Dynamical polarizability of graphene beyond the Dirac cone approximation

Dynamical polarizability of graphene beyond the Dirac cone approximation

T. Stauber,^1,2J. Schliemann,³and N. M. R. Peres¹

1Centro de Física e Departamento de Física, Universidade do Minho, P-4710-057 Braga, Portugal

2Depto. de Física de la Materia Condensada, Universidad Autǿnoma de Madrid, E-28049 Madrid, Spain

3Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany 共Received 7 October 2009; revised manuscript received 16 December 2009; published 4 February 2010兲

We compute the dynamical polarizability of graphene beyond the usual Dirac cone approximation, integrat- ing over the full Brillouin zone. We find deviations atប␻= 2t共wheretis the hopping parameter兲which amount to a logarithmic singularity due to the Van Hove singularity and derive an approximate analytical expression.

Also at low energies, we find deviations from the results obtained from the Dirac cone approximation which manifest themselves in a peak spitting at arbitrary direction of the incoming wave vectorq. Consequences for the plasmon spectrum are discussed.

DOI:10.1103/PhysRevB.81.085409 PACS number共s兲: 63.20.⫺e, 73.20.Mf, 73.21.⫺b

I. INTRODUCTION

Graphene is a novel two-dimensional 共2D兲 system with many outstanding mechanical and electronic properties.¹Es- pecially the early observation of the ambipolar field effect² and of the odd integer quantum Hall effect³have stimulated enormous research on the electronic structure of graphene.

Only recently, the fractional quantum Hall effect was seen in suspended graphene.⁴ For a review of this newly emerging branch of condensed-matter physics, see Ref.5.

To understand the unusual electronic properties of graphene, it often suffices to discuss the charge susceptibil- ity. The static polarizability at kF, e.g., gives the Thomas- Fermi screening length, important for transport properties^6–8 whereas the dynamical polarizability at zero wave number can explain the phonon softening⁹ at the⌫ point. It is also used for the understanding of structural inhomogeneities in graphene, so-called ripples¹⁰ and the van der Waals interac- tion between graphene layers.¹¹

For neutral graphene, the polarizability at zero tempera- ture was first calculated by González et al.,¹² the effect of temperature was discussed by Vafek¹³and vertex corrections were considered in Ref. 14. For a gated system with finite chemical potential, the first expressions were given by Shung¹⁵ in the context of graphite and later by Wunsch et al.¹⁶ and Hwang and Das Sarma.¹⁷ Also the extension to finite temperature has been performed, even though a closed analytical expression is then—as in the neutral case—not possible, anymore.¹⁸Recently, the polarizability was discuss in the presence of a magnetic field¹⁹ and gapped graphene.^20,21

All these results originate from the Dirac cone approxi- mation in which the energy dispersion of the hexagonal lat- tice is linearized around one of the two Dirac points where the valence and conduction bands touch. But corrections to this approximation have to be included to discuss, e.g., the recently measured absorption of suspended graphene in the visible-optics regime,²² which is related to polarizability via the continuity equation. This has been done in a perturbative treatment in Ref.23. The optical properties of graphite were calculated in Ref.24.

Here, we want to extend the previous calculations to the full Brillouin zone of the hexagonal tight-binding model. We

certainly expect deviations at large energies where the Dirac approximation does not hold anymore. But the main purpose is to test whether the diverging density of states at the M point 共Van Hove singularity兲 leads to consequences on the collective excitations of this system.

Our interest is motivated by the recent findings of an ad- ditional plasmon mode that emerges at around 4.7 eV with a linear dispersion²⁵ which was observed on freestanding graphene by electron energy-loss spectroscopy.²⁶ A first guess is to associate this mode to the Van Hove singularity which in the charge susceptibility shows up at 2t⬇5.4 eV,t denoting the tight-binding hopping parameter. Including ex- citonic effects, the prominent absorption peak shifts to 4.5 eV,²⁷thus suggesting that the Van Hove singularity is indeed the origin for this new plasmon mode.

Apart from the high-energy corrections stemming from interband transitions, we also look at the intraband contribu- tion at low energies and find that even there deviations from the Dirac cone approximation occur. By this, we complement a recent work, where the plasmon dispersion is discussed by also considering the full Brillouin zone.²⁸

The paper is organized as follows. In Sec.II, we introduce the model and notation and define the polarizability of graphene. In Sec. III, we discuss the imaginary part of the polarizability which will only involve one numerical integra- tion. We first treat the interband contribution where we espe- cially focus on the behavior around the M point where the Van Hove singularity occurs. We then discuss the intraband contribution and the different behavior at certain directions of the incoming wave vectorq. In Sec.IV, we obtain the real part of the polarizability via the Kramers-Kronig relation and discuss implications on the modified plasmon spectrum due to the inclusion of the whole Brillouin zone. We close with conclusions in Sec. V and give details on the analytical evaluation of the polarizability around the M point in the Appendix.

