Plasmon resonance sensitivity in fine metal particles
The sensitivity of the wavelength position of the surface plasmon resonances in a prolate and oblate fine metal particles to the refractive index of an embedding solution, a particle shape, and electron temperature is studied theoretically in the framework of different analytical methods. All calcul...
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Grigorchuk, N.I. 2019-06-14T10:37:18Z 2019-06-14T10:37:18Z 2014 Plasmon resonance sensitivity in fine metal particles / N.I. Grigorchuk // Condensed Matter Physics. — 2014. — Т. 17, № 1. — С. 13701:1-6. — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 78.67.-n, 65.80.-g, 73.23.-b, 68.49.Jk, 52.25.Os arXiv:1403.1518 DOI:10.5488/CMP.17.13701 https://nasplib.isofts.kiev.ua/handle/123456789/153491 The sensitivity of the wavelength position of the surface plasmon resonances in a prolate and oblate fine metal particles to the refractive index of an embedding solution, a particle shape, and electron temperature is studied theoretically in the framework of different analytical methods. All calculations are illustrated on the single potassium nanoparticle as an example. В рамках рiзних аналiтичних методiв теоретично вивчається чутливiсть положення довжин хвилi резонансiв поверхневого плазмона у витягнутих i сплюснутих малих металевих частинках до показника заломлення оточуючого розчину, форми самої частинки та електронної температури. Всi обчислення проiлюстрованi на прикладi окремої наночастинки калiю. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Plasmon resonance sensitivity in fine metal particles Чутливiсть плазмонних резонансiв у малих металевих частинках Article published earlier |
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Чутливiсть плазмонних резонансiв у малих металевих частинках |
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The sensitivity of the wavelength position of the surface plasmon resonances in a prolate and oblate fine metal particles to the refractive index of an embedding solution, a particle shape, and electron temperature is studied theoretically in the framework of different analytical methods. All calculations are illustrated on the single potassium nanoparticle as an example.
В рамках рiзних аналiтичних методiв теоретично вивчається чутливiсть положення довжин хвилi резонансiв поверхневого плазмона у витягнутих i сплюснутих малих металевих частинках до показника
заломлення оточуючого розчину, форми самої частинки та електронної температури. Всi обчислення
проiлюстрованi на прикладi окремої наночастинки калiю.
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1607-324X |
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https://nasplib.isofts.kiev.ua/handle/123456789/153491 |
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Plasmon resonance sensitivity in fine metal particles / N.I. Grigorchuk // Condensed Matter Physics. — 2014. — Т. 17, № 1. — С. 13701:1-6. — Бібліогр.: 21 назв. — англ. |
| work_keys_str_mv |
AT grigorchukni plasmonresonancesensitivityinfinemetalparticles AT grigorchukni čutlivistʹplazmonnihrezonansivumalihmetalevihčastinkah |
| first_indexed |
2025-11-25T20:31:27Z |
| last_indexed |
2025-11-25T20:31:27Z |
| _version_ |
1850524221419028480 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 1, 13701: 1–6
DOI: 10.5488/CMP.17.13701
http://www.icmp.lviv.ua/journal
Plasmon resonance sensitivity in fine metal particle
N.I. Grigorchuk
Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine,
14-b Metrologichna St., 03680 Kyiv, Ukraine
Received September 26, 2013, in final form November 4, 2013
The sensitivity of the wavelength position of surface plasmon resonances in prolate and oblate fine metal par-
ticles to the refractive index of an embedding solution, a particle shape, and electron temperature is studied
theoretically in the framework of different analytical methods. All calculations are illustrated on the single potas-
sium nanoparticle as an example.
Key words: sensitivity, surface plasmon resonance, fine metal particles
PACS: 78.67.-n, 65.80.-g, 73.23.-b, 68.49.Jk, 52.25.Os
The wavelength, intensity, and shift of the surface plasmon resonance (SPR) [1] of fine metal particles
(MPs) are sensitive to changes in different physical parameters. The possibility to measure the optical ef-
fects with the refractive index variation has been often used to detect changes in the surroundingmedium
for chemical sensing or for probing the dynamics of biological molecules [2]. An important group of op-
tical sensors is based on exploiting SPR to detect small refractive index changes in close proximity to the
sensing surface of MPs [3–11] and are associated with a biomolecule sensing [8, 9]. Several authors have
demonstrated, on the example of noble nanoparticles, how the SPR position depends on the environment
and themorphology of nanoparticles [3–5, 7]. Much less attention has been paid to the study of sensitivity
modes of a spheroidal MP with a small effect of a core polarization, where the contribution of the bound
electrons can be neglected (e.g., K, Na, Be, Al, etc).
