Journal ArticlePolarizability of noninteracting 2D Dirac electrons has a 1 1 / √qv-ω singularity at the boundary of electron-hole excitations. The screening of this singularity by long-range electron-electron interactions is usually treated within the random phase approximation. The latter is exact only in the limit of N →∞, where N is the ‘‘color'' degeneracy.We find that the ladder-type vertex corrections become crucial close to the threshold as the ratio of the nth order ladder term to the same order random phase approximation contribution is lnn|qv - ω|/Nn. We perform an analytical summation of the infinite series of ladder diagrams which describe the excitonic effect. Beyond the threshold, qv > ω, the real part of the polarization operator is found to be positive leading to the appearance of a strong and narrow plasmon resonance

Gangadharaiah, S.

Mishchenko, Eugene

English

The University of Utah: J. Willard Marriott Digital Library

P R L  100, 166802 (2008) P H Y S I C A L  R E V I E W  L E T T E R Sweek ending25 APRIL 2008Charge Response Function and a Novel Plasmon Mode in GrapheneS. Gangadharaiah, A .M . Farid, and E .G . M ishchcnko Department o f  Phvsies, University o f  Utah, Salt Ixike Citv, Utah 84112, USA (Received 13 October 2007; published 23 April 2008)Polarizability of noninteracting 2D Dirac electrons has a 1 /  j q v  — 10 singularity at the boundary of electron-hole excitations. The screening of this singularity by long-range electron-electron interactions is usually treated within the random phase approximation. The latter is exact only in the limit of N  —'* oo, where N  is the “color" degeneracy. We find that the ladder-type vertex corrections become crucial close to the threshold as the ratio of the «th order ladder term to the same order random phase approximation contribution is ln"|gv — co\/N". We perform an analytical summation of the infinite series of ladder diagrams which describe the excitonic effect. Beyond the threshold, qv > co, the real part of the polarization operator is found to be positive leading to the appearance of a strong and narrow plasmon resonance.DOI: 10.1103/PhysRevLett. 100.166802Introduction.—M any properties of interacting tw o­dimensional clcctrons with a linear Dirac spcctrum e =  ± v p ,  found in graphene monolayers, differ sharply from those with the parabolic spcctrum present in conventional semiconductor hctcrostructurcs [1,2]. A vanishing density of states at the Dirac point and less cffcctivc screening of the Coulomb interaction lead to the clcctronic properties of graphene being qualitatively different.One of the distinct signatures of nonintcracting 2D clcctrons in graphene, causcd by the abscncc of spcctrum curvature, is a divergent behavior of the polarization operator (chargc susceptibility) at the threshold for the cxcitation of clcctron-holc pairs [3 -7 ], I I (0)((u, q) =- N q 2/ I 6 y f a h ?  — o r ,  where N  is the degree of dcgcncr- acy (in graphene N  =  4 in the abscncc of magnetic field). The cffccts of clcctron-clcctron interaction on the polar­ization operator arc customarily accounted for by the ran­dom phase approximation (RPA) [5,8-10], which sums the infinite series of clcctron loops,n ^ A =  ( n (0)) - '  -  Vir (1)where V(/ =  l i r e 2/q ,  the diclcctric constant of the sub­strate being incorporated into the chargc e2. In particular, RPA predicts that the imaginary part of the polarization operator (which determines absorption in the system and the density-density correlation function) instead of diverg­ing actually vanishes at the threshold according to— IIj(pA a  g - 2^ r = ~ q v ,  where g =  e2/ v  is the dimcn- sionlcss interaction constant.A notable property of 2D clcctrons, well captured by RPA in a conventional Fermi liquid, is the cxistcncc of a low-frcqucncy collcctivc mode of chargc oscillations whose spcctrum is w 2 =  2e2E Fq, where E F is the Fermi energy. In undoped graphene, E F =  0, the plasmon mode is thus absent within RPA (finite doping or temperature can lead to the usual RPA plasmons [11,12]). M athematically this comcs from the fact that the real part of I I (0) isPACS numbers: 73 .23 .