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Surface plasmon dispersion relation for normal and tangential modes
Physica A 170 (1991) North-Holland
SURFACE
673-681
PLASMON
TANGENTIAL O.K.
DISPERSION
RELATION
FOR NORMAL
AND
MODES
HARSH
Department of Post-Graduate Studies and Research in Physics. Feroze Gandhi College, Rae Bareli, 229001, India
S.P. GODIA Department of Physics, V.S.S. D. College, Kanpur,
B.K.
208002, India
AGARWAL
X-Ray Laboratory,
Physics Department,
Received 7 July 1989 Revised manuscript received
7 June
University of Allahabad,
Allahabad (U. P.). India
1990
A theoretical investigation has been made for the surface plasmon dispersion relations for the case of normal and tangential modes to the surface. The hydrodynamical model has been used. We have obtained the dispersion relation by using the continuity of I#J and d4/dx, where I$ is the scalar potential. The dispersion relations obtained in the present work have been successfully compared with the theoretical and experimental work of several authors.
1. Introduction The studies
of Ritchie
[l], Stern and Ferrell
[2], Powell
[3], Bloch [4,5],
and
Ritchie and Wilems [6] on surface plasmon oscillations are well known. Recently Smithard, Gariere et al. [8] have studied experimentally the surface plasmon
modes
of small
metallic
particles.
Boardman
et al. [9, 101 used
the
double-well model to deal with the problem of surface plasmon oscillations. Several other workers [ll-131 have also experimentally found the dispersion of surface plasmons from electron-energy-loss measurements. Arakawa et al. [14], and Harsh and Agarwal [15,16] have also studied the problem of surface plasmon oscillations. In the present work, the dispersion relation of surface plasmon oscillations for the semi-infinite plane bounded electron gas has been derived for a homogeneous electron density along the x-direction only, using the hydrodynamical model of Bloch [4,5], and Ritchie and Wilems [6].
2. Calculations
of dispersion
relation
the uniform
positive
If we consider
neutralising
background for the electron the velocity V and the electro~1,
gas in a plane-bounded regioin of thickness static potential C$for a hydrodynamic fluid are given
by
and
D,(x, y. z) is the ion density, n and n’ arc where P(n’) is the Fermi pressure, the electron concentrations, and didt is a co-moving time derivative. The Fermi pressure P(n’) may be expressed as
P(n) =
h’(37r’)2~i
The velocity
sm potential
i’? n ly(_r, y, z, t) is given
by
v= -VVf
(3)
Put (3) in (1) and one gets II~~.,.:.I~ m
d(-V’J’)
= eV4 -V
dt
Eq. (4) can transform
dP(n’) I
I?’
(3)
into
(5) The continuity
$
equation
is
(6)
=V.(nVq).
In the process
of linearization.
one can write
615
0. K. Harsh et al. I Surface plasmon dispersion relation
4x9 Y,
2,
t) = ql(x,
4(x, y,
2, t> =
Y,
2) + n,(x, Y,
z,f>+d?
$0(x, y, 2) + 4,(x, Y, 2) + &(x, Y,
?w,y,2, t) = q,:(x,y, 2,
t) + qx,
where no B n, + n2. Substituting (7)-(9) zeroth order.
Wn) =
t) + . . .
23 f)
y,2, t) + . . . ,
5
+ ... >
(7) (8) (9)
in (5) and equating the coefficients of
(10)
&#)
0 .
rl’
i
Y, 2,
0
Eq. (1) turns into ; Pn#y3 = e(bo
(11)
and (2) becomes in zeroth order V2C#J” = 4Tre(n, - 0) .
(12)
In first order, (5) becomes (13) In first order, eq. (2) becomes V*+, = 47rn,e .
(14)
Eq. (6), the equation of continuity in first order, is
an,_ at -V-[n,(x,
Y7 2) VT11
(15)
We take the particle or fluid density as for inside (I) , for outside (II)
(16)
From (13)-( 16)
s_ at
5 cp,+
--WI
2
It, 0
(17)
and
On eliminating
P, between
(17) and (19) and using eq. ( 1X), we can have
(20) where
p’ = Vi.13 and of = 4rrne’lm.
