fulltext

Optical properties of excitons in quantum dots

F. BASSANI(1), R. BUCZKO(2) and G. CZAJKOWSKI(3)

(1_{) Scuola Normale Superiore - Pisa, Italy}

(2_{) Institute of Physics, Polish Academy of Sciences - Warsaw, Poland} (3_{) University of Technology and Agriculture - Bydgoszcz, Poland} (ricevuto il 15 Luglio 1997; approvato il 4 Agosto 1997)

Summary. — We show how to compute the optical spectra of quantum dots (QDs) from the analysis of the scattering of an electromagnetic plane wave. The method used combines the microscopic calculation of quantum dot exciton eigenfunctions and the macroscopic Stahl’s density matrix approach for the optical functions. The eigenfunctions and eigenenergies of excitons are computed in the infinite barrier model for dots of spherical shape using the effective-mass approximation. Numerical results are given and discussed for the case of GaAs/Ga_{1 2x}AlxAs QD.

PACS 73.20.Dx – Electron states in low-dimensional structures (superlattices, quantum well structures and multilayers).

PACS 78.55.Cr – III-V semiconductors.

1. – Introduction

Recently zero-dimensional structures which are ususally called quantum dots (QD) have been extensively investigated. Particular interest has been focused on their linear and nonlinear optical properties, as explained in ref. [1-3]. Optical properties of quantum dots, similarly to those of the other low-dimensional structures, are dominated by excitons, because the confinement of quasi-particles in a dot leads to enhancement of the oscillator strength of electron-hole excitations.

In what follows we discuss optical properties of quantum dots combining a microscopic derivation of quantum dot eigenstates with Stahl’ density matrix approach which relates them to the macroscopic electromagnetic field of the medium [4-5]. This method has been quite succsessful in explaining the optical excitation spectra of semiconductor bulk crystals, semiconductor superlattices and some low-dimensional structures [6-10]. The advantage with respect to the quantum-mechanical calculation is that all coherence effects are automatically included.

In sect. 2 we calculate the one-particle eigenfunctions of electrons and holes in a quantum dot and from them the transition energies and the optical transition probabilities taking into account the electron-hole interaction.

In sect. 3 we adopt the Stahl density matrix approach for calculating the amplitude of an electromagnetic wave scattered by a quantum dot, and thus the optical response.

The results are discussed in sect. 4, in reference to GaAs/Ga_{1 2x}AlxAs quantum

dots.

2. – Electron-hole pair eigenstates in a spherical quantum dot

We consider a spherical quantum dot of radius R. In first approximation we use a simple model of quantum one-particle eigenstates, in which the conduction and the valence band are discussed separately. The confining potentials for the electron and the hole are assumed infinite outside the quantum dot, i.e. hard walls QD are considered. We also neglect interaction between light and heavy holes. The present oversimplified approach will require improvements to take into account the above effects, but it has the advantage of being mathematically very simple. This makes it possible to describe the essential effects due to confinement and electron-hole attraction on the optical properties with a small amount of computer time and computer capacity.

In previous works several methods were used for the study of electronic states, such as numerical matrix diagonalization [2, 11], and variational methods [12-18]. Below we calculate the electronic excitation energies starting from the single-particle eigenfunctions and performing the numerical matrix diagonalization, within the effective-mass approximation. The method has been initiated by Hu et al. [11], and used by others, as reviewed in ref. [2]. The single-particle eigenfunctions for infinite barriers can be written as Bloch functions times envelope functions, which for electrons are [11, 19]

Ce

i(r) 4Aiejli(kir) Ylimi(u , f) ,

(1)

and for holes

Chj(r) 4Ajhjlj(kjr) Yljmj(u , f) ,

(2)

where the indices (i , j) label the single-electron and -hole states, respectively, jl(kr)

are the spherical Bessel functions, Ae_{, A}h_{are the normalization factors:}

Ae i4

o

2 R3j 21 li1 1(kiR) , (3)

with a similar expression for Ah

j, and the values of kj are determined by the boundary

conditions

Ce

i(R) 40 , Chj(R) 40 .

