Screening in the two-dimensional electron gas with spin-orbit coupling

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Screening in the two-dimensional electron gas with spin-orbit coupling

M. Pletyukhov¹and V. Gritsev^2,3

1Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany

2Département de Physique, Université de Fribourg, CH-1700 Fribourg, Switzerland

3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 共Received 9 June 2005; revised manuscript received 1 February 2006; published 12 July 2006兲 We study screening properties of the two-dimensional electron gas with Rashba spin-orbit coupling. Calcu- lating the dielectric function within the random phase approximation, we describe the new features of screen- ing induced by spin-orbit coupling, which are the extension of the region of particle-hole excitations and the spin-orbit-induced suppression of collective modes. The required polarization operator is calculated in an analytic form without any approximations. Carefully deriving its static limit, we prove the absence of a small- qanomaly at zero frequency. On the basis of our results at finite frequencies we establish the new boundaries of the particle-hole continuum and calculate the SO-induced lifetime of collective modes such as plasmons and longitudinal optical phonons. According to our estimates, these effects can be resolved in inelastic Raman scattering. We evaluate the experimentally measurable dynamic structure factor and establish the range of parameters where the described phenomena are mostly pronounced.

I. INTRODUCTION

Fundamental issues of interaction effects in the two- dimensional electron gas共2DEG兲are at the center of discus- sions since the early days of its fabrication.¹Quite generally, screening described by the dielectric function forms the basis for understanding a variety of static and dynamic many-body effects in electron systems.² The dielectric function of the 2DEG was computed a long time ago by Stern³ within the random phase approximation共RPA兲. Different quasiparticle and collective 共plasma兲 properties deduced from those ex- pressions were confirmed experimentally soon after.⁴Recent experiments measuring plasmon dispersion, retardation ef- fects, and damping^5–7 unambiguously show the importance of correlations between electrons.

More recently, the possibility of manipulating spin in 2DEG by nonmagnetic means has generated a lot of activity.⁸ The key ingredient is the Rashba spin-orbit 共SO兲 coupling⁹tunable by an applied electric field.¹⁰A recent ex- ample of SO-induced phenomena that has attracted much attention is the spin-Hall effect.¹¹

In this context the study of interplay between electron- electron correlations and SO coupling in 2DEG becomes an important problem. In the preceding papers it has been al- ready discussed how SO coupling affects static screening,¹² plasmon dispersion and its attenuation,^13–15and Fermi-liquid parameters.¹⁶ However, the approaches of these papers as well as of many other papers on Rashba spin-orbit coupling are based on various approximations. We can outline the most popular ones: a linearization of spectrum, also known as␰-approximation; an expansion in SO coupling parameter up to the lowest nonvanishing contribution; reshuffling the order of momentum integration and evaluation of the zero- frequency limit in the polarization operator; a combination of any of those approximations. In our opinion, their accuracy is not comprehensively discussed. Since its lack might seri- ously affect a description of SO-induced phenomena, this issue should be thoroughly investigated.

In the present paper, we study the effects of the dynamic screening in 2DEG with Rashba SO coupling described by the RPA dielectric function in the whole range of momenta and frequencies. The main part of our paper is devoted to the analytic evaluation of the polarization operator, which does not employ any approximations. The knowledge of the di- electric function allows us to predict new features in the directly observable dynamic structure factor. In particular, we observe a SO-induced extension of the particle-hole ex- citation region and calculate the SO-induced broadening of the collective excitations such as plasmons and longitudinal optical共LO兲phonons. We obtain the values of lifetimes that lie in the range of parameters experimentally accessible by now in the inelastic Raman scattering measurements.

On the basis of our analytic results, we also revisit the earlier approaches, estimating their accuracy and establishing the limits of their applicability. Although the approximations mentioned previously usually work well for a conventional 2DEG共without SO coupling兲, we demonstrate that in case of the Rashba spectrum they should be applied with caution. In our paper we discuss the subtle features of the 2DEG with a SO coupling warning about this. Special attention is focused on a derivation of a zero-frequency共static兲limit of the po- larization operator. We present an analytic result, which, however, does not contain an anomaly at small momenta predicted in Ref.12.

The paper is organized as follows. In Sec. II we introduce the main definitions and notations. In Sec. III we present a detailed analytic calculation of the polarization operator

⌸共q,␻兲 and discuss the most important modifications generated by SO coupling. We also derive an asymptotic value of Im⌸共q,␻兲 at small q and compare it to the approximate expressions known before.^13,14 In Sec. IV we implement a thorough analytic derivation of the static limit lim_␻→0⌸共q,␻兲and make a conclusion about the absence of an anomaly at small momenta. In Sec. V we study the di- rectly observable dynamic structure factor and evaluate SO- induced plasmon broadening. The latter effect also causes a

Published in "Physical Review B 74(4): 045307, 2006"

which should be cited to refer to this work.

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modification of the energy-loss function that is estimated as well. In Sec. VI we calculate the lifetime of LO phonons generated by SO coupling.

II. BASIC DEFINITIONS

We consider a 2DEG with SO coupling of the Rashba type⁹ described by the single-particle Hamiltonian,

H= k²

2m^*+␣Rn共␴⫻k兲, 共1兲 wheren is a unit vector normal to the plane of 2DEG and ប= 1. The dispersion relation is SO split into two subbands labeled by␮^{= ±,}

⑀k␮= k²

2m^*+␮␣Rk⬅共k+␮^kR兲² 2m^* − k_R²

2m^*. 共2兲 These subbands have the distinct Fermi momenta k_␮=kF

−␮^kR and the same Fermi velocityv_F=k_F/m^*. Here we de- note kR=m^*␣R and kF=

冑

2m^*EF+k_R², and assume that 2kR

⬍kF.

