(Junsung Park)
^{1}
(SungMin Hong)
^{1}^{†}

(School of Electrical Engineering and Computer Science, Gwangju Institute of Science
and Technology)
Copyright © The Institute of Electronics and Information Engineers(IEIE)
Index Terms
Multiphonon emission process, first principles calculation, density functional theory
I. INTRODUCTION
Point defects cause unintentional physical effects on semiconductor devices^{[1]}. Especially, in the MOS (metaloxidesemiconductor) structure the bulk and interface
defects degrade device performance and stability^{[2,}^{3]}. These defects act as charged traps or recombination centers. For the siliconbased
electronic devices, the nonradiative recombination is the major mechanism of the
defectinduced trapping process^{[4]}. Despite of practical importance and tremendous research efforts, rigorous analysis
of defects in the semiconductor devices is still a difficult task^{[5]}.
The nonradiative multiphonon emission (MPE) has been widely adopted to describe the
nonradiative trapping process. Many studies have made theoretical contributions over
the last 60 years^{[6}^{8]}. A common problem of these models is that the calculated result sensitively depends
on model formulations^{[9]}. This problem is solved by evaluating electronphonon coupling constants via the
firstprinciples calculation^{[10]}. By first principles calculation based approaches, physical quantities like the formation
energy and the transition level of a charged defects can be obtained^{[11]}.
These methods have been successfully applied to obtain nonradiative recombination
coefficients of IIIV semiconductor systems^{[12}^{[14]}. In this paper, we apply the calculation method to the Si bulk and the SiSiO_{2} interface. In this study, the Si bulk model is used to check the physical soundness
of our first principle based MPE calculation method. The SiSiO_{2} interface structure model is considered to analyze the nonradiative recombination
phenomena when defects are formed at the oxidesemiconductor interface.
This paper is organized as follows. In Sec II, the theoretical formulation based on
the first principle approach is explained in detail. The calculation workflows and
computational tools for the calculation are also introduced. In Sec III, the atomistic
structures with defects for the calculation are introduced. In Sec IV, results of
the MPE calculation are presented. In Sec V, conclusions are drawn.
II. Theoretical Formulation and Calculation Method
First principlesbased calculations are powerful tools for material properties calculations
^{[15]}. The density functional theory (DFT) calculation based on the KohnSham equation
is used as the most popular first principles calculation method with the accuracy
of calculation and the reasonable computational cost ^{[16]}. In this paper, we have calculated the characteristics of defect structures required
for MPE calculation through the DFT approach.
The calculation is performed for the atomistic supercell structures with defects.
All calculations are derived from the results of a pristine structure without a defect
and a charged or neutral structure with defects. The thermodynamic formation energy
of a defect and its transition levels are key factors to the MPE simulation workflow.
The formation energy $E^{f}\left[X^{q}\right]$ of a defect $X$ at charge state $q$
in a charge state $q$ is defined as [11]^{[11]}
where $E_{t}\left[X^{q}\right]$ is the total energy derived from a supercell containing
defect X and $E_{t}[b u l k]$ is the total energy of defectfree crystalline supercell.
The integer $n_{i}$ is the number of each defect type and $\mu_{i}$ is the chemical
potential. $E_{\text { corr }}$ is a correction term for the charge interaction between
neighboring periodic supercells. Fig. 1 schematically shows the formation energy and its transition levels in the bandgap
as functions of the Fermi energy level $E_{F}$.
Fig. 1. Schematic illustration of the formation energy $E^{f}$ versus the Fermi level
$E_{F}$ for a defect X for three charged states ($q :1,0,1$)).
The thermodynamic transition level $\varepsilon\left(q_{1} / q_{2}\right)$ is defined
as the Fermilevel position for which the formation energies of charge states $q_{1}$
and $q_{2}$ are equal (cross point):
The thick blue lines represent the minimum energies that each cell with a defect can
have, depending on the charge state of the defect. The intersecting point is the transition
level according to the pair of charge states.
Fig. 2 shows a schematic diagram illustrating how transitions are made through the MPE process.