(4)

Both particles interact by a screened Coulomb potential, which gives rise to bound excitonic-like states within the dot. The excitonic eigenstates and eigenenergies can then be calculated from the following Schrödinger equation in the effective-mass approximation:

(Hs1 V) Cn4 EnCn,

with

Cn4 Cn(re, rh) c *v(k0, rh) cc(k0, re) ,

(6)

and Hs is the single-particles effective mass Hamiltonian, which on the envelope

function takes the form

Hs4 Eg2 m me ˜2e2 m mh ˜2h1 V(re) 1V(rh) , (7)

where V(r) is the confining potential

V(r) 4./ ´ 0 , Q , for r GR , for r DR ; (8)

V is the dielectrically screened Coulomb potential: V(re, rh) 42

2 Nre2 rhN

, (9)

m is the reduced mass:

1 m 4 1 me 1 1 mh , (10)

and we have used the effective Rydberg energy and the effective Bohr radius as energy and length units. The excitonic eigenstates can be obtained from the Schrödinger equation (5) by the method of numerical matrix diagonalization. To this end we construct the excitonic envelope wave function as a linear combination of products:

Cn(re, rh) 4Cie(re) Chj(rh) ,

(11)

where n 4 (i, j). The new basis is more appropriate for calculations since it takes into account that the total angular momentum of an electron-hole pair is a good quantum number because of the spherical symmetry of the Hamiltonian (7) [11, 13]. If the angular momenta of the electron, the hole, and the pair are denoted by Le, Lh, and L 4Le1 Lh, respectively, this means that

[ (Hs1 V), L] 4 0 .

(12)

Exploiting the symmetry properties, we obtain the following expression for the basis of the envelope exciton functions:

CLM

n (re, rh) 4AieAjh jli(kire) jlj(kjrh) Y

lilj

LM(Ve, Vh) ,

(13)

where n is the index which denotes all possible choices i and j, and

Ylilj LM(Ve, Vh) 4 [Yli(Ve) Ylj(Vh) ]LM4

!

m1, m2 CLM limiljmjYlim1(Ve) Yljmj(Vh) , (14)

where (Ve, Vh) denote the angular variables, ClLMimiljmj are the Clebsch-Gordon

coefficients, and Ylilj

the expansion of the Coulomb potential in terms of spherical harmonics [20], V 42 2 Nre2 rhN 4

.

`

/

`

´

28 p rh l 40

!

Q ₁ 2 l 11

g

re rh

h

l

(

Yl(Ve) Q Yl(Vh)

)

, 28 p re l 40

!

Q ₁ 2 l 11

g

rh re

h

l

(

Yl(Ve) Q Yl(Vh)

)

, reE rh, reF rh, (15)

we arrive at a secular equation for the excitonic eigenenergies in terms of the one-particle eigenstates and of the matrix elements

amNVNnb 4C

!

l 40 Q _{4 p} 2 l 11



_dVedVhYLM * l1l2 (Ve, Vh) Y LM l3l4(Ve, Vh) 3 (16) 3

(

Yl(Ve) Q Yl(Vh) Il1l2l3l4, with C 422AieAjhAi 8eAj 8h, (17) and Il1l2l3l44



0 R



0 rh

g

re rh

h

l r2 erhjl1(k1re) jl2(k2rh) jl3(k3re) jl4(k4rh) dredrh1 (18) 1



0 R



rh R

g

rh re

h

l rh2rejl1(k1re) jl2(k2rh) jl3(k3re) jl4(k4rh) dredrh.

Making use of the symmetry properties of the spherical harmonics the angular integration in coefficients amNVNnb can be performed to give

amNVNnb 4C

!

l 4max (Nl12 l3N , Nl22 l4N) min (l11 l3, l21 l4)

k

( 2 l11 1 )( 2 l21 1 )( 2 l31 1 )( 2 l41 1 ) 3 (19) 3./ ´ l1 l4 l3 l2 l L ˆ ¨ ˜ Q

u

l3 0 l 0 l1 0

v

Q

u

l4 0 l 0 l2 0

v

Il1l2l3l4. Given dot radius and material parameters the integrals Il1l2l3l4 are evaluated

numerically, and the Hamiltonian matrix can be diagonalized.

