COLLECTIVE OSCILLAfiGNS 03:“ A NEARLY DEGENERATE $03139 S‘EATE PLASMA Thesis for flu: Degree of M. S. NJCH!%N STATE Ué‘éiVERSETY lamas 1:. Presses; if. 19.53. SSSSSS IIIIIIIIIIIIIIII III I III II 3129301743 18 86 LIBRARY Michigan State Univqsicy PHYSICS-ASTRC'fiOMY ”MARY ”1.....7 A SEP 1991 COLLECTIVE OSCILLATIONS OF.A NEARLY DEGENERATE SOLID STATE PLASMA By James F. Prosser, Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics and.Astronomy 1963 m “x ‘ q~lv ABSTRACT COLLECTIVE OSCILLATIONS OF.A NEARLY DBEENERATE SOLID STATE PLASMA by James F. Prosser Jr. The spectrum of collective oscillations of a 2-component nearly degenerate plasma which is carrying current has been studied by gen— eralizing a previous zero—temperature theoryl. The case of finite temperatures small compared to the degeneracy temperature has been discussed through an expansion of the plasma dispersion relation in powers of kT/Ef where kT is the thermal-excitation energy and Ef is the relevant Fermi energy. The effects of finite temperature on the stabil— ity of the acoustic modes and the damping of the high—frequency modes are discussed. The thermal effects studied are thoughtlikely to play a role in high-current experiments on semiconductors and semimetals possessing experimentally accessible degeneracy temperatures. 1M. J. Harrison, J. Phys. Chem. Solids 22, 1079 (1962). ACKNOWLEDGMENTS The author is indebted to Dr. M. J. Harrison for the suggestion of this research problem and for his continuous guidance and instruc- tion. ii TABLE OF CONTENTS Page I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . l A. One Component Plasmas . . . . . . . . . . . . . l B. TWO Component Plasmas . . . . . . . . . . . . . 7 II. OSCILLATIONS IN A CURRENT CARRYING DEGENERATE PLASMA . . 9 III. EXPERIMENTAL EVIDENCE FOR PLASMA OSCILLATIONS . . . . . 18 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . 20 iii I. INTRODUCTION A. One Component Plasmas In recent years, a theory has been developed which takes into ac- count the collective aspect of the interaction between electrons in a metal. This theory offers a simple justification for the independent particle model and also allows collective properties to be calculated. This is the theory of Plasma Oscillations in solids developed prim- arily by Bohm and Pines.1 A plasma can be defined as a large number of positive ions embedded in a sea of free electrons with zero total charge. If there is no thermal motion, one can easily see that if a slab of electrons is displaced from its equilibrium position the elec- trons will oscillate about this point. One can easily calculate the frequency of this oscillation by the method of Tonks and Langmuir. If, in two dimensions, there is a small diSplacement '3 (x’t) of the elec- trons originally at x and if the positive ions remain fixed, there will be a net force tending to restore the electrons to their equilibrium position. The total charge density which has crossed the line x +3 is ne13(x’t) where n is the equilibrium charge density. Then by Gauss' theorm the electric field at this point is given by E{x + ‘(x’t),t] = Lume3(t) (1) therefore the equation of motion becomes 1 STE-9 = vii mm: <2> . . the plasma frequency is given by 2 e?- hnn)1/2 COP=+(—-—m—' . ' (3) A more detailed classical theory takes into account the effects of thermal motions. This will give a keener insight into the nature of plasma oscillations. After the method of Bohm and Pines, consider n electrons in a cubic box of unit volume. This box also contains an equal uniform density of positive charge. The Fourier expansion of the potential energy between two electrons in the box is: 1g k If periodic Boundary Conditions are applied, 5 = 2"(n121 + n2§2 + “$33)° (5) and Ck is given by C " l 33 exp[-i}_g_1-(£i—£j)] d3r (6) k s (2") ’u‘ because of the orthogonal properties of the Fourier terms. This integral is equal to Area/hz except when.5 = 0. When E = 0 this integral becomes dr. C0 = 62 _.l . (7) r . 