In this Letter, using the new approach specifying the nanoparticle shape, we propose the expressions
for calculation of the plasmonic sensitivity to the changes in the main factors that significantly affect the
position of the SPR modes on the example of potassium nanoparticle.
1. The MPs considered are small in size relative to the wavelength of light. If we change the refractive
index of the embedding medium n by a definite value, ±n, the sensitivity can be expressed in frequency
units as S
j ,!
(n)Æ �!
j ,res
/�n. The resonance frequency (or the corresponding wavelength) for nonspher-
ical MP is given by [4]
!
j ,res Æ!pl
.
q
"
1
Å (1/L
j
¡1)n
2
, (1)
¸
j ,res Ƹpl
q
"
1
Å (1/L
j
¡1)n
2
, (2)
where the Drude model with Re(") ' "
1
¡!
2
pl/!
2 is used, !pl is the plasma frequency, "
1
is the high
frequency dielectric constant, and L
j
refers to the Osborn’s demagnetizing factors [12] in j -th direction.
We have restricted ourselves to the MPs with a spheroidal shape. Then, variables L
k
and L
?
will
play the roles of longitudinal (directed along the revolution axis of a spheroid) and transverse (directed
across this axis) components of the L factor. Since Osborn’s factors are somewhat cumbersome to be used
directly, they can be changed with a high accuracy by the following relations
L
k
Æ
1
(1ÅR
k
/R
?
)
z
, L
?
Æ
1
2
¡
1¡L
k
¢
, (3)
where z Æ log3/log2, R
k
and R
?
are the spheroid semiaxes directed along and across the revolution axis
of a spheroid, correspondingly.
© N.I. Grigorchuk, 2014 13701-1
http://dx.doi.org/10.5488/CMP.17.13701
http://www.icmp.lviv.ua/journal
N.I. Grigorchuk
Equation (2) describes the change in the SPR wavelength when the factors L
j
or refractive index
are changed. Figure 1 depicts the result of calculations for a rodlike potassium nanoparticle. One can
see that the SPR peak wavelength ¸
res
(normalized by plasma wavelength ¸
pl
Æ 2822 Å for K) linearly
depends on n, and is shifted to the red spectral side as n is increased. Physically, this implies that the
Coulomb attractive force between an electronic cloud and a positive metal lattice is weakened with n
and the energy of SPR is reduced. The linear increase of SPR peak with n retains for different values of
the aspect ratio R
k
/R
?
(´ a/b Æ x). Only the line slope changes: it enhances for MPs with larger aspect
ratios. From the slope of the linear fit, the refractive index sensitivity �¸res/�n was calculated to be 256
nm per refractive index unit (RIU) for S
?
and 380 nm per RIU for S
k
, with “golden” x Æ 1.618, or to be
205 nm/RIU for S
?
and 2100 nm/RIU for S
k
, with x Æ 16.18. A tenfold increase of the potassium particle
aspect ratio enhances the S
k
-sensitivity to the changes in the refractive index roughly 5 times. Higher
spheroid aspect ratio supports both the k-SPR mode at a longer wavelength and the ?-mode at a rather
shorter wavelength. One can see that the k-plasmon mode is much more sensitive to the change in the
refractive index than the transverse mode.
2 4 6 8
0
20
40
60
80
Refractive Index , n
re
s
p
l
λ
/ λ
⊥
||
||
Κ
Figure 1. (Color online) SPR peak wavelength modes (k and?) position as a function of medium refractive
index for prolate potassium nanoparticle with different aspect ratios: R
k
/R
?
= 1.618 (thick lines) and =
16.18 (thin lines).
2. By an analogy with the previous case, the sensitivity to the changes in the aspect ratio of a
spheroidal nanoparticle x can be expressed in frequency units as S
j ,!
(x)Æ �!
j ,res/�x. Using equations (1)
and (2) in the determination of sensitivity, one obtains
S
j ,!
