Electronic structure and optical properties of PbS ... - OSA Publishing

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J. Opt. Soc. Am. B / Vol. 14, No. 7 / July 1997

I. Kang and F. W. Wise

Electronic structure and optical properties of PbS and PbSe quantum dots Inuk Kang and Frank W. Wise Department of Applied Physics, Cornell University, Ithaca, New York 14853 Received September 30, 1996; revised manuscript received February 12, 1997 The electronic structure of spherical PbS and PbSe quantum dots is calculated with a four-band envelopefunction formalism. This calculation accounts for both exciton energies and wave functions with the correct symmetry of the materials. The selection rules and the strength of the dipole transitions of lead-salt quantum dots are derived accounting for the symmetry of the band-edge Bloch functions of the lead salts. The calculated energies of the optically allowed exciton states are found to be in good agreement with experimental data. The effects of many-body perturbations, such as Coulomb interactions and intervalley scattering, are also discussed. © 1997 Optical Society of America [S0740-3224(97)03207-4]

1. INTRODUCTION Nanometer-sized semiconductor quantum dots (QD’s) have been studied extensively as candidate materials for photonics applications as well as a means to probe the transition between bulk and molecule. Compared with well-known II–VI (CdS or CdSe) QD’s, QD’s of the lead salts (PbS, PbSe, and PbTe) are better suited for the study of the strong-confinement limit. For example, the electron radius a e 5 e \ 2 /m e e 2 and hole radius a h 5 e \ 2 /m h e 2 of PbS are both ;10 nm. These are much larger than a e ; 3 nm and a h ; 1 nm of CdSe. Thus the regime of strong confinement of both electron and hole can be more easily accessed in these materials. Nonlinear optical properties of semiconductor QD’s are expected to be greatly enhanced in the strong-confinement regime. Combined with their sparse electronic spectra, this makes lead-salt QD’s interesting for potential applications in nonlinear optics. Finally, the highly dispersive optical phonons of the lead salts make the system ideal for the study of phonon confinement. PbS QD’s have been fabricated in polymer hosts,1–3 porous TiO2 electrodes,4 and glass hosts.5 Large blue shifts in the absorption spectra have been observed in all existing samples,1–5 and a number of monodisperse samples show discrete absorption peaks characteristic of quantum confinement of carriers in QD’s of narrow size distribution.2,3,5 The effects of phonon confinement have been observed in Raman and far-infrared measurements6 and also in the femtosecond dynamics of photoexcited excitons.7 Quite recently, quantum size effects have been observed in chemical-solution-deposited films of nanocrystalline PbSe.8 Knowledge of the electronic states is needed for the study of almost any property of nanocrystals. The electronic structure of PbS QD’s has been studied theoretically using the parabolic effective-mass model,1,9 hyperbolic-band approximation,1 and tight-binding type calculations.1,10 The parabolic effective-mass model is clearly inadequate for lead-salt QD’s because it neglects the nonparabolic band structure of the materials.1,9 The 0740-3224/97/0701632-15$10.00

hyperbolic-band model introduced by Wang et al.1 is an improvement over the parabolic effective-mass model in that it phenomenologically includes the nonparabolicity of the band structure; the model accurately predicts the energies of the lowest exciton states. However, it is incapable of calculating energies of the higher excited states because it models only the lowest interband transition as an electron transfer between lead and sulfur ions. Furthermore, the envelope functions obtained by this approach are simply those of the parabolic effective-mass model and thus lack the correct symmetries. It also neglects the properties of the band-edge Bloch functions that are essential for the calculation of optical transitionmatrix elements. As a consequence, the hyperbolic-band approximation is of limited use for the study of the optical properties of lead-salt QD’s. The tight-binding calculations described in Refs. 1 and 10 provide accurate values of the energy levels of PbS QD’s and useful information about the character of the wave functions. Unfortunately, results are presented only for QD’s with radii smaller than ;2 nm, very small compared with the ;18-nm Bohr radius of the exciton in PbS. The strongconfinement limit of nanocrystals and clusters is investigated in Ref. 10, but the transition from bulk to quantumconfined properties is not addressed. Finally, no attempt is made to calculate optical transition strengths; the absorption spectrum is assumed to be proportional to the electronic density of states.10 It is clearly desirable to have more accurate and detailed information about the electronic structure of leadsalt QD’s. For example, investigation of the linear and the nonlinear optical properties of lead-salt QD’s depends on accurate information on both the energies and the wave functions of excited states as well as the lowest exciton state. Also, the coupling of excitons to optical phonons in QD’s is known to be very sensitive to the details of the electronic wave functions.11 An envelopefunction calculation based on the correct bulk band structure of II–VI semiconductors has explained fine structure in the optical properties of QD’s made of these materials that could not be explained otherwise.12 © 1997 Optical Society of America


Vol. 14, No. 7 / July 1997 / J. Opt. Soc. Am. B

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Here we present a four-band envelope-function calculation of the electronic structure of PbS and PbSe QD’s. The calculation is based on a realistic model of the nonparabolic and the anisotropic band structures of these materials. This is the first treatment, to our knowledge, of the electronic structure of lead-salt QD’s to provide ac-

This model includes the coupling between the highest valence band and the conduction bands and also the coupling between the lowest conduction band and the valence bands in a second-order perturbation approximation, and it treats the resulting four-band problem (including spin) exactly. The spin-orbit interaction is also taken into ac-

curate energies of higher exciton states as well as the lowest-exciton, electron and hole wave functions with the correct symmetries, as well as the interband dipole transition strengths and selection rules. The paper is organized as follows: In Section 2 the four-band envelopefunction formalism is applied to spherical PbS and PbSe QD’s to calculate the electron and hole states of the QD’s. Direct interband transitions between the electron and hole states obtained in Section 2 are discussed in Section 3. Calculated exciton energies and transition strengths are compared with existing experimental data and other theories. In Section 4, perturbations caused by the multivalley band structure, Coulomb interaction, and exchange interaction are discussed.

count in this model. The corresponding Hamiltonian is where E g is the bulk band gap; m is the free-electron mass; z is the coordinate along the ^111& direction; k t 2 5 k x 2 1 k y 2 ; P t and P l represent the transverse and the longitudinal momentum-matrix elements taken between the extremal valence- and conduction-band states; and 6 m6 t and m l are the far-band contributions to the transverse and the longitudinal band-edge effective masses, respectively. All of the parameters of the Hamiltonian can be obtained from experiments17,18 and are listed in Table 1. This Hamiltonian can be diagonalized to yield the following dispersion relation:

F

Eg \ 2k t2 \ 2k z2 1 2 E ~ k! 2 1 2 2m t 2m 2 l

F

2. FOUR-BAND ENVELOPE-FUNCTION FORMALISM The lead salts (PbS, PbSe, and PbTe) have the rock-salt crystal structure and have direct band gaps at four equivalent L points in the Brillouin zone. The bottom of the conduction band has L 62 symmetry in the doublegroup notation, and the top of the valence band has L 61 symmetry.13 Spatially, the valence band-edge Bloch function is s-like, while the conduction band-edge Bloch function is p z -like, where z denotes the ^111& direction of the cubic lattice. The symmetry of the band-edge states is known to underlie the unusual temperature dependence of the band gap of the lead salts and is also responsible for the cubic to rhombohedral phase transitions in these materials.13,14 In an envelope-function calculation of quantumconfined structures, confinement is imposed on wave functions of the electrons and the holes of the bulk material. Thus it is essential to base the calculation on an accurate bulk k • p Hamiltonian. The most accurate model used to describe the band structure near the L point is the k • p model of Mitchell and Wallis15 and Dimmock.16

