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Frequency Mixing Effects in Graphene

Sergey MikhailovUniversity of Augsburg

Germany

1.Introduction

Graphene is a new nanomaterial which has been discovered a few years ago, Novoselov et al.(2004); Novoselov et al. (2005); Zhang et al. (2005) and has demonstrated unique mechanical,electrical, thermal and optical properties, see review articles Katsnelson (2007); Castro Netoet al. (2009); Geim (2009). This is a one-atom-thick layer of carbon atoms arranged in a highlysymmetric two-dimensional honey-comb lattice, Figure 1. Graphene exhibits many interestingfundamental physical properties such as the minimal electrical conductivity Novoselov et al.(2005); Zhang et al. (2005); Katsnelson (2006); Nomura & MacDonald (2007); Tan et al. (2007),unconventional quantum Hall effect Novoselov et al. (2005); Zhang et al. (2005) observableup to room temperatures Novoselov et al. (2007), Klein tunneling Stander et al. (2009); Young& Kim (2009), optical conductivity determined only by the fine structure constant Ando et al.(2002); Kuzmenko et al. (2008); Nair et al. (2008) and many other. Graphene promises manyelectronic applications like terahertz transistors, photodetectors, transparent electrodes fordisplays, gas and strain sensors and so on, Geim (2009).Microscopically, the most distinguished feature of graphene is that, in contrast to other(semiconductor) materials with two-dimensional electron gases, electrons and holes ingraphene have not a parabolic, but a linear energy spectrum near the Fermi level Wallace(1947); McClure (1956); Slonczewski & Weiss (1958). The Brillouin zone of graphene electronshas a hexagonal shape, Figure 2, and near the corners Kj, j = 1, . . . 6, the electron and holeenergy bands Elp touch each other; here p is the quasi-momentum of an electron and l is theband index (l = 1 for holes and l = 2 for electrons). The spectrum Elp near these, so calledDirac points is linear,

Elp = (1)lV|p hKj| = (1)lV|p| = (1)l hV|kKj|, (1)

where k is the quasi-wavevector, p = p hKj and V is the Fermi velocity. In graphene V 108 cm/s, so that electrons and holes behave like massless relativistic particles with theeffective velocity of light V c/300, where c is the real velocity of light. In the intrinsicgraphene the chemical potential (or the Fermi energy EF) goes through Dirac points, = 0.If graphene is doped or if a dc (gate) voltage is applied between the graphene layer and asemiconductor substrate (in a typical experiment the graphene sheet lies on a substrate, e.g.on Si-SiO2) the chemical potential can be shifted to the upper or lower energy band, so thatthe electron or hole density ns can be varied from zero up to 1013 cm2. It is the unusualrelativistic energy dispersion of graphene electrons (1) that leads to the unique physicalproperties of graphene.

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a1a2b

12

12

xa

1

2 3

1

3

3

2

ya

Fig. 1. The honey-comb lattice of graphene. All points of the sublattice A (black circles) aregiven by n1a1 + n2a2, of the sublattice B (open circles) by n1a1 + n2a2 + b. Dashed lines showthe boundaries of the elementary cell. a is the lattice constant.

G2

G1

K1

K2

K3K4

K5

K6

4 3

2 3

2 3

4 3

kxa

2

3

2

3

kya

Fig. 2. The Brillouin zone of graphene. The basis vectors of the reciprocal lattice are G1 andG2. The vectors Kj, j = 1, . . . , 6, correspond to the corners of the Brillouin zone (the Diracpoints). HereK1 = K4 = 2a1(1/3, 1/

3), K2 = K5 = 2a1(2/3, 0),

K3 = K6 = 2a1(1/3,1/3).

In 2007 it was predicted Mikhailov (2007) that the linear spectrum of graphene electrons (1)should lead to a strongly nonlinear electromagnetic response of this material. The physical

520 Physics and Applications of Graphene - Theory

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origin of this effect is very simple. As seen from (1) the graphene electrons cannot stop and canmove only with the velocity V. If such a particle is placed in the uniform oscillating electricfield E(t) = E0 cost, its momentum will oscillate as p(t) sint. The velocity of suchparticles v = Elp/p = Vp/|p| will then take the value of +V or V dependent on the signof the momentum p, i.e. v(t) sgn(sint). The electric current j(t) induced by the externalfield is determined by the velocity, therefore one will have 1

j(t) sgn(sint) =4

(

sint+13sin 3t+

15sin 5t+ . . .

