Substrate carrier concentration
dependent plasmon-phonon coupled
modes at the interface between graphene
and semiconductors
Lei Wang,1,2,3 Wei Cai,1,3,∗ Linyu Niu,1,3 Weiwei Luo,1,3 Zenghong
Ma,1,3 Chenglin Du,1,3 Shuqing Xue,1,3 Xinzheng Zhang,1,3 and
Jingjun Xu1,3,4
1 The Key Laboratory of Weak-Light Nonlinear Photonics, Ministry of Education, TEDA
Applied Physics Institute and School of Physics, Nankai University, Tianjin 300457, China
2 College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang
464000, China
3 Synergetic Innovation Center of Chemical Science and Engineering, Tianjin 300071, China
4 [email protected]
*[email protected]
Abstract: The coupled modes between graphene plasmons and surface
phonons of a semiconductor substrate are investigated, which can be
efficiently controlled by carrier injection of the substrate. A new physical mechanism on tuning plasmon-phonon coupled modes (PPCMs) is
proposed due to the fact that the energy and lifetime of substrate surface
phonons depend a lot on the carrier concentration. Specifically, the change
of dispersion and lifetime of PPCMs can be controlled by the carrier
concentration of the substrate. The energy of PPCMs for a given momentum
increases as the carrier concentration of the substrate increases. On the
other hand, the momentum of PPCMs for a given energy decreases when
the carrier concentration of the substrate increases. The lifetime of PPCMs
is always larger than the intrinsic lifetime of graphene plasmons without
plasmon-phonon coupling.
© 2015 Optical Society of America
OCIS codes: (240.6680) Surface plasmons; (250.5403) Plasmonics; (300.6340) Spectroscopy,
infrared.
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(C) 2015 OSA
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16 Nov 2015 | Vol. 23, No. 23 | DOI:10.1364/OE.23.029533 | OPTICS EXPRESS 29534
1.
Introduction
Recently, graphene plasmons (GPs) have been focused much attention for their extremely large
field enhancement and relatively long propagation length [1, 2]. For these unique optical properties, graphene is believed as one of the best plasmon materials in the infrared and terahertz
regimes. Due to the large momentum mismatch between free light and GPs, GPs cannot be excited by free light along translation invariant directions. Therefore, structured graphene such as
graphene ribbons [3], periodic ribbon arrays [4], graphene disks [5] and graphene rings [6] have
been put forward to break up translation invariance. Meanwhile, several research groups have
used a scattering-type scanning near-field optical microscope (s-SNOM) to generate and mapping GPs in real space [7–9], which takes use of the increased momentum of the incident light.
However, the generation of highly wavelength controllable and long-lived GPs with a relatively
simple experimental method is still the bottleneck for applications of GPs. Therefore, the realization of highly tunable plasmons in graphene is particularly important. It is well-known that
GPs can be actively controlled by bias voltages or chemical doping [10]. Specially, some outstanding works regarding plasmon induced doping of low-dimensional materials open a new
route for controlling the properties of materials in the past few years [11–13]. Despite so many
efforts on doping of graphene itself to controlling the GPs have been employed, how to realize
a relatively simple and effective way to tune GPs is far from solved. As far as we know, little
attention has been paid on how to tuning the properties of GPs by doping of the substrates.
To demonstrate the possibility of doping substrates to tuning the GPs, we turn to see the dispersion relation of GPs. For long wavelengths k kF , where kF is the Fermi momentum. The
dispersion relation of GPs reduces to [2]
q=
2
h̄εeff ω pl
2αcEF
,
(1)
where q is the plasmon momentum, ω pl is the frequency of GPs, εeff = (1 + ε∞ )/2 is the effective average over the dielectric constants of the substrate and air, α ≈1/137 is the fine structure
constant, and EF is the Fermi energy linked to the carrier density of graphene. This equation
means that the properties of substrates can be an equally important factor as the Fermi energy
of graphene to affect the GPs. For example, when a graphene sheet is placed on a polar substrate, the coupling between GPs and surface phonons of the substrate exists [14–19]. These
plasmon-phonon coupled modes (PPCMs) can reduce intrinsic damping of GPs and process
longer lifetime than GPs, which have been verified experimentally [16, 20]. However, the energy of PPCMs at the interface between graphene and a polar substrate usually locates near the
surface phonon energy of substrates, which is determined by the phonon of substrates. As a
result, these PPCMs can hardly be controlled. In this paper, we propose using a semiconductor
material as the substrate of graphene. One can change the surface phonon energy and lifetime
of the substrate, thus realize active control of the properties of PPCMs, which can be explained
by the actively adjusting of the coupling between the free electron oscillations and phonons in
the substrate. On the other hand, the carrier concentration of semiconductor substrates can be
easily modified by electrical fields or light, therefore these PPCMs can also be controlled by
applying external visible or near-infrared optical excitations or electrical fields.
