~q | p q , ω ˆ q p q ! , (32)which in the long-wavelength limit becomes n (3) µ = (0 , ˆ q );(ii) in that limit, Ref. [11] obtains Drude expres-sions for ǫ L = 1 − ( ω e /ω ) and ν L = 1 − ( ω m /ω ).Inserting this into (22) and (23), and using the fact that ω m = 2 ω e , we find ω e − ω as the denominator for bothlongitudinal and transverse plasmon propagators.The collective plasmon excitations correspond tocharge density and current density oscillations. Indeed,the collective field A L ≡ n (3) µ P Lµν A ν = A ν n (3) ν , in Eu-clidean space, is given by A L = − i~q. ( − i~qA + iq ~A ) p q | ~q | = − ~ ∇ . ~E p q | ~q | . (33)Since the field has a longitudinal component, we maydefine an effectice charge density ρ e as ~ ∇ . ~E ≡ ρ e ( q ).Similarly, the collective field A Tµ ≡ P Tµν A ν is given by(0 , ~A T ), where ~A T = A ˆ n (1) + A ˆ n (2) and A i = ~A. ˆ n ( i ) .One then obtains ~A T = i~q ∧ ( i~q ∧ ~A ) | ~q | = ~ ∇ ∧ ~B | ~q | . (34)We may then define an effective current density ~j e through ~ ∇ ∧ ~B = ~j e . Then, if we use (12), (13), and (14),and leave aside a gauge term, the plasmon Lagrangeanmay be written, in Minkowski space, as ρ e ( q ) (cid:18) ǫ L ~q (cid:19) ρ e ( q ) + j ke ( q ) ( ν L + 1)(1 − ω | ~q | )2 ~q j ke ( q ) , (35)where q = ( ω, ~q ). The above expression physically de-scribes the interaction of charge densities induced by thelongitudinal component of the fluctuating electric fieldsand current densities (loops in the plane perpendicularto ˆ q ) induced by the fluctuating magnetic fields. Apartfrom that, whenever ǫ L = 0 and ν L = −
1, we just havethe propagation of an electromagnetic wave with a prop-agator given by (23).An alternative and somewhat complementary analy-sis may be obtained from Maxwell’s equations combinedwith the constitutive relations written out previously.Maxwell’s equations are (we have now restored the speedof light c ) q i D i = 0 , (36a) q i B i = 0 , (36b) ǫ ijk q j E k = ωc B i , (36c)and ǫ ijk q j H k = − ωc D i . (36d)The constitutive equations were defined in the paragraphafter Eq. (15). From Eq. (36a) and the constitutiverelations, we derive ( ~q. ~E ) ǫ L = 0 . (37)If ǫ L = 0, then we must have ~q. ~E = 0, so that Eq. (36d)and the constitutive relations give h τ | ~q | + ǫ ωc i ~E + h ν | ~q | − τ ωc i (ˆ q ∧ ~B ) = 0 , (38) which combined with Eq. (36c) yields [25] a generalizedwave equation for ~E (and an analogous one for ~B ) (cid:20) | ~q | − µǫ ω c − µτ | ~q | ωc (cid:21) ~E = 0 . (39)However, using Eqs. (16) to (19), (20) and (21), Eq. (38)becomes ( ν L + 1)[ ~q ∧ ~B + ωc ~E ] = 0 , (40)whereas Eq. (39) yields( ν L + 1) q = ( ν L + 1) (cid:20) ω c − | ~q | (cid:21) = 0 . (41)We see that Maxwell’s equation (36a) will be satisfied if ǫ L = 0. This coincides with the longitudinal plasmoncondition. If ǫ L = 0, then ~E is transverse and eq. (41)will be satisfied if either ν L = − q = 0 (photons). The fact that the wave equationfactors out into two terms is a consequence of the specificform of the EM responses for the REG.The plasmon modes and the photonic mode obtainedfrom quantum responses to the electromagnetic fields willappear whenever the dispersion relation ω = ω ( | ~q | ) obeysone of the conditions derived on eqs. (24)-(26) ( ǫ L = 0; ν L = − ω = | ~q | , respectively). Otherwise, theelectromagnetic field will propagate with responses givenby ǫ ij ( ω, | ~q | ) and ν ij ( ω, | ~q | ).Before proceeding, we note that A ∗ , B ∗ , C ∗ are givenexplicitly by [11, 19] A ∗ = A α ˜ q − ˜ ω I + (cid:20) −
32 ˜ q − ˜ ω ˜ q (cid:21) B ∗ , (42) B ∗ = B α ˜ q − ˜ ω J , (43)and C ∗ = − C α (cid:26)
13 + (cid:0) γ (cid:1) [ γ arccot( γ ) − (cid:27) , (44)where γ = r ω − ˜ q − , (45) A α = B α = 4 α/π , C α = A α / α is the fine-structure constant, and the functions I and J are theone-dimensional integrals I = Z ∞ dy y p y + 1 F ( y, ˜ β, ˜ ξ ) × (cid:20) − ˜ q + ˜ ω y ˜ q F ( y, ˜ q, ˜ ω ) (cid:21) (46) −2 −1 x ~ b ~ h ~ = 0.01 h ~ = 1 h ~ = 10 0 1 2 3 410 −2 −1 x ~ b ~ h ~ = 0.01 h ~ = 1 h ~ = 10 0 1 2 3 410 −2 −1 x ~ b ~ h ~ = 0.01 h ~ = 1 h ~ = 10 FIG. 1: (Color online) Chemical potential as a function of thegas temperature. Solid, dashed, and dotted lines correspondto ˜ η = 0 .