)

. (2)

The induced current thus contains higher frequency harmonics n, n = 3, 5, . . . and hence asingle graphene sheet should radiate electromagnetic waves not only at the frequency butalso at n with n = 3, 5, 7, . . . Mikhailov (2007; 2008); Mikhailov & Ziegler (2008). Graphenecould thus serve as a simple and inexpensive frequencymultiplierMikhailov (2007; 2009). Thenonlinear electromagnetic response of graphene has been also discussed by Lpez-Rodrguez& Naumis (2008).Apart from the frequency multiplication effect all other known nonlinear electromagneticphenomena should be also observable in this material. For example, irradiation of thegraphene layer by two electromagnetic waves E1(t) and E2(t)with the frequencies 1 and 2should lead to the emission of radiation at the mixed frequencies n11 + n22 with integernumbers n1 and n2. Since the graphene lattice (Figure 1) has a central symmetry, the even ordereffects are forbidden in the infinite and uniform graphene layer, so that n1+ n2 must be an oddinteger. In the third order in the external field amplitudes E1 and E2, apart from the frequencies1,2, 31 and 32, the radiation at the mixed frequencies1 22 and 212 should beobserved. In a recent experiment Hendry et al. (2010) the coherent emission from graphene atthe frequency 212 has indeed been discovered in the near-infrared and visible frequencyrange.In this Chapter we develop a theory of the frequency mixing effect in graphene. We beginwith a discussion of the electronic spectrum and the wave functions of graphene obtainedin the tight-binding approximation (Section 2) and continue by a brief overview of thelinear response theory of graphene in Section 3. Then we study the frequency mixing effectsin graphene within the framework of the quasi-classical approach which works on low(microwave, terahertz) frequencies (Section 4). In Section 5 we introduce a quantum theoryof the nonlinear electromagnetic response of graphene to high (infrared, optical) frequencies.A summary of results and the prospects for future research are discussed in Section 6.

2. Energy spectrum and wave functions of graphene

We calculate the spectrum and the wave functions of graphene electronswithin the frameworkof the tight-binding approximation Wallace (1947); Reich et al. (2002) assuming that both theoverlap and the transfer integrals are nonzero only for the nearest-neighbor atoms. The latticeof graphene, Figure 1, consists of two triangular sublattices A and B. For the basic vectorsof the A lattice we choose the vectors a1 = a(1/2,

3/2) and a2 = a(1/2,

3/2), where

a is the lattice constant (a 2.46 in graphene). The vector b connecting the sublattices is

1 In conventional electron systems with the parabolic spectrum of charge carriers Ep = p2/2m thevelocity v = Ep/p = p/m is proportional to the momentum, therefore j v sint and thehigher frequency harmonics are not generated.

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b = a(0, 1/3). The two-dimensional single-particle Hamiltonian H0 of graphene can then

be written as

H0 =p2

2m+

a

[Ua(r a) +Ua(r a b)], (3)

where p = ih(x , y) is the two-dimensional momentum operator, m is the free electronmass and Ua is the atomic potential. Following the standard procedure of the tight bindingapproximation we get the energy Elk and the wave functions |lk of graphene electrons as

Elk = (1)lt|Sk|, (4)

|lk lk(r) =1Seikrulk(r), (5)

ulk(r) =

A

2 aeik(ra)

[

(1)lka(r a) + a(r a b)]

, (6)

where l = 1, 2, k = (kx, ky) is the quasi-wavevector, t is the transfer integral (in graphenet 3 eV), S and A are the areas of the sample and of the elementary cell, respectively, and ais the atomic wave function. The functions Sk and k in (4) and (6) are defined as

Sk = 1+ eika1 + eika2 = 1+ 2 cos(kxa/2)ei3kya/2, (7)

k = Sk/|Sk|. (8)They are periodic in the k-space and satisfy the equalities

Sk = Sk; Sk+G = Sk; k = k; k+G = k; (9)

where G are the 2D reciprocal lattice vectors. Similar relations are valid for the energies Elkand the wave functions lk(r),

El,k = Elk; l,k(r) =

lk(r); El,k+G = Elk; l,k+G(r) = lk(r). (10)

The basic reciprocal lattice vectors G1 and G2 can be chosen as G1 = (2/a)(1, 1/3), G2 =

(2/a)(1,1/3), see Figure 2.