Eq. (20) represents the volume plasmon dispersion relation. It involves only the space and time in the fluid density tz,. Since there is a rectangular symmetry in our problem, the space and time can be represented as H,(X, y. z, t) = n, (x, y. 2) c ““A’
(21)
and also n,(x.
I’, 2) = c
On substituting
(22) into
d ‘x, i)h-2 = (fP where
X,(X) W( L’. 2)
~ K’)X(x)
K’ = (co:. - w;?,)/p’
(20),
(22)
we get
.
(23)
and K’ is a constant.
Eq. (23) has the solution (24)
where I’ = (K” - K’). d, (x. y. z, I) given by eq. (18) can be separated 4,(x.
y,
2,
t) = 4,(x, I’, 2) c
and #+(x, Y, z) can be expanded
&,(x, y, z) = With
c 4,WY.
‘cvA’
as
(25)
as
2)
the help of eqs. (18) and (26),
(26) we get
0. K. Harsh et al.
d24,
2 dx
/ Surface plasmon dispersion relation
- zP#+(X) = 4?TeX,(x) .
677
(27)
Adding K*4, to both sides of eq. (27) and putting R = 4neA,
2
+ (K* - Kt2)4,(x) = RX,(x) + K24,(x).
(28)
The solution for the interior of (28) may be expressed as 4:“‘(x) = - j$ X,(x) + A’ eiK’x ,
O
For the exterior, the equation becomes V*4,= 0, since there charges, i.e. 11,= 0, thus the solution for the exterior becomes 4,“xt
=
e-iK+
B
,
x>a.
(30)
4:“‘(x) = 4,““‘(x)
(I) >
= a4YW ax
(II) .
From (29)-(31), A’=
are no real
conditions at x = a are
The boundary
a4:“‘H ax
(29)
i2K*K’
(31)
we have
R
VW(a) + Xi (all
eiK’o
(32)
T
where X’(a) = (aX(x)iax),,,. Substituting A’ (eq. (32)) in eq. (29), we get the complete solution for the interior as 4%,
Y, z) = - $
X,(x) + i2;*;:;Y.”
[&IX,(a) + Xi (a)]
(33)
and
n;“’ = c
AX,(x)
W( y, z) .
(34)
To obtain the surface plasmon dispersion relation, we put the hydrodynamical condition of zero electronic velocity normal to the surface. Thus the normal velocity V, and acceleration V, of electrons are given by
V= V,(x) em”“’ ,
p, =
5
v(#),
-
in
On using
(35)
p’vn nil
eqs. (33).
=
I
()
at
(34) and (36).
1x1= n
(36)
we have
(37)
Eq. (37) can be transformed
The dispersion the dispersion
into
relations (37) and (38) represent the normal modes. In deriving relation (38). the boundary conditions at x = a have been used
iis
Here P. is the dielectric constant of the medium in which the rectangular slab is embedded and F_ is the dielectric constant of the dielectric slab embedded in the vacuum.
Now, if we take the corresponding
solutions
of eqs. (24). (29) and
(30) as
In,
4,
=
$
X,(x)
+ A’
e”’ .
o<.r
(41)
and +,“’
then
= B e--li\ .
the new dispersion
.Y> Cl . relation
(42) for tangential
modes
may be derived
as
(43)
0. K. Harsh et al. I Surface plasmon dispersion relation
679
3. Results
(a) If we put E, = +l and K’ = 0 in eqs. (38) and (43), then the well known result of Stern and Ferrell [2] can be obtained as Wk
=
w,/v2.