01, ˜ η = 1, and ˜ η = 10, respectively. and J = Z ∞ dy y p y + 1 F ( y, ˜ β, ˜ ξ ) × (cid:20) y + 1) − ˜ q + ˜ ω y ˜ q F ( y, ˜ q, ˜ ω ) − ˜ ω p y + 12 y ˜ q F ( y, ˜ q, ˜ ω ) , (47)respectively. The functions F , F and F are defined as F (cid:16) y, ˜ β, ˜ ξ (cid:17) = 1e ˜ β (cid:16) √ y +1 − ˜ ξ (cid:17) +1 − ˜ β (cid:16) √ y +1+˜ ξ (cid:17) +1 , (48) F ( y, ˜ q, ˜ ω ) = ln (cid:20) (˜ q − ˜ ω + 2 y ˜ q ) − y + 1)˜ ω (˜ q − ˜ ω − y ˜ q ) − y + 1)˜ ω (cid:21) , (49)and F ( y, ˜ q, ˜ ω ) = ln " ˜ ω − ω p y + 1 + y ˜ q ) ˜ ω − ω p y + 1 − y ˜ q ) . (50)respectively. III. RESULTS AND DISCUSSIONA. The chemical potential
To compute the electromagnetic responses of the REGthrough Eqs.(16)-(19), one needs to obtain the ξ chemi-cal potential which depends on the temperature and car-rier density. The carrier density is η = ∆ N/V , where∆ N = N − − N + is the difference between the N − num-ber of particles and the N + number of antiparticles in x ~ h ~ x ~ h ~ x ~ h ~ FIG. 2: (Color online) Chemical potential as a function ofthe gas density. Solid, dashed, and dotted lines correspond to˜ β = 1, ˜ β = 10, and ˜ β = 100, respectively. −8 −7 −6 −5 −4 −3 −2 −1
1 3 5 7 9 r b ~ −8 −7 −6 −5 −4 −3 −2 −1
1 3 5 7 9 r b ~ −8 −7 −6 −5 −4 −3 −2 −1
1 3 5 7 9 r b ~ FIG. 3: (Color online) The ratio ρ = N + /N − [cf. Eq. (54)]as a function of ˜ β . Solid, dashed, and dotted lines correspondto ˜ η = 0 .