The energy dispersion (4) is shown in Figure 3. At the corners of the Brillouin zone, in theDirac points k = Kj, the function Sk vanishes and at |kKj|a = |kj|a 1 one has

Sk 3a2

[

(1)jkjx + ikjy]

, (11)

k = (1)jkjx + ikjy

(kjx)2 + (k

jy)2

. (12)

The energy (4) then assumes the form (1) with the velocity V =3ta/2h 108 cm/s.

Using the wave functions (5)(6) one can calculate the matrix elements of different physicalquantities. For example, for the function eiqr we get

l k|eiqr|lk = 12k,k+q

[

1+ (1)l +lk+qk]

. (13)


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Fig. 3. The bandstructure of graphene electrons Elk calculated in the tight-bindingapproximation, Eq. (4).

In the limit q 0 we obtain from here the matrix elements of the coordinate

l k|x|lk =i

2(1)l +lk

kk

(14)

and of the velocity operator v = p/m,

lk|v|lk =1h

Elkk

, lk|v|lk =Elk Elk

2hk

kk

. (15)

Here l means not l, i.e. l = 2 if l = 1 and l = 1 if l = 2.

3. Linear response

Before presenting our new results on the nonlinear frequency mixing effects we will brieflyoverview the linear response theory of graphene. We will calculate the linear responsedynamic conductivity of graphene (1)() () which has been studied theoretically inRefs. Gusynin & Sharapov (2006); Gusynin et al. (2006); Nilsson et al. (2006); Abergel & Falko(2007); Falkovsky & Pershoguba (2007); Falkovsky & Varlamov (2007); Mikhailov & Ziegler(2007); Peres et al. (2008); Stauber, Peres & Castro Neto (2008) and experimentally in Refs.Dawlaty et al. (2008); Li et al. (2008); Mak et al. (2008); Stauber, Peres & Geim (2008).

3.1 Quantum kinetic equation

The system response is described by the quantum kinetic (Liouville) equation

ih

t= [H, ] (16)


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for the density matrix , where

H = H0 + H1 = H0 e(r, t), (17)

is the Hamiltonian of graphene in the presence of the external electric field E(r, t) = (r, t)and e > 0 is the electron charge. We will be interested in the response to the uniform electricfield but at this step will describe the electric field by the potential

(r, t) = qeiqrit+t, +0, (18)

and will take the limit q 0 later on. The unperturbed Hamiltonian H0 is given by Eq. (3). Ithas the eigenenergies and eigenfunctions,

H0|lk = Elk|lk, (19)

given by equations (4) and (5), respectively; we have also introduced the spin index here.Expanding the density matrix up to the first order in the electric field,

= 0 + 1, (20)

where 0 satisfies the equation

0|lk = f0(Elk)|lk, (21)

and f0 is the Fermi function, we get

l k|1|lk =f0(El k) f0(Elk)

El k Elk h(+ i0)l k|H1|lk, (22)

l k|H1|lk = eql k|eiqr|lk. (23)Calculating the first order current j(r, z, t) = jqeiqrit+t(z) we obtain

jq = e

2SSp

(

1[v, eiqr]+

)

=e2gs2S

q kkll

lk|[v, eiqr]+|l kf0(El k) f0(Elk)

El k Elk h( + i)l k|eiqr|lk, (24)

where j = (jx, jy) and [. . . ]+ denotes the anti-commutator. Taking the limit q 0 gives thefrequency dependent conductivity ()which describes the linear response of graphene toa uniform external electric field. The conductivity () consists of two contributions, theintra-band (l = l ) and the inter-band (l = l ) conductivities.

