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THz wave generation in cylindrical heterostructure nanowire
Accepted Manuscript Title: THz wave generation in cylindrical heterostructure nanowire Author: Sh. Rahmatallahpur A. Rostami PII: DOI: Reference:
S0030-4026(16)30653-2 http://dx.doi.org/doi:10.1016/j.ijleo.2016.06.031 IJLEO 57817
To appear in: Received date: Accepted date:
16-4-2016 6-6-2016
Please cite this article as: Sh.Rahmatallahpur, A.Rostami, THz wave generation in cylindrical heterostructure nanowire, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.06.031 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
THz wave generation in cylindrical heterostructure nanowire Sh. Rahmatallahpur and A. Rostami
Photonics and Nanocrystal Research Lab. (PNRL), Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz 5166614761, Iran
Abstract- In this paper, we consider THz plasma radiation in ungated cylindrical heterostructure nanowire (NW) or carbon nanotube (CNT) via perturbation approach, considering the effect of viscosity, momentum relaxation time and drift velocity as sufficiently small quantities. The scalar potential, channel charge and current are determined by solving Euler, continuity and Poisson equations in cylindrical coordinates with appropriate boundary conditions using Dyakonov-Shur plasma theory. We obtain a general dispersion relation, plasma frequency, increment and the radiated power. It is shown that in the limit of small radius, the plasma frequency behaves logarithmically while in the limit of large radius, the plasma frequency approaches to the plasma frequency of flat twodimensional electron gas (flat-2DEG). We show that as in the flat-2DEG, drift velocity increases instability of plasma waves while viscosity and momentum relaxation time decreases the instability. Keywords- THz wave, cylindrical heterostructure nanowire, carbon nanotube, plasma waves, viscosity, power radiation.
Introduction An area of technological interest is the creation of cylindrical heterostructure nanowire (CNW) where the material composition and/or doping concentration are modulated in the radial direction of the heterostructure. The cylindrical heterostructure provides a unique advantage compared to homogenous cylindrical nanowire since it allows confinement of two-dimensional electron gases (2DEG) at the semiconductor-semiconductor heterojunction interface. For the specific case of a GaN/AlGaN cylindrical heterostructure nanowire as shown in Figs. 1, carriers are confined within the GaN core by the larger band gap of the AlGaN shell. As illustrated in Figure 1 a constant voltage is applied to the source and a constant current is applied to the drain.
1
FIG. 1. Schematic of the cylindrical heterostructure nanowire with appropriate biasing considered in this work
The hydrodynamic model ,in flat-2DEG, predicts radiation of electromagnetic frequency in the THz range at plasma oscillation frequency which can be used for efficient low cost THz sources [1-3].In twodimensional (2D) electron channels, the collective plasma excitations discovered for the first time by [4, 5]. The generation of plasma waves in low dimensional structures such as 2DEG, have been studied experimentally in [6, 7].In [8] using the hydrodynamic model and within the framework of classical electrodynamics, a transmission line model describing the propagation of electric signals along metallic single wall CNT interconnect , was derived. In [9] a Single wall CNT-based RTD has been analyzed. It has been shown that the oscillation frequency and the output power are greater by one order of magnitude, attaining 16 THz. In [10] Plasma oscillations and terahertz instability has been investigated with Corbino geometry where the oscillations become unstable at symmetric boundary conditions with frequency in the THz ranges .Prior studies have established two distinct contributions to the plasma wave generation regarding the theory of Dyakonov-Shur (DS) plasma instability in flat-2DEG:A constant drain current makes the plasma waves to grow and became unstable while the momentum relaxation time damps out the plasma waves [6-10, 14]. While previous analytical and experiments studies, clearly established the existence of plasma waves in flat-2DEG systems but there is a lack of resource on the theory and applicability of DS plasma theory on the cylindrical systems. The aim of this work is to consider such systems and apply the concept of Dyakonov-Shur theory on the cylindrical heterostructure nanowire and carbon nanotube. This paper applies the concept of DS theory to the cylindrical heterostructure nanowire, considering the effect of viscosity. We show that the viscosity affects the Dispersion law and severely limits the number of modes. The rest of this paper is organized as follows. In Section II we develop hydrodynamic -Poisson model in cylindrical coordinates and formulate 2D plasma in terms of this model. In Section III, this model is used to derive and analyze the plasma dispersion equation including the effect of viscosity for both (CNW) and CNT. We also present in this section the generation
2
of THz plasma waves and emitted power. Our conclusions and a brief summary of the results are presented in Section IV.