1J This is just the contribution due to one charge distributed thru the box. This can be seen if we define P = e/volume, Then lei=efg§=e—§—-d—t (8) in volume r . H but we are considering a unit volume. Therefore equation (8) reduces 3 to equation (7) and the total potential of the ith electron is I n . Ui(r) = E: Z 5-2- eXp[15_-(_I:_i-_I;j)] (9) ifi 5 k The prime over the second summation sign means the §.= 0 term is not to be included. The R = O term is excluded because this term just cancels the contribution due to the positive charge distribution. Therefore for the ith electron, rm}i = “'51 (lO) 1 => 3.5: = M82 :1 X EBXPUEQi-gjfl (11) m k2 The actual charge density can be written as: N P = 215g - rj) (12> J=1 Again going to the Fourier representation F = I: (:95? (13) k ll where F 1 {5. 8—15-1313 dsr (III) k‘ (2n)3 Pk. = 2 e-iE'E-i (15) k Therefore N - - Z. 15-31615Ti (16) :r-1D- I LI 0. N . I"k = - Em my + iriiI 6151111 (17) But ‘7i is given by equation (11) and ?k is given by equation (lb), Therefore ‘9}: = -: (5-392 {1523i 1 (18) - 3:123 E 2 212.1. (3!. 6X10 [ME-ELL] Equation 18 can be split up into two parts, one with k = k' and the other with k 7-! k'. Then the second term in equation 18 becomes: _i__f’_ — if 23 _-_; I; PM. on k k The second term in equation 19 contains the phase factors exp[i(§fk')§£]. These depend on the positions of the particles and tend to cancel out since there are a large number of particles randomly distributed throughout the medium. This is called the Random Phase approximation and is discussed by Bohm and Pines.l Therefore, Pk = - Egg-Xi)?- e—iE'Ei ~05 Pk (20) where qu is the plasma frequency. The first term on the right in equation 20 is due to the thermal motions of the particles. Obviously for small values of k, this term is small compared to the other term and can be neglected. We can then see from equation 20 that there is a normal mode of vibration of the plasma with frequencywp. Equation 20 5 then shows both the collective behavior of the particle and the random motion due to thermal effects. And if we rewrite equation (20) as (I: = - 2 [Ii-2,)?- NJ] 245% (21) i we see that a rough criterion for the normal mode oscillation to be in— dependentexcitations is Up >> 3'2, which implies substances with low thermal velocities but high densities favor collective behavior. The previous example can be extended out of the classical realm by noting that equation (20) implies the plasma can be represented by n harmonic oscillators with frequency(‘)p. But it is well known that a quantum harmonic 05cillator will have energy (N + l/2)‘h). In the case of a solid state plasma, Pines has called this quantum harmonic oscillator a plasmon. The zero point energy is given by N = O and the energy required to excite one plasmon will be ‘W p' This energy hnezn m can be easily calculated using(4)§ = If this is done2 it will be found that at normal temperatures the energy required to ex— cite one plasmon is higher than the energy of an electron at the top of the Fermi Dirac distribution. Therefore a plasmon will not be excited by ordinary thermal motions but will remain in the ground state unless excited by other methods,such as passing fast electrons thru the metal. This is why the independent particle model gives such good results. The plasmons will not be excited by most processes at normal temperatures and therefore the collective behavior can be ignored. 6 Let us now consider the short range interactions of the electron gas. The total potential energy of the electron gas is C! II g3 282/111 3 rij = E‘B‘ 2 Z 2' 21T€2 2 ifi k k QXPUE' (ii-QM (21) (22) but we have seen above that the collective behavior is given by the -l excitation of plasmons for k (g kc, kcc(vO/QA. Therefore if the collective contribution is subtracted out we will be left with the short range residual interactions. In this case equation (22) becomes: 1 _ 211e2 , . Us.R. ‘ g7}: 2}? k2 OM15 (331-ng K7K. Therefore for two electrons at a distance r apart, U _ 2fl€2 lkor “ k—nre“- k>k c If this sum is changed into an integral we can write 2 ‘ l ik~r U = gr}? I E2 8 — — d3k (25) k>kn n d 2 = Ei— J dk y eikrcos 6 sin 6 de G o 49 2e2 sin kr e2 R]? dk = ?