Electromagnetic field quantization in planar absorbing heterostructures
The quantization scheme for the electromagnetic field in planar absorbing heterostructures has been developed. The scheme is based on the field expansion over a complete set of orthonormal modes. We used two types of the field modes. The first one is defined as the field created by a plane wave inci...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2011
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| Cite this: | Electromagnetic field quantization in planar absorbing heterostructures / V.I. Pipa // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 1. — С. 91-97. — Бібліогр.: 15 назв. — англ. |
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| citation_txt | Electromagnetic field quantization in planar absorbing heterostructures / V.I. Pipa // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 1. — С. 91-97. — Бібліогр.: 15 назв. — англ. |
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| description | The quantization scheme for the electromagnetic field in planar absorbing heterostructures has been developed. The scheme is based on the field expansion over a complete set of orthonormal modes. We used two types of the field modes. The first one is defined as the field created by a plane wave incident at the surface of the structure from the non-absorbing half space. The second type of modes corresponds to the field generated by electric current fluctuations in the absorbing media. To normalize the field modes, the following conditions were used: 1) the time-averaged Poynting vector attributed to the incident wave equals the density of energy flow of elementary quanta of the field energy; 2) for the given frequency and polarization, the total time-averaged Poynting vector equals to zero. The theory is applied to calculate the rate of spontaneous transitions between electron subbands in a quantum well placed near the absorbing layer that can support the surface phonon or plasmon polaritons.
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 91-97.
PACS 42.50.Ct, 42.88.+h, 78.20.Ci
Electromagnetic field quantization
in planar absorbing heterostructures
V.I. Pipa
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
41, prospect Nauky, 03028 Kyiv, Ukraine; e-mail: pipa1@bigmir.net
Abstract. The quantization scheme for the electromagnetic field in planar absorbing
heterostructures has been developed. The scheme is based on the field expansion over a
complete set of orthonormal modes. We used two types of the field modes. The first one
is defined as the field created by a plane wave incident at the surface of the structure
from the non-absorbing half space. The second type of modes corresponds to the field
generated by electric current fluctuations in the absorbing media. To normalize the field
modes, the following conditions were used: 1) the time-averaged Poynting vector
attributed to the incident wave equals the density of energy flow of elementary quanta of
the field energy; 2) for the given frequency and polarization, the total time-averaged
Poynting vector equals to zero. The theory is applied to calculate the rate of spontaneous
transitions between electron subbands in a quantum well placed near the absorbing layer
that can support the surface phonon or plasmon polaritons.
Keywords: electromagnetic field, quantization, absorbing heterostructure, spontaneous
emission.
Manuscript received 16.09.10; accepted for publication 02.12.10; published online 28.02.11.
1. Introduction
In the conventional quantization scheme, the
electromagnetic field is expressed in terms of a set of
orthonormal modes; each mode is quantized as a
harmonic oscillator. For free space, the plane-wave
modes that are defined using the periodical boundary
conditions in a fiction quantization box are usually used.
The field quantization in the presence of bounded non-
absorbing media has been carried out for plane-parallel
layers [1, 2] and semi-infinite dielectrics [3, 4]. For such
systems, the eigenmodes of Maxwell’s equations with
the appropriate boundary conditions are used.
The methods of quantization developed for non-
absorbing media fail when losses are present [5]. The
quantization problem in absorbing media has been
considered by many authors, and many different
schemes have been proposed (see, e. g., [6] for a
review). In this paper, we have modified the approach
[7-11] based on the introduction of noise currents into
the Maxwell equations for the macroscopic
electromagnetic field. In the works [7-11], the current
correlation function has been postulated but not derived.
This function is taken in the form that ensures
preservation of the known canonical field commutation
relations in the presence of absorption. Here, for the
electromagnetic field in the presence of absorbing and
dispersing planar structures, we develop the quantization
scheme based on the mode decomposition. In order to
calculate the field generated by some absorbing medium,
the oscillator model is employed. The modes are
normalized with respect to the electromagnetic energy
flow. It is assumed that the total normal electromagnetic
energy flow through an arbitrary plane parallel to the
interfaces is absent. The theory is applied to calculate the
rate of spontaneous photon emission under transitions
between electron subbands in a quantum well placed in
the near field of an absorbing layer that supports the
surface-phonon polaritons or the surface-plasmon
polaritons.
