Sood, Jyoti

Tewari, S P

IACS Institutional Repository

InJianJ. Phys. 76B (I),47 -51  (2002)U P  Ban international journalWavevector and frequency dependent complex dielectric function and the dynamical structure factor of two-dimensional weakly-coupledquantum and classical {hot plasmasS P Tevvari* and Jyoti JoodDepartment of Physics and Astrophysics, University li>f Delhi, Delhi-1 10 007. India E-mail . sptewari( /^)physics dtl.ac.inReceived 1 March 200L accepted 30 October 2001Abstract Complete expressions for the wavevector and frequency dependent complex dielectric function for two-dimensional quantum and classical vvcakly-coupicd onc-component thermal plasma have been obtained and computed in the entire domain of plasma frequency. The d>namical structure factor and other related quantities have been computed to obtain full knowledge about the collective dynamics of the two dimensional plasma. Comparisons have also been made with other models suitably modified for the two dimensional case.Keywords Siirlaces and interfaces, thin films, dielectric responsePAC'S Nos. 52 2S.Mq, 73 50 Mx. 71 45 GmI. IntroductionI wo-dimensional weakly-coupled hot plasma can be realized experimentally, amongst other solid state methods such as, thin semiconductor interfaces, thin films, inversion layers ale , at very low temperatures, around 1 K when the electrons of concentration approximately, 10^  < « < 10’® cm“^ , are (rapped on the surface of a liquid dielectric like He [1,21 Nc. The combination of external applied field and the image potential provide the necessary positive neutralizing background and confine electrons to the lowest energy state perpendicular to the surface [3]. The electrons are completely tree to move on the surface unrestricted. The electrons in such a plasma, unlike three-dimensional plasma, possess only two degrees of freedom. Such a change, results in dramatic changes in the p!.isma behaviour. For instance, while the plasma frequency in three dimensions is independent of the wave vector, in two-dimensional plasma, it turns out to be dependent on the wavevector q with a power law, i.e.,fOrIn a weakly-coupled plasma, the coupling parameter /"((= e~ jkaT ) , where n is surface number density ofelectrons and other symbols have their usual meaning), is'Corresponding Authorequal to or less than one. Here, an electron is assigned a circular area of radius /•,. The change over from classical to quantum behaviour is on the other hand, governed by the value of n A*,, (where A,|,(= h f^ lm kgT )  is the thermal de- Broglic wavelength of electrons carrying mass m at temperature T, h is Planck’s constant and is Boltzmann’s constant). When n «  I. the plasma behaves classically both in its dynamics and statistics, but as n /l^ i, approaches larger values, the statistical behaviour of the system changes to quantum mechanical description and one has to use two particle distribution function [4 -6] in its description instead of one particle distribution function. In the present communication, an attempt is made to study such a one- component quantum plasma which has not been hitherto studied, along with the classical plasma for which, the complete frequency ruand wavevector ^ -dependent dielectric function and collective dynamics have not been worked out. It may be noted that when n )) 1, the plasma is highly degenerate and strongly coupled where quantum Hall effects both integral and fractional, occur. For such a degenerate plasma, wave vector and frequency-dependent dielectric ftinction has already been reported [7].O 2002 lACS48 S P Tewari and Jyoti Sood2. Mathcinalicsil formalismIhc complex (iideclric function tor a two-dimensional, weakly-coupled, one-component quantum plasma, is related to the two-particle quantum distribution function }\{quo) 1^ 1 follows :^ .0 { q .o »  -  1  ^ - ~ [ F ,  U l.fo )  F. ( I)where FA(1,(0) are given as,w V * 2,Te- f (Ip• q ^(' andp are the charge and two dimensional linear momentum of the electron respectively. 