Fine Splitting of Electron States in Silicon Nanocrystal with a Hydrogen-like Shallow Donor

  • Vladimir A Belyakov1Email author and

    Affiliated with

    • Vladimir A Burdov1

      Affiliated with

      Nanoscale Research Letters20072:569

      DOI: 10.1007/s11671-007-9101-5

      Received: 7 July 2007

      Accepted: 16 October 2007

      Published: 2 November 2007

      Abstract

      Electron structure of a silicon quantum dot doped with a shallow hydrogen-like donor has been calculated for the electron states above the optical gap. Within the framework of the envelope-function 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 splitting

      Introduction

      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 V-group shallow donors have been carried out earlier. The microscopic first-principles study of charge distribution and electrostatic fields in bulk silicon [1921] and silicon crystallites [1618] in the presence of V-group donors have shown existence of short-range and long-range components of the electron–ion Coulomb interaction in the system. The short-range potential differs from zero only in a nearest vicinity (about Bohr radius) of the donor nucleus. This extra potential, frequently named a “central-cell correction”, leads to so-called 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 long-range component represents standard ɛ-times 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 central-located 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 one-two order less than that for V-group 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 central-cell effect for Li will be weakened compared to the V-group donors by the same order of magnitude. As was shown for 2–5 nm nanocrystals [10, 11], maximal values of the splitting for the V-group 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 long-range hydrogenic potential turns out to be considerably greater (about 100 meV). This indicates a key role of the hydrogenic potential compared to the central-cell one for Li. On this reason, the hydrogenlike model may be applied to a lithium donor in a silicon nanocrystal. Therefore, when dealing with Li-doped dots, we shall neglect the short-range central-cell field and, consequently, the valley–orbit splitting.

      In turn, for V-group donors in quantum dots, a contribution of the long-range field to the energy splitting is 3–5 times less than that of the short-range field. In particular, in the works of the authors [10, 11] electronic structure of silicon crystallites doped with V-group donors has been already calculated in the presence of both short- and long-range Coulomb interactions. Note, however, that the results of Refs [10, 11] may not be automatically extended to Li-doped nanocrystals by zero setting the terms caused by the short-range field. For V-group donors the central-cell 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 hydrogen-like potential leads to some insignificant corrections only. In case of a lithium donor, the situation is opposite. Precisely the hydrogen-like 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 envelope-function approximation. Of course, an applicability of the http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq1_HTML.gif 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 first-principles 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 wide-band matrix such as SiO2. 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:
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ1_HTML.gif
      (1)
      Here r and h are the position-vectors 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 self-polarization term. It can be represented in the form, see, e.g., Eqs (3.24), (3.26) in Ref. [25]:
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ2_HTML.gif
      (2)
      At last, the third term http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq2_HTML.gif introduced in the Eq. 1, represents an electron–ion interaction. It has the form [25]:
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ3_HTML.gif
      (3)

      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 position-vector 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 single-particle Schrödinger-like equation for the envelope functions http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq3_HTML.gif and the electron energy E:
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ4_HTML.gif
      (4)

      Here, H ij is the matrix http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq4_HTML.gif 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 X-points symmetrically relative to the boundary of the Brillouin zone. At the X-point energy branches intersect, which leads to the double degeneracy. Since there are three physically nonequivalent X-points 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 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq5_HTML.gif where E X and E Δ are, respectively, the energies of the X-point and the point of the energy minimum located at the Δ-direction. 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 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq6_HTML.gif Hamiltonian operator written in a basis of two Bloch states http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq7_HTML.gif http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq8_HTML.gif or http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq9_HTML.gif for each of three nonequivalent X-points in the Brillouin zone. All the Bloch functions belong to the spinless irreducible representation X 1 of an X-point. Let us consider, for definiteness, the X-point along the direction (0, 0, 1). Then the wave function is expanded as http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq10_HTML.gif where http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq11_HTML.gif and http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq12_HTML.gif are slow envelope functions being the expansion coefficients in the Bloch-state basis http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq13_HTML.gif The bulk http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq14_HTML.gif Hamiltonian operator may be written as the sum of isotropic and anisotropic parts: http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq15_HTML.gif where the former is represented by http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq16_HTML.gif with the effective electron mass m e = 3m l m t /(2m l + m t ). Such the explicit form of http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq17_HTML.gif is obtained as the average of H ij over angles in the p-space. The anisotropic part is defined with the following expression:
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ5_HTML.gif
      (5)

      Here http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq18_HTML.gif is the distance from the X-point to any of the two nearest energy minima in the p-space, a 0 = 0.543 nm stands for the lattice constant of silicon. The quasimomentum p and the energy E have the origin at the X-point.

