Dhar, Tapasi

Godia, S P

Harsh, O K

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N O T EIndian J. Phys. 68A (6). 579 -  585 (1994)U P  A- an imemational journalStudy of the behaviour of electron gasO K Harsh"^ , Tapasi Dhar and S P Godia*Department of Past Graduate Studies and Research in Physics, Feroze Gandhi College, Roe Bairli, Uttar Pradesh, IndiaReceived 29 September 1993. accepted 22 July 1994Abstract : Effect of perturbations on plasma oscillations in degenerate electron gas in metals have been studied. By taking acurount of pertuibcd long range forces and the effect of short range interactions, it has been shown that there is a critical wave vector at which the plasma oscillations can be realized The effect of radius parameter r ,^ plasma energy and interatomic distance on the dispersion characteristic's have been discussed The effect of perturbations decreases the critical wave vector for exciting the plasma oscillations. Our studies reveal that in beryllium both first order perturbed and relative first and second order perturbed dispersion curves are qualitatively similar to that of the unperturbed curve, while for aluminium these contributions are oppositeKeywords : Plasma oscillations in metals, critical wave vector, dispersion relationsPACS No. : 52.35-gThe valence electrons in solids may oscillate collectively under certain conditions with a high eigen-frequency Wp— t^he plasma frequency [1-111, and is denoted by the relation2V/2=Annewhere n is the electron concentration and m is the effective mass of an electron. Such oscillations in electron gas of solids play very important role in the study of various propertiese.g. determining the frequency range of transparency and reflection coefficient of electromagnetic waves [5], diagnosing various features of bulk metal in the form of thin metallic films etc.*  Foiiiieriy at Lucknow University and Institute of Technology, Banaras Hindu University. Varana-si, India* Physics Department, V. S. S. D. College, Kanpur, India© 1994 lACS580 O K Harsh, Tapasi Dhar and S P GodiaFor metals, if the ratio of phase velocity to the Fermi velocity of electron gas is not very large, then the electromagnetic waves of frequencies much higher than the electron- plasmon frequency are damped out 16]. At the limit of zero temperature, the responsefunction of electro^ gas in metals is purely real and natural oscillations are sustained, while at non-zero temperatures, the above mentioned velocity ratio decreases and perturbations are initiated. Due to the presence of such perturbations the total energy of the electron gas becomes a negative quantity and the frequency of plasma oscillations decreases. For higher values of the wave vector (A:), the phase velocity of electron gas oscillations is less than the net average electron gas velocities and the response function becomes complex. Under these perturbed conditions, for high k, the damping of natural oscillations is enhanced which results into complete cessation of oscillations at k = k^ ., where kc is the critical wave vector. Thus the study of perturbation on plasma oscillations for degenerate electron gas plays a vital role in understanding the sharp cut-off of reflection and transmission coefficients and hence^n understanding the principle of filters.In the present work, studies have been made for di.spersion characteristics of a degenerate electron gas in the presence of small amplitude perturbations. For such a gas, the unperturbed dispersion equation may be written as [2,7]6An?-= w + — E —   ^ 5 ^ m+ Exchange term, ( 1 )where is the minimum average energy of an electron, which may be approximated to Fermi energy for monovalent metal at a very low temperature.If we include the bilinear and the exchange interaction between electrons which were not considered by Bohm and Pines [2], in eq. (1), then2 2 ^ , 2  2 ^ k ^  r.