- Email: [email protected]
Exchange and correlations effects in small metallic nanoshells
23 November 2001
Chemical Physics Letters 349 (2001) 153±160 www.elsevier.com/locate/cplett
Exchange and correlations eects in small metallic nanoshells E. Prodan, P. Nordlander
*
Department of Physics and Rice Quantum Institute, Rice University, Houston, TX 77251, USA Received 4 June 2001; in ®nal form 4 September 2001
Abstract A method for the calculation of the electronic structure of metallic nanoshells is presented. Using this method, we investigate the role of many-electron eects in small nanoshells. A comparison of the electronic properties calculated using Hartree, Hartree±Fock and local density approximations reveals that the Hartree±Fock approximation leads to unphysical results in the same way as it does for bulk electrons in metallic phases. These anomalies disappear with use of the local density approximation, indicating that the eect of correlations is still strong even for very small metallic nanoshells. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction In the last few years, an increasing amount of research has been dedicated to the physics of nanometer-scale structures, since the fabrication of nanoparticles and nanostructures has become much easier and the experimental techniques at this length scale have become more precise. At the theoretical level, dierent techniques used to investigate very small or very large clusters of atoms have been extended to the study of these systems [1±4]. The results of many of these calculations are in excellent agreement with experimental observations for speci®c systems. In this Letter we investigate the electronic structure of small metal nanoshells, a novel type of nanoparticle [5]. The particles consist of a thin gold or silver shell around a solid dielectric core
*
Corresponding author. Fax: +1-713-348-4150. E-mail address: [email protected] (P. Nordlander).
[6,7]. These nanoparticles exhibit unique optical properties which are determined by their plasmon frequency. By varying the ratio of the thickness of the gold shell with respect to the overall diameter of the particle, the plasmon frequencies of nanoshells can be placed at arbitrary wavelengths between the mid-infrared and the UV [8]. Recent experiments on these nanoparticles have shown that their optical response also can be modi®ed by chemisorption of impurities on their surfaces and by depositing them on metal surfaces [9]. For a microscopic understanding of the electronic and optical properties of metal nanoshells, it is necessary to develop an electronic structure method capable of calculating the electronic structure and polarizabilities of metal nanoshells in vacuum, on surfaces and when interacting with chemisorbed impurities. When applied to nanoparticles, the success of the available computational techniques depends on how important the many-body eects are at this length scale. For the density functional approach, it is crucial to
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 1 1 0 4 - 6
154
E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
understand the role of the correlation terms in order to determine the applicability of this method. In this Letter we present an investigation of the importance of many-electron eects in determining the electronic structure and polarizabilities of metal nanoshells. A comparison of the results using the Hartree (H), Hartree±Fock (HF) and local density (LDA) approximations is presented. The results show that, even for very small nanoparticles, the differences between these three approximations are similar to the dierences that are present for bulk electrons in metals [10], where the HF approximation is known to result in too large bandwidths and in too long a screening length for an impurity point charge. The results show that the correlation terms are important also for very small nanoparticles and suggest that the LDA is an appropriate approach for the electronic structure of nanoshells. 2. Jellium model for metallic nanoshells A metallic nanoshell is composed of a metallic layer (the shell) grown over a spherical dielectric core [5,6]. Since the optical properties are determined by the conduction electrons of the shell, we expect a jellium model to provide an accurate model. Previously, jellium models of shell structures have been successfully used for the investigation of the electronic structure of C60 and endohedral fullerene systems with impurity atoms trapped inside the cage [3]. Also, jellium models have been used to model small spherical metallic clusters [11,12]. We will denote the interior radius of the shell by a and the exterior radius by b. In the metallic shell, the positive charge is considered uniformly distributed over the shell. We model the gold shells using jellium with a free electron radius rs 3 (a.u.). The eect of the dielectric phase and ion cores are taken into account by introducing an external potential: 8 < Vi ; r < a; Vext r Vs ; a < r < b; 1 : 0; b < r:
The value of the potential in the dielectric core is kept the same for all three approximations (Vi 0:0735 hartree). In order to compare the electron distribution for a nanoshell calculated using dierent many-body theories, it is essential that the ionization potential of the shell is the same. To accomplish this, the value of the background potential Vs inside the shell is adjusted so that the ionization potential of the shell is kept constant. To avoid the problem associated with partial ®lling of degenerate electronic levels, the calculations were performed at ®nite temperature. The temperature was chosen smaller than the lowest excitation energy of the shells. For a local potential such as Hartree or LDA, the most suitable approach for solution of the Schroedinger equation would be direct integration. This is not an ecient approach for the HF method. To compare the results of the three approximations, we therefore developed a method that is based on the expansion of the electron wavefunctions in a ®nite basis set. At every step of iteration, we use the same orthonormal basis to calculate the matrix elements of the self-energy. This basis can be chosen as /klm r; r^ gk0 rYlm ^ r, where frgk0 gk is a complete set of orthonormal functions de®ned on the positive real axis. Suppose the spherical symmetry is preserved after the nth iteration. In this case, the eigenvectors of H n 12D Vext R n can be expressed as n gkl rYlm ^r, where R n is the self-energy after nth iteration, calculated in any of the three approxi n mations. Let ekl be the corresponding eigenvalues n and let l be the Fermi energy after the nth it n n eration. We will use the notation nkl for ekl l n . 