nano.tul.cz Inovace a rozvoj studia nanomateriálů na TUL

Transcription

1 Inovace a rozvoj studia nanomateriálů na TUL nano.tul.cz Tyto materiály byly vytvořeny v rámci projektu ESF OP VK: Inovace a rozvoj studia nanomateriálů na Technické univerzitě v Liberci

2 Units for the frequency and energy of electromagnetic waves The various units of frequency and energy are used in the literature. Frequency of light n is given in Hz cycle per second. The angular frequency w is related to n w=2pn. In the light scattering spectroscopy (Raman, Brillouin) and infrared spectroscopy wavenumber is used, which is defined as the reciprocal wavelegth. A wavenumber of 1 cm -1 corresponds to wavelength 1 cm or a frequency Hz.

3 Units for the frequency and energy of electromagnetic waves When we study quantum processes involving electromagnetic waves, we quantized the energy of electromagnetic waves into photons. The energy of the photons is equal to hw where h=h/2p Planck constant. Photon energy is often expressed in electron volts. Another unit an equivalent temperature T (K) is used. Photon energy is equal k B T where k B is Boltzmann`s constant.

4 The conversion factor between ev and 1 ev cm ev Hz. 1 ev mm nm. 1 ev K.

5 Helmholtz's theorem Statement of the theorem Let F be a vector field on a bounded domain V in R 3, which is twice continuously differentiable. Then F can be decomposed into a curl-free component and a divergence-free component: where

6 Potential formulation of Maxwell`s equation On the base of Helmholtz's theorem electromagnetic field can be expressed in a potential form. Also Maxwell's equations can be expressed Maxwell's equations in a 'potential formulation' involving the electric potential (also called scalar potential), φ, and the magnetic potential, A, (also called vector potential). These are defined such that:

7 Potential formulation of Maxwell`s equation These equations, taken together, are as powerful and complete as Maxwell's equations. Moreover, if we work only with the potentials and ignore the fields, the problem has been reduced somewhat, as the electric and magnetic fields each have three components which need to be solved for (six components altogether), while the electric and magnetic potentials have only four components altogether. Many different choices of A and φ are consistent with a given E and B, making these choices physically equivalent a flexibility known as gauge freedom. Suitable choice of A and φ can simplify these equations, or can adapt them to suit a particular situation.

8 Potential formulation of Maxwell`s equation The choice of these potentials is notunique due to gauge invariance, for simplicity we will choose the Coulomb gauge in which: j=0 and div A=0. In this gauge E and B are defined by E=-1/c da/dt and B=div x A

9 Low absorption coefficient 1. Absorption due to isoelectronic traps Isoelectronic centers are formed by replacing substitutionally one atom of the crystal by another atom of the same valence. The most important example is the case of nitrogen substituting for phosphorus in GaP. The nitrogen isoelectronic trap is a very localized potential well which can trap an electron, thus become charged, the resulting Coulomb field attracts a hole. The two trapped carriers form an exciton bound to the isoelectronictrap. When these centres are close together, they may interact with each other in pairs and form

10 1. Absorption due to isoelectronic traps A new set of energy levels for their own bound exciton. The binding energy of an exciton bound to isoelectronic centres depends on the distance between two atoms of a pair. The strongest binding energy occurs for the nearest pairs. The absorption spectrum due to such excitons is shown in fig

11 1. Absorption due to isoelectronic traps

12 1. Absorption due to isoelectronic traps The lowest energy absorption peak correspond to the most strongly bound exciton of a nearest neighbor N- N pairs. The intensity of the absorption depends on the number pairs.

13 2. Transition between band and impurity levels The transition between a neutral donor and the conduction band or between the valence band and the neutral acceptor can occur by the absorption of low energy photons.

14 2. Transition between band and impurity levels

15 2. Transition between band and impurity levels

16 2. Transition between band and impurity levels

17 2. Transition between band and impurity levels No absorption is obtain until the photon energy equals the first excitation energy state. Absorption peaks are obtained for excitation to states n=1, 2 and 3. The higher modes merge in the band corresponding to the complete ionization of donor. Note that although the density of final states increases with energy, the absorption coefficient for complete ionization of impurity decreases with photon energy. This decrease is due to a rapid decrease of the transition probability away from the bottom of the conduction band. At k/=k 0,where k 0 is the momentum at the bottom of the conduction band, where wavefunction of the impurity decreases as 1/(1+[1+(k-k 0 ) 2 ] 2 ).

