The Fano effect in excitonic spectra

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S o l l d S t a t e Comauntcattons, Vol. 68, No. 11, pp.981-984, 1988.

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Prlnted In Great Britain.

0038-1098/88 $3.00 + .00 Pergamon Press plc

THE FANOEFFECTIN EXCITONICSPECTRA L Reining* and I. Egry** (*) Dipartimento di Fisica, 2a Universit*, di Roma, via Orazio Raimondo, 00173 Roma, Italy (**) Institut for Raumsimulation, Deutsche Forschungs- und Versuchsanstalt fOr Luft- und Raumfahrt, 5000 KSIn-Porz90, Fed. Rep. Germany (Received by E. Tosatti 15

3uly 1988)

This paper deals with the extension of a two band model for Wannier excitons to a model which includes three bands. The interaction of the excitons resulting from any pair of bands is taken into account by diagonalizing a truncated eigenvalue equation. Energy bands and eigenstates of excitons are determined, and the imaginary part of the dielectric function ¢2(g.=0,o~) is calculated. For certain choices of parameters, a Fano effect shows up.

1.Introduction

gives the interaction between the 2-band-excitons, with M12~-k') being the Fourier transform of the monopole-dipole-interaction

In a three band model of appropriate symmetry, two different types of interband transitions, and hence excitons, exist. Each pair of bands gives rise to a hydrogen-like series of excitonlc levels. However, due to the interaction between different excitons, these levels are shifted or split, and the corresponding wave functions are modified. The most interesting effect occurs when the effective masses and the distances between the bands are such that discrete lines of one spectrum are situated in the continuum range of another spectrum; this is the classical Fano situation 1 -Fig.la-. However, we had to extend this formalism to include degeneracies in both spectra. With this modification, we calculated the characteristic change in the line shape of the absorption spectrum. We restrict our description to direct semiconductors with all energy band extrema lying at k=0, the bands being parabolic for small k and of rotational symmetry around the energy axis -Fig.lb-. The calculations are done for one conduction and two valence bands; the inverse situation requests only simple changes.

(2) M12~)=e2.(l~12-r)/r3 e is the elementary charge and

2.The Hamlltonlan The Hamiltonian H describing our sy+stem in second quantization contains the operators c(i)Kand c(i)K, which

Fig.la: Excitonic sedes in the 3-band-model

create and destruct, respectively, an electron with wave vector k in the band (i)2; i=I,2,3 as defined in Fig.la. It consists of three parts: (I) H=HB+HE+Hint where (la) HB= ~ ~kE(i)(j~)c(i);c(i) k i=1,2,3 gives the energy bands in Bloch representation with E(')(J~)energy of band (i) at wave vector k.

E 3

(lb) HE~I/N ~-kJ~',g U(-k-k-')lS~.(--qlSk'(-q)+T~_(-q)Tk'(-ql} describes non-interacting excitons in the respective 2-band-model, because -Sk(g)=c(3)l~+gc( -_ 1)k creates an exciton with wave vector g. between the bands (1) and (3), T~(g,),,c(3)j~+gc(2)k between (2) and (3),

Im

0

and U~-k') is the Fourier transform of the effective Coulomb interaction e2/e~r.

k

1

11c) Hint=-l/N ~'k.k',g, H12(k'k')lT~_(-q)Sk'(-q)÷S~(glTk '(-q)}

Fig.1 b: The 3-band-model 981

982

Vol.

THE FANO EFFECT IN EXCITONIC SPECTRA

(3) ~12=j-d3 r al *(~).r.a2(~ is the dipole moment between the Wannier functions of the valence bands. It is treated as a parameter and may contain also screening effects. The matrix elements U and M 12 are the lowest order terms in a multipole expansion of the Coulomb interaction between the localized Wannier functions. Of course some other combinations of the operators c + and c would also give possible contdbutions to the Hamiltonian. Applying the equation of motion method, however, with a linear combination of the

6 8 , No. I I

This expression accounts for the cases E(g~)=~E(2)(g.), given by the principal value expression P{1/(E(g,.)-E (2) (g.))}, and E(g.)=E(2)(g,.), given by the delta function weighted by the factor z(E(g,)). As in the nondegenerate case 1, z does not depend on other quantum numbers. This has been shown by doing the calculations for a quasi-continuum 3 and then performing the continuum limit. Inserting (7) into (6a), we obtain an equation in o¢ which is slightly more complicated than in the work of FanoI : N

(8) (E(1)n(g.)-E(g))O~ns+~n,s,OCns,{j'dE(2)(~tVg(ECZ)t)(ns)Vg.(E(2)t)(ns,))/(E(g)-E(2)(.g)) +z(E(g,))~tV*g(Et)(ns)Vg(Et)(ns')}=0

