Signature of electronic excitations in the Raman spectrum of graphene

0.2 MB·English

by  Oleksiy Kashuba

Signature of electronic excitations in the Raman spectrum of graphene.
Oleksiy Kashuba and Vladimir I. Fal’ko
Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
Inelastic light scattering from Dirac-typeelectrons in graphene is shown to be dominated by the
generation of the inter-band electronic modes which are odd in terms of time-inversion symmetry
andbelongtotheirreduciblerepresentationA ofthepointgroupC ofthehoneycombcrystal. At
2 6v
highmagneticfields,theseelectron-holeexcitationsappearaspeculiarn− n+ inter-Landau-level
modeswithenergiesω =2√2n~v/λ andcharacteristicallycrossedpolari→sationofin/outphotons.
n B
PACSnumbers: 73.63.Bd,71.70.Di,73.43.Cd,81.05.Uw
0 Inelastic(Raman)scatteringoflightis a powerfultool g
1 to study excitations in solids [1]. Recently, Raman spec- 2+
0 troscopyhasbeenusedtostudyphononsingraphene[2], I1 1+
2 where it has become the method of choice for determin- 0

n ing the number of atomic layers in graphitic flakes [2–8].
1−
a In particular, single- and multiple-phonon-emissionlines
2−
J
in the Raman spectrum of graphene and the influence Ω
2Μ Ω1 Ω2 Ω3
5 oftheelectron-phononcouplingonthe phononspectrum
2
havebeeninvestigatedingreatdetail[9–16]. However,no
FIG. 1: Spectral density g(ω)of light inelastically scatterred

experimental observation or comprehensive theoretical fromelectronicexcitationsingrapheneatquantisingmagnetic
]
l analysis has been reported on the Raman spectroscopy fields (solid line) and at B = 0 (dashed line). Here ω Ω
l ≪
a of electronic excitations in graphene, despite extensive is the Raman shift. Sketch illustrates intermediate and final
h studies of optical absorption in this material [17–23]. states of thedominant Raman process.
-
s
In this Letter, we present a theory of inelastic light
e
m scattering in the visible range of photon energies accom- teraction [26, 27] of an electron with two photons is a
paniedbyelectronicexcitationsingraphene. We classify peculiarity of the Dirac-type electrons in graphene. Fill-
.
t therelevantmodesaccordingtotheirsymmetryandpre- ing the conduction band or depleting the valence band,
a
m dict peculiar selectionrules for the Raman-activeexcita- up to the Fermi level αµ, forbids the excitation of inter-
tions of electrons between Landau levels in graphene at band electron-hole pairs with energies ω <2µ leading to
-
d quantizing magnetic fields. Graphene is a gapless semi- the linear Raman spectrum with a 2µ threshold [dashed
n conductor[24,25],withanalmostlinearDirac-typespec- line in Fig. 1]. The quantization of the electronic spec-
o trum, ε = αvp in the conduction (α = +) and valence trum into Landau levels [30] (LL) ε[nα] = α√2n~v/λ
[c (α = −) band, which touch each other in the corners of in a strong magnetic field (λB = ~c/eB is the magB-
the hexagonalBrillouin zone, usually called valleys. The netic length, n=0,1,2,...) makes Raman spectrum dis-
4 band structure of graphene is prescribed by the hexago- creteatlowenergiesω =2√2n~v/pλ ,withpeculiarfor
v nalsymmetryC ofitshoneycomblattice,anditisnat- n B
6v the Dirac-type electrons selection rules, n n+ of the
1 −
uraltorelateRaman-activemodestotheirreduciblerep- →
5 dominant Raman-active transitions [solid line in Fig. 1].