(44)
Eqs. (38) and (43) represent the general characteristics of surface plasmon oscillations at E, = 2 1. If cZ = 0, then there is a true longitudinal collective volume plasmon oscillation. (b) The condition E_ = -1 in (38) and (43) can also be applied to the general volume plasmon dispersion relation [17, 181 for the free electron gas, as 2 +$+>=-l
(45)
w
also gives w = writi. (c) For a fixed electron density, one can vary the wave vector K and observe the change in the surface plasmon dependence on momentum. Calculations have been performed using (38) and (43). For Mg (n = 8.6 x 10” electrons/ cm’) we give the calculated curves because experimental data exist for it. Fig. 1 shows the square of the reduced surface plasmon frequency (w/w,)’ versus K for values of KC 0.5K,. For K = 0,the surface mode approaches w,/fi. The dispersion curve for K' = 1 exhibits the homogeneous material behaviour. The curve is monotonically increasing and is fitted well by Ritchie’s [19] and Kunz’s [13] results, fig. 1. For higher values of K, curves for tangential and normal modes diverges rapidly as was expected theoretically. (d) Comparison
2 OK =w;+-
of (20) and (21) gives
V2,K2 3
.
(46)
which is the volume plasmon dispersion relation for both the normal and tangential modes. Relation (46) has close resemblance with the Pines formula [I71 2 OK
=02,+---
3V;K2 5 .
6X0
1
3 i P
,ol
__----o-------_-_a
-_
-11 e
‘j 0.9
0.8
07
o.l[
0.0
,
,
,
,
,
0.1
0.2
0.3
0.L
0.5
Fig. 1. Plot of (w/q,)’ B: volume
surface plasmon
dispersion
in the pl~mc bounded obtained
as 2~function
plasmon
dispersion
electron
points are the cxperimcntal
of K (wave vector). curve according
curve for tangential gas (eq. (43)):
using the hydrodynamicel
07
0.6
0.8
A: volume plasmon dispersion
to Bohm
modes. obtained
and Pines (ref.
[ 17).
curve (cq.
cq. (47)):
using the hydrodynamical
D: surface plasmon dispersion cur\c
mode in the plane hounded
data of Kunz
__,
(a-‘,
L (46));
-~~“;------_t
clcctron
C:
model
for normal mode
gas (cq. (38)).
The solid
[ 131
Acknowledgement One of us (OKH) is Delhi, India for providing
grateful
to
the financial
the
university grants commission, New assistance to conduct this research.
0. K. Harsh et al. I Surface plasmon dispersion relation
681
References [II R.H. Ritchie, Phys. Rev. 106 (1957) 874. PI E.A. Stern and F.A. Ferrell, Phys. Rev. 120 (1960) 130. [31 C.J. Powell, J. Nucl. Energy C: Plasma Phys. 2 (1961) 1. [41 F. Bloch, Z. Phys. 81 (1933) 363. I51 F. Bloch, Helv. Phys. Acta 1 (1934) 358. (61 R.H. Ritchie and R.E. Wilems, Phys. Rev. 178 (1969) 372. [71 M.A. Smithard, Solid State Commun. 13 (1973) 153. and N.A. Smithard, Solid State Commun. 16 (1975) 113. [81 J.D. Gariere, R. Rechsteiner B.V. Paranjape and R. Teshima, Surf. Sci. 49 (1975) 275. [91 A.D. Boardman, B.V. Paranjape and Y.O. Nakamura, Phys. Stat. Sol. 75 (1976) 347. 1101 A.D. Boardman, [Ill T. Kloos, Z. Phys. 208 (1968) 77. [=I J.B. Swan, A. Otto and H. Fellenzer, Phys. Stat. Sol. 23 (1967) 171. iI31 C. Kunz, Z. Phys. 196 (1966) 311. M.W. Williams, R.N. Hamm and R.H. Ritchie, Phys. Rev. Lett. 31 (1973) [I41 E.T. Arakawa, 1127. O.K. Harsh and B.K. Agarwal, Physica B 144 (1987) 114. O.K. Harsh and B.K. Agarwal, Physica B 150 (1988) 378. D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1964). C. Kittel, Introduction to Solid State Physics, 4th ed. (Wiley-Eastern, New Delhi, 271. 1191 R.H. Ritchie. Prog. Theor. Phys. 29 (1963) 667.