2-band-excitons as an ansatz for the true elementary excitation in our problem, the terms given above are retained as the effective ones. 3.Solution

In order to have non-trivial solutions OCns' the determinant of (8) must vanish. This condition yields an implicit equation for z(E). Simple results are obtained for n=l and for n=2 at g.=0: (9) z(E)=(E-E(1)n-T'tFnt(E))/(~'tlV(Et)n 12)

The eigenstates 14~>g.of the Hamiltonian (1) are linear combinations of eigenstates of the uncoupled Hamiltonian (HB+

(10a) O ( n ( E ) = s i n ( Z ~ ) / ( / t ~ )

(1Oh) ,SE(2)t(E)={sin(A)/(II-(E-E(2)))-cos(Z~).8(E-E(2))}.V(E(2)t)n/./rtlV(EC2)t)n 12 with

HE) : (4) I¢>~=~n,sO~nsl¢l ns>g+~n,s~8nsl¢2ns>g. The indices 1,2 indicate the valence band which is involved, and n,s is the set of all hydrogen quantum numbers. This leads to a system of equations for the coefficients o~ and ~: (5a) E(1)n(_q)O~ns+~m,tV o(mt)(ns)~mt=E(g.)O~ns

(1 1) Fnt(E)=J'dE(2)IV(E¢2)t)nI2/(E-E(2)) (12) A=-atan(/t~'tlV(Et)nl2/(E-E (1)n-T'tFnt(E)) ) Equation (10) defines the coefficients o< and ~] and, therefore, via (4), the corresponding eigenstata. 4.Optlcel Properties

(5b) E(2)m(.~mt+Y'n,sVg.(mt)(ns) O~ns=E(g)~mt E(i)are 2-band exciton energy levels. The indices n,m denote principal quantum numbers, while s,t represent angular quantum numbers. E(g.) is eigenvalue of the full Hamiltonian (1), and Vg,(mt)(ns)=g.<~2mtlM 12 I~lns>.g. is the coupling matrix element, which is calculated analytically. Equation (5) cannot be solved exactly but only in a suitably chosen subspace. It is determined by the region of energy of interest, the relative position of the two unperturbed spectra, and the computational effort one is ready to invest. in the case of low excitation lines in both spectra, eq. (5) is solved by a simple matrix diagonalization. However, we are interested in the Fano situation and consider therefore the case where a single line E(1)n(~) lies in the continuum E(2)(g.). This requests the solution of the system: (6a) E(1)n(g)O~ns+T_tJ'dE(2)VN_q(E(21t)(ns)~ El2)t=E(g)O~ns

It is now possible to calculate the probability of excitation by an arbitrary operator T from the ground state I~0> to a state N'E,u>: (13) I<¢E,ulTI¢0>12= {la+~12/(1+s2)I.{l~tV*Et <~2EtlTI¢0>I2/ITtlVEtl2)} E is the energy of the state and u the set of other quantum numbers charactedzing a possible degeneracy. (14) ~=-l/tan(A) (15) a=(¢lTl~0)/('rl:.~'tV*Et(~2EtlTI4,0)) ¢ is a dimensionless energy parameter1, and a is the fraction of the excitation probability to the modified discrete state J~'> and the continuum state [#2Et >, respectively, with (16) I~>=N,1 >+~tJ'dE(2)(VEC2)t.I#2E(2)t>/(E-E(2))).

(6b) E(2)(.g)~SE(2)t+T's Vg,(EC2)t)(ns) O(ns =E(g),6E(2)t (6b) is satisfied by the ansatz :

The parameter a determines the amount of asymmetry in line shape.

(7) $ E(2)t=T'sO¢nsVg,(E(=)t)(ns)(P{I/(E(Q)-E(2)(.g)))+z(E(g,))$(E(gJ-E(2)(g)))

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THE FANO EFFECT

Apart from these solutions, one obtains also certain linear combinations of unperturbed continuum states accounting for the cases o~=0. To determine the operator T for the case of excitation by light, we performed time-dependent perturbation theory with the operator

IN E X C I T O N I C

983

SPECTRA

1o

'

' 1 ' ' ' 1 '

I

8

%

6

x

(17) Hp(t)=e/me.~jA(r.j,t)- ~

4

where e,m e electron charge and mass, ~j,IZj spacepoint and momentum of the valence electron j, A(Lt) vector potential of the perturbing electromagnetic field. With 2~[H,£], this leads to the identification of T with the operator ~j£j. Therefore, the transition matrix elements from the ground state 14'0> to the unperturbed 2-band exciton states I~i >, appearing in (13), are determined by dipole moments between the bands:

2 0

i

i

~(i)m,t(f.)

is the hydrogenic envelope function,

i

i

I

I

I

I

2.588

I

I

I

I

2.59

2.592

z [eV]

