1. Introduction

In current semiconductor physics literature, the two roles based on the same physical phenomenon-thermal occupation probability-of the midgap defect levels are treated inconsistently. This inconsistency is the root cause of some of the confusions and misrepresentation of semiconductor and device physics problems.

2. Current inconsistent treatment of doping and trapping

In traditional semiconductors, such as Si and GaAs, localized intrinsic (such as vacancy and antisite)/impurity/defects are classified as dopants, with shallow ionization or activation energy level less than 0.05 eV from the band edge, and as deep level traps or Shockley-Read-Hall (SRH) recombination centers otherwise. When states are considered as dopants, the occupation probability $f$ of the donor state $D$ and of the acceptor state $A$ are^[1-4]

$\begin{equation} \begin{cases} f_{\rm D} =\dfrac{N_{\rm D} -N_{\rm D}^+ }{N_{\rm D}}= \dfrac{1}{1+\dfrac{1}{g_{\rm D}}\exp \dfrac{E_{\rm D} -E_{\rm F}}{kT}} \\[10mm] f_{\rm A} =\dfrac{N_{\rm A}^- }{N_{\rm A}}= \dfrac{1}{1+g_{\rm A} \exp \dfrac{E_{\rm A} -E_{\rm F}}{kT}} \\ \end{cases} \end{equation}$

(1)

where $N_{\rm D}$, $N_{\rm D}^{+}$, $N_{\rm A}$, and $N_{\rm A}^{-}$ are the concentration of donors, ionized donors, acceptors, and ionized acceptors, respectively. $E_{\rm F}$, $E_{\rm D}$ and $E_{\rm A}$ are Fermi level, the ionization energy of the donor and the acceptor, respectively. $g_{\rm D}$ and $g_{\rm A}$ are the degeneracy of the donor and acceptor states, calculated by using Boltzmann statistics^{[5, 6]}. For most widely used tetrahedral cubic semiconductors, such as Si, GaAs, and CdTe, $g_{\rm D}$ $=$ 2 due to spin degeneracy, and $g_{\rm A}$ $=$ 4 due to heavy hole and light hole degeneracy in addition to spin degeneracy^[2]. An alternate explanation of the origin of $g_{\rm D}$ and $g_{\rm A}$ is also offered^[3]. $kT=$ 0.0259 eV at room temperature.

When the states are treated as traps or SRH recombination centers, however, without bothering its justification under specified conditions, most current literature of semiconductor physics simply assumes that the degeneracy factors $g_{\rm D}$ and $g_{\rm A}$ negligible, and uses a simplified version of the occupation formulation unified for traps-the same for donors and acceptors^{[1-4, 7]}

$f_{\rm t} =\frac{1}{1+\exp \dfrac{E_{\rm t} -E_{\rm F}}{kT}},$

(2)

where $t$ stands for trap or recombination center, either donor or acceptor. The reason to neglect degeneracy factor $g_{\rm D}$ or $g_{\rm A}$ is: if not neglecting degeneracy, the equation of SRH recombination rate^[1]

$U_{\rm SRH} =\frac{np-n_{\rm i}^2 }{(n+n^\ast )\tau _{\rm P} +(p+p^\ast)\tau _{\rm n}},$

(3)

will have two sets of formulations,

$\begin{equation} \begin{cases} n^\ast =\dfrac{n_{\rm i}}{g_{\rm D} }\exp \dfrac{E_{\rm D} -E_{\rm i}}{kT}, \\[5mm] p^\ast =g_{\rm D} n_{\rm i} \exp \dfrac{E_{\rm i} -E_{\rm D}}{kT}, \\ \end{cases} \end{equation}$

(4)

for the SRH recombination center that is a donor state, and

$\begin{equation} \label{eq5} \begin{cases} n^\ast =g_{\rm A} n_{\rm i} \exp \dfrac{E_{\rm A} -E_{\rm i} }{kT}, \\[5mm] p^\ast =\dfrac{n_{\rm i} }{g_{\rm A} }\exp \dfrac{E_{\rm i} -E_{\rm A} }{kT}, \\ \end{cases} \end{equation}$

(5)

for the center that is an acceptor state. In Eq. (3) the rates of capture of electron and of hole are defined by the carrier life time^[1]

$\begin{equation} \label{eq6} \begin{cases} \tau _{\rm n}^{-1} =\sigma _{\rm n} v_{\rm n} N_{\rm t}, \\[2mm] \tau _{\rm p}^{-1} =\sigma _{\rm p} v_{\rm p} N_{\rm t}, \\ \end{cases} \end{equation}$

(6)

where $\sigma _{\rm n}$, $\sigma _{\rm p}$, $v_{\rm n}$, $v_{\rm p}$, and $N_{\rm t}$ are the cross sections of electron and of hole, the mean thermal velocities of electron and of hole, and the concentration of the SRH centers, respectively. To avoid such complications may be one of the reasons degeneracy is neglected in dealing with localized intrinsic/impurity defect levels as SRH recombination centers.

