Abstract
Electron structure of a silicon quantum dot doped with a shallow hydrogenlike donor has been calculated for the electron states above the optical gap. Within the framework of the envelopefunction approach we have calculated the fine splitting of the ground sixfold degenerate electron state as a function of the donor position inside the quantum dot. Also, dependence of the wave functions and energies on the dot size was obtained.
Keywords:
Silicon nanocrystal; Donor; Energy spectrum; Fine splittingIntroduction
Introduction of shallow impurities into silicon quantum dots is considered [15] as an efficient way to modify optical properties of the dots. In connection to this, calculations of electronic structure [611] and dielectric function [1218] of silicon nanocrystals doped with Vgroup shallow donors have been carried out earlier. The microscopic firstprinciples study of charge distribution and electrostatic fields in bulk silicon [1921] and silicon crystallites [1618] in the presence of Vgroup donors have shown existence of shortrange and longrange components of the electron–ion Coulomb interaction in the system. The shortrange potential differs from zero only in a nearest vicinity (about Bohr radius) of the donor nucleus. This extra potential, frequently named a “centralcell correction”, leads to socalled valley–orbit interaction [22,23]. In turn, the valley–orbit interaction causes the splitting of the electron ground state that is sixfold degenerate if the spin variables have not been taken into account. The degeneracy order exactly coincides with the number of valleys in a conduction band of bulk silicon. The longrange component represents standard ɛtimes weakened Coulomb attraction between the donor ion and electron.
Quantum confinement in nanocrystals considerably strengthens the valley–orbit interaction and level splitting [811] relative to the bulk systems [24]. Provided that the valley–orbit interaction is strong, the sixfold degenerate lowest energy level splits into three (if the donor occupies the dot center) or six (if the donor position is arbitrary) levels. In case of symmetric centrallocated donor position in the nanocrystal the ground state splits into a singlet, doublet, and triplet [8,10,11], as it takes place in the bulk silicon [22,23], the singlet level being strongly split off from the doublet and triplet levels and always turns out to be the lowest one.
However, not all the donors manifest strong valley–orbit coupling. For example lithium, being interstitial donor of the first group, has the splitting of about 1–2 meV [24] which is onetwo order less than that for Vgroup donors such as P, As, Sb, and Bi. It is, therefore, logical to assume also for quantum dots (without resorting to calculations), that the valley–orbit splitting and centralcell effect for Li will be weakened compared to the Vgroup donors by the same order of magnitude. As was shown for 2–5 nm nanocrystals [10,11], maximal values of the splitting for the Vgroup donors are of the order of several tenths of eV. As a consequence, the valley–orbit splitting for Li in the dot does not exceed, presumably, 10 meV. At the same time, as will be shown below, the splitting caused by the longrange hydrogenic potential turns out to be considerably greater (about 100 meV). This indicates a key role of the hydrogenic potential compared to the centralcell one for Li. On this reason, the hydrogenlike model may be applied to a lithium donor in a silicon nanocrystal. Therefore, when dealing with Lidoped dots, we shall neglect the shortrange centralcell field and, consequently, the valley–orbit splitting.
In turn, for Vgroup donors in quantum dots, a contribution of the longrange field to the energy splitting is 3–5 times less than that of the shortrange field. In particular, in the works of the authors [10,11] electronic structure of silicon crystallites doped with Vgroup donors has been already calculated in the presence of both short and longrange Coulomb interactions. Note, however, that the results of Refs [10,11] may not be automatically extended to Lidoped nanocrystals by zero setting the terms caused by the shortrange field. For Vgroup donors the centralcell potential, being stronger than the hydrogenic one, imposes certain symmetry to the Bloch states. The latter are described by the functions possessing A_{1}E, or T_{2} symmetry. Existence of the hydrogenlike potential leads to some insignificant corrections only. In case of a lithium donor, the situation is opposite. Precisely the hydrogenlike potential is a dominant factor, while the valley–orbit interaction is negligibly small. As a result, we have to use different approaches to solve the problem. Consequently, the solutions obtained in these two cases are appreciably different.
In the present article we find the energies and wave functions of the ground and several excited electronic states of silicon quantum dot with a lithium donor that may be treated as a shallow hydrogenic one. We shall also discuss an effect of degeneracy removal caused by the presence of a hydrogenlike center in an arbitrary place inside the dot, and calculate the splitting of the lowest energy level being sixfold degenerate initially.
