7649
J. Phys. Chem. 1995,99, 7649-7653
Quantum Confinement Effects in CdSe Quantum Dots B. Zorman,*Tt M. V. Ramakrishna? and R. A. Friesner*,g The Department of Physics, Columbia University, New York, New York 10027, The Department of Chemistry, New York University, New York, New York 10003-6621, and The Department of Chemistry, Columbia University, New York, New York 10027-6948 Received: August 1, 1994; In Final Form: March 2, 1995@
The energies of the lowest excited state of hexagonal and zinc-blende CdSe spherical quantum dots are calculated using empirical pseudopotentials of the bulk semiconductor. The lowest excited state energies computed for hexagonal clusters are in reasonable agreement with recent experiments. We predict that zincblende and hexagonal dots have nearly the same lowest exciton energy shifts down to very small dot sizes. We have found that small changes in the pseudopotentials can turn the lowest energy transitions of small particles into indirect gaps.
I. Introduction Semiconductor nanocrystals or quantum dots have appeared in some types of colored glass for a historically long period.’ Today, theorists and experimentalistsin various fields are trying to understand in detail the optical properties of quantum dots, which may lead to future applications in small-scale electronics and nonlinear optics. As part of this effort, this paper supplements earlier pseudopotential calculations of the lowest optical absorption peak energy of dots by focusing on spherical CdSe clusters, which have been heavily studied by recent experiments in the hexagonal crystal f ~ r m . ~ . ~ Previously, two of us have applied the empirical pseudopotential method (EPM)to compute the spectral shifts of semiconductor quantum dots using only bulk semiconductor parame t e r ~ .Calculations ~ of the lowest exciton state for zinc-blende CdS spherical quantum dots were in excellent agreement with experiment for a wide range of dot radii. Calculations were also done for wurtzite (or hexagonal) CdS dots and other materials such as GaAs and GaP, but a good set of experimental data to adequately test the model on wurtzite dots was not available then. For this paper, we have calculated the spectral shifts of wurtzite and zinc-blende CdSe quantum dots and have improved our model using 3-D plots of energy dispersion to account for differences in crystal type. While several groups have tried variations of the tight-binding method to compute energies for CdSe quantum dot^,^-^ none of these calculations have used the commonly found wurtzite lattice structure; so our calculations using both wurtzite and zinc-blende lattice types provide a test of the role that the lattice type has on the lowest optical transition. Finally, with recent measurements for wurtzite dots, we are able to make some comparisons of theory and experiment. The organization of the paper is as follows: Section I1 presents our method of calculation and a summary of the energy dispersion plot results. In section 111, we compare our exciton calculations with recent experiments, and section IV presents our conclusions.
start by calculating the energy structure due to a dot’s crystalline nature because we assume that crystal effects dominate the energy splittings near the ground state. To clarify our model, consider the Hamiltonian of an electron in a periodic crystal potential Vp(r) and confined to a spherical well of radius R and magnitude V, expressed in terms of the unit step function.
11. Method of Calculation Instead of calculating the kinetic contribution of an envelope function as in effective mass or multiband approximations,swe
where G are the reciprocal-lattice vectors and Vs and VA are form factors. For both zinc-blende and wurtzite crystal types we have used the form factors empirically determined by Bergstresser and Cohen, which can be found in refs 9 and 10. We have found that the wurtzite form factors from Bergstresser and Cohen’s paper did not adequately reproduce the bulk CdSe crystal field splitting at the top of the valence band with our
+ The Department of Physics, Columbia University.
* New York University. 8 @
The Department of Chemistry, Columbia University. Abstract published in Advance ACS Abstracts, May 1, 1995.
0022-3654/95/2099-7649$09.00/0
For a large confining potential (V, >> V,), the last term can be neglected; and the periodic potential splits the energy states of the confinement potential. In the large radius limit, these splittings become the bulk energy band structure. For bulk semiconductors, the magnitude of the bulk band gap is fixed by the periodic potential, so we approximate the size of the gap between the ground state and the first excited state for our dot using the energy dispersion due to the periodic potential term. To approximate the effect of confinement, we have computed energy levels at wavelengths detennined by the spherical boundary conditions. In effect, we have dropped the bulk states of Bloch wavelengths larger than the size of the dot. With this approximation and the assumption that the confinement potential changes the magnitudes of wave functions near the ground state slowly in comparison to the lattice unit cell’s dimensions, we have computed energy splittings near the ground state using the Hamiltonian
h2
X= - -v* 2m
+ V,(r>
where Vp(r)is the local empirical pseudopotential. Obviously, we expect our method to break down for quantum dots smaller in size than several crystal lattice spacings. For wurtzite and zinc-blende crystal types, Vp(r) can be expanded to the form93’O Vp(r>= ~ [ V S ( G ) S S ( G4) ~V.A.(G)SA(G)I exP(iG*r) (3) G
0 1995 American Chemical Society
7650 J. Phys. Chem., Vol. 99, No. 19, 1995
Zorman et al.