2.
Models and calculation methods
Our proposed scheme is indicated in Fig. 1. A graphene sheet is placed on a doped semiconductor substrate. Gallium arsenide (GaAs), which is a typical semiconductor material, is taken
for example. The carrier concentration of GaAs can be tuned by applying an external electrical
field or an optical wave. The dielectric function contributed from phonons and free electrons in
#248777
(C) 2015 OSA
Received 28 Aug 2015; revised 22 Oct 2015; accepted 23 Oct 2015; published 3 Nov 2015
16 Nov 2015 | Vol. 23, No. 23 | DOI:10.1364/OE.23.029533 | OPTICS EXPRESS 29535
Doped GaAs
Carrier
injection
Fig. 1. The sketch of graphene placed on doped GaAs. The carrier concentration of GaAs
can be tuned by field effect or photon excitation. The dots in substrate indicate the free
electrons injected into GaAs.
polar semiconductor materials is described as follows [21, 22]:
ε(ω) = ε∞ +
2
ε∞ ω p2
(ε0 − ε∞ )ωTO
,
−
2 − ω 2 − iωΓ
ω 2 + iωδ p
ωTO
(2)
where ωTO denotes the frequency of transverse optical phonons, Γ is the damping rate related to phonons, ω p2 = 4πne2 /m∗ ε∞ denotes the plasma frequency of free electrons, δ p = h̄/τ p
is the damping rate related to the electron scattering in GaAs. ε0 and ε∞ are the static and
high frequency dielectric constants, respectively. For GaAs, these parameters [23, 24] are
∗
2
ωTO = 268.7/cm, ε0 = 13,
p ε∞ = 11 , m = 0.067me and µGaAs =∗ 8500 cm /Vs. Thus the derived parameters ωLO = ε∞ /ε0 ωTO = 291.1/cm, τ p = µGaAs m /e = 0.324 ps. On the other
hand, the optical response of graphene is described by the in-plane complex conductivity which
is computed with local random phase approximation (RPA) [25, 26]. The Fermi energy of
graphene is set as 0.2 eV (corresponding to carrier density 2.94 ×1012 cm−2 ), which is a typical
value for graphene grown by chemical vapor deposition. The intrinsic electron relaxation time
of graphene is set as τe = µEF /ev2F , where vF ≈ c/300 is the Fermi velocity, µ = 10000 cm2 /Vs
is the measured DC mobility [27], and the ambient temperature is set as 300 K.