resentations of the point group C . We argue that the
2 6v
dominant electronic modes generated by inelastic scat-
5 The following theory is based upon the tight-binding
6. teringofphotonswithenergyΩ lessthanthe bandwidth model of electron states in graphene expanded into the
ofgraphenearesuperpositionsofthe interbandelectron-
0 Dirac-type Hamiltonian [31],
hole pairs which have symmetry of the representation
9
0 A2 of the group C6v and are odd with respect to the =vΣ P v2 ΛzΣx(ΣP)Σx(ΣP)Σx. (1)
: inversion of time. Their excitation process consists of H · − 6γ
v 0
two steps: the absorption (emission) of a photon with
i
X energy Ω (Ω˜ = Ω ω) transfering an electron from an The latter describes electrons in the conduction and
r occupied state in t−he valence band into a virtual state valence bands around the Brillouin zone (BZ) corners
a in the conduction band, followed by emission (absorp- K and K′. We use notations [32] such that Σ =
tion) of the second photon with energy Ω˜ (Ω). Its am- (ΠzKK′ ⊗ σAxB,ΠzKK′ ⊗ σAyB), Σz = ˆ1KK′ ⊗ σAzB and
plitude is determined by the sum of partial amplitudes Λz = Πz ˆ1 , where σx/y/z and Πx/y/z are Pauli
KK′ ⊗ AB AB KK′
distinguished by the order of absorption and emission of matrices acting on A-B (sublattice) and K-K (valley)
′
photons, and by which carrier in the intermediate state indices of the four-component wave function ψK,ψK′ ,
{ }
(an electron above the Fermi level or hole below it) un- where ψK = [ϕA,ϕB] and ψK′ = [ϕB,ϕA]. While 4-
dergoes the second optical transition. The dominance of spinors ψK,ψK′ realise 4D irreducible representation
{ }
such process over the process involving the contact in- of the full symmetry group of the crystal, the valley-
2
C rep. A B A B E E Ω ε,p
6v 1 1 2 2 1 2
matrix 1 Λz Σz ΛzΣz Σ Λze Σ
z× R = + + + +
t t + + +
→− − − −
Ω−ω ε+ω,
TdiAagBoLnEal oI:peCra6tvorisrrΣediuacnidbleΛzrΣepi.resentations by the valley- p+k RD Rw TVe
FIG.2: FeynmandiagramsdescribingRamanscatteringwith
theexcitations of electron-hole pairs in thefinalstate.
diagonal operators Σi and ΛzΣi can be combined into
irreducible representations [9, 33] of the group C , Ta-
6v
ble I, and Λz is used to describe valley-asymmetry of
A , Table I. As a result, such process excites a ’valley-
2
Dirac electrons. The first term in determines the lin-
symmetric’ electronic mode corresponding to the repre-
ear spectrum αvp with v 108cmH/s and p being the
sentation A of C and odd in terms of time-inversion
≈ 2 6v
in-plane momentum counted from the BZ corner. The
symmetry. ThetermR inEq. (2)describesthecontact
w
second term takes into account weak trigonal warping
interactionbetweenanelectronandtwophotonscharac-
[hoppingparameterγ0 3eVdeterminesthebandwidth, terisedby∂2 /∂p ∂p . Althoughforfreenon-relativistic
≈ i j
6γ ], which has an inverted shape in the opposite H
0 electronscontactinteractionisimportant[26],forDirac-
∼
corners of the BZ [31]. The vector potential of light
type electrons it is absent. It reappears only after devi-
A= ~c lei(qr Ωt)/~b +h.c. isincludedin
l,q,qz √2Ω − q,qz,l ations from the Dirac spectrum are taken into account,
P =Pp − ecA, w(cid:0)here bq,qz,l annihilates a(cid:1)photon char- i.e., the ’valley-antisymmetric’ warping term in Eq. (1),
acterised by the polarisation [l for incident and ˜l for and R generates excitations with the symmetry of the
w
scattered light], in-plane momentum q, energy Ω, and representationE of C . For scattering of photons with
2 6v
q = Ω2/c2 q2. Ω<γ , R R [27]. Finally, V stands for the con-
z − 0 w ≪ D T
tribution of the diagrams containing a ‘triangular’ loop
The amplitude R = R + R + V of the Raman
procepss with the excitationDof anwelectTron-hole (e-h) pair and the RPA-screened electrone-electron interaction
T
V. It accounts for the generation of a virtual e-h pair
in the final state corresponds to the Feeynman diagrams
which recombines creating a real e-h excitation through
shown in Fig. 2. Here, we call an ‘electron’ an excited
quasiparticle above the Fermi level αµ, and a ‘hole’ an tehe electron-electroninteraction,and its effect is negligi-
bly small [34].