01, ˜ η = 1, and ˜ η = 10, respectively. the system. Then, one needs to solve the transcendentalequation [11]∆ N = N − − N + = X ~ p g f ( p, β, ξ ) , (51)where f ( p, β, ξ ) = 1e β (Ω p − ξ ) + 1 − β (Ω p + ξ ) + 1 (52)is the distribution function accounting for the presenceof both particles and antiparticles, Ω p = p p c + m c is the relativistic energy of a carrier with momentum p ,and g = 2 is the degeneracy factor of the electron gas.Eq. (51) reduces to˜ η = ηη = Z + ∞ dy y F (cid:16) y, ˜ β, ˜ ξ (cid:17) (53) · x e ( e V ) T (K)1 2 3 (a) · x e ( e V ) T (K)1 2 3 (a) · x e ( e V ) T (K)1 2 3 (a) · x e ( e V ) T (K)1 2 3 (a) · x e ( e V ) T (K)1 2 3 (a) · x e ( e V ) T (K)1 2 3 (a) x e ( e V ) T (K)1 2 3 (a) x e ( e V ) T (K)1 2 3 (a) x e ( e V ) T (K)1 2 3 (a) · x e ( e V ) T (K) (b) · x e ( e V ) T (K) (b) x e ( e V ) T (K) (b)
FIG. 4: (Color online) Chemical potential, measured with re-spect to the mc rest energy [cf. Eq. (55)], as a functionof the gas temperature for different values of the η gas den-sity. Solid and dashed lines corresponds to numerical resultsobtained from Eqs. (53) and (56), respectively. The set ofcurves 1, 2, and 3 in panel (a) correspond to η = 10 cm − , η = 10 cm − , and η = 10 cm − , respectively. In panel(b), calculations were performed for η corresponding to theelectron density in silicon. The full dot in the left vertical axiscorresponds to the Fermi energy obtained from Eq. (57). where η = g q c / (2 π ) ≈ . × cm − only dependson universal constants and may, therefore, be used asa natural unit to measure the η effective carrier den-sity of the REG. It should be noted [Eq. (48)] that F (cid:16) y, ˜ β, − ˜ ξ (cid:17) = −F (cid:16) y, ˜ β, ˜ ξ (cid:17) , i.e, F = F (cid:16) y, ˜ β, ˜ ξ (cid:17) isan odd function of the chemical potential. Eq. (53) im-plicitly defines the function ˜ ξ = ˜ ξ (cid:16) ˜ β, ˜ η (cid:17) . One sees that˜ ξ = 0 leads to η/η = 0, a case with corresponds tovacuum.The chemical potential as a function of ˜ β is displayedin Fig. 1. Calculations were performed for three differentvalues of the density expressed in units of η . The nu-merical results suggest a weak temperature dependenceof the chemical potential, if compared with mc ≈ .
511 MeV, in the low-temperature limit ( ˜ β → ∞ ). Actually,the chemical potential exhibits variations of a few eV inthe low-temperature limit, a fact which agrees with thenon-relativistic theory of the electron gas (see below).The chemical potential is displayed in Fig. 2 as a func-tion of the density expressed in units of η . Numericalresults were obtained for three different values of the gastemperature. The chemical potential is a growing mono-tonic function of ˜ η . One may note that the chemicalpotentials for ˜ β = 10 and ˜ β = 100 essentially coincide inthe scale of the figure [cf. dashed and dotted lines in Fig.2].The ratio ρ = N + N − = R + ∞ dy y e ˜ β ( √ y ξ ) +1 R + ∞ dy y e ˜ β ( √ y − ˜ ξ ) +1 . (54)as a function of ˜ β is displayed in Fig. 3 for various val-ues of the density. For a given temperature, it is appar-ent that the number of particles exceeds the number ofantiparticles in all cases and the ratio ρ = N + /N − de-creases as the density of particles in the electron gas isincreased, as expected. One may also note that ρ → β → ∞ , since in the low-temperature limit one has N + ≪ N − . In other words, in the limit ˜ β → ∞ , theterm corresponding to the occupation factor of antipar-ticles in the Fermi-Dirac distribution function [cf. thesecond term in the RHS of Eq. (52)] may essentially beneglected ( N + << N − ) and the non-relativistic limit ofthe Fermi-Dirac distribution function is eventually recov-ered.We have also explored the behavior of the chemicalpotential for density and temperature values appropriatefor solid-state materials. In this respect, we have defined ξ e ( T ) = ξ ( T ) − mc (55)as the non-relativistic chemical potential. According tothe non-relativistic theory of the free-electron gas, it iswell known that ξ e ( T ) ≈ E F " − π (cid:18) TT F (cid:19) , (56)where E F = (cid:18) π g η (cid:19) ~ m , (57)is the Fermi energy and T F = E F /k B is the Fermi tem-perature. The ξ e chemical potential is depicted in Fig. 4as a function of the gas temperature. Calculations in Fig.4(a) were performed for three different values of η vary-ing within the range exhibited by most of the solid-statematerials. In Fig. 4(b) we have assumed η ≈ . × cm − corresponding to the electron density in silicon.The solid line corresponds to the result computed by −200−100 0 100 200 0 0.005 0.01 R e ( e L ) w ⁄ w c (a) −200−100 0 100 200 0 0.005 0.01 R e ( e L ) w ⁄ w c (a) −200−100 0 100 200 0 0.005 0.01 R e ( e L ) w ⁄ w c (a) −200−100 0 100 200 0 0.005 0.01 R e ( e L ) w ⁄ w c (a) −30−20−10 0 10 20 30 0 0.005 0.01 R e ( e L ) w ⁄ w c (b) −30−20−10 0 10 20 30 0 0.005 0.01 R e ( e L ) w ⁄ w c (b) −30−20−10 0 10 20 30 0 0.005 0.01 R e ( e L ) w ⁄ w c (b) −30−20−10 0 10 20 30 0 0.005 0.01 R e ( e L ) w ⁄ w c (b) FIG. 5: (Color online) Real part of ǫ L as a function of the ω frequency in units of the ω c Compton frequency, for vari-ous values of the wave vector | ~q | expressed in units of the q c Compton wave vector, and for ˜ η = 0 .
