Optical properties of
semiconductors
Dr. Katarzyna Skorupska
band structure of crystalline solids by solution of Schroedinger equation (one e- approximation)
Solution leads to energy bands separated by an energy band
m*- effective mass (determined by curvature of E-k)
Band structure E(k)
k- wave vector
Band structure of solids k(E) – is a function of the three dimensional wave vector (k) within the
Brillouin zone.
Brillouin zone – depends on crystal structure and corresponds to unit cell of the reciprocal lattice
Forbidden energy region (gap) – no energy states
Energy bands are only permitted above (conduction band) and below (valence band) the gap
ECB and EVB contain several bands
each band has different effective mass (m*)
Light absorption in semiconductor
• Energy conservation
• Momentum conservation
Dispersion relation for quasi free electrons and photons for one dimensional case
photons – linear dispersion
electrons- quadratic dispersion
p= hk
Semiconductor band gap (Eg ) –
the distance between valence band maximum and conduction band minimum.
Direct band gap – ECB minimum and EVB maximum at the same k value
Indirect band gap – ECB minimum and EVB maximum at the different k value
direct vs. indirect semiconductor
Direct (perpendicular) transitions:
dipole-allowed interband transitions
k = ki = kf
Indirect transitions: „inclined“ transitions
within the 1st Brillouin zone: the kconservation can not be realized by a
reciprocal lattice vector . Phonon supplies
the missing momentum to the electron.
Ef - Ei = hν + Eph (phonon absorption)
Ef - Ei = hν - Eph (phonon emission)
kf = ki + kph
Character of optical excitation process
Direct - (perpendicular), dipole-allowed interband transitions
Energy conservation
Momentum conservation
(provided by reciprocal lattice vector)
Indirect - phonon assisted with small probability and weak resulting absorption
(phonon absorption)
(phonon emission)
ABSORPTION COEFFICIENT A AS FUNCTION OF THE ENERGY OF THE IMPINGING LIGHT
Photon energy
E – photon energy
h – Planck’s constant (4.135667516(91)×10−15 eV s)
c – speed of light (299.79 m s-1)
λ - wavelenght
The absorption coefficient α, is a property of a material which defines the amount of light
absorbed by it.
The inverse of the absorption coefficient, α–1, is the average distance traveled by a photon
before it gets absorbed.
direct semiconductors
- square root dependence on photon energy
indirect semiconductors
- quadratic dependence on the photon energy
Optical properties
Relation of absorption coefficient (α) and light intensity (I)
(Lambert-Baer´s Law)
α- absorption coefficient
I0- intensity of incoming light
x- distance to the surface
Exponential decay of intensity profile of absorbed light
Penetration depth and absorption coefficient
The wavelength-dependent value of “α” determines how far the light enters the semiconductor.
the light intensity vs. distance for a few typical examples of absorption behavior.
Penetration depth (x) – the inverse of the absorption coefficient (α-1) – average distance at
which traveled by a photon before it gets absorbed
low α carrier generation
through the material
10-6
10-5
I0 – the intensity of incoming light
10-4
10-3
10-2
10-1 cm
The absorption coefficient of a semiconductor material at a given wavelength determines the
spatial region in which most of the light is absorbed.
For high absorptivity, most of the light is absorbed close to the semiconductor surface.
The low absorption coefficient of indirect semiconductors leads to carrier generation
throughout the material for the curve where α = 10 cm-1.
Semiconductors with direct energy gap are generally characterized by:
•
a high absorption coefficient in the relevant energy range for photovoltaics;
• most of the sunlight is absorbed within a small range beneath the surface
• possibility to fabricate thin film solar cells;
Indirect semiconductors
• need more material to absorb most of the sunlight; (Si, Ge, GaP)
• thicker layers are needed;
• higher material costs and increased demands on purity increase prize
The plot of the absorption coefficient for a series of semiconductors allows identification of thin
film solar cell absorber material:
