Process of an acceptor center μB− formation in diamond crystals

A formation process of an acceptor center μB induced by negative muon in a diamond crystal is considered in this work. It is shown that the process under consideration could be represented as three consecutive stages: a kinetic stage, connected with a capture of track electrons, a chemical stage, connected with establishing of chemical bounds of the radiation defect with lattice atoms, and a formation phase of the ionized negatively charged acceptor center. Quantitative estimates of characteristic times are obtained for every stage. It is shown that the total time necessary for the acceptor center formation is of the order 10−8s. So this fact should be taken into account when experimental results are interpreted.


Introduction
It is well known, that a negative muon is captured on a lattice nuclear forming the so called µ-atom. But just after the muon capture in a lattice arises a positively charged radiation defect. Neutralization process of radiation defects induced by negative muons in crystals with diamond structure (Diamond, Silicon and Germanium) was considered in works [1,2]. There was shown that this process could be considered as four consistent stages. At the first stage positively charged radiation defect with a charge equal to Z − 1, where Z is the charge of a host nuclear, captures electrons of the radiation track. At the second stage a neutral defect restores broken chemical bounds with lattice atoms. At the third stage an ionized acceptor center is formed. And at the final stage is formed a neutral (usual) acceptor center induced by the µ-atom. Only the last stage is studied well for the present. In [2] was considered a neutralization kinetics as a result of a capture of electrons from the radiation trace. There was shown that the capture process strongly differs in diamond crystal and in crystals of silicon and germanium. Namely, in a diamond crystal the radiation defect has not a time to be neutralized in a muon life-time τ µ ≈ 2.2 · 10 −6 s. So, the formation process of a chemically bounded radiation defect differs in crystals under consideration. Chemical bounds are restored in silicon and germanium for the neutral impurity µ A (muonic atom), but in diamond for the positively charged ion µ B + . A chemically bounded impurity µ B + is neutralized in a diamond lattice. Estimation of a formation time of a neutral chemically bounded defect caused by a radiation transition was obtained in [1]. Because of this difference in neutralization processes a formation rate of chemically bounded neutral defect µ A in a diamond almost two order higher than in silicon and germanium. In the following a neutral chemically bounded radiation defect µ A forms an acceptor center in a lattice. At the first step of an acceptor center formation one need to consider a capture of an additional electron to fill an empty chemical bound around µ A. After the capture of an electron an ionized acceptor center µ A − is formed. This process is not studied yet, but the following step of a positive hole capture at the ionized acceptor center is well known. A process of an ionized acceptor center µ B − formation in a diamond crystal is studied in this work.
Experiments with Si and Ge was carried out more than 20 years ago [3]- [7]. Acceptor centers µ Al and µ Ga was observed and hyperfine structure of µ Al was studied in these experiments. Recent experiments with µ − in synthetic diamond [8,9] showed unusual behaviour of a muon spin polarization. A negative shift of a precession frequency and slow temperature dependence of a damping rate at temperature T > 20K were observed. Full polarization (100%) was observed. Thus, in these experiments no neutral acceptor center µ B was observed. The experimental results show that kinetic processes of a formation of the neutral acceptor center µ B demand more detail theoretical examination.
2. Phonon spectrum changing caused by a neutral center µ B A neutral center µ A enters into a chemical bound with nearest atoms of a lattice and forms a cluster in the first coordination sphere. This cluster possesses a local symmetry C 3v and a rather large electric dipole moment [1]. This electric dipole moment gives rise to an interaction necessary to capture a lattice electron and form of an ionized acceptor center. The neutral center µ A has unsaturated chemical bounds because of that a crystal lattice turns out to be deformed. This deformation is a reason to change a phonon spectrum and a local phonon mode appearance. Chemical bounds are saturated after the missing electron capture. The new cluster is an ionized acceptor center and possesses a local crystal symmetry. When the ionized acceptor center is formed an appropriate phonon of the local mode is radiated and crystal deformations disappear. So, the problem is very similar to the problem of a thermalization of molecular ions in molecular crystals and cryocrystals of noble atoms (see e.g. [10,11,12]). An exact solution of the problem taking into account a crystal symmetry is scarcely possible. But we can obtain qualitative results to get an estimation of a formation rate of the ionized acceptor center µ B − in diamond.
To solve the problem we will carry out some simplifications. They will make our model rather crude but describing qualitatively all necessary effects nevertheless. First of all we will accept an approximation of an elastic continuous media to describe a phonon spectrum changing. In this case a displacement vector u satisfies the wave equation in an elastic media at the presence of the electric field [13]: where ρ is the density of a media. A tension tensor will be represented as a sum Here σ el ik is determined by constants of the elastic media as usual. The second term in the expression (2) is the tension tensor of the electromagnetic field in a media [14]: where E and D are the average (macroscopic) electric field and the induction created by the dipole moment of the cluster d respectively. We are taking a simplification to determine the electric field E, namely ε ik = ε(r)δ ik So, for the force acting on the media from the side of the electric field created by the dipole moment we are obtaining Here was taken into account that outside the first coordinative sphere in the lattice divD = 0. Let's place the coordinate beginning at the center of the nuclear µ A and consider that the dielectric susceptibility possesses a spherical symmetry, i.e. ε(r) = 1 + 4πκ(r), where κ(r) is the dielectric susceptibility. As well we will consider the coordinate dependence of the dielectric susceptibility is stipulated for the specific volume v changing as a result of the deformation. So, we obtain κ ∝ v −1 because the dielectric susceptibility is proportional to the media density. The changing of the specific volume as the result of the deformation is determined by the trace of the deformation tensor [13]: where u ll = ∂u l /∂x l = divu. In this case the gradient of the dielectric penetration could be represented in a form Let us rewrite the equation (1) taking into account that in our case rotu = 0 where c is the longitudinal sound velocity. The equation (8) is rather complicated yet and we'll make some additional very serious simplifications to get analytical expressions. Let's consider only the radial component of the displacement vector u and neglect its angular dependence, i.e. u(r) → u r (r). In this approximation we'll keep in the induction vector only a dependence on the modulus of r: Only in this approximation we can solve the problem analytically. After all these simplifications we haveü It is suitable to search a solution for the radial displacements in a form u r (r, t) = χ(r, t) r .
In this case we have the more simple equation for χ: where χ is the second derivative on the radial variable r. For the function χ(r, t) the one-dimensional Fourier-transformation is valid: For the Fourier-amplitudes (12) we obtain an integral equation taking into account that the expression for the force in the equation (8) is valid only for r > R 1 , where R 1 is the radius of the first coordination sphere: Let's make the last serious simplification in the equation (13), changing an integral in the righthand side by a constant before χ ω,k . Really, values of the wave-vector k ≈ k give the main contribution to the second integral. This permits us to take out the first integral the amplitude χ ω ,k at the point k. Values of k ∼ R −1 1 give the main contribution to the integral as well. So, for the estimation of the integral in the right-hand side we have χ ω,k /(10πR 8 1 ). Finally we obtain a simple dispersion relation: 3. Electron capture rate Electric dipole moment creates a scalar potential and an interaction energy with lattice electrons is U = eϕ = e(dr)/(εr 3 ). So, taking into account the displacement u (r → r + u), we can write an electron-phonon interaction operator where the operator of radial displacements is equal to [11] Here b † n and b n are creation and annihilation operators for radial phonons respectively. An electron capture rate could be calculated by means of well known Fermi "golden rule" where an initial state |i = |k |0 ph corresponds to a free (in a valence band) electron with the wave-vector k and the absence of excited radial phonons, and ν f is the density of final states. The final state is determined by the state of the electron captured in the hybridized state of the cluster [1] and an excitation of a radial phonon with the wave-vector k ph : |f = (α|2s + β|2p )|k ph , |α| 2 + |β| 2 = 1.
Eq.(17) takes into account that all electrons with wave-vectors are in the interval from k to k + dk can participate in the process. Energies of the initial and the final states are equal to where ε ac is the acceptor ionization energy and m * is an effective mass of the electron in the valence band.
We can carry out a partial integration of the expression (17) and obtain a rather short formula where k 2 0 = 2m * (∆ − ω ph )/ , ∆ = ε ac / . For us will be comfortable in the future to introduce dimensionless parameters and variables x = Zr/(2a 0 ), k ph = 2k ph a 0 / Z, k 0 = 2k 0 a 0 / Z and ω 0 = c Z/(2a 0 ). Here Z is the effective nuclear charge for electrons at the appropriate atomic energy level and c is the average longitudinal sound velocity. For Boron it is equal to Z = 2.6. In this case we can express a capture cross-section through a dimensionless matrix element: where We can see, that the matrix element (21) has a peculiarity in the point x = 0 and the integrals diverge if the low limit is equal to zero. So we must take into account that expression (15) is valid for r a 0 also. Thus we need to determine the low limit r a 0 . It is reasonable to take it equal to the length of the chemical bound in the lattice. In diamond crystal it is equal to r 0 ≈ 1.54 · 10 −8 cm and the low limit of the integrals (21) is Making rather long calculations and taking into account the dispersion law (14) we obtain the final formula for the capture rate where ∆ = ∆/ω 0 , Ω = Ω/ω 0 , ω = ω/ω 0 .
The following calculations could be carried out numerically.