= VV,, + -* ^fmV J \4 2 4 7)3 113)3^--- log—+ -------- + --------5 P 5 6 603(2)where fgx the exchange interaction between the electrons and p = The results of eq. (2) may further be improved by including the Exchange effect [12] by a proper linearization of the interaction term, known as the generalized random phase approximation (GRPA) [IS­IS]. For this, one may use the jellium model of the electrons in solids. Using this technique, Glezos [12] obtained the following dispersion relation :6QQ ^  —  i\ -  0.068 r )P 5 £ 2(3)where Q = wJE^, 12^  = w^jE^,  Q = kjk^ and = '’o/^o which is a dimensionless parameter, tq being the average interelectronic distance and a^ is the first Bohr radius.Using the firsi-order perturbation correction due to short range Coulomb interactions, the average energy per electron of the oscillating electron gas in metals may be written as [7],Study of the behaviour of electron gas 581-  2.21 V3/J^ 0.916Ry. (4)The value of corresponding to minimum average energy of an electron maybe obtained by neglecting the last term in eq. (4) and taking its first derivative with respect to P equal to zero. In this way the value of P obtained is = 0,353 .The plasma energy for the metal can be computed with the help of formula [8],f Z a£^(0) = ;rw,,= 28.8 1 —1/2(5)where Z is the total number of electrons participating in plasma oscillations (valence electrons), a  is the specific gravity and W is the molecular weight. The dimensionless parameter r, may be expressed as [3]r = ( 47.11 ^ V J (6)where is the calculated plasma frequency as mentioned in eq. (5).With the help of eqs.(2), (4) and (6) the effect of first order perturbation on the dispersion characteristics of plasma oscillations in several metals i.e. Be, Al, Mg, Li and Na have been studied (Figures 1-5).The second order perturbation energies can be obtained by subtracting the contribution due to long range Coulomb interaction from the sum of the long and short range second order perturbation contributions of Coulombic interactions as given by Nozieres and Pines [9J and may be written asAE = 0.0311 logr^ -  0.0003 -  0.096 Ry. (7)On taking the sum of eqs. (4) and (7), we draw graphs (Figures 1-5) to show the relative contribution of the first- and second-order perturbation on the dispersion characteristics of plasma oscillations. With the help of these graphs, the effect of various parameters e.g. Fermi energy, r  ^and plasma energy can be studied (Tables 1 and 2),Critical wave vector:The critical wave vector which limits the plasma excitations, is determined by the intersection of the disp>ersion function of eq. (2) and the following express on for the maximum energy to excite an individual electron68A(6)-10r a y  (frkF + ^2m 2m ’582 O K Harsh, Tapasi Dhar and S P Godia(8)where Hkf is the nioinentuin of an electron at the Fermi surface.The plot in Figures 1 to 5 for metals Be. Al, Mg, Li and Na show the effect of first and second order perturbations over the unperturbed cases. One can see that the value of decreases as we go towards the first and the second order nerturbations respectively. Also.FIG 2FIG 3 FIG 6Figures (1-5). Excitation Energy (£*) vs wave vector (/c) graphs for Be, Al, Mg, Li and Na respectively, representing(a) Maximum energy to excite an individual electron (eq 8)(b) Plasmon dispenion (unperturbed)(c) Plasmon dispersion with first order perturbation(d) Plasmon dispersion with second order perturbation.the value of kc is higher for metals with lower The rate of decrease of critical wave vector for both the first and the second order perturbation increases as the plasma energy decreases or as the value of decreases (Tables 1 and 2).At the higher value of the dispersion characteristics of Li, Na and Mg are divergent downwards in both the first- and second-order perturbations, while in case of Be, theydiverge upwards. For Al, these two perturbation graphs diverge in opposite direction (see the graphs). This abnormal behaviour has only been found in case of Al. This may be due to the fact that aluminium may behave as a superconductor under certain condition.Table 1 . Effect of various parameters on plasmon energy.Study of the behaviour of electron gas 583SINo.Element Total no of electrons participating in plasma oscillationNearest neighbour distance (in A) RadiusruirameterPlasmon Energy (eV)Calculated (eq H)Experimental Ref [10,11]BeAlMguNa2.22 2.86 3.20 3 023 3 6591.882 07 2.65 3.253 9318.45 15 8 10 89 7.99 5 9219 1 15,3 106 7 12 5 71Table 2. Relative contribution ol the first & second order perturbationsSIno,Element