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7
Plasmon Amplication through Stimulated Emission at Terahertz
Frequencies in Graphene
Farhan Rana
1
, Faisal R. Ahmad
2
1
School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853
2
Department 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 amplication through the process of stimulated
emission. We calculate plasmon gain for dierent electron-hole densities and temperatures
and show that the gain values can exceed 10
4
cm
1
in the 1-10 terahertz frequency range,
for electron-hole densities in the 10
9
-10
11
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 ampliers and oscillators that have dimensions in the 1-10 m range.
1
conduction
band
valence
band
E(k)
k
plasmon plasmon
plasmons
E(k)
k
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
( 10
9
10
11
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 nite temperatures as plasmons can
decay by exciting interband electron-hole pairs [1, 2]. In this paper we show that plasmon
amplication 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 ampliers and
oscillators based on plasmon amplication 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],
2
intrasubband plasmons in quantum wells, being longitudinal collective modes, do not couple
with intersubband transitions that require eld polarization perpendicular to the plane of the
quantum wells. The electromagnetic energy in the two-dimensional plasmon mode is conned
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
dierent 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
p
z
-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],
E
s,k
= shv|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 (q) of the longitudinal plasmon modes of wavevector q are given
by the equation,(q, ) = 0, where (q, ) is the longitudinal dielectric function of graphene [2].
In the random phase approximation (RPA) (q, ) can be written as [10],
(q, ) = 1 V (q)(q, ) (2)
Here, V (q) is the bare 2D Coulomb interaction and equals e
2
/2

q.

is the average of
the dielectric constant of the media on either side of the graphene layer. (q, ) is the
electron-hole propagator including both intraband and interband processes and is given by
the expression [1, 2],
(q, ) = 4

s s

k
| <
s

,k+q
|e
iq.r
|
s,k
> |
2
_
f(E
s,k
E
fs
) f(E
s

,k+q
E
fs
)

h + E
s,k
E
s

,k+q
+ i
(3)
3
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 E
f
) is the Fermi distribution function with Fermi energy E
f
.
|
s,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 dierent
Fermi levels to allow for nonequilibrium population inversion. The Bloch functions have the
following matrix elements [8],
| <
s

,k+q
|e
iq.r
|
s,k
> |
2
=
1
2
_
1 + ss

|k| +|q| cos ()

|k +q|
_
(4)
where is the angle between the vectors k and q. The condition v|q| < (q) must be satised
in order to avoid direct intraband absorption of plasmons. Assuming v|q| < , and using the
symmetry between conduction and valence bands, the intraband and interband contributions
to the propagator can be approximated as follows,

intra
(q, )
q
2
K T /h
2
( + i/) v
2
q
2
/2
log
__
e
E
f+
/KT
+ 1
_ _
e
E
f
/KT
+ 1
__
(5)

inter
(q, )
q
2
h
_

0
d
2
[f( h/2 E
f+
) f(h/2 E
f
)]

2
+i
q
2
4 h
[f( h/2 E
f+
) f(h/2 E
f
)] (6)
Here, q = |q|. In Equation (5), the intraband contribution to the propagator is written in the
plasmon-pole approximation that satises the f-sum rule [10]. This approximation is not valid
for large value of the wavevector q when (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 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 modies the eective dielectric constant and leads to a signicant 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
4
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. E
f+
E
f
> h. 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 (q) as a function of q. However, from the point
of view of device design, it is more useful to assume that the frequency is real and the
propagation vector q(), written as a function of , is complex. Since the charge density wave
corresponding to plasmons has the form e
iq.rit
, the imaginary part of the propagation vector
corresponds to net gain or loss. We dene the net plasmon energy gain g() as 2Imag{q()}.
3 Results and Discussion
In simulations we use v = 10
8
cm/s and

= 4.0
o
(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 E
f+
= E
f
.
Such a non-equilibrium situation can be realized experimentally by either carrier injection in
an electrostatically dened graphene pn-junction or through optical pumping [13, 14]. The
value of the scattering time (momentum relaxation time) is also critical for calculations of
the net plasmon gain. Value of 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],
=
e
2
K T
h
2
log
_
e
E
f+
/KT
+ 1
_
(7)
Values of mobility between 20,000 and 60,000 cm
2
/V-s have been experimentally measured at
low temperatures (T77K) in graphene [6, 7, 15]. Assuming a mobility value of 27,000 cm
2
/V-s
, reported in Ref. [15] for an electron density of 3.410
12
cm
2
at T=58K, the value of 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
5
0 0.5 1 1.5 2 2.5 3
0
1
2
3
4
5
6
Wavevector (10
5
cm
1
)
F
r
e
q
u
e
n
c
y

(
T
H
z
)
T = 10K
n=p=1, 3.5, 6, 8.5 10
9
cm
2

increasing
density
= v q
Figure 2: Calculated plasmon dispersion relation in graphene at 10K is plotted for dierent
electron-hole densities (n = p = 1, 3.5, 6, 8.510
9
cm
2
). The condition (q) > hvq is satised
for frequencies that have net gain in the terahertz range. The assumed values of v and are
10
8
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 dierent 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 signicantly large
reaching 1410
4
cm
1
for electron-hole densities in the 10
9
cm
2
range at low temperatures
and 10
11
cm
2
range at room temperature. The calculated plasmon dispersions indicate that
(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
6
0 1 2 3 4 5 6
4
2
0
2
4
6
x 10
4
Frequency (THz)
P
l
a
s
m
o
n

