## Abstract

We find the first, simple, analytical expressions for the bound states of a short-ranged, spherically-symmetric deep center potential in terms of the eigenstates of total (spin plus Bloch plus envelope) angular momentum and the Kane 8×8 $\overrightarrow{\mathrm{k}}$ ·$\overrightarrow{\mathrm{p}}$ Hamiltonian. We find that the spatial extent of the deep center bound state is proportional to the Kane dipole. This physical size of the deep center bound state is in excellent agreement with both scanning-tunneling-microscopy and measured optical dipoles.

© Optical Society of America

Deep centers are technologically important because of their role [1, 2, 3, 4, 5] in compensation mechanisms in semi-insulating semiconductors. Optical studies are a fast, nondestructive way of characterizing this deep center [1, 2]. Moreover, a specific deep center, the arsenic-on-gallium-site (As_{Ga} or arsenic antisite or EL2), has found important uses in high speed light-emitting-diodes (LEDs), photodetectors (PDs), optical modulators, and nonlinear optics [6, 7]. An important part of the design of such devices is an understanding of the optical properties of deep centers.

Previous work [3, 8, 9] on deep center bound states have included Lucovsky’s [8] one-band model in the effective mass approximation. More recently, numerical pseudopotential models of the short-ranged, deep-center potential have been carried out by Jaros [10] and Baraff [4, 5].

In this Letter, we find the first, simple, analytical expressions for the bound states of a short-ranged, spherically-symmetric deep center potential in terms of the eigenstates of total (spin plus Bloch plus envelope) angular momentum and the Kane 8×8 $\overrightarrow{\mathrm{k}}$·$\overrightarrow{\mathrm{p}}$ Hamiltonian. We also find the first, simple, analytical expressions for the optical emission strength from a deep center bound state to the Γ-point valence band maximum.

One significance of our fully analytical model is its ability to resolve a question of current interest: the relative size of direct optical transitions (involving no phonons) and phonon-assisted optical transitions from deep centers. This Letter will compare experimentally measured optical cross sections with our theoretical model of direct optical transition strengths. Another significance of our fully analytical model is the generality of its predictions with respect to different semiconductor host materials. More-over, our fully analytical model is very easy to use because it does not involve any computationally-intensive numerical methods. Finally, an important application of our model of deep center optical properties is the theoretical prediction of both optical quantum efficiency and the role of phonons in the optical devices mentioned above (LEDs, PDs, modulators, nonlinear optical devices) [6, 7].

In the presence of a spherically-symmetric deep-center potential, we now choose to write the energy eigenstates as eigenstates of total angular momentum, $\overrightarrow{F}$=$\overrightarrow{L}$+$\overrightarrow{J}$=$\overrightarrow{L}$+($\overrightarrow{L}$_{B}+$\overrightarrow{S}$), where $\overrightarrow{L}$ is the envelope angular momentum, $\overrightarrow{L}$* _{B}* is the Bloch angular momentum, $\overrightarrow{S}$ is the electron spin, and $\overrightarrow{J}$=$\overrightarrow{L}$

*+$\overrightarrow{S}$. The lowest bound state of the deep center must have the quantum number $F=\frac{1}{2}.$. If this bound state has A*

_{B}_{1}-symmetry (fully symmetric, as experiment shows As

_{Ga}to be), then we may write the eigenfunction of the 8×8 $\overrightarrow{\mathrm{k}}$·$\overrightarrow{\mathrm{p}}$ Hamiltonian [11] as:

$$+\sqrt{\frac{2}{3}}\sqrt{\frac{{E}_{C}-{E}_{d}}{{E}_{G}}}{f}_{L}=1\left(r\right)\mid F=\frac{1}{2},{F}_{z}=\frac{1}{2};L=1,J=\frac{3}{2}\left({L}_{B}=1\right)\u3009+$$

$$+\sqrt{\frac{1}{3}}\sqrt{\frac{{E}_{C}-{E}_{d}}{{E}_{G}}}{f}_{L}=1\left(r\right)\mid F=\frac{1}{2},{F}_{z}=\frac{1}{2};L=1,J=\frac{1}{2}\left({L}_{B}=1\right)\u3009]$$

where *E _{d}, E_{V}, E_{C}, E_{G}, N* are, respectively, the defect bound energy level, the valence and conduction band edges, the band gap energy, and an overall normalization factor. In Eq.(1) above, the eigenstates of total (Bloch plus envelope plus spin) angular momentum are related to the zone-center [11, 12, 13] eigenstates |

