† Corresponding author. E-mail:
A brief history of the impurity theories in semiconductors is provided. A bound exciton model is proposed for both donor- and acceptor-like impurities and point defects, which offers a unified understanding for “shallow” and “deep” impurities and point defects. The underlying physics of computational results using different density-functional theory-based approaches are discussed and interpreted in the framework of the bound exciton model.
Impurities and point defects are very similar in terms of their primary functions in semiconductors. For instance, they both can behave as electron donors or acceptors to change the electrical conductivity of the material. In terms of theoretical treatments, the impurity and defect problem are also very similar, namely, one host atom on one particular lattice site is replaced by another atom that normally should not be there in the perfect lattice. A few typical examples are offered here to illustrate the point that most impurities or point defects behavior either as a donor or an acceptor in a semiconductor. (i) In Si, Al substituting for Si results in an acceptor state, because Al has one less valence electron than Si and Al 3p state is higher than Si 3p state, an unoccupied impurity state is likely to appear above the top of the valence band or valence band maximum (VBM). (ii) In GaAs, a Ga on As anti-site defect is expected to behave as an acceptor, because the As site replaced by Ga is short of two valence electrons and Ga 4p state is higher than As 4p state, an unoccupied defect state is likely to appear above the VBM. Here Ga on the wrong lattice site can be viewed as either an anti-site defect or an impurity. (iii) In GaP, an N impurity substituting for P is another example of an acceptor impurity, which is often known as an isoelectronic impurity, because N and P have the same number of valence electrons. Because N 2s state is lower than P 3s state, an N impurity has the tendency to form an empty level (a s-like anti-bonding state) below the bottom of the conduction band or conduction band minimum (CBM), which can be viewed as a deep acceptor. By deep acceptor, we mean that the acceptor level is very far away from the VBM for the electron in the valence band to make a transition to the acceptor level. (iv) In NaCl, a Cl vacancy behaves like a deep donor, because in this ionic crystal, Na atoms are supposed to give out their valence electrons to Cl atoms; now with one Cl missing, one extra valence electron of the nearby Na atoms has to find a state to occupy. This state turns out to be localized at the Cl vacancy. This defect state is a deep donor state; i.e., the bound electron has a large binding energy with respect to the CBM.
The Cl vacancy, known as a “color center” or “F center”, played a very important role in the history of the impurity and defect theory in semiconductors. The well-known hydrogen model was first proposed by Gurney and Mott[1] and Tibbs[2] to understand the electronic structure of this simple point defect. For the extra valence electron, if somehow the vacancy site can still keep this electron, then the general volume of the crystal, away from the vacancy site, will more or less remain the same as the defect-free crystal. This arrangement is indeed possible and it is normally considered as a neutral state of the vacancy (V0). However, if the electron is released from the vacancy, for instance, being excited into the conduction band, the crystal will exhibit some conductivity and we may say that the Cl vacancy is ionized. In the latter case, relative to the charge distribution of the defect-free crystal, the defect site has a positive charge, thus, the vacancy is said to be in +1 charge state (V+). One could view a vacancy as a virtual atom that has an empty electronic state at the vacuum level, which suggests that the vacancy site tends to push away the electron, acting like an anti-quantum dot. Alternatively, a vacancy could also be viewed as an interface between the vacuum and crystal “surface” with dangling bonds. The dangling bonds in a semiconductor surface are known to often generate a surface state that is highly localized at the surface. Although the vacuum space is quite small, a highly localized state can indeed form in the small cavity to accommodate the “orphan” electron. Because this state is an anti-bonding state in nature (to be explained later), its energy level is mostly likely to be close to the conduction band, which is a rather general phenomenon for an anion vacancy in an ionic crystal.[3] The examples given above illustrate that at least most impurity and defect problems can be understood qualitatively in a similar way by considering the electronic structure difference between the host and “impurity” atom, with the help of the knowledge about the host band structure, to predict if the “impurity” should behave as an acceptor or a donor. This point is particularly important for the introduction of a new and unified theoretical framework[4] for the impurity and point defect problem in this paper. Therefore, in the discussions below, the word “impurity” may be understood as representing either impurity or point defect, unless specifically stated otherwise.
