Introduction

Colloidal semiconductor nanocrystals, also known as quantum dots, have unique optical and electronic properties that are controlled by their size, shape and chemical composition1,2,3,4,5. Doping of nanocrystals with impurity atoms or ions is another method of tuning their physical properties, giving rise to many useful optical phenomena6,7,8,9. The presence of various impurities in semiconductor nanocrystals is responsible e.g. for the Stark effect10, the spin Hall effect11, the Kondo effect12, giant Zeeman splitting13 and light-induced spontaneous magnetization14. This provides the basis for applications of semiconductor nanocrystals in nanodevices such as transistors15,16, light-emitting diodes17, bioimaging labels18, chemical sensors19, solar cells20 and spintronic devices21.

There are several methods of doping nanocrystals based on two different fabrication strategies: nucleation doping and growth doping22,23. Nucleation doping is realized by mixing the dopant and host precursor during nanocrystal formation24, whereas the growth-doping strategy requires absorption of dopant precursors by the nanocrystal surface, followed by encapsulation of the precursors through overgrowth of an isocrystalline or heterocrystalline shell. The radial position of impurities inside a semiconductor nanocrystal can be controlled by adding the precursors at different stages of the nanocrystal formation25. This control can be performed with angstrom precision, enabling extremely fine tuning of the optical and magnetic properties of the nanocrystals26.

Nanocrystal impurities are studied theoretically using two major approaches, based on the effective mass approximation. The first approach, relying on an exact solution of the Schrödinger equation for a given confining potential, yields the impurity's wave functions and energy spectrum in quadratures27,28,29. The second technique, employing the variational method, allows one to find an approximate solution to the Schrödinger equation as a linear sum of the exact wave functions corresponding to the nanocrystals with and without an impurity30,31. Most of the theoretical work based on the effective mass approximation is devoted to studies of size dependencies of the energy levels and binding energies of impurity states in simple nanocrystals27,29,32. A few were focused on modifications of energy spectrum by external electric and magnetic fields33,34, multilayer nanocrystals with different numbers of shells35,36 and nanocrystals with off-center impurities37,38.

Since most of the optical properties of doped nanostructures strongly depend on their size, the study of the size dependencies of nanostructure impurity levels is a problem of both fundamental and applied significance. Precise, yet rather complex and time-consuming methods of single quantum-dot spectroscopy39,40,41 allows one to resolve this problem experimentally. In this paper, we theoretically show that the simultaneous exposure of charge carriers to the Coulomb and confining potentials results in two different contributions to the size dependence of the nanocrystal energy spectrum. The confining potential ceases to determine the structure of the spectrum with an increase of the nanocrystal size while the Coulomb potential starts to dominate. As a consequence, the energies of some impurity states can accidentally coincide in nanocrystals of a particular size. By examining the ordering of the energy levels for two limiting cases of small and large spherical nanocrystals, we show that this accidental degeneracy is inherent to doped nanocrystals regardless of their shape and composition. The degeneracy is removed by any interaction between the degenerate states, leading to the anticrossings in the size dependencies of the respective energy levels. Using an illustrative example of electron–phonon interaction, we show how the accidental degeneracy is removed and identify which anticrossings are easiest to interpret experimentally. We also discuss which anticrossings are convenient to use for the estimation of interaction strengths in semiconductor nanocrystals.

Results

Energy spectrum of hydrogenic impurity

Consider a donor hydrogenic impurity in a spherical semiconductor nanocrystal of radius R, as shown in Fig. 1(a). The impurity comprises a positive ion of charge Ze, which is located at the nanocrystal center and coupled through the Coulomb interaction to an electron of charge −e and effective mass me. By assuming that the nanocrystal boundary is impenetrable to the electron, one can find the wave functions and energy spectrum of the impurity from the Schrödinger equation

in which the confining potential is given by

with α = Ze2/ε0 and ε0 being the low-frequency permittivity of the nanocrystal.

Figure 1
figure 1

(a) Donor impurity at the center of spherical semiconductor nanocrystal of radius R; energy spectra of (b) donor impurity for R = 9 rB, (c) infinitely high spherical potential well without an impurity and (d) hydrogen-like atom; and size dependencies of energies of (e) first excited and (f) higher impurity states. Dash-dotted curve in (e) is the energy of the lowest state in the potential well.