G

\ 2k t2 Eg \ 2k z2 2 2 E ~ k! 1 2 2 2m t 2m 1 l

3 2

5

G

\2

~ P t 2 k t 2 1 P l 2 k z 2 ! . (2) m2 Table 1. Parameters of the k – p Hamiltonians [Eqs. (1) and (3)] of PbS and PbSea Parameters E g (T 5 300 K) (eV) m/m 2 t m/m 2 l m/m 2 m/m 1 t m/m 1 l m/m 1 2P t 2 /m(eV) 2P l 2 /m(eV) 2P 2 /m(eV) a b c

PbSb

PbSec

0.41 1.9 3.7 2.5 2.7 3.7 3.0 3.0 1.6 2.5

0.28 4.3 3.1 3.9 8.7 3.3 6.9 3.0 1.7 2.6

The definitions of the parameters are given in the text. The longitudinal and transverse parameters are from Ref. 17. The longitudinal and transverse parameters are from Ref. 18.

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The dispersion relation is highly nonparabolic owing to the interaction between the extremal conduction and valence bands, and it is anisotropic owing to the difference between the transverse and longitudinal parameters. The degree of anisotropy is greater for PbSe than PbS, as can be seen in Table 1. The band structure (T 5 0 K) of PbS near the L point, calculated by the empirical pseudopotential method,19 is shown with the band structure according to Eq. (2) in Fig. 1. The calculation with the parabolic-band approximation is also shown with the ef* 5 0.09m. 17 The accuracy of an fective masses m * e 5 mh envelope-function calculation depends on how well the true band structure is approximated, especially away from the band edges. From Fig. 1 it is evident that the four-band model [Eq. (2)] is a much better representation of the real band structure of PbS than the parabolic-band model. Similar conclusions can be reached for PbSe. As in II–VI QD’s,12 a spherical approximation of the Hamiltonian in Eq. (1) is necessary for an analytical calculation of the spectrum of spherical lead-salt QD’s. The degree to which the band structures of PbS and PbSe deviate from spherical symmetry will be shown to be small enough to be treated as a perturbation; the anisotropy of the band structure of PbTe is too extreme to be adequately treated by a perturbative method. The Hamilˆ can then be separated into a spherically symtonian H ˆ and the anisotropic perturbation V ˆ , where metric part H 0

ˆ ~ k! 5 H 0

F

S

D

Eg \ k 1 1 2 2m 2 2 2

\P k•s m

2

S

\P k•s m

D

Eg \ 2k 2 1 1 2 2m 1

G

spin matrix. fined as

The parameters, m 2, m 1, and P are de-

3P 2 5 2P t 2 1 P l 2 ,

6 3/m 6 5 2/m 6 t 1 1/m l

(4)

(see Table 1 for their values). They are chosen such that the ellipsoidal constant-energy surface is best approximated by a spherical one, and thus the effect of the perturbation is minimized. Details of the band anisotropy are discussed later in this section and in Appendix A. In the spherical approximation the unperturbed energies and the wave functions of the quantum-confined levels (QCL’s) can be obtained by solving the envelopefunction equation with the boundary condition that the wave function vanish at the surface of the quantum dot: ˆ ~ 2i¹ ! F ~ r! 5 EF ~ r! , H 0

(5)

F ~ u ru 5 a ! 5 0.

(6)

In Eqs. (5) and (6), a is the radius of the quantum dot and F (r) is an eigenvector whose four components are the envelope functions of the total wave function u C(r) & : F ~ r! 5 @ F 1 ~ r! , F 2 ~ r! , F 3 ~ r! , F 4 ~ r!# ,

(7)

u C ~ r! & 5 F 1 ~ r! u L 62↑ & 1 F 2 ~ r! u L 62↓ & 1 F 3 ~ r! u L 61↑ &

1 F 4 ~ r! u L 61↓ & . .

(3)

ˆ is given in Appendix A. The anisotropy perturbation V In Eq. (3), 1 is the 2 3 2 unit matrix and s is the Pauli

The assumption of an infinite potential boundary, although simple, has been effective in explaining the intrinsic optical properties of II–VI QD’s.12 Note that the wave functions of both conduction and valence electrons are expanded with the same set of band-edge Bloch functions. This coupling of the conduction and the valence bandedge states is important for the optical properties of the narrow-band-gap materials. The solution of the eigenvalue problem can be simplified by taking advantage of the symmetry of the Hamilˆ . H ˆ has the same symmetry as the Dirac electonian H 0 0 ˆ commutes with tron in a central potential.20 Since H 0 ˆ , we seek sithe total angular momentum Jˆ and parity P multaneous eigenstates of these operators, where ˆ 1 \ S ˆ, Jˆ 5 L 2

ˆ 5 P

Fig. 1. Band structure of bulk PbS (T 5 0 K) near the L point along the ^111& direction, calculated by the empirical pseudopotential method (dashed curve), the four-band model (solid curve), and the parabolic effective-mass model (dotted curve). The lattice constant a is 5.93 Å. The bulk band gap of PbS at T 5 0 K is 0.28 eV.

(8)

F

ˆ 5 S

ˆ ^1 2P

0

0

ˆ ^1 P

F G

G

s

0

0

s

.

,

(9)

(10)

ˆ is the usual spatial inversion operator. In Eq. (10), P The parity operator is constructed to account for the opposite parities of the conduction and the valence bandedge Bloch functions, and the resulting parity of F (r) coincides with that of the total wave function u C(r) & . The eigenvectors are labeled by angular-momentum quantum numbers j and m and by the parity p. For given j and



m there are two orthogonal eigenvectors of opposite parity because there are two distinct ways to construct an eigenfunction of total angular momentum from eigenfunctions of orbital angular momentum and spin. These are20

F p , j,m ~ r! 5

3 3

f l11 ~ r !

D D D D

4

l 1 m 1 1/2 m21/2 Yl 2l 1 1 l 2 m 1 1/2 m11/2 Yl 2l 1 1 l 2 m 1 3/2 m21/2 Y l11 2l 1 3 l 1 m 1 3/2 m11/2 Y l11 2 2l 1 3

if l ~ r !

p 5 ~ 21 ! l11 ,

j 5 l 1 1/2,

F p , j,m ~ r! 5

S S A SA A SA A A A A

l 2 m 1 3/2 m21/2 Y l11 2l 1 3 l 1 m 1 3/2 m11/2 Y l11 2 2l 1 3 l 1 m 1 1/2 m21/2 Yl 2l 1 1 l 2 m 1 1/2 m11/2 Yl 2l 1 1

ig l11 ~ r !

g l~ r !

p 5 ~ 21 ! .

j 5 ~ l 1 1 ! 2 1/2,

l

f l11 ~ r ! , g l11 ~ r ! 5 cj l11 ~ kr ! 1 di l11 ~ lr ! , ,

(11)

4

H

2

2E2

F

d2

m

S

d dr

2

l r

1

2m 2 dr 2

D

2

r dr

r2

F

G

GJ

f l~ r ! 2

H

Eg

1E2

2

\2

r

F

d2

2m 1 dr 2

~ l 1 1 !~ l 1 2 !

