Second Harmonic Generation in Graphene

0.44 MB·English

by  M. M. Glazov

Second Harmonic Generation in Graphene
M.M. Glazov∗
Ioffe Physical-Technical Institute of the Russian Academy of Sciences, St. Petersburg, 194021 Russia
We report on theoretical study of second harmonic generation in graphene. Phenomenological
analysis based on symmetry arguments is carried out. It is demontrated, that in ideal graphene
samplessecondharmonicgenerationispossibleonlyiftheradiationwavevectororitsmagneticfield
istakenintoaccount. Microscopictheoryisdevelopedfortheclassicalregimeofradiationinteraction
with electrons, where photon energy is much smaller than the charge carriers characteristic energy.
It is demonstrated, that the emitted radiation can be strongly circularly polarized for the linearly
polarized incident wave.
1
1
PACSnumbers: 72.80.Vp,78.67.Wj,68.65.Pq,42.65.Ky
0
2
n Introduction. Graphene, a monolayer of carbon atoms ergyofelectronsandaclassicalregimeisrealizedforthe
a arranged in honeycomb lattice, became recently one of radiation interaction with electrons in graphene, which
J themostpromisingmaterialsowingtoitsintriguingelec- brings about very strong polarization effects. In particu-

4 tronic and transport properties, for recent reviews see lar, secondharmoniccanbestronglycircularlypolarized
1 Refs. [1–4] and references wherein. for linearly polarized incident radiation.

It is well known that non-linear transport and optical Phenomenological theory. We consider an ideal single
]
l phenomena are most sensitive to the fine details of en- layer graphene described by the point symmetry group
l
a ergy spectrum and carrier scattering in condensed mat- D . This point group contains an inversion center,
6h
h ter [5, 6]. One of the non-linear effects is the generation therefore,thesecondharmonicgenerationispossibleonly
-
of dc current under homogeneous excitation caused by if one takes into consideration the light wave vector q
s
e the asymmetry of optical transitions and scattering in transfer to electron ensemble or joint action of electric
m non-centrosymmetric media (photogalvanic effect) or by E and magnetic fields B of the incident radiation [6, 7].
t. the transfer of photon momentum to the electron sys- SincethemagneticfieldcomponentsBα areproportional
a tem (photon drag effect) [7]. The latter one does not to certain combinations q E , where greek subscripts
β γ
m require low symmetry of the structure, but it is masked enumerate Cartesian coordinates, in phenomenological
- by various photogalvanic effects in ordinary semiconduc- considerations it is enough to allow for the contributions
d tor quantum wells which lack an inversion center as a tothesecondorderresponsebeinglinearinthewavevec-
n
rule[8]. Bycontrast, grapheneisacentrosymmetricma- tor components and quadratic in the electric field ampli-
o
terial which allowed unambiguous identification of the tudes.