The effective Coulomb interaction isV_q^{ef f}_␻=V_q/␧q␻, where Vq= 2␲^e2/共q␧_⬁兲, and ␧_⬁ is the 共high-frequency兲 dielectric constant of medium. The dielectric function ␧q␻= Re␧q␻

+iIm␧q␻describes effects of dynamic screening, and in the random phase approximation共RPA兲it is given by²

␧q␻= 1 −V_q⌸q␻, 共3兲 where⌸q␻is a polarization operator. In the presence of SO coupling, the latter is a sum

⌸q␻=_␹,␮=±

兺

^{⌸q␹,␮␻ 共4兲}

of contributions

⌸q␹,␮␻ = lim

␦→0

冕

共2^d␲^2k兲²

nF共⑀k␮兲−nF共⑀k+q␹␮兲

␻⁺ⁱ␦⁺⑀k␮−⑀k+q␹␮ Fk,k+q␹ , 共5兲 where the indices␹^{= + and}␹= − correspond to the intersub- band and intrasubband transitions, respectively. The form factors

F_k,k+q^␹ =1

2关1 +␹cos共␾k−␾k+q兲兴 共6兲 originate from the rotation to the eigenvector basis, and

cos共␾k−␾k+q兲=兩k兩+x兩q兩

兩k+q兩 ,x= cos共␾k−␾q兲 ⬅cos␾^. 共7兲 Throughout the paper we will use the dimensionless units y=kR/kF,z=q/ 2kF,v=k/kF, andw=m^*␻^{/ 2k}F

2.

III. POLARIZATION OPERATOR AT ARBITRARY FREQUENCY

A. Evaluation of the polarization operator

For a calculation of共5兲, it is useful to shiftk→k−q in those terms of共5兲that containnF共⑀k+q␹␮兲. At the same time we

can shift the integration angle␾→␾⁺␲ in the same terms due to the momentum isotropy of the spectrum共2兲. Conve- niently regrouping all contributions in Eq.共4兲, we cast it into the form

⌸q␻=_␮,␭=±

兺

^{⌸q␮␻,␭, 共8兲}

where

⌸q␮␻,␭= lim

␦→0

冕

共2^d␲^2k兲²nF共⑀k␮兲

冉

⑀k␮−⑀k+q␮^Fk,k+q++␭共␻⁺ⁱ␦兲 + F_k,k+q⁻

⑀k␮−⑀k+q−␮ +␭共␻⁺ⁱ␦兲

冊

^{, 共9兲}

and the index␭ effectively labels the contributions from the different共“in” and “out”兲Fermi functions.

In the limit of zero temperature we obtain

⌸q␮␻,␭= 1 8␲²␦^lim→0

冕

0

k_F−␮k_R

k dk

冕

0 2␲

d␾

⫻

冉

⑀^{1 + cos}k␮−⑀k+q␮ ^共␾+^k␭共⁻␻^␾k+q+i␦^兲兲+ 1 − cos共␾k−␾k+q兲

⑀k␮−⑀k+q−␮ +␭共␻⁺ⁱ␦兲

冊

^.

共10兲 After the intermediate steps共A1兲,共A2兲we cast共10兲into the form

− 1

␯^{Im⌸q␮␻,␭=}

冕

0 1−␮y

vg_i共v,z,w,y兲dv, 共11兲

−1

␯^{Re⌸q␮␻,␭=}

冕

0 1−␮y

vfi共v,z,w,y兲dv, 共12兲 where␯⬅␯2D=m^*/共2␲兲is the density of states in 2DEG per each spin component; the indicesi= 1 , 2 , 3 , 4 correspond to 兵␮,␭其=兵− , +其,兵+ , +其,兵− , −其,兵+ , −其, and the functions gi, fi

are defined by gi=␭Ci

2

冕

0 2␲

d␾sign关2vzx−␮^yv+ 2共z²−␭w兲兴

⫻共x+␦i兲␦共x²+␤ix+␥i兲, 共13兲

f_i= C_i 2␲

冕

0

2␲

d␾ ^x+␦i

x²+␤ix+␥i

, 共14兲

with the coefficients Ci=v−␮^y

2v²z , ␦i=共z²−␭w兲−␮^yv 共v−␮y兲z ,

␤i=2共z²−␭w兲−␮y共v+␮y兲

vz ,

␥i=共z²−␭w兲²−␮yv共z²−␭w兲−z²y²

v²z² . 共15兲

In共13兲the Dirac delta function is denoted by␦共. . .兲, as usual.

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The equation x²+␤ix+␥i= 0 has the roots labeled by ␴

= ±,

␭i␴=␮y共v+␮y兲− 2共z²−␭w兲+␴^y

冑

共v+␮y兲²+ 4␭w

2vz .

共16兲 The roots␭_1␴ and␭_2␴ are always real, while ␭_3␴ and ␭_4␴

become complex in the ranges y− 2

冑

_w⬍v⬍y+ 2

冑

w and

−y− 2

冑

w⬍v⬍−y+ 2

冑

w, respectively.

All integrals in共11兲and共12兲receive a contribution from those ranges ofvwhere␭i␴ are real. Additionally, the inte- grals 兰vf3,4dv receive a contribution from the ranges with complex␭_3,4␴. In order to take into account both types of contribution, we represent the functionf_i=f_i^I+f_i^IIas a sum of f_i^I=fi⌰共␤i2− 4␥i兲 andf_i^II=fi⌰共4␥i−␤i2兲. Obviously, f_1,2^II ⬅0.

Let us also introduce Re⌸^I and Re⌸^II that are obtained by integrating the functions 兺ivf_i^I and 兺ivf_i^II, respectively.