The procedure we have here adopted for the exciton-like states does not differ substantially from those recently adopted by other authors, and numerically gives results in agreement with those reported in the literature for the lowest states. Here we have chosen a larger basis set than other authors and have computed the eigenvalues and eigenfunctions of a large number of states as a function of the hole effective mass.

The results for energies are given in fig. 1 for the case of R 4a *B and refer to the

states with L 40; the notation “1, 0” means “the first pair state where both single-particle functions are of the type l 40”, etc. We span the whole interval of mass ratio mh/mebetween 1 and 10 for various excitonic states: previously only the values of

Fig. 1. – Excitonic energies as functions of the hole and electron effective mass ratios for the case of R 4a *B. The case of light hole gives the same figure with the appropriate definition of Bohr

radius and effective Rydberg.

the ground-state energy for a few values of this ratio were reported [11, 13, 14]. Here we have used 40 single-particle functions for electrons and holes, combined into 164 pair state functions. We should note, furthermore, that the relations are defined in terms of effective Rydbergs and effective Bohr radii: when comparing, for example, heavy- and light-hole energies, one should have in mind the changes of both these quantities.

We observe from the results of fig. 1 that in the case of those states which will be relevant for the optical transitions like (1, 0), (1, 1), etc., the energy values are independent of the mass ratio. Only for those states in fact the oscillator strength will be shown to have an appreciable value.

The method is appropriate for small dots (R G 3 excitonic Bohr radii). In the limit of large dots the number of basis functions needed increases enormously and different approaches are required.

3. – Calculation of coherent electron-hole amplitudes

We now wish to adopt the Stahl real density matrix approach to describe the situation in which a quantum dot scatters an external plane electromagnetic wave [4]-[10]. The eigenfunctions derived in the previous section will be used to solve the Stahl constitutive equations for the coherent amplitudes Ycv_{of the electron and the}

hole of coordinates re and rh, respectively; cv label the allowed transitions. The

coherent amplitudes Ycv_{satisfy the equations:}

¯tYcv1 i ˇ HcvY cv 4 i ˇ M cv E 2GcvYcv_, (20)

where E denotes the electric field, and Mcv are couplig functions defined as the

interband transition dipole moments. The coherent amplitudes determine the polarization within the quantum dot:

P(r) 42



d3_(r

e2 rh) Mcv(re2 rh) Ycv(re, rh) ,

(21)

where the center-of-mass r is defined as

r 4 mere1 mhrh

me1 mh

. (22)

To derive the optical properties of a matrix containing a QD, we need the simultaneous solution of the constitutive equations (20) and of Maxwell’s equations. This corresponds to the famous physical problem of diffraction of a plane wave on a dielectric sphere, which leads to the so-called Mie resonances [21]. Note also that our constitutive equations refer to a 6-dimensional configuration space (re, rh), in which the no escape

boundary conditions (4) hold for electrons and holes, whereas the definition of P(r) in Maxwell equations refers to the excitonic center-of-mass coordinate within the QD. This makes our problem extremely complicated and requires simplifying assumptions. To have an insight into the optical properties of quantum dots, we start with a very simple model, which seems to be appropriate in the limit of small dots (dot radius not exceeding a few excitonic Bohr radii). This consists first in the neglect of coherence effects related to a spatial extension of the dipole density so that

Mcv_(r

e2 rh) 4M0 cvd(re2 rh) ,

(23)

where the dipole matrix elements M0 cv are treated as known numbers. The second

assumption is that a plane electromagnetic wave passing through a medium where the dots are homogeneously distributed behaves like a plane wave with an amplitude E0in

a medium characterized by the homogeneous susceptibility:

xcv4 2 M0 cv E0



dot d3_rYcv_{(r , r) .} (24)

Assuming finally a harmonic time dependence of the field and of the coherent amplitude, we reduce the constitutive equation (20) for a quantum dot to the form

(H 2ˇv2iG) Ycv_(r

e, rh) 4M0 cvd(re2 rh) E0,

(25)

where H4Hs1V, V is the dielectrically screened Coulomb potential and Hsthe

single-particles Hamiltonian in the effective-mass approximation. The coherent amplitudes

Ycv_{are appropriate linear combinations of the envelope eigenfunctions}

Ycv 4

!