2. Photons in the presence of bounded non-adsorbing
dielectrics
Let us first consider a structure consisting of two non-
absorbing dielectrics with a common planar interface at
z = 0 that spreads infinitely in x and y directions. The
optical properties of non-magnetic media 1 (z < 0) and
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
91
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 91-97.
2(z > 0) are described by the constant real dielectric
permittivities and , respectively. We set the scalar
potential equal to zero. For the vector potential A , we
use the Coulomb gauge . We imply periodic
boundary conditions in the
1ε 2ε
0=∇A
yx − plane and seek for the
solution in the complex-valued form,
)()(),( tiezt ω−= qrArA , (1)
where is the real propagation vector, and
is the frequency. The function is found
from the equation
)0,,( yx qq=q
0>ω )(zA
( ) 0)( 22
02
2
=−ε+ AA qkz
dz
d , (2)
where , is the Heaviside
step-function,
)()()( 121 zz Θε−ε+ε=ε )(zΘ
ck ω=0 . The electric field and
magnetic field are found from a vector potential:
, . At the interface
E
H
AE 0ik= AH ×∇= 0=z , the
tangential components of E and are continuous. H
The set of modes consists of the fields created by
the plane waves that are incident at the interface 0=z
from either half space [3, 4]. The modes are specified by
the complex quantum number },,,{ ςω=μ vq where
ps,=ν labels polarization of the incident wave and
. The subscripts + and – denote propagation of the
incident wave to the right and left, respectively. Let
and
±=ς
( ) ( ) ( )+
ν
+
ν
+
ν = 1eA ii A ( ) ( ) ( )−
ν
−
ν
−
ν = 2eA ii A be the amplitudes of
the incident waves. The unit vectors of polarization are
defined as
( ) ( )0,,1
xysjs qq
q
−=≡± ee ,
( ) ( 2
0
,1 qk
qk jz
j
jp −±
ε
=± qe
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
) , (3)
where 22
0 qkk jjz −ε= , the subscript denotes
the media. The mode
2,1=j
},,,{ +ω vq are given by
( ) ( ) ( ) ( ) ( )( zikzik
i
zz ereAz 11
1121
−−
νν
+
ν
+
ν
+
ν += eeA ), , (4) 0≤z
( ) ( ) zik
i
zetA 2
212
+
νν
+
ν= e , . 0≥z
Here, and ( ) are the Fresnel
amplitudes of reflection and transmission,
jjr ′ν jjt ′ν jj ′≠
jzjjjz
jzjjjz
jj kk
kk
r
εε
εε
′′
′′
′ν +
−
= , ( jj
j
j
jj rt ′ν
′
′ν += 1
ε
ε ). (5)
Let us now calculate the spectral and angular
density of energy flow associated with the above
introduced modes. To calculate the Poynting vector,
][
4
HES ×
π
=
c , (6)
we express the physical fields and via the real
vector potential. For z-component of the Poynting vector
averaged over time, which corresponds to rightwards–
propagating incident wave, we find
E H
( ) ( )
zizi kS 1
2
2
),( +
ν
+
ν π
ω
=ω Aq . (7)
Replacing with and with zk1 zk2− ( )+
νiA ( )−
νiA in
Eq. (7), we get the classical expression for the density of
the normal flow ( )−
ν ziS associated with the leftwards–
propagating incident wave. The same values associated
with a photon with the energy are given by ωh
( ) ( )
22 L
SS zizi π
ω
=−= −
ν
+
ν
h , (8)
where is the area of normalization in the 2L yx −
plane. Comparing the classical and quantum expressions
for the energy flows, we find
( )
z
i kL
A
1
2
2 h
=+
ν , ( )
z
i kL
A
2
2
2 h
=−
ν . (9)
In the conventional quantization scheme, the
amplitudes ( )±
νiA are determined from the
orthonormalization condition [3] for scalar product
corresponding to the electromagnetic energy density. It
is convenient to write this condition in the following
form
( ) ( ) ( ) ( ) ( ) )ω(ωδδδ
ω
π2zε ςςνν2
2
νω ′−= ′′
∞
∞−
ς′∗
′′
ς
ων∫ L
czzdz hAA .(10)
The amplitude factors ( ) 2±
νiA determined from
Eq. (10) with ς′ν′=ςν ,, coincide with those given by
Eq. (9). Thus, the mode normalizations with respect of
the density of electromagnetic energy and the density of
energy flow lead to the same results.