Section II The most commonly used approach to predict electronic properties in nanowires is a simplistic Eulercontinuity-Poisson formalism where a coupled system of Euler,continuity and Poisson equations are solved under appropriate biasing conditions to obtain electric field, electric potential , and electron densities. In this simple approach, the equations in cylindrical coordinates are (1a) (1b) (1c) where is the two dimensional fluid velocity , the electron sheet density,
the momentum relaxation time,
the scalar potential,
elementary charge , m the effective mass and confined in the surface of cylinder with radius surface of devices and
Fermi velocity , N
the radius of 2DEG in CNW or CNT, e the the viscosity respectively. The 2DEG has been
.
, is the differential operator on the
. Viscosity, similar to resistivity in a conductor, leads to
degradation of waves and fields. Fluids with ideal viscosity cannot exist in nature, so it is natural to include its effect in the above equations. We assume that under the proper bias, there is an average DC velocity of
along the z direction.For simplicity we consider fluid velocity small compared to the
plasma velocity. Section III We consider a wave propagating on the surface of tube with a complex frequency as. [
]
With
[
]
[
constant values and
]
(2)
small perturbations. Then for the AC parts of equations,
we have (3a) (3b) We can rewrite the above equations as 3
(4) (5) Where we have introduced We relate n to , by omitting * (
from Eq. 4as: [ (
)+
)]
,
(6)
Substituting n in Eq.1c, we get (
)
(
)
For simplicity, we e define
.
Considering above equation and for
we have
(7)
(8) The solution of Eq.8, in general, is the Bessel functions of second kind as {
(9)
The coefficient have been written in such a way that the continuity of potential to be ensured at Integrating Eq.7 for
Substituting
.
, we get
and its left and right derivatives from Eq.9 in above relations, and using the relation , and rearranging terms we finally obtain
[
]
(10)
This gives us the final form of dispersion equation, which shows the general dependency of dispersion on number density
, tube radius , Fermi velocity
, drift velocity
, momentum relaxation time
and viscosity .We have plotted the dispersion relation (10), in Fig2 for a typical value of rc=25 nm, n0=1016 m-2. Where figure2a shows this relation for the first mode number (m=0) and figure 2b shows for higher modes. Fig. 2a also shows that the dispersion relation is symmetric for zero drift velocity but always asymmetric for nonzero drift velocity,
4
v =0 , v =1 0 0
f
5
6
m/ s
Frequency f (THz)
v =1 0 , v =1 0 0
f
6
m/ s
5
v =1 0 , v =0 m/ s 0
f
10
5
a m=0 0 -50
Frequency f (THz)
15
5
6
v =10 , v =10 m/s
40
0
f
m=5
30 20
m=1
10
b 0 50 wave vector k (1/ m)
-100
0 100 wave vector k (1/ m)
Fig. 2: Schematic 2D plasma dispersion for a: m=0,b:m#0
THz emission under Dyakonov-Shur instability conditions: As seen from figure.2, for each , the dispersion relation has two dominant roots as: (
[
The parameter
)]
(11)
is the root of
*
+
and
(12)
The boundary conditions are similar to [1], zero ac density at the source and zero ac current at the drain.