Thus, the full real part of the polarization operator is given by the sum of the two, i.e.,

Re⌸= Re⌸^I+ Re⌸^II. 共17兲 Note that in the limitw→0 all the roots␭i␴become real, and it might seem that Re⌸^II vanishes in the static limit. How- ever, it is not the case, and Re⌸^IIdoes give a finite contri- bution asw→0. We will discuss this point in a deep detail in the next section.

For␤i

2⬎4␥iwe have

␦共x²+␤ix+␥i兲=␦共x−␭i+兲+␦共x−␭i−兲

␭i+−␭i−

, 共18兲

x+␦i

x²+␤ix+␥i

= 1

␭i+−␭i−

冉

^␭xi+−+␭^␦i+ⁱ

−␭_i−+␦i

x−␭i−

冊

^{. 共19兲}

Using the integrals 共A8兲 and 共A9兲 we establish that gi

=兺_␴gi␴, f_i^I=兺_␴f_i^I_␴, and g_i␴= ␭C_i

␭i+−␭i−

␭i␴+␦i

冑

1 −␭i2␴⌰共1 −兩␭i␴兩兲

⫻⌰„共v+␮y兲²+ 4␭w…

⫻sign关y+␴

冑

共v+␮^y兲²+ 4␭w兴, 共20兲 f_i^I_␴= − ␴^Ci

␭i+−␭i−

␭i␴+␦i

冑

␭i2␴

− 1⌰共兩␭i␴兩− 1兲sign共␭i␴兲. 共21兲 For␤i

2⬍4␥iwe have

−1

␯ ^Re⌸

II=

冕

_D−^vf3IIdv+

冕

_D+^{vf4IIdv, 共22兲}

whereD_⫿are defined in the following way:

y²⬎4w: D−=关y− 2

冑

w,y+ 2

冑

w兴, 共23兲 4w⬎y²艚1⬎4w:D−=关0,y+ 2

冑

w兴, 共24兲

4w⬎1:D−=关0,1 +y兴. 共25兲

and

y²⬎4w:D+= 쏗, 共26兲 4w⬎y²艚1⬎4w:D₊=关0,−y+ 2

冑

w兴, 共27兲

4w⬎1:D+=关0,1 −y兴. 共28兲

Applying the table integral共A10兲, we obtain f_3,4^II 共v,z,w,y兲= 1

2z

冑

共v₁^␮−v兲共v−v₂^␮兲

⫻

冑

P_␮共v兲+

冑

Q_␮共v兲

冑 _共 冑

P_␮共v兲+

冑

Q_␮共v兲

兲

^{2− 4y2z, 共29兲}

where

P_␮共v兲=共z+␮y兲共v−v₂^␮兲共v₃^␮−v兲, 共30兲 Q_␮共v兲=共z−␮y兲共v−v₁^␮兲共v₄^␮−v兲, 共31兲 v_1,2^␮ = −␮y±共z+w/z兲, 共32兲

v_3,4^␮ = ±z+ w

␮^{y±z, 共33兲} and the indices ␮^{= − and} ␮= + correspond to f₃^II and f₄^II, respectively.

The numerical evaluation of共22兲can be performed with a controlled and sufficiently high accuracy, since the functions 共29兲 may diverge only near the actual integration edges

−␮^{y± 2}

冑

w. On the other hand, one can try to find an analytic expression for 共22兲. An alternative representation of f_3,4^II , which is equivalent to 共29兲 and more suitable for further analytic evaluation, is introduced in Appendix B.

As for Im⌸= −␯兺_i=1⁴ 兰₀^1−␮yvgidv and Re⌸^I=

−␯兺_i=1⁴ 兰₀^1−␮yvf_i^Idv, their analytic evaluation is presented be- low. Making a change of variables,

␶⁼¹

2

共

^{−␮共v+␮y兲+␴}

^冑

^{共v+␮y兲2+ 4␭w}

兲

^{, 共34兲}

we establish the relations共A3兲–共A7兲, and thus deduce

−1

␯ ^Im⌸=

兺

␴,␮␴

冕

␶_␴+共y兲

␶_␴+共␮兲

d␶L⁺共␶兲

−⌰共1 − 4w兲

兺

_␮

冕

_␶^␶₊₋⁻⁻_共␮兲^{共␮兲 d␶L−共␶兲}

+ 2⌰共y²− 4w兲

冕

_␶−−共y兲

␶+−共y兲

d␶L⁻共␶兲, 共35兲 and

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−1

␯ ^Re⌸I=

兺

_␴,␮

冕

␶_␴+共y兲

␶_␴+共␮兲

d␶R⁺共␶兲

+⌰共1 − 4w兲

兺

_␴,␮

冕

_−␮␶^␶␴−₊₊^共␮兲_共0兲 ^{d␶R−共␶兲}

+ 2⌰共y²− 4w兲

兺

_␴

冕

_␶␴−共y兲

␶−+共0兲

d␶R⁻共␶兲, 共36兲 where

␶1,2= ±w/z, ␶3,4= −y±z, 共37兲

␶_␴␭共x兲=1

2

共

^−x+␴

^冑

^{x2+ 4␭w}

兲

^{, 共38兲}

L^±共␶兲=L共␶兲sign关␶²+y␶±w兴, 共39兲 R^±共␶兲=R共␶兲sign关␶^z⫿w共␶⁺y兲/z兴, 共40兲

L共␶兲= 1 2z

共␶−␶3兲共␶−␶4兲

冑 _兿

k=1

4 共␶⁻␶k兲⌰

冉 ^兿

^{k=14 共␶−␶k兲}

冊

^{, 共41兲}

R共␶兲= 1 2z

共␶−␶3兲共␶−␶4兲

冑

⁻

_兿

k=1

4 共␶⁻␶k兲⌰

冉

⁻

^兿

^{k=14 共␶−␶k兲}

冊

^{. 共42兲}

In Fig.1, the quarter-plane 共z⬎0 ,w⬎0兲 is divided into the domains A=兵共z,w兲兩w⬍z共z−y兲其, B=兵共z,w兲兩w⬎z共z