For 0≤z , the time-averaged density of energy
flows ( )±
νzS associated with the modes are given by
( ) ( ) ( )
ziz kARS 1
212ν
2
1
ω
π
−
= +
ν
+
ν , (11)
( ) ( )
ziz kATS 1
221
2
ω
π
−= −
ν
ν−
ν . (12)
Here, 2
1212 νν = rR and 2
2121 νν = tT are the
reflection and transmission coefficients. Using Eqs. (5)
and (9), we obtain
( ) ( ) 0=+ +
ν
−
ν zz SS . (13)
Obviously, equality (13) is valid in dielectric 2 as
well. Thus, for photon modes of each polarization, the
leftwards– and the rightwards–propagating energy flows
compensate each other, and the total energy flow
through an arbitrary plane parallel to the interface is
absent. It is worth to mention that a definition of the
92
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 91-97.
electromagnetic-energy flow, contrary to that of the
electromagnetic energy, remains valid in an absorbing
medium as well. In the next Section, we describe the
field quantization in the presence of an absorbing
medium based on the mode normalization with respect
to the density of electromagnetic energy flow.
3. Photon modes in the presence
of absorbing medium
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
)
)
We now consider the structure consisting of the non-
absorbing dielectric 1 and absorbing dielectric 2
. The material 2 is characterized by the complex
permittivity
( 0<z
( 0>z
)()()( 222 ωε ′′+ωε′=ωε i . We set the scalar
potential equal to zero. For the one-interface structure,
we choose two types of the field modes. The first one
formally coincides with the considered above
modes (Eqs (4), (5) with the complex function ).
The second type of modes corresponds to the
electromagnetic field generated by the noise-current
fluctuations in the absorbing medium 2. It should be
noted that the modes of the first type are defined in the
absence of the noise current, and the field generated by
the current fluctuations is calculated in the absence of
the external incident waves.
},,,{ +ω vq
)(2 ωε
To calculate the field emitted by the absorbing
medium 2, we divide its volume into small cubic cells of
the volume and assume that a point electric dipole
with the moment is placed in the center m of
each cell. The vectors are assumed to have
random orientations; their magnitude does not
depend on position and orientation. This model
corresponds to the isotropic and homogeneous medium.
The vector potential of the field emitted by the
oscillator m satisfies the equation
VΔ
tie ω)( −mp
)(mp
0p
mA
)()(4)()( 0
2
0 zikzk Θ−δπ=ε+∇∇−Δ mrpAAA mmm .
(14)
The solution of Eq. (14) obeying the outgoing-
wave boundary conditions as has been
obtained in [12]. Writing
±∞→z
( )∑ ω−−=
q
mrq
m qArA tii
z emzt ),,(),( , (15)
we have for and 0≤z zmz ′→
( )( ) ( ) zikzki
z
zz ee
k
tkzz 12
12
2
2102),,( −
ν
−
ν
−
ν
′ν∑ π
=′ epeqA . (16)
Assuming that the fields emitted by different
oscillators are incoherent, we replace the time-averaged
Poynting vector by a sum of the partial Poynting vectors
corresponding to radiation of separate cells. Therefore,
the density of flow emitted to the outside of the
absorbing dielectric is given by
( ) ∑π
ω
=
m
mA z
em
z kS 1
2 Re
2
. (17)
This expression has to be averaged over all possible
directions of the dipole moment p . Let )(pf denotes
the orientation average of the function . Replacing
the discreet variable by the continuous vector
)(pf
m r′ , we
calculate the sum by the integration,
∑ ∫ ∫
∞
′′′
Δ
→
m 0 2
1
L
ydxdzd
V
. (18)
Upon the integration over and , the terms
with
x′ y′
qq ≠′ and ν≠ν′ disappear (in Eq. (17), the
averages over the orientations do not depend on m ). We
get
3
(
2
02 p
s =p)e , ( )
( ) 2
2
0
2
2
22
02
2 3
)(
ε
+
=−
qk
kp z
p
q
pe . (19)
The integration over z′ yields
( ) ( )∑
ν
ν ω=
q
q
,
),(em
z
em
z SS , (20)
where
( ) ( 12
2
2
2
0 1
3
2
),( νν −
ε ′′Δ
ωπ
−=ω R
VL
p
S em
z q ) . (21)
To determine the unknown parameter Vp Δ