larger basis of plane waves. Neglecting the effect of spin, the top of the bulk valence band experimentally consists of a doublet which is about 23 meV higher than a singlet.“*’* We found the singlet to have a higher energy using the 1967 form factors, so we changed two of the form factors slightly to bring the doublet above the ~ing1et.I~ We have used both the Bergstresser and Cohen form factors and our modified set in our quantum dot calculations. Zinc-blende has structure factors S,(G) = cos(G.t,)
(4)
SA(G) = sin(G*t,) where ti = (1, 1, l)ad8 and a0 is the bulk lattice constant of 6.05 We use the vectors (-1, 1, 1)2n/uo, (1, -1, 1)2n/u0, and (1, 1, - 1)2n/u0as the reciprocal-lattice basis vectors. For the hexagonal structure, S,(G) = cos(G*tJ cos(2/u0G/2)
where tz = (114, l/&, 1/2/6)ao and uo = 3 / ~ . The reciprocallattice basis consists of the vectors ( 4 2 , - &, 0), (0, 4%, 0), and (0, 0, v ‘%) in units of n&/uo. The bulk hexagonal CdSe lattice constant, ao, is 4.299 A.9 We diagonalized the Hamiltonian in the plane wave exp[i(G k ) ~ expansion ] to obtain the electron or hole energies as a function of pseudomomentum k. For zinc-blende and wurtzite structures we used 360 and 475 G vectors, respectively, to converge the energies to within 0.01 eV and the energy splittings to within 0.001 eV. For the wurtzite crystal, the calculated band gap at the bulk r point is 1.796 eV using the Bergstresser and Cohen form factors and 1.880 eV using our modified form factors. These values lie near the experimental values of 1.829 eV at 293 K and 1.751 eV at 80 K.I4 For zinc-blende, our calculated bulk band gap is 1.84 eV. For dot calculations, we have shifted our transition energies by a small constant to achieve a bulk value limit of 1.79 eV for both crystal types. Assuming a large potential barrier at a spherical quantum dot boundary, one can make the approximation that the lowest energy states of a dot of radius R have a pseudomomentum k of magnitude n/R. To find the ground state and first excited state of a dot, one needs to scan the energy versus the direction of k because one cannot assume that a particular direction in k-space provides the ground state to first excited state transition for all crystal types. The use of plots of energy vs direction of k in this paper provides a more systematic way of calculating the lowest energy gap for semiconductors with bulk gaps at the origin of k-space than the fixed k direction approach used previ~usly.~Although we are considering the discrete states of a giant molecule, we will use the terms “valence” and “conduction” bands instead of molecular orbital terminology for convenience. Structures of effective “top valence” and “bottom conduction” bands for 25 A zinc-blende dots and the “bottom conduction” band of 15 8, wurtzite dots are shown in Figures 1 and 2. For zinc-blende dots, one finds from our plots that the bottom of the conduction band is at the (JSx, Ky, Kz) = (1, 1, l)n/ (dk)direction for all dot sizes, reflecting their cubic structural symmetry. One can find the top valence states near the directions {(e, 4) = (n/4, 0), (n/2, n/4), (n/4, n/2) and their crystal symmetry relatives}, as seen in Figure lb. For zinc-
+
blende, a more accurate calculation using a nonlocal pseudopotential might reveal if the angular shift in Figure 1 between the top of the valence band and the bottom of the conduction band, which implies the existence of a slightly lower energy indirect transition, is actually a defect of the local pseudopotential used. A more significant change in the direction of band extrema was found with one of the two wurtzite pseudopotentials. Finally, we find that the fixed k or direct transition for the zinc-blende structure occurs in the (n/4,0) direction, except for dots smaller than about 10-15 8, in radius, where the transition occurs near the conduction band minimum. Although we had used the (1, 1, l)n/(&R) direction for this transition in previous papers for all radii, we find that the error in exciton energy due to this assumption is less than 0.1 eV for both CdSe and CdS; therefore, a good agreement between our model and experimental results for zinc-blende CdS remains after correcting the direction of k.4 For the wurtzite crystal structure, both sets of form factors give conduction bands with minima along the Kz direction, and Figure 2 shows a typical conduction band plot. In contrast, the two pseudopotentials produce qualitatively different valence band structures. Our modified form factors give top valence bands which have maxima along the Kz direction; thus, the lowest energy transition between the