−y兲艚w⬍z共z+y兲艚w⬎z共y−z兲其, C=兵共z,w兲兩w⬎z共z+y兲其, D

=兵共z,w兲兩w⬍z共y−z兲其, which are specified by an ordering of the roots␶k共37兲:

A:␶4⬍␶2⬍␶1⬍␶3, 共43兲 B:␶4⬍␶2⬍␶3⬍␶1, 共44兲 C:␶2⬍␶4⬍␶3⬍␶1, 共45兲 D:␶4⬍␶3⬍␶2⬍␶1. 共46兲 In each domain one should compare␶kand␶_␴␭共x兲in order to establish actual limits of integration in共35兲and 共36兲. After that, it becomes possible to write down Im⌸and Re⌸^Iin an explicit form. We refer to the Appendix C, where the neces- sary expressions for establishing the explicit form of共35兲are

presented. Similar expressions for共36兲can be easily derived from Ref.17.

B. SO-induced extension of the particle-hole excitation region Let us analyze the most important modifications to the polarization operator induced by SO coupling. First of all, we are interested in establishing the new boundaries of a particle-hole continuum共or Landau damping region兲, which is defined by the condition Im⌸⫽0. They can be deter- mined from a simple consideration of extremes of the de- nominators in共5兲. On the basis of this purely kinematic ar- gument, it is easy to establish that due to SO coupling there appears a new wedge-shaped region of damping共shown in Fig.2兲. It is bounded by the two parabolas −共z−y兲²−共z−y兲

=w₄共z兲⬍w⬍w₁共z兲=共z+y兲²+共z+y兲 and attached to the boundaryw₀共z兲=z²+zof the conventional particle-hole con- tinuum共obtained in the absence of SO coupling according to Ref. 3兲. Another boundary w=z²−zof the latter transforms into w=共z−y兲²−共z−y兲 for nonzero y 共this occurs at z⬎1, and therefore it is not shown in Fig.2兲.

An extension of the particle-hole continuum reflects an opened possibility for transitions between SO-split subbands.

Therefore one can expect new SO-induced effects of damp- ing of various collective excitations共plasmons, LO phonons兲 in the new regions of damping. These issues will be dis- cussed in the subsequent sections. Below we present Im⌸in explicit form for the valuesw⬎w₀共z兲, i.e., above the bound- aryw₀共z兲of the conventional particle-hole continuum. Since w₀共z兲⬎z²+yz, we deal with the case of the roots’ ordering 共45兲 corresponding to the domain C. Assuming that y does not exceed the value共2 −

冑

2兲/ 2⬇0.3, we deduce from 共35兲,

−1

␯ ^{Im⌸= −⌰共w2共z兲−w兲⌰共w−w4共z兲兲}

冕

−z−y z−y

d␶L共␶兲

−⌰共w1共z兲−w兲⌰共w−w₂共z兲兲

⫻

冕

−z−y 1 2关1−冑1+4w兴

d␶L共␶兲+⌰共w3共z兲−w兲 FIG. 1. The domains A, B, C, D 关see Eqs. 共43兲–共46兲兴 corre-

sponding to the different orderings of the roots共37兲.

FIG. 2. SO-induced extension of the particle-hole continuum shown fory= 0.1. It is bounded from above by parabolaw₁共z兲 关Eq.

共49兲兴and bounded from below by the parabolaw₄共z兲 关Eq.共52兲兴and by the parabolaw₀共z兲=z²+zof the conventional particle-hole共Lan- dau damping兲region. The small darkened triangle indicates the re- gion where the approximation␴共z,w兲⬇␴共0 ,w兲is applicable.

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⫻⌰共w−w₄共z兲兲

冕

−z−y 1 2关冑1−4w−1兴

d␶L共␶兲. 共47兲 Applying the table integral 共C5兲 for x₁=␶1, xa=x₂=␶3, xb

=x₃=␶4, andx₄=␶2, we obtain the analytic expression

−1

␯ ^{Im⌸= −⌰共w2共z兲−w兲⌰共w−w4共z兲兲A共z−y兲}

−⌰共w1共z兲−w兲⌰共w−w₂共z兲兲A

冉

^{12关1 −}

^冑

^{1 + 4w兴}

冊

+⌰共w3共z兲−w兲⌰共w−w₄共z兲兲A

冉

^12关

^冑

^{1 − 4w− 1兴}

冊

^,

共48兲 where

w₁共z兲=共z+y兲²+共z+y兲, 共49兲 w₂共z兲=共z−y兲²−共z−y兲, 共50兲 w₃共z兲= −共z+y兲²+共z+y兲, 共51兲 w₄共z兲= −共z−y兲²−共z−y兲, 共52兲 and

A共x兲= 1

2z

冑

^ww⁻+^zxzx关z²−共x+y兲²兴+ k

4z

冑

w兵关共w/z−y兲²−z²兴

⫻F„␸共x兲,k…−关共w/z+z兲²−y²兴E„␸共x兲,k…

+ 2y共w/z−y−z兲⌸„␸共x兲,n,k…其. 共53兲 The argument␸共x兲and the parameterskandnof the elliptic functionsF,E, and⌸are defined in the following way:

␸共x兲= arcsin

冑

^共x+z+2共^yzx^兲共w/z+w⁻兲 ^y+z兲, 共54兲 k= 2

冑

w

冑

共w/z+z兲²−y², n= 2z

w/z+z−y. 共55兲 Note that␸共−z−y兲= 0 and␸共z−y兲=␲/ 2, and henceA共z−y兲 is expressed in terms of the complete elliptic integrals F共␲^{/ 2 ,k}兲,E共␲^{/ 2 ,k}兲, and⌸共␲^{/ 2 ,n,k}兲.