1974) p,
SURFACE
673-681
PLASMON
TANGENTIAL O.K.
DISPERSION
RELATION
FOR NORMAL
AND
MODES
HARSH
Department of Post-Graduate Studies and Research in Physics. Feroze Gandhi College, Rae Bareli, 229001, India
S.P. GODIA Department of Physics, V.S.S. D. College, Kanpur,
B.K.
208002, India
AGARWAL
X-Ray Laboratory,
Physics Department,
Received 7 July 1989 Revised manuscript received
7 June
University of Allahabad,
Allahabad (U. P.). India
1990
A theoretical investigation has been made for the surface plasmon dispersion relations for the case of normal and tangential modes to the surface. The hydrodynamical model has been used. We have obtained the dispersion relation by using the continuity of I#J and d4/dx, where I$ is the scalar potential. The dispersion relations obtained in the present work have been successfully compared with the theoretical and experimental work of several authors.
1. Introduction The studies
of Ritchie
[l], Stern and Ferrell
[2], Powell
[3], Bloch [4,5],
and
Ritchie and Wilems [6] on surface plasmon oscillations are well known. Recently Smithard, Gariere et al. [8] have studied experimentally the surface plasmon
modes
of small
metallic
particles.
Boardman
et al. [9, 101 used
the
double-well model to deal with the problem of surface plasmon oscillations. Several other workers [ll-131 have also experimentally found the dispersion of surface plasmons from electron-energy-loss measurements. Arakawa et al. [14], and Harsh and Agarwal [15,16] have also studied the problem of surface plasmon oscillations. In the present work, the dispersion relation of surface plasmon oscillations for the semi-infinite plane bounded electron gas has been derived for a homogeneous electron density along the x-direction only, using the hydrodynamical model of Bloch [4,5], and Ritchie and Wilems [6].
2. Calculations
of dispersion
relation
the uniform
positive
If we consider
neutralising
background for the electron the velocity V and the electro~1,
gas in a plane-bounded regioin of thickness static potential C$for a hydrodynamic fluid are given
by
and
D,(x, y. z) is the ion density, n and n’ arc where P(n’) is the Fermi pressure, the electron concentrations, and didt is a co-moving time derivative. The Fermi pressure P(n’) may be expressed as
P(n) =
h’(37r’)2~i
The velocity
sm potential
i’? n ly(_r, y, z, t) is given
by
v= -VVf
(3)
Put (3) in (1) and one gets II~~.,.:.I~ m
d(-V’J’)
= eV4 -V
dt
Eq. (4) can transform
dP(n’) I
I?’
(3)
into
(5) The continuity
$
equation
is
(6)
=V.(nVq).
In the process
of linearization.
one can write
615
0. K. Harsh et al. I Surface plasmon dispersion relation
4x9 Y,
2,
t) = ql(x,
4(x, y,
2, t> =
Y,
2) + n,(x, Y,
z,f>+d?
$0(x, y, 2) + 4,(x, Y, 2) + &(x, Y,
?w,y,2, t) = q,:(x,y, 2,
t) + qx,
where no B n, + n2. Substituting (7)-(9) zeroth order.
Wn) =
t) + . . .
23 f)
y,2, t) + . . . ,
5
+ ... >
(7) (8) (9)
in (5) and equating the coefficients of
(10)
&#)
0 .
rl’
i
Y, 2,
0
Eq. (1) turns into ; Pn#y3 = e(bo
(11)
and (2) becomes in zeroth order V2C#J” = 4Tre(n, - 0) .
(12)
In first order, (5) becomes (13) In first order, eq. (2) becomes V*+, = 47rn,e .
(14)
Eq. (6), the equation of continuity in first order, is
an,_ at -V-[n,(x,
Y7 2) VT11
(15)
We take the particle or fluid density as for inside (I) , for outside (II)
(16)
From (13)-( 16)
s_ at
5 cp,+
--WI
2
It, 0
(17)
and
On eliminating
P, between
(17) and (19) and using eq. ( 1X), we can have
(20) where
p’ = Vi.13 and of = 4rrne’lm.