18 2. Transition between band and impurity levels The transition between valence band and ionized donor (it must be empty to allow the transition) or between ionized acceptor and the conduction band occurs at photon energy hw>e g -E i Hence the transition between an impurity and band should manifest themselves by a shoulder in the absorption edge see. 3.31

19 2. Transition between band and impurity levels

20 2. Transition between band and impurity levels The absorption coefficient for transitions involving the impurity levels covers a much smaller range than transition between valence and conduction bands because the density of impurity states is much lower than the density of states in the band. In practice, the shallow impurities are seldom resolved from the background of absorption due to transition involving tails of the state.

21 3. Free carrier absorption By free carriers we mean a carriers free to move inside a band, i.e. carriers which can interact with their ambient. Free carrier absorptionis characterized by a monotonic spectrum which grows as l p where p is from the range To absorb a photon the electron must to make a transition to a higher energy state within the same valley.

22 3. Free carrier absorption

23 3. Free carrier absorption Such a transition requires an additional interaction to conserve momentum. The change in momentum can be provided by interaction with phonons or by scattering from ionized impurities. Because there is no transition between bands we do not need to use QM, we can use Drude classical theory for the oscillation of an electron driven by a periodic electric field in a metal lead to a damping (attenuation) which increase as l 2.

24 3. Free carrier absorption The collision with semiconductor lattice resulting in the scattering by acoustic phonons leads to absorption increasing as l 1.5, the scattering by optical phonons gives the dependence l 2.5, while scattering by ionized impurities lead to increase as l 3 or l 3.5. a fe =A l B l C l 3.5

25 3. Free carrier absorption

26 4. Band Band transition (Val-Val or Con Con)

27 5. Lattice absorption In compound semiconductors (GaAs), the bonding between atoms of different species forms a set of dielectric dipoles. These dipoles can absorb energy from the electromagnetic field, achieving a maximum coupling to the radiation when the frequency of the radiation equals a vibration mode of the dipole. Usually the vibrational mode is complex, consisting of several types of fundamental vibrations (multiphonon emission) The photon has negligible momentum, in comparison with phonon, therefore two or more phonons must be emitted to satisfy momentum conservation.

28 5. Lattice absorption In homopolar semiconductors (Si, Ge) there is no bonding dipole, but lattice vibrational spectra were observed. Apparently the second order process occurs: the radiation induces a dipole which in turn has a strong coupling to the radiation and produces more phonons.

29 5. Lattice absorption Phonons are quantized simple harmonic oscillators (SHO). What is the response of a collection of identical and charged SHO to a radiation in the form of the plane wave E(r,t)=E 0 exp i(kr-wt)? We assume that these SHOsare isotropic and uniformly distributed in the entire space. The mass, charge and natural vibration of each SHO are M, Q and w T.

30 5. Lattice absorption In response to applied field the SHO are displaced from their equilibrium position by vector u. The equation of SHO motion is: M(d 2 u/dt 2 )=-Mw T2 +QE u in study state solution can expressed as u=u 0 exp[i(kr-wt)], where u 0 =QE 0 /[M(w T2 -w 2 )] (3) Since these SHOs are charged and they are all displaced by the same amount u, they produce macroscopic polarization P oscillating also at frequency w. P=Nqu (4) and D=E + 4pP=eE

31 5. Lattice absorption Substituting equation (3) and (4) into (5) we obtaine e: e=1+4pnq 2 /M(w T2 -w 2 ) (6) When equation (6) is generalized toa collection of SHOs with different resonance frequencies wi : e=1+s i 4pN i Q 2 /M(w i2 -w 2 )

32 5. Lattice absorption

33 5. Lattice absorption

34 Experimental determination of optical functions We know from KKRs that from the knowledge of one optical function in all spectral range we can calculate an other optical function in one frequency. If the photon energy is below the electronic band gap and well above any phonon energies, the sample absorption coefficient is either zero or very small. I(d)=I 0 exp (a d), if a is small we need thick crystal for absorption measurement, to receive ad~1. In this case is advantage to measure refractive index, which is real value because k is zero.