weak absorption of crystalline Si (x-Si) in the IR to visible range prohibits the use in thin film
solar cells.
III-V compound sc, the steep increase of the absorption coefficient with the photon energy,
reaching values of α > 104 cm-1 within about 0.2eV beyond the fundamental absorption edge,
makes these materials candidates for thin film applications.
ternary chalcopyrites CuInS2 and its selenide -even steeper increase of α.
amorphous hydrogenated silicon (a-Si:H) has a considerably increased absorption compared to
x-Si and an optical gap shifted by about 0.6eV compared to the crystalline material which allows
application in thin film devices with in principle higher photovoltages.
Absorption coefficient vs absorption length for hν ~ Eg + 0.2eV
semiconductor
α /cm -1
x /µm
penetration
depth
CuInSe2
x-Si
InP
GaAs
a-Si:H
2x105
103
5x104
1.5x104
104
0.05
100
0.2
0.7
1
Excess carriers
We consider here absence of surface or bulk recombinations
Excess carrier concentration in EVB and ECB depends on:
- Carrier life time
- Absorption profile
- Temperature
Excess carriers
Intrinsic carrier concentration similar to Si
ni= pi = 1010 cm-3
For n-type doping with majority carriers concentration
n = 1016 cm-3
Mass action law:
(1010 ) 2
p=
= 104 cm −3
16
10
p=
ni2
n
Optical excitation perturbs this relation
Minority carriers concentration p=104 cm-3
Stationary excess carrier concentration
P- photon flux 1017cm-2s-1
for hν=2eV (red light)
AM 1.5 at 84.4 mWcm-2
τ- carrier lifetime 1µs
Xα- absorption of photons 10-3 cm3
within a volume of 1 cm-3 x 10 µm depth
∆p = ∆n =
Pτ
xα
1017 ⋅ 10 −6 s −1s
[
]
∆p =
−3
−3
10
cm
∆p = 1014 cm −3
ni- intrinsic carriers SC
ni=1010cm-3
n- electrons in doped SC in the dark
n=1016cm-3
p- holes in doped SC in the dark
p=104cm-3
∆n- electrons in doped SC created by illumination
∆p- holes in doped SC created by illumination
∆n=1014cm-3
∆p=1014cm-3
n*- electrons in doped SC under illumination
n*=n + ∆n=1016+1014
n*=1016+1014
p*- holes in doped SC under illumination
p*=p + ∆p=104+1014
p*=104+1014
For majority carriers change by illumination is only 1%
For minority carriers change is illumination is drastical – ten orders of magnitude
For n-type semiconductor:
- concentration of electrons coming from doping and thermal excitation is much higher than concentration
of electrons coming from illumination
- cocentration of holes coming from illumination is much higher than holes coming by thermal excitation
spatially dependent carrier concentration profiles in equilibrium (dark) and under illumination in
comparison with the light absorption profile.
Light intensity decay
Whereas the excess majority carrier profile changes little (the change has been magnified in the
figure), the excess minority carrier concentration p* deviates strongly from the constant dark
concentration (p).
Quasi Fermi levels, definitions
For stationary illumination and sufficiently long carrier life time, excess minority and majority
carriers exist stationary at the respective band edges. Their excess carrier concentration relation
defines a new quasi equilibrium and attempts have been made to describe this situation in
analogy to the dark equilibrium terminology. Therefore one describes the Fermi level for an
illuminated semiconductor in the framework of the equations derived for the non illuminated
semiconductor. For n-type and p-type semiconductors, EF was given by
which can be written, based on the approximations derived as
Carrier concentration for illumination:
n
E F * ( x) = ECB
n*(x) = n + ∆n
N CB
− kT ln
n*
p*(x) = p + ∆p
p
E F * ( x) = EVB
N VB
+ kT ln
p*
because
knowing:
we can write:
We can write:
Quasi Fermi level for e- is energetically located
above above the dark Fermi level
*
p E F ( x ) = EVB + kT ln
NVB
p*
NVB
p
N
= E F − kT ln VB
p
E F = EVB + kT ln
EVB
NVB
N
+ kT ln VB
p
p*
NVB
N
*
)
+ ln VB
p E F ( x ) = E F − kT (ln
p
p*
*
p E F ( x ) = E F − kT ln


NVB

p 
*
p E F ( x ) = E F − kT ln

NVB

p* 


p*
*
p E F ( x ) = E F − kT ln
p
p * = p + ∆p
p + ∆p
p
p ∆p
*
+
n E F ( x ) = E F − kT ln
p
p
∆p
*
]
p E F ( x ) = E F − kT ln[1 +
p
*
p E F ( x ) = E F − kT ln
Quasi Fermi level for h+ is energetically located
below the dark Fermi level