Plasmon Amplification through Stimulated Emission at Terahertz
  Frequencies in Graphene
Plasmon Amplification through Stimulated Emission at Terahertz
Frequencies in Graphene
Farhan Rana1, Faisal R. Ahmad2
1School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853
2Department of Physics, Cornell University, Ithaca, NY 14853
Abstract
We show that plasmons in two-dimensional graphene can have net gain at terahertz
frequencies. The coupling of the plasmons to interband electron-hole transitions in population
inverted graphene layers can lead to plasmon amplification through the process of stimulated
emission. We calculate plasmon gain for different electron-hole densities and temperatures
and show that the gain values can exceed 104 cm−1 in the 1-10 terahertz frequency range,
for electron-hole densities in the 109-1011 cm−2 range, even when plasmon energy loss due
to intraband scattering is considered. Plasmons are found to exhibit net gain for intraband
scattering times shorter than 100 fs. Such high gain values could allow extremely compact
terahertz amplifiers and oscillators that have dimensions in the 1-10 µm range.
http://arxiv.org/abs/0704.0607v2
conduction
valence
plasmon plasmon
plasmons
Figure 1: (LEFT) Energy bands of graphene showing stimulated absorption of plasmons.
(RIGHT) Population inverted graphene bands showing stimulated emission of plasmons.
1 Introduction
Tremendous interest has been generated recently in the electronic properties of two dimensional
(2D) graphene in both experimental and theoretical arenas [1, 2, 3, 5, 6, 7]. Graphene is a single
atomic layer of carbon atoms forming a dense honeycomb crystal lattice [8]. The massless energy
dispersion relation of electrons and holes with zero (or close to zero) bandgap results in novel
behavior of both single-particle and collective excitations [1, 2, 3]. In addition, the high mobility
of electrons in graphene has generated interest in developing novel high speed devices. Recently,
it has been shown that the frequencies of plasma waves in graphene at moderate carrier densities
( 109−1011 cm−2) are in the terahertz range [3]. Electron-hole decay through plasmon emission
has been recently experimentally observed in graphene [4]. The zero bandgap of graphene leads
to strong damping of the plasma waves (plasmons) at finite temperatures as plasmons can
decay by exciting interband electron-hole pairs [1, 2]. In this paper we show that plasmon
amplification through stimulated emission is possible in population inverted graphene layers.
This process is depicted in Fig.1. We show that plasmons in graphene can have a net gain
at frequencies in the 1-10 THz range even if plasmon losses from electron and hole intraband
scattering are considered. A net gain for the plasmons implies that terahertz amplifiers and
oscillators based on plasmon amplification through stimulated emission are possible. The gain
at terahertz frequencies is possible due to the (almost) zero bandgap of graphene. Although
terahertz gain is also achievable in population inverted subbands in 2D quantum wells [9],
intrasubband plasmons in quantum wells, being longitudinal collective modes, do not couple
with intersubband transitions that require field polarization perpendicular to the plane of the
quantum wells. The electromagnetic energy in the two-dimensional plasmon mode is confined
within very small distances of the graphene layer and therefore waveguiding structures with large
dimensions, such as those required in terahertz quantum cascade lasers [9], are not required
for realizing plasmon based terahertz devices. We also present results for plasmon gain under
different population inversion conditions taking into account both intraband and interband
electronic transitions and carrier scattering.
2 Theoretical Model
In this section we discuss the theoretical model used to obtain the values for the plasmon gain
in graphene. In graphene, the valence and conduction bands resulting from the mixing of the
pz-orbitals are degenerate at the inequivalent K and K
′ points of the Brillouin zone [8]. Near
these points, the conduction and valence band dispersion relations can be written compactly
as [2],
Es,k = sh̄v|k| (1)
where s = ±1 stand for conduction (+1) and valence (−1) bands, respectively, and v is the
“light” velocity of the massless electrons and holes. The wavevector k is measured from the
K(K ′) point. The frequencies ω(q) of the longitudinal plasmon modes of wavevector q are given
by the equation,ǫ(q, ω) = 0, where ǫ(q, ω) is the longitudinal dielectric function of graphene [2].
