= (1) Correlations in 2D electron gas at arbitrary temperature and spin polarizations. Abstract. n and n )/n. We will. n ( n

Transcription

1 Correlatios i D electro gas at arbitrary temperature ad spi polarizatios Nguye Quoc Khah Departmet of Theoretical Physics, Natioal Uiversity i Ho Chi Mih City, 7-Nguye Va Cu Str., 5th District, Ho Chi Mih City, Vietam Abstract Usig the Sigwi-Tosi-Lad-Sjolader theory we have studied the may-body effects i the twodimesioal electro gas with arbitrary polarizatio at fiite temperatures. We have calculated the structure factors, pair correlatio fuctios, local-field factors ad the Helmholtz free eergy for differet values of spi polarizatio, temperature ad desity parameter. We have show that the spi polarizatio ad fiite temperature effects are remarkable ad i the low temperature or paramagetic case our results match closely with those obtaied i earlier papers.. Itroductio I the course of the last decades, may theoretical works have bee devoted to quasi-twodimesioal systems such as electros i iversio layers, heterojuctios ad quatum wells []. Most of the theoretical calculatios have bee performed i the framework of the radom phase approximatio (RPA) which gives good results oly i the high-desity limit. I additio, it has bee show that the may-body correlatios are more proouced i two tha i three dimesios []. The beyod-rpa effects i bulk ad low-dimesioal systems have bee studied by several authors through differet approaches[3-5]. However, most of these works referred to the paramagetic state oly ad the extesio to spi-polarized systems is motivated ot oly by fudametal may-body cosideratios, but also by practical experimetal eeds [6]. I fact the umber of theoretical ivestigatios o spipolarized systems is growig i the recet years [7-9]. Davoudi ad Tosi (DT) have studied the spi polarizatio i a two-dimesioal electro gas (DEG) at zero temperature [] usig the self-cosistet approximatio of Sigwi, Tosi, Lad ad Sjolader ( STLS ). The fiite temperature effects have bee studied by Shweg ad Bohm (SB) usig the STLS scheme for upolarized DEG []. It was show that the properties of DEG deped remarkably o the temperature. I this paper we exted the DT s work to the DEG at fiite temperatures. Our model is the DEG iteractig with the e /r law ad movig i a plae over a uiform eutralizig backgroud of positive charge. Cofirmig that our results reduce to those of zero temperature [] or upolarized case [], we aalyze the correlatio characteristics of DEG for differet values of spi-polarizatio, temperature ad desity.. Theory We cosider a DEG at temperature T cotaiig two spi species of surface desities ( ad )/. We will ) with the total desity = + ad spi polarizatio ζ = ( * ε oh use the effective atomic uits with the effective Bohr radius ab = where ε * o is the backgroud m e dielectric costat.the STLS self-cosistet equatios fopi-polarized DEG read [-] δ χ ( k, [ δ + ( ) ψ ( k) χ ( k, ] χ ( k, ( k, = ()

2 G r r r d q k. q k) = k. q ( π ) r r [ S ( k q ) δ ] ( S [ ] h hω ( q) = Im χ ( q, coth( ) π k T B dω () (3) where k, = [ ψ ( k) ][ ψ ( k) ] ψ ( k) ψ ( k) ) (4) ( ω ψ ( k) = Vk[ G ( k, ]. (5) with V k is the bare electro-electro iteractio, χ ( k, ) are the susceptibilities of the ideal DEG ω ad deotes the spi orietatio opposite to. The pair distributio fuctio ( PDF) ca be calculated from the structure factors as g ( r) = + [ S ( q) δ ] d q (π ) e rr iq r (6) The exchage-correlatio ad total Helmholtz free eergy are give by F xc d λ = π dr[ g( λ, r) ] (7) F = F + (8) tot F xc where, λ ad g(λ, r) are the desity parameter, couplig costat ad spi-averaged PDF, respectively [8-9]. Here F F + is the total free eergy of a o-iteractig electro gas ad = F F µ E = (9) E F / T µ = T log[ e ] () E F = = ( ± ζ ) π () π E T = π dx x exp[ x µ / T ] + () E (3) = E + E At T = we have

3 E + ζ = (4) We have solved the STLS self-cosistet equatios umerically ad the results will be discussed i the ext sectio. 3. Results ad discussios : I this sectio we report our umerical results for the structural fuctios ad the free eergy at differet values of spi-polarizatio ζ, temperature T ad desity parameter =. * a B π 3.. Static structure factor : I Figs. ad we show the parallel ad atiparallel spi structure factors S (q) as fuctio of, first o varyig ζ, T at = ad the o varyig ζ, at T = T F. We observe remarkable ifluece of spi-polarizatio ζ, temperature T ad desity parameter o the structure factors...8 =.5. S = ζ =. ζ =.4 ζ = Fig.. Parallel ad atiparallel spi structure factors S (q) T = T F ( right ) ad various ζ at =. versus for T =.5T F ( left) ad 3