-b , 72.30.+qnegative for q v  >  co (i.e., beyond the domain of clcctron-holc excitations). In the present Letter wc dem ­onstrate that this conclusion is an artifact o f the approxi­mation ( 1) and that a collcctivc mode docs exist in undoped graphene. This mode dcscribcs chargc fluctuations in a system of clcctron-holc pairs interacting in a final state ( “ cxcitonic” cffcct), rather than free pairs, as implied by E q .(1 ).Wc begin with noting that RPA is much less justified in the ease of undoped graphene than in the ease of the conventional parabolic spcctrum, where it is formally valid for momenta less than the inverse screening length, q <K k . In undoped graphene the vanishing density of carriers ensures that k —> 0. In particular, direct calculation of the clcctron sclf-cncrgy to the sccond order in interaction demonstrates that non-RPA contributions arc generally smaller than the RPA terms only by virtue of 1 / N  [13]. The RPA is cxact only in the formal limit of N  —> oo, sincc cach RPA term corresponds to the largest number of loops possible in cach order of the perturbation theory.Below wc analyze consccutivc orders of the perturbation series in the bare interaction Vq =  l i r e 2f q  and compare RPA diagrams to the ladder corrections. Wc find that the ratio of the nth ladder term to the corresponding RPA contribution of the same order is N ~ nlan(^ " ^ ) and large closc to the threshold. After extracting the leading singular terms in cach order, wc calculate the infinite ladder series and obtain a nonpcrturbativc expression for the polariza­tion operator near the threshold. Summation of the leading divcrgcncics is possible  oncc wc realize that for q v / \ q v  — o)\ »  1 the main contribution arises from the processes that involve quasi-onc-dimcnsional motion of interacting particlcs along the direction of the external momentum q. In general, when N  ~  the perturbation expan­sion of the polarization operator rcccivcs similar contribu­tions from both the ladder and RPA terms. Thus, the polarization operator is composed of all possible com bi­nations of ladders and loops.0031 - 9 0 0 7 /0 8 /1 00( 16 ) /1 66802(4) 166802-1 ©  2008 The A m erican Physical SocictyP R L  100, 166802 (2008) P H Y S I C A L  R E V I E W  L E T T E R Sweek ending25 APRIL 2008Noninteracting electrons. —To demonstrate our method and to introduce some notations it is instructive to begin with reproducing the zeroth-order polarization operator. Close to the resonance, when \co -  qv\ q v ,  only those transitions are im portant that have mom enta p and q directed almost along the same line. M ore precisely, the electron is taken from the state in the lower cone with m omentum p almost antiparallel to q and placed in the upper cone in the state with momentum p + q, which is almost parallel to q. Obviously, this is possible only as long as p <  q. The relevance of these particular transitions is understood from  the form of the polarization operator,cleJ 2  -77 C e™ .P+qC 6p, (2)where the electron G reen's function consists o f the contri­butions from both cones /3 =  ± 1 ,1ep x ­' /8=+11 + ( 3 & pf i v p  + if ir) '(3)where &p =  & • p / p  is the projection of the pseudospinPauli matrix onto the direction of electron momentum. The energy integral in Eq. (2) yields_____________ - p ^ ) ( l  ~  ffo -p )4 ’ ’T r(l +  /3«x,co -  (3v(p  + |p + q|) + />S77(4)We observe that (i) for positive co the singular denominator appears for /3 =  + 1 , (ii) the trace operation in the numera­tor imposes that p and p + q are antiparallel (for the parallel configuration the trace vanishes). Close to the threshold (ca =  qv)  collinear processes become dom inant allowing us to approximate |p + q| ~  t/ — p  + ~,p{‘j-p) > where 6 denotes the angle between p and - q  and T r(l + o'pt-qXl — tf-p) ~  4; thus, we obtain, from  Eq. (4),fy p d p  f 00 cW(27t)2 q v , _p¥>_ _|_ •2 Uj-p) +  iyl(5)Upon calculating the integrals we recover n !0)(&>,t/)N^ - q W / J q v  -  co. Note that, as expected, the relevantangles are small, 6 ~  ^/| 1 — co /q v | 1.First-order correction. —To the first order in the electron-electron interaction there are three diagrams, Fig. 1. O f these diagrams, the self-energy correction, Fig. 1 (a), is the m ost singular near the threshold,n sr!