Impact Ionization Effects on Propagation of a Millimeter
Wave in GaAs
RAJENDRA PATHAK1, J. PARASHAR2 and S. KATAREY3
1
Department of Mathematics, Samrat Ashok Technological Institute, Vidisha (M.P.) 464001, India
2
Department of Physics, Samrat Ashok Technological Institute, Vidisha (M.P.) 464001, India
E mail: j.p.parashar@gmail.com 3
Department of Mechanical Engineering, Samrat Ashok Technological Institute, Vidisha (M.P.) 464001, India.
ABSTRACT
Effect of impact ionization on propagation of a millimeter (mm) wave with a Gaussian profile is studied under paraxial ray approximation. It is found that early in time the charge density is less and hence the defocusing of mm wave is less however later in time and space as the carrier density builds up due to impact ionization the defocusing of the mm wave is rapid and this results in decrease in the intensity of the wave.
PACS Nos.: 52.40.Db; 52.25.Jm; 52.35.Hr; 41.20.Jb; 42.70.Na
Key Works: Impact ionization, semiconductors, microwaves
Interaction of electromagnetic wave with semiconductor is an active area of research for almost last four decades and several nonlinear effects viz. harmonic generation, instabilities, wave mixing, etc. have been observed1,2. When an electromagnetic wave passes through a semiconductor the valence electron can jump to conduction band at the expense of the wave energy via one of the following processes: impact ionization, tunnel ionization or avalanche effect3. Some notable applications of this transition are in switches, millimeter wave generation, UV lasers etc.4,5. In impact ionization an electron
or hole can gain energy in the presence of electric field and subsequently lose their energy by creation of other charge carriers. This can lead to avalanche breakdown in semiconductors. The electron hole (e-h) plasma created via impact ionization is space time varying and alters the propagation dynamics of the electromagnetic wave.
integrated circuits, infrared light emitting diodes, laser diodes, mobile phones, satellite communications, microwave point to point links, radar systems and solar cells6,7. In the microwave frequency range GaAs is one of the best candidates because of its higher saturated electron velocity, higher electron mobility, less noise as compared to Silicon (Si) and also can be operated at higher powers due to higher breakdown voltages8,9. In this communication we develop a theory to study the effect of impact ionization on propagation of a Gaussian millimeter wave propagating through GaAs. The wave equation is solved by generalizing the paraxial ray theory of nonlinear wave propagation.
Consider the propagation of a millimeter wave with Gaussian intensity profile through a GaAs semiconductor with <100> orientation,
) ( 0( , , )
z k t i
e t r z E
E= − ω −
r r
(1) At z = 0
2 2
0
/
0 0
,
0
0
,
r r
E
A e
for
t
otherwise
τ
−
=
< <
=
(2)where τ is the pulse duration.
The wave creates electron – hole (e – h ) pairs inside semiconductor via impact ionization. The ionization rates for electrons and holes can be written as
, 2
0 2
e i e
n t
n
α
= ∂ ∂(3)
and 2
2 0
,
hi h
n
n
t
β
∂
=
∂
(4)respectively. Here ne(h) is electron (hole) density, ne(h)0 being the equilibrium density, and
α
i(β
i)is electron (hole) ionization rate given by0 /
0
n E E
i
e
α α
=
− , (5)0 /
0
p E E
i
e
β β
=
− , (6)are constants (c.f. Shur9 pp.188).
The electron (hole) current density
J
1 ( )e hr
is governed by
2
( ) 1 ( )
1 ( ) *
( ) e h e E e h
e h
e h
n
dJ
J
dt
+
ν
=
m
r
r
r
, (7)
where ν is the electron – hole collision frequency and m*e(h) is the electron (hole) effective mass.
The millimeter wave field in the space – time evolving electron – hole plasma could be written as
φ
i
e A E
r r
= , (8)
where
A
r
is a slowly varying function of z,t and
φ
is a fast varying function of z, t. The wave equation governing the propagation of millimeter wave is written as10, 11,t J c t E c
E L
∂ ∂ = ∂
∂ − ∇
r r
r
2 2 2
2
ε
4π
, (9)
where εL is lattice permittivity and we have neglected the
( )
Er
.
