16

Reflectance spectra for Reflectance spectra for SiSi

Notice R and εi and εr show considerablestructures in the form of peaks and shoulders.These structures arise from the opticaltransitions between valence bands to theconduction bands.

17

Microscopic TheoryMicroscopic Theory:: semiclassical approach semiclassical approach

We will use a semi-classical approach to derive the Hamiltonian describing theinteraction between an external electromagnetic field and Bloch electrons inside thesemiconductor.In this approach the electromagnetic field is treated classically while electrons aredescribed by quantum mechanical (Bloch) waves.This approach is not rigorous as the fully quantum mechanical treatments in which theelectromagnetic waves are quantized into photons but has the advantage of beingsimpler and easier to understand and generates the same results.We start with the unperturbed one-electron Hamiltonian:

To describe the electromagnetic field we introduce a vector potential A(r,t) and a scalarpotential Φ(r,t). Because of gauge invariance, the choice of these potentials is notunique. For semplicity, we will choose the Coulomb gauge, in which

In this gauge the electric and magnetic fields (E,B) are given by

).(2

2

0 rVm

pH

r+=

0 and 0 =⋅∇=Φ Ar

ABt

A

cE

rrr

r×∇=

∂

∂−= and

1

18

Semiclassical ApproachSemiclassical Approach

The classical Hamiltonian of a charge Q in the presence of an external magnetic fieldcan be obtained from the free-particle Hamiltonian by replacing the momentum P byP-(QA/c). Correspondingly, we obtain the quantum mechanical Hamiltonian describingthe motion of an electron (-e) in an external electromagnetic field by replacing theelectron momentum operator p in p+(eA/c):

The term [p+(eAc]2/2m can be expanded (keeping in mind that p is an operator whichdoes not commutate with A(r)) as

Using the definition of p as the operator

( )[ ]).(

2

/2

rVm

cAepH

rrr

++

=

2

2222

22222

1

mc

AeAp

mc

epA

mc

e

m

p

c

Aep

m+⋅+⋅+=

+

rrrrr

r

( )∇i/h

19

Electron-Electron-radiation radiation InteractionInteraction

From the Coulomb gauge we have thatand therefore [e/(2mc)]p⋅A= [e/(2mc)] A⋅p.

For the purpose of calculating linear optical properties we can also neglect the termwhich depends quadratically on the field. Under this assumption we can approximate Hby

Compared with the unperturbed Hamiltonian (H0) the extra term describes theinteraction between the radiation and a Bloch electron. As a result, this term will bereferred to as the electron-radiation interaction Hamiltonian HeR:

Note that the form of HeR depends on the choosen gauge.

0 =⋅∇ Ar

( ) .)( fAi

fi

ArfAp

⋅∇+

∇⋅=⋅rhhrrr

pAmc

eHH

rr⋅+= 0

pAmc

eHeR

rr⋅=

20

Matrix ElementMatrix Element

There are several ways to calculate the dielectric function of a semiconductor from HeR.We are consider the simplest approach. We first assume that A is weak enough that wecan apply time-dependent perturbation theory (in the form of the Fermi Golden Rule) tocalculate the transition probability (R) per unit volume for an electron in the valence bandstate (v, with energy Ev and wavevector kv) to the conduction band (c, withcorresponding energy Ec and wavevector kc ). To do this we need to evaluate the matrixelement:

We will now write the vector potential A as Aê, where ê is a unit vector parallel to A. Interms of the amplitude of the incident electric field E(q,ω) the amplitude of A can bewritten as

The integration over time of the term [i(-ωt)] and the corresponding factors in theelectron Bloch functions leads formally to

This result means that the electron in the valence bands absorbs the photon energy andis then excited into the conduction band.

( ) ./ 222 vpAcmcevHc eRrr⋅=

( )[ ]{ }..exp2

cctrqiq

EA +−⋅−= ω

rr

))()(()/exp()](exp[)/exp( ωδω hrr

hh −−∝−−∫ vvccvc kEkEdttiEtitiE

21

Matrix ElementMatrix Element

Similarly, the matrix element of the complex conjugate (exp(iωt)) gives rise to

This means that the process describes the transition from electron from the valenceband to the conduction band by an absorption (-) or emission (+) of photons.