The boundary condition (
)(13)
From the boundary condition of
Or
5
gives
, we obtain
Substituting [
from Eq.11, we obtain (
)]
(
(
)
)
Taking logarithm from both sides, we have
*
(
(
))+
After simplifying above relation, we finally obtain
(
)
Where we have defined
(
(
))
, now returning to definition 12, we have
(
[
)
(
] Or
(
Defining √
6
)
(
(
))
we finally obtain √*
+
(14)
(
))
(
(
))(15)
For m=0, (in one dimensional model) the increment is exactly the same as ref [11]. But either the frequency or the increment as seen from Eq14 and 15 behaves very differently in comparison with flat 2DEG. Section IV: Figure 3a shows the plasma spectrum as a function of rc for (a) n=0,m = 0 (i.e. the first mode) and (b) n=0,m=1..5 (i.e. the higher modes) with different VF. It demonstrates that for the first mode (m=n=0), the frequency increases by increasing the tube radius, then it peaks and then it tends to the constant value (as we will show in the next section this constant value is the frequency of a flat 2DEG).For the higher modes, by increasing the radius, the frequency deceases and tends to the constant value (again the frequency of flat 2DEG). 3
10
Fig a L=200 nm 2 m=0 n=0
Frequency f(THz)
Frequency f (THz)
3
6
V =10 m/s F
1
V =0 F
0 0 10
2
m=5
10
Fig b L=200 nm n=0 6 V =10 m/s F
1
10
m=1 0
1
2
10 10 Radius r (nm) c
10
3
10 0 10
1
2
10 10 Radius r (nm) c
3
10
Fig. 3. Plasma spectrum as a function of rc for a: first mode(m,n)=(0,0) b: higher modes (m>0,n=0)
Figure 4a shows the plasma frequency versus tube length for first mode (m=n=0) and for different value of tube radius and figure 4b shows the increment for the first mode and for rc=10 nm, and without viscosity. In case of a sustained plasma spectrum oscillation, we should have for viscose fluids and
7
for non-viscose fluids.
with
10
2
0.5 =h/m
10
1
Fig b
=0
12
10
increment n (10 )
Frequency f (THz)
Fig a
0
r =1 nm c
r =10 nm
L=150 nm
0 L=23.5 nm
c
r =100 nm
L=126 nm
c
10
-1
10
1
2
10 Length L (nm)
10
-0.5
3
2
10 Length L (nm)
10
3
Fig. 4 a: the plasma frequency versus tube length for first mode and for different value of tube radius, b: the increment with and without viscosity
For
, which is the case of a realistic CNT with radius of order 10nm, we have [12]
Again substituting these relations in Eq14, we get √
(16)
(
) (
(
(
))
(17a)
)(17b)
In this case the plasma spectrum logarithmically depends on the 2DEG radius rc. In this stage we can retrieve dispersion relation in a flat 2DEG by letting
and keeping the surface
charge density, bounded. We call this case a flat 2DEG. Again we use the well-known asymptotic expression [12] √
√
Replacing above approximation in Eq. 10,14,15, we obtain 8
√
√
[
](18)
√
(19) (
)(20)
Which are the dispersion relation, plasma frequency and increment for the ungated 2DEG.By increasing the tube radius, the frequency and increment will approach these values (see also fig3) Power estimation: The energy radiated from a source can be obtained in the far zone approximation as [13] ̂ Where
is the Poynting vector,
is the solid angle , r is distance from the source (the tube) to the
point at which the power is evaluated, ̂ is the unit vector having angle
with the tube axis (z direction)
and is the dipole moment defined as ∭
(21)
The total radiated power is the integration of
over a sphere of radius r:
∮ Upon substituting n from Eq13 to Eq21 we obtain for the dipole moment and the total radiated power ∭
∫
∫
(
)
∫
(22) B is the amplitude of density perturbation and A is the amplitude of potential perturbation where from Eq.6 we have an approximation as
9
For typical value of
, we have P=60 Pico Watt,
which is dramatically small and shows that CNT or nano wire with small radius is inefficient for THz generation, and for
the power became P=6 nano Watt for the same electron density. On
the other hand the electron density have a relation with the total charge
as
, to increase
the radius, we should increase the total charge in order to keep the electron density constant. Conclusion- In this paper, we presented the analysis of a THz source which employs the plasma excitation
method in 2DEG. We studied the plasma oscillations in ungated nanowire or CNT in the presence of small drift velocity and viscosity in the hydrodynamic approximation. Based on this theoretical formalism we solved Euler, continuity and Poisson equations and derived analytical expressions for dispersion relation, the discrete frequencies and the “increment”. Our results show that the plasma spectrum and the increment (Power) depend on the radius of tube. We showed that in nano wire and CNT, the plasma frequency and increment differs considerable with respect to a flat 2DEG. We also have a power estimation of the THz radiation.
