Chapter 2 Basic Theory
2.1 The theory of generated THz radiation
As we illuminated the photoconductor antenna and the optical power is greater than the energy gap of semiconductor which is the substrate of the photoconductor antenna. The Energy band structure of GaAs has is shown in Figural 2.1-1. Then, this semiconductor will absorb the energy of incident wave and generate free electron-hole pair at the same time. We give bias to the electrode
of the photoconductor antenna; it can accelerate the carrier which is generated by incident wave. The brief surface current will be generated. This process can find expression in the Current-surge model.
2.1-1 Current-surge model
The first step, there are some time-average parameters which we need to define the THz radiation [22].
Charge density⇒ρ(x,y,z,t) Current density⇒J(x,y,z,t)K
Electric field intensity⇒E(x,y,z,t)JK Magnetic flux density⇒B(x,y,z,t)JK
Then, it is necessary to construct Maxwell’s equation for Current-surge model.
From Maxwell’s equation [28]:
E=- B From Equation (2.1-1) and (2.1-5), we get:
( )
Then, we set a non-vector value V and employ that to substitute the
From Equation (2.1-3)、(2.1-5) and B H=μ
Substitute Equation (2.1-7) into Equation (2.1-8)
( )
Because of Lorentz gauge A+ V=0 εμ ∂t
∇• ∂
JK , Equation (2.1-9) becomes as
2
And, Equation (2.1-10) can be written as
2 2 Equation (2.1-11) and (2.1-12) are the two inhomogeneous wave equations written in terms of AJK
and V. The two wave equations are used to determine a functional
and time dependent of the radiated electric field in the far field.
From Equation (2.1-3), the continuity equation of the free carriers is obtained.
(
H =)
⎛⎜J+∂Dt ⎞⎟⎟= J+∂ρt =0∇• ∇× ∇•⎜⎜⎜⎝ ∂ ⎟⎟⎠ ∇• ∂
JK K JK K
(2.1-13)
In the fact, the current density of the photoconductor antenna which adds the bias is the transverse current, parallel the surface of the photoconductor antenna and perpendicular the direction of propagation. So that
∇•J=0K
(2.1-14) From Equation (2.1-13) and (2.1-14), we can deduce that the charge density does not vary with time and not contribute the time dependent radiated electric field. As the result, Equation (2.1-7) becomes
( )
The solution of the vector potential AJKin Equation (2.1-11) leads into the Equation (2.1-15) to express the time dependent radiated electric field E (r,t)JKrad K
at the displacement rK
from the center of the photoconductor antenna.
s
where is the permittivity of free space, c is the speed of light in vacuum,
is the surface current of the photoconductor antenna in the retarded time, and is the increment of surface area at the displacement
ε0 KJs
da' r'K
from the center of the photoconductor antenna. The integration is covered with whole the optically
illuminated area of the photoconductor antenna. In the far field,
The gap between the electrodes of the photoconductor antenna is assumed to be uniformly illuminated by the optical source. Therefore, the surface current can be set as a constant in space. Then, the radiated electric field in Equation (2.1-16) can be written as where A is the illuminated area of the photoconductor antenna and
2 2 2
r= x +y +z .We considered that the THz radiation is generated on Z axis (i.e. x
= y = 0 ), and let z Beside, we can know the equation (2.1-20) from surface current [29].
0
where n is the refractive index of photoconductor antenna under the wavelength of . is the impedance in free space, and is the surface conductivity which is shown in the equation (2.1-21).
μm η0 σs( )t
where e is the electric charge, R is the reflectivity of photoconductor antenna, is the photon energy, is the time-dependent carrier mobility at time t from created carrier at time ,
=w ( , ')
m t t '
t Iopt is the time-dependent of optical intensity, and is the carrier lifetime of excited carriers. For the present derivation, we τ
assume that carrier mobility is a constant.
( , ')
m t t = (2.1-22) m And assume that the carrier lifetime is long enough, . A Gaussian intensity profile of the optical beam is assumed:
τ → ∞car
From these assumptions, the surface conductivity becomes
2
From the equation (2.1-20) and (2.1-24) lead to the equation (2.1-19).
2 2
Comparing with the result from experiment, it is necessary to rewrite the equation (2.1-25) in term of the experimental parameters which is the bias electric field applied across the photoconductor, and which is the incident optical intensity.
where is the average optical energy and A is the area of the incident optical beam. Then we define the parameter B and D to simplify the equation.
Eopt And then, the electric field in the far field can be written as
2 2
Beside of current surge model, we can also discuss on the point of the carrier
dynamics in semiconductor to analyze the THz generation by Drude-Lorentz model when we need to discuss the factor of material in photoconductive antenna [30-31].
2.1-2 Drude-Lorentz model
For the calculation of carrier transport and THz radiation in a biased semiconductor, the one-dimensional Drude-Lorentz model is used. When a biased semiconductor is pumped by a laser pulse with photon energies greater than the band gap of the semiconductor, electrons and holes will be created in the conduction band and valence band, respectively. The carrier pumped by ultrashort laser pulse is trapped in the mid-gap states with the time constant of the carrier trapping time. The time-dependence of carrier density is given by the following equation.
c
(t) (t) dn n (t)
dt = − τ +G (2.1-30) where n(t) is the density of the carrier, G(t) is the generation rate of the carrier by the laser pulse, and τ is the carrier trapping time. The generated carriers will be c accelerated by the bias electric field. The acceleration of electrons (holes) in the electric field is given by
, , momentum relaxation time, and E is the local electric field. The subscript e and h
, (t)
νe h qe h,
,
me h τe h,
represent electron and hole, respectively. The local electric field E is smaller than the applied bias electric field due to the screening effects of the space charges,
Eb
b
E E P
= −αε (2.1 -32) where P is the polarization induced by the spatial separation of the electron and hole, is the dielectric constant of the substrate and is the geometrical factor of the photoconductive material. The geometrical factor is equal to three for an isotropic dielectric material. It is noted that both, the free and trapped carriers contribute to the screening of the electric field. The time dependence of polarization P can be written as
ε α
where is the recombination time between an electron and hole. In the equation (2.1-33), J is the density of the current contributed by an electron and hole,
τr
J=enν −h enν (2.1-34) e where e is the charge of a proton. The change of electric currents leads to electromagnetic radiation according to Maxwell’s equations. In a simple Hertzian dipole theory, the far-field of the radiation ETHz is given by
THz
E J
t
∝∂
∂ (2.1-35) To simplify the following calculations, we introduce a relative speed ν between an electron and hole,
= h
ν ν −νe (2.1-36) Then the electric field of THz radiation can be expressed as
THz
E e n+en
t t
ν∂ ∂ν
∝ ∂ ∂ (2.1-37)
The first term on the right hand side of the equation (2.1-37) represents the electromagnetic radiation due to the carrier density change, and the second term represents the electromagnetic radiation which is proportional the acceleration of the carrier under the electric field.
2.1-3 Photoconductor antenna with different structure
In our experiment, we used the different structure of PC dipole antennas to radiate THz wave. According to the literature [32] and equation (2.1-35), the radiation ETHz can be written as:
THz
E (t) J(t)
e t l ∂
∝ ∂ (2.1-38)
where l is the effective length of the dipole. From Equation (2.1-38), the e amplitude of THz radiation is influenced by the effect length and time-dependence photocurrent. The effect of current will be discussed on next section. The effect length relate to the structure of PC antenna. The larger effective length of the dipole can get the larger amplitude of THz radiation.