2-1 Terahertz Generation
The THz pulse has been generated by different methods, such as irradiation of
photo-conductive antenna, semiconductor surfaces, or quantum structures with
femtosecond optical pulses. The most common emission mechanisms are surge current
and transient polarization which acts as broadband terahertz sources. As the sample is
illuminated by an ultrashort laser pulse, electron-hole pairs are created in
semiconductors, and then the carriers are accelerated by the external or internal
electrical field to form a transient current which radiate THz waves. In this work, the
THz waves are generated via surge-current model, which will be described in detail in
the next section.
2-1-1 Surge-Current Model
From Maxwell’s equation, we have two inhomogeneous wave equations in terms
of electric potential V and magnetic potential A r
where J r
is the current density and ρ is the charge density. These equations describe the propagation of the electromagnetic disturbances. In terms of these potential, the electric
field E
absorption of an optical pulse is obtained directly from Maxwell’s equation:
( ) J t
arrows should be corrected) imply that the charge density is constant in time and will
not contribute to the time dependent radiated electric field. Therefore, from Eq (2.3) we
have
The solution to the wave equation (2-1-1.1) and hence for the vector potential A r
da'
surface current in emitter evaluated at the retarded time, and da' is the increment of
surface area at a displacement 'rr
from the center of the emitter. In the far field and
assuming Js r
is a constant at all points in the emitter, the radiated field can be written as
J (t)
2-1-2 Surface Depletion Field
In semiconductors with wide bandgap, such as GaAs (1.43 eV) and InP (1.34 eV),
there may exist surface states in the forbidden gaps between valance and conduction
bands due to the discontinuous structures in the surface. These states with occupied or
vacant would affect the equilibrium concentration of electrons and holes in the surface.
Because the Fermi-level is position independent, the energy bands must bend to form a
depletion region where the surface built-in field exists. The strength of
surface-depletion field is a function of Schottky [ 18] barrier potential and dopant
concentration Eq (2.1.2-1)
( ) ( )
d
E x eN W x
= ε − (2-1-2.1)
Where N is dopant concentration, ε is semiconductor permittivity, and W is depletion
width and has the form of Eq (2.1.2-2)
2 [ kT]
W V
eN e
= ε − (2-1-2.2)
Where V is the potential barrier and kT e is the thermal energy. The direction and /
magnitude of the surface depletion field depend on the dopant or impurity species and
the position of the surface states relative to the bulk Fermi level. In general, the energy
band is bent upward and downward for n-type and p-type semiconductors, respectively.
After optical excitation, the electrons and holes are accelerated in opposite directions
under the surface-depletion field, forming a surge current in the direction normal to the
surface as shown in Fig. 2-1. Therefore, when the depletion surface field is the
dominant mechanism for the surge current, the polarity of the emitted THz radiation
waveform is opposite between that of the n-type and that of the p-type semiconductor.
Contrarily, if the dominant mechanism is Photo-Dember effect, the emitted THz
radiation waveform will be same direction either the n-type semiconductor or the
p-type semiconductor, as explained in Section 2.1.3. Thus, by comparing the polarity of
the THz waveforms emitted from an n-type semiconductor and that of the p-type
semiconductor, we can determine which mechanism is dominant in the semiconductor.
Fig 2-1. Band diagram and the schematic flow of drift current in a typical n-type semiconductor.
2-1-3 Photo-Dember Effect
The role of main band structure parameters such as carrier effective mass and
V
W +
-
E
cE
vE
fSurface states
E
surface-
+
-
J
Drift-
-
+
-
surfaces. Some semiconductors such as InAs and InN with small band gap and small
effective electron mass (Table 2-1) would radiate THz via Photo-Dember effect. With
NIR light, the absorption depth is very small(≈100nm) and the excess energy of photo-excited carriers is very large for the excitation of narrow-bandgap
semiconductors. All these conditions in the narrow-bandgap semiconductors enhance
the photo-Dember effect, which is known to generate current or voltage in
semiconductors due to the difference diffusion velocities of the electron and hole.
The band bending in this kind of semiconductors is not obvious resulting in a
relative small surface depletion field because of comparatively small bandgap. The
high absorption coefficient due to a small bandgap causes a large gradient carrier
concentration after excitation by ultrafast laser pulse. The excited electrons and holes
diffuse in the same direction but with different velocities, and therefore the
Photo-Dember effect dominant (). Because the electron mobility is always larger than
hole mobility, the photo-current is always in the same direction either n-type or p-type
semiconductors. The fast photo-current rise and decay time due to semiconductors with
the small electron mass and high mobility often possess fast photo-current rise and
decay time that is helpful for efficient THz generation. In this mechanism the need of
photon energy is comparably small; therefore, the free carriers have large excess energy.