0 For two positions r and r , we denote by r< the minimum and r> the maximum of r and r0 . Then, for the n 1th iteration, the Hartree term of the self-energy is given by n1
h/klm ; RH
/k0 l0 m0 i Z Z 2 dr0 gk0 rgk00 r dll0 dmm0 dr r n n r0
n0 r0 r02 =r> ;
2
where n n is the density of electrons for the nth step of the iteration:
E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
n n r 2s 1
X
2l 1 1 ebnkl
1
k;l
n 2 jgkl rj :
3
The matrix elements of the Fock term are given by n1 h/klm ; Rex /k0 l0 m0 i
dll0 dmm0
X
2l2 1jjj
j;l1 ;l2
1 ebnkl2
r
r>l1 1
1
Z
dr r2
Z
l1 0
l2 0
l 0
2
n
dr0 r02 gk0 rgjl2 r
n
gjl2 r0 gk00 r0 ;
4
where jjj denotes the three j's symbols. For the LDA, the exchange±correlation part of the selfenergy is given by Z n1 h/klm ; Rexc /k0 l0 m0 i dll0 dmm0 dr r2 gk0 r vxc n n rgk00 r:
5
The exchange±correlation potential used here is the one provided by Perdew and Zunger [13]. The main advantage of the present basis is that the matrix elements in Eqs. (2) and (5) do not depend on the angular quantum numbers. The exchange part of the self-energy in the HF approximation given in Eq. (4) depends on the angular quantum number l, which is the main source of CPU cycles in the numerical calculations. The above expressions show that, for a spherically symmetric charge density in the nth iteration, the self-energy at the n 1th iteration is diagonal with respect to the angular quantum numbers l and m. This shows that the spherical symmetry is also preserved after the n 1th iteration and thus at any iteration, for all three approximations. The ®nal result of the iterative process will be a set
155
of eigenvectors of the form gkl rYlm ^ r. The corresponding eigenvalues depend only on the radial quantum number k and angular quantum number l. Each of the eigenvalues has a 2 2l 1-fold degeneracy. In the following two sections, the electronic structure and the screening properties of nanoshells calculated using Hartree, HF and the LDA will be presented. We will investigate nanoshells of two dierent sizes. Shell-I contains 259 electrons with a 10 a.u. and b 20 a.u. Shell-II contains 542 electrons with a 10 and b 25 a.u. The parameters Vs used in the calculations are listed in Table 1.
3. Electronic structure In the numerical implementation of our method, the nanoshell was considered in a ®nite volume, a sphere of radius R b.pAs radial basis functions we chose gn0 r r 1 2=R sin npr=R. The expansion of the wavefunction includes all such radial wavefunctions up to n N and all spherical harmonics up to angular quantum number l L. The present calculation used R 50 a.u., L 15 and N 30. The results does not change for larger values of these parameters. We started with the HF approximation, because it is the most time consuming. We chose a value of Vs equal to )0.3 hartree, such that the resulting ionization potential 0.2 hartree is similar to the experimental value for the work function of gold. For the Hartree and LDA calculations, we varied Vs (see Eq. (1)) to get the same value of the ionization potential. The Vs values are listed in Table 1. The major dierences in the electronic structure calculated using the three approximations appears
Table 1 The calculated occupied bandwidth for Hartree (H), Hartree±Fock (HF) and local density (LDA) approximations (Vs is the background potential de®ned in Eq. (1) used in the calculations) Nanoshell
Shell-I
Shell-II
Approximation
H
HF
LDA
H
HF
LDA
Bandwidth (hartree) Vs (hartree)
0.175 )0.45
0.283 )0.30
0.175 )0.18
0.187 )0.45
0.323 )0.30
0.189 )0.18
156
E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
when the density of states is compared. Qualitatively, all calculated densities of states show a few common features, like the dependence on the radial and angular quantum numbers as shown in Fig. 1. The electronic structure is analogous to the result that was obtained by Puska and Nieminen [3] in their application to C60 . For each radial quantum number k, the eigenstates with dierent angular momentum l 0; . . . ; 1 generate a branch of the DOS. The ®ve branches that show up in Fig. 1 correspond to dierent radial quantum numbers, k 0; . . . ; 4. Quantitatively, there is a major dierence between HF and the other two approximations. For both nanoshells, the bandwidth is much larger in the ®rst case. From Table 1, one can see that the bandwidths are almost the same for Hartree and LDA, while it increased by a factor of 1.62 for shell-I and by a factor of 1.71 for shell-II, in the HF approximation. For homogeneous electron gas, with the same rs 3 a.u., the HF approximation is also known to overestimate the bandwidth by around a factor of two. The electron densities, calculated using the three approximations, are shown in Fig. 2, for the two dierent nanoshells under study. The calculated charge densities were found to be similar in all three many-body descriptions. In all three cases, one can observe that the electrons spread well over the edges of the shell, while the general features in the middle of the shells are preserved. The oscillations of the density of the electrons
Fig. 1. The density of states calculated using the LDA approximation for shell-II. The highest occupied orbital is drawn with the dashed line.
