Addition energies and quasiparticle gap of CdSe nanocrystals Alberto Franceschetti and Alex Zunger Citation: Applied Physics Letters 76, 1731 (2000); doi: 10.1063/1.126150 View online: http://dx.doi.org/10.1063/1.126150 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/76/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Exciton-phonon coupling and disorder in the excited states of CdSe colloidal quantum dots J. Chem. Phys. 125, 184709 (2006); 10.1063/1.2363190 First principles study of CdSe quantum dots: Stability, surface unsaturations, and experimental validation Appl. Phys. Lett. 88, 231910 (2006); 10.1063/1.2209195 Interdot interactions and band gap changes in CdSe nanocrystal arrays at elevated pressure J. Appl. Phys. 89, 8127 (2001); 10.1063/1.1369405 Optical properties of CdSe nanocrystals in a polymer matrix Appl. Phys. Lett. 75, 3120 (1999); 10.1063/1.125250 Electronic properties of CdSe nanocrystals in the absence and presence of a dielectric medium J. Chem. Phys. 110, 5355 (1999); 10.1063/1.478431
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APPLIED PHYSICS LETTERS
VOLUME 76, NUMBER 13
27 MARCH 2000
Addition energies and quasiparticle gap of CdSe nanocrystals Alberto Franceschettia) and Alex Zunger National Renewable Energy Laboratory, Golden, Colorado 80401
共Received 4 January 2000; accepted for publication 28 January 2000兲 Using atomistic pseudopotential wave functions we calculate the quasiparticle gap, the optical gap and the electron and hole addition energies of CdSe nanocrystals. We find that the quasiparticle gap and the addition energies depend strongly on the dielectric constant of the surrounding material, while the optical gap is rather insensitive to the environment. We provide scaling lows for these quantities as a function of the quantum dot size, and compare our results with recent scanning tunneling spectroscopy experiments. © 2000 American Institute of Physics. 关S0003-6951共00兲02413-X兴 Recent developments in the spectroscopy of single semiconductor quantum dots allow one to obtain resolutionlimited spectra by eliminating all sources of inhomogeneous broadening. These experimental techniques include singledot far-field photoluminescence,1 single-electron tunneling,2 and confocal optical microscopy.3,4 In recent single-dot scanning tunneling spectroscopy 共STM兲 experiments,5,6 an STM tip is positioned above a specific quantum dot, and the tunneling current-voltage spectrum is acquired by applying a bias V between the STM tip and the substrate. The conductance dI/dV shows, as a function of the voltage V, a series of sharp peaks which correspond 共possibly via a scaling factor兲 to the electron and hole charging energies N . Figure 1 shows a schematic diagram of the conductance/voltage spectrum. The basic physical quantities that can be measured by this method 共see Fig. 1兲 are: 共i兲 The ‘‘zero-current gap,’’ which is the measured difference between the voltage of the first peak in forward bias and the first peak in reverse bias. It corresponds to the difference between the charging energy 1 for adding the first electron to the quantum dot and the charging energy ⫺1 for removing an electron from the dot. This quantity is also called ‘‘quasiparticle gap,’’ and will be denoted here as qp . Its microscopic meaning is the energy required to re gap move an electron from the highest occupied orbital h 1 of a neutral dot and place this electron in the lowest unoccupied orbital e 1 of an identical dot located at infinite separation from the first. If E N (E ⫺N ) denotes the ground-state total energy of a quantum dot with N electrons in the conduction band (N holes in the valence band兲, the quasiparticle gap is qp gap ⬅ 1 ⫺ ⫺1 ⫽ 关 E 1 ⫺E 0 兴 ⫺ 关 E 0 ⫺E ⫺1 兴 .
共1兲
共iii兲 The ‘‘inter-multiplet spacing,’’ which is the measured difference between the second and third peaks in forward bias. It corresponds to the second electron addition en(e) ergy ⌬ 2,3 ⫽ 3 ⫺ 2 , and is given by (e) ⌬ 2,3 ⬅ 3 ⫺ 2 ⫽ 关 E 3 ⫺E 2 兴 ⫺ 关 E 2 ⫺E 1 兴 .