The photo-Dember voltage can be expressed by [ 19]:
0 0
( ) 1 ( 1)
ln(1 )
1
B e h
D
k T b T b n
V e b n b p
− + ∆
= +
+ +
(2-1-3.1)Where n0 and p0 are the intrinsic concentration of the electrons and holes, b is the
mobility ratio e
p
b µ
= µ , and Te and Th are the temperature of photo-excited electrons and
holes, respectively. From Eq (2-1-3.1) several properties can be concluded: The
conditions to enhance the Photo-Dember voltage are high electron temperature,
mobility ratio b and low intrinsic carrier concentration. The corresponding electric field
(ED=VD/d, d is absorption length) is enhanced by decreasing absorption length. Thus,
the Photo-Dember effect is much stronger in narrow bandgap materials (InAs) than in
wide bandgap materials (GaAs). Recently, InAs has been reported as a strong THz
emitter with intensity at least one order of magnitude higher than other unbiased
semiconductor emitters such as InP and GaAs. Therefore, high conversion efficiency
has made InAs received much attention and be one of the most widely used THz
emitters. InN has also been considered to generate THz waves via Photo-Dember effect
[ 15].In this thesis, we would focus on the properties of terahertz emission from InN
with different lattice structure.
The other band structure parameter we need to concern is the valley energy
separations. If the excited carriers have excess energy higher than valley energy
separation, intervally scatterings will happen. Because of the reduction of the transient
mobility in the L-valley, where the electron mobility is expected to be extremely low,
the terahertz radiation from Photo-Dember effect became relatively small.
Table 2-1. Band structure parameters of InN and InAs. The effective mass m0* at the bottom of the valley and nonparabolicity factor α are given for the central valley. △E1 (△E 2) is the energy separation between the central valley and first (second) satellite valley. The parameters for InN and InAs are taken from [ 20] [ 21], respectively.
Semiconductor Eg (eV) m0*(me) α (eV-1) △E1 (eV) △E2 (eV)
InN 0.78 0.04 1.43 1.775 2.709
InAs 0.354 0.023 1.4 0.73. 1.02
Air + Semiconductor
-
-
- -
+
+ +
+
Surface
J
Diffusion current2-1-4 Optical Rectification
The nonlinear optical process was first suggested by Chuang et al.[ 22][ 23][ 24].
THz radiation field generated by the optical rectification, ETHz(t), is proportional to the
second-order nonlinear polarization in the near field. In the far field, the observed THz
field amplitude, ETHz(ob), is proportional to the projection of the second time derivative
of the nonlinear polarization to the polarization direction of detection, n (a unit vector
normal to the observation direction), at the observation point:
) (t
E
THzob = nnnn˙ETHz ∝ nnnn˙ 2 2( )
t t P
∂
∂
(2-1-4.1)
)
2( Ω ∝ Ω
ob
E
THz nnnn˙P(Ω)
(2-1-4.2)The strong dependence of the emitted THz radiation intensity on the crystal
orientation to the pump polarization is the most unambiguous evidence for the
contribution of the χ(2) process. By rotating a sample about its surface normal, the relative contribution of the azimuthal-angle-dependent component to the total THz
radiation can be estimated.
Taking the surface normal as the X-axis and the reflection plane as the XY-plane
in the laboratory frame, the nonlinear polarization induced in the semiconductor due
(2-1-4.4), respectively [ 25]: For (111) surfaces,
Here, φ is the angle between the pump-laser beam and the surface normal refracted
inside the sample, θ is the azimuthal angle of the sample orientation around the X-axis(See Fig 2-2.) and d14 =χ14(2)/2 is the nonlinear susceptibility coefficient for the difference frequency, Ω , in the contracted notation, and E(Ω) is the
autocorrelation function of E(ω). Using (2-1-4.1) and considering the refraction at
the interface between the semiconductor/air interfaces, the p-polarized THz field
amplitude observed in the direction of optical reflection is given by the following
equation:
( )
generalized Snell’s law as :THz
respectively. The azimuthal angle dependence of THz radiation amplitude from
semiconductors with n-InAs (111) and n-InSb (100) are shown on Fig 2-3.
Fig 2-2 Optical geometry of THz radiation from a semiconductor surface [ 25]
Fig 2-3 Azimuthal angle dependence of THz radiation amplitude from semi-conductors with (a) n-InAs (111) and (b) n-InSb. [ 25]
2-1-5 THz Enhancement by External Magnetic Field
In past years, several groups have observed a large magnetic field induced the
enhancement in surface–field THz emission from a variety of semiconducting
materials (GaAs, InAs, InP, GaSb and InSb). In the paper published by M. B.