Writing the Block functions for the electrons in the conduction and valence bands, as:

and using the espression for A we obtain

Operating with p on uv,kvexp(ikvr) yields two terms:

The integral of the second term multiplied by u*c,kc vanishes because uc,kc and uv,kv areorthogonals.

))()(( ωδ hrr

+− vvcc kEkE

[ )[ ) )](exp[)(

and )](exp[)(

,

,

rkiruv

rkiruc

vkvv

ckcc

rrr

rrr

⋅=

⋅=

).exp()exp()exp( ,,, rkiukuprkirkiup vkvvvkvvvvkvvrrr

hrrrrr

⋅+⋅=⋅

( )[ ] 2,*,22

2)exp()(exp

4 ∫ ⋅⋅⋅−=⋅ drrkiuperkqiuqE

vpAc vkvvckccrrr(rrrrr

22

Matrix ElementMatrix Element

We can split the corresponding integral of the first term

into two parts by writing r=Rj+r’, where r’ lies within one unit cell and Rj is a latticevector.Because of the periodicity of the uc,kc and uv,kv, we find

The summation of exp[i(q-kc+kv) Rj] over all the lattice vectors Rj results in a deltafunction δ(q-kc+kv). This term ensures that wavevector is conserved in the absorptionprocess (q+kv=kc). This is a consequence of the translation symmetry of a perfectcrystal. Using the wavevector conservation the integral over the unit cell simplifies to

This expression can be further simplified if we assume that q (q=0) is much smaller thanthe size of the Brillouin zone, a condition usually satisfied by visible photons, whosewavelengths are of the order of 500nm.

( )[ ]∫ ⋅+− druprkkqiu kvvvckcc ,*, exp rrrrr

( )[ ] ( )[ ] ''expexp ,*, druprkkqiuRkkqi kvvvccellunit

kccj

jvc

rrrrrrrrr⋅+−

⋅+− ∫∑

( )[ ] '''exp ,*,,*, drupudruprkkqiu kvvcellunit

qkvckvvvc

cellunit

kcc

rrrrrr∫∫ +=⋅+−

23

Electric dipole approximationElectric dipole approximation

For small q the wavefunction ukv+q can be expanded into a Taylor series in q:

When q is small enough that all the q-dependent terms can be neglected, the matrixelement is given by

This approximation is know as the electric dipole approximation. Notice that the electricdipole approximation is equivalent to expanding the term exp(iq·r) into a Taylor series:1+i(q·r)+ ..... and nectecting all the q-dependent terms. In this case we have that kv=kcand the transitions are said to be vertical or direct.If the electric dipole matrix element is zero, the optical transition is determined by the qterm (in the top equation) and the matrix element is

gives rise to electric quadrupole and magnetic dipole transitions.

...,,, +∇⋅+=+ kvvkkvvqkvv uquur

2

,*,

2')(

⋅=⋅ ∫ drupeuvpec kvcellunit

kc

r(r(

2

,*,

2')()(

⋅∇⋅=⋅ ∫ + drupeuqvpec kvvcellunit

qkvc

r(rr(

24

Electric Dipole Transition ProbabilityElectric Dipole Transition Probability

From now we shall restrict ourselves to electric dipole transitions and therefore kc=kv=kand the momentum matrix element is not strongly dependent on k so we shall replace itby the constant Pcv2. The equation can by semplified

where A contains the term exp(-iωt) (absorption process) and his complex conjugate(emission) not consider in the following discussion.