共3兲
共iv兲 In addition, optical spectroscopies give access to the opt , which is the minimum energy needed to ‘‘optical gap’’ gap optically excite an interacting electron-hole pair in the quantum dot. It is related to the quasiparticle gap via opt qp tot ⫽ gap ⫺J h1,e1 , gap
共4兲
tot is the total electron-hole Coulomb energy. where J h1,e1 Our purpose here is to compare the calculated and measured quantities indicated in Eqs. 共1兲–共4兲. This will establish a quantitative, microscopic interpretation of the fundamental energetics of quantum dots. If successful, this can be used to predict the scaling laws for such quantities as a function of the quantum dot size. By writing the total energies E N of Eqs. 共1兲–共4兲 in terms of single-particle, Coulomb and polarization energies, and assuming that the single-particle electron and hole levels are occupied in order of increasing single-particle energies, we obtain the following expressions: qp 0 pol pol gap ⫽ gap ⫹⌺ h1 ⫹⌺ e1 ,
共5兲
(e) dir pol tot ⫽J e1,e1 ⫹J e1,e1 ⫽J e1,e1 , ⌬ 1,2
共6兲
(e) 0 0 tot tot ⫽ 共 e2 ⫺ e1 ⫺J e1,e1 ⌬ 2,3 兲 ⫹ 共 2J e1,e2 兲 ⫺K e1,e2 ,
共7兲
共ii兲 The ‘‘intra-doublet spacing,’’ which is the measured difference between the voltages of the first and second peaks in forward bias 共for electrons兲 or in reverse bias 共for holes兲. (e) ⬅ 2 ⫺ 1 for It corresponds to the first addition energy ⌬ 1,2 (h) electrons and ⌬ 1,2 ⬅ ⫺1 ⫺ ⫺2 for holes. In terms of the total energies E N , we have (e) ⬅ 2 ⫺ 1 ⫽ 关 E 2 ⫺E 1 兴 ⫺ 关 E 1 ⫺E 0 兴 . ⌬ 1,2
共2兲
(h) . An analogous equation holds for ⌬ 1,2 a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
FIG. 1. Schematic diagram of the conductance/voltage spectrum of a semiconductor quantum dot.
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opt 0 dir pol pol pol 0 dir gap ⫽ 共 gap ⫺J h1,e1 ⫹⌺ e1 ⫺J h1,e1 ⫺J h1,e1 . 兲 ⫹ 共 ⌺ h1 兲 ⯝ gap
共8兲
0 0 0 ⫽ e1 ⫺ h1 gap
0 is the single-particle gap, and e2 Here 0 ⫺ e1 is the splitting between the two lowest electron levels. ⌺ ␣pol is the polarization self-energy of an electron 共or a hole兲 in the single-particle orbital ␣ which occurs due to the dielectric discontinuity between the dot and the surrounding material,7 and J ␣pol,  is the polarization energy arising from the interaction of an electron in the single-particle orbital ␣ and an electron in the single-particle orbital  mediated by the surface polarization charge.7 Both ⌺ ␣pol and J ␣pol,  vanish when ⑀ out⫽ ⑀ in , and decay monotonically as ⑀ out increases. The quantity J ␣dir,  is the conventional direct Coulomb repulsion between particles in orbitals ␣ and , while K ␣ ,  is the corresponding exchange attraction. We consider here nearly spherical CdSe nanocrystals having the wurtzite lattice structure. The interatomic bondlength is assumed to be the same as in bulk CdSe, and the surface dangling bonds are passivated using ligandlike potentials. As discussed in Ref. 8, we first solve the singleparticle Schro¨dinger equation
关 ⫺ⵜ 2 ⫹V ps共 r兲 ⫹Vˆ nl兴 ␣ 共 r, 兲 ⫽ ␣0 ␣ 共 r, 兲
共9兲
