NGUYEN VAN HIEU and NGUYEN AI VIET: Raman Scattering on Donor Levels

phys. stat. sol. (b) 92, 537 (1979)

Subject classification: 13.1 and 20.1; 22.2

Institute of Physics, National Center for 8cientqic Research of Vietnam, Hanoi

537

Electronic Raman Scattering on Donor Levels in Three-Band Semiconductors with Direct Band Gaps

BY NGUYEN VAN HIEU and NGUYEN AI VIET

A theory is presented of the electronic Raman scattering on shallow donor levels in three-band semi- conductors with direct band gaps. Explicit expressions of the matrix elements of the transitions from the ground state to the first excited state and to the states of the continuous spectrum are derived. It is shown that the most important contribution to these matrix elements comes from the intermediate states in the higher conduction band and in the valence band.

Eine Theorie der elektronischen Ramanstreuung fur den flachen Donator in einem Drei-Band-Halb- leiter mit direkter Bandstruktur wird behandelt. Explizite Formeln der Matrixelemente der tfber- gange vom Grundzustand in den ersten Anregungszustand und zu dem Kontinuumspektrum wer- den angegeben. Es wird gezeigt, da13 die Zwischenzustande in dem hoheren Leitungsband und Valenzband den wichtigsten Beitrag fur diese Matrixelemente liefern.

1. Introduction

Different electronic Raman scattering processes in semiconductors were studied in many experimental and theoretical works [l to 161. The influence of strong electro- magnetic fields on their cross-sections was considered in [17 to 211. If the initial and final states of electrons (OP holes) are states of free electrons (or holes) in the same energy band the scattering process is called intraband; if they are free charge carriers in different energy bands we have the interband scattering process. The initial states of electrons (or holes) may also be donor (or acceptor) levels, and the final state other donor (or acceptor) levels or states of free electrons (or holes) in the same energy band. I n this case we have the Raman scattering on the impurity levels. The theory of electronic Raman scattering on donor levels in the two-band multivalley semi- conductor with indirect band gaps was developed by Colwell and Klein [12]. I n this work we study the electronic Raman scattering on donor levels in the three-band semiconductors with direct band gaps (Fig. 1).

I n the lowest order of the perturbation theory the electronic Raman scattering is represented by the Feynman diagrams in Fig. 2 a and b. Denote the contributions of these diagrams to the scattering ma8trix element by MCf and M!!?f, respectively. I n the unit system with h = c = 1 we have

Here li), If>, and IN) are the initial, final, and intermediate states of electrons or

538 NGUYEN VAN HIEV and NGUYEN AI VIET

Fig. 1 Fig. 2

Fig. 1. Energy bands Fig. 2. Feynman diagrams

holes, Ei, E,, and EN are their energies, k , k' and co, co' are, respectively, the momenta and energies of the incoming and outgoing photons, 5 and 5' are their polarization unit vectors, e and M are the free electron charge and mass, P, with M = 1,2, 3 denote three components of the momentum operator. I n the denominators of the right-hand side of (1) we take +iO for the states in the conduction bands including the donor levels and -iO for the states in the valence band.

We shall study the scattering process in which the initial and final states are donor levels in the lowest conduction band. The intermediate states in the matrix element (1) can be donor levels or free electron states in the same energy band; their contribu- tion to the matrix element (1) is called that of the virtual intraband transitions. They can belong also to the higher conduction band or the valence band; their contribu- tions are called those of the virtual interband transitions. We shall see that in some energy interval the virtual interband transitions give the main contributions to the matrix element. These contributions depend strongly on the ratios of the effective masses of electrons in the corresponding energy bands, and the electronic Raman scattering, in particular, can be used for checking the value of the effective mass of the electron in the higher conduction band.

I n this work we consider only one scattering process in which the initial electron state is the ground state, Is, of the electron in the donor atom, and the final state is the first excited one, 2s. This transition was observed in many experimental works. Our consideration can be generalized to the case of the transitions into other final states.