Let us consider a MPE process shown in Fig. 2(a). The defect energy with respect to the configuration coordinate is shown in Fig. 2(b). $\Delta Q$ is the distance between the defect and the neighboring (electrically
interacting) atom. In Fig. 2(a), $\Delta E$, which means the minimum ionization energy level of the hole, is defined
as the closest transition level to the valence band in the supercell calculation of
the defect. (Position of $\varepsilon(+/ 0)$ in Fig. 1) In Fig. 2(b), the $\Delta Q$ value shows the distance between adjacent atoms of the defect in
the supercell calculation. The curves in Fig. 2(b) are indicate the potential curves in neural (black, $A^{0}$) and negative charged
defect ($A^{0+h^{+}}$). The hole carrier capture process is occurred by a vibrational
relaxation process between the two potential curves^{[8,}^{11]}.
Fig. 2. Illustrations of nonradiative multiphonon emission transition process (a)
band diagram, (b) configuration diagram. In this illustration, the defect is a deep
acceptor with a negative and a neutral charge state. $\Delta E$ is the ionization
energy of the acceptor, and $Q$ is relative defect configuration coordinate.
Recently, several calculation methods are suggested based on the first principlesbased
MPE transition scheme^{[8,}^{17,}^{18]}. In this study, we employ a method based on the adiabatic approximation^{[17]}. The electron transition rate between electron states $s$ and $l$ can be expressed
as
where $\Delta E_{s l}$is the total energy change between the atomic relaxed $s$ and
$l$ electron states. (same as $\Delta E$ in Fig. 2) $k$ is the index of phonon mode with frequency $\omega_{k}$. Also, $D^{2}=\frac{1}{2}
\sum_{J} \omega_{j}^{2} \Delta_{j}^{2}\left(2 \overline{n_{j}}+1\right)$, $\overline{n_{j}}=
\left[\exp \left(\beta \hbar \omega_{j}\right)1\right]^{1}$, $\Delta_{j}=\left(\frac{M_{j}
\omega_{j}}{\hbar}\right)^{\frac{1}{2}}\left(Q_{j}^{0(s)}Q_{j}^{0(l)}\right)$, $y_{k}=
\beta \hbar \omega_{k} / 2$, and $\beta=(k T)^{1}$. The relaxed ground state coordination
of phonon mode $j$ are $Q_{j}^{0(s)}$ and $Q_{j}^{0(l)}$ when electron states $s$
and $l$. Electron states s and l are the lowest charged transition level in the bandgap
and the first three valence bands, respectively. All electron states are evaluated
from the first principle calculation of the structure with the defect. The reorganization
energy of relaxed atom after electronic transition is $E_{M}=\frac{1}{2} \sum_{j}
\hbar \omega_{j} \Delta_{j}^{2}$. All parameters come from the first principlesbased
calculation and its analysis.
The electronphonon coupling constant in (3) between electronic states $s$ and $l$ and phonon mode $k$ is
where $\mu_{k}(R)$ is the kth phonon mode, $\psi_{s}$ and $\psi_{l}$ are electronic
wave functions of electron states $s$ and $l$, and $H$ is the system Hamiltonian of
the supercell with the defect. The capture coefficient is obtained by multiplying
the transition rate $W_{s l}$ and volume of the supercell, $V$. ($C_{p}=V W_{s l}$)
In this work, as a preliminary study, only the gamma point in the momentum space and
the first three phonon modes are used in the calculation. Most of the transitions
in shallow charged traps depend on low energy phonon modes. These phonon dispersions
are distributed near the gamma point so that a physically correct result can be obtained.
The results obtained from more sampling points in the momentum space and phonon modes
will be reported in the future publication.
All of the electronic structure calculations are performed by a DFT simulation package,
Vienna ab initio simulation package (VASP)^{[19]}. The VASPDensity Functional Perturbation Theory (DFPT) calculation is performed
to obtain the exact force constraint. The phonon modes are calculated through the
Phonopy package^{[20]}. In order to perform all calculation procedures in the python environment, we process
data from VASP and Phonopy on the pymatgen library, the python based data processing
and I/O packages^{[21]}. We also use the pycdt package for data normalization for the supercell correction
term calculation. FreysoldNeugebauerVan de Walle (FNV) supercell correction scheme
is used^{[22,}^{23]}.
III. ATOMISTIC MODELS
1. Silicon bulk Model
Fig. 3 shows the atomistic models of silicon bulk structures with or without a defect. A
64atom ($3 \times 3 \times 3$) supercell is used to calculate the formation energies
and the transition levels. The planeaugmented wave (PAW) cutoff energy of $400 eV$
and $3 \times 3 \times 3 k$points are applied for electron structure calculations.