empty state at ε < αµ. The building blocks of the dia-
gramsincludeGreen’sfunctionsfortheelectronsandthe Theprobabilityforaphotontoundergoinelasticscat-
electron-photon interaction vertices: tering from the state (q,qz) with energy Ω into a state
(q˜ =q k,q˜ ) with energy Ω˜ = Ω ω, by exciting an
z
− −
1 1+αΣ n p e-h pair in graphene with Fermi energy αµ at low tem-
- = GRε,/pA = 2 ε αvp· ip0, np =p, perature T <ω, is
α= − ±
X±
R(cid:6)(cid:6)(cid:4)Rs@(cid:0)@(cid:0)(cid:9) = e2v|~Ω|Σ·l, (cid:7)(cid:9)(cid:7)(cid:5)Rs@(cid:0)@(cid:0)(cid:9)= e2v|~Ω|Σ·˜l∗, w = Z 4dπ2~p3fpηα(1−f(p+k)α)δ(εpηα −ε(p+k)α +ω)
R (pe~v)2i(lטl∗)zΣz, p (2) × tr R(1+ηαΣnp)R+(1+αΣnp+k) . (3)
D ≈ 2Ω Ω Xξ (cid:8) (cid:9)
| |
Rw = 3√e22v2Ω~2γ (Λzez ×Σ)·d, Hoferger,apαhe=ne,±η =disti/n+gusisthanesdsbfeotrwetehne enx-caitnadtiopn-doofptinhge
| | 0 −
d = (l ˜l +l ˜l ,l ˜l l ˜l ). inter/intra-bandelectron-holepairs,andspin-degeneracy
x y∗ y x∗ x x∗− y y∗ is taken into account. The probability w describes the
angle-resolvedRamanspectrum,asopposedtotheangle-
IntheamplitudeR,thetermR representsthecontri-
D integrated spectral density,
bution of the firsttwo diagramsin Fig.2. They describe
a photon-assisted transition of an electron with momen-
Ω w(k,ω)d2k
tum p from under the Fermi level into a strongly off- g(ω) . (4)
rvepsona1nωtvirtΩu)a,lfionltloerwmededbiayteansotathteer(ntoratenstihtiaotnv(|pof+eikth|e≈r ≡ (2π~)3c2 Zc|k|<Ω √Ω2−c2k2
≈ 2 ≪ In undoped graphene the inter-band e-h pairs are the
electronorahole)whichreturnsthesystemontothe en-
only allowed electronic excitations. The probability,
ergy shell. The two diagrams in R differ by the order
D
of absorption/emission of the photons with Ω,Ω˜ vp,
≫ ω 1 ω
and, therefore, by the sign of the energy denominator w Ξ ~e4v2 + Ξ ~e4v2 , (5)
in GR/A. A partial cancellation between them deter- 0 ≈ s Ω4 2 o (6γ0Ω)2
mines the effective 2-photon coupling to the electrons Ξ = l ˜l 2, Ξ =1+(l l )(˜l ˜l ),
characterised by the matrix form in the representation s × ∗ o × ∗ × ∗
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3
of their excitation by photons with Ω<γ is dominated Then,electron’sGreenfunctionsandinteractionvertices
0
by the contribution, R of the first two diagrams in leading to optically active inter-LL excitations in mono-
D
Fig. 2. This determines typically crossed linear polar- layer graphene summarised in Table II take the form
isation of in/out photons described by the polarisation
factorΞs,whichisequivalenttosayingthattheyhavethe - =GR/A = δnn′δαα′ ,
samecircularpolarisation,incontrasttoaweakcontribu- ε αε i0
n
− ±
tion of the process enabled by the warping term (second
term in Eq. (5)), with the opposite circular polarisation R(cid:6)(cid:6)(cid:4)s(cid:0)(cid:0)(cid:9)= ev~ J l, (cid:7)s(cid:0)(cid:0)(cid:9)= ev~ J ˜l ,
∗
of in and out photons described by the factor Ξ . R@@ 2√Ω · (cid:7)(cid:9)(cid:5)R@@ 2√Ω ·
o
troInnicdoexpceidtagtiroanpshewnieth, wωit<h2µµ≫areΩbvl/occ,keindt,esro-btahnadt elec- Jnαn′α′ =αiδn′,n−1e−−α′iδn′−1,ne+,
e2v2~2
R = ΛzJ e (le )(˜l e ).