2
0 ,
we use the fact that for the modes of each polarization
for the leftwards– and the rightwards–propagating
energy flows compensate each other. Substituting the
result of Eq. (21) into Eq. (13) for ( )−
νzS , we get
2
22
0 4
3
π
ε ′′Δ
=
Vp h . (22)
Let ( ) ( ) ( )−
ν
−
ν
−
ν ≡ 200 eA A be the amplitude of a wave
incident at the interface 0=z from a volume of the
absorbing medium. The transmitted wave ( )0≤z is
given by
( ) ( ) ( ) ( ) tiiem eAt ω−−
νν
−
ν = rkerqA 1
1,, , (23)
where ( ) ( )
210 ν
−
νν ≡ tAA em , ),( 11 zk−= qk , and
( ) ( )( ) zik
z
zezzd
k
kiA ′−
ν
∞
−
ν ′′ωπ
−= ∫ 2)(2
2
02
0
0 pe . (24)
The orientation-averaged amplitude factors become
( ) ( ) ( )
( )20221
22
222
022
2
22
0 ,
Lkk
qkk
A
Lk
kA
z
zz
p
z
z
s
εε
+′
=
′
= −− hh , (25)
( ) ( ) 22 +
ννν = i
em AEA , (26)
where 121 νν −= RΕ is the emissivity of the semi-infinite
media 2. We will call the plane waves (23) by
},,,{ emvqω modes. In the limit , the amplitudes
(25) coincide with those given by Eq. (9). In this case,
the wave (23) corresponds to mode
02 →ε′′
},,,{ −ω vq ( )0≤z .
93
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 91-97.
Thus, in the half space , the electromagnetic
field (13) can be presented by the modes
(Eqs (4), (9)) and the modes (Eqs (23),
(25), (26)):
0≤z
},,,{ +ω vq
},,,{ emvqω
∑
μ
ω−
μμ += ..)(),( ccezat tiiqrArA (27)
Here Σμ denotes the summation over q , ν ,
em,+=ς and the integration over the frequency
. The quantization of the field is to regard the
coefficients and in Eq. (27) as a photon
annihilation and creation operators, respectively. Note
that the appropriate solutions of Eq. (14) for , with
the magnitude of the dipole moment given by Eq. (22),
provide the quantum description of the electromagnetic
field inside the absorbing dielectric 2 as well. One can
show that the operator-valued (Eq. (27)) satisfies the
commutation relation
( ∞≤ω≤0
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
)
μa ∗
μa
0≥z
Â
)(4),(),,( 2 rrrr ′−δδπ=⎥
⎦
⎤
⎢
⎣
⎡
∂
′∂
lk
k
l ci
t
tAtA h , (28)
which has been postulated but not derived in works
[7-11].
The obtained results may be extended to multilayer
absorbing structures. Consider, for example, a non-
transparent multilayer structure bounded by a
non-absorbing dielectric (
( 0>z )
)0≤z . In the half space
, the field can be presented by the same modes
and , but the amplitudes of
reflection and transmission in Eq. (4) and the emissivity
in Eq. (26) have to be replaced by generalized
expressions taking into account all interfaces. For semi-
transparent absorbing structures ( , the mode
expansion (27) for includes also the mode
– the field created by a wave which is
incident at the interface
0≤z
},,,{ +ω vq },,,{ emvqω
)lz ≤≤0
0≤z
},,,{ −ω vq
lz = from the non-absorbing
half space . In this case, the coefficient in
Eq. (26) is the emissivity of the layer .
lz > νΕ
( )lz ≤≤0
4. Radiative lifetime of an excited two-level system
In this Section, we examine the spontaneous decay of an
excited two-level atomic system placed near a plane
surface of an absorbing medium. It is known (see, e.g.,
[13] and references therein) that the spontaneous decay
is controlled by the configuration and dielectric
properties of atom’s macroscopic surrounding. In
classical electrodynamics, an emitting atom is described
as an oscillating point electric dipole driven by the
reflected part of its own radiation field. In quantum
theory, modification of the spontaneous emission stems
from a dependence of the transition probability on the
position of the atom. For non-absorbing dielectrics,
when the relative emission rate is defined as a ratio of
the emission rate to its bulk value, the classical and
quantum theories (in dipole approximation) gave the
same result [3]. Here, we compare the analogous results
for an atomic system in the presence of an absorbing and
dispersing medium.