*J, J*〉 through Clebsch-Gordan coefficients, $\mid F,{F}_{z};L,J\u3009={\Sigma}_{{L}_{z},{J}_{z}}\mid L,{L}_{z}\u3009\mid J,{J}_{z}\u3009\u3008L,{L}_{z},J,{J}_{z}\mid F,{F}_{z};L,J\u3009$. The radial functions in Eq.(1) are spherical Bessel functions

_{z}*f*(

_{L}*r*)=

*j*(

_{L}*kr*) for states in the energy continuum, and Hankel functions

*f*(

_{L}*r*)=

*h*(

_{L}*iαr*) for localized states. The length scale of these radial functions are determined by the $\overrightarrow{\mathrm{k}}$·$\overrightarrow{\mathrm{p}}$ Hamiltonian [11, 12, 13] in degenerate peturbation theory. When the $\overrightarrow{\mathrm{k}}$·$\overrightarrow{\mathrm{p}}$ Hamiltonian is written explicitly in the conduction band, light-hole, heavy-hole, and split-off hole basis states, then the energy eigenvalues

*E*′ (as measured from the valence band edge) satisfy [11],

where *P*=*E _{G}*〈

*s*|

*z*|

*z*⃚ is the Kane [11] matrix element, and Δ is the spin-orbit splitting [11]. Eq. (2) goes beyond the effective mass approximation to explicitly include nonparabolicity of the energy bands.

Eq. (1) is the first main result of this Letter. Though it is known that the conduction band, light-hole, and split-off hole wave functions have a form very similar to Eq. (1), the deep center bound state wave function has never been written in the form of Eq. (1). A major difference between the deep center wave function and conduction (or valence) band wave functions is that the deep center wave function is localized to a radius of about α^{-1}, where α is given in Eq. (2). Another major difference is that the deep center wave function has approximately 50% conduction-band character and 50% valence-band character for ${E}_{d}\sim \frac{1}{2}{E}_{G}.$. The $\overrightarrow{\mathrm{k}}$·$\overrightarrow{\mathrm{p}}$ Hamiltonian determines the precise fraction of the defect wave function |*d*〉 which has conduction or valence-band character. Significantly, the entire wave function |*d*〉 remains fully symmetric (A1-symmetric) even though |*d*〉 has much valence-band character, because the *L _{B}*=1 Bloch component always occurs with a L=1 envelope component.

$F=\frac{1}{2},{F}_{z}=\frac{1}{2};$
$L=0,J=\frac{1}{2}\left({L}_{B}=0\right)\u3009$ is the part of the defect wave function |*d*〉 which has conduction-band character, and is shown in Fig. 1. The yellow wave function denotes the *L _{B}*=0 Bloch-component. The green wave function denotes the

*L*=0 envelope-component. $\mid F=\frac{1}{2}$,${F}_{z}=\frac{1}{2};L=1$ $J=\frac{3}{2}\left({L}_{B}=1\right)\u3009$ is the part of the defect wave function |

*d*〉 which has light-hole character, and is shown in Fig. 2. The yellow wave function denotes an |

*L*=1,

_{B}*L*=0〉 Bloch-component. The green wave function denotes an |

_{Bz}*L*=1,

*L*=0〉 envelope-component. (For comparison, Fig. 3 shows a linear combination of orbitals used in numerical pseudopotential models. The antisymmetric linear combination |1〉+|2〉-|3〉-|4〉 shown in Fig. 3 has the same symmetry as the |

_{z}*L*=1,

*L*=0〉 envelope function in Fig.2.)