Fig. 2. The density of electrons in the three approximations: H (solid), HF (dotted) and LDA (dashed), for the two nanoshells: shell-I (panel a) and shell-II (panel b).
inside the shell are caused by the nodes of the radial wavefunctions. For shell-I, the k 0 and k 1 states with zero and one node, respectively, are populated and cause the two maxima in the density distribution. For shell-II, also the k 2 state with two nodes contributes to the charge density giving rise to the three maxima. Fig. 2 shows that the electronic charge calculated using HF and LDA shows a slightly larger electron spill out into the dielectric core and the vacuum regions than is calculated with the Hartree approximation. This is caused by the attractive exchange interaction. Although a weak phenomenon, the eect is strongest for the HF approximation. While this eect is barely visible in the density of electrons, it has a major eect on the Coulomb potential, as can be seen in Fig. 3. Since the nanoshell is assumed neutral, the Coulomb potential vanishes to
E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
Fig. 3. The Coulomb potential for shell-II, in the three approximations: H (solid), HF (dotted) and LDA (dashed).
zero outside the shell and at R. However for the HF approximation, the Coulomb potential becomes positive inside the dielectric core. As we already discussed, this is due to the asymmetry of the electron charge relative to the shell. The nonlocalized, attractive nature of the exchange interaction in the HF approximation forces the electrons to spill out more into the dielectric core. Fig. 3 illustrates how unstable this problem can be. Small deviations of the electron charge produce large deviations in the eective potential. This sensitivity can also lead to convergence problems. To overcome these problems, we started the iterations with a screened Coulomb interaction between electrons and we slowly increased the screening length until the results were stabilized.
the comparison of Hartree, HF and LDA, we use the method described in the previous section. This approach provides an accurate description of the bound states but a relatively poor description of the scattering states. For this reason, when comparing the screening properties, we have to limit ourselves to static and spatially localized perturbations. An interesting and also relevant application that ful®lls these two criteria is the calculation of the screening charge due to a impurity point-charge, located in the middle of the metallic shell. Within linear response theory, the screening charge is given by Z Z dn x P x; x0 0 dx0 ; 6 jx x0 j where x0 denotes the position of the impurity (with charge Z) and P is the trace of the polarization function over the spin degrees of freedom. We will compute the polarization operator using two different approximations: the independent electron approximation (Lindhart), P0 and RPA, PRPA . When evaluating the polarization functions, we will use the electronic structures calculated in the previous section using the Hartree, HF and LDA approaches. For ®nite temperatures, the expression for the zero temperature Lindhart polarization function [15] can be generalized and takes the form X P0 x; x0 2s 1 ei x ej x ij
4. Screening charge Another relevant comparison between the three approximations is provided by the linear response of the electron distribution to an external perturbation. In general, the linear response calculations must include the continuum spectrum, especially for high frequencies and when the external perturbation is not spatially localized. For a local potential such as Hartree or LDA, such a calculation could easily be accomplished using direct integration of the Schroedinger equation [14,15]. However, since the purpose of the present Letter is
157
f ni f nj ei x0 ej x0 ; ni nj ie
7
where ej are the wavefunctions calculated in the previous section and f is the Fermi±Dirac distribution. The second approximation for the polarization function is given by the RPA expression: PRPA x; x0 P0 1
V P0 1 x; x0 :
8
For the Hartree and HF models, V represents the Coulomb interaction between electrons. For LDA, a term, dvxc =dn n xd x x0 should be added [14]. Since Eq. (8) involves a matrix inversion, we decided to expand all the functions in the basis fgk0 rYlm ^rgklm discussed in the previous section.