Unperturbed> Pertuibed kfRelative decrease of kf. (percent)1 si order 2nd order 1 St order 2nd order1 Be 1 223 1 0154 1 002 16 97 1..322 . Al 1.14 0 94 0.929 17.54 1.173 Mg 0 9784 0.8015 0 7803 18,08 2.644 b 0 8570 0 6949 0.6753 1891 2 825 Na 0 747 0 5905 0.577 20.96 2 26Graphs have also been drawn in Figure 6 showing the variation of excitation energy vs wave vector, by using the eqs. (3) and (8) for Be for which experimental results exist [12]From these graphs (unperturbed, first- and second-order), it can be concluded that the inclusion of effect of exchange interaction in the Be reduces the value of critical wave vector (it J  for exciting the plasmon oscillations. The value of the critical wave vector for Be have been found as 1.0096 A“‘, 0.9947 A“‘ and 0.9796 A"' for the unperturbed, first- and second- order perturbations respectively. These values are in excellent agreement with the work of other authors [ 16,17],The values of the critical wave vectors obtained for Be in latter case may be applied to explain the minimum plasmon scattering angle, as experimentally observed by Suzuki and Tanokura [11] and Papademetriou et al [18] using incident CrK^  ^ and CuK„^^ radiation for studying the inelastic scattering phenomenon in Be. In our present results the values for the critical wave vectors (using GRPA) are 1.0096 A 0.9947 A ' and 0.9796 A for584 O K Harsh, Tapasi Dhar and S P Godiaunperturbed, first- and second-order perturbations respectively. Using the slow-fastk.interaction effects of Langreth [ 19] as -  0.1, (where kp being the Fermi waveFigure 6. Excitation energy ^  vs reduced wave vector Q = graphs for Be, lepiesentmg ^(a) Limit of partide-hole spectrum [12] \(b) Plasmon dispersion [ 12] (unperturbed)(c) Plasmon dispersion with first-order perturbation(d) Plasmon dispersion with second order perturbation, while asterisks stand for experimental values 1 12]vector), the values of minimum critical angles have been calculated as 15.57°, 15.28° and 14.9° for unperturbed, first- and second-order perturbations respectively which are close to 10° and 16° as observed by Suzuki and Tanokura 111] and Papadcmctriou etal [18].The propagation of electromagnetic waves and their optical properties are determined by the above mentioned features of electron gas oscillations. It appears that the precise measurements of optical proporties at various temperature and different fractions of impurities would help in diagnosing the electron gas in metals. The role played by the mode of preparation of thin films for various electron densities can be analysed by having such features of electron gas oscillations.References[1 ] D Pines in Solid State Physics cd. F Seitz and D Turnbull (New York : Academic) Vol. 1 p 368 (1955)[2] D Dohm and D Pines Phys. Rev. 92 609 (1953)[3] D Pines in Elementarv Excitations in Solids (New York ■ Benjamin) (1964)[4] O K Harsh in DSc Thesis (Kanpur University . Kanpur, India) (1987)[5] R H Ritchie Prog. Theo. Phys. 29 603 (1963)[6] A A Maraduddin und D L Mills Phys. Rev B7 2787 (1973)[7] S Roimes Many Electron Theory (Amsterdam North Holland) (1972)[8] L Morton, L B Leder and H Mcndlowitz Advances in Electronics and Electron Physics edited by L Maiton V ol. 7 (New York . Academic) p 225 (1955)[9] P Nozieres and D Pines P/iyr. Rev. I l l  442(1958)[10] C  Kittel Introduction to Solid State Physics 4th edn. p 277[11] T Suzuki and A Tanokura J. Phys. Soc. Jpn. 29 972 (1970)Study o f the behaviour o f electron gas 585[12] N M Glezos Phys. Rev. B43 7538 (1991)[13] D Pines and P Noziers The Theory o f  Quantum Lujutds (New York ; Benjamin) (1966)[14] PM Platzmairand X Wolff Solid State Physics Supplement 13 (New York : Academic) (1966)[15] S Ichimarv Rev. M od. Phys. 34 1017 (1982)[16] FBrosens.JTDevreeseandLFLemmens P/iyx. B21 1363 (1980). 21 1349(1980); PAy.v. 5rar. So/idi B74 45 (1976)[17] H Kanazawa and Sho-ichiro Tam Prog. Theo. Phys 19 153 (1958)[18] D K Papademetriou, D Katsanos and A G Doukas Phys Stat Sol. (b) 133 223 (1986)[19] DCLangrcth Phys. Rev. B1 471 (1970)

Study of the Behavior of Electron Gas

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Study of the Behavior of Electron Gas

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