Johnston et. al.[ 30], they build up a Monte Carlo dynamics model to explain the
phenomenon of magnetic-field enhancement of THz emission and the schematic
diagram of the experiment geometry is shown on Fig 2-4. The carrier motion is:
]
rotational symmetry about the z axis. Hence, a simple linear THz dipole is formed in
the z direction. The magnetic field rotates the dipole, producing x and y components
of similar magnitude to the z component.
The THz radiation emitted by the dipole is transmitted through the
semiconductor surface, and the enhanced power recorded in the experiments is a
result of a dramatic increase in transmission when the dipole is rotated. The TE and
TM fields inside the semiconductor are obtained from J(t), and the external fields
computed using the Fresnel transmission coefficients for the two polarizations :
with ne , ni the external and internal refractive indices, respectively. At B=0T, there isa strong suppression of the TM polarized bow-tie dipole pattern due to the index of
semiconductor (GaAs: ni=3.5). It is less than 17° of the internal angle, so, only few
fraction of the emission close to the dipole axis (z-axis) can pass through the surface.
For B=8T, the dipole is rotated by the magnetic field , thus, the magnetic field
enhancement can be seen to be in reality a reduction of the suppression of the
emission from the z polarized dipole. The theory predicts a TM power enhancement
of ~10-fold, and a total enhancement (TE+TM) of ~15-fold. The corresponding
experimental enhancements are ~20 and 30 times.
Fig 2-4. Schematic diagram of the experimental geometry and coordinate system.[ 30]
2-2 Terahertz Detection: Electro-Optical Crystal and Free Space Electro-Optic Sampling (FS-EOS)
The coherent detection of a THz-pulse beam with EO crystals is based on the
linear Electro-Optical effect (Pockels effect). It is a phase-sensitive detection of
electromagnetic (EM) radiation based on a birefringence in an EO crystal induced by
the incident EM radiation. Under the external field, the refractive index of a certain
material can be changed with the field intensity. In 1996, X. C. Zhang found that THz
pulse can be detected by electro-optic sampling methods [ 6]. The THz electric field
would modulate the birefringence of the EO crystals and then change the polarization
ellipticity of the optical probe beam passing through the crystal with the time delay. The
refractive-index-ellipsoid modulation of the optical beam can then be analyzed to
provide the information of the amplitude and phase of the applied electric field.
When using an amplified laser system, the THz pulses are best detected via free
space electro-optic sampling rather than photoconductive antenna, because the need of
focused beam is not necessary that decrease the potential for damaging the material. It
is a non-resonant method and is suitable for broadband detection [ 26]. The THz beam
is focused onto an EO crystal and modify the index ellipsoid transiently via Pockels
effect. The linearly polarized laser probe beam propagate inside the crystal combined
with the THz beam, and its phase is modulated by the refractive index change induced
by the electric field of the THz pulse as shown in Fig 2-5. The signal difference is
proportional to the phase change Γ induced by THz field, for EO crystal, the phase
retardation induced in the EO crystal is given by the following equation change term Γ can be expressed by
3
d n 4 1
π γ E
Γ = λ (2-2.1)
Where d is the thickness of the crystal, n is the refractive index of the crystal at the
wavelength of the near-infrared (NIR) probe beam, λ is the probe beam wavelength,
γ41 is the Electric-Optic coefficient, and E is the electric field of the THz wave. Thus, we can obtain the entire THz time-domain waveform by measuring the signal
difference via a balance detector as a function of delay time between the THz pulse and
the probe pulse. The total frequency response of the EO sampling technique is given
by the product of (2-2.1) and the following equation:
)
T (ω) = 2/[nTHz(ω)+1], is the Fresnel transmission coefficient with the refractive index of the EO crystal nTHz(ω) in the range of the target THz frequency. Therefore, a thicker crystal produces a greater interaction length, but on the other hand it reduces
the detection bandwidth due to group-velocity mismatch.
For the time being, there are many different materials for EO sampling such as
ZnTe, GaP, GaSe and InP. In addition, ZnTe and GaP are the most commonly used
material for EO detection from sub-THz to several tens of THz, because of their relative
large EO coefficient and they are transparent at the wavelength of the incident THz
radiation. Furthermore, for high chopping frequency, EO sampling has a high SNR as
well as photoconductive sampling, but it is more sensitive to laser noise.
Fig 2-5 Schematic figure of Electro-optic sampling EO Crystal
Wollaston Balance Detector
Quarter-Wave Plate