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Owing to the spherical symmetry of the confining potential, Eq. (1) allows an analytical solution27,28,29 (see Methods), which enables a simple analysis of the absorption and photoluminescence spectra of nanocrystals with donor impurities42,43,44,45,46. It is possible, in particular, to estimate the nanocrystal size by finding the number of absorption peaks lying below the fundamental absorption band. The peaks are centered at the energies of the lowest states in the nanocrystal energy spectrum En,l, corresponding to different principal quantum numbers and angular momenta, n = 1, 2, 3, … and l = 0, 1, 2, … A more practical application is the problem of calculating the size of a nanocrystal with a certain number of impurity states within the band gap. For instance, a nanocrystal of radius R = 9 rB, where rB = ħ2/(meα) is the effective Bohr radius, has five impurity states with negative energies E1,0, E1,1 and E2,0 (the state with l = 1 is three-fold degenerate), whereas the energies of the rest of its states are positive [see Fig. 1(b)].

To qualitatively understand how the energies and ordering of impurity states depend on the nanocrystal size, we note that the impurity spectrum transforms into the spectrum of an infinitely high potential well without an impurity when α = 0 and becomes similar to the hydrogen-like spectrum for . The ordering of energy levels in the first case, illustrated by Fig. 1(c), is determined by the zeros ξn,l of the spherical Bessel functions of the first kind jl(z). The hydrogen-like spectrum, shown in Fig. 1(d), consists of negative- and positive-energy domains47: the negative-energy spectrum , where Ry = meα2/(2ħ2) is the effective Rydberg, is discrete; and the positive-energy spectrum E(i)(k) = Ry k2 (k > 0) is continuous.

These limiting cases show that impurity centers should exhibit discrete energy spectra which may extend to negative energies. The negative-energy states are absent in small nanocrystals, emerging one by one and growing in numbers with the nanocrystal radius. The emergence is caused by the weakening of spatial confinement and the reduction of kinetic energy of the impurity electron and its resulting distancing from the ion29. The descent of quantum state (n, l) from the positive-energy domain to the negative-energy domain occurs when its energy passes through zero in the nanocrystal of radius

where ζn,2l+1 is the nth zero of the cylindrical Bessel function of the first kind J2l+1(x).

Figures 1(e) and 1(f) show how the energies of the lowest four impurity states vary with R. All the energies are seen to be positive for R < R1,0 ≈ 1.8 rB. The first state of negative energy appears in the spectrum when the nanocrystal radius exceeds R1,0. It is then followed by the higher-energy states with quantum numbers (1, 1), (2, 0), … and (3, 0), whose energies turn zero for R ≈ 5.1 rB, 6.2 rB, … and 12.9 rB. As the nanocrystal becomes larger, some of these states start forming multiplets due to the ‘accidental’ degeneracy of the Coulomb interaction47. An unlimited increase in R leads to a transformation of all orbitally nondegenerate impurity states {n, l} to hydrogen-like states {ν}, which are ν-fold degenerate in angular momentum, so that for R → ∞.

To find the number of negative-energy impurity states in the nanocrystal of radius R, one needs to determine the pair of quantum numbers (nR, lR) such that . Some algebra shows that there are

states of different energies within the interval −Ry ≤ En,l < 0. These states are (2l+1)-fold degenerate with respect to the angular momentum projections, making the total number of negative-energy states equal to (nR + lR)(nR + lR + 1)(2nR + 2lR + 1)/6 − (lR + 1)2. Figure 2(a) shows the staircase function N(R), the steps of which correspond to new negative-energy levels successively emerging in the impurity spectrum. The figure enables one to estimate the nanocrystal radius for a known number of negative-energy states, as well as to find the radius required for a desired number of states.

Figure 2
figure 2

(a) Number of negative-energy impurity states versus nanocrystal radius R and (b) variation of state ordering with R, illustrating formation of accidental-degeneracy points A, B, C, etc. in the impurity energy spectrum. Double arrows and underbraces show how the impurity spectrum En,l (shown for R > R4,0) transforms to the hydrogen-like spectrum when and to the spectrum of the infinite quantum well without an impurity when . Note that in an actual energy scale.