(x)Æ
n!pl[1/L
j
(x)¡1℄
©
"
1
Ån
2
[1/L
j
(x)¡1℄
ª
3/2
, (4)
or
S
j ,¸
(x)Æ
n¸
pl
[1/L
j
(x)¡1℄
q
"
1
Ån
2
[1/L
j
(x)¡1℄
. (5)
We can see that the sensitivity of the SPR to the medium can be enhanced by tuning the morphology
of the nanoparticle [3–5, 7]. The shape factor L
j
(x), which governs the geometric tunability of the SPR,
also determines the plasmon sensitivity of nanoparticles. It is easy to find those values of L
j
for which
the sensitivity S
j
(x) becomes maximal. They can be expressed in the wavelength or frequency units by
means of expressions
L
k,¸
Æ
n
2
n
2
¡2"
1
, L
k,!
Æ
n
2
n
2
Å2"
1
(6)
for a maximal longitudinal component of the S
k
, or
L
?,!
Æ
2"
1
¡n
2
2"
1
Ån
2
(7)
13701-2
Plasmon resonance sensitivity in fine metal particle
for a maximal transverse component of the S
?
.
Only a few MPs and dielectric media conform to the first of the two equation (6). For this reason, we
propose somewhat different approach. The derivative �¸res/�n can be represented as a derivative of a
composite function
�¸res
�n
Æ
�¸res
�"
0
�"
0
�n
, (8)
where "0 is the real part of a particle dielectric function. In the high frequency limit of dielectric function,
¸res is
¸res Ƹpl
p
"
1
¡"
0
. (9)
From the plasmon resonance condition
"
0
Æ¡(1/L
j
¡1)n
2
, (10)
and finally, we get
�¸
j ,res
�n
Æ
¸
2
pl
¸
j ,res
(1/L
j
¡1)n. (11)
In the case of MP having a spherical form, the factor L
j
should be put equal to 1/3 in all the above
equations.
3. Some experimental studies [3–5, 10, 13–16] were carried out on metal nanoparticles with different
geometry to find the best nanoparticle configuration to enhance the sensitivity of the SPR response. Let
us theoretically find the best geometrical factors L
j
to enhance the sensitivity of the SPR response for
MPs with a spheroidal shape. To define the spheroid shape which provides the maximal sensitivity, one
can use the following relations:
¡
R
k
/R
?
¢
long Æ
¡
L
k,!
¢
¡1/z
¡1, (12)
¡
R
k
/R
?
¢
trans Æ
¡
L
?,!
¢
¡1/z
¡1, (13)
for longitudinal and transverse components, correspondingly.
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
A x i a l r a t i o , R / R
Sω
(
eV
/
A
R
U
)
K||
⊥
⊥ ||
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
A x i a l r a t io , R / R
S ω
(e
V
/
A
R
U
)
||
⊥
Au
⊥ ||
Figure 2. (Color online) Dependence of the sensitivity of the plasmon resonance components (longitudinal — solid
line and transverse — dashed line) vs semiaxial ratio for spheroidal potassium nanoparticle embedded in the SiO
2
matrix (n Æ 1.56). The inset shows the same dependence for Au nanoparticle embedded in the water.
For instance, if we take the Au nanoparticle with "
1
Æ 9.84 [10] embedded in water (n Æ 1.33), then
from the second equation (6) and equation (7), we obtain L
!
' 0.082 for k and L
!
' 0.835 for ? com-
ponents, which enable us to get from equations (12) and (13) the maxima at (R
k
/R
?
)long Æ 3.76 and at
(R
k
/R
?
)trans Æ 0.12.
13701-3
N.I. Grigorchuk
Figure 2 shows the dependence of plasmon sensitivity components on the axial ratio (in axial ratio
units (ARU)) calculated for the potassium nanoparticle with "
1
Æ 1.24 [17]. We have found that the max-
imal sensitivity can be realized at the axial ratio (R
?
/R
k
)long ' 2 for S
k
, and at (R
?
/R
k
)trans ' 0 for S
?
.
The inset in figure 2 displays the same dependence S
E
(x) for Au nanoparticle. Maximal sensitivities are
attained for Au at (R
?
/R
k
)long ' 0.26 and (R
?
/R
k
)trans ' 8.5 for S
k
and S
?
, correspondingly. Such quanti-
ties of maximal sensitivities for potassium and Au nanoparticles can be obtained from equations (12) and
(13) as well. The data in figure 2 for prolate and oblate potassium and Au nanoparticles display a clear
evidence of the spectral shift of the both components S
k
and S
?
as a function of spheroidal axial ratios.
Such a behavior of both S(x) components is far from the linear one predicted, e.g., in [3] for the single Au
nanorod.