in a plane-wave basis set. Here V ps(r) is the total pseudopotential of the system 共nanocrystal⫹ligands兲, and Vˆ nl is a short-range operator that accounts for the nonlocal part of the potential, including spin-orbit coupling. The local pseudopotential V ps(r) is calculated from the superposition of screened atomic pseudopotentials, which are fitted to reproduce the measured bulk transition energies, deformation potentials, and effective masses, as well as the bulk singleparticle wave functions calculated using density-functional theory in the local-density approximation. These pseudopotentials were previously used to calculate the first eight excitonic transitions of CdSe nanocrystals.8 The single-particle wave functions ␣ (r, ) obtained from Eq. 共9兲 are then used to calculate the Coulomb and polarization integrals that occur in Eqs. 共5兲–共8兲. We assume that the macroscopic dielectric constant ⑀ (r) changes smoothly from ⑀ in inside the dot to ⑀ out outside the dot, with a transition region of the order of the interatomic bondlength. We use a modified Penn model9 to calculate ⑀ in(D), while ⑀ out is treated as a parameter. The Coulomb energies J ␣tot,  are calculated as J ␣tot,  ⫽e
兺
冕 兩
␣ 共 r, 兲 兩
2
⌽  共 r兲 dr,
共10兲
where ⌽  (r) satisfies the Poisson equation ⵜ• ⑀ 共 r兲 ⵜ⌽  共 r兲 ⫽⫺4 e
兺 兩 共 r, 兲 兩 2 .
共11兲
The Coulomb energies J ␣tot,  can be further decomposed into a direct contribution and a polarization contribution. The polarization self-energies ⌺ ␣pol are calculated as e ⌺ ␣pol⫽ 2
兺
冕 *
␣ 共 r, 兲
V S 共 r兲 ␣ 共 r, 兲 dr,
共12兲
qp TABLE I. Quasiparticle gap gap , intra-doublet electron splitting ⌬ (e) 1,2 , (e) opt 共all in eV兲 of CdSe inter-multiplet splitting ⌬ 2,3 , and optical gap gap nanocrystals calculated for a few values of the effective dielectric constant ⑀ out . For each nanocrystal, the diameter D, the dielectric constant ⑀ in com0 puted at the diameter D 共Ref. 9兲, the single-particle gap gap , and the split0 0 ting e2 ⫺ e1 are given in the first column.
⑀ out
qp gap
⌬ (e) 1,2
⌬ (e) 2,3
opt gap
D⫽20.6 Å ⑀ in⫽5.2 0 ⫽3.11 eV gap 0 0 e2 ⫺ e1 ⫽0.62 eV
1 3 6 12
4.41 3.42 3.05 2.77
1.40 0.56 0.33 0.21
1.99 1.14 0.90 0.77
2.98 2.83 2.69 2.53
D⫽29.3 Å ⑀ in⫽5.9 0 gap ⫽2.62 eV 0 0 e2 ⫺ e1 ⫽0.41 eV
1 3 6 12
3.54 2.85 2.62 2.45
1.00 0.39 0.23 0.15
1.39 0.79 0.62 0.52
2.51 2.43 2.36 2.27
D⫽38.5 Å ⑀ in⫽6.3 0 ⫽2.35 eV gap 0 0 e2 ⫺ e1 ⫽0.30 eV
1 3 6 12
3.06 2.53 2.36 2.24
0.77 0.30 0.18 0.11
1.06 0.59 0.45 0.38
2.27 2.22 2.17 2.11
System
and G bulk(r,r⬘ ) is the bulk Green’s function. The exchange energies, such as K e1,e2 in Eq. 共7兲, are small, and will be neglected in the following. Table I shows our results for three different sizes and a range of values of ⑀ out . We see that: qp , and the addition energies 共i兲 The quasiparticle gap gap (e) (e) ⌬ 1,2 and ⌬ 2,3 depend strongly on the effective dielectric constant ⑀ out of the environment. opt depends weakly on ⑀ out . The 共ii兲 The optical gap gap pol pol ⫹⌺ e1 and reason is that as shown in Eq. 共8兲, the terms ⌺ h1 pol J h1,e1 , which depend strongly on ⑀ out , nearly cancel each other. 0 qp (e) 共iii兲 The formula gap ⫽ gap ⫺⌬ 1,2 , used by Banin et al.5 6 and Alperson et al. to extract the single-particle gap from conductance measurements is incorrect. From Eqs. 共5兲 and 共6兲 we see that qp (e) 0 dir pol pol pol ⫺⌬ 1,2 ⫽ 共 gap ⫺J e1,e1 ⫹⌺ e1 ⫺J e1,e1 gap 兲 ⫹ 共 ⌺ h1 兲.