2.594

Fig.2a: Imaginary part of the dielectric constant for an n=l exciton peak, small perturbation

'

(18) g<+im,tlZ~14,0>~s.g.,0{4~(i)m,t*(o)-g3i*Tn4~(0m,t"(.~n)-;z3'

i

2.586

I

'

'

'

I

'

'

'

I

'

'

'

I

'

n} -1

with

quantum numbers m,t, for an exciton state including the valence band (i). Since it disappears at the origin for certain quantum numbers, also the second term is taken into account, where ~ is the next neighbour distance, g.3i is defined by analogy with (3), and (19) p.3in=Id3[ a3*~-.r.n)-r.ai(E) The transilion matdx elements (18) allow to calculate, via (13), optical properties as e.g. the imaginary part of the dielectric constant z2(¢0).

r.=l v =:

-2: n=t -3

,

I, 2.586

,

J

I , , ,

I,,,

2.588

I ,

2.59

i

,

z (ev]

, 2.594

2.592

Fig.2b: Energy-dependent Fano factor for an n=l exciton peak 5.Numericel Results end Discussion In the simplest case, i.e. a single line n=l lying in a continuum, the result consists of two parts: A peak of the form (13) arising from transitions to the discrete level, which is shown in Fig.2a, and the usual b a c k g r o u n d 4 arising from transitions to the unperturbed continuum as indicated above. For this figure, the resonance enregy is assumed to be ER=2.5gl eV and the gap EG=2.583 eV, the latter being the value for CdS, from which also all other parameters are taken to represent typical values. It shows an asymmetry characteristic of the Fano effect. The dipole moments, which will usually be treated as fit parameters, are chosen as: e.1231 =2.10 "10m as a typical 1-center-integral e.R32n=2-10 -11 m as a typical 2-center-integral, and e.i;Z12 =2.10"13m much smaller. 9. is the unit polarization vector of the vector potential A. Unlike the usual simple Fano fit 5, our Fano parameter a is not constant but energy dependent, as can be seen in Fig.2b. The clearly shown antirasonance in Fig.2a will be difficult to detect, because the absolute variation in ~2 is small, and the background superimposed by the transitions to the continuum is comparatively strong: For 9.-E32=6"10 -1 l m it has the order of magnitude of 10 "2. However, R 12 can become much larger, looking for example at the interaction between a core exciton level and a valence band continuum. With increasing .g, the slope at the right hand side of the peak becomes flattened; the result is an increase of ~2 on the high

.005

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,

I 2.59

+ 2.592

I , 2.594-

,

,

I 2.596

,

~

,

I , 2.598

E [+v]

Fig.2c: Imaginary part of the dielectric constant for an n=l exciton peak, large perturbation energy side of the peak. In addition, tne peak broadens and shifts to higher energies up to several meV. (The line shape of ~2 would then look like Fig.2c. Note the changes in scale.) This can be expected to be seen, also because the background, being of a simple form4, can be easily subtracted. In the case n=2, (13) gives also contributions where transitions to the continuum dominate, due to the degeneracy. These contributions should be considered as a part of the background, which hence becomes too complicated to be subtracted from the peak, since the involved dipole moments are not known.

984

V o l . 6 8 , No. ii

THE FANO EFFECT IN EXCITONIC SPECTRA 10

~0

'o

s

2.586

;2.588

2.59 E [,v]

2.592:

2.594

Fig.3: Imaginary part of the dielectric constant for an n=2 exciton peak The order of magnitude of the background is, however, entirely determined by the parameter e.lg.°2. By changing the polarization 9. of the incident light, one has effectively picked new bands 1 and 2 out of the p-type valence bands of our model. It can be assumed that their dipole moments will be different. Therefore, by changing 9., one can manipulate 9..g.32. In this way, one can make the contribution of the continuum very small. The result, i.e. the full dielectric constant including the background, is shown in Fig.3. Again the parameters of CdS are used, with its true exciton energy ER=2.5913 eV for the n=2 peak. Here the antiresonance is situated on the low-energy side of the peak. The dipole moments used are:

9..~31 =6.10 -11 m e-,g32 =6-10 "13m e.~31n=e.~32n=1.10-13m e.~12 =1-10 "13m 6.Conclusions The interaction of excitons can produce an asymmetry in the line shape of the dielectric constant. It can be detected in slightly anisotropic band structures or for core excitons, where at the same time a shift of the exciton energy should Occur.

7.References /1/ U.Fano: Phys.Rev. 124 (6), 1866 (1961) 121 I.Egri: Excitons and Plasmons Phys.Reports 119 (6), 363-402 (1985)

/31 L. Reining: Diplomarbeit RWTH Aachen, 1985 14/ R.Elliott: Phys.Rev. 108 (6), 1384 (1957) 151 U.Plekhanov: Opt.Spectrosc. 32 (1)(1972)