C. Comparison with the approximate result

It is also worthwhile to compare our exact result共48兲with an approximation for smallzcommonly used in the literature 共cf., e.g., Refs.13,14, and 16兲. It neglects the square of the transferred momentum q² in the denominator of 共5兲, and therefore leads to the kinematic extension of the conven- tional particle-hole continuum to a strip y−y²⬍w⬍y+y² parallel to thezaxis. This approximation can be effectively expressed in the form

⌸共q,␻兲 ⬇−i q²

e²␻␴共0,␻兲, 共56兲 and considered as stemming from the identity

⌸共q,␻兲= −i q²

e²␻␴共q,␻兲, 共57兲 with ␴共q,␻兲 replaced by ␴共0 ,␻兲. The optical conductivity

␴共0 ,␻兲⬅␴␻can be easily found.¹³For example, its real part equals

Re␴␻= e²

16␲^{⌰共y2−兩w−y兩兲. 共58兲} Thus, Eq.共56兲yields

−1

␯^Im⌸⬇ 4␲^z2

e²w Re␴␻=␲^z2

4w⌰共y²−兩w−y兩兲. 共59兲 We would like to argue that the approximation共59兲works well only in a quite small region restricted by the conditions w₃共z兲⬍w⬍w₂共z兲 共see Fig.2兲. Since the parabolasw₂共z兲and w₃共z兲intersect atz=1

2关1 −

冑

1 − 4y²兴⬇y², this region is repre- sented by a small triangle between the points 共0 ,y−y²兲, 共0 ,y+y²兲, 共y²,y兲. Inside this triangle −Im⌸/␯ is entirely determined by −A共z−y兲. Observing thatk²⬇2n⬇4z²/wfor small z⬍y² and expanding the complete elliptic integrals with respect toz, we find an asymptotic value,

−A共z−y兲 ⬇␲^z2

4w, 共60兲

which coincides with共59兲, except for the domain of applica- bility.

In Fig.3we compare the curves 16␲Re关␴共zc,w兲兴/e²cal- culated at the different values ofzcandy= 0.1, and plotted as a function of w. The unit step between y−y²= 0.09 and y +y²= 1.01 corresponds to the optical conductivity 共58兲, which is recovered atzc= 0. One can observe that forzy² the approximation ␴共z,w兲⬇␴共0 ,w兲 leading to 共59兲 works quite well, while forzy² it completely breaks down.

IV. STATIC LIMIT OF THE POLARIZATION OPERATOR A. Careful derivation of the static limit

In this section we derive the static limit

FIG. 3. The real part of conductivity for the values ofz_c= 0.0, 0.025, 0.005, 0.01, 0.02, 0.03, 0.04 calculated at y= 0.1. The unit step corresponds to z_c= 0. The dashed line corresponds to z_c=y²

= 0.01, which is the limiting value for the applicability of the ap- proximation␴共z,w兲⬇␴共0 ,w兲.

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lim

w→0Re⌸共z,w兲= lim

w→0Re⌸^I共z,w兲+ lim

w→0Re⌸^II共z,w兲.

共61兲 After Eq. 共17兲 we have already made an observation that lim_w→0Re⌸^II共z,w兲 might seem to vanish. Below we prove that, in fact, it does not vanish, but rather produces a finite and a very important contribution to limw→0Re⌸共z,w兲 for smallz⬍y.

Let us first identify the quantity limw→0Re⌸^I共z,w兲. One can see that if we putw= 0 in共16兲, all the roots␭i␴become real for all values of the integration variablev. Tracing back the derivation of the expression 共14兲, it is easy to see that limw→0Re⌸^I共z,w兲 is obtained, in fact, from the definition 共cf. Ref.12兲

⌸_q,␻+i␦

=0=

兺

␹,␮=±

冕

共2^d2␲^k兲²

nF共⑀k␮兲−nF共⑀k+q␹␮兲

⑀k␮−⑀k+q␹␮ F_k,k+q^␹ . 共62兲 We would like to emphasize that, in general,⌸_q,␻+i␦

=0 is not always the same as lim_␻→0⌸q␻, and there might occur a specific situation when

␻→0lim⌸q␻⫽⌸_q,␻+i␦

=0. 共63兲

Once limw→0Re⌸^II共z,w兲 is nonzero, the subtle property 共63兲, for example, holds for the 2DEG with Rashba SO cou- pling described by the Hamiltonian共1兲.

The correct static limit is, of course, given by lim_␻→0⌸q␻. However, it is very tempting to put␻^{= 0 from} the very beginning. In principle, one can do that. But then, in order to protect oneself from a possible mistake, it is neces- sary to keep small ␦⫽0 until the very end, even during a calculation of a real part of the polarization operator. More formally, the sequences of operations,

␻→lim0

冕

^d2klim␦→0共¯兲 and lim

␦→0

冕

^d2k␻→^lim0共¯兲 共64兲 always lead to a correct static limit, while the sequence

冕

^d2k␻,^lim␦→0共¯兲 共65兲 might give in some specific cases an unphysical result break- ing causality and violating analytic properties of a共retarded兲 response function.

Let us rigorously prove the statements that have been made previously. In studying the limit w→0 of Re⌸

= Re⌸^I+ Re⌸^II, it is sufficient to consider how we approach the axis w= 0 from the domains D and A 共see Fig.1兲that cover the ranges of momentaz苸共0 ,y兲 andz苸共y, +⬁兲, re- spectively, as w→0. In turn, the domains C and B in this limit shrink to the points z= 0 and z=y, and therefore their consideration is not important for our current purpose.