We first consider a two-level atom located at the
position ( )00 ,0,0 z−=r in the non-absorbing
dielectric 1, is the distance to the absorbing dielectric 0z
( )0≥z . The interaction of electrons with a weak
electromagnetic field is describes by
( ∇−= A
cm
eiH h
int ) , (29)
where and are the charge and mass of electron. For
transitions between the first excited state 2 and the
ground state 1, the spontaneous emission rate
e m
211 W≡τ
( τ is the radiative lifetime of the excited state and
is the transition probability) is determined using the
Fermi golden rule:
21W
((∑
μ
μ −−ωδ ))π
= 12
2)(
1221
2 EEMW h
h
. (30)
Here, is the matrix element of the transition,
and are the electron energies. The wavelengths
of emitted photons are assumed to be large as compared
to the atom size. So, the vector potential can be
replaced with (dipole approximation). Replacing
the summation over
)(
12
μ
M
2E 1E
)(rA
)( 0rA −
q by the integration, we obtain
( ) ( )( )∑ ∫
ςν
ν −
π
=ω
ς
,
2
2102
2
0
21
)(
2
)( qdA dzLkW
h
, (31)
where is the dipole matrix element, 21d ck ω=0 , and
h)( 12 EE −=ω . Let 0τ denotes the radiative lifetime
of the atom placed in a boundless dielectric with the real
permittivity 1ε . Using the bulk modes },,,{ ±ω vq
( ,012 =νr )121 =νt , we get
h3
41 1
2
21
3
0
0
ε
=
τ
dk
. (32)
This rate coincides with that for emission of
conventional bulk photons introduced using the periodic
boundary conditions [13]. Substituting the functions
given in Eqs (4), (9) and Eqs (23), (26) into Eq. (31) for
the photon modes, we obtain the following expression
after straightforward calculation:
∫
∞ φ
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
ξ−ε
ξ
ε
+=
τ
τ
0
11
0 1Re
8
31 ierd . (33)
Here, ( )0/ kq=ξ , ξ−ε=φ 1001 2 zk , and r is the
effective amplitude of reflection:
ϑ
ε
ξ
+ϑ
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
ε
ξ
+= 2
12
1
2
12
1
12 cos2sin1 pps rrrr , (34)
94
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 91-97.
where is the angle between the vector and the
interface normal. Note that the relative spontaneous
emission rate determined by Eqs (33), (34) coincides
with that given by the classical theory [13]. Our
calculations show that such agreement takes place also
for an atom placed near a multilayer absorbing structures
and semi-transparent slab. The obtained agreement
between the classical and quantum theories (in the dipole
approximation) forms a check on the validity of our
quantization scheme.
ϑ 21d
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
)
We now consider the spontaneous emission for
intersubband transitions in a QW. The infinite deep
rectangular QW of the dimension is placed in the
non-absorbing dielectric 1 parallel to the absorbing
cladding layer ( . The QW center is at the
distance from the interface . We consider the
direct transitions from the first excited state to the
ground state (parabolic electron subbands). Note that
only photons with polarization provide the
transitions. Our estimations show that
wL
lz ≤≤0
0z 0=z
p
11 <<wz Lk , so
that one may use the dipole approximation. The bulk rate
of the spontaneous emission is given by Eq. (32) where
now
),0,0( 2121 d=d , weLd
2
21 3
4
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
π
= , 2
2
2
3
wLm
hπ
=ω , (35)
m is the electron effective mass. For the relative
emission rate, one gets Eq. (33) where r is determined
by Eq. (34) with . 0=ϑ
We will study two types of absorbing media that
can support the surface-phonon polaritons (polar
dielectric) and the surface-plasmon polaritons (doped
semiconductor or metal). For polar dielectric, the
dielectric permittivity is described by an oscillator
model:
ωΓ−ω−ω
ωΓ−ω−ω
ε=ωε ∞ i
i
t
l
22
22
)( , (36)
where is the permittivity for high frequencies, ∞ε lω
and are the frequencies of longitudinal and
transverse optical phonons, and is the damping. For a
medium with a free electron gas, the dielectric
permittivity is given by the Drude model:
tω
Γ
)(
)(
2
τ+ωω
ω
−∞ε=ωε
i
p , (37)
where is the plasma frequency, and pω τ is the
plasmon relaxation time.
