Otical roerties of semiconductors Dr. Katarzyna Skoruska
band structure of crystalline solids by solution of Schroedinger equation (one e - aroximation) Solution leads to energy bands searated by an energy band m * - effective mass (determined by curvature of -k) Band structure (k) k- wave vector
Band structure of solids k() is a function of the three dimensional wave vector (k) within the Brillouin zone. Brillouin zone deends on crystal structure and corresonds to unit cell of the recirocal lattice orbidden energy region (ga) no energy states nergy bands are only ermitted above (conduction band) and below (valence band) the ga CB and VB contain several bands each band has different effective mass (m * )
Light absortion in semiconductor nergy conservation Momentum conservation
Disersion relation for quasi free electrons and hotons for one dimensional case hotons linear disersion electrons- quadratic disersion hk
Semiconductor band ga ( g ) the distance between valence band maximum and conduction band minimum. Direct band ga CB minimum and VB maximum at the same k value Indirect band ga CB minimum and VB maximum at the different k value
direct vs. indirect semiconductor Direct (erendicular) transitions: diole-allowed interband transitions Indirect transitions: inclined transitions within the 1 st Brillouin zone: the k- conservation can not be realized by a recirocal lattice vector. Phonon sulies the missing momentum to the electron. k k i k f f - i hν + h (honon absortion) f - i hν - h (honon emission) k f k i + k h
Character of otical excitation rocess Direct - (erendicular), diole-allowed interband transitions nergy conservation Momentum conservation (rovided by recirocal lattice vector) Indirect - honon assisted with small robability and weak resulting absortion (honon absortion) (honon emission)
ABSORPTION COICINT A AS UNCTION O TH NRGY O TH IMPINGING LIGHT Photon energy hoton energy h Planck s constant (4.135667516(91) 10 15 ev s) c seed of light (299.79 m s -1 ) λ - wavelenght The absortion coefficient α, is a roerty of a material which defines the amount of light absorbed by it. The inverse of the absortion coefficient, α 1, is the average distance traveled by a hoton before it gets absorbed.
direct semiconductors - square root deendence on hoton energy indirect semiconductors - quadratic deendence on the hoton energy
Otical roerties
Relation of absortion coefficient (α) and light intensity (I) (Lambert-Baer s Law) α- absortion coefficient I 0 - intensity of incoming light x- distance to the surface xonential decay of intensity rofile of absorbed light
Penetration deth and absortion coefficient The wavelength-deendent value of α determines how far the light enters the semiconductor. the light intensity vs. distance for a few tyical examles of absortion behavior. Penetration deth (x) the inverse of the absortion coefficient (α -1 ) average distance at which traveled by a hoton before it gets absorbed
low α carrier generation through the material 10-6 10-5 10-4 10-3 10-2 10-1 cm I 0 the intensity of incoming light
The absortion coefficient of a semiconductor material at a given wavelength determines the satial region in which most of the light is absorbed. or high absortivity, most of the light is absorbed close to the semiconductor surface. The low absortion coefficient of indirect semiconductors leads to carrier generation throughout the material for the curve where α 10 cm -1.
Semiconductors with direct energy ga are generally characterized by: a high absortion coefficient in the relevant energy range for hotovoltaics; most of the sunlight is absorbed within a small range beneath the surface ossibility 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 urity increase rize
The lot of the absortion coefficient for a series of semiconductors allows identification of thin film solar cell absorber material: weak absortion of crystalline Si (x-si) in the IR to visible range rohibits the use in thin film solar cells. III-V comound sc, the stee increase of the absortion coefficient with the hoton energy, reaching values of α > 10 4 cm -1 within about 0.2eV beyond the fundamental absortion edge, makes these materials candidates for thin film alications. ternary chalcoyrites CuInS 2 and its selenide -even steeer increase of α. amorhous hydrogenated silicon (a-si:h) has a considerably increased absortion comared to x-si and an otical ga shifted by about 0.6eV comared to the crystalline material which allows alication in thin film devices with in rincile higher hotovoltages.
Absortion coefficient vs absortion length for hν ~ g + 0.2eV semiconductor CuInSe 2 x-si InP GaAs a-si:h α /cm -1 2x10 5 10 3 5x10 4 1.5x10 4 10 4 x /µm enetration deth 0.05 100 0.2 0.7 1
xcess carriers We consider here absence of surface or bulk recombinations xcess carrier concentration in VB and CB deends on: - Carrier life time - Absortion rofile - Temerature
xcess carriers Intrinsic carrier concentration similar to Si n i i 10 10 cm -3 or n-tye doing with majority carriers concentration n 10 16 cm -3 Mass action law: 10 2 ( 10 ) 4 3 10 16 cm 10 n 2 i n Otical excitation erturbs this relation Minority carriers concentration 10 4 cm -3 Stationary excess carrier concentration P- hoton flux 10 17 cm -2 s -1 for hν2ev (red light) AM 1.5 at 84.4 mwcm -2 τ- carrier lifetime 1µs X α - absortion of hotons 10-3 cm 3 within a volume of 1 cm -3 x 10 µm deth n 17 10 10 3 10 14 10 cm Pτ x 3 α 6 1 s s [ cm 3 ]
n i - intrinsic carriers SC n i 10 10 cm -3 n- electrons in doed SC in the dark n10 16 cm -3 - holes in doed SC in the dark 10 4 cm -3 n- electrons in doed SC created by illumination n10 14 cm -3 - holes in doed SC created by illumination 10 14 cm -3 n*- electrons in doed SC under illumination n*n + n10 16 +10 14 n*10 16 +10 14 *- holes in doed SC under illumination * + 10 4 +10 14 *10 4 +10 14 or majority carriers change by illumination is only 1% or minority carriers change is illumination is drastical ten orders of magnitude or n-tye semiconductor: - concentration of electrons coming from doing 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
satially deendent carrier concentration rofiles in equilibrium (dark) and under illumination in comarison with the light absortion rofile. Light intensity decay Whereas the excess majority carrier rofile changes little (the change has been magnified in the figure), the excess minority carrier concentration * deviates strongly from the constant dark concentration ().
Quasi ermi levels, definitions or stationary illumination and sufficiently long carrier life time, excess minority and majority carriers exist stationary at the resective band edges. Their excess carrier concentration relation defines a new quasi equilibrium and attemts have been made to describe this situation in analogy to the dark equilibrium terminology. Therefore one describes the ermi level for an illuminated semiconductor in the framework of the equations derived for the non illuminated semiconductor. or n-tye and -tye semiconductors, was given by which can be written, based on the aroximations derived as Carrier concentration for illumination: n*(x) n + n *(x) + n * ( x) CB kt ln N CB n * * ( x) VB + kt ln N VB *
because knowing: we can write: We can write: Quasi ermi level for e - is energetically located above above the dark ermi level
] ln[1 ) ( ln ) ( ln ) ( ln ) ( ln ) ( ) ln (ln ) ( ln ln ) ( ln ln ln ) ( * * * * * * * * * * * * * * kt x kt x kt x kt x N N kt x N N kt x N kt N kt x N kt N kt N kt x n VB VB VB VB VB VB VB VB VB VB VB VB + + + + + + + + Quasi ermi level for h + is energetically located below the dark ermi level