Let us first consider Re⌸^I in the domainDat very small w. In the range of integration关␶2,␶1兴we can replace

R^±共␶兲 ⬇R1共␶兲sign共␶⫿wy/z²兲= ⫿R1共␶兲, 共66兲 where

R1共␶兲= 1 2z

冑

y²−z²

冑

共w/z兲²−␶^{2. 共67兲}

Meanwhile, in the range of integration关␶4,␶3兴, we have R^±共␶兲 ⬇R2共␶兲= − 1

2z␶

冑

z²−共␶⁺y兲². 共68兲 We observe that ␶++共−兲= −␶−+共+兲⬇1, ␶++共+兲= −␶−+共−兲⬇w,

␶++共y兲⬇w/y, ␶−+共y兲⬇−y; ␶+−共−兲= −␶−−共+兲⬇1, ␶−−共−兲=

−␶+−共+兲⬇w, ␶+−共y兲⬇−w/y, ␶−−共y兲⬇−y; ␶++共0兲= −␶−+共0兲

=

冑

w. Taking into account thatz⬍y, we carefully arrange the limits of integration, and thus obtain

−1

␯^wlim→0Re⌸^I=

冉 ^冕

^{−yz−y +}

^冕

^{−y−z−y +}

^冕

^{z−y−z−y + 2}

^冕

^−yz−y

冊

^{d␶R2共␶兲}

− lim

w→0

冉 ^冕

^{w/yw/z +}

^冕

^{w/yw +}

^冕

^−w/z−w

冊

^{d␶R1共␶兲}

+ lim

w→0

冉 ^冕

^{w/zw +}

^冕

^{−w/z−w + 2}

^冕

^−w/y−w/z

冊

^{d␶R1共␶兲}

= 2

冉 ^冕

^{−yz−y +}

^冕

^−y−z−y

冊

^{d␶R2共␶兲}

− lim

w→04

冕

w/y w/z

d␶R1共␶兲= −

冕

0

z2␶

冑

z²−␶²d␶ z共␶^2−y2兲

−

冕

z/y

1 2

冑

y²−z²d␶ z

冑

1 −␶^{2 = 2 −}

␲

z

冑

y²−z², 共69兲 by virtue of the table integral共A11兲.

Let us now consider Re⌸^Iin the domainAat very small w. Taking into account that nowz⬎y, we obtain

−1

␯^wlim→0Re⌸^I= lim

w→0

冉 ^冕

^{w/ymin共1,z−y兲 +}

^冕

^{w/ymax共w,w/z兲}

冊

−

冕

−y

−max共w,w/z兲

−

冕

−y

−min共1,z+y兲

+

冕

冑w min共1,z−y兲

+

冕

冑w max共w,w/z兲

−

冕

−冑w

−max共w,w/z兲

−

冕

−冑w

−min共1,z+y兲

−共2

冕

−w/y

−冑w

− 2

冉 ^冕

^−y−冑w

冊

^{d␶R共␶兲}

= lim

w→0

冉 ^冕

^{w/ymax共w,w/z兲 −}

^冕

^{−w/y−max共w,w/z兲}

−

冕

−y

−min共1,z+y兲

冊

⁺

冉 ^冕

^{冑min共1,z−yw 兲}

−

冕

−y

−冑w

冊

^{d␶2R共␶兲. 共70兲}

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In the first and the second integrals of the last part of共70兲, we can replace

R共␶兲 ⬇R˜

1共␶兲= − 1 2z

冑

z²−y²

冑

␶²−共w/z兲², 共71兲 while in the third integral

R共␶兲 ⬇R2共␶兲sign共␶兲. 共72兲 The fourth and fifth integrals require a more careful consid- eration. We note that for smallw there holds the inequality

冑

w⬎w/z, which allows us to neglect w inR共␶兲. Therefore these two integrals can be rewritten as

2 lim

w→0

冉 ^冕

^{−ymin共1,z−y兲 −}

^冕

^−冑冑ww

冊

^{d␶R2共␶兲= 2}

^冕

^{−ymin共1,z−y兲d␶R2共␶兲,}

共73兲 where the integral on the rhs of 共73兲 is understood in the sense of a principal value.

Thus we obtain forz⬎y

−1

␯^wlim→0Re⌸^I= −

冕

z/y

max共1,z兲2

冑

z²−y²d␶ z

冑

␶^{2− 1}

−

冕

0

min共1−y,z兲

冑

z²−␶^2d␶ z共␶^+y兲

−

冕

0

min共1+y,z兲

冑

z²−␶^2d␶

z共␶^−y兲 . 共74兲 It is easy to calculate this expression using the table integral 共A12兲. The result is expressed in共A13兲and共A14兲.

Let us now consider the static limit of Re⌸^II. In Appen- dix D we show that it equals

−1

␯^wlim→0Re⌸^II=␲

冑

y²−z²

z ⌰共y−z兲. 共75兲 We note that this term is nonzero only forz⬍y共orq⬍2kR兲.

One can observe that 共75兲is exactly canceled by the coun- terterm from共69兲.

Collecting all contributions and introducing sin␺=y/zfor y⬍zand sin␺±=共1 ±y兲/zfor 1 ±y⬍z, we present the static limit of the polarization operator in the form

−1

␯^{Re⌸共z,0兲= 2⌰共1 −y−z兲+⌰共y−兩z− 1兩兲}

冉

^{1 +␲}2sin␺

冊

− 2⌰共z− 1兲arccoshzcos␺

+_␮=±

兺

^{⌰„z−共1 +␮y兲…}

冉

^{1 +␮␺␮sin␺}

− cos␺_␮^{− 2 cos}␺^{ln 1 +zsin共}␺_␮⁻␮␺兲 2

冑

2zcos¹₂␺␮cos¹₂␺

冊

^,

共76兲 where Re⌸共z, 0兲⬅limw→0Re⌸共z,w兲. In what follows we also use the abbreviations Re⌸^I共z, 0兲 and Re⌸^II共z, 0兲 for limw→0Re⌸^I共z,w兲 and limw→0Re⌸^II共z,w兲.