In the random phase approximation (RPA) ǫ(q, ω) can be written as [10],
ǫ(q, ω) = 1− V (q)Π(q, ω) (2)
Here, V (q) is the bare 2D Coulomb interaction and equals e2/2ǫ∞q. ǫ∞ is the average of
the dielectric constant of the media on either side of the graphene layer. Π(q, ω) is the
electron-hole propagator including both intraband and interband processes and is given by
the expression [1, 2],
Π(q, ω) = 4
s s′ k
| < ψs′,k+q|e
iq.r|ψs,k > |
f(Es,k − Efs)− f(Es′,k+q − Efs′)
h̄ω + Es,k − Es′,k+q + iη
The factor of 4 outside in the above equation comes from the degenerate two spins and the
two valleys at K and K ′. f(E − Ef ) is the Fermi distribution function with Fermi energy Ef .
|ψs,k) > are the Bloch functions for the conduction and valence bands near the K(K
′) point.
The occupancy of electrons in the conduction and valence bands are described by different
Fermi levels to allow for nonequilibrium population inversion. The Bloch functions have the
following matrix elements [8],
| < ψs′,k+q|e
iq.r|ψs,k > |
1 + ss′
|k|+ |q| cos (θ)
|k+ q|
where θ is the angle between the vectors k and q. The condition v|q| < ω(q) must be satisfied
in order to avoid direct intraband absorption of plasmons. Assuming v|q| < ω, and using the
symmetry between conduction and valence bands, the intraband and interband contributions
to the propagator can be approximated as follows,
Πintra(q, ω) ≈
q2K T /πh̄2
ω(ω + i/τ)− v2q2/2
eEf+/KT + 1
e−Ef−/KT + 1
Πinter(q, ω) ≈
[f(h̄ω/2− Ef+)− f(−h̄ω/2− Ef−)]
ω2 − ω2
[f(h̄ω/2− Ef+)− f(−h̄ω/2− Ef−)] (6)
Here, q = |q|. In Equation (5), the intraband contribution to the propagator is written in the
plasmon-pole approximation that satisfies the f-sum rule [10]. This approximation is not valid
for large value of the wavevector q when ω(q) → vq. However, in this paper we will be concerned
with small values of the wavevector for which the plasmons have net gain, and therefore
the approximation used in Equation (5) is adequate. Plasmon energy loss due to intraband
scattering has been included with a scattering time τ in the number-conserving relaxation-time
approximation which assumes that as a result of scattering the carrier distribution relaxes
to the local equilibrium distribution [11]. The real part of the interband contribution to the
propagator modifies the effective dielectric constant and leads to a significant reduction in the
plasmon frequency under population inversion conditions. The imaginary part of the interband
contribution to the propagator incorporates plasmon loss or gain due to stimulated interband
transitions. A necessary condition for plasmon gain from stimulated interband transitions is
that the splitting of the Fermi levels of the conduction and valence electrons exceed the plasmon
energy, i.e. Ef+−Ef− > h̄ω. But the plasmons will gave net gain only if the plasmon gain from
stimulated interband transitions exceed the plasmon loss due to intraband scattering. The real
and imaginary parts of the propagator in Equations (5) and (6) satisfy the Kramers-Kronig
relations. Equations (5) and (6) can be used with Equation (2) to calculate the real and
imaginary parts of the plasmon frequency ω(q) as a function of q. However, from the point
of view of device design, it is more useful to assume that the frequency ω is real and the
propagation vector q(ω), written as a function of ω, is complex. Since the charge density wave
corresponding to plasmons has the form eiq.r−iωt, the imaginary part of the propagation vector
corresponds to net gain or loss. We define the net plasmon energy gain g(ω) as −2Imag{q(ω)}.
