Optical properties of semiconductors

Dr. Katarzyna Skorupska

Band structure E(k) k- wave vector

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 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

electrons- quadratic dispersion p= hk

photons – linear dispersion

Dispersion relation for quasi free electrons and photons for one dimensional case

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

Indirect transitions: „inclined“ transitions within the 1st Brillouin zone: the k-conservation can not be realized by a reciprocal lattice vector . Phonon supplies the missing momentum to the electron.

Direct (perpendicular) transitions: dipole-allowed interband transitions

k = ki = kf Ef - Ei = hν + Eph (phonon absorption) Ef - Ei = hν - Eph (phonon emission) kf = ki + kph

Direct - (perpendicular), dipole-allowed interband transitions

Character of optical excitation process

Indirect - phonon assisted with small probability and weak resulting absorption

Momentum conservation (provided by reciprocal lattice vector)

Energy conservation

(phonon absorption)

(phonon emission)

ABSORPTION COEFFICIENT A AS FUNCTION OF THE ENERGY OF THE IMPINGING LIGHT

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.

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

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

I0 – the intensity of incoming light

low α carrier generation through the material

10-6 10-5 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.

semiconductor CuInSe2 x-Si InP GaAs a-Si:H

α /cm -1 2x105 103 5x104 1.5x104 104

x /µm penetration

depth

0.05 100 0.2 0.7 1

Absorption coefficient vs absorption length for hν ~ Eg + 0.2eV

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: 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

nnp i

2=

3416

210

1010

)10( −== cmp

α

τxPnp =∆=∆

314

3

1

3

617

10

][10

1010

−

−

−

−

−

=∆

⋅=∆

cmpcm

ssp

Optical excitation perturbs this relation

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 ∆n=1014cm-3

∆p- holes in doped SC created by illumination ∆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.

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).

Light intensity decay

Quasi Fermi levels, definitions

Carrier concentration for illumination: n*(x) = n + ∆n p*(x) = p + ∆p

*ln)(*

nN

kTExE CBCBFn −= *

ln)(*p

NkTExE VB

VBFp +=

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

knowing:

We can write:

because

we can write:

Quasi Fermi level for e- is energetically located above above the dark Fermi level

]1ln[)(

ln)(

ln)(

ln)(

ln)(

)ln(ln)(

lnln)(

ln

ln

ln)(

*

*

*

*

**

*

*

**

**

**

ppkTExE

pp

ppkTExE

pppkTExE

pppppkTExE

pN

pN

kTExE

pN

pNkTExE

pNkT

pNkTExE

pNkTEE

pNkTEE

pNkTExE

FFp

FFn

FFp

FFp

VB

VB

FFp

VBVBFFp

VBVBFFp

VBFVB

VBVBF

VBVBFp

∆+−=

∆+−=

∆+−=

∆+=

−=

−=

+−=

+−=

−=

+=

+=

Quasi Fermi level for h+ is energetically located below the dark Fermi level