158
E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
In this case, Eq. (8) is reduced to a simple algebraic calculation. The expression of the polarization function in the independent electrons approximation can be found from Eq. (7) if one plugs in the orbitals fgkl rYlm ^ rgklm : r; r0 ; r^0 P0 r; ^ 2s 1
X l1;2 ;k1;2 ;m1;2
f nk1 l1 f nk2 l2 nk1 l1 nk2 l2 ie
gk1 l1 rgk2 l2 rgk1 l1 r0 gk2 l2 r0 Yl1 m1 ^ rYl2 m2 ^ rYl1 mm ^ r0 Yl2 m2 ^ r0 :
9
One can simplify the above expression by using the identity X Fl1 l2 Yl1 m1 ^ rYl2 m2 ^ rYl1 mm ^ r0 Yl2 m2 ^ r0
X
dn r; h
l
Z
Pl cos h
X kk 0
dr0 r02 gk00 r0
l
gk0 rPkk0
r<0l : r>0l1
13
The symbols r< and r> denote the maximum and minimum of r and R0 . Qualitatively, the screening charge shows similar features in all three models. In Figs. 4 and 5, examples of the spatial dependence of the calculated screening charges are shown. The induced charge is very large around the impurity and decays as the distance to the impurity is increased. At the inner and outer surfaces of the nanoshell, a strong spatial dependence of dn is induced. These surface ef-
l1;2 ;m1;2
X 2l2 1 2l1 1 F l1 l2 4p l1;2 X l1 l2 l 2 jjj Ylm ^ rYlm ^ r0 : 0 0 0 l;m
(a)
10
With this simpli®cation, the matrix elements of the polarization function in the independent electron approximation become X 2l1 1 2l2 1 P0klm;k0 l0 m0 2s 1dll0 dmm0 4p l1;2 ;k1;2 2 f nk1 l1 f nk2 l2 l1 l2 l jjj 0 0 0 nk1 l1 nk2 l2 ie Z dr r2 gk0 rgk1 l1 rgk2 l2 r Z dr0 r02 gk00 r0 gk1 l1 r0 gk2 l2 r0 : 11 From the above, it follows that the polarization functions are diagonal with respect to the quantum numbers l and m and that they do not depend on m. Thus, for both the independent electron and RPA, we can use the notation l
hgk0 Ylm jPjgk00 Yl0 m0 i dll0 dmm0 Pkk0 :
12
With these matrix elements, for an impurity of unit charge at R0 12 a b and h 0, the expression for the screening charge reduces to
(b)
Fig. 4. Calculated screening charge induced by a positive point charge for shell-II in the independent electrons approximation (panel a) and in the RPA (panel b). The center of the nanoshell is placed at the origin of the Cartesian coordinate system. The impurity charge is placed at 0; 0; 12:5 a.u. The screening charge is shown as a function of x and z in the plane y 0. The electronic structure of the shell was calculated using LDA. The contour spacing in panel (a) is 0.005 a.u. The contour spacing in panel (b) is 0.002 a.u.
E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
(a)
(b)
Fig. 5. Comparison of the radial dependence of the screening charge calculated using the RPA approach Eq. (8) using H (solid line), HF (dotted line) and LDA (dashed line) models for the electronic structure. Panel (a) shows the result for shell-I and panel (b) shows the result for shell-II. The insets show the screening charge on a ®ner scale.
fects are large in the independent electron approximation while they are almost negligible for RPA. This is because, in the latter case, the screening charge is con®ned within a volume of a radius approximately equal to the Thomas± Fermi screening length, which is smaller than the distance from the impurity to the shell surface. A quantitative comparison of the results from the dierent approximations is shown in Fig. 5. The insets show charge density oscillations similar to Friedel oscillations in bulk systems. As can be seen, the Hartree and LDA lead to similar screening charges, while the HF model leads to a very dierent result. A similar behavior can be observed for the homogeneous electron gas and in small atoms [14].
159
5. Conclusions Our numerical results show that correlation eects are still strong even for very small metallic nanoshells. The neglect of correlation corrections in the HF approximation has the same eects as for the bulk electrons in an in®nite metallic phase: an unphysical increase of the conduction electron bandwidth and inecient screening of an impurity charge. The correlation correction in LDA makes the results more similar to the Hartree approximation, as for bulk metals. For larger nanoshells, we expect the analogy, from this point of view, between nanoshells and in®nite metallic phases to be even stronger, a fact that favors the LDA among the three approximations considered here. Work on implementing a more ecient numerical method for the LDA using direct integration of the Schroedinger equation is in progress. This will enable the calculation of the electronic structure and ®nite frequency polarizabilities of larger nanoshells. Acknowledgements This work was supported by the Robert A. Welch foundation and by the Multi-University Research Initiative of the Army Research Oce. References [1] Refs. in D. Tomanek, R.J. Enbody (Eds.), Science and Applications of Nanotubes, Kluwer Academic/Plenum Publishers, New York, 2000. [2] Refs. in R. Saito, G. Dresselhaus, M.S. Dresselhaus (Eds.), Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998. [3] M.J. Puska, R.M. Nieminen, Phys. Rev. A 47 (1993) 1181. [4] D. Ostling, D. Tomanek, A. Rosen, Phys. Rev. B 55 (1997) 13980. [5] R.D. Averitt, D. Sarkar, N.J. Halas, Phys. Rev. Lett. 78 (1997) 4217. [6] S. Oldenburg, R.D. Averitt, S. Westcott, N.J. Halas, Chem. Phys. Lett. B 288 (1998) 243. [7] J.B. Jackson, N.J. Halas, J. Phys. Chem. 105 (2001) 2743. [8] S.J. Oldenburg, J.B. Jackson, S.L. Westcott, N.J. Halas, Appl. Phys. Lett. 75 (1999) 2897. [9] S.L. Westcott, R.D. Averitt, J.A. Wolfgang, P. Nordlander, N.J. Halas, J. Phys. Chem. B 105 (2001) 9913.
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E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
[10] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt Rinehart and Winston, New York, 1976. [11] M.J. Puska, R.M. Nieminen, M. Manninen, Phys. Rev. B 31 (1985) 3486.