valence band and the conduction band is a direct transition. The original Bergstresser and Cohen (BC) pseudopotentials produced valence band plots with maxima along the Kz = 0 plane, which implies that the lowest energy transition changes the direction of k or is indirect. The lowest direct or fixed k transitions with the BC pseudopotentials occur in the Kz = 0 plane of k-space, except for dots smaller than about 10-15 8, in radius, where the transition switches to the Kz axis. The indirect valence top to conduction bottom transitions with the BC form factors have slightly smaller energies than their direct counterparts. The energy difference between the indirect and direct transitions is greater than 0.01 eV for dots smaller than 30 8, in radius but remains less than 0.15 eV down to 7.5 8, dots. Given the direct gaps from the pseudopotential method, we added an electron-hole Coulomb interaction term to obtain the exciton energies, E,, as a function of radius.22
1.786 E, = Egap- ER 6 is the high-frequency dielectric constant. For both crystal types we set E to 6.25, the average of €11 and E L for hexagonal CdSe.I4 We did not include an electron correlation energy term used previously which is smaller than 0.01 eV.4
111. Results and Discussion Results for wurtzite and zinc-blende dots are plotted in Figures 3 and 4 along with experimental lowest absorption peaks. Figure 3 shows that the two pseudopotentials used for the wurtzite dots give very similar exciton energy shifts, and one obtains a reasonable agreement between the model with both sets of bulk form factors and the experiments. It is important to emphasize that neither potential was adjusted to yield agreement with cluster band gaps; our modification of the Bergstresser and Cohen form factors was fit only to bulk properties. Since the appropriate dielectric constant of a sphere is expected to increase to the zero frequency dielectric constant as the radius increase^,^^^*^ our model slightly underestimates the high-quality data of Murray et al. for radii larger than 30 A. The wide radial distributions in older experiments and
Quantum Confinement Effects in CdSe Quantum Dots
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J. Phys. Chem., Vol. 99, No. 19, 1995 7651
ENERGY vs. k DIRECTION LOWEST CONDUCTION BAND o
;
ANGLE FROM Kz AXISTHETA 1.5 2
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b
ANGLE FROM Kx AXIS:PHI ENERGY vs. k DIRECTION HIGHEST VALENCE BAND o
ENERGY, eV
3.905
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2.5 ANGL.E F'ROM KZ AXISlTHETA 3 0
".=
ANGLE FROM Kx AXIS:PHI Figure 1. Relationship between energy and the direction of the pseudomomentum k at a fixed magnitude of z / R for a 25 plotted points are not necessarily the allowed states. 200 G vectors were used at each point.
differences in temperature have contributed to much of the scatter in the experimental data. The dot samples of Borrelli et ~ 1 . grown ' ~ in glass have standard deviations in particle diameter in excess of 15% while the techniques described in Murray et a1.* have produced samples with deviations in diameter smaller than 6%. The deviations in ref 2 were error limited to one atomic plane by transmission electron microscopy (TEM)measurements. Finally, the most noticeable discrepancy in Figure 3 between our model and experiment comes from a reference which did not report any TEM measurements to confirm the reported mean particle radii for its glass grown samples.8
A zinc-blende dot. The
Due to the instabilities of the zinc-blende crystal structure,26 the three experimental samples plotted in Figure 4 probably have significantly more irregularities than the best samples of the wurtzite dot studies and do not, in our view, provide an adequate set of energies to compare with theory; however, our calculation can be viewed as a prediction which can be c o n f i e d or denied if it becomes possible to make more accurate measurements at a future date. In Figure 5, one can see only a small difference between the zinc-blende and wurtzite energy shifts from the bulk, unlike our previous pseudopotential dot calculations that used the same k direction for the wurtzite and zinc-blende structures and
Zorman et al.
1652 J. Phys. Chem., Vol. 99, No. 19, 1995 ENERGY vs. k DIRECTION LOWEST CONDUCTION BAND o
ENERGY, eV
Q
4% Q
006
6.59 6.57
3
ANGLE FROM Kz AXISTHETA
0.5 ANGLE FROM Kx AXIS:PHI
-.-
Figure 2. Relationship between energy and the direction of the pseudomomentum k at a fixed magnitude of d R for a 15 8, wurtzite dot. The plotted points are not necessarily the allowed states. 475 G vectors were used at each point.