2

,

H

2

1E2

\2 2m

1

F

2 \P m

S

d dr

2

l r

D

2

1

\ 2k 2 2m

F

Eg 2

2

2

GJ

d2 dr

2

\P m

g l~ r ! 2

2

H

2 d

1

F

r dr

d dr

Eg 2

1

2

l~ l 1 1 ! r2

~l 1 2!

2E2

r \2

r dr (13)

r

F

d2

GJ

Eg

2

2

2

\ 2k 2 2m 1

2 E~ k !

2m

2 E~ l ! 2

GF

Eg

2

2

1

\ 2l 2 2m 1

G

\2 m2

2 E~ l !

P 2k 2,

(16)

G (17)

Note that the dispersion relation for E(k) is identical to that obtained in reciprocal space [Eq. (2)] if we make the 6 substitutions m 6 5 m 6 t 5 m l and P 5 P t 5 P l in Eq. (2). E(k) can be written explicitly for the conductionelectron (1) and valence-electron (2) levels as E 6~ k ! 5

1 @gk2 6 2

A~ E g 1 a k 2 ! 2 1 b 2 k 2 # ,

(18)

where the parameters a, b, and g are defined as

S S

D D

a5

\2 1 1 1 , 2 m2 m1

g5

\2 1 1 2 . 2 m2 m1

b5

2\P , m (19)

l 6~ k ! 5

A

2 a E g 1 b 2 1 ~ a 2 2 g 2 ! k 2 1 4 g E 6~ k !

a2 2 g 2

.

(20)

g l~ r !

Finally, by subjecting the radial functions [Eq. (15)] to the boundary condition [Eq. (6)], we get the conditions that k and l must satisfy for confinement in a QD of radius a. They are % 6~ k ! j l11 ~ ka ! i l ~ l 6a ! 2 m 6~ k ! j l ~ ka ! i l11 ~ l 6a ! 5 0,

g l11 ~ r ! 5 0,

2m 2 dr 2

~ l 1 1 !~ l 1 2 ! 2

G

GJ

GF

The requirement that E(k) 5 E(l) for a given state gives the relation between k and l,

2 d

f l11 ~ r ! 5 0.

2 E~ k !

\ 2l 2

f l~ r !

1

2

\2 5 2 2 P 2l 2. m

The functions g l (r) and g l11 (r) satisfy a similar set of equations: Eg

Eg

5

(12)

l~ l 1 1 !

F

\P d ~l 1 2! 1 f l11 ~ r ! 5 0, m dr r

2 \P

2 d

(15)

where a, b, c, d, k, and l are to be determined by Eqs. (13) and (14), wave-function normalization, and the boundary condition [Eq. (6)]. Insertion of solution (15) in Eqs. (13) and (14) yields the dispersion relations E(k) and E(l):

In the equations above, are the spherical harmonics, and f l (r) and g l (r) are the radial functions to be determined by substituting Eqs. (11) and (12) into the eigenvalue equation [Eq. (5)]. The substitution separates the variables and the resulting equations for the radial functions f l (r) and f l11 (r) are \2

The most general solution to these coupled second-order differential equations that is also nonsingular at the origin can be written as a linear combination of a spherical Bessel function j l (kr) and a modified spherical Bessel function i l (lr): f l ~ r ! , g l ~ r ! 5 aj l ~ kr ! 1 bi l ~ lr ! ,

Ym l

Eg

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1

j 5 l 1 1/2,

p 5 ~ 21 ! l11 ,

(21)

2 d

% 6~ k ! j l ~ ka ! i l11 ~ l 6a ! 1 m 6~ k ! j l11 ~ ka ! i l ~ l 6a ! 5 0,

r dr

j 5 ~ l 1 1 ! 2 1/2,

g l11 ~ r ! 5 0.

(14)

p 5 ~ 21 ! l .

(22)

The functions % 6(k) and m 6(k) are defined as % 6~ k ! 5 @ E g 1 ~ a 1 g ! k 2 2 2E 6~ k !# / b k,

(23)

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f l ~ r ! 5 j l ~ kr ! 2

j l ~ ka ! i ~ lr ! , i l ~ la ! l

F

f l11 ~ r ! 5 % ~ k ! j l11 ~ kr ! 2

g l11 ~ r ! 5 j l11 ~ kr ! 2

F

G

j l11 ~ ka ! i ~ lr ! , i l11 ~ la ! l11

(25)

j l11 ~ ka ! i ~ lr ! , i l11 ~ la ! l

g l ~ r ! 5 2% ~ k ! j l ~ kr ! 2

G

j l ~ ka ! i ~ lr ! . i l ~ la ! l

(26)

It can be seen from Eqs. (8), (11), and (12) that %(k) is a measure of the relative contributions of the two different orbital angular-momentum components to the total wave function of a QCL, and thus it determines the relative contributions of the components for the valence bandedge Bloch function, u L 61& , and those for the conduction band-edge Bloch function, u L 62& , to the total wave function. To summarize, Eqs. (18)–(24) determine the energies of the QCL’s of a spherical PbS or PbSe QD without the perturbation owing to the band anisotropy. Because of the

Fig. 2. Unperturbed energy levels of conduction and valence electrons of (a) PbS QD’s and (b) PbSe QD’s.

Fig. 3. Plot of %(k) for conduction electrons of PbS QD’s with j 5 1/2, p 5 1 (solid curve) and j 5 3/2, p 5 1 (dashed curve). The definition and explanation of %(k) are given in the text.

m 6~ k ! 5 @ E g 2 ~ a 1 g ! l 6~ k ! 2 2 2E 6~ k !# / b l 6~ k ! . (24) The radial components of the envelope functions [see Eqs. (11), (12), and (15)] have the following form neglecting an overall normalization constant:

Fig. 4. Shifts of the energies of the conduction electrons with j 5 1/2, p 5 21 (solid curve) and the valence electrons with j 5 1/2, p 5 1 (dashed curve) of (a) PbS QD’s and (b) PbSe QD’s owing to band anisotropy.