B. Analysis of Eq. (76)

Let us analyze the expression 共76兲. For the valuesz⬍1

−ywe obtain Re⌸共z, 0兲= −2␯, which ensures the fulfillment of the compressibility sum rule. Forz⬎1 −y, Re⌸共z, 0兲de- viates from the value −2␯^.

Fory= 0, Eq. 共76兲reproduces the conventional result of Stern,³

−1

␯ ^{Re⌸y=0共z,0兲= 2 − 2}

冑

1 − 1/z²⌰共z− 1兲. 共77兲 In Fig.4we compare −Re⌸共z, 0兲/␯^{fory= 0.1 andy= 0. We} demonstrate that −Re⌸y⫽0共z, 0兲/␯ is always larger than

−Re⌸y=0共z, 0兲/␯, although the difference between the two curves共shown in the inset兲is quite small. In particular, the maximal value of −Re⌸共z, 0兲/␯ ^atz= 1 scales withylike

−1

␯ ^{Re⌸共1,0兲 ⬇2 +} 2

冑

2

3 y^3/2, 共78兲 while at largez the asymptotic behavior of共76兲is⬃^1+y_z2².

An important test of our results is provided by the Kramers-Kronig relations and the sum rules. For example, using the analytic expressions for Re⌸共z, 0兲and Im⌸共z,w兲 we have checked the zero-frequency Kramers-Kronig rela- tion

Re⌸共z,0兲= 2

␲

冕

0

⬁dw

w Im⌸共z,w兲. 共79兲 The actual limits of the integration are finite and given by the boundaries of the particle-hole continuum. This has allowed us to confirm共79兲 numerically with a sufficiently good ac- curacy共the deviation isⱕ10⁻⁶ even for a quite simple rou- tine兲.

It is instructive to compare Re⌸共z, 0兲and Re⌸^I共z, 0兲, FIG. 4. The behavior of −Re⌸共z, 0兲/␯ near z= 1 for y= 0.1 共solid line兲and y= 0共dotted line兲. The difference between the two curves is shown in the inset.

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−1

␯ ^Re⌸

I共z,0兲= −1

␯ ^{Re⌸共z,0兲−}

␲

冑

y²−z²

z ⌰共y−z兲.

共80兲

It is obvious that Re⌸does not have any anomaly at z=y 共cf. Ref.12兲, while Re⌸^Idoes have it共see Fig. 5兲. One can also observe that Re⌸^I diverges in the limit z→0 and changes the sign at some value ofz. Therefore, it cannot be regarded itself as a correct static limit. Otherwise, it would have violated the compressibility sum rule and generated an instability of the medium by virtue of SO coupling.

Thus, we have demonstrated that in order to obtain the correct static limit of the polarization operator for the 2DEG with Rashba SO coupling, it is crucially important to follow the thorough definition that implies共64兲, but not共65兲. Oth- erwise, the contribution Re⌸^II共z, 0兲 is missing. Being the difference between Re⌸共z, 0兲 and Re⌸^I共z, 0兲, it shows up for z⬍y 共or q⬍2kR兲, and disappears only when kR goes to zero. Therefore, for the conventional 2DEG共k_R= 0兲 there is no difference between⌸q,0and⌸˜

q,0关and between共64兲and 共65兲as well兴. So the property共63兲is a peculiar feature of the 2DEG withkR⫽0. We suppose that it might be related to the singularity of the spectrum共2兲atk= 0.

C. Effective interaction: Modification of the Friedel 2k_F-oscillations

Let us conveniently rewrite the RPA expression共3兲for the dielectric function in terms of˜r_s=r_s/共2

冑

2兲,

␧共z,w兲= 1 −˜r_s

z␯^{⌸共z,w兲, 共81兲} wherers=

冑

2m^*e²

Ⲑ

^{共kF␧⬁兲} is the 2D Wigner-Seitz parameter.

The latter controls the accuracy of RPA, which becomes bet- ter with decreasingr_s. In the limitw→0, Eq.共81兲describes the static screening of the Coulomb interaction. It is well known³ that the singular behavior of the derivative,

−1

␯ d

dz⌸y=0共z,0兲

冏

^{z=1+␣⬇−}

^冑

␣^{2, 共82兲} atz= 1 +␣ with small␣⬎0 gives rise to the Friedel oscilla- tions of a screening potential, their leading asymptotic term being

V_osc^y=0共␳兲= −2e²kF

␧_⬁ · 2r˜_se^−2kFd 关␧y=0共1,0兲兴²

sin 2kF␳

共2kF␳兲², 共83兲 where␳⁼

冑

x₁²+x₂², anddis a distance from a probe charge to the 2DEG plane. The SO modification to共83兲follows from

−1

␯ d

dz⌸共z,0兲 ⬇−

冑

_␣^{2 ·}

^冑

^{1 −y2, 共84兲}

and results in

Vosc共␳兲= −2e²kF

␧_⬁ · 2r˜_se^−2kFd 关␧y=0共1,0兲兴²

sin 2kF␳

共2kF␳兲²Q共y兲, 共85兲 where the factor

Q共y兲=

冑

1 −y²

冉

^␧y=0␧共1,0兲^共1,0兲

冊

^{2 共86兲}

is always smaller than 1. It means that due to the SO cou- pling the amplitude of the Friedel oscillations is diminished 共cf. Ref.12兲, although the amount of such a decrease is quite small共⬃0.5% for y= 0.07 and rs= 0.2兲. We note that due to 共78兲it is possible to approximate

冉

^{␧y=0␧共1,0兲共1,0兲}

冊

^{2⬇1 −}3共1 + 2r⁴

^冑

^2y3/2˜_s兲. 共87兲 Rigorously speaking, it is necessary to take into account in 共85兲the dependence ofkF onkR as well.