In the literature, impurities are typically classified into two categories: “shallow” and “deep”. In the early days, an impurity is deemed as “shallow” when the separation of its ground state energy level from the relevant band edge, i.e., “impurity binding energy”, is comparable to the thermal energy kT corresponding to room temperature (the usual device operating temperature), and as “deep” otherwise.[5] This intuitive classification is practically useful because from the device operation point of view, the exchange of electrons between the impurity levels and the bulk band states depends sensitively on the “impurity binding energy”. The exact meaning of the term “impurity binding energy” will be the subject of later discussions. However, another criterion of classification of “shallow” and “deep” impurities has also been widely used. This emphasizes the difference in the degree of impurity potential localization by recognizing the fact that an impurity level despite being generated by a highly localized impurity potential can be energetically very close to the band edge but its properties can be very different from the specific band edge.[6] The best example may be the electron bound state of an isolated N impurity in GaP, GaP: N, that has an impurity level very close to the lowest conduction band state near the X point but the pressure response of the exciton bound to the N center is found to be very different from that of the X point. Another well-cited example is a resonant state of an isolated N impurity in GaAs, GaAs: N, that has an impurity level well above the conduction band edge at the Γ point and somewhat close to the conduction band L point but the pressure response of this impurity level does not follow any of the three critical points at Γ, L, and X. The qualitative explanations for these two examples are relatively simple: an impurity state associated with a highly localized potential will require states throughout the BZ, maybe even from different bands, to serve as a basis for its wave function expansion using the host band states, thus, one cannot expect the impurity state to behave like one particular band edge state, even they could be incidentally close to each other. Because of the examples like these two isoelectronic impurity systems, it was proposed to classify the impurities as “shallow” and “deep” based on the degree of impurity potential localization.[6] Of course, this classification scheme was introduced with respect to the well-established theory for “shallow” impurities, primarily, those typically referred to as donors and acceptors that were generally believed to have a screened Coulombic impurity potential that was much more extended than the impurity potential of the isoelectronic impurity. In the new framework of the impurity model to be described in this paper, the distinction of the “shallow” and “deep” impurity will essentially disappear, at least on the qualitative level.
The best known and most widely used theory for a donor or an acceptor-like impurity is the so-called hydrogen model with a screened Coulomb potential and an effective mass, respectively, for either electron or hole. In this model, for a non-degenerate conduction band with parabolic dispersion near a special k point, the donor binding energy ED > 0 is the solution of the equation below[7]
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![]() | Fig. 1. Hydrogen model for a Cl vacancy in NaCl given by Tibbs (1939).[2] Above: potential energy of an electron in the field of a vacant Cl lattice point (full line). The broken line represents −e2/(ε0r). Below: the envelope wave functions of an electron in a Cl vacancy. + for Na+ ions, • for Cl− ions. Source: Mott and Gurney.[8] |
Today, with the vast improvement in both computation power and theoretical methodology, in principle, we should be able to examine more closely various concepts and models proposed intuitively in the early stage of the semiconductor research, using first-principles based techniques. Indeed, recent advances in first-principles density functional theory (DFT) have made it the tool of choice for studying the properties of defects in semiconductors.[15,16] However, when coming to compare the computational results with experimental data, one will find it not at all a straightforward task. Ultimately, a correct conceptual understanding is important in interpreting the first-principles results.
At first glance, the model proposed by Gurney and Mott, and Tibbs is quite reasonable. It was stated by Tibbs in his paper:[2] “Suppose that a negative ion is removed from the interior of such a crystal, leaving a vacant, negative-ion lattice point. This is equivalent to putting a positive charge at the point in the crystal from which the negative ion is removed. … the potential field in the crystal due to this positive charge is e/K.r, where K is the dielectric constant for static fields . . .”. Tibbs then treated the electronic states in the field of the positive charge using an effective-mass hydrogen model. Note that the crystal with a missing ion, a charged system, usually is unstable, thus will be neutralized by an electron in the environment. As stated by Tibbs, “if we introduce an electron into the lowest state of this potential hole the crystal is again electrically neutral”. Actually, for the crystal to remain charge neutral, the “added-back” electron does not have to go into the lowest state but can be anywhere in the crystal. Therefore, it makes more sense to consider an alternative scheme; i.e., removing one (neutral) Cl atom as a whole. In this way, the crystal always remains charge neutral. One could view the system with such a vacancy to be equivalent to a system with a substitutional impurity of a virtual “empty” atom; i.e., a small vacuum space, replacing the Cl atom. Either way, in real space, one will need to decide where to put the “added-back” electron or the extra valence electron of the nearby Na atoms that would have been transferred to the removed Cl atom. In energy space, the question will be: which energy level should this electron go into? There are two distinct options of allocating the extra electron: one is to place it back to the vacancy site; another is to let it be away from the defect site, although it might still sense the attractive force of the potential hole. For the latter option, if the electron is able to escape the attraction all together, it will become a free conducting electron in the conduction band. The situation is more complicated for the former option. One would need to specify an energy state with its wave function being largely localized at the vacancy site. What kind of state is it expected to be and where should its energy level be? Let us do a simple electron counting. Assuming that the defect-free crystal has N0 occupied valence states for hosting 2N0 valence electrons, the defected crystal has N0 − 4 valence states due to missing one Cl atom (one 3s orbital, and three 3p orbitals), and can accommodate 2N0 − 8 valence electrons. In the meantime, the system has 2N0 − 7 valence electrons, which implies that one valence electron (the one from the Na atoms next to the vacancy) will need to occupy an excited state above the fully occupied valence band. One of the occupied valence states is in fact a singlet bonding-state of the vacancy, which could be understood as a bonding state of the Na “dangling bonds”. This bonding state is expected to lie deeply in the valence band, known as a “hyper-deep” defect state, which is localized in the vicinity of but not exactly at the vacancy site, because the vacancy site (vacuum) has the highest electron potential energy. There will correspondingly be a singlet anti-bonding state somewhere close to the conduction band.[3] The defect state that can host the orphan electron is exactly this anti-bonding state that is expected to be more localized at the defect site than the “hyper-deep” bonding state. With this physical picture in mind, one may now realize that the exact position of this energy level should depend on the difference between the vacuum level and the atomic orbitals of the atoms involved. The question would be: is it at all reasonable to expect this defect state to be like a hydrogen ground state? One could already see that the formation of this defect state is quite complicated such that the possibility for it to be a hydrogen-like state is rather small.