c
[ photon drag effect by its remarkable polarization depen- Itisconvenienttostudytheelectriccurrent,j(2ω),in-

dence [9]. ducedatthedoublefrequency2ω asaresponsetothein-
1
Here we consider theoretically light wave vector in- cident radiation with the fundamental frequency ω. The
v
duced second harmonic generation in graphene which electric current is related with the oscillating polariza-
2
4 consistsintheformationoftheac response: electriccur- tion, P(2ω), as j(2ω) = −2iωP(2ω). The general phe-
8 rent or dielectric polarization oscillating at the double nomenologicalexpressiondescribingthesecondharmonic
2 frequencyascomparedwiththefrequencyoftheincident generation reads
1. radiation. Experimentally second harmonic in graphene
0 wasobservedin[10,11],butmicroscopictheoryisabsent jλ(2ω)=SλµνηqµEνEη. (1)
1 so far. Alike the photon drag effect, in centrosymmetric
1 media like graphene second harmonic generation is al- HereS isthematerialtensorandtheelectricfieldof
λµνη
v: lowed only provided that the photon momentum trans- the fundamental wave is taken in the complex form
i fer to electron system is taken into account. This is in
X contrast to the third (and other odd) harmonic genera- E(r,t)=Eei(qr−ωt)+c.c. . (2)
r tion [12, 13] as well as coherent injection of ballistic cur-
a
rents [14, 15] which do not require photon momentum Thesamecomplexform,Eq.(2),isassumedforallother
transfer, but appear in higher orders in incident electro- quantities in question: the magnetic field, B, and the
magnetic field. current density, j. It is worth mentioning that, in con-
The second harmonic response is calculated here for trast to the photon drag effect where the dc current is
low frequencies, e.g. terahertz radiation. In such case, proportionalto thebilinear combinations ofelectric field
thephotonenergyismuchsmallerthancharacteristicen- anditsconjugate,EνEη∗,thesecondharmonicgeneration
isdescribedbythequadraticcombinationsE E . Hence,
ν η
tensorS isinvariantwithrespecttopermutationsof
λµνη
the two last subscripts. The elements S are, in gen-
λµνη
∗[email protected]ffe.ru eral, complex, the presence of both real and imaginary
2
(a) (b) becomes non-centrosymmetric, described by the point
groupC ,andsecondharmonicgenerationbecomespos-
6v
sible for q = 0. It can be shown that additional contri-
butions have the following form
FIG. 1: Schematic illustration ofthe second harmonic gener-
ationgeometry,Eqs.(4). Itisseenthattherearebothlongi- j (2ω)=S(cid:48)E E , j (2ω)=S(cid:48)E E , (5a)
x 3 x z y 3 y z
tudinal [panel (a)] and transverse [panel (b)] components of
the second order response.
j (2ω)=S(cid:48)E2+S(cid:48)(E2+E2), (5b)
z 5 z 6 x y
parts describes the phase shift between the fundamental where S(cid:48), S(cid:48) and S(cid:48) are some coefficients directly pro-
3 5 6
and second harmonics. portional to parameters ∂t , ∂t and ∂t introduced in
15 31 33
There are 7 independent parameters, S ...S , which Eq. (10) of Ref. [11]. As one can see from Eqs. (5) these
1 7
describe second harmonic generation in the system of contributionsrequiretransitionsinzpolarizationandare
point group D : strongly suppressed in classical frequency range [16] but
6h
they may take over for higher frequencies [11].
j (2ω)=S q (E2+E2)+S [q (E2−E2)+2q E E ] Microscopic theory. We consider here a doped
x 1 x x y 2 x x y y x y
graphene layer and assume that the characteristic elec-
+S q E E +S q E2, (3a)
3 z x z 4 x z tron energy (being the Fermi energy, ε , for degenerate
F
andthetemperature, k T, fornon-degenerateelectrons)
B
j (2ω)=S q (E2+E2)+S [q (E2−E2)+2q E E ] is much larger than the photon energy (cid:126)ω. This condi-
y 1 y x y 2 y y x x x y tion allows one to treat electron dynamics in the frame-
+S3qzEyEz+S4qyEz2, (3b) work of Boltzmann equation. For typical carrier densi-
ties on the order of 1012 cm−2 the electron Fermi energy
ε ∼ 100 meV, therefore carriers are degenerate even
j (2ω)=S q E2+S q (E2+E2)+S (q E E +q E E ), F
z 5 z z 6 z x y 7 x x z y y z at room temperature and classical treatment is valid for
(3c)
microwave or terahertz radiation [9].
where we assumed that the graphene layer occupies (xy)
We introduce time, t, position, r, and momentum, p,
plane. Expressions (3a) and (3b) are similar to phe-
dependent electron distribution function f(p,r,t) which
nomenological equations describing linear drag effect in
satisfies standard kinetic equation [9, 16]
graphene [16].
The specifics of the graphene band structure imposes (cid:18) (cid:19)
∂f ∂f 1 ∂f
strong restrictions on values of the constants S1,...,S7 ∂t +vp∂r +e E+ c[vp×B] ∂p =Q{f}. (6)
in Eqs. (3). Indeed, conduction and valence band of
graphene are formed from π orbitals of carbon atoms.