- F(kCr) 1T (23) (2h) (26) (27) 7 The integral in equation (27) cannot be evaluated analytically, but it is a well known function and there exist numerical tables to evaluate it. If this is donéLit can be shown that except for slight deviations r which need not concern us here, F(kCr) 3.". e—kc . So that equation (27) now has the form U = e2 e-kc r (28) and one immediately sees that the effect of the short range interaction is to screen the coulomb potential. B. Two Component Plasmas Consider now, a two component plasma in which there are two dis- tinct carriers such as electrons and holes in semi conductors, or s and p electrons in a semi metal. If there were no interactions between particles one would expect to get two independent frequencies, anJL p for the heavy particles and an LL$ for the light particles. Because of the interaction, however, the spectrum is altered. Since there are two kinds of particles. one will expect two branches of the spectrum. One mode will correspond to the two different type particles moving out of phase and the other mode will correspond to the two different type particles moving in phase.3 In analogy to the two different branches in the spectrum of crystal vibrations we shall call the modes of plasma oscillations the optic mode and the acoustic mode respectively. In the acoustic mode, we will show that the frequency is low and therefore these excitations could possibly play a role in certain transport phenomenon. In the acoustic mode, the light particles tend to screen 8 out the field of the heavy particles since they are free to move and can easily follow the heavy particles. We saw in the theory of the one component plasma that for the cou— lomb interaction V1,.(r) =EEE: expliij-(Ei-Ej)] = E‘Vk eXpli§°(§_i-§_j)] (29) However, since the light electrons tend to screen the heavier ones, -k (r.-r.) 2 ViJ-(r) =6: C 1 J (3o) 1.1 Therefore Vk = l/k2 + kc2 where kC is the screening number.( See equation 28). Therefore k2 a 1 A. __ o 2 Vk __ kcz vk for k <:<:kC (31) Therefore the plasma frequency will be.“p =JL§ {E (32) c If 2— kc is the Debye length, = = 15.! E V Vclassical In n (33) where v is the velocity of the acoustic plasma and N and n are the densities of the heavy and light carriers respectively. If l/kC is the Thomas Fermi screening length, ——"-°—— (3t) v m = 9 4P§ —> -4> I+> ii: +V.Vr fi :(—:1:)B.vai=0 (35) where the top set of signs refer to the equation for the holes and the bottom set of signs refer to the equation for the electrons. E in this equation is E and is determined by Poisson's Equation: internal -‘>'-> 56 ‘7- E = An e(N+ - N_) + An (Imp. (36) where Pimp is the compensating background charge to assure neutrality. The densities N+ and.N_ are given by 2 N3: = WP [dsk r: (I: E: t) (37) The dispersion relation can be found by solving equations (35), (36) and (37) as coupled equations. These equations will be solved to first order. In other words, we will linearize the Boltzman equation by writing f: = fio + r+,1 . (38) where f:.o is the unperturbed current distribution function and fi-1 is the linear term in a departure from f:.0° If this is put into equations (35 and 36), we find: ~Vv ' = O (39) and 60 V E = $123er [mg Sofivrhl - m3 (oh/rd] (no) 11 We now assume that the current carrying distribution function can be written as fi-o = fO (v - Voi’ Ni)' In other words we assume the Fermi Dirac distribution function for a steady state current is a displaced sphere in velocity Space. We then seek solutions of the form: —>~ —> £351 = A exp [ik . r -(-0t] exp at. where the a is a small positive number. The a is chosen to satisfy the initial condition that the system was quiescent some time in the past.6 The a will go to zero at the end of the calculation. Then solving the equations we find for the dispersion relationship: ‘d3 R>'_f 1+2(-—m—i)3%£{f V V: +1 211‘ W... 3’. ?