Is there any problem with the idea that the Cl vacancy in NaCl or more general a donor atom in a semiconductor (e.g., P in Si) would generate a hydrogen-like long-range potential away from the defect site? For the case of Cl vacancy, if the extra electron is taken away from the defect site, then there will be indeed a positive charge at the vacancy site with respect to the defect-free ionic crystal. For a hydrogen atom, whether or not the electron is present, the Coulomb potential generated by the core remains the same—i.e., −e/r—and the electron energy levels are determined by this potential. However, for a more complex atom, even He, the single particle potential for a He ion (He+) is significantly different from that of H+. Therefore, even though both H+ and He+ have the same positive charge, the ionization energies of H and He are very different (13.6 eV vs. 24.6 eV). Based on this consideration alone, one cannot really expect that the electronic structure of a donor would be something close to what predicted by the hydrogen model because of the many-particle effect and/or the variation in the detailed bonding situation with the host. With this understanding, we can say that the model potential depicted in Fig.
To help readers visualize the ground state electron distribution in NaCl with a vacancy, a one-dimensional (1D) model is provided in Fig.
![]() | Fig. 2. (color online) 1D electron occupation model for NaCl crystal with one Cl vacancy for the ground state of the system or the neutral vacancy state V0. |
Now let us accept that a Cl vacancy does introduce a bound state below the CBM with a binding energy EI, which will be referred to as impurity binding energy. In the single electron picture, the conduction band states represent the electronic states free from the interaction with the defect center. However, when the electron tries to escape from the defect site or is excited from the lowest defect level, a Coulombic attractive potential arises, which qualitatively corresponds to the situation described in Fig.
We will next discuss acceptor-like impurities to further help the conceptual understanding offered above. The case of the acceptor seems to be somewhat more transparent in physics than that of the donor.
When an impurity with one or more valence electron(s) less than that of the replaced host atom is introduced into an otherwise perfect semiconductor, it typically introduces a partially occupied state near the top of the valence band. Such an impurity is often referred to as an acceptor, because it can accept one or more electron(s) from the valence band by thermal excitation, assuming these states are relatively close to the VBM. Here we do not consider the trivial case where the impurity level turns out to be below the VBM, and thus the acceptor will be self-ionized (i.e., generating a hole in the valence band), effectively resulting in a metallic material. In a typical textbook description, an acceptor is a negative charged center that has an attractive Coulomb potential for the hole, which introduces a hole bound state or an empty electron level at energy EA above the VBM. EA is known as acceptor binding energy, and understood as the energy needed to promote an electron from the VBM to the impurity level and thus generate a free hole in the valence band. Consequently, the transition energy for an electron in the conduction band to the acceptor level, known as a free-to-bound transition, would be EF−B = Eg − EA, where Eg is the bandgap. With this understanding, the energy diagram of an acceptor center and the related transition energies are illustrated in Fig.
Again the previously-mentioned standard textbook description about acceptor is conceptually problematic. Despite having fewer valence electrons, the acceptor impurity is in fact charge neutral if the sample temperature is sufficiently low. It is important to understand that a neutral acceptor impurity does not have a long-range Coulomb potential centered at the impurity site. The attractive Coulomb potential in Eq. (
![]() | Fig. 6. (color online) Comparison of valence atomic energy levels with the acceptor impurity binding energies in Si. (a) Valence atomic energy levels calculated using a density functional theory within a local density approximation for most group II–VI elements (provided by Wei[19]). (b) Left: p valence electron energy levels of Group III elements with respect to that of Si 3p energy level (from the graph in panel (a)); right: experimental values of the acceptor impurity binding energies of Group III elements in Si. |
Till now we have implicitly assumed that the acceptor level to be occupied by the electron taken from the valence band is related to the p valence state or the first ionization energy of the acceptor atom. Conceptually, because we are adding an additional electron to the impurity atom, this level should resemble more of the second ionization level or correlate with the electron affinity of the impurity atom. In a freestanding atom, this level tends to be much higher than the valence state due to the screening effect of the core electron(s). In a crystal, because of the delocalization of the valence electrons, the screening effect is expected to be much weaker, so we may not need to emphasize this subtle issue at least in the qualitative level. However, for B its 2p level is very close to the Si 3p. Although shown to be somewhat higher in Fig.