3.2 Intra-band conductivity

The intra-band conductivity reads

intra () =ie2gs

h2( + i)Slk

Elkk

f0(Elk)

E

Elkk

, (25)

where gs = 2 is the spin degeneracy and the summation over k is performed over the wholeBrillouin zone. If the chemical potential lies within 1 eV from the Dirac points (which istypically the case in the experiments) and if the photon energy h does not exceed 1 2 eV


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the main contribution to the integrals is given by the vicinity of Dirac points and one can usethe linear (Dirac) approximation (11), (1). Then one gets intra () =

intra() and

intra() =ie2gsgvT

2h2( + i)ln

[

2 cosh(

2T

)]

, (26)

where T is the temperature and gv = 2 is the valley degeneracy factor. At low temperaturesT || the formula (26) gives

intra() =e2gsgv||4h2

i

+ i=

nse2V2

||i

+ i=

nse2

mi

+ i, (27)

where the last equalities are written in the Drude form with the phenomenological scatteringrate , the effective mass of graphene quasiparticles at the Fermi level m = ||/V2 and thecharge carrier density

ns =gsgv4

2

h2V2=

gsgv4

k2F

. (28)

The value kF in (28) is the Fermi wavevector.The intra-band conductivity has a standard Drude form. In the collisionless approximation it is an imaginary function which falls down with the growing frequency as 1/. Inthe currently available graphene samples with the mobility e 200000 cm2/Vs Orlita et al.(2008); Geim (2009) and the electron density ns 1012 cm2 the condition is satisfiedat /2 0.1 THz.

3.3 Inter-band conductivity

For the inter-band conductivity inter () which is dominant at high (infrared, optical)frequencies we get in the limit q 0

inter () =ie2hgs

S k,l =l

f (El k) f (Elk)El k Elk h(+ i)

lk|v|l kl k|v|lkEl k Elk

. (29)

Using the matrix elements (15), assuming that 0 and considering again only the vicinityof Dirac points one gets inter () =

inter(), where

inter() =ie2gsgv16h

0dk

sinh(hVk/T)cosh(/T) + cosh(hVk/T)

(

1k+/2V + i0

1k/2V i0

)

.

(30)The real part of the inter-band conductivity (30) is calculated analytically at all values of , and T,

Re inter() =e2gsgv16h

sinh(h||/2T)cosh(/T) + cosh(h||/2T) . (31)

At high frequencies h ||,T this gives the remarkable result of the universal opticalconductivity

opt() =e2gsgv16h

=e2

4h, h ||,T, (32)


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which depends only on the fundamental constants e and h, Ando et al. (2002); Kuzmenko et al.(2008); Nair et al. (2008). The imaginary part is determined by the principal value integral

Im inter() =e2gsgv16h

0dx

sinh xcosh(||/T) + cosh x

(

1x+ h/2T

1x h/2T

)

, (33)

which can be analytically calculated at low temperatures T ||. Together with the real partthis gives

inter() =e2gsgv16h

(

(h 2) + i

ln

1 h/2||1+ h/2||

)

, (34)

see Figure 4. At high frequencies the inter-band conductivity tends to a (universal) constantand exceeds the intra-band contribution if h ||. Finally, the total conductivity () =intra() + inter() in the collisionless limit has the form

() =e2gsgv16h

(

4i||h

+ (h 2||) + i

ln

1 h/2||1+ h/2||

)

(35)

and is shown in Figure 5.Summarizing the linear response results on the dynamic conductivity of graphene one seesthat at low frequencies h || the conductivity () is imaginary and is described bythe classical Drude formula. It corresponds to the intra-band response of the system. At highfrequencies h || the real part of the quantum inter-band conductivity dominates. Atthe typical charge carrier densities of ns 1011 1013 cm2 the transition between thetwo regimes h || lies in graphene at the frequencies /2 10 100 THz. Thelow-frequency limit h || thus corresponds to the radio, microwave and terahertzfrequencies, while the high-frequency limit to the infrared and optical frequencies. Thecollisions can be ignored, in the high quality samples, at /2 0.1 THz.

4. Frequency mixing: Quasi-classical theory

In this Section we consider the frequency mixing effect using the quasi-classical approachbased on the solution of the kinetic Boltzmann equation. The quasi-classical solution is simplerand, within the collisionless approximation, can be obtained non-perturbatively, at arbitraryvalues of the external electric field amplitudes Mikhailov (2007). The quasi-classical theory isvalid at h ||, which corresponds to the very broad and technologically important rangeof radio, microwave and terahertz frequencies.