Such an inconsistent way of dealing with a localized intrinsic/impurity/defect as a dopant and as a recombination center is particularly awkward and incorrect for the states that are not shallow as desired for dopants, nor deep as most recombination centers considered, since for those states the difference between Eqs. (4) and (5) is significant, and may not be negligible. A well known example is Cu$_{\rm Cd}$, the Cu substitute of Cd in CdTe polycrystalline thin film widely used in solar panels. Its activation energy is calculated as 0.22 eV^[8], and experimentally measured as 0.3-0.4 eV^[9]. Cu$_{\rm Cd}$ is widely recognized as a non-shallow p-type dopant^{[10, 11]}, as well as a non-deep majority carrier recombination center^[12]. Besides, it is not necessary to invoke such an inconsistent and awkward way of dealing the same state with two different sets of formulation of occupation probability.

3. Introduction of TGFEL in semiconductors

Instead of the widely used term of ionization or activation energy, Wei and Zhang^[8] used the term transition energy level, which is defined as the Fermi energy at which the energies of formation of the two electronic charge states of an intrinsic/impurity defect, such as $\Delta H_{\rm D}^{(+)}$ and $\Delta H_{\rm D}^{\rm (o)}$ for a donor, and $\Delta H_{\rm A}^{\rm (o)}$ and $\Delta H_{\rm A}^{(-)}$ for an acceptor, are equal, and the transition occurs between the two charge states. Therefore, the ionization energies or transition energy levels of the acceptor and donor are

$\begin{equation} \label{eq7} \begin{cases} E_{\rm A} =E_{\rm A}^{\rm (o/-)} =\Delta E_{\rm A}^{(-)} -\Delta E_{\rm A}^{\rm (o)} =E_{\rm F}, \\[3mm] \quad\quad\quad {\rm when}\, \Delta E_{\rm A}^{\rm (o)} =\Delta E_{\rm A}^{(-)} (E_{\rm F} )=\Delta E_{\rm A}^{(-)} -E_{\rm F}, \\[4mm] E_{\rm D} =E_{\rm D}^{\rm (+/o)}=\Delta E_{\rm D}^{\rm (o)} -\Delta E_{\rm D}^{(+)} =E_{\rm F}, \\[3mm] \quad\quad\quad {\rm when}\, \Delta E_{\rm D}^{\rm (o)} =\Delta E_{\rm D}^{(+)} (E_{\rm F} )=\Delta E_{\rm D}^{(+)} +E_{\rm F}, \\ \end{cases} \end{equation}$

(7)

where the definition and calculation of $\Delta E_{\rm D}^{(+)}$ and its like can be found in Ref. [8]. However, if we consider the transition between two electronic charge states of a defect as a phase transition in general, the condition of transition is not equality of energies of formation, but equality of Gibbs free energies. Thus, we define the transition Gibbs free energy level (TGFEL) as the Fermi energy at which the Gibbs free energies of formation of the two charge states are equal

$\begin{equation} \label{eq8} \begin{cases} G_{\rm D}=G_{\rm D}^{\rm (+/o)} =\Delta G_{\rm D}^{\rm (o)} -\Delta G_{\rm D}^{(+)} \\[2mm] \quad\quad =\left( {\Delta E_{\rm D}^{\rm (o)} -T\Delta S_{\rm D}^{\rm (o)} } \right)-\left( {\Delta E_{\rm D}^{(+)} -T\Delta S_{\rm D}^{(+)} } \right) \\[2mm] \quad\quad =E_{\rm D} -kT\ln g_{\rm D}, \\[4mm] G_{\rm A} =G_{\rm A}^{\rm (o/-)} =\Delta G_{\rm A}^{(-)} -\Delta G_{\rm A}^{({\rm o})} \\[2mm] \quad\quad =\left( {\Delta E_{\rm A}^{(-)} -T\Delta S_{\rm A}^{(-)} } \right)-\left( {\Delta E_{\rm A}^{\rm (o)} -T\Delta S_{\rm A}^{\rm (o)} } \right) \\[2mm] \quad\quad =E_{\rm A} +kT\ln g_{\rm A}, \\ \end{cases} \end{equation}$

(8)

where, if only considering the spin and heavy/light hole degeneracy, $g_{\rm D}$ $=$ 2 and $g_{\rm A}$ $=$ 4 for most widely used semiconductors of cubic tetrahedral structure. Substituting Eq. (8) to the two equations of Eq. (1), the two equations are unified as one

$f =\frac{1}{1+\exp \dfrac{G-E_{\rm F}}{kT}},$

(9)

where $G$ is the TGFEL of any SRH recombination center, whether it is donor or acceptor. It is interesting to note that the two occupation probability equations (1) are obtained by Boltzmann statistics, while their exact, not approximate, equivalent and single equation (9) has the form of Fermi-Dirac distribution.