For the accurate bandgap calculation, the HeydScuseriaErnzerhof (HSE) screened hybrid
functional is used^{[24]}. Two types of are considered: the silicon ($Si$) vacancy ($V_{Si}$) and the boron
(B) substitutional ($Si : B$).
Fig. 3. Supercells of the Si bulk structures (a) A defectfree supercell, (b) a silicon
vacancy ($V_{Si}$) defect supercell, (c) a boron substitutional ($Si : B$) supercell.
Blue circles are corresponding to defect sites.
2. SiSiO_{2} Interface Model
Fig. 4 shows the atomistic slab structure of SiSiO_{2} interface models. In the previous simulation studies, αquartz or βcristobalite
structures were used for interface simulations^{[25,}^{26]}. In this study, we use αquartz (1000)/silicon(100) slab model (see Fig. 4(a)). The optimized lattice constants of atomistic models are obtained from the materials
projects database^{[27]}. Native vacancy defects are assumed to be the major interface defects which are formed
at the semiconductoroxide interface, so the Si and oxygen ($O$) vacancy ($V_{Si}$,
$V_{o}$) defect models are used for the simulation. Since the surrouding enviroment
of the defect may affect its electrical properties considerably, the calculation of
the interface defect is performed independently from the bulk model. The defect is
formed near the SiSiO_{2} interface. Fig. 4(b) shows the location of defects. The PAW cutoff energy of 400 eV and $3 \times 3 \times
3 k$points are applied for electron structure calculations. The PerdewBurkeErnzerhof
(PBE) potential is used for geometry optimization and electronic structure calculation^{[28]}. Structural minimization is performed until all the atomic forces are less than $0.01
eV Å$. To compensate for bandgap underestimation problems by using PBE functional,
band gap and defect energy correction method is used^{[29]}.
Fig. 4. The SiSiO_{2} interface structure (a) The minimum structure cells building the interface structure,
(b) The position where the defect is formed on the interface structure. The defect
is located at SiSiO_{2} molecules bond.
IV. CALCULATION RESULTS
1. Silicon Bulk Model
Fig. 5(a) shows the formation energies for a silicon vacancy ($V_{Si}$) and a boron substitutional
($Si : B$). The thick lines indicate the energetically minimal charge states for a
given Fermi level. Fig. 5(b) shows the charged transition levels, which are intersections in Fig. 5(a). The ($+/ 0$)) transition level of the $V_{Si}$ structure and the ($0 /$)) transition
level of the $S i : B$ structure are close to the valence band maximum. The energy
distances from the valence band maximum are $15 meV$ for the $V_{S i}$ structure and
$117 meV$ for the $Si : B$ structure, respectively.
Fig. 5. (a) Calculated formation energy versus the Fermi energy level for Si vacancy
( $V_{Si}$ ) and substitutional B ( $Si : B$ ) in Si supercell, (b) Charged transition
levels.
In Fig. 6, the configuration coordinate (CC) diagram is plotted for the $Si : B$ system. The
generalized coordinate $Q$ is the massweighted CC of the defect for the Cartesian
coordinate system $R$^{[18]}.
Fig. 6. Calculated configuration coordinate (CC) diagram for the hole captured substitutional
B ( $Si : B$ ) in the Si supercell system. Symbols are calculated values from the
DFT simulation. Quadratic lines are fitted to the calculated values. Each curve represents
the neutral ground state (black line) and the hole captured excited state (purple
line). $\Delta E_{Si}$ is the energy difference between minima of two curves. It represents
the same theoretical concept as minimal transition levels in bandgap (Fig. 2) and energy difference of two electronic states (Eq. (3)).
The ground state (bottom black line and symbol) is the noncharged neutral state for
the $Si: B$ supercell. The excited state (top purple line and symbol) is corresponding
to the defect in the negatively charged state ($ / 0$) and a hole in the valence
band. $\delta Q$ is the total sum of relaxations of neighbor atoms around the defect
site. The energy difference between two minimum points of the curve is equivalent
to the difference between the charged transition level and the valence band maximum
in the formation energy diagram (see Fig. 5). $\Delta E_{s l}(Si : B)=118 meV$ from the supercell calculation, and the CC offset
of two states is $\Delta Q=2.36 a m u^{\frac{1}{2}} Å$ . The quadratic fitting model
(lines) comes from a reference paper^{[30]}.