4.1 Boltzmann kinetic equation

Consider the classical motion of massless particles (1) under the action of the external electricfield E(t). The evolution of the electron distribution function fp(t) is determined by theBoltzmann kinetic equation which has the form

fp

t eE(t) fp

p= 0 (36)

in the collisionless approximation. Its exact solution is

f (p, t) =[

1+ exp(

V|p p0(t)| T

)]1, (37)


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0 1 2 3 4

/

-2

-1.5

-1

-0.5

0

0.5

1

inte

r-b

an

d c

on

du

ctivity

/=0.01

0.03 0.05real

imaginary

Fig. 4. The inter-band conductivity of graphene at / = 0.01 and three differenttemperatures as shown in the Figure. The conductivity is measured in units e2gsgs/16h.

0 1 2 3 4

/

-2

0

2

4

6

tota

l co

nd

uctivity

/=0.01

0.03 0.05

real

imaginary

Fig. 5. The total conductivity of graphene at / = 0.01 and three different temperatures asshown in the Figure. The conductivity is measured in units e2gsgs/16h. At low frequenciesh || the conductivity is imaginary and corresponds to the intra-band classicalcontribution. At high frequencies h || the quantum inter-band contribution dominates.where p0(t) satisfies the classical equation of motion dp0(t)/dt = eE(t). The current canthen be calculated as

j(t) = e gsgvV(2h)2

dpxdpyp

p2x + p2y

1

1+ exp(

V|pp0(t)|T

) . (38)


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Assuming now that the temperature is low, T , and expanding the right-hand side of Eq.(38) we get

j(t)ensV

=P(t)

P2(t) + 1

[

1+38

P2(t)

[P2(t) + 1]2

]

, (39)

where

P(t) =p0(t)

pF. (40)

Notice that the expansion parameter in Eq. (39) is |P(t)/(P2(t) + 1)| 1, i.e. the result (39) isvalid both at small and large |P(t)|. In the regime of low electric fields |P(t)| 1 the currentis

j(t)ensV

P(t)(

1 18P2(t)

)

. (41)

4.2 Frequency mixing response

If the graphene layer is irradiated by two waves with the frequencies 1, 2 and the bothwaves are linearly polarized in the same direction,

E(t) = E1 cos1t+ E2 cos2t, E1 E2, (42)

then

p0(t) = e(

E11

sin1t+E22

sin2t)

, (43)

the current is parallel to the electric fields and equals

j(t)

ensV= 1

(

1 332

21 316

22

)

sin1t

+3132

sin 31t+321232

[

sin(21 +2)t sin(21 2)t]

+ . . . , (44)

where

j =eEj

pFj=

eEjV

||j, j = 1, 2. (45)

The omitted terms marked by the dots are obtained from the present ones by replacing 1 2 and 1 2. We will call j the field parameters. The field parameter = eEV/ is thework done by the electric field during one oscillation period (eEV/) divided by the averageenergy of electrons , Mikhailov (2007).If the field parameters are small, j 1, Eq. (44) describes the low-frequency linear response,since ensVj = intra(j)Ej, see Eq. (27). If j are not negligible, the first line in (44) representsthe second order corrections to the linear conductivity, the first term in the second line givesthe third harmonics generation effect, Mikhailov (2007), and the last terms in the secondline the frequency mixing. The amplitudes of the third-order mixed frequency current

j(3)(212)(t) = j

(3)(212) sin(2 21)t can be rewritten as

j(3)(212) =

332

ensV

(

eE1pF1

)2 eE2pF2

=332

intra(2)E221, (46)


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i.e., up to a numerical factor, the third order current is the product of the low-frequencyDrude conductivity (27), the electric field E2 and the squared field parameter 21. One seesthat the amplitude of the third order mixed-frequency current will be comparable with thelinear response current if the field parameter is of order unity (or larger). This means thatthe required electric field is determined by the inequality

E pF

e=

hnse

. (47)

The lower the charge carrier density ns and the radiation frequency the smaller is the electricfield needed for the observation of the nonlinear effects. If, for example, the density is 1011cm2 and the frequency is 0.5 THz, Eq. (47) gives E 1 kV/cm which corresponds to theincident wave power 2 kW/cm2.If the linear polarizations of the two incident waves1 and 2 are perpendicular to each other,E1 E2 = 0, the current at the mixed frequencies 21 2 is three times smaller than for theparallel polarization. This is a general result which is also valid in the quantum regime.

