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Surface plasmon polariton amplification in
metalsemiconductor structures
Dmitry Yu. Fedyanin* and Aleksey V. Arsenin
Laboratory of Nanooptics and Femtosecond Electronics, Department of General Physics, Moscow Institute of
Physics and Technology (State University), 9, Institutsky lane, Dolgoprudny, 141700, Russia
*feddu@mail.ru
Abstract: We propose a novel scheme of surface plasmon polariton (SPP)
amplification that is based on a minority carrier injection in a Schottky
diode. This scheme uses compact electrical pumping instead of bulky
optical pumping. Compact size and a planar structure of the proposed
amplifier allow one to utilize it in integrated plasmonic circuits and couple it
easily to passive plasmonic devices. Moreover, this technique can be used to
obtain surface plasmon lasing.
©2011 Optical Society of America
OCIS codes: (250.5403) Plasmonics; (250.4480) Optical amplifiers.
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1. Introduction
Operation frequency of modern microprocessors does not exceed a few gigahertz due to high
heat generation and interconnect delays. SPPs, which are surface electromagnetic waves
propagating at the interface between a metal and an insulator, are considered as very
promising information carriers that can replace electrons in integrated circuits [1–3]. An
exceedingly short wavelength and a very high spatial localization of the electromagnetic field
near the interface allow to get over the usual diffraction limit and design ultracompact
interconnects with the transverse size of the order of 100 nm [1,4] that is comparable with
electronic components. Unfortunately, high propagation losses due to Joule heating restrict the
application of SPPs. Thus, one should increase the SPP propagation length, i.e. partially or
fully compensate Joule losses. This can be done by using an active media placed near a metal
surface [5]. In recent years, a number of paper devoted to the SPP amplification have been
published [6–14] and several methods have been proposed. Despite the advantages of these
methods, the necessity of an external high power pump laser prevents us to use them in
nanoscale circuits.
In this paper, we propose a different technique that is based on a minority carrier injection
effect in metalsemiconductor contacts that gives one a possibility to use compact electrical
pumping instead of a bulky optical approach.
2. Schottky barrier diode
Usually, Schottky diodes are treated as majority carrier devices. For instance, if one has an n
type semiconductormetal contact, the electron concentration is much greater than the
concentration of holes all over the semiconductor at all bias voltages (here and below, only
the case of forward bias is considered) and the hole current is much less than the electron one.
However, the situation changes drastically when the metal work function
2/
ge
E
, where
e
and
g
E are the electron affinity and the band gap of the semiconductor,
respectively. In this case, the concentration of holes (minority carriers) near the metal
semiconductor contact becomes greater than the concentration of electrons (majority carriers)
and it is said that an inversion layer is formed. Under forward bias, holes are injected into the
M
exceeds
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bulk of the semiconductor and recombine with electrons that results in light emission [15]. So,
Schottky barriers can be used to design efficient and compact light [15] and plasmon
emitting diodes [16], but what about lasers and amplifiers? To design a laser, one should
satisfy the condition for net stimulated emission or gain [17,18]
.
ghe
EFF
(1)
Here, is the SPP frequency,
respectively.
How can we satisfy inequality (1)? Firstly, if we use a degenerate semiconductor,
is positive nearly everywhere inside the semiconductor under sufficient forward bias. Hence,
one should only maximize the difference
E
semiconductor contact,
FE
by increasing the metal work function or decreasing the electron affinity of the
semiconductor. Inside the semiconductor,
v
E
Schottky contact condition (1) is still satisfied. Thus, one should maximize
make it positive.
To demonstrate the principle of operation of the proposed device, consider a structure
depicted in Fig. 1(a). For simplicity, assume the back contact to be an ideal ohmic contact, i.e.
e F ,
h
F are quasiFermi levels for electron and holes,
ce
EF
vh.
F
It is obvious that, near the metal
0
F
and one can increase
h F is very close to the metal Fermi level
m
hv
h
F
will decrease but in the region near the
geM
E
and
VVFVEFF
V

LzLz
m0 fshe

(2)
where
fs
E is the Fermi level under zero bias. The boundary conditions at
0
z
are [19–22]
n0
nr00
p
p0pr00
( 
( 
p
)
)
zz
zz
Je
e
nn
J
(3)
where,
n
0
electron and hole concentrations at
effective recombination or collection velocities. We neglect the effect of surface states and
image forces on the barrier height, assume the donor concentration to be independent on z
and suppose mobilities and diffusion constants to obey the Einstein relation.