Here E and B are the electric and magnetic fields of the
As a result, the transitions in z polarization are possible
incident wave taken in the form Eq. (2), e is the electron
onlywithallowanceforother,σ,orbitalswhicharesepa-
charge,e=−|e|,cisthespeedoflight,v istheelectron
p
rated from π orbitals by several electronvolts. It implies
velocity in the given state p,
that in the model where only π orbitals are taken into
account (e.g. in the classical frequency range of interest) ∂ε p
v = p =v . (7)
constants Si = 0 for i > 2 and second harmonic genera- p ∂p |p|
tion is described by the simplified expressions with only
two independent parameters: Here ε = v|p| is the electron dispersion, v ≈ c/300 is
p
effectiveelectronspeedingraphene,andQ{f}inEq.(6)
jx(2ω)=S1qx(Ex2+Ey2)+S2[qx(Ex2−Ey2)+2qyExEy], is the collision integral. In Eq. (6) it is assumed that
(4a) f(p,r,t)describestheelectrondistributioninanyoftwo
equivalent degenerate valleys, K and K(cid:48). It is clearly
j (2ω)=S q (E2+E2)+S [q (E2−E2)+2q E E ].
y 1 y x y 2 y y x x x y seen from Eq. (6) that electron motion is affected by the
(4b)
in-plane components of electric field E and by z compo-
Figure 1 schematically shows the geometry of the sec-
nent of magnetic field B.
ond harmonic generation and the response at a dou-
The distribution function is sought in a form [16]
ble frequency, 2ω. For instance, for q (cid:54)= 0, q = 0,
x y
there is a component of the in-plane oscillating current
f(k,r,t)=f (ε)+[fω(k)ei(qr−ωt)+c.c.]+
j(2ω) parallel to the light incidence plane described by 0 1
(S1+S2)qxEx2+(S1−S2)qxEy2 [Fig. 1(a)]. Additionally, f20(k)+[f22ω(k)e2i(qr−ωt)+c.c.] , (8)
thereisatransversecontribution, 2S q E E , Fig.1(b).
2 x x y
Microscopic theory for these constants S and S is de- wheref (ε)istheequilibrium(Fermi-Dirac)distribution
1 2 0
veloped below. function,fω(k)isthefirstordercorrection,andfunctions
1
Graphenesamplesareusuallydepositedonsubstrates, f0,f2ω arethesecondordercorrectionscorrespondingto
2 2
hencez →−z symmetryisbroken. Asaresult,structure the responses at zero frequency (photon drag effect) and
3
atthefrequency2ω(secondharmonic),respectively. The (a) (b)
terms containing higher frequencies (and, hence, higher
powers of E, B) are disregarded here because they do
not contribute to the second harmonic generation. The
dc responsewascalculatedinRefs.[9,16], belowwefind
the function f2ω(k).
2
In order to determine f2ω(k) we solve Eq. (6) by iter-
2
ations. Tothatendonehastofindfirstordercorrection,
FIG. 2: Frequency dependence of the contributions to the
f1ω(k),whichcanbeexpresseduptothelinearinqterms secondharmonicresponse: qS1isshowninpanel(a),andqS2
as [16] is shown in panel (b). Solid blue lines depict real part and
dashedredlinesdepictimaginarypartoftheresponse. Values
(cid:26) of current are given in arbitrary units, radiation frequency is
f (k)=−eτ f(cid:48) (E v )− presented in units of ωτ , where τ is momentum relaxation
1 1,ω 0 0 p 1 1
time. Calculationiscarriedoutfortheshort-rangescattering
v2(qE )(cid:27) where τ =2τ ∝ε−1 for degenerate electrons.