Fig. 1 provides the related emission rate as a
function of the wavelength of emitted light ω=λ cπ2
for the QW placed near the SiC layer ( ) . The
different can correspond to the QWs with different
thicknesses
λ<<0z
λ
( )2
wL∝λ . The transition wavelength can
also be changed by an applied electric field. For
numerical evaluation, the GaAs quantum well was
chosen ( , , where is the free
electron mass) and the following parameters of SiC [14]
were used:
111 =ε 0067.0 mm = 0m
7.6=ε∞ , 112 s105.149 −×=ω t
( )μm6.12=λ t , ,
and . The curves 1 to 3 show how the
thickness l of SiC layer affects the emission rate. Note
that emission of the photon modes
provides the leading contribution to the rate. The
enhancement of the radiative decay in a narrow spectral
interval is caused by emission of the evanescent waves
with the large wave vector
112 s107.182 −×=ω l ( )μm3.10=λ l
112 s109.0 −×=Γ
},,,{ empqω
q . The wave vector becomes
very large for a wavelength such that
0)(Re 1res =ε+λε . The solution of this equation
( μm57.11res =λ ) determines the surface-polariton
resonance in the semi-infinite SiC. For , the
SiC layer may be considered as a semi-infinite sample,
and the position of the maximal magnitude of the
emission rate is approximately equal (curves
1, 2, and 4). For curve 1,
μm5.0≥l
)(1 λτ− resλ
μm6.11max ≈λ corresponds to
the QW thickness . As the thickness of
SiC layer is reduced, the dispersion relation for polariton
starts to depend upon l , and increases with a
decreasing of (curves 2 and 3). Comparing curves 1
and 4, we see that for QW placed in the near field of the
polar layer
o
A125≈wL l
maxλ
l
( )λ<<0z the emission rate fast decreases
when the distance of QW to the layer increases. From
Eqs (33) and (34) for small distances ( )100 <<kz and
large in-plane wave vectors ( )2
01
2 kq ε>> , we get the
dependence 3
0z0 1∝ττ . Note that the 3
01 z
dependence of the spontaneous rate corresponds to the
variation of the electromagnetic energy density with the
distance from the surface of absorbing medium [15].
11,2 11,6 12,0 12,4
0
40
80
120
160
200
4 3
2
1
τ 0
/
τ
λ, μm
Fig. 1. Relative rate τ0/τ of spontaneous photon emission from
QW versus the wavelength of the emitted light for different
thicknesses l of SiC layer and distances z0 between QW and the
interface. For z0 = 0.2 μm, l = 1 μm (1), 0.25 μm (2), and
0.05 μm (3); for z0 = 0.3 μm, l = 1 μm (4).
95
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 1. P. 91-97.
4 5 6 7
0
20
40
60
80
100
120
3
2
1
τ 0
/
τ
Ne, 1018cm-3
Fig. 2. Relative rate τ0/τ of spontaneous photon emission from
QW placed at the distance z0 = 0.3 μm from a layer of InAs
with the thickness l versus the electron concentration Ne in the
layer: l = 1 μm (1), 0.3 μm (2), and 0.15 μm (3).
The related spontaneous emission rate for
intersubband transitions in the QW placed near the InAs
layer is presented in Fig. 2 as a function of the electron
concentration in the layer. The calculations were
carried out for the GaAs quantum well with the
thickness (the wavelength of emitted light
equals ). For InAs, the following parameters
were used: , , .
