Physics / 2. Solid State Physics

Chkhartishvili L.S.*^, ** and Darchiashvili M.D.**

* Georgian Technical University, Georgia

** Ferdinand Tavadze Institute of Metallurgy and Materials Science, Georgia

Binding energies and electron energy levels

of impurity atoms in crystallographic voids

1. Introduction

Doping is known to be the main method of tuning the electronic properties of semiconductors. In conventional crystalline semiconductors, like Ge, Si, GaAs etc, atoms constituting a substance are substituted with foreign dopants. In β-rhombohedral boron (β-B), which is a promising high-temperature material, the attempt to make the substitutional doping with non-metallic elements faces an obstacle. The ideal structure of β‑B is the electron-deficient and, therefore, unstable. Real crystals are stabilized by a large number of point structural defects – vacancies and boron atoms in irregular sites – determining electronic properties of the undoped material. For this reason, β‑B is almost insensitive to doping: to achieve a desired effect it is necessary to introduce impurities with concentrations comparable with that of intrinsic defects. But, any attempt to replace such high number of boron atoms in their regular sites leads to the lattice destruction and / or formation of the phase inclusions. But, in β‑B a fundamentally different mechanism of doping can be released. Its crystalline structure is characterized by the different types of crystallographic voids accommodating metal atoms with very slight structural distortions. Corresponding mechanism of doping has several advantages compared with the traditional one: (1) Metallic impurities in crystallographic voids of β‑B can be introduced to the very high concentrations, up to a several atomic percents; (2) Because of the diversity of void types, such kind of doping can simultaneously affect various physical properties of the material; (3) Since atoms of various chemical elements in different ways can be distributed between the voids of different types, a similar effect can be achieved by double, ternary, etc doping of β‑B combining several dopants.

Nowadays, many metallic elements are found experimentally in different crystallographic voids of the β‑B lattice. Let only list papers determining some of their structural, ground-state and electronic parameters: ₃Li [1‑5], ₁₂Mg [6‑8], ₁₃Al [1, 9, 10], ₂₁Sc [11‑13], ₂₂Ti [11, 12, 14‑16], ₂₃V [2, 3, 11, 12, 17‑27], ₂₄Cr [11, 12, 22, 25, 28], ₂₅Mn [11, 12, 29‑36], ₂₆Fe [10‑12, 14, 16, 20, 22‑24, 30, 33, 34, 37‑46], ₂₇Co [11, 12, 16, 24, 25, 38, 47], ₂₈Ni [3, 11, 12, 16, 18, 22, 30, 38, 44, 48‑53], ₂₉Cu [1, 3, 9, 11, 12, 19, 20, 22, 30, 54], ₄₀Zr [11, 12, 18, 19, 25, 30, 55‑57], ₄₁Nb [56], ₇₂Hf [11, 18, 55, 56], ₇₃Ta [11, 13, 56, 58‑60], ₇₅Re [44, 56, 57, 61]. Because the real crystals of β‑B contain a very high concentration of intrinsic point structural defects, pure samples are characterized by the p-type hopping conduction mechanism. A typical picture of the metal-doping influence on the electrical conduction in β‑B is as follows. These impurities occupying the crystallographic voids, freely supply the boron lattice with electrons from their outer valence shells. Initially, these electrons fill the deep levels related with dangled bonds at the intrinsic defects. This leads to an increase in the concentration of the empty centers of localization of hopping holes. As a result the p-type hopping conduction increases. But, when more than half of the centers are filled with electrons concentration of the localized holes begins to decrease and increasing in the p-type conductivity is replaced by its decreasing. At a certain level of doping, it comes the inversion of the carrier’s charge sign – further increase in the concentration of metallic impurities yields the increase in the n-type conductivity, which is realized by the hopping of localized electrons between the impurity centers and / or delocalized electrons transmitted from these centers in the β‑B conduction band.

There is a lack of theoretical studies on in-void doping mechanism, which largely hinders the purposeful design of the high-temperature semiconductor materials based on the metals solid solutions of β‑B. This paper is an attempt to partially fill this gap. It is the first part of the study focused on general theoretical basis developed for obtaining the binding energies and electron energy levels of impurities localized in the crystallographic voids. In future, we intend to calculate these parameters for various metal atoms doped in different crystallographic voids in β‑B lattice. Suggested approach is based on the quasi-classical-type method [62], which was successfully utilized for calculating the ground-state parameters and electron energy spectra of boron nitrides BN [63‑73], the compounds of boron, boron nanotubes [74, 75], as well as for treatment of boron isotopic effects [76‑79]. Quasi-classical parameters of the constituent atoms should be assumed as given as in [80] we have pre-calculated and tabulated their values.

2. Bonding energy

Let, is the coordination number of the impurity (metal M) atom, i.e., number of (boron B) atoms surrounding the atom localized in the void, and denotes the average length of (M – B) bonds. Then, the desired energy of binding between impurity atom and lattice can be written as

,

where and are the static energy of interaction (except for the non-physical self-energy contribution) and the vibrational energy of the impurity atom relative to the surrounding atoms, respectively. In the initial quasi-classical approximation, these values are calculated from the relations

,

.