-
4.0
--t
A
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+
\*
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Y
-
.'-I
Pseudopctential A Pseudopolential B Effective Mass Model
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Ref. 2 Ref. 8 Ref. 15 Ref. 16 Ref. 20 Ref. 21
3
@
Theoretical Effective Mass Model Ref. 17 0 Ref. 18,lQ
3.0-
2.5
w
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-
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1.o
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RADIUS, A Figure 3. Exciton energy as a function of wurtzite cluster radius. Pseudopotential A involves our modified form factors, and pseudopotential B involves the original Bergstresser and Cohen form factors. Our results are compared to the effective mass model, which is described in ref 4. Effective masses were taken from ref 25. We used the TEM radii for plotting data from ref 15.
showed a larger disagreement between crystal types? The closeness in exciton energy between the two lattice types helps to explain why a recent tight-binding calculation using an effective face-centered cubic lattice yields band gaps close to our hexagonal results.6 To show the effect of a 1% lattice contraction, we have computed the exciton energies of 7.5 and 10 8,lattice contracted dots for Figure 6 using our modified wurtzite form factors. The reduction of the lattice constant lowers the exciton energies, an effect also seen in pseudopotential calculations for zinc-blende CdS quantum dots? For CdS dots smaller than 15 A, calculations using a reduced lattice constant based upon X-ray diffraction line shifts produced a better agreement with experim e ~ ~however, t;~ Murray et al. report that, for their samples of
0
10
20
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40
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RADIUS, A Figure 4. Exciton energy as a function of zinc-blende cluster radius. Our results are compared to the effective mass model. The -35 8, sample of ref 17 was reported to have a band gap between 2.0 and 2.1 eV.
CdSe dots, apparent shifts in the first X-ray diffraction lines can be fitted better by changing the shape of the dots than by using a lattice contraction.2
IV. Conclusion In summary, we have shown that calculations using empirical pseudopotentials give the lowest exciton energies for wurtzite CdSe dots in reasonable agreement with experimental values. We have used only known bulk semiconductor parameters. Second, while reliable data are not available now for zinc-blende CdSe dots, we predict that they will have nearly the same spectral shifts as wurtzite dots down to small dot radii. Also, we found that slight differences in the pseudopotentials could change a direct gap into an indirect gap. To date, our implementation of the EPM has proven to be more accurate than effective mass models in estimating the band
Quantum Confinement Effects in CdSe Quantum Dots
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J. Phys. Chem., Vol. 99, No. 19, 1995 7653 critical in validating the accuracy of the method. The obvious explanation for the success of the method is that the terms neglected in the approximations have a small impact on the computed band gap. Nevertheless, it would be useful to improve on the present methodology, for example by carrying out calculations in real space rather than in k-space so that the off-diagonal couplings between k states due to confinement can be included. One example of such a real space calculation has already appeared in the l i t e r a t ~ r e .We ~ ~ intend to carry out such calculations in the near future. One important application of such calculations will be to porous silicon, where it is crucial to determine the selection rule violation by which the lowest energy excitation. apparently acquires oscillator strength.
Zinc-Blende Wuluite A Wunzite B
1.5
0
10
20
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50
RADIUS, A Figure 5. Comparison of wurtzite and zinc-blende exciton energies with our modified pseudopotential A and Bergstresser and Cohen’s pseudopotential B.
-
Unconlracled 1% contracted 0
Ref. 2
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Figure 6. Comparison of exciton energies of wurtzite clusters and small clusters with a 1 % lattice contraction.