2. Calculation of the Matrix Element

We assume that in the semiconductor there are three spherically symmetric parabolic energy bands: the valence band v with the effective mass p', the lowest conduction band c with the effective mass p, and the higher conduction band C with the effective mass p. Their extrema are supposed to be located a t the centre of the Rrillouin zone (Fig. 1). I n the Coulomb field of a donor ion an electron has a spectrum of discrete levels with quantum numbers n, 1, m and a continuous spectrum of states with asymp- totic momenta q. For simplicity the set n, I, m or the momentum q will be also sub- stituted by a unique symbol v. The total state vectors of electrons in the bands c and c" are denoted by

respectively. I n the field of the positive ion a hole has only a continuous energy spec- trum with state vectors

Iv; n> -

Electronic Raman Scattering on Donor Levels in Three-Band Semiconductors 539

First we consider the contribution of the virtual intraband transitions to the matrix element (1). Suppose that the wave functions of the electrons (Bloch functions) a t the extrema of the energy bands have definite parities, as in the case of a crystal with inversion cent,re, and denote by cpv(r) the envelope wave function of the donor states. Neglecting the local field corrections, we have

The r.h.s. of this relation is the matrix element in the Hilbert space of the wave functions of an electron with mass ,u in the Coulomb field. Therefore, the contribution

l+f of the virtual intraband transitions to the matrix element (1) equals the ampli- tude of the Raman scattering on the electron with mass p in a hydrogen-like atom. The latters was calculated by Zon et al. [22] , Granovskii [23] , Saslow and Mills [24], and others [25, 261for the transition 1s --f 2s. According to the results of these works we have

Mfntra

2 e2 p 3 M M MEtfaz, = - - - ( ( '"5) P1(w); (4)

where i% is the incident photon energy in units of the binding energy El , of the donor atom in its ground state:

where E is the dielectric constant, a the fine-structure constant, and

n3 exp [-2n (artanh 1/n + artanh 2 / n ) ] + nz3 (n2 - 1) (n2 -- 4)2 (1 - l/n2 - cc) - - i 0 )

} dx . x3 exp [--2x(arctan l/x + arctan 2/x)

(7) 0

Now we compute the contribution M E : of the virtual interband transition to the matrix element (1). We denote by y),,(r), q v ( r ) , and yv(r) the envelope wave functions of the electrons in the conduction bands c, 'E and the valence band v, respectively. Neglecting again the local field corrections, we obtain

( E ; v'l eikr Pa Ic; v) = IT$ J eikr @?pV d3r, (v; v ' I eikr Pa Ic; v) = IT;" J eikr y3rpy d3r,

where II$ and lI2v are the matrix elements of the electrical dipole transitions from the lowest conduction band c to the higher conduction band E or the valence band v. These transitions are assumed t o be allowed. Substituting the matrix elements (8)

540 NGUYEN VAN HIEU and KCUYEN AI VIET

into t'he r.h.s. of (l) , we obtain

where

1 2(1 - 4t + t2 ) =

(1 + t )3 (2 + t ) 4 ' Q2 = (1 + 2t)4 (1 + t ) 5 9

Qn = - (1 - t ) X I n5(4 - 4n2t + n2t2) exp [-2n(artanh l/nt + artanh 2 / 4 1

(n2t2 - 1)2 (n2t2 - 4)3 X

= 1/ii28e3(1 + e ) 2 ,

x6(402 - 4x28 - z2) exp [2z(arctan (0/2) + arctan ( 2 0 / x ) ) ] dz -- i ( 2 2 + e 2 ) 2 ( ~ 2 + 4872 [i - e l x 2 - (6 + 6) + io] [exp ( 2 ~ 4 - I]'

F4(0) = F,(+ El, - 0) 3

0

aB is the Bohr radius of the hydrogen-like donor &

aB = - v'

P' 0 = --, P

atom,

0" and A are the distances from the bottom of the conduction band c to the bottom of the conduction band E or to the top of the valence band v, respectively.

Expressions (13) and (14) were obtained in the case when the valence band is non- degenerate and, therefore, there is only one kind of holes. However, they can be ex- tended immediately to semiconductors with degenerate valence bands. I n the first approximation we consider separately the contributions from each kind of holes and then take their sum.

The matrix element (2) can also be calculated in a similar manner. It vanishes in the dipole approximation, and we consider the lowest non-vanishing order in the ex- pansion of the exponential. We have

2%' e2

36 M M(b) ls+2s = f ___ - (5'*5) a i ( k - k ' ) 2 .

Electronic Raman Scattering on Donor Levels in Three-Band Semiconductors 54 1

3. Comparison of the Contributions of Different Scattering Mechanisms

I n the preceding section we expressed the scattering matrix element in terms of the amplitudes Fl(w) - F5(w). I n order to compare the contributions of different scat- tering mechanisms to the matrix element and to study their energy dependence first we consider some general properties of these amplitudes.

The amplitudes li;(co) determine the contribution of the intraband virtual transi- tions. It has significant values only when the incident photon energy w is comparable with Els, and in this region there is a series of poles of F,(o) corresponding to the discrete energy levels of the donor atom. As a function of the dimensionless variable 6, IF,(co)lz is represented in Fig. 3.