35 Experimental determination of optical functions If a is large ~10 6 cm -1 we need very thin sample d~ 10-6 cm, which is difficult to prepare. An alternative for determination of the complex dielectric function of strongly absorbing samples is to use reflection measurements. The major drawback of these measurements is their sensitivity to the sample surface quality.

36 Reflection measurements

37 Reflection measurements Determining the dielectric function from reflection measurement is quite straightforward. It involves irradiating the sample at either normal or oblique incidence.

38 Reflection measurements

39 In oblique incident techniques the reflectance R S and R P of the s- and p-polarized components of the incident light are measured, component of the incident perpendicular and parallel to the plane of incidence are labeled, respectively, as s- and p- polarized.

40 Reflection measurements These reflectances are related to the complex refractive index by Fresnel formulae Where r s and r p are the complex reflectivity for s- and p-polarized light, respectively, and f is angle of incidence.

41 Reflection measurements The complex refractive index can be deduced by measuring both R s and R p at a fixed f. An oblique angle of incident technique which has became very popular in the past decade is ellipsometry. This name is derived from the fact that when linearly polarized light that is neither s- or p-polarized is incident on a medium at an oblique angle, the reflected light I ellipticaly polarized.

42 Ellipsometry The ratio (s) of the complex reflectivities r p /r s can be determined by measuring the orientation and the ratio of the axes of the polarization ellipse corresponding the reflect light. The complex dielectric function can be determined from s and f using: e=sin 2 f + sin 2 f tan 2 f (1-s/1+s) 2

43 Ellipsometry In principle, it is necessary to measure both the normal incidence reflectance R and the absorption coefficient a in order to determine the complex refractive index and hence the dielectric function. In practice it is sufficient to simply measure R over a wide range of photon frequencies and than deduce the absorption coefficient using the KKRs.

44 Dielectric function We are able to determine the dielectric function from the reflection measurements, but what is a nature of dielectric function? e(w)=e r (w)+i e i (w) e(w)= n~ 2 and n~(w)= n(w)+ik(w) e(w)= n 2 (w)-k 2 (w) +i 2nk e r (w)= n 2 (w)-k 2 (w) e i (w)= 2nk

45 Microscopic theory of dielectric function The same procedure which was used to determine the absorption coefficient we can obtain the dielectric function: Where hw cv =E c (k)-e v (k)

46 Microscopic theory of dielectric function We know the dielectric function for collection of classical charged harmonic oscillators with frequency wi (6.50). If we compared it with relation (6.48) 2 P cv 2 /mhw cv is essentially the number of oscillators with frequency w cv. This is called the oscillator strength of the optical function.

47 Join density of states and Van Hove singularities Note that most of the dispersion in e i comes from the summation over k the delta function. From dispersion of optical functions we want to determine E c (k) and E v (k). But as you can se in 6.48 we have only difference E c (k) - E v (k). From the difference E c (k) - E v (k) we cannot determine E c (k) and E v (k). We can only determine extreme points because absorption is non zero only for hw E g <hw<e m

48 Join density of states and Van Hove singularities Dependence of E c (k) and E v (k) on k Dependence of difference E c (k) - E v (k) on k.

49 Join density of states and Van Hove singularities Extreme are for: div k (E c (k) - E v (k))=0 In the three dimensional space are four kind of singularities labeled M 0, M 1, M 2, and M 3. M 0 represent a minimum in the band separation. M 3 represent a maximum in the band separation. M 1, M 2 represent saddle points.

50 Join density of states and Van Hove singularities

51 Join density of states and Van Hove singularities These points are known as critical points and corresponding the singularities in joint density of states are known as Van Hove singularities. The join density of for doubly degenerate conduction and valence band is defined as: D j (E cv )=1/4p Integral ds k /div k (E cv ), where E cv =E c -E v and S k is the constant energy surface defined by E cv =const.