With Eq. (9), the two sets of Eqs. (4) and (5) are also unified as the same one set of equations

$\begin{equation} \label{eq10} \begin{cases} n^\ast =n_{\rm i} \exp \dfrac{G-E_{\rm i} }{kT}, \\[3mm] p^\ast =n_{\rm i} \exp \dfrac{E_{\rm i} -G}{kT}. \\ \end{cases} \end{equation}$

(10)

Thus, the SRH recombination rate has only one formulation, independent of the trap being donor or acceptor.

It is interesting to note that the concept of

$\begin{equation} \label{eq11} \begin{cases} E_{\rm D} -kT\ln g_{\rm D}, \\[2mm] E_{\rm A} +kT\ln g_{\rm A}, \\ \end{cases} \end{equation}$

(11)

was first introduced by Kittel and Kroemer^[1] as a mathematical tool for the plotting of LCN condition without exploring its physical meaning until this work.

4. Implication of TGFEL and discussion

Apparently, the introduction of $G_{\rm D}$, $G_{\rm A}$ and $G_{\rm t}$ to replace $E_{\rm D}$ (with $g_{\rm D})$, $E_{\rm A}$ (with $g_{\rm A})$ and $E_{\rm t}$ (without $g_{\rm t}$, inconsistently) (Fig. 1) is not only a mathematical simplification and convenience, as discussed before in the derivation of the expression of SRH recombination rate. The other implications of the introduction of TGFET include:

(1) Derivation of Eq. (9), the transition level's equilibrium occupation probability, from the steady state occupation probability

$f =\frac{c_{\rm n} +e_{\rm p} }{c_{\rm n} +e_{\rm p} +c_{\rm p} +e_{\rm n}},$

(12)

where

$\begin{equation} \label{eq13} \begin{cases} c_{\rm n} =\sigma _{\rm n} v_{\rm n} n=\sigma _{\rm n} v_{\rm n} n_{\rm i} \exp \dfrac{E_{\rm Fn} -E_{\rm i} }{kT}, \\[3mm] c_{\rm p} =\sigma _{\rm p} v_{\rm p} p=\sigma _{\rm p} v_{\rm p} n_{\rm i} \exp \dfrac{E_{\rm i} -E_{\rm Fp} }{kT}, \\[3mm] e_{\rm n} =\sigma _{\rm n} v_{\rm n} n^\ast =\sigma _{\rm n} v_{\rm n} n_{\rm i} \exp \dfrac{G-E_{\rm i} }{kT}, \\[3mm] e_{\rm p} =\sigma _{\rm p} v_{\rm p} p^\ast =\sigma _{\rm p} v_{\rm p} n_{\rm i} \exp \dfrac{E_{\rm i} -G}{kT}, \\ \end{cases} \end{equation}$

(13)

where $c_{\rm n}$, $c_{\rm p}$, $e_{\rm n}$, and $e_{\rm p}$ are the transition level's capture rate of electron, of hole, and the emission rate of electron and of hole, respectively. $E_{\rm Fn}$, $E_{\rm Fp}$, and $E_{\rm i}$ are the electron's quasi Fermi level, the hole's quasi Fermi level, and intrinsic Fermi level, respectively. In view of Eq. (10), and by making $E_{\rm Fn} =E_{\rm Fp} =E_{\rm F}$, steady state equation (12) is readily reduced to Eq. (9) under equilibrium. Without Eq. (10), the derivation from Eq. (9) to Eq. (12) will be extremely messy, to say the least.

(2) With a unified and consistent expression of all the defect transition levels' occupation probability, we can visualize the ratio of the level's cross section of capturing electron and hole, as shown in Fig. 2.