iτ [(qv )(E v )−v2(qE )/2]+ 0 . (9) 1 2 k
2,ω p 0 p 0 2ω
where the factor 4 takes into account spin and valley
Here we introduced f(cid:48) =df /dε,
0 0 degeneracy. Solving Eq. (11), retaining in f2ω(p) only
2
τ linear in q or linear in B terms and expressing B =
τ = n , (n=1,2),
n,ω 1−iωτ [(q/q)×E] (for the sample in vacuum) we arrive at the
n
following expressions for the constants S and S :
1 2
whereτ andτ aretherelaxationtimesforthefirstand
1 2 e3v4 (cid:88)
second angular harmonics of the distribution function S =− τ f(cid:48) × (13a)
which describe the momentum (first angular harmonic) 1 2ω 1,ω 0
k
andmomentumalignment(secondangularharmonic)re- (cid:20) (cid:21)
τ dτ
laxationofelectrons,respectively. Theserelaxationtimes 1ε,2ω(3+iωτ2,ω)+(1−iωτ2,ω) d1ε,2ω ,
can be determined in Born approximation as [17] k k
1 1 e3v4 (cid:88)
τ ≡ τ (ε ) = (10) S2 = 2ω τ1,ωf0(cid:48) × (13b)
n n k k
2(cid:126)πnimp(cid:88)|Vk−p|21+2cosϑ[1−cos(nϑ)]δ(εk−εp), (cid:26)τ1ε,2ω(1+4iωτ2,2ω)− ddε [τ1,2ω(1−2iωτ2,2ω)](cid:27).
p k k
It is noteworthy, that the physical mechanisms govern-
wheren isthe(two-dimensional)impuritydensity,V
imp q ing second harmonic generation are the same as in the
is the Fourier-transform of the impurity potential, ϑ is
photon drag effect [16]. First one is related to the joint
thescatteringangle,(1+cosϑ)/2comesfromtheoverlap
action of electric and magnetic fields of the incident ra-
of Bloch amplitudes and the normalization area is set
diation: electric field induces ac current with the funda-
to unity hereafter. Intervalley scattering processes are
mental frequency ω, while magnetic field oscillating at
disregarded in Eq. (10).
thesamefrequencyinducesbothdc currentandthecur-
The energy relaxation processes are neglected assum-
rent component with the double frequency, 2ω. This is
ing that ωτ (cid:29) 1, τ /τ (cid:29) 1, where τ is the en-
ε ε 1,2 ε EB mechanism of the second harmonic generation. An-
ergy relaxation time. Equation (9) is valid as long as
other qE2 mechanism is connected with the spatial de-
qvτ ,qvτ ,qv/ω (cid:28) 1, that is if spatial variations of the
1 2 pendence of the electric field, its qualitative description
distribution function occur on the scale exceeding by far
can be carried out along the same lines as in Ref. [16].
the mean free path, this condition is readily fulfilled in
In quantum language the microscopic mechanisms gov-
the relevant frequency range.
erning these effects are due to the magneto-dipole and
The second order correction satisfies the following
quadrupole transitions, respectively.