∇
∇ term for transverse waves. For
E
=
y
ˆ
E
yr
form Eq.(8) we have p
nandE
E0 0
φ φ φ
φ
φ
φ
i i i y e z A z i e A z e A z i z E ∂ ∂ ∂ ∂ + ∂ ∂ − ∂ ∂ ≈ ∂ ∂ 2 2 2 2 2 2, (10)
, 2 2 2 2 2 2 φ φ φ
φ
φ
φ
i i i y e t A t i e A t e A t i t E ∂ ∂ ∂ ∂ + ∂ ∂ − ∂ ∂ ≈ ∂ ∂ (11) φ i yA
e
E
22
⊥ ⊥
=
∇
∇
, (12)where we have neglected ∂2A/ z∂ 2 terms. Defining
ω
,
k
asω
=
−
∂
φ
/
∂
z
andt
k
=
∂
φ
/
∂
, withω
2=
ω
2p(
r
=
0
)
+
k
2c
2, and using Eqs.(7), (11) & (12) in Eq.(9) we get(
)
[
]
0,(13)1 2 0 2 2 2 2 2 2 2 2 = + − + − ∂ ∂ + ∂ ∂ + ∂ ∂ + ∇ + ∂ ∂ = ⊥ A c A t c i z k i t A c i A z A r ph pe ph pe L L ω ω ω ω ω ε ωε
where
ω
pe2 (h)=
4
π
n
e(h)/
m
*e(h)is the electron (hole) plasma frequency andm
e*(h)is the electron (hole) effective mass.Multiplying Eq.(13) by A, we obtain
(
)
2 2
2 2 2 2
2
2
2 2 2 2 2
0 ( ) 2 0.(14) pe ph g
pe ph pe ph r
A A ic A
v A A
t z t
i
A
ω ω
ω ω
ω ω ω ω
ω ⊥ = ∂ + ∂ + ∂ − ∇ + ∂ ∂ ∂ + + − + =
Later we will recast Eq.(14) in terms of new variables
z
'
,
r
,
t
'
where t'=t−z/vg,z'= z. We assume a Gaussian ansatz for the r- profile of laser intensity12,2 2 0 2 / 2 2 00 2 0 f r r
e
f
E
E
=
− , (15)with f as beam width parameter. Using Eq.(15) we expand Eqs.(3) and (4) around
r=0 and obtain
∂ ∂ + = = ∂ ∂ = 2 0 2 2 0 0 2 ) 0 ( r r Q r Q t r e e pe pe
α
ω
ω
, (16) and ∂ ∂ + = = ∂ ∂ = 2 0 2 2 0 0 2 ) 0 ( r r Q r Q t r h h ph phβ
ω
ω
, (17) respectively. Here )] ( ) / (exp[ 2/202 2
00 ) ( 0 ) ( f r r p n h
e E E f e
Q = − .
On integrating Eqn.(16) and (17) we get
(
)
∫
+ − = g g v z t v zpe r r dt
/ / 2 0 2 '' 0 ' 0 2 ' /
α
α
ω
, (18)and
(
)
∫
+ − = g g v z t v zph r r dt
/ / 2 0 2 '' 0 ' 0 2 ' /
β
β
ω
, (19)respectively. Here, 00 0 / 0 '' 0
2
E
E
f
nα
α
=
, pe2e
(En/E )f 0 0 ' 0 00 0 −=
α
ω
α
, 00 0 / 0 '' 02
E
E
f
nβ
β
=
,f E E ph
n
e
( / )2 0 0 ' 0 00 0 −
=
β
ω
β
.Using Eqs.(18) & (19) in Eq.(14) we get
(
)
0
'
2
2 0 2 22
+
+
=
∇
+
∂
∂
⊥A
r
r
c
A
z
A
ik
p pβ
α
. (20) Introducing an eikonal
)] , ( exp[ ) , (
0 r z S r z
A
A= − 13
2 2 0 2 0 2 0 2
)
(
1
1
'
2
c
k
r
r
A
A
k
r
S
k
z
S
α
p+
β
p+
∇
=
∂
∂
+
∂
∂
⊥ (21) and( )
1
0
1
'
2 0 2 0 2 2 0=
∂
∂
∂
∂
+
∇
+
∂
∂
⊥r
A
r
S
k
A
S
k
z
A
(22) respectively.Following Eq.(15) we write
)
/
exp(
2 02 22 2 00 2
0
r
r
f
f
A
A
=
−
, (23)and expand S as
)
'
(
)
'
(
2
2z
r
z
k
S
=
β
+
φ
, (24)in the paraxial ray approximation. Using Eqs.(23) &(24) in Eqs.(21) & (22) we get
' 1 dz df f =
β
, (25)and the equation governing the beam width parameter f as
2 2 2 0 3 2 2 2
)
(
1
'
R
c
r
f
f
R
dz
f
d
d p p dβ
α
+
+
=
, (26)where Rd2=kr02.