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This point is illustrated in Fig. 3, where %(k) for the QCL’s of the conduction electrons of PbS QD’s with j 5 1/2, p 5 1 and with j 5 3/2, p 5 1 are plotted as a function of dot size. For the QCL with j5 1/2, p 5 1 ( j 5 3/2, p 5 1), %(k) is the amplitude of the l 5 0 (l 5 2) envelope-function component relative to the l 5 1 envelope-function component. These two states with very different symmetries have nearly identical energies, as shown in Fig. 2. This is in contrast to the results of Refs. 1 and 9 where energies are determined by the spherical symmetry of the envelope functions only. In the remainder of this section, the effects of band anisotropy on the unperturbed QCL’s will be examined. The degeneracy of the unperturbed states in a given L valley is lifted by the anisotropy. The level shift owing to the anisotropy is calculated with first-order degenerate perturbation theory. The calculation of the perturbationmatrix elements can be simplified by exploiting the symmetry that is not broken by the anisotropy. First, since ˆ [Eq. (1)] has axial symmetry the total Hamiltonian H around the z axis, the perturbation commutes with the operator J z and can only mix states with the same magnetic quantum number m. Second, because of the time-

Fig. 5. Shifts D u m u 51/2 (solid curve) and u D u m u 53/2u (dashed curve) of (a) the conduction-electron levels, u j 5 1/2, p 5 1 & and u j 5 3/2, p 5 1 & , and (b) the valence-electron levels, u j 5 1/2, p 5 21 & and u j 5 3/2, p 5 21 & , of PbS QD’s. See the text for the meaning of D u m u 51/2 and D u m u 53/2 . The cusplike feature of the dashed curve in (b) results from the zero crossing of the shift D u m u 53/2 .

spherical symmetry, the unperturbed energies depend only on the total angular-momentum quantum number j and parity p. Also, since the lead salts have four equivalent L valleys, each level with quantum numbers j, m, and p is fourfold degenerate. The calculated energies of the unperturbed QCL’s of PbS and PbSe QD’s (T 5 300 K) are shown up to j 5 5/2 in Fig. 2. The bulk band gaps at this temperature are 0.41 eV for PbS and 0.28 eV for PbSe. It should be noted that the larger exciton Bohr radius of PbSe results in stronger quantumconfinement effects. In addition to the degeneracy arising from spherical symmetry, some of the levels with the same parity but with total angular momenta that differ by one unit are nearly degenerate; the envelope functions for u L 62& ,( u L 61& ) of these degenerate quantum-confined conduction (valence) levels have the same orbital angularmomentum quantum number. This implies that the energy of a quantum-confined conduction (valence) level of a lead-salt QD is determined primarily by the orbital angular momentum of the envelope functions for u L 62& ( u L 61& ).

Fig. 6. Shifts D u m u 51/2 (solid curve) and u D u m u 53/2u (dashed curve) of (a) the conduction-electron levels, u j 5 1/2, p 5 1 & and u j 5 3/2, p 5 1 & , and (b) the valence-electron levels, u j5 1/2, p 5 21 & and u j 5 3/2, p 5 21 & , of PbSe QD’s.

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reversal symmetry, all levels should remain doubly degenerate even after the perturbation is considered. Moreover, the following relations,

^ p , j, m u Vˆ u p , j,m & 5 ^ p , j, 2m u Vˆ u p , j, 2m & , (27) ^ p , j, m u Vˆ u p , j 2 1, m & 5 2^ p , j,2m u Vˆ u p , j 2 1,2m & , (28) which are consequences of time-reversal symmetry,21 reduce the number of matrix elements to be evaluated by half. In Eqs. (27) and (28), u p , j, m & is used to denote the ˆ u p , j, m & deenvelope function F p ,j,m (r), and ^ p , j, m u V † ˆ notes the integral * drF p ,j,m (r)V F p , j,m (r), for example. With the help of these simplifications the matrix elements are explicitly calculated and the results are presented in Appendix A. The effects of the anisotropy on individual QCL’s are as follows: Because of the time-reversal symmetry, the lowest j 5 1/2 conduction-electron states and the highest j 5 1/2 valence-electron states, both of which are doubly degenerate, remain degenerate. The energy shifts of those states of PbS and PbSe QD’s are shown in Fig. 4. The shifts are less than ;5 meV for all sizes of PbS and PbSe QD’s and are negligible for most purposes. The perturbation mixes the states u p 5 61, j 5 1/2, u m u 5 1/2& and u p 5 61, j 5 3/2, u m u 5 1/2& (plus is for the QCL’s of conduction electrons, and minus is for those of valence electrons). This mixing gives rise to two doubly degenerate levels in which one is pushed up and the other is pushed down by the same amount, D u m u 51/2 , from the average energy of the original unperturbed levels. On the other hand, the perturbation just shifts the energy of the state u p 5 61, j 5 3/2, u m u 5 3/2& by the amount D u m u 53/2 . Calculated values of the shifts D u m u 51/2 and u D u m u 53/2u of PbS and PbSe QD’s are shown in Figs. 5 and 6, respectively. While the effect of the band anisotropy is quite negligible for PbS, the more-anisotropic band structure of PbSe results in a much larger splitting of the energy levels, especially in the QCL’s of the valence electrons. The discussion could be continued in the same fashion, and similar conclusions would be reached for the higher excited states.

3. DIPOLE TRANSITIONS AND EXCITON STATES The most easily measurable consequences of quantum confinement are a large blue shift and the appearance of discrete peaks in optical absorption spectra. Absorption spectra of QD’s result from the dipole transitions between quantum-confined conduction and valence levels. There are four equivalent valleys in the lead salts, so indirect transitions between different valleys mediated by an X-point phonon must be considered. Only direct transitions within the same valley will be discussed in this section; the effects of intervalley scattering will be discussed in Section 4. For simplicity, only the transitions between eigenstates of the spherically symmetric Hamiltonian [Eq. (3)] will be considered; the effects of the perturbation can always be added later once the transitions between the unperturbed states are calculated.


The selection rules and the strength of the dipole transitions are determined by the transition-matrix element M c,v 5 u ^ C c ~ r! u e • pu C v ~ r! & u 2 ,

(29)

where e represents the polarization of light and u C c (r) & and u C v (r) & are the total wave functions for the conduction and the valence electrons in a QD, respectively. Using the form of the wave function given in Eq. (8) and the fact that the envelope functions are approximately constant over a unit cell, we can write the matrix element for direct dipole transitions within a given L valley as

M c,v 5

UE 3

drF

E

† p c , j c ,m c ~ r !~ e

drF

• p! F p v , j v ,m v ~ r! 1 ~ e • zˆ ! P l

† p c , j c ,m c ~ r !~ s x ^

U

2

s z ! F p v , j v ,m v ~ r! . (30)

This expression is derived in Appendix B. The symmetry of the band-edge Bloch functions has important consequences for the direct interband transitions of lead-salt QD’s. In this expression, zˆ is one of the four equivalent ^111& directions in the face-centered cubic lattice; s x ^ s z is the direct product of s x and s z ; and P l is the longitudinal Kane momentum-matrix element between the conduction and valence band-edge Bloch functions. After summing over the transitions in all four equivalent valleys, the transition element becomes isotropic, as expected from the cubic symmetry of the lattice. Since p and s x ^ s z are vector operators, selection rules for the direct transitions can be stated immediately: Dj 5 0, 61,

Dm 5 0, 61,

and

p c p v 5 21. (31)