equation
Figure 2 shows frequency dependence of electric cur-
rent components at double frequency j(2ω) proportional
−2iωf22ω(p)+2iqvpf22ω(p)+ to coefficients S1 [panel (a)] and S2 [panel (b)], respec-
(cid:18) 1 (cid:19)∂fω(p) tively. Calculationiscarriedoutfortheshort-rangescat-
e E+ [v ×B] 1 =Q{f2ω}, (11)
c p ∂p 2 tering where Vk−p in Eq. (10) is constant and relaxation
times are interrelated as τ = 2τ ∝ ε−1. The elec-
1 2 k
and the electric current density is given by: tron gas is assumed to be degenerate. It can be seen
that the response at double frequency has complex non-
j =4e(cid:88)v f2ω(p), (12) monotonous behavior. In the static limit, ω →0, the co-
p 2
efficients S and S are real and diverge as 1/ω, but the
k 1 2
4
net current remains finite due to factors ∝q in Eqs. (4). (a) (b)
Coefficients S and S become, up to common factor,
1 2
equaltotheconstantsT andT describinglinearphoton
1 2
drageffect,seeEqs. (26)ofRef.[16],becauseatω =0re-
sponses at zero and double frequencies are indistinguish-
able. Athighfrequencies,ωτ (cid:29)1,ωτ (cid:29)1,parameters
1 2
S and S are proportional to 1/ω3, hence, current den-
1 2
sity decays as 1/ω2. The imaginary part of current is
FIG. 3: Frequency dependence of Stokes parameters of sec-
determined by the retardation between the driving elec-
ond harmonic: P is shown by black, P(cid:48) by blue and P by
tricfieldandelectronvelocity[9,16]. Therefore,itiszero l l c
red curves, respectively. It is assumed that the fundamen-
in the limit of static fields, shows maximum at ωτ ∼ 1
1 tal harmonic is fully linearly polarized in x(cid:48)y(cid:48) axis frame:
and decreases for ωτ (cid:29) 1. Typical values of j(2ω) for
1 |E |2 = |E |2, 2Re{E E∗} = |E |2 +|E |2. Calculation is
x y x y x y
ωτ ∼1 are close to those of dc current. For the experi-
1 carriedoutfordegenerateelectronsinthemodeloftheshort-
mental parameters of Ref. [9]: j(2ω)/P ∼0.5 nAcm/W, range scattering where τ =2τ ∝ε−1 [panel (a)] and of the
1 2 k
where P is the power density. Coulomb scattering where τ =3τ ∝ε [panel (b)].
1 2 k
Polarization of the second harmonic. The electromag-
netic field emitted at the double frequency can be found
bymeansofMaxwellequations. Itisconvenienttodeter- q (cid:28) q , and considered the emitted wave propagating
(cid:107) z
minethevectorpotentialA(r,t)oscillatingat2ω,which along positive z axis.
satisfies the following equation [18] We assume that the in-plane wave vector is directed
along x axis, q (cid:54)= 0, q = 0. Non-trivial polarization
x y
∆A(r,t)+ 4ω2A(2ω)=−4πe2iq(cid:107)ρ−2iωtδ(z)j(2ω)+c.c. effectstakeplaceifbothjx(2ω)andjy(2ω)arenon-zero.
c2 c Such situation may arise for the linearly polarized inci-
(14) dent radiation in x(cid:48)y(cid:48) axis frame, where |E |2 = |E |2,
x y
whereq(cid:107) =(qx,qy)istheprojectionoftheincidentradia- 2Re{E E∗} = |E |2 + |E |2 [P (ω) = 0, P(cid:48)(ω) = 1,
tionwavevectorontothegraphenelayerandweassumed x y x y l l
P (ω) = 0] which is considered in the following. Stokes
c
that the sample is positioned in vacuum at z = 0 plane.
parameters for the second harmonic are plotted in Fig. 3
Using one-dimensional Green’s function [7], the solution
asfunctionsofthefundamentalradiationfrequency. Cal-
of Eq. (14) can be written as
culationpresentedinFig.3(a)iscarriedoutfortheshort-
range scattering where τ = 2τ ∝ ε−1 [panel (a)]. For
iπ (cid:104) 1 2 k
A(r,t)= e2iq(cid:107)ρ−2iωt j(2ω)e2iqz|z| (15) comparison Fig. 3(b) shows calculation results for the
cq
z unscreened Coulomb scattering potential, where V ∝
−e j(2ω)q(cid:107)(e2iqz|z|−1)signz(cid:21)+c.c. |k−p|−1inEq.(10)andonecanshowthatτ1 =3τ2k∝−pεk.
z q In all calculations it is assumed that the electron gas is
z
degenerate.