The electric dipole transition probability R for photon absorption per unit time is:

is thus given by

The power lost by the field due to absorption in unit volume of the medium is simply thetransition probability per unit volume multiply by the energy in each photon:

( ) ( ) .// 222222 cveR PAmcevpAcmcevHc =⋅=rr

))()(()/2(,

2ωδπ h

rrh −−= ∑ kEkEvHcR v

kvkcceR

))()((2

)()/2(

,

222

ωδω

ωπ h

rrh −−

= ∑ kEkEPE

m

eR v

kvkcccv

ωhRPowerloss =

25

Dielectric FunctionsDielectric Functions

This power lost by the field can also be expressed in terms of α and εi of the medium bynoting that the rate of decrease in the energy of the incident beam per unit volume isgiven by –dI/dt, where I is the intensity of the incident beam:

we obtain

By using the KKR we can obtain the espression for the real part:

where

ωωε

α hRn

II

n

c

dt

dx

dx

dI

dt

dI i ===

−=−2

))()((2

)(2

2

ωδωπ

ωε hrr−−

= ∑ kEkEPme

vk

ccvi

−

+= ∑

k

cv

cvr

cv

P

mm

e22

22 241)(

ωωωπ

ωεh

)()( kEkE vccvrr

h −=ω

cncn ii

ωεκωακε ===

2 2

26

Joint Joint Density ofDensity of States States

The dispersion in εi comes from the summation over the delta function of the energyconservation. The matrix elements Pcv2 between a given couple of valence andconduction bands are shown to be not strongly dependent from k, except near special kvectors where Pcv vanishes because of symmetry. Neclecting such situation and takingPcv as a constant, we find that the contribution to the dielectric function from a pair ofbands is proportional to 1/ω2 and to the quantity:

which is called joint density of states because it gives the density of pair states (oneoccupied and the other one empty, separated by an energy). The integration can beperformed by using the properties of the δ function. We know that

in which x0 represents a zero of the function f(x) contained in the interval (a,b). In 3D wehave

where Ecv is the abbreviation for Ec-Ev, andSk is the constant energy surface definedby Ecv(k)=const.

∫ −−=BZ

vcj kEkEdkD ))()((4

13

ωδπ

hrr

[ ] ∑∫−

=

=0

1

0

)()()(x xx

o

b

a dx

dfxgdxxfxg δ

∫ ∇= )(413

cvk

kj E

dSD

π

27

Van Hove singularitiesVan Hove singularities

The joint of density for interband transitions as a function of Ecv shows strong variationsin the neighbourhood of paricular values of Ecv which are called critical point energies.From the espression of Dj we see that singularities in the joint density of states areexpected when

This point are know as critical points and occur in general at high simmetry points of theBrillouin zone and the corresponding singularities in the density of states are known asVan Hove singularities.The analytic behaviour of Dj near a singularity may be found by expanding Ec(k)-Ev (k)in a Taylor series about the crytical point. In the expansion the linear terms are zerobecause of the above condition. Limiting the expansion to quadratic terms and denotingthe wave vectors along the principal axes with the origin at the crytical point by kx, ky,kz,

with mx,my,mz are positive quantities and εx, εy, εz equal to +1 and -1.

0)()(

whengenerally moreor 0)()(

=∇−∇

=∇=∇

kEkE

kEkE

vkck

vkckrr

rr

+++=−

z

zz

y

yy

x

xxovc m

k

m

k

m

kEkEkE

2222

2)()( εεε

hrr

28

Critical PointsCritical Points

We obtain four type of singularities, depending on the signs of εx, εy, εz. The criticalpoints are called:

M0 when all coefficients of the quadratic expansion are positive (minimum);M1 when two coefficients of the quadratic expansion are positive and one negative(saddle point);M2 when two coefficients of the quadratic expansion are negative and one positive(saddle point);M3 when all coefficents of the quadratic expansion are negative (maximum);where the subscripts attached to M indicate the number of negative coefficients in theexpansion of energy differences.

The analytic behaviour of the joint density of states near critical points can be obtainusing the espression of Dj and the energy expansion. We can notice that there are sharpdiscontinuities at the critical points.

+++=−

z

zz

y

yy

x

xxovc m

k

m

k

m

kEkEkE

2222

2)()( εεε

hrr