01. Results of panels (a)and (b) were obtained for ˜ β = 1000 and ˜ β = 1, respectively.Solid, dashed, dotted, and dot-dashed lines in panel (a) [(b)]correspond to ˜ q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . combining Eqs. (53) and (55), whereas the dashed linewas obtained from Eq. (56). The Fermi energy computedfrom the non-relativistic electron-gas model [cf. full dotin the vertical axis of Fig. 4(b)and Eq. (57)] essentiallycoincides with the numerical result obtained from Eqs.(53) and (55) in the limit T →
0. Low-temperature re-sults obtained from the non-relativistic model agree withthose derived from the relativistic theory, as expected.
B. Longitudinal plasmon modes of the REG
Now we focus our attention on the ǫ ij electric permit-tivity tensor. According to Eq. (20) one has ǫ L = ǫ + ǫ ′ = 1 + C ∗ + (cid:18) − ˜ ω ˜ q (cid:19) B ∗ , (58) FIG. 6: (Color online) Upper ( ω Lp longitudinal plasmon fre-quency) and lower frequency zeros (solid and dashed lines,respectively) of the real part of ǫ L as functions of the | ~q | wavevector. Calculations were performed for ˜ β = 1 and ˜ β = 1000,by taking ˜ η = 0 .
01. Open circles correspond to the analyticalresults obtained from Eq. (65). Both ω and | ~q | are expressedin units of ω c and q c , respectively.FIG. 7: (Color online) Upper ( ω Lp plasmon frequency) andlower frequency zeroes of ǫ L (solid and dashed lines, respec-tively) as functions of the | ~q | wave vector. Calculations wereperformed for ˜ β = 100, ˜ β = 10 and ˜ β = 1 by taking ˜ η = 0 . q max ) beyond which the longitudinal plasmondecays into particle-antiparticle pairs. Both the frequencyand wave vector are given in units of ω c and q c , respectively. The real part of ǫ L is depicted in Fig. 5 as a functionof ω in units of ω c . Results were computed for ˜ η = 0 . | ~q | expressed in units of q c . InFigs. 5(a) and 5(b) we have set ˜ β = 1000 and ˜ β = 1,respectively. The real part of the longitudinal electricpermittivity exhibits two zeros for a given value of ˜ q ata given temperature. The lower zero lies within a regionwhere Im[ ǫ L ] = 0. Therefore, in spite of the fact thatRe[ ǫ L ] = 0 in this case, such a zero cannot be considered −4−2 0 2 0.005 0.01 0.015 0.02 0.025 R e ( n L ) w ⁄ w c (a) −4−2 0 2 0.005 0.01 0.015 0.02 0.025 R e ( n L ) w ⁄ w c (a) −4−2 0 2 0.005 0.01 0.015 0.02 0.025 R e ( n L ) w ⁄ w c (a) −4−2 0 2 0.005 0.01 0.015 0.02 0.025 R e ( n L ) w ⁄ w c (a) −4−2 0 2 0.002 0.004 0.006 0.008 0.01 R e ( n L ) w ⁄ w c (b) −4−2 0 2 0.002 0.004 0.006 0.008 0.01 R e ( n L ) w ⁄ w c (b) −4−2 0 2 0.002 0.004 0.006 0.008 0.01 R e ( n L ) w ⁄ w c (b) −4−2 0 2 0.002 0.004 0.006 0.008 0.01 R e ( n L ) w ⁄ w c (b) FIG. 8: (Color online) Real parts of ν L as a function of the ω frequency in units of the ω c Compton frequency, for variousvalues of the wave vector | ~q | given in units of the q c Comptonwave vector. Calculations were performed for ˜ η = 0 .