Thus, we have four boundary conditions (Eqs. (1) and (2)) and six nonlinear first order
differential equations that describe the carrier behavior within the semiconductor:
n J and
F
2/ 1
p J are the electron and hole current densities, e is the electron charge,
Tk
BeM
/ )
and
FNp
M2/ 1v0
(
0
z (
2
is the FermiDirac integral),
N
c
(
TkE
Bge
/ )
are quasiequilibrium
and
/ 1 F
nr
pr
are
p
eUdzdJ
eUdz dJ
E
D
p
J
eDdz
dp
E
D
n
J
eDdz
dn
Nn
edzdE
Edzd
z
z
z
z
p
n
p
p
p
p
n
n
n
n
std
1
1
4
(4)
where, symbols have their usual meaning [20]. Two missing boundary conditions are certainly
0
z
.
V
Lz
0
and
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The carrier recombination rate consists of three components
long as there are three recombination processes: spontaneous emission
emission
)(
stim
U
and nonradiative SchockleyReadHall and Auger recombination
directbandgap semiconductors,
nr
U is usually much less than
discussed in the present work.
spont
(
U
stim
),
nr
UUUU
, as
spont
stimulated
(
nr
U
).
In
,
spont
U
therefore it is not
Fig. 1. (a) Sketch of the ideal Schottky barrier diode and schematic band diagram under zero
bias, L = 400 nm, ψM = 5.4 eV [34,35], εst = 13.94 [36], χe = 4.5 eV [36], Eg = 0.75 eV that
correspond to Ga0.47In0.53As at T = 300 K and donor concentration Nd = 9.3 × 1017 cm3. (b)
Carrier density distribution and (c) band diagram under zero bias.
To begin with, we demonstrate that it is possible to satisfy inequality (1). For this purpose,
U
be zero, while
)(
eq eq spont
pn npBU
concentrations and
1.43 10 cm s
B
for In0.53Ga0.47As [23]. In the presence of
degeneracy and in the case of high minority carrier injection, system of Eqs. (4) cannot be
solved analytically and we have to implement the NewtonRaphson method. Under zero bias,
namely at thermal equilibrium,
mhe
FFF
holes are injected into the bulk of the semiconductor and
time, the concentration of electrons changes slightly and
high forward bias (Fig. 2(b)), the difference between quasiFermi levels exceeds
satisfy inequality (1). This clearly demonstrates that it is possible to realize a SPP amplifier
based on a Schottky barrier diode [24].
let
stim
, where
eq
n and
eq
p are the equilibrium
1031
0
(Figs. 1(b) and 1(c)). As the bias increases,
h F shifts downward. At the same
e F remains constant (Fig. 2). Under
g
E and we
Fig. 2. QuasiFermi levels at V = 0.8 V (a) and V = 0.85 V (b).
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3. SPP dispersion
The dispersion relation for SPPs propagating along the planar interface between a metal and a
semiconductor (Fig. 3(a)) with permittivities
1 and
2 , respectively, has the form
i
2 1 1 2,
SPP wavevector). Assuming the semiconductor to be lossless and taking into account losses in
the metal [25], we calculate the propagation length of the SPP at the interface between Au and
Ga0.47In0.53As. At a light wavelength of 1.7 µm (0.73 eV),
corresponds to a wavelength of 436 nm and a propagation length of 6.2 µm. The imaginary
part of is much less than the real one and losses almost do not affect the field distribution
and SPP wavelength, therefore we will use the power flow approach. The essence of the
method is that only the real parts of permittivities are used to determine the real part of the
wavevector
(Re 144018cm ),
while Im is found from the power flow equation
[26,27]. This method provides a simple treatment and gives a clear physical interpretation of
the SPP attenuation or amplification. Let us denote by P the power flow per unit guide width
where
22
( / )
ii
c
is the penetration constant (
1,2
and is the
1
(143926 806i)cm
that
1
(
x
PS dz
, where
x S is the xcomponent of the complex Poynting vector) and by R the
Joule loss power per unit guide length per unit guide width
0
2
1
( /8Im ).