+ ia (Al) m d3 k>oii> f + 2(211-1‘ 335; g V ->V:> “0 _ (k) _ v- k + id The small imaginary term id comes from the choice of a retarded solu- tion of equation (bl). This term usually leads to damping of the plasma oscillations due to the individual electron excitations. This result is equivalent to the result of Landau treating the problem as an initial value one, using Laplace transforms.7 We shall see later however, that for a current carrying plasma, the term that leads to damping may change sign and the oscillations will grow instead of die out. The wp's in equation (Lil) are the plasma frequencies introduced earlier. We can write this equation as k3; k3 l--—F+-—-—F_=O (n2) 1:2 k2 12 where (k:)2 = 3(ESLEE)2 Vfi'. vf is the Fermi velocity and the k’s are the Thomas—Fermi screening numbers. Note that the Thomas—Fermi screening numbers are slightly different from the screening number talked about in the introduction. This is because the wave numbers are derived from different theories. One is from the Thomas—Fermi method, the other is from a more exact plasma theory. The integration over angles of these integrals is well known and if we put VI = (V>— V3) and note that l/2 mVZ = B, we find I. —> -> _ ._ -V .k “fodBZV: v (E) f0 2 + u- v2. k> log 1" kVEE) E kV(E) ->’—> I I _713-To VD k I * kV E + l (w - {In ' k ) 61“” :3 l. 1 x < l where aPIx) = O l x l > 1 Equation (AB) is true for finite temperature. The integrals can be immediately evaluated in the zero temperature limit since in this case Of TEQ =‘S (E - Bf): This was done by M. J. Harrison.8 However, instead of citing his results now, we will discuss them as a limiting case of the finite temperature theory. Let us now define Il, IZ and Is as <: A (7'1 Q; lab fl Ezra Ti 1 0 2V E —> -+> .—> —> .9 (‘)-Vb- k Ll- 60"VP. k f —>- —> (I) - V“. k 1 + kV(E) '9 («0— v0.10 91:0 L.) - v0. k 0 I3 can be integrated immediately '0 c) r tg-'v - k D 13 = in dB 3 E °~( —2k-V——- (Ml) (_w: VII-.6 7.)? E f _ k 2 —> —> I = _ l‘IT Q— V11. k (Ll-S) 2ka -> -> 2 + eXp “" VD° k E-- E + 1 XT I is Boltzman's constant. 11 and 12 can be evaluated using the well known Somerfeld Bethe method for solving integrals of the form.9 I = {$41 F(B) dB. Since g4? is a sharply peaked function for IT << Ef, F(B) in a Taylors series around BI and keep only lower order terms. we can expand F(E) may not converge rapidly itself but since it is multiplied by a sharply peaked function, the product will converge rapidly. So fol- 9 lowing the method of Kittel, we expand the function around Ef. = _ I _ 2 n ... F(E) F(Ef) + (E Ef)F (Bf) + l/2(E Bf) F (Bf) + then I = L0 F(Ef) + LlF'(Ef) + L2 mgr) + M where .9 ‘ LO= Ingmar; L1= {(E-Ef)rr(E)oB 0 It i5iell known that "2 Loz‘l3Ll=03L2=-7(XT)2 If we apply this method to 11 and I2 we find for 11 V n2 IT 2 I1 = _ _ 1- .—_ + .... (hé) vf '21 Sf J + _. Am 1/2 s 1" —gf1/2 12 = " % log + r Am 1/2 1* 7671/2 f A2m+i/e 3E _ A2m+ n2 2 + l/Ll V 3/2 + E‘O‘T) ()4?) {Br (E - Azm )2 —> -> w- VD - k A — 2ka To carry the analysis any further, we will have to consider the two modes separately and make suitable approximations. We will consider the acoustic mode first. Fbr the acoustic mode we will assume the phase velocity of the excitation is large compared to the Fermi velocity of holes but small compared to the Fermi velocity of the electrons, ie. LaJee >> 1 5; (~é- V0. k W“ kV - << 1 (t8) f+ f Here we have assumed for simplicity that in the coordinate system which we study the properties of the collective oscillations, the 15 holes have either vanishing drift velocity or their drift velocity is perpendicular to R). Then expanding in appropriate small quantities we find for the dispersion relationship k new: “I” 2” 1/2 o9) k2+k2[l-" __ 2. [ _ mgr-)3] (SO) and E2 . “4(2fi‘q‘N—V/3 f Mimi + m3) We note that equation (A9) implies that the effect of temperature is to reduce the screening contribution to the frequency. In other words k3 is replaced by k - [i - (15;) ("g—:- )2]. We also note that it is possible for the imaginary part of on to change sign and therefore we have the possibility of an instability. To find the critical drift velocity, above which the waves will begin to grow, —> -> we simply set Vo' k - Re 00 = 0. We find _1UP+V 2), 2 -— i-Ii+"(T>J (51) 3UP- ESE; And.we immediately see that the effect of temperature on the critical VDc l6 drift velocity is to require higher drift velocities for an instability. We also note that to keep v as small as possible, one should chose a DC material with L‘)p+/“Jp- as small as possible . This finishes the di3cussion of the acoustic mode. If we go on to the optic mode, again using the Sommerfeld Bethe method, only this time under the approximation kV kV \ f+ <1 __£: <1 w, w, 1‘: we find the diSpersion relation becomes 2 k2 Cue 1-5.).LL-" _BEQL(%?)