Reference
[1] M. Dyakonov, M. Shur, Shallow water analogy for a ballistic field effect transistor: New mechanism of plasma wave generation by dc current, Phys.Rev. Lett. 71 (1993) 2465–2468. [2] M. Dyakonov and M. Shur, “Detection, mixing, and frequency multiplication of terahertz radiation by twodimensional electronic fluid”, IEEE Trans. Electron. Devices, vol. 43, no. 3, pp. 380 – 387 (1996). *3+ M. I. Dyakonov, “Boundary Instability of a Two-Dimensional Electron Fluid”, Semiconductors vol. 42 8, pp. 984 – 988 (2008). [4] S. J. Allen, D. C. Tsui, and R. A. Logan, Phys. Rev. Lett. 38, 980 (1977). [5] D.C. Tsui, E. Gornik, R.A. Logan, Far infrared emission from plasma oscillations of Si inversion layers, Solid State Communications 35 (1980) 875–877. [6] W. Knap, J. Lusakowski, T. Parenty, S. Bollaert, A. Cappy, V.V. Popov, M.S. Shur, "Terahertz emission by plasma waves in 60 nm gate high electron mobility transistors," Appl. Phys. Lett. 84 (2004) 2331–2333. [7] J. Lusakowski, W. Knap, N. Dyakonova, L. Varani, Voltage tunable terahertz emission from a ballistic nanometer InGaAs/InAlAs transistor, J. Appl. Phys. 97 (2005) 064307. [8] A. Maffucci,G.Miamo and F.Villone, "A transmission line model for metallic carbon nanotube interconnects,” International Journal of circuit theory and applications int.j.circ.theor.appl.2008;36:31-51
10
[9] D. Dragoman, M. Dragoman “Terahertz oscillations in semiconducting carbon nanotube resonant-
tunnelling diodes.” Physica E 24 (2004) 282–289. [10]O. Sydoruk, R. R. A. Syms, and L. Solymar, “Plasma oscillations and terahertz instability in field-effect transistors with Corbino geometry,”Appl. Phys. Lett. 97, 263504 (2010). *11+ P. Dmitriev et al “Nonlinear theory of the current instability in a ballistic field-effect transistor,” PRB,1996. [12] G.B Arfken and H.J.Weber, "Mathematical methods for physics,”fifths edition,Academic press. [13]John David Jackson “Classical Electrodynamics,” John Wiley & Sons Ltd. 1962.