II. EFFECTIVE MODEL AND THE POLARIZABILITY OF GRAPHENE

The Hamiltonian of a hexagonal graphene sheet in Bloch spinor representation is given by

H=

兺

_k ^{共−tHk−␮12⫻2兲, Hk=␺k†}

冉

␾⁰kⴱ ^␾0^k

冊

^{␺k 共1兲}

with t⬇2.7 eV the tight-binding hopping parameter,␮ ^the chemical potential, and␺k=共a_k,b_k兲^T, wherea_kandb_kbeing the destruction operators of the Bloch states of the two tri- angular sublattices, respectively. Further, we have ␾k

=兺_␦ie^{−i共␦i−␦3兲·k}, where ␦i denote the three nearest-neighbor vectors. Here we choose them to be

␦1=a

2共− 1,

冑

3兲, ␦2=a

2共− 1,−

冑

3兲, ␦3=a共1,0兲 共2兲 with a= 1.42 Å being the nearest carbon-carbon distance.

The Brillouin zone is then defined by the two vectors b₁

= 2␲/共3a兲共1 ,

冑

3兲 and b₂= 2␲/共3a兲共1 , −

冑

3兲, see Fig. 1共a兲.

The dimensionless eigenenergies thus read

兩␾k兩=

冑

3 + 2 cos共

冑

3kya兲+ 4 cos共

冑

3kya/2兲cos共3kxa/2兲.

共3兲 In terms of the bosonic Matsubara frequencies ␻n= 2␲ⁿ/␤ 共␤= 1/kBT and ប= 1兲, the polarizability in first order is de- fined as

P^共1兲共q,i␻n兲= 1 A

冕

0

␤

d␶e^i␻n␶具␳共q,␶兲␳共−q,0兲典, 共4兲

where A denotes the area of the graphene sample and the density operator is defined as ␳q=␳q

a+e^iq·␦3␳q

b with ␳q c

=兺k,␴c_k,^†_␴c_k+q,␴ 共c=a andb兲.

Hence, we obtain the general expression for the polariz- ability

P^共1兲共q,i␻n兲= −gs

共2␲兲²

冕

1BZ

d²k

兺

s,s⬘^=⫾

f_s·s⬘共k,q兲

⫻ nF关E^s共k兲兴−nF关E^s⬘共k+q兲兴 E^s共k兲−E^s⬘共k+q兲+i␻n

共5兲 with E^⫾共k兲=⫾t兩␾k兩−␮ ^{and n}F共E兲 the Fermi function. For the band overlap, we have

f_⫾共k,q兲=1

2

冉

^1⫾Re

冋

^{eiq·␦3兩␾␾kk兩兩␾␾k+qⴱk+q兩}

册 ^冊

^{. 共6兲}

Note that since we are summing over the entire Brillouin zone, only the spin degeneracy gs= 2 has to be taken into account.

For neutral graphene, ␮= 0, there is no intraband contri- bution due to the canceling Fermi functions in the numerator of Eq. 共5兲. Due to f₋共k,q→0兲→0, we further expect no interband contribution forq= 0. We finally note that for high energies␻⬎t, the phase factor between the particle densities of the two sublattices, e^iq·␦3, is crucial even in the long- wavelength limit q→0.

III. IMAGINARY PART OF THE POLARIZABILITY With the substitution i␻n→␻⁺i0, the imaginary part of the retarded susceptibility is written in terms of a delta func- tion in the usual way. Determining the zeros of the argument of the delta function allows us to perform the integration over kx analytically. The subsequent integration over ky is then done numerically. We have also performed the direct summation of Eq.共5兲of a finite system to check our results.

The integration over the Brillouin zone can be split up into separate parts with slight modifications of the integrand 关see below and Fig.1共b兲兴. The final domain is then given by 0⬍3kxa/2⬍␲/2 and 0⬍

冑

3kya/2⬍␲/2 and the substitu- tion x= sin共3kxa/2兲 and y= sin共

冑

3kya/2兲 can be performed.

The resulting expression关see Eq.共8兲兴explicitly displays the inversion symmetry ofq with respect to theqxandqyaxes.