      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 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq19_HTML.gif it is possible to classify its eigenstates similarly to atomic systems as the states of s-, p-, d-type, etc. Accordingly, one may expand the envelope functions over these eigenstates as:
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ6_HTML.gif
      (6)

      where http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq20_HTML.gif denote the s-, p-, d-,... states, and C j α 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 p-states, so that http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq21_HTML.gif becomes equal to http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq22_HTML.gif with a = x y,z.

      Substitution of the expansion (6) into Eq. 4 yields algebraic equations for C j α:
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ7_HTML.gif
      (7)

      Here http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq23_HTML.gif and http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq24_HTML.gif are the energies of the s- and p-states, μ = 4.4934 is the first root of the spherical Bessel function j 1(x). Explicit form of the matrix elements of the operators http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq26_HTML.gif 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 p-type envelope states, which is used here, one can solve Eq. 7 analytically [26] if neglect s−p a Coulomb matrix elements http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq27_HTML.gif and anisotropic components of p a −p b type http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq28_HTML.gif both for a = b and ab. Notice that the diagonal p a −p a Coulomb matrix elements consist of two parts [10]: isotropic V pp (h), and anisotropic http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq29_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq30_HTML.gif and isotropic ones http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq31_HTML.gif 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 .
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Fig1_HTML.jpg
      Figure 1

      Coulomb matrix elements versus dimensionless donor displacement from the dot center. Lower solid line— V ss ; Upper solid line— V pp ; Long-dashed line— V z for h x = h y = 0; Short-dashed 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 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq32_HTML.gif has been obtained in Ref. [26]. There was shown that the twofold (s-type) and sixfold (p-type) 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:
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ8_HTML.gif
      (8)

      where the matrix elements of the band anisotropy http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq33_HTML.gif http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq35_HTML.gif 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 two-level 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 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq36_HTML.gif in Eq. 7, which reflect an asymmetry of the donor position inside the nanocrystal. At the same time, one may neglect, apparently, the terms http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq37_HTML.gif and http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq38_HTML.gif in Eq. 7 because of their small magnitudes, see Fig. 1.
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Fig2_HTML.jpg
      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 X-point energy

      The presence of nonzero http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq39_HTML.gif hampers solving Eq. 7. However, relative smallness of the off-diagonal Coulomb interaction with respect to the energies E je allows one to apply a perturbation theory. It is important to emphasize that only the off-diagonal matrix elements http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq40_HTML.gif are treated as perturbation in this case, but not the Coulomb interaction http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq41_HTML.gif 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 X-points and write down the energies of all the six lowest levels originated from the energy E 0e in the following form:
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ9_HTML.gif
      (9)
      Here
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ10_HTML.gif
      (10)
      is the second-order shift of the energy E 0e , and the term W a , defining the splitting, is
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ11_HTML.gif
      (11)
      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 X-point located at a-direction in the k-space. At last, the angle λ is defined by the following relationships:
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ12_HTML.gif
      (12)
      Notice that the level splitting appears only in the second order in http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq42_HTML.gif Contrary to this, wave functions, corresponding to the energies E a (±), have the first-order corrections in http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq43_HTML.gif
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ13_HTML.gif
      (13)

      Here, the capital letter A runs the values X, Y, Z and defines the Bloch state of an X-point. It is important to note that index A describes the Bloch function of the X-point situated precisely at the k a -axis in the Brillouin zone, i.e., in a certain sense, the small and big indices coincide.

      The ground-level splitting is shown in Fig. 3. If the impurity position-vectorh has 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 ground-state energy splits into the six different levels as is seen in Fig. 3.
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Fig3_HTML.jpg
      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 off-diagonal s−p type matrix elements http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq44_HTML.gif For example, in case h x > h y > h z > 0 (this is the case shown in Fig. 3) the energy http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq45_HTML.gif 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 position-vector 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 X-point, 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 hR, 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 hidrogen-like donor. Because of the donor existence inside the dot, the ground-state wave function http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq46_HTML.gif acquires some first-order correction that can be represented as the product of two factors. The first one is the correction to the envelope-function
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Equ14_HTML.gif
      (14)

      while the second factor is the Bloch function http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq47_HTML.gif of the irreducible representation http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq48_HTML.gif It is convenient to introduce the unit vector http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq49_HTML.gif along the donor’s position-vector. 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 envelope-function correction http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq50_HTML.gif as a function of the electron position (i) at the radial axis http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq51_HTML.gif and (ii) on the sphere r = h, see Figs. 4 and 5, respectively.