_{z}The second main result of this Letter is the spatial extent of the deep center bound state. Eq. (2) shows the deep center bound state to have a radius of approximately,

$$\approx 2\mid <s\mid z\mid z\u3009\mid \mathrm{for}\mathrm{deep}\phantom{\rule{.2em}{0ex}}\mathrm{center},$$

where the last approximation is a result of the depth of the deep center bound energy ($({E}_{d}\sim \frac{1}{2}{E}_{G})$). In GaAs, we can calculate the radius of the deep center bound state to be *α*
^{-1}≈2〈*s*|*z*|*z*〉=13Å. This is in excellent agreement with scanning tunneling microscopy images made by Feenstra [14, 15] of the As_{Ga} antisite in GaAs. Such measurements show the bound electron of AsGa to extend over a radius of about 11–12Å.

If the defect state |*d*〉 has the wave function shown in Eq. (1), then the optical dipole 〈*d*|*z*|*v*〉 is nonzero when the Γ-point valence band (final) state |*v*〉 has the form,

$$+\sqrt{\frac{2}{3}}\sqrt{\frac{{E}_{C}-{E}^{\prime}}{{E}_{G}}}{f}_{L=2}\left(r\right)\mid F\mid =\frac{1}{2},{F}_{z}=\frac{1}{2};L=2,J=\frac{3}{2}\left({L}_{B}=1\right)\u3009+$$

$$+\sqrt{\frac{1}{3}}\sqrt{\frac{{E}_{C}-{E}^{\prime}}{{E}_{G}}}{f}_{L=0}\left(r\right)\mid F=\frac{1}{2},{F}_{z}\mid =\frac{1}{2};L=0,J=\frac{1}{2}\left({L}_{B}=1\right)\u3009]$$

or

$$+\sqrt{\frac{1}{3}}\sqrt{\frac{{E}_{C}-{E}^{\prime}}{{E}_{G}}}{f}_{L=0}\left(r\right)\mid F=\frac{3}{2},{F}_{z}=\frac{1}{2};L=0,J=\frac{3}{2}\left({L}_{B}=1\right)\u3009+$$

$$+\sqrt{\frac{1}{3}}\sqrt{\frac{{E}_{C}-{E}^{\prime}}{{E}_{G}}}{f}_{L=2}\left(r\right)\mid F=\frac{3}{2},{F}_{z}=\frac{1}{2};L=\mid 2,J=\frac{3}{2}\left({L}_{B}=1\right)\u3009+$$

$$+\sqrt{\frac{1}{3}}\sqrt{\frac{{E}_{C}-{E}^{\prime}}{{E}_{G}}}{f}_{L=2}\left(r\right)\mid F=\frac{3}{2},{F}_{z}=\frac{1}{2};L=2,J=\frac{1}{2}\left({L}_{B}=1\right)\u3009]$$

It is easy to show [13] that $\mid F=\frac{1}{2}$,${F}_{z}=\frac{1}{2}$; *L*=2, $J=\frac{3}{2}\left({L}_{B}=1\right)\u3009$ has light-hole symmetry, and that $F=\frac{3}{2}$,${F}_{z}=\frac{1}{2}$; *L*=0, $J=\frac{3}{2}\left({L}_{B}=1\right)\u3009$,$\mid F=\frac{3}{2}$,${F}_{z}=\frac{1}{2}$; *L*=2, $J=\frac{3}{2}\left({L}_{B}=1\right)\u3009$ each have 50% light-hole character and 50% heavy-hole character. It is also easy to show that the reduced matrix elements [16] which appear in the Wigner-Eckart theorem satisfy: $\sqrt{2}\mid <{J}^{\prime}=\frac{1}{2}\left({L}_{B}=0\right)\parallel \ell =1\parallel J=\frac{1}{2}\left({L}_{B}=1\right)\u3009\mid =\mid <{J}^{\prime}=\frac{1}{2}\left({L}_{B}=0\right)\parallel \ell =1\parallel J=\frac{3}{2}\left({L}_{B}=1\right)\u3009$. Summing over the relevant Clebsch-Gordan coefficients with Racah coefficients, we can now calculate the oscillator strength *f*=2*m*
_{0}(*E _{d}*-