Note that, in contrast to the situation in wide-band-gap systems, the dipole moment in lead-salt QD’s consists of two distinct contributions. The first term in Eq. (30) comes purely from the dipole moment of the envelope functions and is isotropic in its polarization. This term, which is not found in II–VI QD’s, arises because the total wave functions of both conduction and valence electrons are expanded in the same set of band-edge Bloch functions [see Eq. (8)]. For PbS QD’s this contribution nearly vanishes for Dj 5 0 transitions since p is odd under time reversal and the electron and hole effective masses of PbS are almost identical. The second term contains the dipole moment of the Bloch functions, P l , and is polarized along one of the four ^111& directions before being summed over the four equivalent valleys. Since this term is proportional to the overlap between electron and hole envelope functions, it becomes important only for Dj 5 0 transitions in lead-salt QD’s. The energies of the optically excited confined exciton states can be calculated with the selection rules [Eq. (31)] and the calculated energies of the QCL’s. The exciton levels can be modified by several many-body perturbations: Coulomb and exchange interactions between electron and hole and intervalley scattering, the most important of which is the Coulomb attraction between electron and hole. Fine structure in the spectrum owing to these


Fig. 7. Comparison of the experimental values of the lowest exciton energy of PbS QD’s (triangles, solid circles, and squares) with the calculation by the four-band envelope-function formalism (solid curve), the hyperbolic-band model (dashed curve), the tight-binding calculation (crosses),and the parabolic effectivemass model (dotted curve).


1639

We measured the optical absorption spectra and determined the size of the nanocrystals by transmission electron microscopy. The average diameter of the spherical nanocrystals is 2.9 nm in the poly(vinyl alcohol) host and 4.8 nm in the glass host; the data are denoted as triangles. Other data points (solid circles and squares) are adapted from Refs. 1, 4, and 5. All of the data agree well with the four-band envelope-function calculation. The samples described in Refs. 1 and 4 contain QD’s with broad size distributions; the dot-size distribution causes uncertainties in the dot size as well as the first exciton energy. The predictions of the parabolic effective-mass model,1,9 hyperbolic-band model,1 and semi-empirical tight-binding calculation10 are also shown in Fig. 7. For the parabolic effective-mass and hyperbolic-band models, * 5 0.11m, and m * 5 0.12m for we used m * e 5 0.12m, m h the effective masses of PbS at T 5 300 K, where the inverse of m * is the average of the inverses of electron and hole effective masses.17 The discrepancy between the parabolic approximation and the experimental data can be easily explained by the fact that the band structure of the lead salts near the L point is not parabolic owing to the interband coupling (see Fig. 1). The hyperbolic-band model predicts the lowest exciton energies quite well.

Fig. 8. Comparison of the energies of the first and second absorption peaks of PbS QD’s measured in this work (triangles) and Ref. 5 (squares) with the theoretical calculations by the fourband envelope-function formalism (solid curve), the hyperbolicband model (dashed curve), and the parabolic effective-mass model (dotted curve). For the hyperbolic-band model, only the prediction of the lowest exciton energy can be given.

perturbations will be discussed in the next section. The lowest direct exciton level is composed of ( j c 5 1/2, p c 5 21) electron and ( j v 5 1/2, p v 5 1) hole states and is fourfold degenerate for a given L valley, and the level is degenerate even if the anisotropy perturbation is included. The energy of the first exciton level of PbS QD’s is calculated and compared with experimental data in Fig. 7. For the comparison, data were taken from a sample with PbS QD’s grown in a poly(vinyl alcohol) host (fabricated by the recipe in Refs. 2 and 3) and another sample with a glass host (processed by Corning, Inc., following a procedure similar to that explained in Ref. 5).

Fig. 9. Absorption spectra and calculated transition strengths of PbS QD’s with diameters (a) 4.8 nm and (b) 7.6 nm.

1640



* 5 0.070m, and m * 5 0.077m (m * m* e 5 0.084m, m h 5 0.022m, which is only half of the effective mass of PbSe at T 5 4 K, was incorrectly used in Ref. 8).17 It is somewhat surprising that the parabolic approximation agrees better with the experimental data than the envelope-function calculation. Further investigations on samples with a better characterization should clarify this apparent discrepancy.

4. FINE STRUCTURE IN THE ENERGY SPECTRA

Fig. 10. Comparison of the experimental values of the lowest exciton energy of PbSe QD’s (bars) with the calculation by the four-band envelope-function formalism (solid curve), the hyperbolic-band model (dashed curve), and the parabolic effective-mass model (dotted curve).

The tight-binding calculation in Ref. 10 predicts the firstexciton energy of small dots accurately; a simpler cluster model by Wang et al.1 also gives similar predictions. The merits of the four-band envelope-function formalism can be recognized clearly by comparing its predictions for the higher exciton states with experimental data and the other theories. In Fig. 8, experimental values of the second as well as the first exciton peaks in the absorption spectra of PbS QD’s are shown along with the theoretical predictions. The experimental data were obtained from samples that show discrete absorption peaks characteristic of monodisperse QD’s. The calculation with the fourband envelope-function formalism predicts both the first and second exciton peaks accurately. The hyperbolicband model is incapable of calculating the second and higher exciton energies, and the tight-binding calculations in Refs. 1 and 10 are done only for small clusters (d & 5 nm). According to the four-band envelopefunction calculation, there are four strong dipole transitions associated with the second peak: u j v 5 3/2, p v 5 21 & → u j c 5 3/2, p c 5 1 & , u j v 5 3/2, p v 5 21 & → u j c 5 1/2, p c 5 1 & , u j v 5 1/2, p v 5 21 & → u j c 5 3/2, p c 5 1 & , and u j v 5 1/2, p v 5 21 & → u j c 5 1/2, p c 5 1 & , in order of decreasing transition strengths. The strengths of the dipole transitions of two of the samples whose exciton energies are plotted in Fig. 8 are calculated with Eq. (30), and they agree well with the actual absorption spectra of these samples as shown in Fig. 9. The results shown in Figs. 8 and 9 will be modified only slightly by band anisotropy since the magnitude of the anisotropy perturbation is only ;10 meV for the PbS QD’s of the sizes shown in the figures. The theoretical calculations and experimental data8 for the lowest dipole transitions in PbSe QD’s are shown in Fig. 10. The fabrication of PbSe QD’s has just been reported;8 the samples have extremely broad size distributions, and the uncertainties in the dot sizes are shown explicitly. The masses used for the parabolic effectivemass and hyperbolic-band calculations for T 5 300 K are