      Figure 4 represents the dependence of http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq52_HTML.gif (dashed and dotted lines) on the electron position at the axis drawn through the donor and the dot center. In this case electron position-vector r is strictly parallel or antiparallel to h. For comparison, the zeroth-order envelope function of s-type has been also plotted in the figure with solid line.
      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Fig4_HTML.jpg
      Figure 4

      The first-order corrections (dashed and dotted lines) to the envelope function of the s-type (solid line) in arbitrary units. n x = 0.8, n y = 0.5, n z = 0.33.e = n. R = 1.5 nm

      http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_Fig5_HTML.jpg
      Figure 5

      Contour plot of the first-order correction http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq53_HTML.gif at n x = 0.8, n y = 0.5, n z = 0.33 for 3 nm quantum dot. The value of http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq54_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq55_HTML.gif 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 first-order 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 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq56_HTML.gif Such the behavior of http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq57_HTML.gif has the simple explanation. The first-order correction http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq58_HTML.gif is directly proportional to the off-diagonal 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 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq59_HTML.gif 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 “two-level” approximation accepted in Eq. 7. If we take into account not only the lowest s- and p-states, the dependence of the first-order correction http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq60_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq61_HTML.gif as a function of r. In particular, http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq62_HTML.gif is always positive when http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq63_HTML.gif 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 electron-density distribution becomes asymmetric. It rises along the vectorh, and reduces along the opposite direction.

      We have also plotted in Fig. 5 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq64_HTML.gif 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 a-component of the unit vector http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq65_HTML.gif Because the envelope functions http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq66_HTML.gif and Coulomb matrix elements http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq67_HTML.gif are directly proportional to e a and n a , respectively, the angle dependence of http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq68_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1007%2Fs11671-007-9101-5/MediaObjects/11671_2007_Article_9101_IEq69_HTML.gif (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 “two-level” 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 central-cell potential, permits of only numerical calculations. Second, it has been shown that, the wave functions in Li-doped nanocrystals have already no the symmetry of tetrahedral group T d , or close to that, as it took place for V-group 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 short-range 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 V-group donors creating the central-cell 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.

      Declarations

      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 05-02-16762, respectively.