G
a
i
n

g
(

)

(
c
m

1
)
T = 10K
n=p=1, 3.5, 6, 8.5 10
9
cm
2

increasing
density
Figure 3: Net plasmon gain in graphene at 10K is plotted for dierent electron-hole densities
(n = p = 1, 3.5, 6, 8.5 10
9
cm
2
). The assumed values of v and are 10
8
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 10
9
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 = 10
10
cm
2
and values of the intraband scattering time
varying from 0.1 to 0.5 ps. The net gain decreases as the plasmon losses increase with a
decrease in the value of and the maximum gain value equals zero for = 0.15 ps. However,
it should not be concluded from Fig. 8 that plasmons cannot have net gain for less than 0.15
ps since electron-hole density can always be increased to achieve net gain for smaller values of
7
0 2 4 6 8
0
2
4
6
8
10
12
14
16
Wavevector (10
5
cm
1
)
F
r
e
q
u
e
n
c
y

(
T
H
z
)
increasing
density
= v q
T = 77K
n=p=1, 2, 3, 4 10
10
cm
2

Figure 4: Calculated plasmon dispersion relation in graphene at 77K is plotted for dierent
electron-hole densities (n = p = 1, 2, 3, 4 10
10
cm
2
). The condition (q) > hvq is satised
for frequencies that have net gain in the terahertz range. The assumed values of v and are
10
8
cm/s and 0.5 ps, respectively.
0 2 4 6 8 10
4
2
0
2
4
6
x 10
4
Frequency (THz)
P
l
a
s
m
o
n

G
a
i
n

g
(

)

(
c
m

1
)
T = 77K
n=p=1, 2, 3, 4 10
10
cm
2

increasing
density
Figure 5: Net plasmon gain in graphene at 77K is plotted for dierent electron-hole densities
(n = p = 1, 2, 3, 4 10
10
cm
2
). The assumed values of v and are 10
8
cm/s and 0.5 ps,
respectively.
8
0 5 10 15 20
0
5
10
15
20
25
30
35
40
Wavevector (10
5
cm
1
)
F
r
e
q
u
e
n
c
y

(
T
H
z
)
T = 300K
n=p=1, 1.5, 2, 2.5 10
11
cm
2

increasing
density
= v q
Figure 6: Calculated plasmon dispersion relation in graphene at 300K is plotted for dierent
electron-hole densities (n = p = 1, 1.5, 2, 2.510
11
cm
2
). The condition (q) > hvq is satised
for frequencies that have net gain in the terahertz range. The assumed values of v and are
10
8
cm/s and 0.5 ps, respectively.
0 5 10 15 20
4
2
0
2
4
6
x 10
4
Frequency (THz)
P
l
a
s
m
o
n

G
a
i
n

g
(

)

(
c
m

1
)
T = 300K
n=p=1, 1.5, 2, 2.5 10
11
cm
2

increasing
density
Figure 7: Net plasmon gain in graphene at 300K is plotted for dierent electron-hole densities
(n = p = 1, 1.5, 2, 2.5 10
11
cm
2
). The assumed values of v and are 10
8
cm/s and 0.5 ps,
respectively.
9
0 2 4 6 8
4
2
0
2
4
6
8
x 10
4
Frequency (THz)
P
l
a
s
m
o
n

G
a
i
n

g
(

)

(
c
m

1
)
T = 10K
n = p = 10
10
cm
2

= 0.5, 0.4, 0.3, 0.2, 0.15, 0.1 ps
increasing
Figure 8: Net plasmon gain in graphene at 10K is plotted for dierent intraband scattering times
( = 0.5, 0.4, 0.3, 0.2, 0.15, 0.1 ps). The assumed value of v is 10
8
cm/s and the electron-hole
density is 10
10
cm
2
.
0 2 4 6 8 10 12
4
2
0
2
4
6
8
x 10
4
Frequency (THz)
P
l
a
s
m
o
n

G
a
i
n

g
(

)

(
c
m

1
)
increasing
T = 10K
n = p = 3 10
10
cm
2

= 150, 125, 100, 75 fs
Figure 9: Net plasmon gain in graphene at 10K is plotted for dierent scattering times
( = 150, 125, 100, 75 fs). The assumed value of v is 10
8
cm/s and the electron-hole density is
3 10
10
cm
2
.
10
. Fig. 9 shows the net gain at T=10K for n = p = 3 10
10
cm
2
and values of the intraband
scattering time 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 (> 10
4
cm
1
) in graphene implies that
terahertz plasmon oscillators only a few microns long in length could have sucient gain to
overcome both intrinsic losses and losses associated with external radiation coupling. Plasmon
elds 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
10
5
cm
1
at terahertz frequencies. Therefore, the electromagnetic energy associated with the
terahertz plasmons is conned within 100 nm of the graphene layer. Strong eld connement
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 10
12
cm
2
[16]. This suggests that population inversion can be experimentally
achieved in graphene via current injection in electrostatically dened 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 10
11
and 10
12
cm
2
[17]. Therefore, at low
electron-hole densities (less than 10
11
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 (> 10
11
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
11
even for carrier intraband scattering times shorter than 100 fs. The high gain values and
the strong plasmon eld connement near the graphene layer could enable compact terahertz
ampliers and oscillators. The authors would like to thank Edwin Kan and Sandip Tiwari for
helpful discussions.
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13