[12] [13] [14] [15]
W. Ekardt, Phys. Rev. B 29 (1984) 1588. J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. M.J. Stott, E. Zaremba, Phys. Rev. A 21 (1980) 12. A. Zangwill, P. Soven, Phys. Rev. A 21 (1980) 1561.
Chemical Physics Letters 349 (2001) 153±160 www.elsevier.com/locate/cplett
Exchange and correlations eects in small metallic nanoshells E. Prodan, P. Nordlander
*
Department of Physics and Rice Quantum Institute, Rice University, Houston, TX 77251, USA Received 4 June 2001; in ®nal form 4 September 2001
Abstract A method for the calculation of the electronic structure of metallic nanoshells is presented. Using this method, we investigate the role of many-electron eects in small nanoshells. A comparison of the electronic properties calculated using Hartree, Hartree±Fock and local density approximations reveals that the Hartree±Fock approximation leads to unphysical results in the same way as it does for bulk electrons in metallic phases. These anomalies disappear with use of the local density approximation, indicating that the eect of correlations is still strong even for very small metallic nanoshells. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction In the last few years, an increasing amount of research has been dedicated to the physics of nanometer-scale structures, since the fabrication of nanoparticles and nanostructures has become much easier and the experimental techniques at this length scale have become more precise. At the theoretical level, dierent techniques used to investigate very small or very large clusters of atoms have been extended to the study of these systems [1±4]. The results of many of these calculations are in excellent agreement with experimental observations for speci®c systems. In this Letter we investigate the electronic structure of small metal nanoshells, a novel type of nanoparticle [5]. The particles consist of a thin gold or silver shell around a solid dielectric core
*
Corresponding author. Fax: +1-713-348-4150. E-mail address: [email protected] (P. Nordlander).
[6,7]. These nanoparticles exhibit unique optical properties which are determined by their plasmon frequency. By varying the ratio of the thickness of the gold shell with respect to the overall diameter of the particle, the plasmon frequencies of nanoshells can be placed at arbitrary wavelengths between the mid-infrared and the UV [8]. Recent experiments on these nanoparticles have shown that their optical response also can be modi®ed by chemisorption of impurities on their surfaces and by depositing them on metal surfaces [9]. For a microscopic understanding of the electronic and optical properties of metal nanoshells, it is necessary to develop an electronic structure method capable of calculating the electronic structure and polarizabilities of metal nanoshells in vacuum, on surfaces and when interacting with chemisorbed impurities. When applied to nanoparticles, the success of the available computational techniques depends on how important the many-body eects are at this length scale. For the density functional approach, it is crucial to
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 1 1 0 4 - 6
154
E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
understand the role of the correlation terms in order to determine the applicability of this method. In this Letter we present an investigation of the importance of many-electron eects in determining the electronic structure and polarizabilities of metal nanoshells. A comparison of the results using the Hartree (H), Hartree±Fock (HF) and local density (LDA) approximations is presented. The results show that, even for very small nanoparticles, the differences between these three approximations are similar to the dierences that are present for bulk electrons in metals [10], where the HF approximation is known to result in too large bandwidths and in too long a screening length for an impurity point charge. The results show that the correlation terms are important also for very small nanoparticles and suggest that the LDA is an appropriate approach for the electronic structure of nanoshells. 2. Jellium model for metallic nanoshells A metallic nanoshell is composed of a metallic layer (the shell) grown over a spherical dielectric core [5,6]. Since the optical properties are determined by the conduction electrons of the shell, we expect a jellium model to provide an accurate model. Previously, jellium models of shell structures have been successfully used for the investigation of the electronic structure of C60 and endohedral fullerene systems with impurity atoms trapped inside the cage [3]. Also, jellium models have been used to model small spherical metallic clusters [11,12]. We will denote the interior radius of the shell by a and the exterior radius by b. In the metallic shell, the positive charge is considered uniformly distributed over the shell. We model the gold shells using jellium with a free electron radius rs 3 (a.u.). The eect of the dielectric phase and ion cores are taken into account by introducing an external potential: 8 < Vi ; r < a; Vext r Vs ; a < r < b; 1 : 0; b < r:
The value of the potential in the dielectric core is kept the same for all three approximations (Vi 0:0735 hartree). In order to compare the electron distribution for a nanoshell calculated using dierent many-body theories, it is essential that the ionization potential of the shell is the same. To accomplish this, the value of the background potential Vs inside the shell is adjusted so that the ionization potential of the shell is kept constant. To avoid the problem associated with partial ®lling of degenerate electronic levels, the calculations were performed at ®nite temperature. The temperature was chosen smaller than the lowest excitation energy of the shells. For a local potential such as Hartree or LDA, the most suitable approach for solution of the Schroedinger equation would be direct integration. This is not an ecient approach for the HF method. To compare the results of the three approximations, we therefore developed a method that is based on the expansion of the electron wavefunctions in a ®nite basis set. At every step of iteration, we use the same orthonormal basis to calculate the matrix elements of the self-energy. This basis can be chosen as /klm r; r^ gk0 rYlm ^ r, where frgk0 gk is a complete set of orthonormal functions de®ned on the positive real axis. Suppose the spherical symmetry is preserved after the nth iteration. In this case, the eigenvectors of H n 12D Vext R n can be expressed as n gkl rYlm ^r, where R n is the self-energy after nth iteration, calculated in any of the three approxi n mations. Let ekl be the corresponding eigenvalues n and let l be the Fermi energy after the nth it n n eration. We will use the notation nkl for ekl l n . 0 For two positions r and r , we denote by r< the minimum and r> the maximum of r and r0 . Then, for the n 1th iteration, the Hartree term of the self-energy is given by n1
h/klm ; RH
/k0 l0 m0 i Z Z 2 dr0 gk0 rgk00 r dll0 dmm0 dr r n n r0
n0 r0 r02 =r> ;
2
where n n is the density of electrons for the nth step of the iteration:
E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
n n r 2s 1
X
2l 1 1 ebnkl
1
k;l
n 2 jgkl rj :
3
The matrix elements of the Fock term are given by n1 h/klm ; Rex /k0 l0 m0 i
dll0 dmm0
X
2l2 1jjj
j;l1 ;l2
1 ebnkl2
r
r>l1 1
1
Z
dr r2
Z
l1 0
l2 0
l 0
2
n
dr0 r02 gk0 rgjl2 r
n
gjl2 r0 gk00 r0 ;
4
where jjj denotes the three j's symbols. For the LDA, the exchange±correlation part of the selfenergy is given by Z n1 h/klm ; Rexc /k0 l0 m0 i dll0 dmm0 dr r2 gk0 r vxc n n rgk00 r:
5
The exchange±correlation potential used here is the one provided by Perdew and Zunger [13]. The main advantage of the present basis is that the matrix elements in Eqs. (2) and (5) do not depend on the angular quantum numbers. The exchange part of the self-energy in the HF approximation given in Eq. (4) depends on the angular quantum number l, which is the main source of CPU cycles in the numerical calculations. The above expressions show that, for a spherically symmetric charge density in the nth iteration, the self-energy at the n 1th iteration is diagonal with respect to the angular quantum numbers l and m. This shows that the spherical symmetry is also preserved after the n 1th iteration and thus at any iteration, for all three approximations. The ®nal result of the iterative process will be a set
155
of eigenvectors of the form gkl rYlm ^ r. The corresponding eigenvalues depend only on the radial quantum number k and angular quantum number l. Each of the eigenvalues has a 2 2l 1-fold degeneracy. In the following two sections, the electronic structure and the screening properties of nanoshells calculated using Hartree, HF and the LDA will be presented. We will investigate nanoshells of two dierent sizes. Shell-I contains 259 electrons with a 10 a.u. and b 20 a.u. Shell-II contains 542 electrons with a 10 and b 25 a.u. The parameters Vs used in the calculations are listed in Table 1.
3. Electronic structure In the numerical implementation of our method, the nanoshell was considered in a ®nite volume, a sphere of radius R b.pAs radial basis functions we chose gn0 r r 1 2=R sin npr=R. The expansion of the wavefunction includes all such radial wavefunctions up to n N and all spherical harmonics up to angular quantum number l L. The present calculation used R 50 a.u., L 15 and N 30. The results does not change for larger values of these parameters. We started with the HF approximation, because it is the most time consuming. We chose a value of Vs equal to )0.3 hartree, such that the resulting ionization potential 0.2 hartree is similar to the experimental value for the work function of gold. For the Hartree and LDA calculations, we varied Vs (see Eq. (1)) to get the same value of the ionization potential. The Vs values are listed in Table 1. The major dierences in the electronic structure calculated using the three approximations appears
Table 1 The calculated occupied bandwidth for Hartree (H), Hartree±Fock (HF) and local density (LDA) approximations (Vs is the background potential de®ned in Eq. (1) used in the calculations) Nanoshell
Shell-I
Shell-II
Approximation
H
HF
LDA
H
HF
LDA
Bandwidth (hartree) Vs (hartree)
0.175 )0.45
0.283 )0.30
0.175 )0.18
0.187 )0.45
0.323 )0.30
0.189 )0.18
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when the density of states is compared. Qualitatively, all calculated densities of states show a few common features, like the dependence on the radial and angular quantum numbers as shown in Fig. 1. The electronic structure is analogous to the result that was obtained by Puska and Nieminen [3] in their application to C60 . For each radial quantum number k, the eigenstates with dierent angular momentum l 0; . . . ; 1 generate a branch of the DOS. The ®ve branches that show up in Fig. 1 correspond to dierent radial quantum numbers, k 0; . . . ; 4. Quantitatively, there is a major dierence between HF and the other two approximations. For both nanoshells, the bandwidth is much larger in the ®rst case. From Table 1, one can see that the bandwidths are almost the same for Hartree and LDA, while it increased by a factor of 1.62 for shell-I and by a factor of 1.71 for shell-II, in the HF approximation. For homogeneous electron gas, with the same rs 3 a.u., the HF approximation is also known to overestimate the bandwidth by around a factor of two. The electron densities, calculated using the three approximations, are shown in Fig. 2, for the two dierent nanoshells under study. The calculated charge densities were found to be similar in all three many-body descriptions. In all three cases, one can observe that the electrons spread well over the edges of the shell, while the general features in the middle of the shells are preserved. The oscillations of the density of the electrons
Fig. 1. The density of states calculated using the LDA approximation for shell-II. The highest occupied orbital is drawn with the dashed line.