Conclusion
We shall choose hydrogen-like 2s and 2p wave functions like in Ref. [1] to calculate matrix elements (21). The value of the rate (22) is proportional to the unknown parameter d, but numerical estimates show that d 1D (Debye, 1D = 10 −18 CGSE). We will regard m * = m (the free electron mass) and use the following numerical values for diamond parameters [15,16]: ρ = 3.51g/cm 3 , ε = 5.75, c = 1.83 · 10 6 cm/s, ε ac = 0.37eV, the lattice constant is R 1 = 3.57 · 10 −8 cm. For these parameter values we calculated the value of the rate (22) in two cases: d = 1.0D an d = 0.5D.
Numerical results are shown in table 1, where w s and w p are the capture rates at 2s and 2p-states respectively. In spite of serious approximations we obtained a numerical values which give an expected interval for a formation-time of the negatively charged acceptor center as an intermediate state of a radiation defect in crystals.
Negatively charged cluster ( µ BC 4 ) − is the ionized boron acceptor in a diamond lattice, which captures a hole from the valence band of a crystal because of Coulomb interaction and forms a neutral acceptor. This last stage of a radiation defect formation is well known and strongly depends on temperature of a crystal. Thus we can write a formula to determine the total formation rate of the radiation defect induced by negative muon in diamond crystals as a sum of rates of all stages: W = w tr + w cl + w 0 + w − + w h .
Definitions of all items are obvious (w − = w s + w p ).
The numerical results show that the lowest stage is connected with the stage of the negatively charged cluster ( µ BC 4 ) − formation. From the table 1 we see that 0.5 · 10 8 s −1 < w − < 2 · 10 8 s −1 . So, we can see that a formation-time of the neutral acceptor center µ B is rather large and must be taken into account in the interpretation of experimental results [8,9]. Namely, a computing of experimental results must be carried out on the base of kinetic equations like in chemistry reactions. This kind of the analysis could give us unknown parameters of the total process. But, this is the other problem.