Response Functions
Josef Humlíček
humlicek@physics.muni.cz
• Electromagnetic spectrum;
• waves, photons, energy and momentum, flux;
• polarization of matter, optical functions and their relationships;
• example: polar (/doped) semiconducting crystal – GaAs.
1
F8150, 2024
Optical frequencies of the electromagnetic spectrum
10
3
10
6
10
9
10
12
10
15
10
18
10
21
frequency (Hz)
kilo mega giga tera peta exa zeta
visible
THz, IR UV
electronics photonics
1PHz  1 fs  0.3 m  33000 cm
-1
 4.1 eV  48000 K
2
~5 orders
of magnitude
Electromagnetic waves
Harmonic oscillations at a fixed point in space,
2
0 0( ) ,i ft i t
E t E e E e − −
= =
0( 2 / ) ( )
0 0( , ) .i t k r i t kr
E r t E e E e   − − − −
= =
wavy pattern in space-time, like the simple plane wave, propagating along
the unit vector k0:
The basic charateristics:
electric intensity E (V/m),
frequency f = /2 (Hz),
wavelength  = c/f (m, nm),
wavenumber W = 1/ (cm-1).
Quantum behavior (Planck + Einstein):
photon energy ħ (eV, K - from the corresponding temperature T, = kBT),
momentum ħ/c (eVs/m). 3
Electromagnetic waves
Wavelength - photon energy - wavenumber conversions:
-11.239852
(eV) , (cm ) 8065.48 (eV).
(μm)
W 

= =
The light-matter interaction is governed by the electric field E of the wave;
however, the signals measured by detectors are proportional to intensity,
the time-averaged Poynting vector:
20
0| ( ) ( ) | | | .
2
c
I E t H t E

=   =
Using the value c0 = 0.002654 A/V, we find the following relations:
a wave with the electric field amplitude of 1V/m has the intensity of 1.33 mW/m2;
a wave with the intensity of 1 mW/m2 has the amplitude of the electric intensity
of 8.68 105 V/m.
This can be compared with the magnitude of the intensity of the field of one
elementary charge at the distance of 0.1 nm, which is 1.441011 V/m.
4
Electromagnetic waves - quantum behavior
The classical picture of the electromagnetic wave fails on many occasions.
The flow of energy has to be treated as a train of quanta (photons); the intensity I
and power P of a monochromatic beam are
(number of photons per units of area and time),
(number of photons per unit of time).
I
P


= 
= 
For example, the red HeNe laser line of =632.8 nm is composed of the 1.959 eV
quanta. The power of 1 mW corresponds to the flux of
3.191018 photons per second.
State-of-the art detectors operate at the dark noise level of about 10 elementary
charges (that can be produced by a slightly higher number of photons) →
it is fairly easy to observe the linear response of matter (the number of absorbed
photons proportional to the number of incoming photons).
5
Polarization of matter
The electric field of the light wave moves the atomic nuclei and electrons in our samples;
the movements are typically asynchronous, with “phase shifts” between the external force
(proportional to the electric intensity E) and the induced oscillating dipole moments
(or, equivalently, induced currents), with pronounced dependence on the (angular)
frequency ;
complex numbers keep track of the amplitudes and phases
(ellipsometry provides two real numbers at each frequency);
in weak fields, the response is linear (D is induction, P is polarization, c is susceptibility,
 is permittivity, j is induced current, s is conductivity):
6
0 0 0
ˆ ˆ(1 )D E P E E c   + = + =
j i P E s= − =
Relationships between optical constants
7
The real part of conductivity (imaginary part of dielectric function) is
a measure of absorbed energy;
the imaginary part of the refractive index is a measure of the attenuation of
the light wave.
SI units
8
D (F/m)ˑ(V/m)=C/m2
P C/m2
j A/m2
ED (V/m)ˑ(Aˑs/m2)=J/m3
I (V/m)ˑ(A/m)= W/m2
c,  dimensionless
s (A/m2)/(V/m)=1/(Wm)
Example: polar (/doped) semiconducting crystal - GaAs
Electron density in the ground state
(Cohen and Chellikowsky, Electronic Structure and Optical Properties of Semiconductors, nonlocal pseudopotential)
suggests a difference in the IR response due to lattice vibrations
Si
GaAs
Doped GaAs – dielectric function
0.00 0.02 0.04 0.06 0.08 0.10
-200
-100
0
100
200
2.7x10
18
cm
-3
Im
Re
n-GaAs
DIELECTRICFUNCTION
PHOTON ENERGY (eV)
5 10 15 20
-10
0
10
20
Im
Re
GaAs
DIELECTRICFUNCTION
PHOTON ENERGY (eV)
Doped GaAs - conductivity
0.00 0.02 0.04 0.06 0.08 0.10
-200
0
200
400
600
800
2.7x10
18
cm
-3
Im
Re
n-GaAs
CONDUCTIVITY(W
-1
cm
-1
)
PHOTON ENERGY (eV)
5 10 15 20
-10000
-5000
0
5000
10000
15000
Im
Re GaAs
CONDUCTIVITY(W
-1
cm
-1
)
PHOTON ENERGY (eV)
Doped GaAs – refractive index
0.00 0.02 0.04 0.06 0.08 0.10
0
5
10
15
2.7x10
18
cm
-3Im
Re
n-GaAs
REFRACTIVEINDEX
PHOTON ENERGY (eV)
5 10 15 20
0
1
2
3
4
5
Im
Re
GaAs
REFRACTIVEINDEX
PHOTON ENERGY (eV)
Doped GaAs – inverse dielectric function
0.00 0.02 0.04 0.06 0.08 0.10
-0.5
0.0
0.5
2.7x10
18
cm
-3
Im
Re
n-GaAs
-1/
PHOTON ENERGY (eV)
5 10 15 20
-0.5
0.0
0.5
Im
Re
GaAs
-1/
PHOTON ENERGY (eV)
Doped GaAs – penetration depth
0.00 0.02 0.04 0.06 0.08 0.10
1
10
2.7x10
18
cm
-3
n-GaAs
PENETRATIONDEPTH(m)
PHOTON ENERGY (eV)
5 10 15 20
0.01
0.1
1
10
GaAs
PENETRATIONDEPTH(m)
PHOTON ENERGY (eV)
Doped GaAs – normal-incidence reflectivity
0.00 0.02 0.04 0.06 0.08 0.10
0.0
0.2
0.4
0.6
0.8
1.0
2.7x10
18
cm
-3
n-GaAs
REFLECTIVITY
PHOTON ENERGY (eV)
5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
GaAs
REFLECTIVITY
PHOTON ENERGY (eV)