To this point, the electronic excitations in QD’s have been modeled as noninteracting electron and hole pair states excited by direct dipole transitions. This simple picture will be modified by the direct Coulomb interaction and the exchange interaction between the electron and the hole, as well as by indirect optical transitions involving intervalley scattering of the carriers by X-point phonons: Indirect transitions will create indirect excitons made of an electron and a hole from different L valleys, and these transitions will be shifted by a phonon energy from the direct transitions that were discussed in the previous section. The Coulomb interaction between the electron and the hole will lower the overall energy levels. The exchange interaction, as well as the Coulomb attraction, will lift the degeneracy or near-degeneracy of those transitions shown in Fig. 9. The interactions will mix not only direct excitons in the same valley but also direct exciton states of two different valleys or direct and indirect excitons. It will be shown that the magnitude of these many-body perturbations is negligible compared with the confinement energy in lead-salt QD’s: 10–100 meV for Coulomb attraction and 1–10 meV for exchange interaction and intervalley scattering, compared with ;1 eV of confinement energy. The degeneracy of the electron–hole-pair energy spectra will be greatly increased if direct and indirect excitons are considered simultaneously. It is very complicated, if not impossible, to distinguish the consequences of one perturbation from those of another since the effects of the perturbations are interrelated and comparable in magnitude. A proper way to treat the many-body perturbations would be to generate both indirect and direct excitons and to couple them via Coulomb and exchange interactions. The major obstacle to this approach is the absence of a theory of the intervalley-electron–phonon coupling in QD’s. The effects of quantum confinement on phonons and exciton–phonon coupling have been investigated only recently, and existing studies emphasize the zone-center phonons in II–VI materials.22,23 Thus a complete description of the effects of both Coulomb and exchange interactions and intervalley scattering on the electronic structure of lead salt QD’s will be left for future studies. We content ourselves in considering only the effects of the Coulomb and the exchange interactions on the lowest direct exciton states of PbS QD’s. Calculations are performed to illustrate the salient features and order of magnitude of the perturbations. The direct Coulomb interaction in lead-salt QD’s is expected to be small because of the large dielectric constant of the materials [e ` (PbS) 5 17, e ` (PbSe) 5 23]. How-



1641

ever, the magnitude of the Coulomb interaction in leadsalt QD’s has not yet been systematically calculated. We can start from the direct Coulomb-interaction Hamiltonian24,25

ˆ H Coulomb 5 2

( 8

†

c,c ,v,v 8

a c 8 b ˜v† b ˜v 8 a c

†

E E dr1

dr2

†

3 C c 8 ~ r1 ! C v 8 ~ r2 ! 3

e2 C ~ r !C ~ r !, e u r1 2 r2 u v 2 c 1

(32)

where c, v, and ˜v denote the collective quantum numbers for conduction-electron, valence-electron, and valencehole states, respectively. ˜v is related to v by the timereversal operation; a c , a †c (b ˜v , b ˜v† ) are the annihilation and creation operators for conduction electrons (valence holes); C(r) is the wave function for conduction and valence electrons given by Eq. (8); and e is the dynamic dielectric function. The interaction can be separated into the short-range (intra-unit-cell) and the long-range (interunit-cell) components and the dominant term of the two is the long-range component.25–27 When the matrix element of this long-range component is evaluated between the exciton state u c, v & and the state u c 8 , v 8 & , it can be written in terms of the envelope functions defined in Eqs. (11) and (12):

ˆ LR ^ c 8, v 8u H Coulombu c, v & 5 2

E E

3

dr1

†

dr2 F c 8 ~ r1 ! F †v ~ r2 !

e2 F ~ r !F ~ r !. e u r1 2 r2 u v 8 2 c 1 (33)

This can be interpreted as the Coulomb attraction between ‘‘envelope’’ electron and ‘‘envelope’’ hole. The nature of the long-range Coulomb interaction is most easily illustrated by considering its effect on the lowest direct exciton states, where electron and hole states both have j 5 1/2 and are doubly degenerate. The matrix element [Eq. (33)] evaluated for this fourfold degenerate exciton state does not depend on the magnetic quantum numbers m c or m v . Therefore the lowest exciton energy level will just be shifted downward without the degeneracy being lifted. This energy shift is calculated for PbS QD’s and is shown in Fig. 11(a) as a function of the dot size. As shown in Fig. 11(b), the effect of the Coulomb interaction is much smaller than the confinement energy in this strongly confined system. The effects of the Coulomb interaction on those states that contribute to the second absorption peak in Fig. 9 are more complicated; the magnitude of the perturbation is large enough (;50 meV) to couple the four nearby exciton states and significantly redistribute the transition strengths. The short-range component of the Coulomb interaction also just shifts down the overall energy level of the first exciton states. The magnitude of the short-range component relative to that of the long-range one is proportional to V/V, where V is the unit-cell volume and V is the vol-

Fig. 11. (a) Coulomb energy shift in the lowest direct exciton states of PbS QD’s calculated with the four-band formalism. (b) Comparison of the experimental values of the first exciton energy (squares) with calculations that include (solid curve) and neglect (dotted curve) the energy of the Coulomb interaction.

ume of the region occupied by the exciton wave function. Thus this term is enhanced compared with the bulk value when the radius of the QD is smaller than the bulk exciton radius. However, the shift owing to the short-range component is still negligible (&1 meV) for all but the smallest clusters (d & 2 nm). On the basis of similar reasoning, the short-range component has been neglected previously.25–27 Recently, the enhancement of the electron–hole exchange interaction in CdSe QD’s and its role in the bandedge emission have been given considerable attention.28,29 The exchange interaction manifests itself in lead-salt QD’s differently than in CdSe QD’s owing to the narrow band gap of the lead salts. The Hamiltonian for the electron–hole exchange interaction is

ˆ H Exch 5

( 8

c,c ,v,v 8

3

†

a c 8 b ˜v† b ˜v 8 a c

E E dr1

†

†

dr2 C c 8 ~ r1 ! C v 8 ~ r2 !

e2 C ~ r !C ~ r !, e u r1 2 r2 u c 2 v 1

(34)

1642



where the symbols have the same meaning as in Eq. (32). In II–VI QD’s the long-range component of the exchange interaction almost vanishes mainly because the electron and hole wave functions consist of orthogonal Bloch functions.25–27 In lead-salt QD’s, both the electron and hole wave functions have the same Bloch components, and thus even the leading monopole term is nonvanishing in the multipole expansion; this is shown in more detail in Appendix C. Thus the conduction- and valence-band mixing in this narrow-band-gap material leads to the long-range exchange interaction with matrix elements

ˆ LR u c, v & 5 ^ c 8, v 8u H Exch

E E dr1

3

†

dr2 F c 8 ~ r1 ! F †v ~ r2 !

e2 F ~ r !F ~ r !. e u r1 2 r2 u c 2 v 8 1 (35)

As the long-range component couples direct excitons in different valleys or direct and indirect excitons, the consequences of the perturbation are quite complicated in general. However, its qualitative features can be described if one considers its effect on the quadruply degenerate lowest direct exciton level. The long-range exchange interaction splits the degeneracy by lowering the energy of the singlet state while leaving the triplet energy level unchanged. The magnitude of the calculated splitting is plotted in Fig. 12. The short-range exchange interaction, which causes the splitting between the longitudinal and the transverse exciton levels in bulk,27 is amplified in QD’s by the enhanced overlap between the confined electron and hole wave functions.25,26,28,29 This general conclusion also applies to lead-salt QD’s. Unfortunately, the exchangestrength constants of the bulk materials, which are needed for a quantitative estimate of the effect, are unavailable for PbS or PbSe. Once again, qualitative understanding of the perturbation can be obtained by considering its effect on the lowest direct exciton states in the same valley. When the short-range interaction Hamil-

Fig. 12. Calculated energy of the triplet–singlet splitting in the lowest direct exciton states of PbS QD’s owing to the long-range exchange interaction.