The amplitudes F2(w) and F3(o) determine the contribution of the intermediate states in the higher conduction band E . At the energies w of the incident photon such that w - 0” is comparable with Els, the amplitude Fz(w) is not small and has a series of poles corresponding to the discrete bound states of the electrons in t h e higher con- duction band c” in the Coulomb field of the donor ion. At large values of Iw - 21, F2(w) tends to zero faster than

- 10 - A\-1,

as it can be shown using the completeness of the system of donor levels in each con- duction band. Similarly, F3(w) tends to zero faster than

(0 + i i - 1

a t large values of w + 3. All the amplitudes F 2 ( o ) and F3(w) depend essentially on the ratios of the masses of the electrons in two conduction bands

t = - P _ . P

They vanish identically if t = 1. The curves in Fig. 4 represent jF2(w)12 for some A1I1Bv semiconductors as functions of (6 - 0”). The values of the mass ratios are taken from [27].

Similarly, the amplitudes F4(w) and F5(w) determine the contributions of the inter- mediate states in the valence band. The amplitude F4(w) is not small a t the incident photon energies w such that w - A is comparable with El,. It tends to zero faster than

at large values of (o - A / . The amplitude F 5 ( o ) also tends to zero faster than

10 - A/-1

(0 + 4- l at large values of w + A. The two amplitudes F4(w) and F5(w) depend on the ratios of

542 NGUYEN VAN HIEU and NQUYEN AI VIET

&-- Fig. 4 -

Fig. 4. IPJ2 as a function of iij - A Fig. 5. lF4[z as a function of 65 - d

Fig. 5

the masses of the electron in t'he conduction band c and of the hole

The values of IF4(w)12 were given in Fig. 5 for the heavy holes of some semiconductors. To compare the contributions from different scattering mechanisms we use the

above-mentioned properties of the amplitudes Fl(o) to F5(w) and expressions (4), (9), (17) of the matrix elements. Note that beside the amplitudes Fi(w) the common factor e2/M and the mass ratio p / M , the contribution MEFZ, of the interband virtual transitions and the matrix element of the diagram with the two-photon vertex contain also the factors agD2 and ag(Ak)2, respectively. For the scattering of the pho- ton in the forward direction

lAkl z w - W' s El,,

aB l ~ k l s & 0.5 x 10-3.

If we take p x O.06M and F z 11, as in the case of GaAs, we have aB x 5 x (eV)-l .

Then a t reasonable values of the incident photon energy o & 2 eV and for the scat- tering in the backward direction

lAkl s 4 eV , CZB IAkI s 2 x 10-l.

The values of IIcv for many semiconductors were given in [28] IDcv\ 2 2 x 103 eV .

With these data

aB lUcvl 2 l o2 .

Electronic Raman Scattering on Donor Levels in Three-Band Semiconductors 543

Prom the values of the magnitudes of the amplitudes represented in Fig. 3 to 5 and the above estimations of the factors aB IAkl and aB IDcv‘( it follows that the contribu- tion ME!+zs of the diagram with a two-photon vertex is always negligible in comparison with that of the interband virtual transitions and the contributions of the intraband virtual transitions M:?zs are significant only in a narrow interval of the incident photon energies

s El, >

where the virtual transitions to the intermediate states are resonant ones. Note that El , is rather small. For GaAs this energy is about 6 meV. At the values of the incident photon energy larger than this binding energy the interband virtual transitions will dominate.

4. Discussion

The experimental study of the electronic Raman scattering on donor levels in three- band semiconductors with direct band gaps can provide very useful information on the band structure of the semiconductors. At values of the incident photon energy considerably larger than the binding energy of the ground state of the electron in the donor atom (several meV) the interband virtual transitions give the main contribu- tion. This contribution M:yZzs depends essentially on the ratios of the masses of elec- trons and holes in the three bands c, C, v and on the matrix elements I l c z and Ilcv of the allowed electrical dipole transitions between the corresponding energy bands. They exhibit also resonance behaviour when w - 3 is comparable with the binding energy of the electron in the higher conduction band E in the Coulomb field of the donor ion. Therefore, the experimental stJudy of the electronic Raman scattering on donor levels would permit t o determine the distance n” of the bottoms of the two conduction bands, the magnitude of the electrical dipole transition matrix element T l c z , and the mass ratios j l lp.

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NGUYEN VAN HIEU and NGUYEN AI VIET: Raman Scattering on Donor Levels

Nauka Dumka, Kiev 1975.

(Received October 6, 1978)