Fig. 2 explores the variation of the emission rate with the
thickness l of the absorbing layer, the distance of the
QW from the nearest interface . If ,
then the InAs layer can be treated as a semi-infinite
medium, and the wavelength corresponding to
surface plasmon resonance is determined by the equation
. Taking into account that the peak
(curve 1) corresponds to the concentration
, we find . For
, the far side of the layer affects the emission
rate (curves 2 and 3). We see that the spontaneous
emission rate can be controlled not only by the variation
of the electron concentration (due to doping or injection)
but by changing the thickness of the absorbing top layer
as well.
eN
o
A140≈wL
μm6.14
3.12=ε∞ 004.0 mm =∗ s108 13−×=τ
μm3.00 =z μm1≥l
resλ
0ε)(εRe 1res =+λ
318 cm105 −×=eN μm4.14res =λ
μm1<l
5. Conclusion
We have presented formalism for the electromagnetic
field quantization in the presence of a planar multilayer
absorbing structure. Field quantization was carried out
by computing the complete set of orthonormal modes
that are the solutions of the Maxwell equations for the
macroscopic electromagnetic field with the appropriate
boundary conditions. We have used the following two
types of the solutions. One type defines the
electromagnetic field related to plane waves that are
incident from the non-absorbing dielectric half space for
both orthogonal polarizations. Another type is the field
generated by the current fluctuations in the absorbing
medium. The modes are normalized with respect to the
electromagnetic energy flow. It is assumed that the
averaged over time normal to the surface component of
the Poynting vector, which is associated with the
incident wave of the frequency ω , equals to the density
of energy flow of the quanta . This condition
determines the amplitudes of the incident waves. To
determine the amplitudes of the waves emitted by the
absorbing medium, we assume that the total
electromagnetic energy flow for the given frequency,
direction, and polarization is absent.
ωh
To demonstrate the applicability of the theory, we
have calculated the rate of the spontaneous photon
emission of an excited two-level atom located in a non-
absorbing dielectric near an absorbing and dispersing
medium. We have proved that the relative spontaneous
emission rate determined in the dipole approximation
coincides with that given by the classical point-dipole
theory. We have also calculated the rate of photon
emission for transitions between electron subbands in a
quantum well placed nearby an absorbing layer
supporting surface phonon or plasmon polaritons. When
the separation between the quantum well and layer is
much smaller than the radiation wavelength (the near-
field zone) the rate was found to be resonant increased.
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© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
97
|
| id | nasplib_isofts_kiev_ua-123456789-117629 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2025-12-07T17:03:46Z |
| publishDate | 2011 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Pipa, V.I. 2017-05-25T17:53:46Z 2017-05-25T17:53:46Z 2011 Electromagnetic field quantization in planar absorbing heterostructures / V.I. Pipa // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 1. — С. 91-97. — Бібліогр.: 15 назв. — англ. 1560-8034 PACS 42.50.Ct, 42.88.+h, 78.20.Ci https://nasplib.isofts.kiev.ua/handle/123456789/117629 The quantization scheme for the electromagnetic field in planar absorbing heterostructures has been developed. The scheme is based on the field expansion over a complete set of orthonormal modes. We used two types of the field modes. The first one is defined as the field created by a plane wave incident at the surface of the structure from the non-absorbing half space. The second type of modes corresponds to the field generated by electric current fluctuations in the absorbing media. To normalize the field modes, the following conditions were used: 1) the time-averaged Poynting vector attributed to the incident wave equals the density of energy flow of elementary quanta of the field energy; 2) for the given frequency and polarization, the total time-averaged Poynting vector equals to zero. The theory is applied to calculate the rate of spontaneous transitions between electron subbands in a quantum well placed near the absorbing layer that can support the surface phonon or plasmon polaritons. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Electromagnetic field quantization in planar absorbing heterostructures Article published earlier |
| spellingShingle | Electromagnetic field quantization in planar absorbing heterostructures Pipa, V.I. |
| title | Electromagnetic field quantization in planar absorbing heterostructures |
| title_full | Electromagnetic field quantization in planar absorbing heterostructures |
| title_fullStr | Electromagnetic field quantization in planar absorbing heterostructures |
| title_full_unstemmed | Electromagnetic field quantization in planar absorbing heterostructures |
| title_short | Electromagnetic field quantization in planar absorbing heterostructures |
| title_sort | electromagnetic field quantization in planar absorbing heterostructures |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/117629 |
| work_keys_str_mv | AT pipavi electromagneticfieldquantizationinplanarabsorbingheterostructures |