Fig. 2. The density of electrons in the three approximations: H (solid), HF (dotted) and LDA (dashed), for the two nanoshells: shell-I (panel a) and shell-II (panel b).
inside the shell are caused by the nodes of the radial wavefunctions. For shell-I, the k 0 and k 1 states with zero and one node, respectively, are populated and cause the two maxima in the density distribution. For shell-II, also the k 2 state with two nodes contributes to the charge density giving rise to the three maxima. Fig. 2 shows that the electronic charge calculated using HF and LDA shows a slightly larger electron spill out into the dielectric core and the vacuum regions than is calculated with the Hartree approximation. This is caused by the attractive exchange interaction. Although a weak phenomenon, the eect is strongest for the HF approximation. While this eect is barely visible in the density of electrons, it has a major eect on the Coulomb potential, as can be seen in Fig. 3. Since the nanoshell is assumed neutral, the Coulomb potential vanishes to
E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
Fig. 3. The Coulomb potential for shell-II, in the three approximations: H (solid), HF (dotted) and LDA (dashed).
zero outside the shell and at R. However for the HF approximation, the Coulomb potential becomes positive inside the dielectric core. As we already discussed, this is due to the asymmetry of the electron charge relative to the shell. The nonlocalized, attractive nature of the exchange interaction in the HF approximation forces the electrons to spill out more into the dielectric core. Fig. 3 illustrates how unstable this problem can be. Small deviations of the electron charge produce large deviations in the eective potential. This sensitivity can also lead to convergence problems. To overcome these problems, we started the iterations with a screened Coulomb interaction between electrons and we slowly increased the screening length until the results were stabilized.
the comparison of Hartree, HF and LDA, we use the method described in the previous section. This approach provides an accurate description of the bound states but a relatively poor description of the scattering states. For this reason, when comparing the screening properties, we have to limit ourselves to static and spatially localized perturbations. An interesting and also relevant application that ful®lls these two criteria is the calculation of the screening charge due to a impurity point-charge, located in the middle of the metallic shell. Within linear response theory, the screening charge is given by Z Z dn x P x; x0 0 dx0 ; 6 jx x0 j where x0 denotes the position of the impurity (with charge Z) and P is the trace of the polarization function over the spin degrees of freedom. We will compute the polarization operator using two different approximations: the independent electron approximation (Lindhart), P0 and RPA, PRPA . When evaluating the polarization functions, we will use the electronic structures calculated in the previous section using the Hartree, HF and LDA approaches. For ®nite temperatures, the expression for the zero temperature Lindhart polarization function [15] can be generalized and takes the form X P0 x; x0 2s 1 ei x ej x ij
4. Screening charge Another relevant comparison between the three approximations is provided by the linear response of the electron distribution to an external perturbation. In general, the linear response calculations must include the continuum spectrum, especially for high frequencies and when the external perturbation is not spatially localized. For a local potential such as Hartree or LDA, such a calculation could easily be accomplished using direct integration of the Schroedinger equation [14,15]. However, since the purpose of the present Letter is
157
f ni f nj ei x0 ej x0 ; ni nj ie
7
where ej are the wavefunctions calculated in the previous section and f is the Fermi±Dirac distribution. The second approximation for the polarization function is given by the RPA expression: PRPA x; x0 P0 1
V P0 1 x; x0 :
8
For the Hartree and HF models, V represents the Coulomb interaction between electrons. For LDA, a term, dvxc =dn n xd x x0 should be added [14]. Since Eq. (8) involves a matrix inversion, we decided to expand all the functions in the basis fgk0 rYlm ^rgklm discussed in the previous section.