11
S0030-4026(16)30653-2 http://dx.doi.org/doi:10.1016/j.ijleo.2016.06.031 IJLEO 57817
To appear in: Received date: Accepted date:
16-4-2016 6-6-2016
Please cite this article as: Sh.Rahmatallahpur, A.Rostami, THz wave generation in cylindrical heterostructure nanowire, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.06.031 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
THz wave generation in cylindrical heterostructure nanowire Sh. Rahmatallahpur and A. Rostami
Photonics and Nanocrystal Research Lab. (PNRL), Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz 5166614761, Iran
Abstract- In this paper, we consider THz plasma radiation in ungated cylindrical heterostructure nanowire (NW) or carbon nanotube (CNT) via perturbation approach, considering the effect of viscosity, momentum relaxation time and drift velocity as sufficiently small quantities. The scalar potential, channel charge and current are determined by solving Euler, continuity and Poisson equations in cylindrical coordinates with appropriate boundary conditions using Dyakonov-Shur plasma theory. We obtain a general dispersion relation, plasma frequency, increment and the radiated power. It is shown that in the limit of small radius, the plasma frequency behaves logarithmically while in the limit of large radius, the plasma frequency approaches to the plasma frequency of flat twodimensional electron gas (flat-2DEG). We show that as in the flat-2DEG, drift velocity increases instability of plasma waves while viscosity and momentum relaxation time decreases the instability. Keywords- THz wave, cylindrical heterostructure nanowire, carbon nanotube, plasma waves, viscosity, power radiation.
Introduction An area of technological interest is the creation of cylindrical heterostructure nanowire (CNW) where the material composition and/or doping concentration are modulated in the radial direction of the heterostructure. The cylindrical heterostructure provides a unique advantage compared to homogenous cylindrical nanowire since it allows confinement of two-dimensional electron gases (2DEG) at the semiconductor-semiconductor heterojunction interface. For the specific case of a GaN/AlGaN cylindrical heterostructure nanowire as shown in Figs. 1, carriers are confined within the GaN core by the larger band gap of the AlGaN shell. As illustrated in Figure 1 a constant voltage is applied to the source and a constant current is applied to the drain.
1
FIG. 1. Schematic of the cylindrical heterostructure nanowire with appropriate biasing considered in this work
The hydrodynamic model ,in flat-2DEG, predicts radiation of electromagnetic frequency in the THz range at plasma oscillation frequency which can be used for efficient low cost THz sources [1-3].In twodimensional (2D) electron channels, the collective plasma excitations discovered for the first time by [4, 5]. The generation of plasma waves in low dimensional structures such as 2DEG, have been studied experimentally in [6, 7].In [8] using the hydrodynamic model and within the framework of classical electrodynamics, a transmission line model describing the propagation of electric signals along metallic single wall CNT interconnect , was derived. In [9] a Single wall CNT-based RTD has been analyzed. It has been shown that the oscillation frequency and the output power are greater by one order of magnitude, attaining 16 THz. In [10] Plasma oscillations and terahertz instability has been investigated with Corbino geometry where the oscillations become unstable at symmetric boundary conditions with frequency in the THz ranges .Prior studies have established two distinct contributions to the plasma wave generation regarding the theory of Dyakonov-Shur (DS) plasma instability in flat-2DEG:A constant drain current makes the plasma waves to grow and became unstable while the momentum relaxation time damps out the plasma waves [6-10, 14]. While previous analytical and experiments studies, clearly established the existence of plasma waves in flat-2DEG systems but there is a lack of resource on the theory and applicability of DS plasma theory on the cylindrical systems. The aim of this work is to consider such systems and apply the concept of Dyakonov-Shur theory on the cylindrical heterostructure nanowire and carbon nanotube. This paper applies the concept of DS theory to the cylindrical heterostructure nanowire, considering the effect of viscosity. We show that the viscosity affects the Dispersion law and severely limits the number of modes. The rest of this paper is organized as follows. In Section II we develop hydrodynamic -Poisson model in cylindrical coordinates and formulate 2D plasma in terms of this model. In Section III, this model is used to derive and analyze the plasma dispersion equation including the effect of viscosity for both (CNW) and CNT. We also present in this section the generation
2
of THz plasma waves and emitted power. Our conclusions and a brief summary of the results are presented in Section IV.