The dispersion relation of eigenmodes in graphene sandwiched by two dielectric materials
can be obtained from the pole of p-polarized Fresnel reflection coefficients [2]:
q
q
ε/ εk02 − q2 + 1/ k02 − q2 = −4πσ (ω)/ω,
(3)
where ε is the dielectric function of the substrate, the superstrate is vacuum, and σ (ω) is frequency dependent conductivity of graphene. In order to get the lifetime of PPCMs, the coupling
mechanism between graphene and GaAs can be considered as follows. The
p coupling strength
between GPs and surface polar phonons is given by M(q) = h̄ge−qz0 πcαωso /q, where
g = [1/(1 + ε∞ ) − 1/(1 + ε0 )]1/2 is the coupling constant, ωso = [(ε0 + 1)/(ε∞ + 1)]1/2 ωTO
is the original surface phonon frequency, and z0 is the distance between graphene and substrate,
which is set to zero throughout this paper. The exchange potential due to the phonon coupling
2]
can be written as vso = |M(k)|2 G(0) (ω, τso ), where G(0) (ω, τso ) = 2ωso /h̄[(ω + i/τso )2 − ωso
is the surface phonon propagator [28, 29]. The total effective carrier interaction results from
#248777
(C) 2015 OSA
Received 28 Aug 2015; revised 22 Oct 2015; accepted 23 Oct 2015; published 3 Nov 2015
16 Nov 2015 | Vol. 23, No. 23 | DOI:10.1364/OE.23.029533 | OPTICS EXPRESS 29536
the Coulomb interaction vc = 2πe2 /qεeff , the screened phonon exchange potential vsc−so and
electron-electron interaction potential from free carrier of GaAs v p , it reads [30]:
Veff = vc /εrpa ≈
1−
M(k)2
vc
ε + ε 2 D(q, ω)
0
M(k)2
1 vc
0
vc ( ε + ε 2 D(q, ω))vc (q)Πs (q, ω)
(4)
where ε = 1 − vc Π0g (q, ω) is the purely graphene electronic dielectric function, Π0g (q, ω) ≈
EF q2 /π h̄2 (ω + iδe )2 is the noninteracting polarizability of graphene, depending on δe = h̄/τe .
D(q, ω) = D(0) (q, ω)/(1 − |M(k)|2 D(0) Π0g (q, ω)/ε) is the renormalized surface phonon propagator. Π0s (q, ω) ≈ nq2 /m∗ (ω + iδ p )2 is the polarizability for free carrier of GaAs, depending on
0
δ p =h̄/τ p . vc (q) = 4πe2 /ε∞ q2 is the 3D Coulomb potential of GaAs. It is worth mentioning that
the coulomb drag effect between graphene and 3D electron gas is ignored because the graphene
layer and semiconductor touch each other and the drag resistivity is usually far less than layer
resistivity [31–34]. From Eq. (4), we can obtain:
ε∞
2 (q)
2
2
ω pl
εrpa
2εeff g2 ωso
1+ε∞ ω p
−
= 1−
−
2
εeff
(ω + iδe )2 (ω + iδso )2 − (1 − 2εeff g2 )ωso
(ω + iδ p )2
(5)
the dispersion relation of PPCMs can be obtained by using Eq. (5) with εrpa (q, ω −i/τ) = 0, and
the carrier concentration dependent lifetime τ of PPCMs can be obtained from the imaginary
part of the energy.
3.
Carrier concentration dependent plasmon-phonon coupled modes
First of all, the dispersion of PPCMs is calculated by using Eq. (3) without considering the
damping of the substrate ( Γ and δ p are set to 0) for simplicity. The carrier concentration dependent dispersion curves of PPCMs are shown in Fig. 2(a). It is worth noting that the intrinsic
carrier concentration of GaAs without additional doping is n = 2 × 106 cm−3 at room temperature. Compared with the dispersion of GPs when a graphene sheet is placed on a nonpolar
substrate, one can find that the dispersion of PPCMs splits into two branches due to plasmonphonon coupling, and these two branches are labeled as A and B, respectively. This splitting
phenomenon is the result of coupling between graphene plasmons and the optical phonon mode
in the GaAs. Furthermore, one can find that there are two cut-off frequencies with two branches
of the surface phonon mode which are result of the coupling between free charges and the
optical phonon mode in GaAs, while there is only one branch and one cut-off frequency for
normal polar material such as SiO2 [35]. Moreover, several unique features for our proposed
system can be illustrated compared with the case where a graphene sheet is laid on a normal
polar material [29]. Firstly, for both branches A and B, the plasmon energy increases as the carrier concentration increases, which provides us an active method to tune the PPCMs. Secondly,
there is a low energy cutoff frequency for the branch B, and the energy range of the branch
B becomes narrower when the carrier concentration increases. Due to the fact that the upper
limit ωTO exists and the branch B only exists near ωTO , near-zero group velocity PPCMs can
be realized, which can be applied in slow light propagation and optical storage. Thirdly, the
branch A becomes nearly linear rather than quadratic (in the no coupling limit) when the carrier concentration is over 5 × 1017 cm−3 , which leads to group velocity dispersiveness PPCMs.
