Screening in the two-dimensional electron gas with spin-orbit coupling
a r X i v :c o n d -m a t /0506227v 2 [c o n d -m a t .m e s -h a l l ] 15 D e c 2005
Screening in the two-dimensional electron gas with spin-orbit coupling
M.Pletyukhov 1and V.Gritsev 2,3
1
Institut f¨u r Theoretische Festk¨o rperphysik,Universit¨a t Karlsruhe,D-76128Karlsruhe,Germany
2
D′e partement de Physique,Universit′e de Fribourg,CH-1700Fribourg,Switzerland 3
Department of Physics,Harvard University,Cambridge,MA 02138,USA
We study the interplay between electron-electron interaction and Rashba spin-orbit coupling in the two-dimensional electron https://www.360docs.net/doc/039752880.html,ing the random phase approximation we predict new screening properties of the electron gas which result in extension of the region of particle-hole excitations and in spin-orbit induced suppression of collective modes.In particular,we observe that the spin-orbit coupling gives a ?nite lifetime to plasmons and longitudinal optical phonons,which can be resolved in inelastic Raman scattering.We evaluate the experimentally measurable dynamic structure factor and estimate the range of parameters where the described phenomena are mostly pronounced.
PACS numbers:71.70.Ej,73.20.Mf,73.21.-b
Fundamental issues of interaction e?ects in the two-dimensional electron gas (2DEG)are at the center of dis-cussions since the early days of its fabrication [1].Quite generally,screening described by the dielectric function forms the basis for understanding a variety of static and dynamic many-body e?ects in electron systems [2].The dielectric function of the 2DEG was computed a long time ago by F.Stern [3]within the random phase ap-proximation (RPA).Di?erent quasiparticle and collective (plasma)properties deduced from these expressions were con?rmed experimentally soon after [4].Recent experi-ments measuring plasmon dispersion,retardation e?ects and damping [5,6,7]unambiguously show the impor-tance of correlations between electrons.
More recently,the possibility of manipulating spin in 2DEG by nonmagnetic means has generated a lot of ac-tivity [8].The key ingredient is spin-orbit (SO)coupling tunable by an applied electric ?eld [9].A recent example for SO-induced phenomena is the spin-Hall e?ect [10].In this context understanding of the combined ef-fects of electron-electron correlations and SO coupling in 2DEG becomes vitally important.In the present paper we study the RPA dielectric function at ?nite momenta and frequencies treating the SO coupling exactly.We observe two interesting phenomena:an extension of the continuum of particle-hole excitations and SO-induced damping of collective modes.New features in the dy-namic structure factor are predicted,and SO-induced widths of both plasmon and longitudinal optical phonon spectra are evaluated.Their values lie in the ranges which are experimentally accessible by now in the inelas-tic Raman scattering measurements.We also revisit the derivation of zero-frequency limit of the dielectric func-tion and present an analytic result which does not contain an anomaly at small momenta [11].
We consider a 2DEG with SO coupling of the Rashba type [12]described by the single-particle Hamiltonian
H =k
2
2m ?
±αR k with two
distinct Fermi momenta k ±=k F ?m ?αR ≡k F ?k R ,but with the same Fermi velocity v F =k F /m ?.The
e?ective Coulomb interaction is V eff
qω=V q /εqω,where V q =2πe 2/(qε∞),ε∞is the (high-frequency)dielectric constant of medium,and the dielectric function εqω=ε1,qω+iε2,qωdescribes e?ects of dynamic screening.In RPA,εqω=1?V q Πqω,where Πqωis the polarization op-erator.In the presence of SO coupling Πqω=Π+qω+Π?
qωincludes contributions from intersubband (+)and intra-subband (?)transitions:Π±qω=lim δ→0
μ=±
d 2k
ω+iδ+
?μk
?
?±μ
k +q
F ±
k ,k +q ,(1)
where the form factors F ±
k ,k +q =
1
nπa ?2B which is de?ned by an electron density n and e?ec-tive Bohr radius a ?B =ε∞m e a B /m ?
.Our main results are compactly summarized in Fig.1which shows the contour plots and cross sections of the structure factor S (z,w )=?Im [1/ε(z,w )]for di?erent values of y and r s .
The introduced notations z =q/2k F and w =m ?ω/2k 2
F ,are,respectively,the dimensionless wave vector and fre-quency.