The polarizability P^共1兲共q,␻兲 is also invariant under rotation of ␲/3, displaying the underlying lattice symmetry. We thus find the following symmetry:

P^共1兲共兩q兩,␲/6 +␸^˜,␻兲=P^共1兲共兩q兩,␲/6 −␸^˜,␻兲. 共7兲 The subsequent plots thus only show four representative curves with 0ⱕ␸ⱕ␲/6.

A. Interband transitions

We shall first discuss the contribution of the interband transitions to the imaginary part of the polarizability. As ex- plained above, we first perform the integral overkx→x, thus eliminating the delta function. For neutral graphene,␮^{= 0, at} zero temperature T= 0 this yields the following expression:

ImP^共1兲共q,␻兲

= 2 sgn共␻兲 共2␲兲²

␲

冑

3

t

冉

3a²

冊

²

冕

0

1 dy

冑

1 −y²

⫻

兺

j=⫾

兺

s,s⬘^=⫾

兺

0⬍x_i⬍1

Fj共xi,y;sqx,s

⬘

^qy兲

冑

1 −x_i²

冏

^{dxdhj共x,y;sqx,s}

^⬘

^qy兲兩xi

冏

^,

共8兲 where we defined

h_j共x,y;q_x,q_y兲=兩␾j共x,y;0,0兲兩+兩␾j共x,y;q_x,q_y兲兩−兩␻兩 共9兲 with

k_x ky

M

M K

K

−

M0 , +

Γ b₁

b2

M

+,+

−,+

−,− +,−

+

+,− +,− −,− −,−

−,+

−,+

+,+

+,+

M₀ K

K’

M

Γ Γ

M−

+

M

−

a) b)

FIG. 1.共a兲The hexagonal and共b兲rhombical Brillouin zone. The symmetrized rhombical Brillouin zone and its segmentation. The inner square refers to j= −, the outer triangles refer to j= +; addi- tionally the values ofsands⬘^{are given.}

兩␾j共x,y;q_x,q_y兲兩=兵3 + 2共1 −y²兲cos共2q˜_y兲− 4

冑

1 −y²ysin共2q˜_y兲 +j4关

冑

1 −x²cos共˜q_x兲−xsin共˜q_x兲兴

⫻关

冑

1 −y²cos共q˜_y兲−ysin共q˜_y兲兴其^1/2 共10兲 and

Fj共x,y;qx,qy兲=1

2

冋

^{1 −兩␾}共x,y;0,0兲兩兩^F˜j共x,y;q␾共x,y;^x,qy兲qx,qy兲兩

册

共11兲 with

˜F

j共x,y;qx,qy兲= cos共2q˜_x/3兲+j2

冑

1 −y²⫻关

冑

1 −x²cos共2q˜_x/3兲

−xsin共2q˜_x/3兲兴+ 2兵2

冑

1 −y²cos共q˜_x/3兲 +j关

冑

1 −x²cos共q˜_x/3兲−xsin共˜q_x/3兲兴其

⫻关

冑

1 −y²cos共q˜_y兲−ysin共q˜_y兲兴. 共12兲 Above, we also introduced ˜q_x= 3q_xa/2 and ˜q_y=

冑

3q_ya/2.

Furthermore, the sum over xiis over all zeros which satisfy hj共xi,y;qx,qy兲= 0, 共13兲 which can be written as a polynomial of fourth order. The zerosxican thus be obtained analytically such that only the subsequent integration over y has to be performed numeri- cally.

On the left-hand side of Fig.2, the imaginary part of the polarizability ImP^共1兲共兩q兩,␸^,␻兲as a function of the energy␻ is shown for different directions of the incoming wave vector qwith兩q兩a= 0.1, where the usual parametrization in terms of the polar angle␸ ^withqx=兩q兩cos␸is used. There is no ap- parent angle dependence except for the region around the

Van Hove singularity which is highlighted in the inset. The result obtained from the Dirac cone approximation is also shown共dashed line兲, which is given by¹²

ImP_0,Dirac^共1兲 共兩q兩,␻兲=1 4

兩q兩²

冑

_␻²−共3t兩q兩a/2兲². 共14兲 For low energies, there is good agreement with the above formula but especially for energies close to the Van Hove singularity,␻= 2t, strong deviations are seen which shall be discussed in the following in more detail.

1. Expansion around the Van Hove singularity

The new feature compared to the Dirac cone approxima- tion comes from the region around the Van Hove singularity, located at the M points of the Brillouin zone. For the Bril- louin zone defined above, the M points are located at M₀

= 2␲/共3a兲共1 , 0兲 andM_⫾=␲/共3a兲共1 ,⫾

冑

3兲.