共13兲
qp The second bracketed term is nearly zero. As a result, gap (e) 0 ⫺⌬ 1,2 is smaller than gap by approximately the direct Coudir dir . In our calculations we find J e1,e1 lomb energy J e1,e2 ⫽0.37, 0.24, and 0.17 eV for D⫽20.6, 29.3, and 38.5 Å, respectively 共where D is the nanocrystal diameter兲. In order to compare our results with the experimental data of Alperson et al.6 we need to know the effective dielectric constant ⑀ out of the surrounding material. Table I illustrates our results for ⑀ out⫽1, 3, 6, and 12. We find that the effective dielectric constant ⑀ out⬃3 provides a good fit to the qp (e) (e) , ⌬ 1,2 , and ⌬ 2,3 for the 30-Å-diam nanocrystal measured ⑀ gap 共see Table II兲. We will thus use ⑀ out⫽3 in the following
TABLE II. Comparison between calculated and measured properties. The qp (e) , ⌬ (e) experimental values of gap 1,2 , and ⌬ 2,3 are taken from Ref. 6, while opt gap is taken from Ref. 10. All values are in eV. qp opt ⌬ (e) gap gap ⌬ (e) 2,3 1,2 Dot diameter 共Å兲 Calc. Expt. Calc. Expt. Calc. Expt. Calc. Expt.
20 3.49 3.13 0.58 0.34 1.18 1.00 2.88 30 2.81 2.88 0.38 0.33 0.77 0.85 2.40 2.54 where V S (r)⫽limr⬘ →r关 G(r,r⬘ )⫺G bulk(r,r⬘ ) 兴 . Here G(r,r⬘ ) 45 terms at:2.41 2.44 0.25 0.22 0.50 0.50 Downloaded 2.14 2.24 to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the http://scitation.aip.org/termsconditions.
is the Green’s function associated with the Poisson Eq. 共11兲,
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A. Franceschetti and A. Zunger
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calculations. We then proceed to determine the scaling laws of the calculated quantities as a function of the quantum dot diameter. Assuming a single power law, we find: qp ⫽1.83⫹82.47⫻D ⫺1.30 eV, gap
共14兲
(e) ⫽11.96⫻D ⫺1.01 eV, ⌬ 1,2
共15兲
(e) ⫽27.93⫻D ⫺1.06 eV, ⌬ 2,3
共16兲
opt ⫽1.83⫹92.75⫻D ⫺1.50 eV, gap
共17兲
where the diameter D is expressed in Å. Note that 共i兲 In the bulk limit (D→⬁) the quasiparticle gap and the optical gap approach the bulk band gap 共1.83 eV兲, while the addition energies approach zero. 共ii兲 The optical gap and the quasiparticle gap decay faster than the addition energies. Using Eqs. 共14兲–共17兲 we are able to extrapolate our calculated quantities to the experimentally determined quantum dot sizes.6 The results are compared in Table II with the experimental data of Alperson et al.6 for single nanocrystals, and of Norris and Bawendi10 for ensembles of nanocrystals. qp (e) (e) , ⌬ 1,2 , and ⌬ 2,3 are in good We see from Table II that gap agreement with the experimental results of Alperson et al.6 The largest discrepancies occur for the D⫽20 Å nanocrystal. opt is somewhat underestimated compared The optical gap gap to the results of Norris and Bawendi.10 We note, however, that the nanocrystal size is difficult to determine experimentally, and is subject to a significant uncertainty. (e) As shown by Eq. 