+.._+in—L‘—J—Y(k) LO 2 FLO 2 Ef+ k2 2kvf-,l + _Q2 2 _2 k3 WV - Nat-”3:572 I? 2 i? 1" 2‘3 Jiffy-(1") ‘52) where B + _ . Mk) " exp [‘ IT— (“23% M) ] And again we have assumed the vo-k term vanishes for the holes. To solve this equation we use approximate methods. Since the imaginary part of equation (52) is much smaller than the real part we neglect it to find the real part of\‘) . Therefore Rel-OZ 205+[1 ‘84:?- -l;2)]+‘~> 2 _[l + 2W)1/21[1T(—::)23 Then putting these results back into equation (52) we find _ - (11?: 1‘8 -V °k Rein.) -v -k2 LO — ReU [l-1fi(;[ 2:31.12” 18 e-xp [:7 ( 2kvf_ D ) ] ‘k- (1.22:: )2 e—Xp [go—.221 > J - 1” (k2 2ka+ 17 where Beh>=[t.);[ rig—I )]+k)2[1+gl°—k——][l+3-(+2)TJH/ p+‘4)p— We see that the imaginary part will again produce damping but this damping will be small because of the exponential term. We see that as T-—<> O or k—+> O (infinite wavelength) there is no damping. It is also interesting to note that even for infinite wavelength there is a temperature dependence in the equation for the real part of (Q . In the classical theory, when k equal zero there is no temperature depend- ence in the plasma frequency. It would be interesting to experimentally check the temperature dependence of the plasma frequency, however, for most materials which satisfy the conditions we have placed on our plasma, the Fermi temperature is extremely high and the electron plasma is essentially a zero temperature plasma. Therefore to look for the temperature effects one would have to make a judicious choice of sample. III. EXPERIMENTAL EVIDENCE FOR PLASMA OSCILLATIONS For a one component plasma we mentioned in the introduction that fast electrons passing through the metal may excite plasmons. Therefore if a beam of high energy electrons are shot through a metal it might be expected that some electrons would lose energy twp others 2134*?) and so on. Experiments of this type have been carried out. Usually several kilovolt electrons were passed through thin films about 100 A thick. For some metals such as beryllium, magnesium and aluminum the results are exactly what is expected, if one uses the valence electrons to calculateL~) p' However, for elements which have large ion cores and tightly bound valence electrons there is little or no agreement with experiment. This is not surprising since now the electrons are bound and the materials do not satisfy the initial assumptions of the theory.3 If one goes into the dielectric theory of plasma oscillations, one of the elementary results is that the frequency dependent dielec- tric constant vanishes at the plasma frequency. This implies that the metal should become transparent to light of this angular frequency. Not too many measurements of this type have been made but for the alkali metals excellent agreement has been found.10 M. Glicksman and W. A. Hicinbothem" Jr. have studyied hot electron effects in indium antimonide at about 770K?1 They have observed drift velocities as high as 9 x 107 cm/sec. giving ratios of drift to thermal velocities as high as 1.5. An anomaly in the drift velocity is ob- served for high electric field. The investigators think this may be 18 19 due to generation of a two stream instability in the plasma. IO. 11. BIBLIOGRAPHY Bohm, D. and D. Pines. Phys. Rev; 85, 338 (1952). Raimes, S. The Wave Mechanics of Electrons in Metals,(Interscience Publishers, New York, 1961). Chapter 11. Pines, D. Can. J. Phys. 2Q, 1379 (1956). Ehrenreich, H. and M. Cohen. Phys. Rev. 115, 786 (1959). Goidstone, J. and K. Gottfried. Il Nuovo Cimento 1;, 8h9 (1959). Pradhan, T. Ann. of Phys., IL hl8 (1962). Pines, D. and J. Schrieffer. Phys. Rev. _1_2_i,, 1387 (1961). Harrison, M. J. J. Phys. Chem. Solids 23, 1079 (1962). Kittel, C. Introduction to Solid State Physics,(John Wiley and Sons, Inc., New York, 1961). P. 256. Pines, D. Revs. Mod. Phys. 28, 18h (1956). Glicksman, M. and W. Hicinbothem. Phys. Rev., to be published. 20 MICHIGAN STATE UNIV. LIBRQRIES IIIIIIllIIllII||||II|lllIIllIIllllllllHllIIIIIIIIlIlll||||||| 31293017430186