In the following, we introduce the substitutions ˜p_x

= 3pxa/2 and ˜p_y=

冑

3pya/2 and shall assume ˜p_x,˜p_yⰆ1 for p=kand q. Expanding around the M₀ point, the dispersion then simplifies to

␾k

M₀⬇− 1 −i2k˜

x+˜k

x 2+˜k

y

2, 共15兲

兩␾k

M₀兩 ⬇1 +˜k

x 2−˜k

y

2 共16兲

and the band overlap yields

f₋^M0共k,q兲 ⬇4q˜_x²/9. 共17兲 For the M_⫾point, we obtain

␾k

M_⫾⬇1⫿i2k˜

y⫾2k˜

x˜k

y, 共18兲

兩␾k

M_⫾兩 ⬇1⫾2k˜

x˜k

y+ 2k˜

y

2 共19兲

and the band overlap yields

f₋^M⫾共k,q兲 ⬇ 共q˜_x/3⫾˜q_y兲². 共20兲 Forqy= 0共␸= 0兲, an analytical approximation similar to that presented in Ref. 29is possible for the M₀ point expansion since the polynomial in the delta function is quadratic. This yields a logarithmic divergence at␻M/t=␻^˜M= 2 +q˜_x²/2 which can be approximated by the following expression:

ImP^共1兲,M0共q_x,␻兲 ⬇2 sgn共␻兲 共2␲兲²

␲

冑

3

t

冉

^3a2

冊

^2˜18qx2

⫻

再

^ln

^冉

^8⌳˜qx22

^冊

^{+ ln}

^冋

^{共␻˜2⌳−2␻˜˜qMx2兲2}

^册 冎

^,

共21兲 where⌳denotes a suitable cutoff. Details on the calculation are given in the Appendix. For generalqor for theM_⫾point expansion, we expect a similar behavior.

In the inset of Fig. 2, the region around the Van Hove singularity which is highlighted. For a general angle ␸^{, all} threeM points contribute and there is a prominent double or

0 1 2 3 4

ω/t

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

ta2 ImP(|q|,ϕ,ω)

ϕ=0 ϕ=10^o ϕ=20^o ϕ=30⁰ Dirac

0 1 2 3 4 5 6

ω/t

0 0.5 1 1.5 2

ta2 ImP(q,ω)

q=K q=q1 M0

q=q₂^M0 q=b1

1.98 2 2.02

ω/t 0.004

0.006 0.008

FIG. 2. 共Color online兲Left-hand side: the imaginary part of the polarizability ImP^共1兲共兩q兩,␸,␻兲 as a function of the energy ␻ at k_BT/t= 0.01 for different angles ␸ with 兩q兩a= 0.1. The result ob- tained from the Dirac cone approximation is also shown 共dashed line兲. Inset: energy region around the Van Hove singularity of the same curves. Right-hand side: the imaginary part of the polarizabil- ity ImP^共1兲共q,␻兲 as a function of the energy␻atk_BT/t= 0.01 for various wave vectorsqdefined in the text.

even triple peak structure. But for␸⁼␲/6 ,␲/2 , . . ., the over- lap function of one M point vanishes and the peaks merge.

2. Behavior at largeq

Let us now discuss the behavior for generalqat␮^{= 0. For} that we expand the energy dispersion 兩␾k+q兩 around the points of high symmetryk=S=⌫,K, and M, and determine theqvector for which兩␾q

S兩= 0. To discuss the dielectric func- tion at large wave vectors 兩q兩⬃1/a, also local-field effects have to be taken into account,³⁰ as was done in Ref.31.

Expanding the dispersion around ⌫=共0 , 0兲 and K

= 2␲/共3a兲共1 , 1/

冑

3兲, we find that for q=K, 兩␾q⌫,K兩= 0. The spectrum of ImP^共1兲 thus starts at ␻= 0. Expanding the dis- persion around the M₀ point, the wave vectors q₁^M0

= 2␲/共3a兲共0 , 1/

冑

3兲 and q₂^M0= 2␲/共3a兲共1 , 2/

冑

3兲 yield 兩␾q M₀兩

= 0 and the spectrum of ImP^共1兲 thus starts at ␻^{=t. For the} M_⫾-point expansion, we obtain兩␾q

M_⫾兩= 0 forq=M_⫿. On the right-hand side of Fig.2, the imaginary part of the polarizability is shown for the above wave vectors q. The curves for theM_⫾-point expansion yield the same curves as the ones for the M₀-point expansion and are not listed. For comparison, we also show the behavior for one of the vectors which define the Brillouin zone, q=b₁, which 共for T= 0兲is identical to the density of states by rescaling ␻→␻/2.