(«, q)N g j v , 5 / 2162\/2  (qv  -  a>y/2In (JC/q), (6)where JC is the upper momentum cutoff. This singularity indicates that the infinite summation of the self-energy diagrams have to be performed first, thus yielding thea) b)o  0 - 0s.+co p,+q P,+q ___ I 2 w e , + wC  I T >PiC)FIG. I. First-order interaction correction: self-energy correc­tion (a), subleading RPA correction (b). leading vertex correction(c). Figure (d) illustrates the origin of the singular behavior. ~ (q v  — « )_l \n\qv — &)|. from almost collinear electron propa­gation. Both states P). p2 belong to the lower ( — ) cone, while Pi +  Q- P2 + Q correspond to the upper ( +  ) cone.renormalization of the electron spectrum. To the lowest order in the interaction this gives v p =  w[l + J  In(JC/ /?)][4]. In the case of a conventional Fermi liquid the corre­sponding velocity renorm alization is trivial. In case of a linear spectrum considered here, however, logarithmic terms lead to a slight curvature of the electron spectrum which can smear the threshold singularities. In what fol­lows we assume the renormalized velocity but neglect this smearing by taking the velocity at a typical momentum, determined by the external momentum v  =  v q.The next term, Fig. 1 (b), is the RPA correction which can be readily written asn RPA(«- Ci)77.■N2g16“ q v O )It can now be verified that the vertex correction, Fig. 1(c), yields a contribution which is opposite in sign and more singular than Eq. (7) as a result of the long-range character of the Coulomb interaction. The energy integra­tions are performed independently in each half of the diagram, resulting inH T r y  y f f d  - f r T p , ^ ) ( l  +  f r T Pl) 16 7~r co + f3 v (p  ^ +  |p1 +  q|)r  1 r  2 PXP P « p ' ................q (  1 +  a<rpi)(\. -  acrp^ q) co + a v ( p 2 + |p2 +  q|) P|-P2 (8)To extract the leading contribution we note that it comes from almost collinear processes with a  =  /3 =  + 1 , and both p + q, p' + q are alm ost parallel to q and almost antiparallel to both p, p'. Using the small-angle expansion introduced in the previous section, and approximatingV.p-p <?2/ l P ~  P*\i we obtainn  ^ ( c o .q )N e 2 2* n -/=1 CO1I/’1 -  P2\ p i d p id d iq v p,-qdr2 (q-pj)(9)The in tegrals over the two angles are independen t and yield166802-2P R L  100, 166802 (2008) P H Y S I C A L  R E V I E W  L E T T E R Sweek ending25 APRIL 2008the same resonant denominator as in Eq. (7). However, the integrals over the magnitude of electron mom enta contain a logarithmic divergence which is due to the long-range nature of the Coulom b interaction. To cut off thisdivergence we note that ]px — p 2l ~  l/’ i — /’2I +\ 0 l2 is the angle between two momenta. Recalling that relevant angles 0 \2 ~  Iqv/co — 1| and that typical pj ~  q, we observe that small-angle expansion fails for |pi -  p 21 ~  q J \q v /c o  — 1|, which should be used as the lower cutoff. As a result we obtain2477 co — q v  \|<w — qv\(10)We, therefore, find that RPA loops exceed the vertex terms only when N  »  ^ e  opposite limit, close tothe resonance, N  the ladder diagrams dom i­nate. We now turn to the summation of the ladder series.Summation o f  the infinite ladder. —In the second order in electron-electron interaction the m ost dominant contribu­tion again comes from the ladder diagram. Fig. 2(a), with three pairs o f electron lines each yielding a 1 / yjqv -  co singularity, and two Coulomb interactions providing two additional powers o f the logarithm. The other diagrams are less singular at co =  q v .  