Here and , and and are the volume averages of the electric charge density and electric field potential in the th and th radial layers of lattice and impurity atoms, respectively, and are the numbers of layers of the quasi-classical averaging in these atoms, and is the impurity atom mass. denotes the volume of intersection of the lattice atom th layer with the impurity atom th layer. It can be calculated as a linear combination of intersection volumes of pairs of spheres,

.

is the partial derivative of this function along the average bonds length. Parameters and are external radii of the th and th layers of lattice and impurity atoms, respectively (inner radii are and , assuming ). As for the function , it is an universal function of the geometric meaning expressing the intersection volume of two spheres with radii and whose centers from each other are at the distance . It is an analytic (algebraic) piece-wise function. Its partial derivative is also continuous, but not continuously differentiable at the points and . Explicit forms of these functions one can find in [62]. Expression with quadratic root standing in the vibrational energy term is the frequency of the mode localized at the impurity atom. When value of the expression under the square root is negative, it must be replaced by zero.

Under the equilibrium conditions, the resultant force acting on the impurity atom from the surrounding atoms has to be zero. This means that to determine the value of binding energy of the impurity atom in different crystallographic voids one should find extremes of the function . We emphasize that one should consider both kinds of extremes, not only the binding energy maxima, which correspond to the stable equilibrium in the subsystem “impurity atom–surrounding cluster of lattice atoms”, but also the minima defining the inter-atomic distances in unstable equilibrium. The fact that such cluster is not a closed physical system: actually it is embedded into the crystalline lattice. Therefore, an impurity atom localized at an unstable equilibrium position with respect to the subsystem “impurity atom–surrounding atoms” is kinetically incapable to reach a stable equilibrium position. Of course, it should be taken into account only the extremes, for which values of the binding energy are positive. Furthermore, one should confine itself to the extremes for which the deviations of the average distances from the geometric centers of voids, where it is supposed location of the impurity atoms nuclei, to the surrounding atoms from the predicted (M - B) bonds lengths are not too large.

One should not think that the calculated average bond length corresponding to an extreme should coincide with the value actually realized in the crystal. It is likely the equilibrium bond length, to which the subsystem tends. Based on its value one can judge whether the crystal will expand or in contrary compress by the doping. In the crystal, there is established certain average bond length, for which the loss (gain) in the binding energy of the impurity with the lattice associated with the deviation of the cluster (including the impurity-atom) from a stable (unstable) equilibrium is compensated by the gain (loss) in the deformation energy of the rest of the crystal as a result of doping. Therefore, the binding energy values at its extremes may be well used to estimate the binding energy of an impurity atom with the crystal as a whole. The fact is that by definition, the latter is the difference between the total energies of doped and, consequently, a locally deformed crystal and pure ideal crystal.

3. Electron energy levels

Determination of the electron energy levels of various dopant elements accommodated in voids of various types in crystalline lattice needs elaboration of the special model of the impurity center. Situation when a dopant atom embedded in crystallographic void is surrounded by the atoms already bonded together essentially differs from that when the lattice constituent atom tightly bonded with neighbors is substituted with a foreign atom. So, standard models are useless for our purposes.

Taking into account that impurity atom placed inside the crystallographic void only slightly affects the ideal crystalline structure, within the first approximation the outer valence-shell electron’s energy levels for neutral and negatively charged impurity atom can be found from the electron ionization potential and electron affinity of the same chemical element in isolated state, respectively, shifting them in internal crystalline field. The mentioned chemical shifts can be directly calculated assuming that changes in electric field inside the impurity atom introduced in crystallographic void mainly are determined by the lattice atoms surrounding the void of given type. Then, based on the semiconductor work function and band gap width values, from the determined electron energy spectrum of the impurity atom one can clarify whether such dopant forms discrete energy level within the band gap, if so what is its energy, is it a donor or acceptor? etc. In this way, one will be able to predict character of influence of doping with given element on the semiconductor conductivity.

The depth of the donor level formed by an impurity atom embedded in certain crystallographic void is determined from the relation

,

where is the ionization potential correction in the crystalline field. It is clear that the result obtained for the will have a physical sense only if .

So to estimate the donor level location it is necessary to calculate only one quantity, . It can be found using perturbation theory, as the absolute value of first-order correction to the energy level. In the initial quasi-classical approximation, we obtain

 .

Here and represent the principal and orbital quantum numbers of the outer valence shell electron of the impurity atom, respectively. In this expression denotes the intersection volume of a lattice atom’s th radial layer with the impurity atom’s -layer, while and are the inner and outer classical turning points radii for the outer valence shell electron of the impurity atom, respectively:

,

.

The value of is calculated like that of in previous section.

As for the depth of the acceptor level formed by the impurity atom embedded in certain crystallographic void, it is determined from the relation

,

where is the electron affinity correction in the crystalline field. It is clear that the result found for the will have a physical sense only if .