Here e is the unit vector along z-axis. It follows from
z Figure 3 clearly demonstrates that all Stokes param-
Eq. (15) that the emitted radiation propagates with the
eters are non-zero, in general. In the limit of static
in-plane wave vector 2q . The dependence of the vec-
(cid:107) fields, ωτ1 = 0, ellipticity is forbidden, Pc(2ω) = 0 in
tor potential on z given by the factor exp(2iq |z|) corre-
z agreement with Fig. 3. In the limit of high frequencies
sponds to the wave diverging from the quantum well. ωτ ,ωτ (cid:29) 1 (but (cid:126)ω (cid:28) ε ) constants S and S are
1 2 F 1 2
Theelectricfieldoftheemittedsecondharmonicisre-
related as S =S /2 which yields
2 1
latedwithitsvectorpotentialbythestandardexpression
E(2ω) = −c−1∂A(2ω)/∂t. Making use of Eq. (15) one 3 4
P = , P(cid:48) = , P =0, at ωτ ,ωτ (cid:29)1.
can see that the Stokes parameters of emitted radiation l 5 l 5 c 1 2
are given by
It follows from Fig. 3 that second harmonic radiation
|j (2ω)|2−|j (2ω)|2 is, in general, elliptically polarized. The remarkable de-
P (2ω)= x y ,
l |jx(2ω)|2+|jy(2ω)|2 gree of circular polarization appears at ωτ1 ∼1. For the
short-range scattering considered in panel (a) of Fig. 3
2Re{j (2ω)j∗(2ω)}
P(cid:48)(2ω)= x y , the maximum absolute value of the circular polarization
l |j (2ω)|2+|j (2ω)|2
x y |P | is about 90% at ωτ ≈1.78. The maximum value of
c 1
2Im{j (2ω)j∗(2ω)} the circular polarization degree for the Coulomb scatter-
P (2ω)=− x y . (16)
c |jx(2ω)|2+|jy(2ω)|2 ingissomewhatsmaller,|Pc|≈45%atωτ1 ≈1.31. Phys-
ically, the circularly polarized second harmonic emission
Here P and P(cid:48) are the degrees of linear polarization in is a result of retardation: j (2ω) and j (2ω) are phase
l l x y
two coordinate frames: the xy frame and in the x(cid:48)y(cid:48) shifted with respect to each other. This phase differ-
frame which is rotated by 45o with respect to xy frame. encecanbesignificantresultinginalmostfullycircularly
QuantityP isthecircularpolarizationdegree. Inderiva- polarized second harmonic emission. In agreement with
c
tion of Eqs. (16) we assumed small incidence angles, qualitativeconsiderationsandFig.2,thephaseshiftvan-
5
ishes both for static and high frequency fields, where S ters. The microscopic theory is presented here for the
1
andS arereal. Circularpolarizationapproacheszeroin classical frequency range, where the photon energy is
2
these two limits. much smaller than the characteristic energy of charge
The appearance of the linear polarization in xy axis carriers in graphene. In this regime effects of other or-
frame, P (2ω) (cid:54)= 0, and of the circular polarization, bitals are negligible and the second harmonic generation
l
P (2ω)(cid:54)=0canbeconsideredasanon-linearpolarization is fully determined by the specifics of the radiation mo-
c
conversionprocess. Inversionofinitiallinearpolarization mentum transfer to “Dirac” electrons in graphene.
P(cid:48)(ω)→−1 results in the reversal of the circular polar-
l We have analyzed the polarization of the emitted ra-
ization, P , and of the linear polarization in x(cid:48)y(cid:48) axis
c diation. It is demonstrated, that even for linearly polar-
frame, P(cid:48)(2ω). Linearpolarization in xy axis framedoes
l ized fundamental harmonic, the second harmonic can be
not change its sign.
strongly circularly polarized. The degree of circular po-
Concluding remarks. We have developed theory of
larization can be controlled by the radiation frequency,
second harmonic generation in graphene. The analy-
it can be reversed by changing the linear polarization of
sis shows that in the simplest model of conduction and
incident wave.
valence bands formed by π orbitals of carbon atoms,
the response at the double frequency appears with al- AuthorisgratefultoE.L.IvchenkoandS.A.Tarasenko
lowance for the photon wave vector transfer to electron forvaluablediscussions. ThesupportbyRFBRand“Dy-
system and it is described by two independent parame- nasty” Foundation—ICFPM is gratefully acknowledged.
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