*E*′)∑|

_{v}〉〈|

*d*|

*z*|

*v*〉|

^{2}/

*ħ*

^{2}for emission from the defect |

*d*〉 to the valence band |

*v*〉 to be,

Where

$$\times \left[\frac{{E}_{G}(\alpha {R}_{C}/2)}{{E}_{d}(\alpha {R}_{C}/2)+\left({E}_{G}-{E}_{d}\right)}\right],$$

$$\times \left[\frac{{E}_{G}(\alpha {R}_{C}/2)}{{E}_{d}(\alpha {R}_{C}/2)+\left({E}_{G}-{E}_{d}\right)}\right].$$

where the Kane matrix element appears in *E _{P}*=2

*m*

_{0}

*P*

^{2}/

*ħ*

^{2},

*g*(

_{hh}*E*′),

*g*(

_{lh}*E*′) are the heavy-hole and light-hole density of states (which explicitly use an energy dependent effective mass through Eq. (2)),

*E*′ is the final (valence band) energy,

*E*′ is the emitted photon energy, and

_{d}-E*R*is the radius of the short-ranged, deep-center potential. The emission cross section is calculated from the oscillator strength through,

_{C}*σ*=[

*πe*

^{2}

*ħ*/(2

*cn*

_{r}∊_{0}

*m*

_{0})]

*f g*(

*E*′), where

*n*

_{r},m_{0}are, respectively, the semiconductor refractive index and the free electron mass. Eq. (6) is the third main result of this Letter.

An example of a well-known [3, 8, 9] short-ranged, spherically-symmetric, deep center potential is EL2 in GaAs. Using [17] *E _{G}*=1.42eV,

*E*=0.85eV,

_{d}*E*=22.62eV,

_{P}*R*=0.17

_{C}*α*

^{-1},

*n*=3.3 for EL2 in GaAs, Eq. (6) yields the calculated cross section (solid line) in Fig. 4. The open circles in Fig.4 denote the deep-level-optical-spectroscopic (DLOS) cross sections measured by Silverberg [2], and are in agreement with cathodoluminescence, photoluminescence, and DLOS measurements made by Viturro [18] and Chantre [3]. Silverberg’s data are the well-known calibration standard for EL2 concentration.

_{r}A current research topic is the understanding of the energy dependence of the optical emission in terms of the details of the deep center bound state wave function. Fig. 4 shows that optical transitions from the deep center state to the valence band are strong only over a narrow range of transition energies. This is unlike the situation of optical transitions from the deep center state to the conduction band, which show sizeable strength for all energies from the threshold of *E _{G}-E_{d}* to the band gap energy

*E*. The results of this Letter easily explain this difference between transitions from the deep state to the conduction band and transitions from the deep state to the valence band. The explanation is that the term in braces in the oscillator strengths in Eqs. (7) and (8) is proportional to the Fourier transform of the deep center bound state probability distribution. This deep state probability distribution is dominated by Fourier components (wave vectors) less than

_{G}*α*. For the typically large heavy hole masses (≈0.5

*m*

_{0}), such wave vectors correspond to a narrow range of energies very near the valence band edge. Transitions to energies further away from the valence band edge involve hole wave vectors larger than α, for which the oscillator strengths in Eqs. (7) and (8) are small.

This Letter has three main results. We find the first, simple, analytical expressions (Eq. (1)) for deep center bound states in terms of the eigenstates of total (spin plus Bloch plus envelope) angular momentum and the 8×8 $\overrightarrow{\mathrm{k}}$·$\overrightarrow{\mathrm{p}}$ Hamiltonian. We find that the spatial extent (Eq. (3)) of the deep center bound state is proportional to the Kane dipole. This physical size of the deep center bound state is in excellent agreement with both scanning-tunneling-microscopy and measured optical emission strengths (Eq. (6) and Fig. 4).

## References and links

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**17. **We have also assumed the deep center bound states to have a Gaussian distribution with a standard deviation of 0.078eV, as is consistent with the scanning-tunneling-spectroscopy measurements of Feenstra [15].

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