Defining

$\beta =\sqrt {\frac{\sigma _{\rm n} v_{\rm n}}{\sigma _{\rm p} v_{\rm p}}},$

(14)

as shown in Fig. 2, we have

$\begin{equation} \label{eq15} \begin{cases} \sigma _{\rm n}^{(++/+)} >\sigma _{\rm n}^{\rm (+/o)} >\sigma _{\rm n}^{\rm (o/-)} >\sigma _{\rm n}^{(-/-)}, \\[2mm] \sigma _{\rm p}^{(++/+)} <\sigma _{\rm p}^{\rm (+/o)} >\sigma _{\rm p}^{\rm (o/-)} >\sigma _{\rm n}^{(-/-)}, \\ \end{cases} \end{equation}$

(15)

and

$\beta ^{(++/+)} \gg \beta ^{(+/o)}\gg \beta^{(o/-)} \gg \beta^{(-/-)} .$

(16)

So far, the largest reported experimental value of $\sigma_{\rm n}$/$\sigma_{\rm p}$ is 17000 from double donor $V_{\rm i}$, valium interstitial in Si, and the smallest measured $\sigma_{\rm n}$/$\sigma_{\rm p}$ is 2 $\times$ 10$^{-5}$ from double acceptor Zn$_{\rm Si}$, zinc substitute in Si^[16], which seem to support Eqs. (15) and (16). Equation (15) shows the effect of defect charging state on its carrier capture cross sections. Equation (16) compares the ratios of electron capture cross section versus hole capture cross section of various types of transition Gibbs free energy levels due to Coulomb interaction between the carrier and the defect. Nevertheless, Equations (15) and (16) are by no means exact. They only reflect one factor-the Coulomb attraction or repulsion between the free carrier and the SRH center-that may affect the carrier capture cross section by the defect level. When other factors, such as multiphonon process, play more important roles than Coulomb interaction, Equations (15) and (16) may not be valid.

(3) In addition, TGFEL has its implications in physics. The introduction of TGFEL suggests that the basic chapter of doping and the chapter of generation-recombination of semiconductor physics may need to be modified. First, conceptually, the energy level associated with a defect, either donor, double donor, acceptor, or double acceptor, is more phase transition referenced to the Fermi level, rather than ionization referenced to the band edge, as shown in the LCN condition equations. Second, the electronic properties experimentally measured, such as the ionization or doping level shown in Fig. 1, as well as the recombination rate shown in Eq. (3), are based on Boltzmann statistics, from which only one parameter $G_{\rm D}$, $G_{\rm A}$, or $G_{\rm t}$, not a pair of parameters $E_{\rm D}$ and $g_{\rm D}$ (or $E_{\rm A}$ and $g_{\rm A}$, $E_{\rm t}$ and $g_{\rm t})$ that can be extracted. To place in an equation some parameters that cannot be measured physically may not be appreciated. Third, the validity of Eq. (8) is not restricted to the case of $g_{\rm D}$ $=$ 2 and $g_{\rm A}$ $=$ 4. Other sources of degeneracy, not due to spin and heavy/light hole, but due to the atomic configuration or phonon coupling of the defect, can also be included in $g_{\rm D}$ and $g_{\rm A}$. Furthermore, $E_{\rm D}$ and $E_{\rm A}$ are defined as temperature independent energy of ionization, while $G_{\rm D}$ and $G_{\rm A}$ defined by Eq. (8) are explicitly temperature dependent. This does not make transition Gibbs free energy level physically less meaningful. On the contrary, the temperature dependence of $G_{\rm D}$, $G_{\rm A}$, and $G_{\rm t}$ may be associated with some not well understood phenomena, such as the Meyer-Neldel rule^[13], which is used to explain the deviation of the expected Arrhenius plot in the cross section measurement of the states^[13-15]. Whether Meyer-Neldel rule is valid for the defect cross section and activation energy is still being debated. Some current literature considers only symmetry induced degeneracy for shallow dopants, while others consider multiple phonon effect for deep levels^{[13, 14]}. Such an inconsistent, even incorrect treatment of the same physical phenomenon may contribute to the controversy.

In summary, in this work we treat the ionization of a localized state, whether it is shallow or deep, donor or acceptor, as a phase transition, and thus introduce the concept of TGFEL. Note that only single level defects are dealt with. Multi-level defects are dealt with elsewhere^{[17, 18]}.

Acknowledgement: The author would like to acknowledge the support from CNBM (China National Building Materials) Group for its partial financial support of the work. The author also acknowledges the inspiring and helpful discussion he has had with the faculty and students at NJIT's CNBM Center, Professors J. S. Liu and Z. Zheng of Beihang University, Beijing, China, and Professors X. C. Shen and W. Lu of the Shanghai Institute of Technical Physics, Academy of Science of China, where he is a visiting professor.