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Anticrossing of impurity energy levels

Comparison of the impurity spectrum with the spectrum of the nanocrystal without an ion reveals essentially different orderings of their states. This is seen from the order of the first ten levels of the two spectra shown in Fig. 2(b) for R > R4,0. By noticing that the ordering of levels coincides with that in small nanocrystals (with ) and recalling that the impurity states form multiplets in large nanocrystals (with ), we conclude that the energies of the orbitally nondegenerate impurity states will sometimes coincide as they shift with the nanocrystal radius. Such an ‘accidental’ degeneracy occurs e.g. for a pair of states (2, 0) and (1, 2) in the nanocrystal with R = 2rB. The crossing of the respective energy levels in ZnSe nanocrystals is shown in Fig. 3(a).

Figure 3
figure 3

(a) Size dependencies of impurity-state energies E1,2 and E2,0 crossing (see the magnifying inset) at R = 2rB ≈ 6.4 nm and (b) level anticrossing in the presence of electron–LO-phonon interaction. [(c) and (d)] are the same as [(a) and (b)] but for levels E1,3, E2,1 and E3,0. The energies in (b)–(d) are measured from levels E1,2 and E1,3. For material parameters refer to Methods.

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Accidental degeneracy resulting from different size dependencies of impurity states is removed by the interaction of the impurity with various excitations of the nanocrystal (phonons48,49,50,51, plasmons52, excitons53, etc.), external electromagnetic fields54, or interaction with the environment55,56,57,58,59,60. This causes splitting and anticrossing of the degenerate energy levels. The degree of degeneracy of the impurity states modified by the interaction is determined by the quantum numbers of the initial states and the nature of the interaction. One of the simplest level-anticrossing problems, arising for the lowest-energy accidentally degenerate states (2, 0) and (1, 2), requires solving the sixth-order secular equation. If an interaction of strength V coupling these states does not couple the states of momentum projections m = 0, ±1 and ±2, then this equation has three different roots, two of which,

correspond to nondegenerate states and one which corresponds to a four-fold degenerate state of energy E1,2 (see Methods). Thus the interaction partially removes the six-fold degeneracy of the impurity states at point A. It should be noted that the degeneracy removal in real nanocrystals is always complete due to the existence of coupling between the states of different momentum projections, splitting the four-fold degenerate state into four components.

Figure 3(b) shows the anticrossing of energy levels and , caused by the polar interaction of the states in Fig. 3(a) with dispersionless longitudinal optical (LO) phonons61,62,63,64,65 confined to the ZnSe nanocrystal. Removal of the accidental degeneracy at point A is seen to result in energy splitting of two out of the six initially degenerate states. This splitting is a natural measure of the electron–LO-phonon interaction strength in the nanocrystal. According to the diagram in Fig. 2(b), the energies and of the split-off states slowly approach Ry/9 ≈ −2.7 meV and Ry/4 ≈ −6.2 meV with the increase in the nanocrystal radius.

Discussion

Strictly speaking, the accidentally degenerate states are coupled by the electron–phonon interaction not only to each other, but also to all other impurity states of the nanocrystal. Since the interaction strength V approximately scales as ∝ R−1/2 (see Methods) and the energy-level spacing in small nanocrystals grows as ∝ R−2, the relative contribution of the latter coupling decreases with the reduction of the nanocrystal radius as ∝ R3/2. This implies that considering a pair of accidentally degenerate states independent of the rest of the nanocrystal states is most justified for states (2, 0) and (1, 2), which are degenerate in the smallest nanocrystal. If an impurity-state level becomes accidentally degenerate several times upon variation of the nanocrystal radius, then the resulting anticrossings can be considered independently as long as the interaction of the crossing levels in pairs is much stronger than their interaction with the rest of the impurity levels. Otherwise, an accurate anticrossing description requires taking into account interactions between all the strongly coupled impurity states. This leads to the secular equation of order , where lj (j = 1, 2, … η) are the angular momenta of the strongly coupled states. If lz is the largest angular momentum and , then both the accidental degeneracy of the states and their degeneracy in momentum projections are fully removed. The degeneracy removal is partial for μ − 4lz < 2, with at least one (4lz + 2 − μ)-fold degenerate impurity state of energy left.