B. Intraband transitions

For finite chemical potential ␮⬎0, there are also intra- band transitions. The extension of Eq.共8兲to finite chemical

potential ␮ ^{and finiteT} is straightforward. The main differ- ence is that the function of Eq.共9兲now reads

hj共x,y;qx,qy兲=兩␾j共x,y;0,0兲兩−兩␾j共x,y;qx,qy兲兩⫾兩␻兩. 共22兲 This expression suggests that there might be differences to the Dirac cone approximation also for small␻. In fact, even for wave vectors兩q兩and chemical potential␮for which the Dirac cone approximation holds 共e.g., 兩q兩a= 0.1 and ␮/t

= 0.05兲, we find deviations from the Dirac cone result.

Let us first start the discussion by summarizing the results coming from the Dirac cone approximation from Ref.16for which we introduce the functions

f共兩q兩,␻兲= 1 4␲

兩q兩²

冑

兩␻²⁻共3t兩q兩a/2兲²兩,

G_⬎共x兲=x

冑

x²− 1 − cosh⁻¹共x兲, x⬎1,

G_⬍共x兲=x

冑

1 −x²− cos⁻¹共x兲, 兩x兩⬍1. 共23兲 Due to the finite chemical potential, the polarizability is now given by ImP_␮,Dirac^共1兲 = ImP_0,Dirac^共1兲 + Im⌬P_␮,Dirac^共1兲 and with ␻q D

= 3t兩q兩a/2 the additional term reads

Im⌬P_␮,Dirac^共1兲 共兩q兩,␻兲=f共兩q兩,␻兲⫻

冦

^{G−GG0,⬎⬎⬍␲}

^{冉 冉 冉}

^{,22␻␮␮␻␻␻− 2++qDqDqD␻␻␮}

^{冊 冊 冊}

^,,−G⬎

^冉

^{2␮␻−qD␻}

^冊

^{, ␻␻␻␻otherwise⬍⬎⬍⬎␻␻␻␻qDqDqDqD∧∧∧∧␻␻␻␻⬍⬍⬎⬍22␻␻␮␮qDqD−− 2+ 2−␻␻␮␮qDqD}

冧

^{. 共24兲}

A distinct signature of noninteracting 2D electrons in graphene is a divergent behavior of the polarizability or charge susceptibility at the threshold for the excitation of electron-hole pairs at ␻q

D, see Eq. 共14兲. This divergence is also present for gated or doped graphene with␮⬎0 and has been usually attributed to the absence of curvature in the spectrum. But even in the regime where the Dirac cone ap- proximation does not hold, i.e., curvature in form of trigonal warping has to be taken into account, we find a divergent behavior at ␻q

Dfor q=qx. For arbitrary direction, we find a peak splitting even in the Dirac cone regime.

Let us discuss the polarizability using the full Brillouin zone in more detail. As stated above, P^共1兲共q,␻兲 is invariant under rotation of ␲/3—independent of the chemical poten-

tial and we also find the symmetry of Eq.共7兲. The numerical results moreover suggest that for moderate chemical poten- tial␮⬍tthe angle-dependent polarizability can be described by a single function where the angle␸^˜ only enters as param- eter.

In Fig.3, ImP^共1兲共q,␸^,␻兲with兩q兩a= 0.01 as a function of the energy␻is shown for various angles␸and two chemical potentials ␮/t= 0.05 共left兲 and ␮/t= 0.5 共right兲 at kBT/t

= 0.01. Only in the direction of qx, i.e.,␸= 0, there is agree- ment with the analytical result of Eq. 共24兲coming from the Dirac cone approximation at zero temperature. Interestingly, this is also the case for a large chemical potential ␮/t= 0.5, where trigonal warping effects should come into play. For arbitrary direction, a double-peak structure appears even for

small chemical potential␮/t= 0.05 for which the Dirac cone approximation should hold.

In Fig.4, the same quantities are shown for larger chemi- cal potentials ␮/t= 1 共left兲 and␮/t= 1.5共right兲. The differ- ences to the analytical result coming from the Dirac cone approximation 共dashed line兲 now become apparent also for q=qx共␸= 0兲. For␮/t= 1 they manifest themselves at lower energies ␻⬍␻q

D and for ␮/t= 1.5, a double-peak structure emerges. But interestingly, the divergence still occurs at ␻

⬇␻q

Din both cases.