Eq. (20) represents the volume plasmon dispersion relation. It involves only the space and time in the fluid density tz,. Since there is a rectangular symmetry in our problem, the space and time can be represented as H,(X, y. z, t) = n, (x, y. 2) c ““A’
(21)
and also n,(x.
I’, 2) = c
On substituting
(22) into
d ‘x, i)h-2 = (fP where
X,(X) W( L’. 2)
~ K’)X(x)
K’ = (co:. - w;?,)/p’
(20),
(22)
we get
.
(23)
and K’ is a constant.
Eq. (23) has the solution (24)
where I’ = (K” - K’). d, (x. y. z, I) given by eq. (18) can be separated 4,(x.
y,
2,
t) = 4,(x, I’, 2) c
and #+(x, Y, z) can be expanded
&,(x, y, z) = With
c 4,WY.
‘cvA’
as
(25)
as
2)
the help of eqs. (18) and (26),
(26) we get
0. K. Harsh et al.
d24,
2 dx
/ Surface plasmon dispersion relation
- zP#+(X) = 4?TeX,(x) .
677
(27)
Adding K*4, to both sides of eq. (27) and putting R = 4neA,
2
+ (K* - Kt2)4,(x) = RX,(x) + K24,(x).
(28)
The solution for the interior of (28) may be expressed as 4:“‘(x) = - j$ X,(x) + A’ eiK’x ,
O
For the exterior, the equation becomes V*4,= 0, since there charges, i.e. 11,= 0, thus the solution for the exterior becomes 4,“xt
=
e-iK+
B
,
x>a.
(30)
4:“‘(x) = 4,““‘(x)
(I) >
= a4YW ax
(II) .
From (29)-(31), A’=
are no real
conditions at x = a are
The boundary
a4:“‘H ax
(29)
i2K*K’
(31)
we have
R
VW(a) + Xi (all
eiK’o
(32)
T
where X’(a) = (aX(x)iax),,,. Substituting A’ (eq. (32)) in eq. (29), we get the complete solution for the interior as 4%,
Y, z) = - $
X,(x) + i2;*;:;Y.”
[&IX,(a) + Xi (a)]
(33)
and
n;“’ = c
AX,(x)
W( y, z) .
(34)
To obtain the surface plasmon dispersion relation, we put the hydrodynamical condition of zero electronic velocity normal to the surface. Thus the normal velocity V, and acceleration V, of electrons are given by
V= V,(x) em”“’ ,
p, =
5
v(#),
-
in
On using
(35)
p’vn nil
eqs. (33).
=
I
()
at
(34) and (36).
1x1= n
(36)
we have
(37)
Eq. (37) can be transformed
The dispersion the dispersion
into
relations (37) and (38) represent the normal modes. In deriving relation (38). the boundary conditions at x = a have been used
iis
Here P. is the dielectric constant of the medium in which the rectangular slab is embedded and F_ is the dielectric constant of the dielectric slab embedded in the vacuum.
Now, if we take the corresponding
solutions
of eqs. (24). (29) and
(30) as
In,
4,
=
$
X,(x)
+ A’
e”’ .
o<.r
(41)
and +,“’
then
= B e--li\ .
the new dispersion
.Y> Cl . relation
(42) for tangential
modes
may be derived
as
(43)
0. K. Harsh et al. I Surface plasmon dispersion relation
679
3. Results
(a) If we put E, = +l and K’ = 0 in eqs. (38) and (43), then the well known result of Stern and Ferrell [2] can be obtained as Wk
=
w,/v2.