3 Results and Discussion
In simulations we use v = 108 cm/s and ǫ∞ = 4.0ǫo (assuming silicon-dioxide on both sides
of the graphene layer) [1]. We assume a nonequilibrium situation, as in a semiconductor
interband laser [12], in which the electron and hole densities are equal and Ef+ = −Ef−.
Such a non-equilibrium situation can be realized experimentally by either carrier injection in
an electrostatically defined graphene pn-junction or through optical pumping [13, 14]. The
value of the scattering time τ (momentum relaxation time) is also critical for calculations of
the net plasmon gain. Value of τ can be estimated from the experimentally reported values
of mobility using the following expression for the graphene conductivity (assuming that only
electrons are present) [17],
e2 τ K T
eEf+/KT + 1
Values of mobility between 20,000 and 60,000 cm2/V-s have been experimentally measured at
low temperatures (T¡77K) in graphene [6, 7, 15]. Assuming a mobility value of 27,000 cm2/V-s
, reported in Ref. [15] for an electron density of 3.4×1012 cm−2 at T=58K, the value of τ comes
out to be approximately 0.6 ps. The phonon scattering time was experimentally determined to
be close to 4 ps at T=300K [15]. Therefore, impurity or defect scattering is expected to be the
0 0.5 1 1.5 2 2.5 3
Wavevector (105 cm−1)
T = 10K
n=p=1, 3.5, 6, 8.5 × 109 cm−2 
increasing
density 
 ω =  v q 
Figure 2: Calculated plasmon dispersion relation in graphene at 10K is plotted for different
electron-hole densities (n = p = 1, 3.5, 6, 8.5×109 cm−2). The condition ω(q) > h̄vq is satisfied
for frequencies that have net gain in the terahertz range. The assumed values of v and τ are
108 cm/s and 0.5 ps, respectively.
dominant momentum relaxation mechanism in graphene, and the scattering time is expected
to be relatively independent of temperature [17]. In the results presented below, unless stated
otherwise, we have used a temperature independent scattering time of 0.5 ps.
Figs. 2-7 show the calculated dispersion relation of the plasmons and the net plasmon gain
at T=10K, 77K, and 300K for different electron-hole densities. At very low frequencies the
losses from intraband scattering dominate. At frequencies ranging from 1 to 15 THz, the
plasmons can have net gain. The values of the net gain are found to be significantly large
reaching 1−4×104 cm−1 for electron-hole densities in the 109 cm−2 range at low temperatures
and 1011 cm−2 range at room temperature. The calculated plasmon dispersions indicate that
ω(q) > vq at all frequencies for which the plasmons have net gain. Therefore, direct intraband
absorption of plasmons is not possible at these frequencies and will not reduce the calculated
gain values. Plasmons acquire net gain for smaller electron-hole densities at lower temperatures.
At higher temperatures the distribution of electrons and holes in energy is broader and the
0 1 2 3 4 5 6
Frequency (THz)
T = 10K
n=p=1, 3.5, 6, 8.5 × 109 cm−2 increasing
density 
Figure 3: Net plasmon gain in graphene at 10K is plotted for different electron-hole densities
(n = p = 1, 3.5, 6, 8.5 × 109 cm−2). The assumed values of v and τ are 108 cm/s and 0.5 ps,
respectively.
gain at any particular frequency is therefore smaller. At T=10K, the plasmons have net gain
for electron-hole densities as small as 2 × 109 cm−2. Almost an order of magnitude larger
electron-hole densities are required to achieve the same net gain values at T=77K compared
to T=10K. The linear energy dependence of the density of states associated with the massless
dispersion relation of electrons and holes in graphene results in the maximum plasmon gain
values to increase with the electron-hole density. The peak gain values shift to higher frequencies
with the increase in the electron-hole density for the same reason.