Section II The most commonly used approach to predict electronic properties in nanowires is a simplistic Eulercontinuity-Poisson formalism where a coupled system of Euler,continuity and Poisson equations are solved under appropriate biasing conditions to obtain electric field, electric potential , and electron densities. In this simple approach, the equations in cylindrical coordinates are (1a) (1b) (1c) where is the two dimensional fluid velocity , the electron sheet density,
the momentum relaxation time,
the scalar potential,
elementary charge , m the effective mass and confined in the surface of cylinder with radius surface of devices and
Fermi velocity , N
the radius of 2DEG in CNW or CNT, e the the viscosity respectively. The 2DEG has been
.
, is the differential operator on the
. Viscosity, similar to resistivity in a conductor, leads to
degradation of waves and fields. Fluids with ideal viscosity cannot exist in nature, so it is natural to include its effect in the above equations. We assume that under the proper bias, there is an average DC velocity of
along the z direction.For simplicity we consider fluid velocity small compared to the
plasma velocity. Section III We consider a wave propagating on the surface of tube with a complex frequency as. [
]
With
[
]
[
constant values and
]
(2)
small perturbations. Then for the AC parts of equations,
we have (3a) (3b) We can rewrite the above equations as 3
(4) (5) Where we have introduced We relate n to , by omitting * (
from Eq. 4as: [ (
)+
)]
,
(6)
Substituting n in Eq.1c, we get (
)
(
)
For simplicity, we e define
.
Considering above equation and for
we have
(7)
(8) The solution of Eq.8, in general, is the Bessel functions of second kind as {
(9)
The coefficient have been written in such a way that the continuity of potential to be ensured at Integrating Eq.7 for
Substituting
.
, we get
and its left and right derivatives from Eq.9 in above relations, and using the relation , and rearranging terms we finally obtain
[
]
(10)
This gives us the final form of dispersion equation, which shows the general dependency of dispersion on number density
, tube radius , Fermi velocity
, drift velocity
, momentum relaxation time
and viscosity .We have plotted the dispersion relation (10), in Fig2 for a typical value of rc=25 nm, n0=1016 m-2. Where figure2a shows this relation for the first mode number (m=0) and figure 2b shows for higher modes. Fig. 2a also shows that the dispersion relation is symmetric for zero drift velocity but always asymmetric for nonzero drift velocity,
4
v =0 , v =1 0 0
f
5
6
m/ s
Frequency f (THz)
v =1 0 , v =1 0 0
f
6
m/ s
5
v =1 0 , v =0 m/ s 0
f
10
5
a m=0 0 -50
Frequency f (THz)
15
5
6
v =10 , v =10 m/s
40
0
f
m=5
30 20
m=1
10
b 0 50 wave vector k (1/ m)
-100
0 100 wave vector k (1/ m)
Fig. 2: Schematic 2D plasma dispersion for a: m=0,b:m#0
THz emission under Dyakonov-Shur instability conditions: As seen from figure.2, for each , the dispersion relation has two dominant roots as: (
[
The parameter
)]
(11)
is the root of
*
+
and
(12)
The boundary conditions are similar to [1], zero ac density at the source and zero ac current at the drain.