For this purpose we employ envelopefunction approximation. Of course, an applicability of the method to quantum dots is justified when the dot size considerably exceeds the size of the unit cell. The latter approximately coincides with the distance between adjacent atoms in silicon lattice (2.35 Ǻ). We suppose this requirement to be fulfilled for 2–5 nm nanocrystals. In this case, keeping ourselves within the framework of a macroscopic picture, one can use bulk static dielectric constants ɛ_{s} and ɛ_{d} for materials inside and outside the dot, respectively. As a result, the standard Coulomb potential is modified due to appearance of polarization charges at the nanocrystal boundary [13,14]. Existence of an excess positive charge near the dot boundary has been directly confirmed by the microscopic firstprinciples calculations of Delerue et al. [16] and Trani et al. [18].
The Model and Basic Equations of the Problem
Let us consider a silicon quantum dot of radiusR, embedded into a wideband matrix such as SiO_{2}. The potential barriers for electrons caused by the band discontinuity at the dot boundary are of the order of several eV. Since the typical energies we shall further consider do not exceed a few tenths of eV, the barriers may be treated as infinitely high.
Within the frames of macroscopic treatment the total electron potential energy U(r) consists of three parts:
Here r and h are the positionvectors of the electron and impurity ion, respectively. U_{0}(r) is the potential of an infinitely deep well that is assumed to be zero inside, and infinity outside, the dot. The second part V_{sp}(r) describes an interaction between the electron and its own image arising due to the charge polarization on the boundary between silicon and silicon dioxide. Since the electron interacts with its own image, V_{sp}(r) is frequently referred to as a selfpolarization term. It can be represented in the form, see, e.g., Eqs (3.24), (3.26) in Ref. [25]:
At last, the third term introduced in the Eq. 1, represents an electron–ion interaction. It has the form [25]:
where θ is the angle between h and r. The first term in the expression (3) corresponds to the direct Coulomb attraction between the donor and electron, while the second term, represented by the sum over l, describes an interaction between the ion image and electron. This term disappears when ɛ_{s} and ɛ_{d} become equal.
Notice that U_{0} (r) and V_{sp} (r) are isotropic functions independent of the electron positionvector direction. On the contrary, the electron–ion interaction strongly depends on the positional relationship of the ion and electron. As a consequence, the direction of r influences the magnitude of V_{ie}.
In order to determine the electron states we have to solve the singleparticle Schrödingerlike equation for the envelope functions and the electron energy E:
Here, H_{ij} is the matrix Hamiltonian operator for bulk Si, and the Einstein convention has been applied for summing overj. It is well known that energy minima in the conduction band of silicon are located nearby Xpoints symmetrically relative to the boundary of the Brillouin zone. At the Xpoint energy branches intersect, which leads to the double degeneracy. Since there are three physically nonequivalent Xpoints in the conduction band, the spectrum is sixfold degenerate on the whole (without spin) as was already mentioned above.
Frequently, when analyzing electron phenomena in silicon, the model of parabolic energy band with longitudinal m_{l} = 0.92m_{0} and transverse m_{t} = 0.19m_{0} effective masses is used. However, such a representation is correct for electron energies obeying the inequality where E_{X} and E_{Δ} are, respectively, the energies of the Xpoint and the point of the energy minimum located at the Δdirection. It is not so in our case. Due to the strong quantum confinement, typical electron energies are of the order of, or even greater than [2628], the energy difference E_{X} − E_{Δ} = 0.115 eV [29]. Therefore, interplay between the two crossing branches must be taken into account. This requires more accurate consideration of the electronic dispersion low, outgoing the frames of parabolic approximation.
This has been done by Kopylov [29] for bulk semiconductors. We use here for the quantum dot the Kopylov’s Hamiltonian operator written in a basis of two Bloch states or for each of three nonequivalent Xpoints in the Brillouin zone. All the Bloch functions belong to the spinless irreducible representation X_{1} of an Xpoint. Let us consider, for definiteness, the Xpoint along the direction (0, 0, 1). Then the wave function is expanded as where and are slow envelope functions being the expansion coefficients in the Blochstate basis The bulk Hamiltonian operator may be written as the sum of isotropic and anisotropic parts: where the former is represented by with the effective electron mass m_{e} = 3m_{l}m_{t} /(2m_{l} + m_{t}). Such the explicit form of is obtained as the average of H_{ij} over angles in the pspace. The anisotropic part is defined with the following expression:
Here is the distance from the Xpoint to any of the two nearest energy minima in the pspace, a_{0} = 0.543 nm stands for the lattice constant of silicon. The quasimomentum p and the energy E have the origin at the Xpoint.