Consequently, to estimate the acceptor level it is necessary to calculate . Using perturbation theory in the initial quasi-classical approximation we get:

.

and represent the principal and orbital quantum numbers of the extra electron in the impurity atom, respectively, while denotes the intersection volume of the lattice atom’s radial th layer with the impurity atom’s -layer. and are the inner and outer classical turning points radii for the extra electron, respectively:

,

.

4. Concluding remarks

The relative error of the quasi-classical method itself when it determines the quantities of dimension of energy does not exceed a few percents. However, the model used to describe a cluster of the impurity and surrounding lattice atoms is based on simplifications, which will serve as additional sources of calculation errors: (1) Even in the perfect crystal, atoms surrounding a void are located in non-equivalent crystallographic positions at different distances from the geometric center; (2) In some cases, the location of the void’s geometric center is poorly defined and consequently the concept of the average length of bonds is poorly defined too; (3) Because of vacancies in real crystals the number of neighboring lattice atoms are not well defined; (4) Introducing of an impurity atom in the void deforms the adjacent region of the crystal – such local deformation itself may contribute to the multiplication of vacancies; (5) The role of the nearest neighbors of impurity atoms in real crystals can play not only atoms in regular lattice sites, but also some irregular sites, which are partially occupied by host atoms, as well as other impurity atoms located in neighboring voids or lattice sites; (6) The nearest neighbors of the impurity atom are counted according to some criterion (see below: the Lundström criterion) and therefore their number is arbitrary to some extent; (7) Model neglects the interaction of an impurity atom with other (i.e., next to the nearest neighbors) atoms of the crystal.

Traditionally, according to the Lundström criterion [31], atoms in the β‑B lattice are considered to be the nearest neighbors if the corresponding bond length does not exceed 2.80 Å. This value is at 0.42 Å larger than the mean radius 2.38 Å of the largest voids (of E type) in β‑B. By reason of this, as a possible lower limit of the M - B bond length we can use 1.65 Å, which is approximately by the same magnitude smaller than the mean radius 2.10 Å of the smallest voids (of A(2) type). Note that optimization of the boron icosahedron B₁₂ geometry carried out by semi-empirical version of the molecular-orbital-method leads to the equilibrium binding energy value of 4.2 eV / at [4], while another estimate obtained theoretically within the local-density-approximation using the factorizable potentials and plane-wave basis set is of 6.8 eV / at [81]. Since the icosahedron is the main structural motif of the β‑B structure it follows that the value of binding energy of an impurity atom heavier than boron with the boron lattice of ~ 10 eV / at and more seems to be quite reasonable.

For determination of an impurity level position in the β-B band gap within the frames of proposed model, it is necessary to know the band gap and electron work function of this material. The b-B band gap width is known to be of 1.50 eV [82]. As for its work function, we intend to use the value of 4.45 eV. The evaluation of this value is presented below. There are only very old measurements of this parameter, which are contradictory in many respects. For this reason, let get this issue in more detail. By photoelectric measurements it was found [83] that the work function of different boron samples varies in range of 4.40 - 4.60 eV. However, there is no certainty that just the β-rhombohedral modification of boron had been investigated. From the density-of-states at the top of the β-B valence band determined by measuring the quantum yield and energy distribution of photoelectrons, another interval of work function was found: 4.50 - 4.65 eV [84]. However, the work function of β-B defined in [85] again from the quantum yield spectrum was lower: 4.30 eV. Differences in measured values of the β-B boron photoelectric work function can be, at least partially, explained by the dependence of this parameter on the incident photons energy. For example, in [86] approximately linear increase in work function value depending on the photon energy is associated with strong polaron effect characteristic for boron: the photon energy must be divided between the photon-capturing electron itself and other quasi-particle, in this case – a phonon tightly bonded with electron. Thermoelectric work function of single-crystalline boron at high temperatures (> 1300 K), calculated from the slope of the Richardson experimental curve was found to be of 4.30 eV [87]. In [85], a much lower value of the thermoelectric work function of 3.80 eV had reported, as it obtained from the β-B current–voltage characteristic. It was noted that the decrease in β‑B thermoelectric work function by a few tenths of eV possibly is due to the Fermi level location above the valence band edge in the measured crystals (the “tail” of the valence band penetrating into the band gap is generated due to the high content of structural defects and foreign impurities in β-B real crystals). In [88], the work function of β-B was estimated at 6.13 eV semi-empirically based on measurements of the β-B inner potential by the high-energy electrons diffraction on a vacuum cleaved single-crystalline structurally perfect surface. Thus, there is a fairly wide spread in the values of the β-B work function measured by different methods: 3.80 - 6.13 eV. And there are explanations for both under‑ and overestimating of the true value, which is defined as the energy separation between the semiconductor valence band edge and vacuum level. In our calculations, as a reference quantity we intend to use 4.45 eV, the value recommended by the reference books [89, 90]. However, in evaluating the calculated energy levels we should take into account the spread in this input parameter.

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