Figure 2(b) shows the lifetime of carrier concentration dependent PPCMs by using Eq. (5). For
simplicity, the free electron damping in GaAs is not considered (in order to keep consistent
with the dispersion of PPCMs in Fig. 2(a)) and the surface-phonon damping τso is set as 1 ps as
#248777
(C) 2015 OSA
Received 28 Aug 2015; revised 22 Oct 2015; accepted 23 Oct 2015; published 3 Nov 2015
16 Nov 2015 | Vol. 23, No. 23 | DOI:10.1364/OE.23.029533 | OPTICS EXPRESS 29537
n = 0, 0.2, 1, 2, 5, 10, 20, 50 (×10
17
-3
cm )
(a)
Branch A
Branch B
Laudau intraband loss
n = 0, 0.2, 1, 2, 5, 10, 20, 50 (×10
17
cm -3)
(b)
Fig. 2. (a) The dispersion of plasmon-phonon coupled modes (PPCMs) for graphene laid on
GaAs substrate with various carrier concentrations. The labelled modes A and B describe
the coupled higher and lower energy branches, respectively. The black arrows mean the
increasing of the carrier concentration in GaAs. The carrier concentrations are 0, 2 × 1016 ,
1×1017 , 2×1017 , 5×1017 , 1×1018 , 2×1018 and 5×1018 cm−3 , respectively. The shadow
triangle area indicates the Landau intraband loss. The parameters of graphene are assumed
to be EF = 0.2 eV, and relaxation time τe = 0.2 ps (correspond to a DC mobility of 10000
cm2 /Vs) in the whole paper. (b) Plasmon lifetime of the two branches A and B with different carrier concentrations. The free electron scattering of substrate is not included in the
calculation.
reported in SiO2 [20]. We can find that the lifetime of PPCMs becomes longer when the carrier
concentration becomes larger.
To further understand the dispersion and lifetime of PPCMs, the surface phonon of the GaAs
substrate is analyzed, which can be obtained by solving [ε(ω̃so − i/τ̃so ) + 1]/2 = 0 with Eq.
(2). The calculated results are shown in Fig. 3(a). One can find that there are two branches for
ω̃so . The higher and lower energies ω̃so are the approximate cutoff frequencies for branches
A and B in Fig. 2(a), respectively. Moreover, one can find that the cutoff frequency of branch
A monotonously increases from the originally value ωso and does not have upper limit in our
calculated carrier concentration range, while the cutoff frequency of branch B monotonously
increases from 0 (no cutoff frequency) to ωTO . On the other hand, the energy of branch A of
PPCMs is larger than the high energy ω̃so , and the energy of branch B locates in the middle
of the low energy ω̃so and ωTO . Thus, the behavior of carrier concentration dependent PPCMs
shown in Fig. 2(a) can be understood, which results from the change of the surface phonon of
substrate. Then we turn to analyze the strength of surface phonons. From the surface phonon
strength definition Sm = |hm|φk |0i|2 , where |mi is the one-phonon excited state in the mth
#248777
(C) 2015 OSA
Received 28 Aug 2015; revised 22 Oct 2015; accepted 23 Oct 2015; published 3 Nov 2015
16 Nov 2015 | Vol. 23, No. 23 | DOI:10.1364/OE.23.029533 | OPTICS EXPRESS 29538
Branch
Branch
Branch
Branch
(b)
A
B
A
B
Energy
Energy
Lifetime
Lifetime
Branch A
Branch B
5×
Fig. 3. (a) The energy and lifetime of surface phonons in doping GaAs, The thick (thin)
line indicates the high (low) energy branch, and the dashed lines indicate their lifetime. (b)
The surface phonon strength of these branches in Fig. 3(a).
level, φk = b−k + b†k and bk is surface phonon creation operator. Surface phonon strength is
given by [22]
p
3
ε∞ /ε0 (ε∞ /ε0 − 1)xm
Sm =
,
(6)
4 + y2 (1 − x2 )2
(ε∞ /ε0 )xm
m
where x = ω̃so /ωTO is the normalized coupled surface phonon energy, and y = ω p /ωTO is normalized bulk plasmon of GaAs. The value Sm describes the weight of the two surface phonons,
which contributes to the properties of total surface phonons. The strength of surface phonons
is calculated and shown in the Fig. 3(b), we know that the phonon strength of branch A (B) decreases (increases) dramatically as the carrier concentration increases, similar to the behavior
of phonon lifetime shown in Fig. 3(a).
