Non-linear electromagnetic response of graphene
Non-linear electromagnetic response of graphene
S. A. Mikhailov
Institute for Theoretical Physics II, University of Augsburg, D-86135 Augsburg, Germany
(Dated: October 22, 2018)
It is shown that the massless energy spectrum of electrons and holes in graphene leads to the
strongly non-linear electromagnetic response of this system. We predict that the graphene layer,
irradiated by electromagnetic waves, emits radiation at higher frequency harmonics and can work
as a frequency multiplier. The operating frequency of the graphene frequency multiplier can lie in
a broad range from microwaves to the infrared.
In the past two years a great deal of attention has been attracted by a recently discovered, new two-dimensional (2D)
electronic system – graphene, built out of a single monolayer of carbon atoms with a honeycomb 2D crystal structure
[1, 2]. The band structure of the charge carriers in this system consists of six Dirac cones at the corners of the hexagon-
shaped Brillouin zone [3, 4], with the massless, linear electron/hole dispersion. The massless electron spectrum leads
to unusual transport and electrodynamic properties, which have been intensively studied in the literature, see e.g.
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] and for review [32, 33].
Electrodynamic properties of graphene have been theoretically studied in Refs. [16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31]. The frequency dependent conductivity[16, 17, 20, 21, 22], as well as plasmon [23, 25, 27, 29, 30],
plasmon-polariton [24], and transverse electromagnetic wave spectra [31] have been investigated. In all these papers
electrodynamic response of the system has been studied within the linear response theory (for instance, using the
Kubo formalism, or the random phase approximation, or the self-consistent-field approach). In this Letter we show
that, apart from all the fascinating and non-trivial properties of graphene predicted and observed so far, this material
should also demonstrate strongly non-linear electrodynamic behavior. In particular, irradiation of the graphene sheet
by a harmonic electromagnetic wave with the frequency Ω should lead to the emission of the higher harmonics with
the frequencies mΩ, m = 3, 5, . . ., from the system. The operating frequency of such a frequency multiplier can vary
from microwaves up to infrared, and the required ac electric field is rather low, especially at low carrier densities
and low temperatures. The predicted non-linear electrodynamic properties of graphene may open up new exciting
opportunities for building electronic and optoelectronic devices based on this material.
To qualitatively demonstrate the non-linear behavior of graphene electrons consider a classical 2D particle with
the charge −e and the energy spectrum ǫp = V p = V
p2x + p
y in the external electric field Ex(t) = E0 cosΩt. Here
V is the velocity of 2D electrons in the energy band (in graphene V ≈ 108 cm/s [1, 2]). According to the classical
equations of motion dpx/dt = −eEx(t) the momentum px will then be equal to px(t) = −(eE0/Ω) sinΩt, and the
velocity vx = ∂ǫp/∂px is then vx(t) = −V sgn(sinΩt). If there are ns particles per unit area, the corresponding ac
electric current
jx(t) = ensV sgn(sinΩt) = ensV
sinΩt+
sin 3Ωt+
sin 5Ωt+ . . .
contains all odd Fourier harmonics, with the amplitudes jm, m = 1, 3, 5 . . ., falling down very slowly with the harmonics
number, jm ∼ 1/m. Notice that at the density ns = 6 · 1012 cm−2 and at V ≃ 108 cm/s (parameters of Refs. [1, 2])
the current amplitude j0 = ensV in our simple estimate gives a giant value of j0 ≃ 100 A/cm.
The above consideration does not take into account the Fermi distribution of charge carriers over the quantum states
in the conduction and valence bands of graphene. To get a more accurate description of the non-linear phenomena
in the considered system we use the kinetic Boltzmann theory, which allows one to get an exact response of the
system not imposing any restrictions on the amplitude of the external ac electric field E(t). Using this quasi-classical
approach we take into account the intra-band contribution to the ac electric current. The inter-band contribution to
the electric current, due to the transitions between the hole and the electron bands, is ignored. This imposes certain
restrictions on the frequency of radiation Ω, which will be discussed below.
Consider a 2D electron/hole gas with the energy spectrum ǫp± = ±V
p2x + p
y under the action of the field
E = (Ex, 0), where the sign + (−) corresponds to the electron (hole) band, Ex(t) = E0eαt cos(Ωt), and α → +0
describes an adiabatic switching on of the electric field. Assume that the Fermi energy ǫF lies in the electron (or
the hole) band and that the temperature is small as compared to ǫF , T ≪ ǫF . The momentum distribution function
of electrons fp+(t) ≡ fp(t) (we omit the sign + for brevity) in the collisionless approximation is described by the
http://arxiv.org/abs/0704.1909v1
Boltzmann equation
∂fp(t)
− ∂fp(t)
αt cos(Ωt) = 0, (2)
which has the exact solution
fp(t) = F0 (px − p0(t), py) , (3)
where
F0(px, py) =
1 + exp
p2x + p
y − ǫF
is the electronic Fermi function, and p0(t) = −(eE0/Ω)eαt sinΩt is the solution of the single particle equation of
motion. The electric current j(t) = −egsgvS−1
vfp(t) then assumes the form
jx(t) = −
gsgveV
(2πh̄)2
dpxdpy
p2x + p
F0 (px − p0(t), py) , (5)
jy = 0, where gs = gv = 2 are the spin and valley degeneracies in graphene, and S is the sample area. After some
lengthy but simple transformation, Eq. (5) can be rewritten as
jx(t) = ensV
2QF (t)
1 +Q2F (t)
∫ π/2
cos2 xdx
2QF (t)
cosx+
1− 2QF (t)
where
ns ≡ ne =
gsgvp
4πh̄2
gsgvǫ
4πh̄2V 2
is the density of electrons, pF = ǫF /V is the Fermi momentum, and
QF (t) = −
p0(t)
sin(Ωt) ≡ QF0 sin(Ωt) (8)
is the field parameter, proportional to the ac electric field E0.