Fig. 13. Calculated energy splitting owing to the short-range exchange interaction in the lowest direct exciton states of PbS QD’s is plotted in units of the bulk exchange interaction strength constant J.

tonian is diagonalized for these states, the short-range exchange interaction is found to further split the degeneracy of the triplet state: the state with m 5 m c 1 m ˜v 5 1 is split from the remaining triplet states. The magnitude and the sign of the shift are proportional to the exchange integral

J5

1 V

E

3

e2 ^ L 61~ r1 ! u L 61~ r2 ! & , e u r1 2 r2 u

unit cell

dr1

1 V

E

unit cell

dr2 ^ L 62~ r1 ! u L 62~ r2 ! &

where V is the volume of a unit cell and u L 62& ( u L 61& ) is the conduction (valence) band-edge Bloch function. In Fig. 13 the magnitude of the splitting in PbS QD’s is plotted in units of the bulk exchange interaction strength J and shows the typical 1/a 3 dependence.25,26,28,29 If we assume that the bulk exchange strength is 10–100 meV, which is quite reasonable, we conclude that the effect of the long-range component and the short-range component should be comparable to each other. It was already pointed out that there should be indirect exciton states owing to intervalley scattering. The analysis of intervalley scattering in QD’s is quite complicated because the indirect transition from the valence band at one L point to the conduction band at another L point, mediated by the X-point phonon in bulk materials, becomes modified in QD’s. In QD’s the properties of phonons are modified by confinement.22,23 Phonon confinement results in new phonon modes and modifications of the electron–phonon coupling. The selection rules for intervalley scattering should be softened by the confinement, and the scattering amplitude will be different from the bulk value. In contrast to the electronic structure the vibrational properties of QD’s have been investigated only recently,6,22,23 and very little is known about the intervalley electron–phonon interaction in QD’s. Qualita-



tively, the energy of an indirect exciton is expected to be shifted from that of the direct exciton. The energy shift owing to intervalley scattering depends on the energy of the confined X-point phonon, which is expected to be ;10 meV in PbS based on the bulk phonon spectrum of PbS.30

5. CONCLUSIONS AND REMARKS In conclusion, the electronic structures of spherical PbS and PbSe QD’s have been calculated with a four-band envelope-function formalism. The calculation is based on the k • p Hamiltonian, which accurately models the bulk band structure of the lead salts near the L point of the Brillouin zone. The Hamiltonian is spherically symmetrized for the treatment of spherical QD’s, and the small anisotropy in the band structure is treated as a perturbation. The correct account of the unique features of the lead salts, such as strong coupling between the conduction and the valence bands, results in an accurate description of the exciton states in these QD’s. The predictions of the model on the exciton states are in good agreement with experimental data. The present calculation provides wave functions and transition strengths that cannot be obtained from prior theoretical treatments. This information is expected to prove useful in future studies of the electronic and vibrational properties of lead-salt QD’s. For example, investigation of the electron–phonon coupling and the nonlinear optical properties of QD’s requires the knowledge of the correct electronic states. Aspects of the fine structures caused by Coulomb and exchange interactions and intervalley scattering are also discussed. The magnitude of these many-body perturbations are shown to be negligible compared with the confinement energy. Aside from the dominant Coulomb interaction, we find competing effects owing to the long- and short-range exchange interactions and intervalley scattering.

APPENDIX A: PERTURBATIVE CALCULATION OF THE BAND ANISOTROPY The anisotropy of the band structures of the lead salts originates in the differences between the transverse parameters, m 6 t and P t , and the longitudinal parameters, m6 l and P l [see Eq. (2)]. Thus the natural choice of the ˆ would parameters for the symmetrized Hamiltonian H 0

be that made in Eq. (4), where the parameters are averaged such that the transverse parameters are weighted twice as much as longitudinal ones since there are two transverse directions. Surely, this choice is not the only one that can be made; however, we checked that the calculated energy with the perturbation is insensitive to a particular choice of the parameters and also found that the perturbation is minimized with the parameters defined in Eq. (4). ˆ is With these parameters, the perturbation operator V written as

ˆ 5 V

F

u L 62↑ &

1643

u L 62↓ &

u L 61↑ &

u L 61↓ &

e 2T ~02 !

0

d l T ~01 !

0

e 2T ~02 !

2A2 d t T ~11 !

1! A2 d t T ~21

d l T ~01 !

1! A2 d t T ~21

e 1T ~02 !

0

2d l T ~01 !

0

e 1T ~02 !

2A2 d t T ~11 !

2d l T ~01 !

G

,

(A1)

where T q( j ) are the operators that transform as the qth component of a spherical tensor of rank j and are defined as T ~02 ! 5 ¹ 2 2 3 ] 2 / ] z 2 . T ~01 ! 5 ] / ] z,

(A2)

]/]x 6 i]/]y 1! . T ~61 57 A2

(A3)

The parameters e 6 and d l,t are

D

S

\2 1 1 , 6 2 6 ml m6 t

e6 5 7

d l,t 5

(A4)

\ ~ P l,t 2 P ! . m

(A5)

The numerical values of these parameters for PbS and PbSe are listed in Table 2. The effect of the perturbation on the degenerate states discussed in Section 2 can be handled with degenerate ˆ is evaluated and diagonalperturbation theory, where V ized in the degenerate subspaces. Because the perturbation couples only the states with the same magnetic quantum number m, the matrix elements that need to be calculated are

^ p , j, m u Vˆ u p , j, m & ,

(A6)

^ p , j, m u Vˆ u p , j 2 1, m&,

(A7)

where u p , j, m & denotes the envelope-function vector, F p , j, m (r) of the QCL with the quantum numbers p, j, and m [see Eqs. (11) and (12)]. ˆ are spherical tensors, Since the terms in the matrix V the evaluation of the matrix elements can be simplified by the use of the Wigner–Eckart theorem. The matrix element in Eq. (A6) is evaluated for the states with j 5 l 1 1/2, p 5 (21) l11 , and l > 1 as Table 2. Parameters of the Operator of the Anisotropy Perturbation of PbS and PbSe Parameters

PbS

PbSe

e 2(meV•nm2) e 1(meV•nm2) d t (meV•nm) d l (meV•nm)

23 213 27 264

215 69 25 258

1644



^ p , j, m u Vˆ u p , j, m & 5 e2

~ 2l 2 1 !@ l ~ l 1 2 ! 1 3/4 2 3m 2 # l ~ l 1 1 !~ 2l 1 1 !

3 ^ f l ~ r ! Y 0l u T ~02 ! u f l ~ r ! Y 0l & 1 e1 3

~ 2l 1 5 !@ l ~ l 1 2 ! 1 3/4 2 3m 2 # ~ l 1 1 !~ l 1 2 !~ 2l 1 3 !

APPENDIX B: DERIVATION OF THE INTERBAND DIPOLE MOMENT OF LEADSALT QUANTUM DOTS In this section the derivation of the explicit expression for the direct transition dipole moment of lead-salt QD’s [Eq. (30)] from the general definition of the transition dipole moment [Eq. (29)] is given. We write the total wave function u C c,v (r) & as [see Eq. (8)] u C c,v ~ r! & 5 F 1c,v ~ r! u L 62↑ & 1 F 2c,v ~ r! u L 62↓ &

0 0 u T ~02 ! u f l11 ~ r ! Y l11 ^ f l11 ~ r ! Y l11 &

1 F 3c,v ~ r! u L 61↑ & 1 F 4c,v ~ r! u L 61↓ &

4 @ l ~ l 1 2 ! 1 3/4 2 m # 2

1 dl

3 ^ f l11 ~ r ! Y 1 dt

4

~ l 1 1 ! A~ 2l 1 1 !~ 2l 1 3 !