2 4 6 8
2
3
4
5
A x i a l r a t io , R / R
re
s
p
l
⊥
||
||⊥
λ
/ λ
K
0.0 0.2 0.4 0.6 0.8 1.0
1.5
2.0
2.5
3.0
T /
F
re
s
p
l
λ
/ λ
T
⊥
||
Κ
Figure 3. (Color online) The SPR peak wavelength components (k and ?) as a function of the spheroid
axial ratio for potassium nanoparticle embedded in the SiO
2
matrix. The inset shows ¸res(T ) for potas-
sium nanoparticle (with x Æ 1.62) embedded in the SiO
2
matrix. For potassium the Fermi temperature is
TF Æ 2.46£10
4 K [17] and electron concentration is n
0
=1.4£1022 [17].
In figure 3 the dependence of the ratio ¸res/¸pl versus spheroid axial ratio is depicted. As it is seen
in this figure, when the aspect ratio for rodlike potassium nanoparticle increases, the k-component of the
peak of the SPR wavelength ¸res exhibits a blue shift, whereas its ?-component exhibits a red shift. For
oblate potassium nanoparticle, vice versa, with an increase of the semiaxial ratioR
?
/R
k
, the k-component
of the SPR wavelength shifts to the red side, whereas its ?-component shifts to the blue side. From the
slope of the linear dependence of ¸
?,res on the axial ratio, the shape sensitivity �¸res/�x becomes roughly
as high as 84 nm/ARU for S
?
.
4. To characterize the thermal stability of the photonic crystal devices, the study of their thermal
sensitivity is very important [18–20]. It can be expressed in wavelength units as S
j ,¸
(T )Æ �¸
j ,res/�T . The
temperature dependence of ¸res(T ) comes from the T -dependence of the electron concentration inside
the MP [21]
ne(T )Æn
0
·
1Å
¼
2
8
µ
kBT
¹
0
¶
2
¸
, (14)
where ¹
0
Æ ¹(T Æ 0) is chemical potential at zero temperature and n
0
Æ (2m¹
0
)
3/2
/3¼
2
ß
3. Combining
these equations with ¸pl(T )Æ 2¼
/
p
4¼ne(T )e
2
/m and equation (2), we obtain
¸
j ,res(T )'¸
pl
(0)
v
u
u
t
"
1
Å (1/L
j
¡1)n
2
1Å (¼
2
/8)
¡
kBT /¹
0
¢
2
. (15)
13701-4
Plasmon resonance sensitivity in fine metal particle
Therefore,
S
j ,¸
(T )'¸
pl
(0)
¼
2
8
q
"
1
Å (1/L
j
¡1)n
2
h
1Å (¼
2
/8)
¡
kBT /¹
0
¢
2
i
3/2
µ
kB
¹
0
¶
2
T. (16)
Equations (15) and (16) allow one to estimate the shift of SPR wavelength position with temperature and
the temperature sensitivity for any ellipsoidal MPs. It is easy to see from equation (15) and from the inset
in figure 3 that the value of ¸res is shifted with T to the shorter wavelength side, which agrees well with
the results of works [19, 20].
In summary, we propose the formulae that enable one to evaluate the plasmonic sensitivity to the
changes in the main factors considerably affected the SPR mode position in MPs. For prolate (oblate)
potassium and Au nanoparticles, there was found a nonlinear behavior of both sensitivity components
(S
k
and S
?
) as a function of an axial ratio. The effect of the temperature variation on the plasmon sensi-
tivity was demonstrated on the example of a potassium nanoparticle.
References
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http://dx.doi.org/10.1021/nn7003734
http://dx.doi.org/10.1021/jp062536y
http://dx.doi.org/10.1007/s11468-010-9130-2
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http://dx.doi.org/10.1063/1.1334362
N.I. Grigorchuk
Чутливiсть плазмонних резонансiв у малих металевих
частинках
М.I. Григорчук
Iнститут теоретичної фiзики iм. М.М. Боголюбова НАН України,
вул. Метрологiчна, 14-б, 03680 Київ, Україна
В рамках рiзних аналiтичних методiв теоретично вивчається чутливiсть положення довжин хвилi ре-
зонансiв поверхневого плазмона у витягнутих i сплюснутих малих металевих частинках до показника
заломлення оточуючого розчину, форми самої частинки та електронної температури. Всi обчислення
проiлюстрованi на прикладi окремої наночастинки калiю.
Ключовi слова: чутливiсть, поверхневий плазмонний резонанс, малi металевi частинки
13701-6
|