4 ..8 =3 S.6.4. ζ =. ζ =.4 ζ = Fig.. Parallel ad atiparallel spi structure factors S (q) versus for = ( left) ad = 3 ( right ) ad various ζ at T = T F. The Fig. 3 shows the temperature ad spi-polarizatio depedece of the partial spi structure factors S (q) at =. It is see that the temperature effect is cosiderable ad i the paramagetic case our results are almost idetical to those of SB []. Our SSFs for = at low temperature T =.5T F show similar behaviour as it is give i the DT s work for T =...8 ζ=. ζ=.4 S.6.4. =.5 =. = Fig. 3. Parallel ad atiparallel spi structure factors S (q) versus for ζ = ( left) ad ζ =.4 ( right ) ad various T at =. The spi-averaged structure factor S(q) for = ad = ad various T at ζ =.4 is plotted i Fg. 4 which shows the strog temperature depedece of SSF at high T. This behavior of SSF is similar to that show i the SB s work for the upolarized DEG. 4

5 ..8 S(q).6.4. ζ=.4 =.5 =. = 5. = Fig. 4. Spi-averaged structure factor S(q) versus various T at ζ =.4. for = ( left) ad = ( right ) ad 3.. Pair correlatio fuctio The pair correlatio fuctio of a DEG ca be calculated from the SSF usig the relatio (6) ad the results are displayed i Figs. 5 ad 6. The behaviours of the Pauli ad Coulomb holes are directly see i the PDF g (r). We observe that the Coulomb hole shows weak spi-polarizatio depedece but appreciable depedece o the desity parameter as i the zero-temperature case give i DT s work. From Fig. 6 we ca coclude that the temperature effects o the PDF are cosiderable for all spi-polarizatios...8 g.6.4. ζ =. ζ =.4 ζ =.7 = Fig. 5. Pair distributio fuctios g (r) for = ( left) ad = ( right ) ad various ζ at T = T F. 5

6 ..8 g (r).6.4 ζ=. ζ=.7.. =. = Fig. 6. Pair distributio fuctios g (r) for ζ = ( left) ad ζ =.7 ad various T at =. The bold lies deote the spi-averaged PDFs g(r) Local-field factor Local field factors (LFFs) G(q) of upolarized (ζ = ) ad polarized (ζ = ) DEG at = for various T are displayed i Fig. 7. We observe appreciable depedece of the LFCs o the temperature. We otice that i the paramagetic case our results for the LFCs are similar to those give i SB s work.. ζ = ζ =.8 = G(q).6.4. = = 5 = Fig. 7. Local field factors G(q) versus for ζ = (left) ad ζ = ( right) ad various T at =. 6

7 3.4. Free eergy We have calculated the exchage-correlatio ad total free eergy of a DEG by usig Eqs. (7) ad (8). The results i the effective Rydberg ( Ryd * * 4 = m e / ε o h ) are give i Table. We ote that i the paramagetic case our results are idetical to those of SB. For polarized case preset results are similar to those give i DT s work for T = ad i our previous paper for fiite temperatures [9]. We observe from the Table that the DEG with temperature ad desity i the rage studied i this paper ( 5, 3 ) remais i the paramagetic phase. Table Exchage-correlatio ad total free eergy per electro : -F xc ad -F for various T, ad ζ. ζ F xc F Coclusios I this paper usig the STLS approximatio we have studied the correlatio characteristics of the spi polarized, fiite temperature DEG such as the structure factor, pair correlatio fuctio, localfield factor ad the Helmholtz free eergy. We have show that at low temperatures our results reduce to those of DT ad i the case of fiite temperature, upolarized DEG our results match closely with those obtaied by SB. Our results idicate that the correlatio characteristics of DEG deped remarkably o the spi-polarizatio, temperature ad desity. We fid that withi the STLS approximatio the DEG with 5 ad 3 remais i the paramagetic phase. The results of other properties such as the plasmo dispersio ad spi susceptibility will be reported i a future publicatio. Ackowledgemet We gratefully ackowledge the fiacial support from the Natioal Program for Basic Research of Miistry of Sciece ad Techology. Refereces: [] T. Ado, A.B. Fowler, F. Ster, Rev. Mod. Phys. 54,437 (98). [] M. Joso, J. Phys. C 9, 355 (976). [3] G.D. Maha, May-Particle Physics, Pleum Press, New York, 98. [4] K. S. Sigwi, M. P. Tosi, R. H. Lad, A. Sjolader, Phys.Rev.76, 589 (968). [5] K.S. Sigwi, M.P. Tosi, Solid State Physics, Academic, New York, 98, Vol. 36. [6] Y. W. Ta, J. Zhu, H. I. Stormer, Phys. Rev. Lett. 94, 645 ( 5). [7] C. Attaccalite, S. Moroi, P. Gori-Giorgi, G. B. Bachelet, Phys. Rev. Lett. 88, 566 ( ). [8] F. Perrot, M. W. C. Dharma-wardaa, Phys. Rev. Lett. 87 () 644. [9] Nguye Quoc Khah, H. Totsuji, Physical Revue B69, 65 (4). [] B. Davoudi, M. P. Tosi, Physica B 3, 4 (). [] H. K. Schweg ad H. M. Böhm, Z. Phys. B 95, 48 ( 994). 7