[

0 0 ~1! l11 u T 0 u f l ~ r ! Y l &

4 @ l ~ l 1 2 ! 1 3/4 1 m 2 # ~ l 1 1 ! A~ 2l 1 1 !~ 2l 1 3 !

(A8)

UE S ( D U( H ( 4

M c,v 5

i51

S ( DU 4

~ e • p!

5

j51

@ F ci ~ rk !# * ~ e • p! F vj ~ rk !

V

i, j51

3

3

1 V

k cell

E

unit cell

(

dru †i ~ r! u j ~ r!

@ F ci ~ rk !# * F vj ~ rk !

l ~ l 1 1 ! A~ 2l 2 1 !~ 2l 1 3 !

E

1 V

0 0 u T ~02 ! u g l21 ~ r ! Y l21 ^ f l11 ~ r ! Y l11 &

2m A~ l 2 m 1 1/2!~ l 1 m 1 1/2! l A~ 2l 2 1 !~ 2l 1 1 !

1 V

3 ^ f l ~ r ! Y 0l u T ~01 ! u g l21 ~ r ! Y 0l21 & 2m A~ l 2 m 1 1/2!~ l 1 m 1 1/2! ~ l 1 1 ! A~ 2l 1 1 !~ 2l 1 3 !

0 3 ^ f l11 ~ r ! Y l11 u T ~01 ! u g l ~ r ! Y 0l & .

1 V

E

unit cell

dru †i ~ r!~ e

• p! u j ~ r!

where V is the volume of a unit cell. properties15

4m A~ l 2 m 1 1/2!~ l 1 m 1 1/2!

2

F vj u j

k cell

6m ^ f ~ r ! Y 0l u T ~02 ! u g l ~ r ! Y 0l & l ~ l 1 1 !~ 2l 1 1 ! l

1 ~dt 2 dl!

†

4

1V

1 ~dt 2 dl!

F ci u i

dr

^ p , j, m u Vˆ u p , j 2 1, m &

3

(B1)

with c,v meaning the collective quantum numbers of quantum-confined conduction and valence electrons, respectively. Substituting Eq. (B1) into Eq. (29), and using the property of the envelope functions that they vary slowly over a unit cell, one obtains

The functions f l (r) and f l11 (r) are defined in Eq. (25). In the case of l 5 0 the first term in the above equation should be identically zero. If the states have the same j but the opposite parity, one simply makes the substitutions 2g l (r) → f l (r), g l11 (r) → f l11 (r), and e 6 → e 7 in the above equation, where the functions g l (r) and g l11 (r) are found in Eq. (26). The matrix elements of Eq. (A7) can be calculated similarly. For the case of j 5 l 1 1/2, p 5 (21) l11 , and l > 1,

1 e1

c,v i ~ r! u i ~ r! ,

i51

0 3 ^ f l11 ~ r ! Y l11 u T ~01 ! u f l ~ r ! Y 0l & .

5 e2

(F

1 V

unit cell

E

unit cell

E

JU

2

,

(B2)

Using the

dru †i ~ r! u j ~ r! 5 d i, j ,

dru 1† ~ r! pu 3 ~ r! 5

1 V

E

unit cell

dru 3† ~ r! pu 1 ~ r!

5 P l zˆ ,

unit cell

dru 2† ~ r! pu 4 ~ r! 5

1 V

E

unit cell

dru 4† ~ r! pu 2 ~ r!

5 2P l zˆ ,

(A9)

(B3)

Equation (B2) can be simplified as

If the parity of the states in Eq. (A9) is p 5 (21) instead of p 5 (21) l11 , the matrix element for this case can be obtained by the substitutions, ( d l 2 d t ) → ( d t 2 d l ) and e 6 → e 7 , in the above equation. The identities in Eqs. (27) and (28) are verified by explicit calculation of the matrix elements using Eqs. (A8) and (A9). l

M c,v 5

UE

X( 4

dr

i51

@ F ci ~ r!# * ~ e • p! F vi ~ r!

1 ~ e • zˆ ! P l $ @ F 1c ~ r!# * F 3v ~ r! 1 @ F 3c ~ r!# * F 1v ~ r! 2

@ F 2c ~ r!# * F 4v ~ r!

2

@ F 4c ~ r!# * F 2v ~ r! %

CU

2

,

(B4)



which is equivalent to Eq. (30). It should be noted that the first term of Eq. (B4) does not exist in II–VI materials because of the orthogonality between the conduction and the valence band-edge Bloch functions as explained in Section 3; this term is incorrectly neglected in Refs. 1 and 9.

2.

3. 4.

APPENDIX C: DERIVATION OF THE LONGRANGE COMPONENT OF THE EXCHANGE INTERACTION The matrix element of the exchange interaction operator [Eq. (34)] taken between the exciton states u c, v & and u c 8 , v 8 & is ˆ ^ c 8, v 8u H Exchu c, v & 5

E E dr1

3

†

dr2 C c 8 ~ r1 ! C †v ~ r2 !

5. 6. 7. 8.

e2 C ~ r !C ~ r !. e u r1 2 r2 u c 2 v 8 1

9.

(C1) The integral can be broken into the summation of integrals in a unit cell over all the unit cells in a QD. The terms for which the unit cells for the coordinates r1 and r2 are different are called the long-range components. The long-range components are further expanded with the multipole expansion, and the monopole term is25–27 ˆ LR u c, v & 5 ^ c 8, v 8u H Exch

(

ijkl

3 3

3

E E dr1

11.

dr2 @ F ci 8 ~ r1 !# * F vj 8 ~ r1 !

e2

e u r1 2 r2 u 1 V 1 V

E E

10.

unit cell

unit cell

12.

@ F kv ~ r2 !# * F cl ~ r2 !

dru †i ~ r! u j ~ r! 13.

dru k† ~ r! u l ~ r! .

(C2)

Note that the total wave function C(r) is expanded in terms of the envelope functions F i (r) and the band-edge Bloch functions u i (r) [see Eq. (B1)]. For materials in which the Bloch functions for conduction and valence electrons are orthogonal, the above term vanishes. However, for lead-salt QD’s, the expression reduces to Eq. (35) by use of the orthogonality relation of Eq. (B3).

14. 15. 16. 17. 18.

ACKNOWLEDGMENTS The authors thank M. Thomas and J. Silcox for transmission electron microscopy measurements, N. F. Borrelli for supplying samples, and Al. L. Efros for useful discussions. This work was supported by the National Science Foundation under grant DMR-9321259.

19.

20. 21. 22.

REFERENCES 1.

Y. Wang, A. Suna, W. Mahler, and R. Kasowski, ‘‘PbS in polymers. From molecules to bulk solids,’’ J. Chem. Phys. 87, 7315–7322 (1987).

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