The fact that plasmons can acquire net gain for relatively small carrier densities suggests
that plasmon gain is relatively robust with respect to intraband scattering losses. Fig. 8 shows
the net gain at T=10K for n = p = 1010 cm−2 and values of the intraband scattering time
τ varying from 0.1 to 0.5 ps. The net gain decreases as the plasmon losses increase with a
decrease in the value of τ and the maximum gain value equals zero for τ = 0.15 ps. However,
it should not be concluded from Fig. 8 that plasmons cannot have net gain for τ less than 0.15
ps since electron-hole density can always be increased to achieve net gain for smaller values of
0 2 4 6 8
Wavevector (105 cm−1)
increasing
density 
 ω =  v q 
T = 77K
n=p=1, 2, 3, 4 × 1010 cm−2 
Figure 4: Calculated plasmon dispersion relation in graphene at 77K is plotted for different
electron-hole densities (n = p = 1, 2, 3, 4 × 1010 cm−2). The condition ω(q) > h̄vq is satisfied
for frequencies that have net gain in the terahertz range. The assumed values of v and τ are
108 cm/s and 0.5 ps, respectively.
0 2 4 6 8 10
Frequency (THz)
T = 77K
n=p=1, 2, 3, 4 × 1010 cm−2 
increasing
density 
Figure 5: Net plasmon gain in graphene at 77K is plotted for different electron-hole densities
(n = p = 1, 2, 3, 4 × 1010 cm−2). The assumed values of v and τ are 108 cm/s and 0.5 ps,
respectively.
0 5 10 15 20
Wavevector (105 cm−1)
T = 300K
n=p=1, 1.5, 2, 2.5 × 1011 cm−2 
increasing
density 
 ω =  v q 
Figure 6: Calculated plasmon dispersion relation in graphene at 300K is plotted for different
electron-hole densities (n = p = 1, 1.5, 2, 2.5×1011 cm−2). The condition ω(q) > h̄vq is satisfied
for frequencies that have net gain in the terahertz range. The assumed values of v and τ are
108 cm/s and 0.5 ps, respectively.
0 5 10 15 20
Frequency (THz)
T = 300K
n=p=1, 1.5, 2, 2.5 × 1011 cm−2 
increasing
density 
Figure 7: Net plasmon gain in graphene at 300K is plotted for different electron-hole densities
(n = p = 1, 1.5, 2, 2.5 × 1011 cm−2). The assumed values of v and τ are 108 cm/s and 0.5 ps,
respectively.
0 2 4 6 8
Frequency (THz)
T = 10K
n = p = 1010 cm−2 
τ = 0.5, 0.4, 0.3, 0.2, 0.15, 0.1 ps
increasing τ 
Figure 8: Net plasmon gain in graphene at 10K is plotted for different intraband scattering times
τ (τ = 0.5, 0.4, 0.3, 0.2, 0.15, 0.1 ps). The assumed value of v is 108 cm/s and the electron-hole
density is 1010 cm−2.
0 2 4 6 8 10 12
Frequency (THz)
increasing τ 
T = 10K
n = p = 3 × 1010 cm−2 
τ = 150, 125, 100, 75 fs
Figure 9: Net plasmon gain in graphene at 10K is plotted for different scattering times τ
(τ = 150, 125, 100, 75 fs). The assumed value of v is 108 cm/s and the electron-hole density is
3× 1010 cm−2.
τ . Fig. 9 shows the net gain at T=10K for n = p = 3× 1010 cm−2 and values of the intraband
scattering time τ varying from 75 to 150 fs. It can be seen that at these larger carrier densities
plasmons have net gain for scattering times that are sub-100 fs.