The above curves were obtained for kBT/t= 0.01, thus slightly larger than room temperature, but we have also in- vestigated the effect of different T. We find that the curves

for ␸= 0 are basically unaffected by temperature but that for arbitrary direction the algebraic divergences seen for ␮/t

= 0.5 or␮/t= 1 are smeared out at larger temperature as it is the case for ␮/t= 0.05. On the contrary, the curves for ␮/t

= 0.05 and ␸⫽0 develop the algebraic divergence for de- creasing temperature. Generally, we can say that the alge- braic divergences become broadened when the energy set by the temperature is much larger that maximal peak splitting at

␸⁼␲/6. Nevertheless, the peak splitting in directions of lower symmetry prevails also at elevated temperatures.

IV. REAL PART OF THE POLARIZABILITY The real part of the polarizability shall be obtained nu- merically via the Kramers-Kronig relation

ReP^共1兲共q,␻兲= 1

␲

冕

0 6t

d␻

⬘

^ImP共1兲共q,␻

⬘

兲 2␻

⬘

␻

⬘

²⁻␻^2. 共25兲 The left-hand side of Fig. 5shows ReP^共1兲共兩q兩,␸^,␻兲 for en- ergies close to the Van Hove singularity with 兩q兩a= 0.1 and

␮= 0 for different angles ␸. As expected, there are strong deviations with respect to the result coming from the Dirac cone approximation and the functions become negative. This opens up the possibility of the emergence of an additional plasmon mode since the plasmon dispersion in the random- phase approximation 共RPA兲 approximation is given by the relation

⑀_⬁^+vqP^共1兲共q,␻兲= 0, 共26兲 where ⑀_⬁ denotes the effective dielectric constant including high-energy screening processes. Since the experiments in Ref.26were done on suspended graphene, we set⑀_⬁= 1. For the Coulomb interaction, we set 兩q兩vq=e²/2⑀0= 90 eV Å

⬇16បvF⬇24ta. For 兩q兩a= 0.1, ta²/vq= 0.004 never crosses

0.013 0.014 0.015 0.016

ω/t

0 0.1 0.2 0.3

ta2 ImP(|q|,ϕ,ω)

ϕ=0 ϕ=10^o ϕ=20^o ϕ=30^o Dirac

|q|a=0.01 ,µ=0.05t

0 0.005 0.01 0.015 0.02

ω/t

0 1 2 3

ta2 ImP(|q|,ϕ,ω)

ϕ=0 ϕ=10^o ϕ=20^o ϕ=30^o Dirac

|q|a=0.01 ,µ=0.5t

FIG. 3. 共Color online兲The imaginary part of the polarizability ImP^共1兲共兩q兩,␸,␻兲 with兩q兩a= 0.01 as a function of the energy␻for various angles␸ and two chemical potentials␮/t= 0.05共left兲and

␮/t= 0.5共right兲atk_BT/t= 0.01. Also shown is the analytical result coming from the Dirac cone approximation at zero temperature 共dashed line兲.

0 0.005 0.01 0.015 0.02

ω/t

0 1 2 3 4 5

ta2 ImP(|q|,ϕ,ω)

ϕ=0 ϕ=10^o ϕ=20^o ϕ=30^o Dirac

|q|a=0.01 ,µ=t

0 0.005 0.01 0.015 0.02

ω/t

0 2 4 6 8 10

ta2 ImP(|q|,ϕ,ω)

ϕ=0 ϕ=10^o ϕ=20^o ϕ=30^o Dirac

|q|a=0.01 ,µ=1.5t

FIG. 4. 共Color online兲The imaginary part of the polarizability ImP^共1兲共兩q兩,␸,␻兲 with兩q兩a= 0.01 as a function of the energy␻for various angles ␸ and two chemical potentials ␮/t= 1 共left兲 and

␮/t= 1.5共right兲atk_BT/t= 0.01. Also shown is the analytical result coming from the Dirac cone approximation at zero temperature 共dashed line兲.

1.98 2 2.02 2.04

ω/t

-0.003 -0.002 -0.001 0

ta2 ReP(|q|,ϕ,ω)

ϕ=0 ϕ=10⁰ ϕ=20^o ϕ=30^o Dirac

|q|a=0.1,µ=0

0.013 0.014 0.015 0.016 0.017

ω/t

-0.2 -0.1 0

ta2 ReP(|q|,ϕ,ω)

ϕ=0 ϕ=10⁰ ϕ=20^o ϕ=30^o Dirac

|q|a=0.01,µ=0.05t

FIG. 5. 共Color online兲 The real part of the polarizability ReP^共1兲共兩q兩,␸,␻兲as a function of the energy␻for various angles␸ atk_BT/t= 0.01. Left: for energies close to the Van Hove singularity with兩q兩a= 0.1 and␮= 0. Right: for low energies with兩q兩a= 0.01 and

␮/t= 0.05. Also shown is the analytical result coming from the Dirac cone approximation at zero temperature共dashed line兲.

one of the several curves which all tend to zero for larger energies. With the bare hopping amplitudet⬇2.7 eV, we do thus not find an additional pole in the RPA susceptibility.