gaps of spherical semiconductor quantum dots. Results for both zinc-blende CdS and wurtzite CdSe agree with most of the experimental data points to within 0.25 eV down to 10 A in radius. Furthermore, recent experiments of Brus and co-workers have reported luminescence for silicon quantum dots of roughly 10 8, in core radii at 0.5 eV above the bulk indirect gap.27This is near our earlier predicted value of 0.43 eV using empirical pseudopotentials.28 Other workers have investigated improvements in prediction of quantum dot band gaps via tight-binding model^,^^^^^ as opposed to the empirical pseudopotential approach that we employ here. One of these results is in good agreement with our calculations for CdSe with what appears to be utilization of bulk parametrization.6 There is no conflict between the two methods, as both are parametrized one-electron Hamiltonians; indeed, it is likely that a formal relation between the two potentials could be derived. There are a number of significant approximations not only in the empirical pseudopotential method itself but also in our procedure. For example, by not allowing blocks of k states to mix, we neglect terms which become nonzero for a finite cluster. Also, for properties such as surface states, accurate treatment of the surface is no doubt crucial; however, our contention is that one can achieve reasonable results for exciton spectral shifts as a function of size by a simple hard wall boundary treatment. The agreement with experiment that we have demonstrated is
Acknowledgment. We thank A. Paul Alivisatos and Sarah Tolbert for providing experimental data before publication. This work was supported by the US.Department of Energy (Grant DE-FGOZ9OERH 162). References and Notes (1) Banyai, L.; Koch, S. W. Semiconductor Quantum Dots; World Scientific: Singapore, 1993. (2) Murray, C. B.; Noms, D. J.; Bawendi, M. G. J , Am. Chem. Soc. 1993, 115, 8706. (3) Mittleman, D. M.; Shoenlein, R. W.; Shiang, J. J.: Colvin, V. A,; Alivisatos, A. P.; Shank, C. V. Phys. Rev. B 1994, 49, 14435. (4) Ramakrishna, M. V.; Freisner, R. A. J. Chem. Phys. 1991,95, 8309; Phys. Rev. Lett. 1991, 67, 629; SPIE Proc. 1991, 1599. (5) Lippens, P. E.; Lannoo, M. Phys. Rev. B 1990, 41, 6079. (6) Ramaniah, L. M.; Nair, S. V. Phys. Rev. B 1993, 47, 7132. (7) Hill, N. A,; Whaley, K. B. J . Chem. Phys. 1994, 100, 2831. (8) Ekimov, A. I.; Hauch, F.; Schanne-Klein, M. C.; Ricard, D.; Flytzanis, C.; Kudryavtsev, I. A.: Yazeva, T. V.; Rodina, A. V.; Efros, Al. L. J. Opt. Soc. Am. B 1993, 10, 100. (9) Cohen, M. L.; Chelikowsky, J. R. Electronic Structure and Optical Properties o j Semiconductors; Springer: Berlin, 1989. (10) Bergstresser, T. K.: Cohen, M. L. Phys. Rev. 1967, 164, 1069. (11) Glasser, M. L. J. Phys. Chem. Solids 1959, 10, 229. (12) Logothetidis, S.; Cardona, M.; Lautenschlager, P.; Ganiga, M. Phys. Rev. B 1986, 34, 2458. (13) We used V s ( m ) = 0.015 35 au and V,(v%) = 0.022 au instead of 0.015 and 0.025 au, respectively, from ref 10. The other form factors match those of ref IO. (14) Data in Science and Technology: Semiconductors Other than Group IV elements and III-V Compounds; Madelung, O., Ed.; Springer: Berlin, 1992. (15) Bowen Katari, J. E.; Colvin, V. L.; Alivisatos, A. P. J . Phys. Chem. 1994, 98, 4109. (16) Borrelli, N. F.; Hall, D. W.; Holland, H. J.; Smith, D. W. J . Appl. Phys. 1987, 51, 5399. (17) Hodes, G.; Albu-Yaron, A.; Decker, F.; Motisuke, P. Phys. Rev. B 1987, 36, 4215. (18) Alivisatos, A. P.; Harris, T. D.; Carroll, P. J.; Steigerwald, M. L.; Brus, L. E. J. Chem. Phys. 1989, 90, 3463. (19) Alivisatos, A. P.; Harris, A. L.: Levino, N. J.; Steigerwald, M. L.; Brus, L. E. J. Chem. Phys. 1988, 89, 4001. (20) Peyghambarian, N.; Fluegel, B.: Hulin, D.; Migus, A,; Joffre, M.; Antonetti, A.; Koch, S. W.; Lindberg, M. IEEE J . Quantum Electron. 1989, 25, 2516. (21) Bawendi, M. G.: Wilson, W. L.; Rothberg, L.; Carroll, P. J.; Jedju, T. M.; Steigerwald, M. L.; Brus, L. E. Phys. Rev. Lett. 1990, 65, 1623. (22) Brus, L. E. J . Chem. Phys. 1984, 80, 4403. (23) Haken, H. Nuovo Cimento 1956, 10, 1230. (24) Nomura, S.; Kobayashi, T. Phys. Rev. B 1992, 45, 1305. ( 2 5 ) Sze, S. M. Physics ojSemiconductor Devices, 2nd ed.; Wiley: New York, 1981. (26) Brus, L. E. Personal communication. . (27) Littau, K. A,; Szajowski, P. J.; Muller, A. J.; Kortan, A. R.; Brus, L. E. J. Phys. Chem. 1993, 97, 1224. (28) Ramakrishna, M. V.; Friesner, R. A. J. Chem. Phys. 1992,96,873. (29) Wang, L.; Zunger, A. J . Phys. Chem. 1994, 98, 2158. Yeh, C.: Zhang, S. B.; Zunger, A. Phys. Rev. B 1994, 50, 14405. JP94205OU