JM YJM_(r e, rh) . (26)

Inserting this expansion into the constitutive equation (25), we obtain the following set of equations for the functions YJM_:

(H 2ˇv2iG) YJM_(r

e, rh) 4E0M0 cvdJ0dM0,

(27)

which means that only components with J 40 and M40

Y00(re, rh) 4

!

ncnC

00

n(re, rh)

(28)

contribute to the optical transitions. The coefficients cnare determined by the following

set of algebraic equations:

!

_n[Hmn2 (ˇv 1 iG) dmn] cn4 E0M0 cvm,

(29)

where

Hmn4 Em1 amNVNnb ,

(30)

Fig. 2. – Imaginary part of the susceptibility of QD with R 40.5a *B. Calculations are made for the

GaAs heavy hole data (m_h/m_eC 5), and for G 4 0.5Ry *. The one-electron transition energies are indicated by inset lines following the sequence of J 40 values; EJ 404 Eg1 Ee(n , l) 1Eh(n , l). It

as given in (19), and M0 cvm4 M0 cvAieAjh



0 R jl1(k1r) jl2(k2r) r 2 dr



Yl1l2 LM(V , V) dV 4 (31) 4 M0 cvk2 l11 1 dL0dM0dl1l2dk1k2(21) l1_.

From the above equations (29) we compute the coefficients cn and the contribution to

the susceptibility of our medium due to the QDs:

xcv(v) 42



QD Y00_{(r , r) d}3 r 4 2 E0

!

_n cnM0 cvn, (32)

which is the only relevant susceptibility in the frequency region close to resonances.

4. – Results and discussion

We have performed the calculations in the frequency region where heavy-hole excitons are relevant, having in mind the case of GaAs/Ga_{1 2x}AlxAs QDs. Using the

scheme of sect. 3, we calculated the susceptibility as a function of the dot radius, and pay particular attention to its imaginary part proportional to the absorption coefficient.

Fig. 3. – The same as fig. 1, for R 41a *B. Also in this case the first peak with l 45 coincides with

Fig. 4. – The same as fig. 1, for R 42a *B.

We use 19 single-particle functions for both valence and conduction band which produce ca. 49 combinations with L40, relevant for the optical transitions. This number of basis functions is demonstrated to be sufficient for obtaining convergence for R G3a *B. The

results for the imaginary part of the susceptibility are reported in fig. 2, 3, and 4 for the cases of R 40.5a *B, R 4a *B, and R 42a *B, respectively. We only consider the case of

heavy holes because the lowest optical transitions refer to them due to the prevalence of the confining energy over the Coulomb attraction for a small QD. The insets show the band-to-band energies where the notation “1, 0” means “the first pair state with

J40, and equal indices n41, l40 for the single-particle eigenfunctions”, etc. For the

states with J 40, l40 we observe the shift of resonance peaks with respect to the one electron transition energies. This gives the excitonic binding energies for all the states considered. The values obtained for the ground state agree well with those computed previously [11, 13, 16] and is in a fair agreement with that observed in experi-ment [22-24]. We notice from the above figures, that the excitonic binding energies of some resonances (those with l c 0) vanish. This is probably due to a strong reduction of the effect of the Coulomb attraction because of central nodes in the wave function with l c 0.