共7兲, the electron addition energy ⌬ 2,3 0 depends both on the single-particle energy difference e2 0 ⫺ e1 and on the Coulomb and exchange energies. The e2 ⫺e1 splitting cannot be directly measured. Alperson et al. (e) from estimated the e2⫺e1 splitting by subtracting ⌬ 1,2 (e) 0 0 ⌬ 2,3 . For a 30-Å-diam dot they obtain e2 ⫺ e1 ⯝0.52 eV. The e2⫺e1 splitting of CdSe nanocrystals was independently derived by Guyot-Sionnest and Hines11 using infrared spectroscopy. For an ensemble of nanocrystals having a mean diameter of 31.5 Å they found the first infrared absorption peak at 0.50 eV. If one assumes that the electron-hole Coulomb energy is nearly the same for an electron in the e1 state and in the e2 state, one can identify the infrared absorption energy with the e2⫺e1 splitting. This value should 0 0 ⫺ e1 be compared with our calculated splitting e2 ⫽0.41 eV at D⫽29.3 Å 共Table I兲. Finally, we have calculated the addition energies ⌬ N,N⫹1 ⬅ N⫹1 ⫺ N of CdSe nanocrystals for N up to 7 electrons 共or holes兲. The electron and hole addition energies of a D⫽29.3 Å CdSe nanocrystal calculated for ⑀ out⫽3 are shown in Fig. 2 as a function of N. The pronounced peak in the electron addition spectrum for N⫽2, corresponding to (e) , is due to the filling of the e1 electron shell: adding a ⌬ 2,3 third electron to a dot already containing two electrons in the 0 e1 shell requires investing the single-particle energy e2 0 ⫺ e1 , as shown by Eq. 共7兲. The second electron shell is p like, and consists of three nearly degenerate single-particle levels which can be occupied by up to six electrons. Thus,
FIG. 2. Electron and hole addition energies of a 29.3-Å-diam CdSe nanocrystal, as a function of the number of particles N, calculated for ⑀ out⫽3.
the next few addition energies are almost entirely determined by the electron Coulomb repulsion, and depend rather weakly on N. The addition energies of the holes are approximately constant up to N⫽4. This is due to the fact that the energy difference between the hole single-particle states is relatively small, and is comparable with the variations of the direct Coulomb energies between different hole states. In conclusion, we have calculated the electron and hole addition energies, the quasi-particle gap, and the optical gap of CdSe quantum dots in the strong confinement regime. Atomistic pseudopotential wave functions are used as input for the calculation of Coulomb and polarization integrals. Our results are compared with recent experimental data obtained by scanning tunneling spectroscopy, and provide a microscopic interpretation of the experimentally measured quantities. This work was supported by the U.S. DOE, OER-SC, Division of Materials Science, under Grant No. DE-AC3698-GO10337. 1
S. A. Empedocles, D. J. Norris, and M. G. Bawendi, Phys. Rev. Lett. 77, 3873 共1996兲. 2 D. L. Klein, R. Roth, A. K. L. Lim, A. P. Alivisatos, and P. L. McEuen, Nature 共London兲 389, 699 共1997兲. 3 E. Dekel, D. Gershoni, E. Ehrenfreund, D. Spektor, J. M. Garcia, and P. M. Petroff, Phys. Rev. Lett. 80, 4991 共1998兲. 4 L. Landin, M. S. Miller, M.-E. Pistol, C. E. Pryor, and L. Samuelson, Science 280, 262 共1998兲. 5 U. Banin, Y. Cao, D. Katz, and O. Millo, Nature 共London兲 400, 542 共1999兲. 6 B. Alperson, I. Rubinstein, G. Hodes, D. Porath, and O. Millo, Appl. Phys. Lett. 75, 1751 共1999兲. 7 L. E. Brus, J. Chem. Phys. 79, 5566 共1983兲; 80, 4403 共1984兲. 8 L. W. Wang and A. Zunger, J. Phys. Chem. 102, 6449 共1998兲. 9 A. Franceschetti, L. W. Wang, H. Fu, and A. Zunger, Phys. Rev. B 58, 13367 共1998兲. 10 D. J. Norris and M. G. Bawendi, Phys. Rev. B 53, 16338 共1996兲. 11 P. Guyot-Sionnest and M. A. Hines, Appl. Phys. Lett. 72, 686 共1998兲.
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