Diagram 2(b) requires that at least one of the interaction lines carries a large momentum, ~ q ,  while both momenta in the ladder-type diagram 2(a) are small. Exchange-type contributions, 2 (c )-2 (e ), though providing the main power-law singularity, involve trans­ferred mom enta which are ~ q  and lack the additional logarithms (while also lacking extra factors N).Repeating the arguments from the preceding section we realize that to each order in the interaction the leading contribution comes from the ladder diagram. Fig. 2(f),C D  0 = C >c) d) e)Pi+qe,+(»p2+qe,+coPn.1+Qe „+<+CQC LPi E1 P2 e2FIG. 2. Leading, (a), and subleading, (b)-(e), non-RPA dia­grams of the second order. The ladder diagram (f) explains the origin of the most singular, (qv — co) "ln"\qv — w|, contribu­tion of the arbitrary order 11. All states propagating along the upper or lower part of the ladder belong to the upper or lower cone.TTn<», q) =  —A'TrX G■ n+1dei—l€| +<t>p| +q2 it ■ ■ G .1/ 1/ v P1- P 2 • • • v p „ - p „ VX ^e„-iP„-i • • • ^ i P rAll the m omenta are almost collinear, with those propagat­ing along the lower part o f the ladder, p (, being antiparallel to the external wave vector q and to all the momenta p, +  q along the upper part o f the ladder. Energy integrations are independent in each step and result in the singularity being strongest when all the propagators in the upper or lower parts of the ladder belong to the upper (+ )  or lower (—) cones. The subsequent trace operation is easily per­formed asTr (1 -  <xPl) . . . ( !  -  <xPl,^ ) ( l  +  <xPn^ q) . . .  (1 +  <Xp^q)yielding the nth  order contribution to the polarization operator in the formW f( c o .q )  =  ( - i Y N  I VtP1-P 2H-t-1x n —a 1 —dp,i—l co -  v(p i  + Ip, +  q |) +  ir)'(11)Integrating over the angles and then evaluating the inte­grals over the absolute values of mom enta in the leading logarithmic approximation, we obtainTT ■"'(&), q) =  —1 T W )^ J lq v iq v  — ca)(12)where we introduced the following dimensionless variable.u l ~ F — ln(i qu i)- The summation o f the infinite) V q v — (o x \ q v — (aY2v2 ~<DloM <ov series can now be readily performed.TT(&>, q) =  —N q1 +  -  arcsin.r — (1 +  - x ) V  1 — .____7T__________ 2____7T 7______4 v s H j ^ U ^ i - x 2(13)The above result (13) for the polarization ladder is sufficient to calculate the polarization operator in a more general case when N  ~  ln ( ]^ r^ |)  ^  L  In this regime both vertex and loop diagrams have to be summed sim ulta­neously. Such a summation can be performed by realizing that it yields a geometric series sim ilar to the usual RPA one ( 1) but with the bare polarization operator TT(0) re­placed by the polarization ladder TTV/ given by Eq. (13),'Pico, q) =TTV/(<w, q)1 — V(q)W v (co, q ) ’(14)Collective mode .—The nonzero imaginary part of the polarization operator determines those values o f co and q for which the dissipation of the external field is possible.166802-3P R L  100, 166802 (2008) P H Y S I C A L  R E V I E W  L E T T E R Sweek ending25 APRIL 2008FIG. 3 (color online). Imaginary part of the polarization op­erator in units of q/4  for N  =  4 and g = 0.3: nonintcracting value lly (dashed line), RPA value IIrPA from (1) (dot-dashed line), and T "  from Eq. (16) that describes collcctivc response of excitons (solid line). The inset illustrates the relative height of the cxcitonic plasmon line to the background of individual clcctron-holc absorption. The position of plasmon qv  =  1.006&), the absorption threshold qv = 1.081&).The imaginary part o f Eq. (13), and as a result o f Eq. (14), arises in two cases: (i) Imaginary x  corresponds to the conventional domain, co >  q v ,  but modified by interaction o f electrons and holes, (ii) Positive 1 <  x  <  oo corresponds to the absorption below  the threshold and represents a nonperturbative effect which arises from lowering the en­ergy o f electron-hole excitations with finite q from attrac­tive Coulomb interaction. The new threshold of absorption io =  qu  is determined from the condition x  =  1, which for weak interactions g <5C 1 givesu =  u [ l — (g]ng)2/2 \.  (15)An even more striking property o fE q . (13) is the behavior o f its real part, which describes the polarizability of an interacting electron-hole pair. It is easy to see that the real part is positive for x  >  1 [14]. This change in sign, as compared with the real part o f the RPA polarizability (1), can already be traced to the first-order vertex correction(10), which is opposite to the first-order RPA correction(7). This fact has immediate consequences for the collec­tive response o f excitons, given by Eq. (14), which devel­ops a pole for a certain value o f frequency co{q). This new mode describes the propagation o f coherent oscillating charge density in a system o f interacting electron-hole pairs. The corresponding peak in the absorption has a finite but small width which is due to the fact that the collective mode falls within the range of decay via individual electron-hole pairs. Its behavior is illustrated in Fig. 3 for N  =  4.Interestingly, the width of the plasmon is maximal around g ~  0.3. For smaller values o f the coupling strength the peak narrows by virtue of the narrowing o f the whole exciton domain, Eq. (15). For larger values, g »  1, the peak width decreases *  g ~ 2 while the height increases only '*■ g. The position of the peak in this limit tends to the value determ ined from the equation [recovered from(13) and (14) when x  —> oo] I n ( ^ r ^ )  =  N,  which yields co{q) =  q v (  1 — e~N). (16)Conclusion .—The usual RPA formalism adequately de­scribes dynamical charge fluctuations only in the limit N  »  In, qv i . Close to the Dirac cone we find that the\q v — co\nth order term in the ladder series is larger than the corresponding term in the RPA series by a factor o f \nn{ q v / \q v  — a>\)/Nn. This stronger singularity in the ladder series arises due to the long-range interaction be­tween electrons which move almost collinearly to the external momentum q. The series is summed up analyti­cally yielding a nonperturbative result: the density and spin response functions acquire nonzero imaginary parts in the additional frequency range qu <  co <  qv .  This extension is a manifestation of the cxcitonic effect. The reversal o f the sign o f the electron polarizability in this new domain gives rise to a sharp plasmonic mode which is absent in the conventional RPA. Interestingly, a similar sign reversal o f the polarizability due to interactions has been reported to yield a surface plasmon in a different case o f a tw o­dimensional Anderson insulator [15].Useful discussions with A. Chubukov, M. Raikh, andO. Starykh are gratefully acknowledged. This work was supported by DOE, Grant No. DE-FG 02-06ER46313.[1] M. Wilson, Phys. Today 59, No. 1, 21 (2006).[2] A.H. Castro Neto, F. Guinea, N.M. R. Peres, K.S. Novoselov, and A. K. Gcim, arXiv:0709.1163 [Rev. Mod. Phys. (to be published)].[3] K.W. Shung, Phys. Rev. B 34, 1264 (1986).[4] J. Gonzalez, F. Guinea, and M. A.H. Vozmediano, Nucl. Phys. B424, 595 (1994).[5] J. Gonzalez, F. Guinea, and M. A.H. Vozmediano, Phys. Rev. B 59, R2474 (1999).[6] A. V. Chubukov and A. M. Tsvelik, Phys. Rev. B 73, 220503(R) (2006).[7] D. V. Khveshchenko, Phys. Rev. B 74, 161402(R) (2006).[8] D. V. Khveshchenko, Phys. Rev. Lett. 87, 246802 (2001).[9] S. Das Sarma, E. H. Hwang, and W.-K. Tse, Phys. Rev. B 75, 121406(R) (2007).[10] Y. Barlas, T. Pcreg-Barnea, M. Polini, R. Asgari, and A. H. MacDonald, Phys. Rev. Lett. 98, 236601 (2007).[11] E.H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418(2007).[12] O. Vafek, Phys. Rev. Lett. 97, 266406 (2006).[13] E.G. Mishchenko, Phys. Rev. Lett. 98, 216801 (2007).[14] For x  >  1 the analytical continuation should be performed from x  £  [0,1) to (I, oo). From the requirement that 11" is negative we observe that it should yield Vl — x2 —» - i -Jx2 — 1. This implies that the analytical continuation must be performed in the upper half-plane of complex x, so that arcsinx —» ir/2 +  /ln(x + Vx2 — l).[15] T. V. Shahbazyan and M.E. Raikh, Phys. Rev. B 53, 7299(1996).166802-4

Charge response function and a novel plasmon mode in graphene

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Charge response function and a novel plasmon mode in graphene

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