1he single particle momentumdistribution function here, corresponds to theMaKwellian and is given by the expressionfW ? )2mk,/rA can be evaluated using the sum rule I (t) Im /;(<:/, (o) da) -(3)(4)where cOp is the angular plasma frequency of the two- dimensional plasma.Substituting expression (3) in (2) and solving the two- dimensional integral, F\(cp a>) can be obtained. Using the derived ro) in the expression ( I), one gets the complete wavevector and frequency dependent complex dielectric function for one-component quantum plasma as(t/, 0)) -  (c/, ) 4- (<y, = I + *>/2 hqh c ]0)42qv 2^ 2 mvI +3 [ 2.42myhqX I4-:( iti h(f 'i VV «/5</v 2^ rtmv ,1 \ - J < L ^V -J 2 q vCO hqT . - -4 2 q v  2 y /2 m v j ^hq-]. pF  m\ 2  h-I - V f . 4- .. V, I Vlf/v \>n.qv 2 j l m v )mvhqw'hcre v = is the thermal velocity.The complex dielectric function for classical plasma can be obtained from expression (5) when h -  0 and is given as :£•' * (c/, (o) ~ e \ ‘ (q , (o) -f {q , (o)col14- co^q^v^ q^v^ q^v^-^Q>^  Ilq^v(1 + )  i0{2q^v^- )fO~/2q^ v^qv (6)Expressions (5) and (6) appear to be similar to the corresponding expressions for three-dimensional plasma [5], but here cOp  ^yj2mw- q/m  and q is a two-dimensional vector and therefore, quantities related to these dielectric functions would have to be evaluated keeping this fact in mind [8,9],The collective modes of the plasma can be obtained from the relation i'i(q, aj) -  0. Such an exercise, strictly speaking, is quite involved as /;(</, aj) is a polynomial of very high order in rtr as can be seen from the analysis of expressions(5) and (6), though it is possible to obtain such modes approximately [10]. Further, one cannot get the detailed knowldcgc about how well-defined these collective modes are, which one can obtain by studying the related dynamical structure factor S(q, a?) of the plasma. One can make use oi' the fluctuation dissipation theorem [5,6,11] to obtain the following expression for S^\q, co) :SL\q,(o)^ 1 4 '(< /.")mo q([c \ '(q .(o )f  j (7)wherecoi"yT * ^ being the two dimensional Debyescreening length. tp*(q,(o)&nd c^{q,a)) are the real and imaginary parts of the complex dielectric function given in expression (5). The singularity in S{q, to) yields the value of the collective modes which corresponds exactly to the value given by €\{q, co) ~ 0. ci{q , (o) gives essentially the damping accompanying the mode. It is also clear from the expression that S(q, co) may not be close to delta function in which case, it describes the presence of extremely well defined collective mode. Sijcj, co) can also be experimentally observed by appropriate electron inelastic scattering experiment from the plasma.The static structure factor S{q) is the zero-th sum rule(5) .% )= (8)Wavevector and frequency dependent complex dielectric /unction etc which is also given by the expression4 9(9)0) is the zero frequency dielectric function which can be obtained from expression (5).The zero frequency dynamic structure factor S^(q, 0) which can be obtained from expression (7), is given as lollows :9 ^ ( 0  O') = - J = --------- -------------Lz,-(/H //2V lm v)i( 10)1 he corresponding expressions for the classical plasma are:w) - 1 *^2( 12)q i  ^S^Hq.O)^ V2/r (1+ ^7^2)2  ^v. (13)In computing S(q, co), the value of 5(^ 7) should be consistent from expressions (8) and (9) for the quantum case and (8) and (12) for the classical case. Like S{q, co), S(q) and S(q^ 0) are also experimentally observable [12,13].3. Results and discussionIn order to compute the dielectric function given by expression (5), we take the surface number density of electrons n = 7.65 X 10*^ env^ at r  1 K. Computations have been made for co) and /;2( ,^ co) for different values of the quantum parameter, R{~ hq fmv). When /? = 0, /? = 0 and the expression (5) reduces to the expression (6) of the classical plasma, I ’he computed values c\(q,co) have been plotted in Figure 1for different values of R. When R is 0.0001, the computed values from expression (5) for c\(q, a>) are very close to the ones cafculated using expression (6) of the