First of all,we observe an exstension of the contin-uum of particle-hole excitations (or Landau damping region)de?ned by ε2=0.A new wedge-shaped re-gion of damping is generated by the SO coupling.It is bounded by two parabolas ?(z ?y )2?(z ?y ) FIG.1:Contour plots of S (z,w )showing the SO-induced wedge-shaped damping region (bounded by the dashed lines).The plasmon mode is depicted by the bold line.Insets show the cross-sections S (z =0.1,w )as a function of w .Left panel:y =0.07,r s =0.2.Right panel:y =0.04,r s =0.6. into w =(z ?y )2?(z ?y )for nonzero y (this occurs at z >1which are not shown in Fig.1).The new regions of damping re?ect an opened possibility for transitions be-tween SO-split subbands,and S (z,w )strongly depends on y inside these regions.Their boundaries can be de-termined from a simple consideration of extremes of the denominators in (1).On the other hand,within the con-ventional boundaries w =z 2±z the function S (z,w )is modi?ed by SO coupling only slightly and can be approx-imated by Stern’s expressions [3]. The second SO-induced modi?cation is the broadening of a plasmon collective mode within the new particle-hole continuum.We determine numerically the position of the plasmon spectrum from the equation ε1(z,w )=0and depict it in Fig.1by the bold line,in black where ε2=0(undamped plasmon,the structure factor is delta-peaked)and in gray where ε2=0(SO-damped the structure factor has ?nite height and width).di?erent cases are possible:(I)the plasmon enters once into the SO-induced damping region (Fig.1,panel);(II)it enters twice escaping for a while after ?rst entrance (Fig.1,right panel).We observe that position of the plasmon in the presence of SO coupling almost indistinguishable (at least,on the scale of Fig.from the RPA result derived in the absence of SO pling [13]w pl (z )= z (z +2?r s ) 4?r 2s +4?r s z 3+z 4 8and z ?is the real positive root the equation z 2(z +4?r s )=4?r 2 s .Later we will more on how small might be the di?erence between exact SO-modi?ed plasmon dispersion and that given (2).At the moment we will use (2)in order to critical values of y and r s which separate the cases and (II).The result is represented in Fig.2where the plane (y,r s )is divided into the respective domains I and II.Changing y and r s ,we can tune the relative position of the new particle-hole region and w pl (z ). In the new region of damping the dynamic structure factor is very well approximated by the Lorentzian S (z,w )SO ?damp pl = α(z ) (w ?w pl (z ))2+γ2(z ) ,(3) which describes the SO-damped plasmon with the width γ(z ).The weight factor α(z )=π(dε1/dw |w =w pl (z ))?1is practically independent of y and given by its value at y =0 α(z )=π√2?r s γ(z )=γ(z )×104for the values of parameters same as in Fig.1:(I)y =0.07,r s =0.2;(II)y =0.04,r s =0.6.The function γ(z )is de?ned for z ≤z ?and vanishes in the region where the plasmon is undamped.In the latter case S (z,w )pl =α(z )δ(w ?w pl (z )). Comparing positions of peaks in constant-z cross-sections of the exact S (z,w )and of the approximate S (z,w )SO ?damp pl (3),we have observed that the shift of the plasmon dispersion due to SO coupling from w pl (z )(2)is one or two orders of magnitude smaller than γ(z )(unless γ(z )=0),and therefore can hardly be resolved experimentally. It is also worthwhile to compare our exact results with an approximation commonly used in the litera-ture.It is based on the formula Πqω≈?iσωq 2/(e 2ω)for small q ,where σωis the ac conductivity (cf.,e.g.,tively (cf.Fig.1).Insets show SO-induced plasmon width and therefore leads to the kinematic extension of the con-ventional particle-hole continuum to a strip y ?y 2 larger z . For example, according to [14] the plas- mon spectrum does not cross the border of their damp-ing region but rather approaches it exponentially from both sides of the strip.This approximation also misses the possibility,shown in the left panel of Fig.1,that the plasmon does not emerge undamped for intermedi-ate momenta. An important test of our results is provided by the Kramers-Kronig relations and the sum rules.In partic-ular,we have checked numerically that our expressions for ε(z,w )?1and 1/ε(z,w )?1both satisfy the zero-frequency Kramers-Kronig relations,which link ε2(z,w )and S (z,w )to the static dielectric function ε1(z,0)=1?(?r s /z )·(2πΠ1(z,0)/m ?).Our exact result for zero-frequency polarization operator is ? 2π 2 sin ψ)?2Θ(z ?1)arccosh z cos ψ + ν=± Θ(z ?(1+νy )) 1+νψνsin ψ?cos ψν?2cos ψln 1+z sin(ψν?νψ) 2z cos 12ψ , (5) where sin ψ=y/z ≤1and sin ψν=(1+νy )/z ≤1 are de?ned in the corresponding ranges of z restricted by the unit-step function Θ.It is assumed that y <1/2(or 2k R ω2?ω2LO ?2ωLO M 2q Πqω/εqω ,(6) where M 2 q = 1 2wε21+Aw 20(?ε1/?w ) w =w ph (z ) ,(7) where w 0=(m ?/2k 2F )ωLO and w ph (z )is the renormal-ized phonon spectrum.Like in the case of plasmons,we may neglect the dependence of w ph (z )on SO cou-pling,and ?nd the phonon spectrum from the equation w 2=w 2 0[1+A (1/ε1?1)],where ε1is taken at y =0. The lifetime e?ects of the LO phonons can be very im-portant for a coupled dynamics of carriers and phonons.An interesting possibility arises when ωLO ～2αR k F (or w 0～y ).This,for example,holds in InAs,where the value ωLO ≈28meV gives w 0≈0.07.Changing the Rashba coupling strength αR by an applied electric ?eld one can manipulate the lifetime of an optical phonon which would result in a modi?cation of transport prop-erties by virtue of the electron-phonon coupling.In Fig.3we show the function γLO (z )= γLO (z )×104 for w 0=0.07,r s =0.4,ε∞=12and ε0=15.Solid and dashed lines correspond to the cases y =0.07(|w 0?y | a number of new screening properties:the region of particle-hole excitations is extended,and the plasmon mode experiences additional broadening.The same mechanism leads to the generation of the SO-induced lifetime for the longitudinal optical phonons.Thus,spin-orbit coupling tends to suppress collective excitations.At the same time,it does not a?ect much the position of the collective mode https://www.360docs.net/doc/039752880.html,parison of our theo-retical estimates for SO-induced lifetimes with the values obtained in the recent Raman scattering measurements suggests that the e?ects described here can be observed experimentally. 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