01. Re-sults of panels (a) [(b)] were obtained for ˜ β = 1000 ( ˜ β = 1).Solid, dashed, dotted, and dot-dashed lines in panels (a) [(b)]correspond to ˜ q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . as a plasmon frequency. In the vicinity of the upper zero,on the other hand, one may have Im[ ǫ L ] = 0 (see discus-sion below), so that it corresponds to the ω Lp longitudinalplasmon frequency. We then show in Fig. 6 the upper(solid lines) and lower (dashed lines) frequency zeros ofthe real part of ǫ L as functions of ˜ q . Numerical calcu-lations were performed for ˜ η = 0 .
01 and two differentvalues of ˜ β .For sufficiently small values of the ˜ q wave vector, thelongitudinal-plasmon dispersion relation may be approx-imately described by the expression(˜ ω Lp ) = (˜ ω L p ) + v T c ˜ q , (59)where the frequencies and wave vector are given in unitsof ω c and q c , respectively. In the above expression ˜ ω L p is the temperature-dependent longitudinal plasmon fre- quency in the long-wavelength limit and v T plays the roleof the carrier thermal speed, except for a numeric factor.The low-temperature non-relativistic case v T = 3 / ( mβ )leads to the well-known Bohm-Gross dispersion relation[26]. To estimate the carrier thermal speed correspond-ing to the REG, we have computed the average of p fromthe expression h p i = P ~ p g p f ( p, β, ξ ) P ~ p g f ( p, β, ξ ) , (60)which reduces to h p i = m c K ( ˜ β, ˜ ξ ) , (61)where K ( ˜ β, ˜ ξ ) = 1˜ η + ∞ Z dy y F (cid:16) y, ˜ β, ˜ ξ (cid:17) . (62)The v T carrier thermal speed is given by h p i = m − v T c v T , (63)which results in v T = c K ( ˜ β, ˜ ξ )1 + K ( ˜ β, ˜ ξ ) . (64)Then Eq. (59) becomes(˜ ω Lp ) = (˜ ω L p ) + K ( ˜ β, ˜ ξ )1 + K ( ˜ β, ˜ ξ ) ˜ q . (65)The above equation is a generalization of the Bohm-Grossdispersion relation to the case of a REG at finite tem-peratures and is valid in the limit ˜ q ≪ ˜ q max , i.e, farfrom the region where particle-antiparticle excitations oc-cur. In condensed matter, that corresponds to electron-hole pairs whereas in a relativistic gas that correspondsto electron-positron pairs. Theoretical results obtainedfrom Eq. (65) are displayed in Fig. 6 as open circles, forboth ˜ β = 1 and ˜ β = 1000, and for ˜ η = 0 .
01. In the long-wavelength regime, the agreement with the curve fromEq. (24) is quite good.We display in Fig. 7 the upper (solid lines) and lower(dashed lines) frequency zeros of the real part of ǫ L asfunctions of ˜ q = | ~q | /q c . Numerical results were computedfor ˜ η = 0 .
01 and different values of ˜ β . Fig. 7 clearly indi-cates that there is a maximum value of the wave vector(˜ q max ) beyond which the longitudinal plasmon decaysinto particle-antiparticle pairs. FIG. 9: (Color online) Transverse plasmon frequency (zeroof ν L + 1 = 0) as a function of the | ~q | wave vector. The ω Tp transversal plasmon frequency and the | ~q | wave vector aregiven in units of the ω c Compton frequency and q c Comptonwave vector, respectively. Solid lines correspond to theoreticalresults obtained from ν L +1 = 0. Calculations were performedfor ˜ β = 1 and ˜ β = 1000, by taking ˜ η = 0.01. Open circlescorrespond to the analytical results obtained from Eq. (66). C. Transverse plasmon modes of the REG
The condition ν L = − ω Tp fre-quency of the REG transverse plasmon modes. We dis-play in Fig. 8 the real part of ν L as a function of the ω frequency in units of ω c , obtained for ˜ η = 0 .