classical plasma, As R irtcreases, the deviation from the classical case starts showing up as shown in the figure for = 1,2 and 3, The computHpd Ciiq^  ty) for various values of R are shown in the inset of|the figure. For R ^  0.0001, the computed values of eiiq, ft>)^atch very closely with the computations of cii/q, co) from e: ipression (6). Here too, the deviation from the classical result becomes increasingly different as R increases from 1 ro 3 as is evident from the figure. The collective mode frequencies coc exist for R < 2. As for R 2, o\{q, co) does not shefw any intersection with c:\(q, co) ^  0. Thus, the presenci of collective modes is restricted to weakly quantum and cla|isical case. Further, the Landau damping given by e2(q, co% keeps on increasing with the increase in the value of R as shown in the inset. It may be mentioned that no analytical expression which covers the entire co/qv was available even for the classical plasma. But here, due care has to be taken to ensure the convergence of the series.In Figure 2 have been plotted, the computed values of S{q, co) using expressions (7) and (11) for the quantum andHgure 1 . Variation of the computed values of real <u) and imaginary(o) parts of quantum two dimensional plasma dielectric function with angular plasma frequency o>, expressed in units of qv for different values of quantum parameter R when the surface number density of electrons « ® 7.65 X l(i< cm’’^  temperature T * 1 K and wave vector q -  1.87 x 10^  tint-’. Cl refers to the classical variation of the diiectric function.Figure 2. Variation of the dynamical structure factor S(q, o) with the angular plasma frequency cu, for different values of the wavevector q and surface number density n at 7’== 1 K for the two dimensional plasma. Forn “ 7.65 X 10^  cm”^  (------) curve (i) is shown for <7 * 3.0 x 10^  cm^‘,and ( - ^ )  curve (ii) for q =* 1,87 x 10  ^ cm~‘. Forn ~ 7 65 x 10^  ( - • # - )  curve (ill) is plotted for q = 1.87 x 10^  cmIn the inset are shown, the details of the /y) and 5 '^(q, (o) by (-----and (— ) curves respectively, near the peak values of the collective modes for n » 7 65 x 10^  cm  ^ at q ~ 3.0 x 10^  cm"'50 S P Tewari and Jyoti Soodclassical plasma respectively when n ~ 7.65 x lO"* cm" .^ The computed value of S(c/, (o) for q - 1.87x10^ cm ‘ is shown by curve (iii). As the density is increased to 7.65 x 10^  cm % the S\q. fo) turns out to be that given by curve (ii), which shows the presence of much more well defined collective modes at higher energy, in comparison with that given for the low'er density. In both these cases, the difference between S^Aq. (^ ) and S^\q, w) is very small as not to be evident in the figure, 'fhe values of S{q) computed using the expressions (8) and (0), (8) and (12) for quantum and classical plasma respectively, are consistent with one another. ALSO, the values of S(q, 0) plotted in Figure 2 are same as those calculated from (10) and (13). In Figure 2 is also shown the S(q, co) for q ^ 3.0 x 10- enr  ^ and n 7.65 x 10" cm - by line (i) Here too, the collective mode is very well defined. The difference between the classical and quantum collective mode forms have been shown in the inset of the figure by curve I for quantum plasma and curve 2 for classical plasma The collective mode in the quantum case occurs at somewhat lower frequency and is more damped as is clear from the figure. The computations have also been done for larger values of R by changing tempei’ature and wavevector, in which case, the dilTcrence between classical and quantum plasma increases but the absolute value of iS’({/, w) turns out to be very small. It may by pointed out, that the frequency (o, at which the collective mode occurs, (in all S{q, ro)) studied, it has been checked that these also satisfy the relation ^*|(^ , ~ 0 numerically.In Figure 3 have been shown, the computed values of %/, (o) given by expression (11) by curve (i) for different values of <7 when n -= 7.65 x cm  ^ and 7 '~ 1 K. As the value of is increased from 2.079 x 10^  