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E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
In this case, Eq. (8) is reduced to a simple algebraic calculation. The expression of the polarization function in the independent electrons approximation can be found from Eq. (7) if one plugs in the orbitals fgkl rYlm ^ rgklm : r; r0 ; r^0 P0 r; ^ 2s 1
X l1;2 ;k1;2 ;m1;2
f nk1 l1 f nk2 l2 nk1 l1 nk2 l2 ie
gk1 l1 rgk2 l2 rgk1 l1 r0 gk2 l2 r0 Yl1 m1 ^ rYl2 m2 ^ rYl1 mm ^ r0 Yl2 m2 ^ r0 :
9
One can simplify the above expression by using the identity X Fl1 l2 Yl1 m1 ^ rYl2 m2 ^ rYl1 mm ^ r0 Yl2 m2 ^ r0
X
dn r; h
l
Z
Pl cos h
X kk 0
dr0 r02 gk00 r0
l
gk0 rPkk0
r<0l : r>0l1
13
The symbols r< and r> denote the maximum and minimum of r and R0 . Qualitatively, the screening charge shows similar features in all three models. In Figs. 4 and 5, examples of the spatial dependence of the calculated screening charges are shown. The induced charge is very large around the impurity and decays as the distance to the impurity is increased. At the inner and outer surfaces of the nanoshell, a strong spatial dependence of dn is induced. These surface ef-
l1;2 ;m1;2
X 2l2 1 2l1 1 F l1 l2 4p l1;2 X l1 l2 l 2 jjj Ylm ^ rYlm ^ r0 : 0 0 0 l;m
(a)
10
With this simpli®cation, the matrix elements of the polarization function in the independent electron approximation become X 2l1 1 2l2 1 P0klm;k0 l0 m0 2s 1dll0 dmm0 4p l1;2 ;k1;2 2 f nk1 l1 f nk2 l2 l1 l2 l jjj 0 0 0 nk1 l1 nk2 l2 ie Z dr r2 gk0 rgk1 l1 rgk2 l2 r Z dr0 r02 gk00 r0 gk1 l1 r0 gk2 l2 r0 : 11 From the above, it follows that the polarization functions are diagonal with respect to the quantum numbers l and m and that they do not depend on m. Thus, for both the independent electron and RPA, we can use the notation l
hgk0 Ylm jPjgk00 Yl0 m0 i dll0 dmm0 Pkk0 :
12
With these matrix elements, for an impurity of unit charge at R0 12 a b and h 0, the expression for the screening charge reduces to
(b)
Fig. 4. Calculated screening charge induced by a positive point charge for shell-II in the independent electrons approximation (panel a) and in the RPA (panel b). The center of the nanoshell is placed at the origin of the Cartesian coordinate system. The impurity charge is placed at 0; 0; 12:5 a.u. The screening charge is shown as a function of x and z in the plane y 0. The electronic structure of the shell was calculated using LDA. The contour spacing in panel (a) is 0.005 a.u. The contour spacing in panel (b) is 0.002 a.u.
E. Prodan, P. Nordlander / Chemical Physics Letters 349 (2001) 153±160
(a)
(b)
Fig. 5. Comparison of the radial dependence of the screening charge calculated using the RPA approach Eq. (8) using H (solid line), HF (dotted line) and LDA (dashed line) models for the electronic structure. Panel (a) shows the result for shell-I and panel (b) shows the result for shell-II. The insets show the screening charge on a ®ner scale.
fects are large in the independent electron approximation while they are almost negligible for RPA. This is because, in the latter case, the screening charge is con®ned within a volume of a radius approximately equal to the Thomas± Fermi screening length, which is smaller than the distance from the impurity to the shell surface. A quantitative comparison of the results from the dierent approximations is shown in Fig. 5. The insets show charge density oscillations similar to Friedel oscillations in bulk systems. As can be seen, the Hartree and LDA lead to similar screening charges, while the HF model leads to a very dierent result. A similar behavior can be observed for the homogeneous electron gas and in small atoms [14].
159
5. Conclusions Our numerical results show that correlation eects are still strong even for very small metallic nanoshells. The neglect of correlation corrections in the HF approximation has the same eects as for the bulk electrons in an in®nite metallic phase: an unphysical increase of the conduction electron bandwidth and inecient screening of an impurity charge. The correlation correction in LDA makes the results more similar to the Hartree approximation, as for bulk metals. For larger nanoshells, we expect the analogy, from this point of view, between nanoshells and in®nite metallic phases to be even stronger, a fact that favors the LDA among the three approximations considered here. Work on implementing a more ecient numerical method for the LDA using direct integration of the Schroedinger equation is in progress. This will enable the calculation of the electronic structure and ®nite frequency polarizabilities of larger nanoshells. Acknowledgements This work was supported by the Robert A. Welch foundation and by the Multi-University Research Initiative of the Army Research Oce. References [1] Refs. in D. Tomanek, R.J. Enbody (Eds.), Science and Applications of Nanotubes, Kluwer Academic/Plenum Publishers, New York, 2000. [2] Refs. in R. Saito, G. Dresselhaus, M.S. Dresselhaus (Eds.), Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998. [3] M.J. Puska, R.M. Nieminen, Phys. Rev. A 47 (1993) 1181. [4] D. Ostling, D. Tomanek, A. Rosen, Phys. Rev. B 55 (1997) 13980. [5] R.D. Averitt, D. Sarkar, N.J. Halas, Phys. Rev. Lett. 78 (1997) 4217. [6] S. Oldenburg, R.D. Averitt, S. Westcott, N.J. Halas, Chem. Phys. Lett. B 288 (1998) 243. [7] J.B. Jackson, N.J. Halas, J. Phys. Chem. 105 (2001) 2743. [8] S.J. Oldenburg, J.B. Jackson, S.L. Westcott, N.J. Halas, Appl. Phys. Lett. 75 (1999) 2897. [9] S.L. Westcott, R.D. Averitt, J.A. Wolfgang, P. Nordlander, N.J. Halas, J. Phys. Chem. B 105 (2001) 9913.
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[10] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt Rinehart and Winston, New York, 1976. [11] M.J. Puska, R.M. Nieminen, M. Manninen, Phys. Rev. B 31 (1985) 3486.
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