Equation 4 for an undoped nanocrystal has been already solved earlier [26]. In the following we shall employ, in fact, the solutions obtained in Ref. [26] as the zeroth approximation of the problem with a doped dot.
Because of the isotropic and diagonal form of the operator it is possible to classify its eigenstates similarly to atomic systems as the states of s, p, dtype, etc. Accordingly, one may expand the envelope functions over these eigenstates as:
where denote the s, p, d,... states, and C_{jα} are the expansion coefficients. As was shown in Ref. [26], in order to find energies and wave functions of a few lower states with an accuracy of about several percent, it is sufficient to keep in the expansion (6) only s and pstates, so that becomes equal to with a = xy,z.
Substitution of the expansion (6) into Eq. 4 yields algebraic equations for C_{jα}:
Here and are the energies of the s and pstates, μ = 4.4934 is the first root of the spherical Bessel function j_{1}(x). Explicit form of the matrix elements of the operators and can be found, e.g., in Ref. [10]. The energy E_{s} is doubly degenerate while the energy E_{p} is the sixfold level. Thus, solving Eq. 7 we should obtain, in general case, eight electron states.
Within the restricted basis of s and ptype envelope states, which is used here, one can solve Eq. 7 analytically [26] if neglect s−p_{a} Coulomb matrix elements and anisotropic components of p_{a}−p_{b} type both for a = b and a ≠ b. Notice that the diagonal p_{a}−p_{a} Coulomb matrix elements consist of two parts [10]: isotropic V_{pp}(h), and anisotropic As our numerical estimations show, the latter is much less than the former. For comparison we have plotted in Fig. 1 for 3 nm quantum dot both the anisotropic Coulomb matrix elements and isotropic ones V_{pp}(h) versus h, in case the anisotropic matrix elements have their highest possible values. For instance, V_{z} and V_{zz} achieve their maximum when h_{x} = h_{y} = 0. On the contrary, V_{xy} has the greatest value for h_{z} = 0, h_{x} = h_{y}. As is seen in the figure, the anisotropic elements V_{xy} and V_{zz} are small compared to V_{z}. In turn, V_{z} is less than the diagonal isotropic matrix elements V_{ss} and V_{pp}. At last, all the Coulomb matrix elements are substantially smaller than the difference E_{p} − E_{s}.
Figure 1. Coulomb matrix elements versus dimensionless donor displacement from the dot center. Lower solid line— V_{ss}; Upper solid line— V_{pp}; Longdashed line— V_{z} for h_{x} = h_{y} = 0; Shortdashed line— V_{zz} for h_{x} = h_{y} = 0; Dotted line— V_{xy} for h_{x} = h_{y}, h_{z} = 0
Results and Discussion
Solution of the simplified Eq. 7 for has been obtained in Ref. [26]. There was shown that the twofold (stype) and sixfold (ptype) levels split into four doubly degenerate levels due to the band anisotropy leading to the s−p_{z} and p_{x}−p_{y} hybridization of the envelope states. The s−s and p−p diagonal Coulomb matrix elements in Eq. 7 contribute only to the shift of the unperturbed energy values: E_{s}(h) = E_{s} + V_{ss}(h), E_{p}(h) = E_{p} + V_{pp}(h). As a result, the twofold energies are written in the form:
where the matrix elements of the band anisotropy and have been introduced.
The energy E_{3e} of the third excited doublet turns out to be strongly split off from the lower energies as is shown in Fig. 2. In the following we do not take this level into account because the twolevel approximation, accepted in Eq. 7, is explicitly insufficient to describe correctly the upper electron states and their energies. The energies of the three lower levels E_{0e}, E_{1e}, and E_{2e} are also presented in Fig. 2 as functions of the dot radiusR. One can see that, the level splitting due to the band anisotropy is great enough. The energy of the splitting turns out to be of the same order as the unperturbed energies E_{s} and E_{p}. However, in spite of such the strong splitting, the double degeneracy of all the levels is conserved. In order to lift it, the symmetry of the system must be violated. To this goal, one needs to introduce nonzero matrix elements in Eq. 7, which reflect an asymmetry of the donor position inside the nanocrystal. At the same time, one may neglect, apparently, the terms and in Eq. 7 because of their small magnitudes, see Fig. 1.