R E dz
Then
2Im
dP dx
in the metal is much less than in the semiconductor that is due to the great difference in
penetration depths of the SPP field inside the metal (22 nm) and semiconductor (223 nm).
Despite that, the absorption in the metal is high enough and the material gain of the order of
1
2Im1630 cm
is required in the semiconductor medium to compensate losses.
PR
[27] and consequently
1
Im /2815 cm .
RP
The power flow
Fig. 3. (a) Schematic view of the SPP propagation along the metalsemiconductor interface. (b)
Dependence of the material gain on the minority carrier density in In0.53Ga0.47As at a light
wavelength of 1.7 µm. (c) Gain spectra of In0.53Ga0.47As.
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4. SPP amplification by stimulated emission of radiation
Stimulated emission recombination rate is given as
,/
stim
gSU
(5)
where S is the local optical power density and g is the local optical gain. In a oneelectron
model, the optical gain connected with bandtoband transitions is given as [18]
,
/ )
h
F
( exp1
1
/ )
e
F
( exp1
1
)()(),(
4
n
),(
BB
vvcc
2
env
2
b
2
e0
22
he
dE
TkETkE
EEEEEEMM
mc
e
FFg
(6)
where
and valence bands, n is the refractive index of the semiconductor,
e0
m is the freeelectron mass,
c
and
v
are the densities of states in the conduction
M is the average matrix
b
element connecting Bloch states near the band edges (
[28]) and
env
M
is the envelope matrix element. When the semiconductor is heavily doped, the
parabolic band approach becomes inapplicable and band tails must be taken into account
[29,30]. We follow Stern's [31,32] approach to calculate the envelope matrix element and use
the Gaussian HalperinLax bandtail (GHLBT) model to calculate the densities of states.
Finally, we fit Eq. (6) to a linear function and use the obtained expression in our solver
substituting it into Eqs. (4) and (5). In a heavily doped Ga0.47In0.53As
300K
T
and
0.73 eV
,
( , ) 8.76 10
g n p
possible to achieve the material gain greater than
required for the SPP amplification.
eV 7 .12/6
0e
2
b
mM
for Ga0.47In0.53As
183
d
(4.3 10 cm )
and it is
N
at
1616
(min( , ) 3.7 10 )
n p
1630 cm
(Figs. 3(b) and (3c)) that is
1
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Fig. 4. Gain, current density (dashed lines correspond to the electron current and solid lines to
the hole current) and carrier density distribution across the active region of the plasmonic
waveguide at four different biases for two different values of the power per unit guide width.
Taking into account stimulated emission, we solve Eq. (4) numerically in the same way as
it was done in section 2 (Fig. 4). For a small signal of 5 mW/µm or less (Fig. 4(a)), the
stimulated emission recombination rate is greater than the spontaneous but its absolute value
is nevertheless quite small
np
(()/ )
eUJJL
(Fig. 4(a)). In this case, the net SPP gain G at V 1.07 V is positive and equals
0
LJJ eU
/ )(
p
in the region near the metalsemiconductor interface that affects the carrier
density distribution. The material gain is smaller (Fig. 4(b)) and the net SPP gain becomes
negative (
cm 620
G
at V 1.07 V and
P
and does not affect the carrier distribution
1
1/ ( )
g z S z dz
( ) 1780 1630 150cm .
L
x
PR
At a high signal power (Fig. 4(b)),
n
1
50 mW/μm).
5. Conclusion
To conclude, we have proposed a novel SPP amplification scheme that utilizes a compact
electrical pumping and gives an ability to design really nanoscale amplifiers and spasers [33].
For the analysis of the scheme, we have developed a fully selfconsistent onedimensional
steadystate model of the amplifier and presented an accurate numerical solution.
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Acknowledgments
This work was supported in part by the Russian Foundation for Basic Research (grants no. 09
0700285, 100700618 and 110700505), by the Ministry of Education and Science of the
Russian Federation (grants no. P513 and P1144) and by the grant MK334.2011.9 of the
President of the Russian Federation.
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