In summary, we have constructed a simple model for calculation of the optical properties of a medium with distributed QDs. In this model we treat the medium as a crystal with an effective susceptibility which is determined by electronic transitions in the QD. Numerical examples appropriate to GaAs/Ga_{1 2x}AlxAs QDs for various dot

Improvements on the above-described model, which consider finite confinement barriers and heavy-light hole mixing are in progress. Preliminary results on the hole one-electron wave functions are reported in ref. [25]. When increasing the dot radius, one also should take into account the structure of the electromagnetic wave within the dot (no more a plane wave) and a realistic shape of the transition dipole density

Mcv_(r

e2 rh) which will indicate the role of the coherence of the carriers with the

electromagnetic field. This will be done in a more detailed future work.

* * *

This paper is based on work supported by INFM (Istituto Nazionale di Fisica

della Materia). The authors wish to acknowledge illuminating conversations with G.

BASTARD. Two of them (RB and GC) wish to thank the Scuola Normale Superiore for

the hospitality.

R E F E R E N C E S

[1] WEISBUCH C. and VINTER B., Quantum Semiconductor Structures: Fundamentals and Applications (Academic Press, New York) 1991.

[2] BANYAI L. and KOCH S. W., Semiconductor Quantum Dots (World Scientific, Singapore) 1993.

[3] WOGGONU. and GAPONENKOS. V., Phys. Status Solidi B, 159 (1995) 285.

[4] STAHL A. and BALSLEV I., Electrodynamics of the Semiconductor Band Edge (Springer-Verlag, Berlin, Heidelberg, New York) 1987.

[5] VICTORK., AXTV. M., and STAHLA., Phys. Rev. B, 51 (1995) 14164. [6] CZAJKOWSKIG. and TREDICUCCIA., Nuovo Cimento D, 14 (1992) 1203.

[7] BASSANIF., CHENY., CZAJKOWSKIG., and TREDICUCCIA., Phys. Status Solidi B, 180 (1993) 115.

[8] BASSANIF., CZAJKOWSKIG., and TREDICUCCIA., Z. Phys. B, 98 (1995) 39. [9] CZAJKOWSKIG., BASSANIF., and TREDICUCCIA., Phys. Rev. B, 54 (1996) 2035. [10] CZAJKOWSKIG., DRESSLERM., and BASSANIF., Phys. Rev. B, 55 (1997) 5243. [11] HUY. Z., LINDBERGM., and KOCHS. W., Phys. Rev. B, 42 (1990) 1713. [12] KAYANUMAY., Solid State Commun., 59 (1986) 405.

[13] TAKAGAHARAT., Phys. Rev. B, 47 (1993) 4569.

[14] NAIRS. V. and TAKAGAHARAT., Phys. Rev. B, 55 (1997) 5153.

[15] LAHELDU. E. H., PEDERSENF. B. and HEMMERP. C., Phys. Rev. B, 48 (1993) 4659. [16] LELONGPH. and BASTARDG., Solid State Commun., 98 (1996) 819.

[17] LAHELDU. E. H. and EINEVOLLG. T., Phys. Rev. B, 55 (1997) 5184. [18] GRU¨NBERGH. H.VON, Phys. Rev. B, 55 (1997) 2293.

[19] BUCZKOR. and BASSANIF., Phys. Rev. B, 54 (1996) 2667.

[20] VARSHALOVICHD. A., MOSKALEVA. N., and KHERSONSKIIV. M., Quantum Theory of Angular Momentum (World Scientific, Singapore) 1988.

[21] BORNM. and WOLFE., Principles of Optics, 6th edition (Pergamon, Oxford) 1986, Chapt. XIII.

[22] NORRISD. J., SACRAA., MURRAYC. B. and BAWENDIG., Phys. Rev. Lett., 72 (1994) 2612. [23] MARZINJ.-Y., GE´RARDJ. M., IZRAE¨LA., BARRIERD. and BASTARD G., Phys. Rev. Lett., 73

(1994) 716.

[24] VERSCHURENC. A., BESTWICKT. D., DAWSONM. D., KEANA. H. and DUGGANG., Phys. Rev. B, 52 (1995) R8640.