The calculated lifetime of PPCMs in Fig. 2(b) does not include the contribution from the
scattering of free electrons in GaAs, in other words, the lifetime of surface phonon is set to
a constant value 1 ps in our pervious calculations. However, from Fig. 3(a) we know that the
lifetime of coupled surface phonon will decrease (increase) for branch A (B) with electron
scattering. So the electron scattering effect must be considered in further calculations if one
wants to compare the theoretical results with actual experiments. By substituting the δ p into Eq.
(5), the total lifetime of GPs is calculated and shown in Fig. 4. The largest lifetime for the branch
A of PPCMs decreases apparently compared to the case without considering electron scattering
in substrate (shown in Fig. 2(b)). In contrast, for the branch B of PPCMs, the largest lifetime is
still long enough for proper carrier concentration. In other words, although the electron doping
in substrate will provide a new pathway for plasmon damping and result in the lifetime decrease
of the PPCMs, the lifetime of PPCMs can also be accepted if the doping concentration is not
so large. Moreover, the Coulomb drag effect between electrons in graphene and substrate can
#248777
(C) 2015 OSA
Received 28 Aug 2015; revised 22 Oct 2015; accepted 23 Oct 2015; published 3 Nov 2015
16 Nov 2015 | Vol. 23, No. 23 | DOI:10.1364/OE.23.029533 | OPTICS EXPRESS 29539
n=0
16
-3
n=2×10 cm
17
-3
n=1×10 cm
17
-3
n=2×10 cm
17
-3
n=5×10 cm
18
-3
n=1×10 cm
18
-3
n=2×10 cm
18
-3
n=5×10 cm
Fig. 4. The plasmon lifetime of the coupled plasmon-phonon modes with the free electron
scattering of the substrate GaAs. It is notable that the curve n = 0 is the same as it in Fig.
2(b).
also be a damping mechanism for PPCMs, however, this is not included in our present physical
model because of the touching between graphene and substrate.
For actual experiments about GPs, we usually have a given light source with certain wavelengths or a given momentum (∼ 1/a) determined by the structures, where a is the structure
size for a tip or a grating, due to the fact that the efficient excitation of GPs always requires to
fulfill momentum conservation conditions. So several discrete wavelengths and momenta are
utilized to realize GPs excitation and propagation in our proposed graphene/GaAs system. First,
we analyze the dispersion of PPCMs with a given wavelength. Without loss of generality, two
energies from branch A (h̄ω = 0.06 eV) and branch B (h̄ω = 0.03 eV) are chosen respectively.
The momentum and lifetime of the carrier concentration dependent PPCMs are displayed in
Fig. 5(a). One can find that the momenta of branches A and B (the solid lines) decrease to
zero as the concentration increases. This result indicates that for a given monochromatic wave,
we can always find a proper concentration to match the momentum out of the light cone. This
effect opens a door for that all kinds of confined modes or evanescent waves can be coupled to
PPCMs. The corresponding lifetime of these branches is indicated by the dashed lines. And the
lifetime increases dramatically as the momentum decreasing. This comes both from the effect
of phonon coupling and lower mode confinement as the increase of carrier concentration. For
the photon energy 0.06 eV, a critical point (8.12 × 1017 cm−3 , 0.203 ps) appears and shown in
Fig. 5(a). In this point, the lifetime of PPCMs is the same as the case with zero concentration of
the substrate. Because of the absence of GaAs electron scattering damping pathway, the lifetime
of PPCMs without substrate carrier doping is longer than low concentration substrate doping.