We introduce dimensionless variables
d
R z /'
=
ξ
and 22 0 2 0 2 2 ' 10 c t r pe
ω
η
= − , thenEq.(26) could be rewritten as
.
10
.
1
2 3 2 2f
f
d
f
d
+
=
ξ
η ω ω β α η d e E E e E E f f E E pe ph p f E E n p n∫
+ − − 0 2 0 2 0 00 0 0 00 0 0 00 0 00 0 1 (27) We have solved Eq.(27) numerically for27 . 6 / 00
0 E =
E n ,E0p/E00 = 4.82 ,
, 11 ~ , 10
5 6 3 L
h
e n cm
n ≈ ≈ × −
ε
, 1 ,
1cm r0 = mm
=
λ
0 * 0 * 063 . 0 , 063 .0 m m m
me = h = ,
kg
m0 =9.1×10−31 ,
α
0 =2.19×106, 60 = 2.47×10
β
with the boundaryconditions: f =1 and
df
/
d
ξ
=
0
at ξ=0 for all η. We have chosen time step size01
.
0
=
∆
η
and the space step size01
.
0
=
∆
ξ
. We write Eq.(27) asf P f
f ''= 13+ (28) where . 102 = P η ω ω β α η d e E E e E E f f E E pe ph p f E E n p n
∫
+ − − 0 2 0 2 0 00 0 0 00 0 0 00 0 00 0 1and the prime represents differentiation with respect to ξ. We begin by evaluating P at ξ=0 for all values of η. We solve Eq.(28) with the Runge-Kutta method using the value of P obtained at ξ=0. Using these values of f, we evaluate P at
ξ
=
∆
ξ
at different values of η. This way we advance in ξ. In Fig. 1 we have plotted the beam width parameter f and in Fig.2 the axial intensity2 0/ f
I I=
0.2 0.4 0.6 0.8 1.0 1.0
1.5 2.0 2.5 3.0 3.5
η=0
η=.25
η=.50
η=.75
f
ξξξξ
Fig. 1. Variation of beam width parameter f with ξ.
0.0 0.2 0.4 0.6 0.8 1.0 0.0
0.2 0.4 0.6 0.8 1.0
η=0 η=0.25 η=0.50 η=0.75
I
=
I
0/f
2
ξξξξ
Fig.2. Variation of axial intensity I=I0/ f 2
with ξ.
REFERENCES
1. Shen Y. R.: The Principles of Nonlinear
Optics (Wiley, New York 1984).
2. Singh L., Konar S. and Sharma A. K., J.
Phys. D : Appl. Phys. 34, 2237 (2001).
3. Vavilov V. S., Sov. Phys. Usp. 37, 269 (1994).
4. Basov N. G., Grasyuk A.Z., Zubarev I.G., Katulin V. A., and Krokhin O. N.,
Sov. Phys. JETP 23, 366 (1966).
5. Liu C. S. and Tripathi V. K., J.Appl.
Phys. 83, 15 (1998).
6. Bhattacharya P: Semiconductor
Optoele-ctronic Devices (Pearson Education,
New Delhi, 2001).
7. Tyagi M. S.: Introduction to
Semicond-uctor Materials and Devices (Wiley,
New York, 2000).
8. Sze S. M.: Physics of Semiconductor
Devices (Wiley, New York, 2001).
9. Shur M.: Physics of Semiconuctor
Devices (Prentice Hall India, New Delhi,
2004).
10. Jackson J. D.: Classical
Electrodyn-amics 3ed (John Wiley, New York,
2004).
12. Akhmanov S. A., Sukhorukov A. P., and Khokholov R.V., Sov. Phys. Usp. 10, 669 (1968).
13. Liu C.S., and Triptahi V. K.: Interaction
of Electromagnetic Waves with Electron Beams and Plasmas (World Scientific,