Figure 2. Energies of the “isotropic” model (V_{a} = 0 in Eq. 7) as functions of the dot radius. From top to bottom: E_{3e}—dots; E_{2e}—short dash; E_{1e}—long dash; E_{0e}—solid line. All the energies are counted from the Xpoint energy
The presence of nonzero hampers solving Eq. 7. However, relative smallness of the offdiagonal Coulomb interaction with respect to the energies E_{je} allows one to apply a perturbation theory. It is important to emphasize that only the offdiagonal matrix elements are treated as perturbation in this case, but not the Coulomb interaction on the whole.
An introduction of an asymmetry in the system leads to the total splitting of the energy levels. In particular, the lowest level splits into two energies E_{z}^{(±)} equal to E_{0e} − S_{z} ± W_{z}. Generally speaking, it is now possible to combine the results for all the three Xpoints and write down the energies of all the six lowest levels originated from the energy E_{0e} in the following form:
Here
is the secondorder shift of the energy E_{0e}, and the term W_{a}, defining the splitting, is
This term leads to symmetric splitting of the energy E_{0e} − S_{a} into two levels. Indices a, b, c enumerate the spatial axes. In Eqs. 10, 11 they do not coincide with each other. The notation “E_{a}^{(±)}” implies the solution obtained for the Xpoint located at adirection in the kspace. At last, the angle λ is defined by the following relationships:
Notice that the level splitting appears only in the second order in Contrary to this, wave functions, corresponding to the energies E_{a}^{(±)}, have the firstorder corrections in
Here, the capital letter A runs the values X, Y, Z and defines the Bloch state of an Xpoint. It is important to note that index A describes the Bloch function of the Xpoint situated precisely at the k_{a}axis in the Brillouin zone, i.e., in a certain sense, the small and big indices coincide.
The groundlevel splitting is shown in Fig. 3. If the impurity positionvectorhhas not directed to any symmetric axis of the lattice, the energy splitting leads to the complete degeneracy removal except for the spin degeneracy. Conformably, the groundstate energy splits into the six different levels as is seen in Fig. 3.
Figure 3. Fine structure of the energy spectrum at h_{x}/h = 0.8, h_{y}/h = 0.5, h_{z}/h = 0.33 with respect to the unperturbed sixfold degenerate energy level E_{0e}. Solid lines— E_{z}^{(±)}; Dashed lines— E_{y}^{(±)}; Dots— E_{x}^{(±)}; The “+”sign corresponds to the upper curves
What energy level of the six ones written in the Eq. 9is the lowest? It depends on the relationship between h_{x}, h_{y}, and h_{z} which define the offdiagonal s−p type matrix elements For example, in case h_{x} > h_{y} > h_{z} > 0 (this is the case shown in Fig. 3) the energy becomes the lowest. Evidently, there are six different groups of relationships between the componentsh_{a}(each group contains eight relationships), every one of which defines new ground state from the set (13). Each of the 48 relationships describes the spherical sector in space of the impurity positionvector within the quantum dot. In a certain sense one may say that belonging of the vector h to one of these sectors fixes one of the states (13) to be the ground.
As is seen in the figure, the splitting of each the twofold level E_{0e}, corresponding to a certain Xpoint, is sufficiently great except for the cases when the donor is situated near the dot center or the interface. For intermediate values of the ratio h/R, the splitting caused by the system asymmetry achieves several tens (or, even, hundred) of meV. At least for lithium, this is expected to be considerably greater than the valley–orbit splitting. As has been shown earlier [10,11] the valley–orbit splitting in quantum dots sharply decreases if h → R, and equals zero at the dot boundary. At the same time, at h → 0, it has some nonzero value. Therefore, the curves presented in Fig. 3 should be slightly corrected at small h/R. However, such a correction does not exceed, apparently, several meV for 3 nm nanocrystal doped with Li, and may be neglected.
Let us now discuss the modification of the ground electron state due to the hidrogenlike donor. Because of the donor existence inside the dot, the groundstate wave function acquires some firstorder correction that can be represented as the product of two factors. The first one is the correction to the envelopefunction
while the second factor is the Bloch function of the irreducible representation It is convenient to introduce the unit vector along the donor’s positionvector. Then the donor site can be defined with n_{x}, n_{y}, n_{z}, and h. To illustrate the role of donors in a reconstruction of the electron wave function and, as a consequence, in a redistribution of the electron density, we choose n_{x} = 0.8, n_{y} = 0.5, n_{z} = 0.33 and plot the envelopefunction correction as a function of the electron position (i) at the radial axis and (ii) on the sphere r = h, see Figs. 4 and 5, respectively.