However, when the concentration is larger than 8.12 × 1017 cm−3 , the lifetime of PPCMs can
be longer than zero concentration lifetime τ = 0.203 ps. For the energy h̄ω = 0.03 eV the lifetime is always larger than zero concentration lifetime (τ = 0.215 ps) in our calculation carrier
concentration range. On the other hand, for a given momentum q = 5 µm−1 , the energy and
lifetime of PPCMs are shown in Fig. 5(b). From the figure, we know that the branches A and
B show behaviors similar with h̄ωso (q → ∞) (namely Fig. 3 (a)). If a wide spectrum pulse is
given, one can tune the output frequency of PPCMs by changing the carrier concentration in a
fixed experimental scheme.
Finally, although GaAs is taken for example in our above analysis, our results can be extended to various semiconductor materials. To verify this point, the coupled surface phonon
energy changing with carrier concentration for several different semiconductor materials is in#248777
(C) 2015 OSA
Received 28 Aug 2015; revised 22 Oct 2015; accepted 23 Oct 2015; published 3 Nov 2015
16 Nov 2015 | Vol. 23, No. 23 | DOI:10.1364/OE.23.029533 | OPTICS EXPRESS 29540
(8.12×10 17 cm -3,0.203ps)
Energy , k=5
Energy , k=5
Lifetime , k=5
Lifetime , k=5
(a)
m -1 Branch A
m -1 Branch B
m -1 Branch A
m -1 Branch B
(b)
10 17
10 18
5×10 18
Fig. 5. (a) Carrier concentration dependent plasmon momentum and lifetime for given
photon energies. The thick line (h̄ω = 0.06 eV) is chosen from branch A, and the thin
line (h̄ω = 0.03 eV) from branch B. The dashed lines show the corresponding lifetime of
the branches. (b) Concentration dependent energy for a given momentum 5 µm−1 , the thick
and thin lines indicate the branch A and B, respectively.
10 17
10 18
5×10 18
Fig. 6. Carrier concentration dependent surface phonon energy in different semiconductors,
GaAs, AlAs, InAs, GaP and InP.
vestigated and shown in Fig. 6. The carrier concentration dependent surface phonon energies
for GaAs, AlAs, InAs, GaP and InP are compared. And the changeable energy range is mainly
determined by the effective carrier mass of semiconductor. For InAs, surface phonon energy
can reach 0.2 eV for concentration near 1019 cm−3 , which means that all the energy below intrinsic graphene phonon can lead to PPCMs. Beyond that, we reaffirm that the direct Coulomb
drag effect has been ignored in this paper. When the effect is considered, extra blue shift will
be pull-in [31] in the system.
#248777
(C) 2015 OSA
Received 28 Aug 2015; revised 22 Oct 2015; accepted 23 Oct 2015; published 3 Nov 2015
16 Nov 2015 | Vol. 23, No. 23 | DOI:10.1364/OE.23.029533 | OPTICS EXPRESS 29541
4.
Conclusions
In summary, in this paper we investigated analytically the effect of substrate carrier concentration on the properties of PPCMs. By using the semiconductor substrate GaAs, we found
the coupled modes can be effectively controlled by changing the carrier density of the substrate. Specifically, the dispersion and lifetime of PPCMs can be controlled by the carrier density of GaAs. In further, the effect can be understood by the changing of surface phonons of
substrate. Our proposed physical mechanism to tune PPCMs can also be extended in other
graphene/semiconductor systems. In addition, although the doping in the substrate will result
in additional damping for GPs, the lifetime of PPCMs can still be longer than uncoupled plasmons in graphene under proper doping concentration. The controllable long-live and easily
excited PPCMs can find applications using GPs. Moreover, the long-range interaction between
electrons of substrate and graphene may lead to additional damping for plasmons, which still
an open question and will be worked out in future works.
Acknowledgments
This work was financially supported by the National Basic Research Program of China
(2013CB328702), Program for Changjiang Scholars and Innovative Research Team in University (IRT0149), the National Natural Science Foundation of China (11374006) and the 111
Project (B07013).
#248777
(C) 2015 OSA
Received 28 Aug 2015; revised 22 Oct 2015; accepted 23 Oct 2015; published 3 Nov 2015
16 Nov 2015 | Vol. 23, No. 23 | DOI:10.1364/OE.23.029533 | OPTICS EXPRESS 29542