Still, there is a renormalization of the hopping amplitude which comes from the wave-function renormalization of the

␲electrons. Near the Van Hove singularity, this renormaliza- tion will be large and the matrix element will be reduced.

With a renormalization of t→2t/3, we would find an addi- tional pole in the plasmon dispersion, consistent with experi- ments.

We also expect deviations from the simple one-particle spectrum around ␻^{= 2t} because expanding the effective screened Coulomb potential within the RPA approximation, we have

Im vq

⑀共q,␻兲= Im vq

⑀0+v_qP^共1兲共q,␻兲, 共27兲

⬇−共v_q/⑀0兲²ImP^共1兲共q,␻兲. 共28兲 As can be seen from Eq. 共21兲, there is a logarithmic diver- gence at␻M/t= 2 +共3qxa兲²/8 even forqx→0 since the pref- actor from the band overlap is canceled by v_q².

The right-hand side of Fig. 5 shows the real part of the polarizability ReP^共1兲共兩q兩,␸^,␻兲 with 兩q兩a= 0.01 and ␮/t

= 0.05 for different angles␸. For these parameters, the Dirac cone approximation is supposed to hold but strong deviations are seen for ␸⬎0 as in the case of the imaginary part. This opens up the possibility of a modified plasmon dispersion as discussed in Ref.28. But for the present parameters, we do not find an additional zero in the RPA dielectric function, i.e., ta²⑀_⬁/vq= 0.0004⑀_⬁ crosses all curves at the same energy 共choosing, e.g., the high-frequency dielectric constant of sili- con ⑀_⬁^{= 2}兲. Nevertheless, for larger wave numbers 兩q兩a

⬇␮/t, deviations are seen, i.e., the plasmon dispersion is more strongly damped and eventually vanishes since the square-root singularity is smeared out.

In Fig. 6, the real part of the polarizability is shown for two large chemical potentials ␮/t= 1 共left兲 and ␮/t= 1.5 共right兲. As for the imaginary part, large deviations compared to the results coming from the Dirac cone approximation 共dashed line兲 are seen. First, the static value P^共1兲共兩q兩,␸^,␻

= 0兲is larger thanta²ReP_␮,Dirac^共1兲 共兩q兩,␻^{= 0}兲=₉⁸_␲␮/t. Since the static value of the polarizability enters in the expression of the screened Coulomb potential, it is independent of the po- lar angle␸, consistent with group theory. Second, there are additional zeros of the real part of the RPA dielectric func- tion, i.e., ta²⑀_⬁/v_q= 0.0004⑀_⬁ 共set, e.g., ⑀_⬁⬇2兲 crosses the curves at various energies different from the one associated with the Dirac cone approximation. Nevertheless, they lie in the region where ImP^共1兲 is finite and the new plasmon modes are thus damped.

For the present parameters, the undamped solution occurs at slightly larger energy compared to the Dirac cone approxi- mation but is independent of the direction␸. For larger wave numbers 兩q兩a⬇␮/t, stronger deviations are seen, i.e., the plasmon dispersion depends on ␸ is more strongly damped and eventually vanishes.

V. CONCLUSIONS

We discussed the polarizability of graphene using the full band structure of the ␲electrons. We especially focused on the features around the Van Hove singularity since they might be responsible for the newly found plasmon disper- sion. We find that there are no plasmon modes coming from the Van Hove singularity within the RPA approximation with the bare hopping amplitudet. But with a renormalization of t→2t/3, there are additional plasmon modes, consistent with experiment.²⁶ We also find a logarithmic divergence of the imaginary part of the effective Coulomb interaction. This will lead to prominent electron-hole interactions as was re- cently found in Ref. 27.

We also looked at the intraband contribution to the polar- izability. Forqin the⌫-M direction, we find basic agreement with the results of the Dirac cone approximation even for rather large chemical potential␮⬇t/2, i.e., when corrections to the linear Dirac spectrum are large. For arbitrary direction of the incoming wave vectorq, we surprisingly found strong deviations from the results coming from the Dirac cone ap- proximation where a double-peak structure emerges. The peak splitting occurs for all values of ␮⬎0 and is a direct consequence of the energy dispersion. For fixedqand␮, it is largest for ␸⁼␲/2 and tends to zero for 兩q兩, ␮→0. As a consequence, the plasmon dispersion is more strongly damped for兩q兩a⬇␮/tand eventually vanishes at larger tem- peratures since the square-root singularities are smeared out.