Figure 4 represents the dependence of (dashed and dotted lines) on the electron position at the axis drawn through the donor and the dot center. In this case electron positionvector r is strictly parallel or antiparallel to h. For comparison, the zerothorder envelope function of stype has been also plotted in the figure with solid line.
Figure 4. The firstorder corrections (dashed and dotted lines) to the envelope function of the stype (solid line) in arbitrary units. n_{x} = 0.8, n_{y} = 0.5, n_{z} = 0.33.e = n. R = 1.5 nm
Figure 5. Contour plot of the firstorder correction at n_{x} = 0.8, n_{y} = 0.5, n_{z} = 0.33 for 3 nm quantum dot. The value of rises from dark to light. The cross indicates the donor position
Since we direct the radial axis parallel to h, the donor is always situated somewhere at the right half of this axis within the range 0 < h/R < 1. We have calculated for three different positions of the donor ion inside the dot. When the donor is close to the dot center (h/R = 0.1), the firstorder correction is small enough, as was already pointed out earlier. At h/R = 0.46, the correction becomes the greatest. Further increase of h/R leads to the general reduce of Such the behavior of has the simple explanation. The firstorder correction is directly proportional to the offdiagonal Coulomb matrix elements of s−p type, see Eq. 14. Meanwhile, these matrix elements rise from zero at h = 0 to their maximum taking place exactly at h/R = 0.46. Then, V_{a} decreases as h increases, as it is shown in Fig. 1. Thus, the correction qualitatively follows, in fact, the dependence V_{a} on h.
It is also seen in Fig. 4 that the maximum of all three curves takes place approximately at r/R = 0.46 and does not depend on h. The latter is a consequence of the “twolevel” approximation accepted in Eq. 7. If we take into account not only the lowest s and pstates, the dependence of the firstorder correction on h appears at once. Nevertheless, such the rough approximation turns out to be quite correct and sufficient to describe the general trend in behavior of as a function of r. In particular, is always positive when i.e., the donor and electron are situated at the same half of the axis. This means that the probability to find the electron near the donor site rises, while on the other side relative to the dot center the probability reduces. Thus, the electrondensity distribution becomes asymmetric. It rises along the vectorh, and reduces along the opposite direction.
We have also plotted in Fig. 5 as a function of the angles θ and φ on the spherical surface r = h for the former values of n_{a}. The angles θ and φ are introduced in the standard form: e_{x} = sinθcosφ, e_{y} = sinθsinφ, and e_{z} = cosθ, where e_{a} is an acomponent of the unit vector Because the envelope functions and Coulomb matrix elements are directly proportional to e_{a} and n_{a}, respectively, the angle dependence of should be sensitive to the donor position on the sphere. This is completely confirmed by our calculations presented in the figure. As is seen, maximal values of (light areas in the figure) are located around the donor site marked with the cross. However, it is also seen that the cross does not fully coincide with the center of the brightest spot. The nature of this discrepancy, apparently, may be explained by the use of the “twolevel” approximation as well.
Conclusion
Let us now briefly describe the obtained results. First, we have found analytical expressions for the electron energies and wave functions in case of arbitrary donor position inside the quantum dot. Note for comparison that more general treatment [10,11], taking into account dominant role of the centralcell potential, permits of only numerical calculations. Second, it has been shown that, the wave functions in Lidoped nanocrystals have already no the symmetry of tetrahedral group T_{d}, or close to that, as it took place for Vgroup donors in the bulk, or nanocrystals, respectively, even in the case of asymmetric donor position inside the nanocrystal [10,11]. This is due to disappearance of the shortrange Coulomb field that symmetrizes the Bloch functions according to the symmetry transformations of the point group T_{d}. Third, energy splitting for a hydrogenlike lithium donor essentially differs from that for Vgroup donors creating the centralcell field. Provided that the valley–orbit interaction is taken into account, the splitting occurs even for the case of central located donor inside the dot. On the contrary, if we deal with the lithium donor, the splitting is absent in case h = 0 as is seen in Fig. 3. At last, fourth, the presence of a donor inside the nanocrystal leads to the substantial relocation of the electron density (up to 10%, see Fig. 4) towards the donor. This, in turn, leads to the reconstruction of the electron wave functions and subsequent polarization of the electron subsystem in the dot. Such the polarization, undoubtedly, should influence the values of electron–photon matrix elements and the transition probabilities on the whole.
Acknowledgments
The authors thank the Russian Ministry of Education and Science, and the Russian Foundation for Basic Research for the financial support of this work through the program “Development of a scientific potential of the high school” (the project No 2.1.1.2363), and the project No 050216762, respectively.
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