5. Frequency mixing: Quantum theory

The full quantum theory of the nonlinear frequencymixing effects in graphene is substantiallymore complicated and is yet to be developed. In this paper we only consider the frequencymixing response at the frequency e 21 2 and calculate it under the conditions

h1, h2, he ||, (48)

relevant for the experiment of Hendry et al. (2010).

5.1 Quantum kinetic (Liouville) equation

In order to investigate the nonlinear response problem in the quantum regime (48) we haveto solve the quantum kinetic equation (16) in, at least, the third order in the external fieldamplitudes Ej, j = 1, 2. We do this using the perturbation theory. Expanding the densitymatrix in powers of the electric fields, = 0 + 1 + 2 + 3 + . . . , we get from (16) a set ofrecurrent equations

ihnt

= [H0, n] + [H1, n1], n = 2, 3, . . . (49)

for n. At high frequencies (48) we can write the Hamiltonian H1 in the form

H1 = (eE1x cos1t+ eE2x cos2t) et = h1 ei(1i0)t + h2 e

i(2i0)t + {j j}, (50)

where h1 = h1 = eE1x/2, h2 = h2 = eE2x/2 and it is assumed that 0. In the firstorder in Ej we get

|1| = ={1,2}

f fE E h+ i0

|h |eit {1,2}

1eit, (51)

where we have used a short notation | for the set of three quantum numbers |lk. The righthand side of Eq. (51) contains the oscillating exponents with the frequencies1 and 2.For the matrix elements of 2 we obtain, similarly,

|2| = a,b

|[ha , 1b ]|E E ha hb + i0

ei(a+b)t. (52)


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The summation here is performed over the sets of frequencies a = {1,2} andb = {1,2}, i.e. the right hand side in (52) contains the oscillating terms with thefrequencies 21, 22, (1 +2), (1 2) and a time independent term with = 0.We will denote this set of frequencies as c = {21,22,(1+2),(12), 0}. Thecommutator [ha , 1b ] is calculated as

|[ha , 1b ]| =

(

|ha ||1b | |1b ||ha |)

,

where the matrix elements |h | and |1 | are known from (14) and (51).Then, for the matrix elements of 3 we get

|3| = a,c

|[ha , 2c ]|E E ha hc + i0

ei(a+c)t. (53)

Now the right hand side of Eq. (53) contains the terms with the frequencies1, 2, 31,32, (21 2) and (22 1).The general formulas (51), (52) and (53) allows one, in principle, to calculate the timedependence of the matrix elements of the density matrix in the third order in Ej. Then, usingthe formula

j(t) = e

S

|v||1 + 2 + 3 + . . . | (54)

one can find the time dependence of the current in the same order of the perturbation theory.This general expression for the current will contain the termswith the frequencies1, 2, 31,32, as well as (21 2) and (22 1).

5.2 Optical frequency mixing at e = 21 2Being interested in this work only in the response at the frequency of the emitted light e 21 2 (see Hendry et al. (2010)) we get, after quite lengthy calculations,

j(3)(e)

(t) =e4gs

8h2E21E2

21 21(1 2)2

eiet1S

k

( f2k f1k)1k|vx|2k1k|x|2k3

(

2E2k E1k + h1 i0

+2

E2k E1k h1 + i0

+1

E2k E1k + h2 i0 1

E2k E1k h2 + i0

1E2k E1k he + i0

+1

E2k E1k + he i0

)

+ (j j). (55)

The current j(3)(e)

(t) contains the resonant terms corresponding to the vertical opticaltransitions at E2k E1k = h1, h2 and he. As it follows from the linear response theory(Section 3), the largest contribution to the current at the optical frequencies h || is givenby the absorption terms proportional to (E2k E1k h). The same is valid in the nonlinear


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regime too. Taking into account in (55) only the terms (E2k E1k h1,2,e) we finally get

j(3)(e)

(t) 3e4gsgvV

2

256h3E21E2

21 21(1 2)2

[

221

122

1(21 2)2

]

cos(21 2)t

= 98opt(2)E2

21F

(

21

)

cos(21 2)t, (56)

where1 =

eE1V

h21(57)

is the optical field parameter and

F(x) =2+ 2x x23x2(2 x) . (58)