w ∗
θ(ω 2µ), ω 2µ >vk; 6γ0Ω · ± ∓ ∓
w =w0×(1 ar−ccos2µ−ω, |ω−2µ|<vk. (6) X±
π vk | − | Heree = 1 (e ie )isusedtostressthatacircularly
± √2 x± y
After integrating over all directions of the propagation polarised photon carries angular momentum m= 1.
±
of scattered photons, we find the spectral density of the The excitation of the e-h pairs by Raman scattering
angle-integratedRaman signal, in graphene at strong magnetic fields characterised by
the first two Feynmann diagrams in Fig. 2 produces the
1 e2 v 2 ω ω 2µ electronictransitionn− n+betweenLLs,withangular
g(ω) Ξ F − . (7) →
≈ 4 s π~c c Ω2 Ωv/c momentum transfer ∆m = 0 and excitation energy ω =
(cid:18) (cid:19) (cid:18) (cid:19) 2ε [Fig. 1], and transitions (n 1) (n+1)+ and
n −
Here F(x < 1) = 1(1 + x) and F(x > 1) = θ(x), (n+1)− (n 1)+,with∆m= −2and→ω =εn 1+εn+1.
step func|ti|on. In und2oped graphene (|µ|= 0), spectral The amp→litud−es of these two pro±cesses, −
̟
density g(ω) corresponds to the yield I = g(ω)dω
2 0 0 ∼ 1(ev~)2
he2c vc̟Ω such that I0(̟ ∼ 21Ω)∼10−10. R Rn−→n+ = 4 c2Ω ×
(cid:16) In dop(cid:17)ed graphene one may also expect to see some (le )(˜l e ) (le )(˜l e )
manifestation of the intra-band e-h excitations in the + ∗ − + ∗ −
vicinity of the Fermi level, with a small energy transfer ×αX=±"Ω−εn−αεn+1 − εn−Ω−αεn−1−
ω < Ωv/c. Their analysis requires taking into account (le )(˜l e ) (le )(˜l e )
alldiagramsinFig. 2,inparticular,duetoanadditional − ∗ + + − ∗ + ,
asymmetry between the conduction and valence bands − Ωεn αεn+1 εn Ω αεn 1#
− − − −
caused by the difference of their filling which increases
the value of the triangular loop [34],
1(ev~)2
(µ) = (ev~)2(l ˜l )µΠ; V = 2π~e2 ; R(n∓1)−→(n±1)+ =∓4 c2Ω (le±)(˜l∗e±)×
T − · ∗ Ω3 k 2π~e2Π α α
− + ,
Π = π~22µv2ω−(ω(ω−i0)i20)2e−v2vk22k2. ×αX=±"Ω−εn+1−αεn εn−1−Ω−αεn#
p
Then, we find that, fopr ω 6−(v/c)Ω− Ω<γ , adrueestuocahptahrattiaRlnca−n→cne+lla≫tionRo(nf∓t1h)e−→tw(on±d1i)a+grfaomr sωco≪nsΩti-,
0
≪ tuting R . Notice that these inter-LL modes n n+
D −
→
1 e2 2 vµ3ω v2 Ω2 have the symmetry of the representation A2 in Table I
δg = Ω2 ω2 Ξ + and the same circular polarisationof in and out photons
2(cid:18)π~c(cid:19) c Ω5 "(cid:18)c2 − (cid:19)8µ4 s involved in its excitation. Finally, the contact term Rw
Ω4 inFig.2allowsforaweaktransitionn− (n 1)+,with
→ ±
+Ξo(cid:18)1+ (6γ0µ)2(cid:19)#. tthraenasmitipolnit,uwdiethR∆wm≪=Rn−1→rens+em[3b7l]e.sStuhpeeirnfitceira-lLlyL,tsruacnhsia-
±
tioninvolvedinthefar-infrared(FIR)absorption[17,21].