01 anddifferent values of the wave vector | ~q | in units of q c . Re-sults depicted in Figs. 8(a) and 8(b) where computedfor ˜ β = 1000 and ˜ β = 1, respectively. We would liketo stress that numerical results (not shown here) indi-cate that Im [ ν L ] = 0 within the respective frequencyranges considered in both Fig. 8(a) and 8(b). Therefore,in the present cases, the transversal plasmon frequenciesmay be obtained by solving the transcendental equationRe [ ν L ] = − ω Tp transverse plasmon frequency is displayed inFig. 9 as a function of the ˜ q = | ~q | /q c wave vector for dif-ferent values of the gas temperature. Calculations wereperformed for ˜ η = 0.01. Solid lines correspond to thenumerical results obtained from ν L = − ν L + 1 for a given value of ˜ q at a given gas tem-perature. Therefore, the dispersion exhibits only a singlebranch. We also note that, at a given temperature, thewave vector dependence of the resonance frequency maybe approximated by(˜ ω Tp ) ( q ) = (˜ ω T p ) + ˜ q , (66)which fits quite well the numerical results obtained fromEq. (25). In the above equation ˜ ω T p is the temperature-dependent transverse plasmon frequency numerically ob-tained from Eq. (25) in the limit ˜ q →
0. Results obtained
FIG. 10: (Color online) The dispersion curves for transverseand longitudinal plasmon modes at T = 0 and ˜ η = 0 .
01 [cf.Eqs. (24) and (25)]. In the shaded area, Im [ ǫ L ] = 0, in-dicating decay of the longitudinal mode. The dashed lineis discarded as a solution for the longitudinal dispersion as itlies entirely in the region of nonzero imaginary part of ǫ L . Wehave also shown the dispersion (dashed line) for the photonmode ˜ ω γ = ˜ q [see Eq. (26)]. from Eq. (66) are displayed in Fig. 9 as open circles. D. Decays and responses of the REG
We display in Fig. 10 the dispersion curves for thetransverse and longitudinal plasmon modes at T = 0 [cf.Eqs. (24) and (25)]. The shaded area corresponds to theregion where the excitation of particle-antiparticle pairsoccurs. The dashed line is discarded as a solution forthe longitudinal plasmon dispersion as it lies entirely inthe region of nonzero imaginary part of ǫ L . We havealso shown the dispersion for the photon mode ˜ ω γ = ˜ q [cf. (26)]. Although not shown in the figure, the straightdotted line for the photon dispersion will eventually reachthe upper region where the excitation of electron-positronpairs will take place [11].Finally, we display in Fig. 11 the different relevant re-gions of the (˜ q, ˜ ω ) plane where the real parts of ǫ L and ν L have different signs [27].We would like to stress there isa region where both ǫ L and ν L are simultaneously nega-tive, indicating that the REG exhibits a behavior has notbeen experimentally observed in natural materials. Thisfact was previously remarked by one of the authors [11],as mentioned. It is important to note that, in contrastto the non-relativistic case where the refractive index isdefined as n = √ ǫ √ µ , simultaneous negative values of ǫ L and ν L observed in the present relativistic case do notimply negative refraction.A figure similar to Fig. 11 can be obtained using thevalues of densities and temperatures encountered in as-trophysical scenarios, as in a superdense electron-plasma(e-p) in Gamma-ray bursts (GRBs) [5, 28], where the e-p0 ν L ν L ν LL LL
FIG. 11: (Color online) Regions of the (˜ q, ˜ ω ) plane accordingto the signs of the real parts of ǫ L and ν L . Outside the shadedregions the real parts of ǫ L and ν L are positive. Results wereobtained for T = 0 and ˜ η = 0 . density is in the range of η = (10 − )cm − .According to ref.[5], in Condensed Matter, e-p plas-mas will eventually be produced in the laboratory withlaser systems. Laser pulses with focal densities I =10 W cm − incident on material targets could lead toe-p plasmas with the densities in the range of (10 − )cm − . We have used the upper limit of that densityrange in our calculations.A study of the refractive index of the REG will be the subject of a further investigation, in which we intend togeneralize the results of Lepine and Lakhatakia [29, 30]. IV. CONCLUSIONS
Summing up, we have presented a theoretical study ofthe EM propagation and responses of a REG for varioustemperatures and carrier densities. Using linear responseand RPA, we have identified the propagation modes andtheir dispersion relations from the QED propagators aswell as from Maxwell’s equations with the added inputof the constitutive relations obtained from the QED re-sponses. We found a longitudinal plasmon mode, twotransverse plasmon modes, and a photonic mode whichpropagates with the speed of light in vacuum, i.e., forwhich the medium is transparent thanks to the specificform of its relativistic electromagnetic responses. In de-riving dispersion relations, we were able to identify sta-ble solutions and regions of instability where the plasmonmodes decay. Finally, we have also identified the regionsin the ( | ~q | , ω ) plane where the longitudinal permittivity ǫ L and longitudinal inverse permeability ν L are both si-multaneously negative. Acknowledgments
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