ACKNOWLEDGMENTS

We thank F. Guinea for useful discussions. This work has been supported by FCT under Grants No. PTDC/FIS/64404/

2006 and No. PTDC/FIS/101434/2008, and by Deutsche Forschungsgemeinschaft via GRK 1570.

0 0.01 0.02 0.03

ω/t

-3 -2 -1 0 1

ta2 ReP(|q|,ϕ,ω)

ϕ=0 ϕ=10⁰ ϕ=20^o ϕ=30^o Dirac

|q|a=0.01,µ=t

0 0.01 0.02 0.03

ω/t

-4 -3 -2 -1 0 1 2

ta2 ReP(|q|,ϕ,ω)

ϕ=0 ϕ=10⁰ ϕ=20^o ϕ=30^o Dirac

|q|a=0.01,µ=1.5t

FIG. 6. 共Color online兲 The real part of the polarizability ReP^共1兲共兩q兩,␸,␻兲 for兩q兩a= 0.01 as a function of the energy␻ for various angles ␸ at k_BT/t= 0.01. Left: for the chemical potential

␮/t= 1. Right: for the chemical potential ␮/t= 1.5. Also shown is the analytical result coming from the Dirac cone approximation at zero temperature共dashed line兲.

APPENDIX

Forq_y= 0 and using the saddle-point approximation of Eq.

共16兲, an analytical calculation of the imaginary part of the polarizability around theM₀point is possible. The integral is preformed in polar coordinates and with x= cos␸^{, we have} to solve the following quadratic equation with respect to x:

兩␾k

M₀兩+␾k+q_x

M₀ 兩−␻^{˜ = 4k2x2}⫾2kq˜_xx+⑀^{= 0} 共A1兲 with ⑀= 2共1 −k²−␻^˜/2 +˜q_x²兲, ␻^˜=兩␻/t兩, and ˜q_x= 3qxa/2. The integral over ␸eliminates the delta function and yields

ImP^共1兲,M0共q_x,␻兲=g_ssgn共␻兲 共2␲兲²

␲

冑

3

t

冉

^3a2

冊

^289˜qx2

⫻关I+共q˜_x,␻^˜兲+I−共q˜_x,␻^˜兲兴, 共A2兲 where we included the spin degeneracygs= 2 and defined the following integrals:

I_⫾共q˜_x,␻^˜兲=

冕

_D⫾共˜q_x,␻˜兲

dkI_⫾共k;˜q_x,␻^˜兲 共A3兲 with

I_⫾共k;q˜_x,␻^˜兲= k

冑

8共k²−k_min² 兲⫻ 1

冑

8共k²−␰²兲⫾2q˜_x

冑

8共k²−k_min² 兲 共A4兲 and the integration domains

D+共˜q_x,␻^˜兲=

再

^{关k关kmin−,⌳兴;,⌳兴; ␻␻˜˜ ⬍⬎2 + 5q2 + 5q˜˜x2x2//88}

冎

^共A5兲

and

D−共q˜_x,␻^˜兲=

冦

^{关k关关kkminmin+,⌳兴;,⌳兴;,k−兴艛关k+,⌳兴; 2 +␻⬍␻˜˜⬍⬎2 + 5q˜q2 +2 + 5qx2/2˜˜⬍qx2/8x2˜/2␻˜x2/8}

冧

^.

共A6兲 We further defined ␰^{2= −q˜}x

2/4 − 1 +␻^˜/2, k_min² =q²/8 −␰^{2, k}_⫾

=兩␰⫾q˜_x/2兩, and⌳denotes a suitable cutoff.

The indefinite integral has the following solution:

冕

^{dkI⫾共k;˜qx,␻˜兲=18ln兩⫾˜qx+}

^冑

^{8共k2−kmin2 兲}

+

冑

8共k²−␰²兲⫾2q˜_x

冑

8共k²−k_min² 兲兩.

共A7兲 There is a logarithmic singularity at ␻^˜M= 2 +q˜_x²/2. Expand- ing the above result around␻^˜Myields the simple expression

ImP^共1兲,M0共qx,␻兲 ⬇gssgn共␻兲 共2␲兲²

␲

冑

3

t

冉

^3a2

冊

^2˜18qx2

⫻

再

^ln

^冉

^8⌳˜qx22

^冊

^{+ ln}

^冋

^{共␻˜2−⌳2␻˜˜qMx2兲2}

^册 冎

^.

共A8兲

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