cm * to 1.87 x 10^  cm the collective mode frequency co^ shifts to higher values as shown in Figure 3 and also plotted in Figure 4(b) by curve (i). The values from an earlier study [10] where onl> rUt have been worked out, arc close to our results. For the sake of comparison, we have also shown in Figure 4(b) the dispersion relation of the cold plasma by curve (ii) and for two-dimensional Wigner solid by curve (iii). By curve (iv), is shown the dispersion relation followed by neutral Maxw'cllian electron gas. The damping of the modes given by the full w idth at half maximum (FWHM) of S{q, (o), increases with the increase in q and is also plotted in Figure 4(c) by curve (i). The collective modes therefore, become increasingly ill-defined with increase in q. The variation in the value of the peak in the collective modes is also evident from Figure 4(a). S{q, 0) keeps shifting to higher values with q and its variation is plotted in Figure 4(d) by curve (i) which matches with the values computed using expression (13). Finally, the static structure factor S(q), as computed from expression (8), is found to be same as that given byexpression (12) and have been plotted by curve (i) in Figure 4(e). Therefore, from this study one gets consistently both the dynamical structure factor and static structure factor and hence, can determine all relevant statistical thermodynamic properties of the plasma.Figure 3. Varialion ot'ihe computed values of dynamical structure lactor S{c/, fo) with (0 for (wo dimensional classical plasma lor n -  1 65 v cm - and T -  \ k lor diflercMit values of q and dilVcrcnt modelsCurve (i) (------) prcsetu calculations (d S{q. (o) from expression (6), curve(ii) (— )S{q, w) loi ftMicar and small damping, curve (iii) {-m~)S{q, (o) calculalcd from mean field theory, and cuivc (iv) ( - • • - )  S(q, (o) lor charged tree electron gasFor the sake of comparison, we have also made studies on sonic ot'the co) suggested for classical plasma. When one con.sidcrs the Coulomb interaction to be zero and appropriately takes account of the two dimensional nature of charged system, one gets an expression for co) [ 14,15], which has been worked out, computed and plotted in the Figure 3 by curves (iv) for q -  1.2 x 10- cm ' and 1.87 X 10^  cm-'.E xpress ion  fo r Sicf, (o) n e a r  co =  <u, an d  sm all d am p in g  [ 16 j h a s  been  used  to  c o m p u te  S{q, co) an d  is p lo tte d  b y  c u rv e(ii) in F igure  3 and  the  v a rio u s  c lia rac te rs tic s  o f  S{q, co) h av e  been  p lo tted  in F igu re  4  b y  c u rv e  (ii) . T h e  c o m p u te d  va lu e  o f  S (^ ) is n o t c o n sis ten t w ith  co m p u ta tio n  o f  S{q) from  ex p ressio n  (1 2 ) as is e v id en t from  F igu re  4 (e ) . T h is  w as  ex p ec ted  as s tric tly  sp eak in g , th e  su g g es ted  S{q, co) is va lid  fo r CO n ea r cO( . B u t as p o s tu la ted  [16], th e  d a m p in g  d o es  no tWavevector and frequency dependent complex dielectric function etc 51uirn out to be small, particularly for ^ 2 0.8 x 10' cm * and IS infact, more than that given by full S(q, eo), as is evident from Figure 4(c). The values of <u, however, are not very Jiffcrent from actual one excepting for large values of q, as shown in the inset of Figure 4(b) by curve (ii). The values o i ' S{q, (Oi) become increasingly different from the present values for larger q as can be seen from Figure 4(a).niIf tI la) ^ (w76b«i0(\vV's( n )  ^') '^T /r 1^'u ■ (I)(III)(1) T• (ill), (li)<e)10-^ -I '( I V )  I /  / /1 / ' /li . i 1 1 -i_i_i--L.j.Ajl .iiLA _J_I_L„i 1r ''I, PB 1? : f, 0^ fl ti ofl 1.? i 6 ?CMi' ') ()l» io'ciii'’’«htiiirt* 4. Variation (^ 1 vaiious characteristic quantities, Peak value of ^ 7  >'t). j c  S'U/. collective mode frequency full width at half iruiMmum (I WHM). SO/, (o) at a>-(), / e , S(q, 0) and static structure factor S(c/) with q tor lv\(t dimensional classical plasma. In Figures (a), (c), (d)  ^iiiul (e), cur\e (i) ( ) corresponds to the prc.sent calculations fromLsptession (6 ). curve (li) (— ) repiesenl the calculations done for SO/, ru) neat to ,^7 and curve (in) (-♦  ) for the expression of mean field theory.In ! iguie (b). the dittcrent curves represent ■ (i) (™ '-) calculations done using (o) fiom expression (6), (li) (• - -) cold plasma, (iii) (-*•■) iw'o (imicnsioiul Wigner solid, and (iv) ( ) neutral Maxwellian electron‘•dsIII ihc mscl ot the Figure (b). present calculations shown by ( ---- ) curve<0 have been compared with the calculations done for the expression given in iclerence 16 showm by (- - -) curve (ii).Computations have also been made using the mean field expression for S{q, fo) [14,15], appropriately modified for the two dimensional case and the results have been plotted hy curve (iii) for various studied quantities in Figures 3 and4. As in the earlier study, for low values of q there is a little difference I'rom the present calculations but, as q increases significant differences start appearing. While, the collective mode frequency occurs at almost the same value, the damping is somewhat larger and increases sharply for </ i  1.4 X 10^  cm"' as shown in the Figure 4(c). Both, S(q, 0) and S(q, (o,.), are much larger than that given by present calculations, as is evident from Figures 4(a) and 4(d).Moreover, values of the zeroth sum rule from this form of S(q, ai) are quite different, as shown in Figure 4(e).4. ConclusionOne may conclude that it is not only possible to obtain complete analytical expressions for both quantum and classical two-din|»nsional plasmas, but also to compute these for the entire r^g e  of frequency, unlike earlier studies. Further, it turns ou| that as the degree of quantum nature of classical plasma increased by either increasing the surface number density Jpf electrons or decreasing the temperature, the collective mode frequency changes and plasma becomes more dinped. There is no collective mode for quantum parametitr R>2. The dynamical structure factor which yields the con^lete dynamics of the collective modes, have been computad using the real and imaginary parts of the dielectric function. Explicit calculations show that though the value of collective mode frequency from some earlier approximate studies are close to the exact values from the present work, the details like the damping, the zero frequency dynamical structure factor etc. arc quite different besides the detailed spectrum of the dynamical structure factor. The results of the present sutdy can be checked by performing appropriate electron inelastic scattering experiment from such two- dimensional plasmas.ReferencesII) T R Brown and C C Cirimes P/iys. Rev Lett, 29 1233 (1972)[2] C C Cirimes and T R Brown Phys Rev Lett. 32 280 (1974)(31 J H Davies The Physics of Low Dimensional Semiconductors (Cambridge Cambridge University Press) (1998)[4] B A I rubnikov and V F Elcsin Soviet Phys JETP 20 866 (1965)[5] S P Tewari, II Joshi and K Hera7 Phys Condens. Matter 1  8405 (1995)[6] S P Tewan, II Joshi, K Bcra and J Sood Pramana-J Phys 52 195 (1999)[7] I* Stern Phys. Rev, Lett 18 546 (1967)|8J S P Tewari, K Beta and J Sood Solid Stat Commim 106 397 (1998)S P lewari and H Joshi Phys, U tt A 173 160 (1993)[10] M Plal/.man and N Tzoar Phys Rev B13 3197 (1976)[ l lj  S Ichimaru Statistical Plasma Physiqs (New York : Addison Wesley) p i07 (1992)[12] R Winter. C Pilgrim. F Hcnsel, C Morkcl and W Glaser Non- Crys. Solids 9 156 (1993)[13] S P Tewari and Surya P Tewari Phys, Lett. A56 99 (1976)[14] J P Hansen. I R Mcdonald and E L Pollock Phys. Rev. A ll  1025 (1975)[15] A A Kuglcr J Stat, Phys, 8 107 (1973)[16] A L Fetter Phys. Rev. BlQ 3739 (1974)

Wavevector and Frequency Dependent Complex Dielectric Function and the Dynamical Structure Factor of Two-dimensional Weakly-coupled Quantum and Classical Hot Plasmas

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Wavevector and Frequency Dependent Complex Dielectric Function and the Dynamical Structure Factor of Two-dimensional Weakly-coupled Quantum and Classical Hot Plasmas

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