The yieldof this low-energyfeature is δI = δg(ω)dω
10 15 for Ω 1eV [36]. ∼ However, the FIR-active excitation is ’valley-symmetric’
− ∼ R [21] and corresponds to the representation E1, whereas
Electronic spectrum of graphene in a strong magnetic the Raman-active n (n 1)+ mode corresponds to
field can be described as a sequence nα of Landau levels − → ±
E , allowing the latter to couple to the Γ-point optical
(LLs),ε[nα]=αε withε =√2n~v/λ ,corresponding 2
n n B phonon and, thus, leading to the magneto-phonon res-
to [21, 30] the states nα = 1 (Φ ,iαΦ ) for n 1
| i √2 n n−1 ≥ onance feature in the Raman spectrum [13]. Also, Rw
and 0 = (Φ ,0) (where λ = ~c/eB and Φ are originatesfromthe trigonalwarpingtermin whichvi-
0 B n
| i H
the normalisedLL wavefunctions in the Landaugauge). olatesthe rotationalsymmetryofthe DiracHamiltonian
p
4
C6v rep transition intensity polarisation broadening which increases with the LL number, ωn =
weak in Raman, 2ε =2√2n~v/λ ,andthefactorΞ = l ˜l 2inEq.(8)
E2 n(n−+→1)(−n+→1n)++ strong in magneto- σ± →σ∓ indnicates that inBandoutphotons hasve t|h×e s∗a|me circular
phonon resonance polarisation.
E (n−1)−→(n+1)+ weak in Raman σ± σ∓
1 (n+1)−→(n−1)+ →
A2 n−→n+ dominant in Raman σ± →σ± The n n+ inter-LL transitions are specific for
−
→
Dirac-type electrons in graphene and representthe most
TABLE II:Raman-active inter-LL excitations in graphene.
pronounced signature of its electronic excitations in the
Raman spectrum. The quantum efficiency of the low-
by transferringangular momentum 3 fromelectrons to est, ω1 = 2√2~v/λB peak in the spectrum in Fig. 1 is
the lattice, so that initial and final±state photons in it I v2 e2/λB 2 per incoming photon. For B = 20T,
have opposite circular polarisations. 1 ∼ c2 πΩ
The dominant inter-LL transitions n− n+ deter- we est(cid:16)imate I1 ∼(cid:17) 10−12 for photons with energies in the
→ visible range, which is feasible to detect in the inelastic
mines the spectral density of light scattered from elec-
light scattering experiments.
tronic excitations in graphene at high magnetic fields:
v2e2/λ 2
gn−→n+(ω)≈Ξs(cid:18)c2 πΩB(cid:19) n 1γn(ω−ωn). (8) A.WPeintchzaunkk, Ia.nAdleMine.r,PDot.emBaskskiof,oAr .uFseefruralrid,isAcu.sGsieoinms,.
X≥ We acknowledge financial support from EPSRC grants
Here γ (x) = π 1Γ /[x2 +Γ2], and Γ is inelastic LL EP/G014787,EP/G035954and EP/G041954.
n − n n n
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~v
thedominantRamanscatteringmechanismingraphene. V e4sa/Ωe R (ev~/Ω)2. Note that no reso-
Such an assumption is justified for scattering of soft X- T ∼ ≪ 0 ∼
e
5
nantlyenhancedcontributiontowards comesfromvir- count theplasma pole of thepropagator V(ω,k) in V,
tualstates with p 1Ω, since,after thTeintegration over we estimated the probability of the plasmeon emissioTneas
itnhteerpmroedduiacttessotfaGters≈e,etn2h’sefcuonncttriiobnustiionnsocfapnacierlseoafchpooltehseirn. wpl = ~e4v2|l·˜l∗|2|Ωµ|63vω2kpl2δ(ω−ωpl) and quantum effi-
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