(44)
Eqs. (38) and (43) represent the general characteristics of surface plasmon oscillations at E, = 2 1. If cZ = 0, then there is a true longitudinal collective volume plasmon oscillation. (b) The condition E_ = -1 in (38) and (43) can also be applied to the general volume plasmon dispersion relation [17, 181 for the free electron gas, as 2 +$+>=-l
(45)
w
also gives w = writi. (c) For a fixed electron density, one can vary the wave vector K and observe the change in the surface plasmon dependence on momentum. Calculations have been performed using (38) and (43). For Mg (n = 8.6 x 10” electrons/ cm’) we give the calculated curves because experimental data exist for it. Fig. 1 shows the square of the reduced surface plasmon frequency (w/w,)’ versus K for values of KC 0.5K,. For K = 0,the surface mode approaches w,/fi. The dispersion curve for K' = 1 exhibits the homogeneous material behaviour. The curve is monotonically increasing and is fitted well by Ritchie’s [19] and Kunz’s [13] results, fig. 1. For higher values of K, curves for tangential and normal modes diverges rapidly as was expected theoretically. (d) Comparison
2 OK =w;+-
of (20) and (21) gives
V2,K2 3
.
(46)
which is the volume plasmon dispersion relation for both the normal and tangential modes. Relation (46) has close resemblance with the Pines formula [I71 2 OK
=02,+---
3V;K2 5 .
6X0
1
3 i P
,ol
__----o-------_-_a
-_
-11 e
‘j 0.9
0.8
07
o.l[
0.0
,
,
,
,
,
0.1
0.2
0.3
0.L
0.5
Fig. 1. Plot of (w/q,)’ B: volume
surface plasmon
dispersion
in the pl~mc bounded obtained
as 2~function
plasmon
dispersion
electron
points are the cxperimcntal
of K (wave vector). curve according
curve for tangential gas (eq. (43)):
using the hydrodynamicel
07
0.6
0.8
A: volume plasmon dispersion
to Bohm
modes. obtained
and Pines (ref.
[ 17).
curve (cq.
cq. (47)):
using the hydrodynamical
D: surface plasmon dispersion cur\c
mode in the plane hounded
data of Kunz
__,
(a-‘,
L (46));
-~~“;------_t
clcctron
C:
model
for normal mode
gas (cq. (38)).
The solid
[ 131
Acknowledgement One of us (OKH) is Delhi, India for providing
grateful
to
the financial
the
university grants commission, New assistance to conduct this research.
0. K. Harsh et al. I Surface plasmon dispersion relation
681
References [II R.H. Ritchie, Phys. Rev. 106 (1957) 874. PI E.A. Stern and F.A. Ferrell, Phys. Rev. 120 (1960) 130. [31 C.J. Powell, J. Nucl. Energy C: Plasma Phys. 2 (1961) 1. [41 F. Bloch, Z. Phys. 81 (1933) 363. I51 F. Bloch, Helv. Phys. Acta 1 (1934) 358. (61 R.H. Ritchie and R.E. Wilems, Phys. Rev. 178 (1969) 372. [71 M.A. Smithard, Solid State Commun. 13 (1973) 153. and N.A. Smithard, Solid State Commun. 16 (1975) 113. [81 J.D. Gariere, R. Rechsteiner B.V. Paranjape and R. Teshima, Surf. Sci. 49 (1975) 275. [91 A.D. Boardman, B.V. Paranjape and Y.O. Nakamura, Phys. Stat. Sol. 75 (1976) 347. 1101 A.D. Boardman, [Ill T. Kloos, Z. Phys. 208 (1968) 77. [=I J.B. Swan, A. Otto and H. Fellenzer, Phys. Stat. Sol. 23 (1967) 171. iI31 C. Kunz, Z. Phys. 196 (1966) 311. M.W. Williams, R.N. Hamm and R.H. Ritchie, Phys. Rev. Lett. 31 (1973) [I41 E.T. Arakawa, 1127. O.K. Harsh and B.K. Agarwal, Physica B 144 (1987) 114. O.K. Harsh and B.K. Agarwal, Physica B 150 (1988) 378. D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1964). C. Kittel, Introduction to Solid State Physics, 4th ed. (Wiley-Eastern, New Delhi, 271. 1191 R.H. Ritchie. Prog. Theor. Phys. 29 (1963) 667.
1974) p,