The exceedingly large values of the net plasmon gain (> 104 cm−1) in graphene implies that
terahertz plasmon oscillators only a few microns long in length could have sufficient gain to
overcome both intrinsic losses and losses associated with external radiation coupling. Plasmon
fields with in-plane wavevector magnitude q decay as e−q |z| away from the graphene layer where
|z| is the distance from the graphene layer. Figs. 2, 4, and 6 show that q has values exceeding
105 cm−1 at terahertz frequencies. Therefore, the electromagnetic energy associated with the
terahertz plasmons is confined within 100 nm of the graphene layer. Strong field confinement
and low plasmon losses at terahertz frequencies are both partly responsible for the high net
gain values in graphene. Recent theoretical predictions for electron-hole recombination rates
in graphene due to Auger scattering indicate that electron-hole recombination times can be
much longer than 1 ps at temperatures ranging from 10K to 300K for electron-hole densities
smaller than 1012 cm−2 [16]. This suggests that population inversion can be experimentally
achieved in graphene via current injection in electrostatically defined pn-junctions or via
optical pumping [13, 14]. It also needs to be pointed out here that graphene monolayers
and multilayers produced from currently available experimental techniques are estimated to
have defect/impurity densities anywhere between 1011 and 1012 cm−2 [17]. Therefore, at low
electron-hole densities (less than 1011 cm−2) graphene is expected to exhibit localized electron
and hole puddles rather than continuous electron and/or hole sheet charge densities [17].
This implies that with the currently available techniques graphene based terahertz plasmon
oscillators might only be realizable with higher electron-hole densities (> 1011 cm−2) for
operation at higher frequencies (> 5 THz).
4 conclusion
In conclusion, we have shown that high gain values for plasmons are possible in population
inverted graphene layers in the 1-10 THz frequency range. The plasmon gain remains positive
even for carrier intraband scattering times shorter than 100 fs. The high gain values and
the strong plasmon field confinement near the graphene layer could enable compact terahertz
amplifiers and oscillators. The authors would like to thank Edwin Kan and Sandip Tiwari for
helpful discussions.
References
[1] X. F. Wang, T. Chakraborty, Phys. Rev. B, 75, 033408 (2007).
[2] E. H. Hwang, S. D. Darma, cond-mat/0610561.
[3] V. Ryzhii, A. Satou, J. Appl. Phys., 101, 024509 (2007).
[4] A. Bostwick, T. Ohta, T. Seyller, Karsten Horn AND E. Rotenberg, Nature, 3, 36, (2007).
[5] K. S. Novoselov et. al., Nature, 438, 197 (2005).
[6] K. S. Novoselov et. al., Science, 306, 666 (2004).
[7] Y. Zhang et. al., Nature, 438, 201 (2005).
[8] R. Saito, G. Dresselhaus, M. S. Dresselhaus, Physical Properties of Carbon Nanotubes,
Imperial College Press, London, UK (1999).
[9] B. Williams, H. Callebaut, S. Kumar, Q. Hu, Appl. Phys. Letts., 82, 1015 (2003).
[10] H. Huag, S. W. Koch, Quantum Theory of the Optical and Electronic Properties of
Semiconductors, World Sientific, NJ (1994).
[11] N. D. Mermin, Phys. Rev. B, 1, 2362 (1969).
[12] L. A. Coldren, S. W. Corzine, Diode Lasers and Photonics Integrated Circuits, Wiley, NY
(1995).
[13] J. R. Williams, L. DiCarlo, C. M. Marcus, cond-mat/0704.3487 (2007).
[14] B. zyilmaz, P. Jarillo-Herrero, D. Efetov, D. A. Abanin, L. S. Levitov, Philip Kim,
cond-mat/0705.3044.
http://arxiv.org/abs/cond-mat/0610561
[15] W. De Heer et. al., Science, 312, 1191 (2006).
[16] F. Rana, cond-mat/0705.1204.
[17] E. H. Hwang, S. Adam, S. Das Sarma, Phys. Rev. Letts., 98, 186806 (2007).
	Introduction
	Theoretical Model
	Results and Discussion
	conclusion