      Authors’ Affiliations

      (1)
      Department of Theoretical Physics, University of Nizhniy Novgorod

      References

      1. Fujii M, Mimura A, Hayashi S, Yamamoto K: Appl. Phys. Lett.. 1999, 75: 184. COI number [1:CAS:528:DyaK1MXktFyjtbs%3D] COI number [1:CAS:528:DyaK1MXktFyjtbs%3D] 10.1063/1.124313View Article
      2. Mimura A, Fujii M, Hayashi S, Kovalev D, Koch F: Phys. Rev. B. 2000, 62: 12625. COI number [1:CAS:528:DC%2BD3cXnvFanurc%3D] COI number [1:CAS:528:DC%2BD3cXnvFanurc%3D] 10.1103/PhysRevB.62.12625View Article
      3. Fujii M, Yamaguchi Y, Takase Y, Ninomiya K, Hayashi S: Appl. Phys. Lett.. 2004, 85: 1158. COI number [1:CAS:528:DC%2BD2cXms1GrtL4%3D] COI number [1:CAS:528:DC%2BD2cXms1GrtL4%3D] 10.1063/1.1779955View Article
      4. Tetelbaum DI, Trushin SA, Burdov VA, Golovanov AI, Revin DG, Gaponova DM: Nucl. Instr. Meth. B. 2001, 174: 123. COI number [1:CAS:528:DC%2BD3MXhtlWgsLo%3D] COI number [1:CAS:528:DC%2BD3MXhtlWgsLo%3D] 10.1016/S0168-583X(00)00457-2View Article
      5. Kachurin GA, Cherkova SG, Volodin VA, Kesler VG, Gunakovsky AK, Cherkov AG, Bublikov AV, Tetelbaum DI: Nucl. Instr. Meth. B. 2004, 222: 497. COI number [1:CAS:528:DC%2BD2cXlslyqs7o%3D] COI number [1:CAS:528:DC%2BD2cXlslyqs7o%3D] 10.1016/j.nimb.2004.03.076View Article
      6. Hada Y, Eto M: Phys. Rev. B. 2003, 68: 155322. 10.1103/PhysRevB.68.155322View Article
      7. Melnikov DV, Chelikowsky JR: Phys. Rev. Lett.. 2004, 92: 046802. 10.1103/PhysRevLett.92.046802View Article
      8. Zhou Z, Steigerwald ML, Friesner RA, Brus L, Hybertsen MS: Phys. Rev. B. 2005, 71: 245308. 10.1103/PhysRevB.71.245308View Article
      9. Xu Q, Luo J-W, Li S-S, Xia J-B, Li J, Wei S-H: Phys. Rev. B. 2007, 75: 235304. 10.1103/PhysRevB.75.235304View Article
      10. Belyakov VA, Burdov VA: Phys. Rev. B. 2007, 76: 045335. 10.1103/PhysRevB.76.045335View Article
      11. Belyakov VA, Burdov VA: Phys. Lett. A. 2007, 367: 128. COI number [1:CAS:528:DC%2BD2sXntl2qu7g%3D] COI number [1:CAS:528:DC%2BD2sXntl2qu7g%3D] 10.1016/j.physleta.2007.02.065View Article
      12. Tsu R, Babic D: Appl. Phys. Lett.. 1994, 64: 1806. COI number [1:CAS:528:DyaK2cXisFymu7Y%3D] COI number [1:CAS:528:DyaK2cXisFymu7Y%3D] 10.1063/1.111788View Article
      13. Lannoo M, Delerue C, Allan G: Phys. Rev. Lett.. 1995, 74: 3415. COI number [1:CAS:528:DyaK2MXlt1Wrt7w%3D] COI number [1:CAS:528:DyaK2MXlt1Wrt7w%3D] 10.1103/PhysRevLett.74.3415View Article
      14. Tsu R, Babic D, Ioriatti L: J. Appl. Phys.. 1997, 82: 1327. COI number [1:CAS:528:DyaK2sXkvFCmsLc%3D] COI number [1:CAS:528:DyaK2sXkvFCmsLc%3D] 10.1063/1.365762View Article
      15. Ogut S, Burdick R, Saad Y, Chelikowsky JR: Phys. Rev. Lett.. 2003, 90: 127401. 10.1103/PhysRevLett.90.127401View Article
      16. Delerue C, Lannoo M, Allan G: Phys. Rev. B. 2003, 68: 115411. 10.1103/PhysRevB.68.115411View Article
      17. Cartoixa X, Wang L-W: Phys. Rev. Lett.. 2005, 94: 236804. 10.1103/PhysRevLett.94.236804View Article
      18. Trani F, Ninno D, Cantele G, Iadonisi G, Hameeuw K, Degoli E, Ossicini S: Phys. Rev. B. 2006, 73: 245430. 10.1103/PhysRevB.73.245430View Article
      19. Pantelides ST, Sah CT: Phys. Rev. B. 1974, 10: 621. COI number [1:CAS:528:DyaE2cXlt1ymtro%3D] COI number [1:CAS:528:DyaE2cXlt1ymtro%3D] 10.1103/PhysRevB.10.621View Article
      20. Bassani F, Iadonisi G, Preziosi B: Rep. Prog. Phys.. 1974, 37: 1099. COI number [1:CAS:528:DyaE2MXms1Omug%3D%3D] COI number [1:CAS:528:DyaE2MXms1Omug%3D%3D] 10.1088/0034-4885/37/9/001View Article
      21. Pantelides ST: Rev. Mod. Phys.. 1978, 50: 797. COI number [1:CAS:528:DyaE1MXjt1equw%3D%3D] COI number [1:CAS:528:DyaE1MXjt1equw%3D%3D] 10.1103/RevModPhys.50.797View Article
      22. Kohn W, Luttinger JM: Phys. Rev.. 1955, 97: 1721. COI number [1:CAS:528:DyaG2MXktFOktQ%3D%3D] COI number [1:CAS:528:DyaG2MXktFOktQ%3D%3D] 10.1103/PhysRev.97.1721View Article
      23. Kohn W, Luttinger JM: Phys. Rev.. 1955, 98: 915. COI number [1:CAS:528:DyaG2MXlvFejuw%3D%3D] COI number [1:CAS:528:DyaG2MXlvFejuw%3D%3D] 10.1103/PhysRev.98.915View Article
      24. Faulkner RA: Phys. Rev.. 1969, 184: 713. COI number [1:CAS:528:DyaF1MXltFOmt7c%3D] COI number [1:CAS:528:DyaF1MXltFOmt7c%3D] 10.1103/PhysRev.184.713View Article
      25. Delerue C, Lannoo M: Nanostructures. Theory and Modelling.. Springer-Verlag, Berlin; 2004.View Article
      26. V.A. Burdov, Zh. Eksp. Teor. Fiz. 121, 480 (2002) [JETP 94, 411 (2002)]
      27. Feng DH, Xu ZZ, Jia TQ, Li XX, Gong SQ: Phys. Rev. B. 2003, 68: 035334. 10.1103/PhysRevB.68.035334View Article
      28. A.S. Moskalenko, I.N. Yassievich, Fiz. Tverd. Tela (St. Petersburg) 46, 1465 (2004); Phys. Solid State 46, 1508 (2004)
      29. A.A. Kopylov, Fiz. Tekh. Poluprovodn. (Leningrad) 16, 2141 (1982); Sov. Phys. Semicond. 16, 1380 (1982)

      Copyright

      © to the authors 2007