Figures 3(c) and 3(d) illustrate the anticrossings of three levels E1,3, E2,1 and E3,0 coupled through the electron–phonon interaction considered earlier. One can see that the interaction leads to a complex anticrossing pattern, which is not the mere sum of two simple patterns similar to those in Fig. 3(b). Since lz = 3 and μ = 11 in this case, the original energy states transform into eight nondegenerate and one three-fold degenerate states. This example shows that only isolated pairs of impurity states that are strongly coupled to each other and weakly coupled to other states are of interest from an experimental viewpoint owing to their simple anticrossing patterns. Since the coupling between nondegenerate impurity states weakens with nanocrystal size, the most ‘isolated’ anticrossing occurs for states (2, 0) and (1, 2). This anticrossing is the most advantageous for the estimation of the strength of various interactions inside semiconductor nanocrystals.

In conclusion, we have shown that different size dependencies of impurity energy levels in semiconductor nanocrystals result in accidental level degeneracies, the removal of which leads to anticrossings in the size dependencies. Such anticrossings are an important feature inherent to doped nanocrystals, which enables measuring the strength of various interactions of the impurity states with the internal and external elementary excitations of the nanocrystals. We illustrated the removal of the accidental degeneracy of the lowest-energy states by the example of their interaction with optical phonons confined to spherical nanocrystals and showed that these states are most suited for experimental studies of the nanocrystal anticrossings.

Methods

Wave functions and energy spectrum of hydrogenic impurity

The solution to Eq. (1) is a product of the radial wave function and a spherical harmonic, Ψ(r) = F(r)Ylm(ϑ, φ), where r, ϑ and φ are the spherical coordinates and l and m are the angular momentum and its projection, respectively. Inside the nanocrystal, the radial wave function obeys the equation

where radius r and energy E are expressed in units of effective Bohr radius and effective Rydberg constant. The finite solution to this equation at r = 0 is proportional to the Kummer function66M(a, b, c),

where N = (−E)−1/2 and K = E1/2. The uniform Dirichlet boundary condition F = 0 at the nanocrystal surface r = R gives the impurity's energy spectrum

where Nn,l and Kn,l are the nth positive roots of the equations

with d = R/rB. The radial wave function of the impurity states can be finally written as47

where Anl is the normalization constant, anl = −Nnl and cnl = −2d/Nn,l for Enl < 0 and anl = i/Knl and cnl = 2iKnl d for Enl ≥ 0.

Degeneracy removal of impurity states

To determine how a pair of accidentally degenerate impurity states (n1, l1) and (n2, l2) of energies and split due to an interaction that does not couple the states differing only by the momentum projections, one needs to solve the secular equation

where , Il is the unit matrix of size 2l + 1,

is the (2l1 + 1) × (2l2 + 1) interaction matrix built of matrix elements , in which qαβ denotes the quantum numbers of the interaction quasiparticle coupling the states of momentum projections m1 and m2. The μ = 2(l1 + l2 + 1) roots of Eq. (11), some of which may coincide, are the new energy levels resulting from the splitting. The problem of degeneracy removal for more than two degenerate states is solved in a similar fashion.

The model of longitudinal optical (LO) phonons confined to a spherical semiconductor nanocrystal with a high degree of ionicity yields61,64:

where ε* = ε0ε∞/(ε∞ − ε0), ε∞ and ε0 are the high- and low-frequency permittivities, is the Clebsch–Gordan coefficient and

Equation (13) shows that the five-fold degenerate impurity state (1, 2, m) is coupled to the nondegenerate impurity state (2, 0, 0) through the five-fold degenerate phonon mode of quantum numbers (1, 2, m). Equation (11) in this case reduces to

with

Material paramseters

Figure 3 was plotted using Eqs. (13), (14) and (16), by assuming that the nanocrystal is made of ZnSe and using the following parameters: Z = 1, me = 0.15m0, ε0 = 9.1, ε∞ = 6.3, ħΩ = 31.7 meV (the energy of the bulk LO phonon at the Brillouin zone center), rB≈ 3.21 nm and Ry ≈ 24.6 meV67.