The boundary condition (
)(13)
From the boundary condition of
Or
5
gives
, we obtain
Substituting [
from Eq.11, we obtain (
)]
(
(
)
)
Taking logarithm from both sides, we have
*
(
(
))+
After simplifying above relation, we finally obtain
(
)
Where we have defined
(
(
))
, now returning to definition 12, we have
(
[
)
(
] Or
(
Defining √
6
)
(
(
))
we finally obtain √*
+
(14)
(
))
(
(
))(15)
For m=0, (in one dimensional model) the increment is exactly the same as ref [11]. But either the frequency or the increment as seen from Eq14 and 15 behaves very differently in comparison with flat 2DEG. Section IV: Figure 3a shows the plasma spectrum as a function of rc for (a) n=0,m = 0 (i.e. the first mode) and (b) n=0,m=1..5 (i.e. the higher modes) with different VF. It demonstrates that for the first mode (m=n=0), the frequency increases by increasing the tube radius, then it peaks and then it tends to the constant value (as we will show in the next section this constant value is the frequency of a flat 2DEG).For the higher modes, by increasing the radius, the frequency deceases and tends to the constant value (again the frequency of flat 2DEG). 3
10
Fig a L=200 nm 2 m=0 n=0
Frequency f(THz)
Frequency f (THz)
3
6
V =10 m/s F
1
V =0 F
0 0 10
2
m=5
10
Fig b L=200 nm n=0 6 V =10 m/s F
1
10
m=1 0
1
2
10 10 Radius r (nm) c
10
3
10 0 10
1
2
10 10 Radius r (nm) c
3
10
Fig. 3. Plasma spectrum as a function of rc for a: first mode(m,n)=(0,0) b: higher modes (m>0,n=0)
Figure 4a shows the plasma frequency versus tube length for first mode (m=n=0) and for different value of tube radius and figure 4b shows the increment for the first mode and for rc=10 nm, and without viscosity. In case of a sustained plasma spectrum oscillation, we should have for viscose fluids and
7
for non-viscose fluids.
with
10
2
0.5 =h/m
10
1
Fig b
=0
12
10
increment n (10 )
Frequency f (THz)
Fig a
0
r =1 nm c
r =10 nm
L=150 nm
0 L=23.5 nm
c
r =100 nm
L=126 nm
c
10
-1
10
1
2
10 Length L (nm)
10
-0.5
3
2
10 Length L (nm)
10
3
Fig. 4 a: the plasma frequency versus tube length for first mode and for different value of tube radius, b: the increment with and without viscosity
For
, which is the case of a realistic CNT with radius of order 10nm, we have [12]
Again substituting these relations in Eq14, we get √
(16)
(
) (
(
(
))
(17a)
)(17b)
In this case the plasma spectrum logarithmically depends on the 2DEG radius rc. In this stage we can retrieve dispersion relation in a flat 2DEG by letting
and keeping the surface
charge density, bounded. We call this case a flat 2DEG. Again we use the well-known asymptotic expression [12] √
√
Replacing above approximation in Eq. 10,14,15, we obtain 8
√
√
[
](18)
√
(19) (
)(20)
Which are the dispersion relation, plasma frequency and increment for the ungated 2DEG.By increasing the tube radius, the frequency and increment will approach these values (see also fig3) Power estimation: The energy radiated from a source can be obtained in the far zone approximation as [13] ̂ Where
is the Poynting vector,
is the solid angle , r is distance from the source (the tube) to the
point at which the power is evaluated, ̂ is the unit vector having angle
with the tube axis (z direction)
and is the dipole moment defined as ∭
(21)
The total radiated power is the integration of
over a sphere of radius r:
∮ Upon substituting n from Eq13 to Eq21 we obtain for the dipole moment and the total radiated power ∭
∫
∫
(
)
∫
(22) B is the amplitude of density perturbation and A is the amplitude of potential perturbation where from Eq.6 we have an approximation as
9
For typical value of
, we have P=60 Pico Watt,
which is dramatically small and shows that CNT or nano wire with small radius is inefficient for THz generation, and for
the power became P=6 nano Watt for the same electron density. On
the other hand the electron density have a relation with the total charge
as
, to increase
the radius, we should increase the total charge in order to keep the electron density constant. Conclusion- In this paper, we presented the analysis of a THz source which employs the plasma excitation
method in 2DEG. We studied the plasma oscillations in ungated nanowire or CNT in the presence of small drift velocity and viscosity in the hydrodynamic approximation. Based on this theoretical formalism we solved Euler, continuity and Poisson equations and derived analytical expressions for dispersion relation, the discrete frequencies and the “increment”. Our results show that the plasma spectrum and the increment (Power) depend on the radius of tube. We showed that in nano wire and CNT, the plasma frequency and increment differs considerable with respect to a flat 2DEG. We also have a power estimation of the THz radiation.
Reference
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