Figure 1a shows the current (6) as a function of time Ωt. One sees that in the low-field limit the response is linear.
Expanding the current (6), we get at QF0 ≪ 1
jx(t) ≈ ensV QF0
sinΩt+
Q2F0 sin 3Ωt
, (9)
so that the linear response conductivity (in the collisionless approximation) is
σǫF ,T=0(Ω) =
. (10)
The expression (10) coincides with the intra-band Drude conductivity, which can be obtained from the linear-response
theory [16, 17, 20, 21, 22, 31]. As the inter-band conductivity is of order of e2/h̄ [16, 17, 20, 21, 22, 31], our quasi-
classical approach is valid at h̄Ω <∼ ǫF . At the electron density ≃ 1011 − 1012 cm−2 this restricts the frequency by the
value of 10− 30 THz.
In the strong-field limit QF0 >∼ 1 Eq. (6) results in the formula (1). From the condition QF0 >∼ 1, rewritten as
E0 >∼
, (11)
−4 −2 0 2 4
−4 −2 0 2 4
FIG. 1: (Color online) The time dependence of the ac electric current, measured in units ensV , at harmonic excitation of the
system at the frequency Ω. (a) The temperature is zero, T/ǫF = 0; the curves are labeled by the values of the electric field
parameter QF0 = eE0V/ΩǫF . (b) The temperature is finite, the Fermi energy is zero, ǫF = 0; the curves are labeled by the
values of the parameter QT0 = eE0V/ΩT .
one sees that the required ac electric field grows linearly with the electromagnetic wave frequency and is proportional
to the square root of the electron density. At f ≃ 50 GHz and ns ≃ 1011 cm−2, the inequality (11) is fulfilled at
E0 >∼ 100 V/cm. This value can be reduced in systems with lower electron/hole density. Therefore, we consider now
an opposite limiting case with ǫF = 0, but finite temperature T .
At finite T and the vanishing ǫF = 0 both electrons and holes contribute to the charge carrier density
ns = ne + nh =
πgsgvT
12h̄2V 2
and to the current. Starting again from Eq. (5) but accounting for the hole contribution and putting ǫF = 0, we get
jx(t) = ensV
cos θ
1 + exp
x2 +Q2T (t)− 2xQT (t) cos θ
) , (13)
where
QT (t) = −
V p0(t)
sinΩt ≡ QT0 sinΩt. (14)
Figure 1b shows the current (13) as a function of time Ωt. In the low-field limit QT0 ≪ 1 we get from (13) the
current
jx(t) ≈ ensV QT (t)
6 ln 2
, (15)
and the correct expression for the linear-response intra-band dynamic conductivity [17],
σǫF=0,T (Ω) =
6 ln 2
gsgvT
. (16)
One sees that the quasi-classical approach is now valid at h̄Ω <∼ T . This restricts the frequency by the value of ≃ 200
GHz at T ∼ 10 K and ≃ 6 THz at room temperature. In the strong field regime QT0 >∼ 1 Eq. (13) is reduced, again,
to (1). Figure 2 shows the Fourier components of the ac electric current, for m = 1, 3 and 5, as a function of the field
parameter eE0V/ΩT at ǫF = 0. The strong-field condition now assumes the form
E0 >∼
. (17)
0 2 4 6 8 10
eE0V/ΩT
FIG. 2: (Color online) The Fourier components of the current (13), in arbitrary units, as a function of QT0 = eE0V/ΩT at
ǫF /T = 0.
At T ≃ 10 K and f ≃ 100 GHz this gives a moderate value of the required electric field E0 ≃ 5 V/cm. The efficiency
of the predicted frequency multiplication effect can be increased further by using the resonance response of the system
at the plasmon, the cyclotron, or the magnetoplasmon frequency.
To summarize, we have investigated the non-linear electrodynamic response of 2D electrons and holes in graphene.
We have shown that irradiation of graphene by the electromagnetic wave with the frequency Ω should lead to the
higher harmonics generation at frequencies 3Ω, 5Ω, et cetera. The efficiency of the frequency up-conversion is rather
high: the amplitudes of the higher harmonics of the ac electric current fall down slowly (as 1/m) with the harmonics
index m. The presented quasi-classical theory is valid at h̄Ω <∼ max{ǫF , T }. This estimate shows that the effect
works at frequencies up to 5− 10 THz, which opens up exciting opportunities for building new graphene devices for
terahertz and sub-terahertz electronics.
I wish to thank Klaus Ziegler for stimulating discussion. The work was partly supported by the Swedish Research
Council and INTAS.
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