Eq. (56) is the main result of this work. It represents the ac electric current induced in thegraphene layer at the frequency e = 21 2 by the two incident waves (42) polarized inthe same direction (the current direction coincides with that of the fields). If the two incidentwaves are perpendicularly polarized, the numerical prefactor in (56) is reduced (9/8 3/8).As seen from (56), at high frequencies the current depends neither on the chemical potential nor on the temperature T as it should be under the conditions when the vertical inter-bandtransitions play the main role. The second line of (56) is written in the form similar to (46):at the optical frequencies the current is the product of the high-frequency, universal opticalconductivity (32), the electric field E2 and the squared field parameter 21. The optical fieldparameter = eEV/h2 is the work done by the electric field during one oscillation period(eEV/) divided by the photon energy h (instead of the Fermi energy at low frequencies,Eq. (45)). In addition to the mentioned parameters, the current (56) weakly depends on theratio of the two optical frequencies 2/1 which is described by the function (58) shown inFigure 6. The function F(x) is of order unity if the difference 2 1 is not very large. If 1,2 or e tend to zero the current (56) has a strong tendency to grow. These limits are veryinteresting for future studies but have been excluded from the current consideration by theconditions (48).The formula (56) is in good quantitative agreement with the experimental results of Hendryet al. (2010). The comparison with other materials made in that paper showed that graphenehas much stronger nonlinear properties than typical nonlinear insulators and some metals(Au). Comparing the experimental results of Hendry et al. (2010) with those of Erokhinet al. (1987) shows that graphene is also a stronger nonlinear material than a typicalnonlinear semiconductor InSb. Further theoretical and experimental studies of the nonlinearelectrodynamic and optical properties of graphene are therefore highly desirable.

6. Summary and conclusions

Due to the massless energy spectrum of the charge carriers graphene demonstrates stronglynonlinear electromagnetic properties. In this work we have developed a theory of thenonlinear frequency mixing effect in graphene. The two physically different regimes havebeen considered. At low frequencies, corresponding to the radio, microwave and terahertzrange, the problem is solved within the quasi-classical approach which takes into account theintra-band response of the material. At high frequencies, corresponding to the infrared and


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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8x

0

2

4

6

8

F(x

)

Fig. 6. The function F(x) from Eq. (58).

visible light, a quantum theory is developed which takes into account the inter-band opticaltransitions.At the optical frequencies our results quantitatively agree with the recent experimentalfindings of Hendry et al. (2010) who have observed the nonlinear electromagnetic responseof graphene for the first time. The results of Hendry et al. (2010) show that in the visibleand near-infrared frequency range the nonlinear parameters of graphene are much strongerthan in many other nonlinear materials. Even more important conclusion is that, according toour theoretical predictions, the nonlinear response of graphene substantially grows at lowerfrequencies. One should expect therefore even stronger nonlinear properties of grapheneat the mid-infrared, terahertz and microwave frequencies which would be of extremeimportance for the future progress of the nonlinear terahertz- and optoelectronics.

The author thanks the Deutsche Forschungsgemeinschaft for support of this work.

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Physics and Applications of Graphene - TheoryEdited by Dr. Sergey Mikhailov

ISBN 978-953-307-152-7Hard cover, 534 pagesPublisher InTechPublished online 22, March, 2011Published in print edition March, 2011

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The Stone Age, the Bronze Age, the Iron Age... Every global epoch in the history of the mankind ischaracterized by materials used in it. In 2004 a new era in material science was opened: the era of grapheneor, more generally, of two-dimensional materials. Graphene is the strongest and the most stretchable knownmaterial, it has the record thermal conductivity and the very high mobility of charge carriers. It demonstratesmany interesting fundamental physical effects and promises a lot of applications, among which are conductiveink, terahertz transistors, ultrafast photodetectors and bendable touch screens. In 2010 Andre Geim andKonstantin Novoselov were awarded the Nobel Prize in Physics "for groundbreaking experiments regarding thetwo-dimensional material graphene". The two volumes Physics and Applications of Graphene - Experimentsand Physics and Applications of Graphene - Theory contain a collection of research articles reporting ondifferent aspects of experimental and theoretical studies of this new material.

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Sergey Mikhailov (2011). Frequency Mixing Effects in Graphene, Physics and Applications of Graphene -Theory, Dr. Sergey Mikhailov (Ed.), ISBN: 978-953-307-152-7, InTech, Available from:http://www.intechopen.com/books/physics-and-applications-of-graphene-theory/frequency-mixing-effects-in-graphene

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