Fig. 7 shows calculated capture cross sections and experimental values for the Si bulk supercell.
The hole capture cross section is obtained by $\sigma_{p}=C_{p} / v_{p}$ ( $v_{p}$
is the mean thermal velocity of holes). When the temperature is increased, the MPE
transition is increased because more energetic phonon branches are activated. Both
of Fig. 7(a) and Fig. 7(b) show an inverse correlation between the decrease in temperature and the value of
the hole capture cross sections ( $\sigma_{p}$ ). At T=300K, we get $\sigma_{p}$=2.26×10^{18} cm^{2} for $V _{Si}$ and $\sigma_{p}$=2.01×10^{18} cm^{2} for $Si : B$. The calculated cross section ( $\sigma_{p}$ ) is about one orderofmagnitude
smaller than experimental results ^{[31}^{33]}. One possible cause of such discrepancy is the gamma point sampling employed in this
work. Further investigation with additional sampling points would be required. It
is also noted that similar results are also found in some calculation results in the
existing reference of first principles based MPE calculations^{[8,}^{17,}^{18]}.
Fig. 7. Calculated nonradiative hole capture cross sections as functions of 1000/T
(K^{1}) for (a) $V _{Si}$ , (b) $Si : B$ defects in Si supercell. The stars are experimental
data taken from [3133]^{[31}^{33]}.
2. SiSiO_{2} Interface Model
Fig. 8 shows the calculated formation energies in the SiSiO_{2} interface structure. Since the charged transition levels of Si vacancy are below
the valence band maximum, we cannot use these transition levels for our calculations.
Instead, in the case of the SiSiO_{2} model, the capture parameter is obtained as a function of charged transition level
($\Delta E$).
Fig. 8. Calculated formation energy versus Fermi energy level for Si and O vacancies
($V_{o}$, $V_{Si}$) in the SiSiO_{2} interface structure.
Fig. 9 shows the hole capture cross section of each defect type according to the value of
the charged transition level ($\Delta E$). As $\Delta E$ increases, the hole capture
cross sections are decreased logarithmically. As the $\Delta E$ value increases, the
phonon energy required for the MPE increases, so the capture cross section decreases.
The total MPE is defined as a sum of hole capture cross sections of two defect models,
$V_{o}$ and $V_{Si}$. The calculated capture cross section values are smaller than
the experimental values^{[34,}^{35]}. This tendency is the same as the result of the Si bulk supercell cases.
Fig. 9. Calculated nonradiative hole capture cross section of O and Si vacancy ($V_{o}$,
$V_{Si}$) defects in SiSiO_{2} interface structure.
V. CONCLUSION
We have performed the first principlesbased nonradiative MPE calculations for the
Si bulk and the SiSiO_{2} interface supercell structures. Using the method based on the adiabatic approximation,
the hole capture cross section is calculated in those structures. Comparison with
the experimental results reveals that the calculated capture cross section shows physically
sound behaviors with significantly smaller magnitude. Considering more kpoints sampling
and phonon modes, better agreement with the experimental results is expected.
It is stressed that our calculation does not depend on the parameters obtained from
the experiments for the trap characterization. The authors hope that this approach
can be further developed to analyze the physical effects of defects in semiconductor
materials and semiconductor device interfaces.
ACKNOWLEDGMENTS
This work was supported by the Basic Science Research Program (Grant No. 2016R1C1B1014852)
through the National Research Foundation of Korea funded by the Ministry of Science,
ICT, and Future Planning.
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Author
received the B.S. degree in electrical engineering and the Ph.D. degree in electrical
engineering and computer science from Seoul National University, Seoul, Korea, in
2001 and 2007, respectively.
He is currently an Assistant Professor with the School of Electrical Engineering and
Computer Science, Gwangju Institute of Science and Technology, Gwangju, Korea.
His main research interests include physicsbased device modeling.
received the B.Eng. degree in electrical engineering from Kyungpook National University,
Daegu, Korea, in 2015 and is currently working toward the Ph.D. degree in electrical
engineering from Gwangju Institute of Science and Technology (GIST), Gwangju, Korea.
His main research interests are defectrelated physical phenomena in MOS interface
of modern semiconductor devices.