The second derivative共d²/dz²兲⌸共z, 0兲also diverges at the points z= 1 ±y+ 0⁺, thus contributing to the subleading asymptotic terms of the oscillating potential.¹²

V. STRUCTURE FACTOR AND SO-INDUCED DAMPING OF PLASMONS

An important quantity that either can be directly observed or enters into expressions for other observable quantities is the structure factor²

S共z,w兲= − Im关1/␧共z,w兲兴. 共88兲 It depends on the Wigner-Seitz parameter rs and contains information about both particle-hole excitations and collec- tive excitations 共plasmons兲. The spectrum of the latter is found from the equation Re␧= 0. For the conventional 2DEG it can be derived on the basis of Ref.3and equals¹⁸

wpl共z兲=z共z+ 2r˜_s兲

2r˜_s

冑

^4r˜s2_z共z^{+ 4r}+ 4r^˜sz˜³_s⁺兲 ^z4⌰共z^*−z兲, 共89兲 where the endpoint of the spectrum z^* is the real positive root of the equation z²共z+ 4r˜_s兲= 4r˜_s². Provided Im␧= 0, the plasmon spectrum is undamped, and the structure factor is

␦^{-peaked atw=w}pl共z兲, FIG. 5. The behavior of −Re⌸共z, 0兲/␯ 共solid line兲 and

−Re⌸^I共z, 0兲/␯共dashed line兲nearz=yfory= 0.1.

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Spl共z,w兲=␣共z兲␦„w−wpl共z兲…, 共90兲 with the weight factor ␣共z兲=␲关Re⳵␧/⳵w兩w=w_pl共z兲兴⁻¹, given by

␣共z兲= ␲

冑

z关16r˜_s⁴−z⁴共z+ 4r˜_s兲²兴

2r˜_s

冑

共4r˜_s²+ 4r˜_sz³+z⁴兲共z+ 4r˜_s兲³. 共91兲 The plasmon spectrum can be visualized on a contour plot of S共z,w兲 by adding an artificial infinitesimal damping ␦ ^to Im␧. It becomes very helpful when an exact analytic expres- sion similar to共89兲is not known.

In the presence of SO coupling, the structure factor de- pends on y as well. Solving approximately the equation Re␧= 0 aty⫽0, we can establish how the plasmon spectrum is modified by SO coupling. In fact, we observe that the SO modification of共89兲 is quite small. Later on, we will com- ment on how small it is, and explain why this effect is not really important共cf. Ref.15兲.

A much more important effect is a damping of plasmons generated by SO coupling. As it has been already discussed, SO coupling extends the continuum of the particle-hole ex- citations up to the new boundaries shown in Fig.2. There- fore, the plasmon spectrum is expected to acquire a finite width, whenever it enters into the SO-induced region of damping.

In Fig.6we show the contour-plots of the structure factor S共z,w兲 depicting the plasmon spectrum by the bold line, in black, where Im␧= 0共undamped plasmon, the structure fac- tor is ␦-peaked兲 and in gray, where Im␧⫽0 共SO-damped plasmon, the structure factor has a finite height and width兲.

Depending on the values ofrsandy, the two different cases are possible:共I兲 the plasmon enters only once into the SO- induced damping region 共upper panel兲; 共II兲 it enters twice, escaping for a while after the first entrance共lower panel兲.

Within the conventional boundaries w=z²±z of the particle-hole continuum, the structure factorS共z,w兲is modi- fied by SO coupling only slightly and can be approximated by the conventional expressions.³

In the SO-induced region of damping,S共z,w兲is very well approximated by the Lorentzian function describing the SO- damped plasmon with the width␥共z兲,

S共z,w兲SO-damp pl=␣共z兲

␲

␥共z兲

关w−wpl共z兲兴²+␥²共z兲, 共92兲 wherewpl共z兲 and ␣共z兲 are supposed to be practically inde- pendent ofyand therefore given by共89兲and共91兲, while

␥共z兲=␣共z兲

␲ ^{Im␧共z,wpl共z兲兲 共93兲} essentially depends on y via Im␧⫽0. In Fig.7 we present the enlarged plot with the cross sectionS共z= 0.1,w兲from the inset of the upper panel of Fig. 6 and compare it with the approximate S共0.1,w兲SO-damp pl given by共92兲. The inset of Fig.7 shows both curves on a more fine scale. Comparing the positions of their peaks, we conclude that the shift of the plasmon dispersion due to SO coupling from w_pl共z兲 共89兲 is one or two orders of magnitude smaller than ␥共z兲 关unless

␥共z兲= 0兴, and therefore can hardly be resolved experimen- tally. The almost equal height of the peaks confirms that共91兲 is also a very good approximation for the weight factor at y⫽0.

Let us find the values of yandrsthat correspond to the typical cases共I兲and共II兲shown in the upper and lower panels of Fig. 6, respectively. For this purpose, it is necessary to FIG. 6. Contour plots of S共z,w兲 showing the SO-induced wedge-shaped damping region共bounded by the dashed lines兲. The plasmon mode is depicted by the bold line. Insets show the cross sectionsS共z= 0.1,w兲 as a function of w. Upper panel:y= 0.07,r_s

= 0.2.Lower panel:y= 0.04,r_s= 0.6.