In the next step, after the acceptor level is occupied by an electron from the valence band, the hole left behind is not free but still attracted to the ionized center A−. It is this potential that gives rise to those hole bound states shown in Fig.
One complication should be noted, which is that an acceptor typically introduces multiple impurity states of which are either fully or partially occupied, instead of merely one empty state. For instance, an element with three p electrons replaces one host atom in Si, the impurity states will be p-like occupied by five p-like valence electrons (of which 4 from the nearby Si atoms and one from the impurity) with one empty state. If a spin-orbit interaction is taken into account, these p-like impurity states will split into two states with the lower one fully occupied. When drawing a band diagram, one typically ignores the existence of occupied impurity states.
What exactly is the difference between the above-described acceptor model and the conventional one? It is interesting to read the description about the acceptor in a classic book (published in 1950) entitled “Electrons and Holes in Semiconductors” by Shockley,[21] which is quoted here: “The hole in one of the bonds to the boron atom can be filled by an electron from an adjacent bond, and the hole can thus migrate away, as described in Fig. 1.5(d). The boron thus becomes an immobile, localized negative charge. Because of the symmetry between the behavior of holes and electrons, we can describe the situation shown in Fig. 1.8 by saying that the negative boron atom attracts the positively charged hole but that thermal agitation shakes the latter off at room temperature so that it is free to wander about and contribute to the conductivity.” Although the conventional model does recognize the transfer of the additional electron from the adjacent Si, it is implicitly assumed the transfer occurs spontaneously without costing any energy. Therefore, the conventional acceptor model somehow skips the first step or fails to recognize the independent identity of the neutral impurity state. There, this state is either non-existent or implied to be the same as the hole bound state with a binding energy EA. After understanding this subtle point, we may conclude that the acceptor binding energy described by the hydrogen model in the conventional acceptor theory is in fact equivalent to the hole binding energy in the bound exciton model for the acceptor. The underlying physics of the bound exciton model can be further clarified after the acceptor-bound exciton is compared with an isoelectronic impurity bound exciton in the next section.
We first describe the electronic structure of an exciton bound to an isoelectronic impurity of an electron trap, such as GaP: N, where the bound exciton is known as an “acceptor-like bound exciton” based on the model proposed by Hopfield, Thomas, and Lynch (HTL model).[22,23] This bound exciton problem is viewed as a classic example of the “deep” impurity that is thought to be profoundly different from the “shallow” impurity, either an acceptor or a donor, in terms of the extension of the impurity potential.[6] As illustrated in Fig.
We now explicitly compare the absorption spectra of an isoelectronic bound exciton and an acceptor bound exciton to illustrate the similarity in their electronic structures. Figure
![]() | Fig. 8. (color online) Photoluminescence excitation spectrum of NN1 center in GaP: N (from Cohen and Sturge[26]). Vertical dashed lines and labels in red color are added to indicate the primary transition energies. |
The bound exciton formation is ultimately a many-electron problem that should be treated as the transition between two states of the whole system. Figure
We now compare the absorption spectrum of an acceptor with the example of the isoelectronic bound exciton. Figure
![]() | Fig. 9. (color online) Absorption spectrum of Ga in Si (from Fischer and Rome[29]). The inset shows the transitions between hole bound states (from Onton, Fisher, and Ramdas[18]). Vertical dashed lines and labels in red color are added to indicate the primary transition energies. |
In many cases, different interpretations might not have any practical consequence. For instance, to generate free holes in a p-type semiconductor, in the conventional model, electrons are thermally excited to the acceptor level with an activation energy of EA, as shown in Fig.
All of these examples involve relatively simple impurities or defects. For more complex situations, such as transition metal impurities, also known as magnetic impurities, the analyses are less straightforward.[32,33] Nevertheless, the excitonic effect is expected to also play the similar role in related optical transitions. This effect has so far been neglected in the literature.
Next we discuss how the electronic structure of an impurity can be calculated using first-principles theories. Different schemes have been developed for this purpose, but their results are not necessarily equivalent, not only because they involve different approximations but also because what they calculate can be different things. In virtually all density-functional theory (DFT) calculations for impurities or point defects, the calculated transition energies were either explicitly or implicitly treated as EA or were compared to experimental results that have been interpreted as EA. Based on the discussions given in the previous sections, this practice is problematic. We will clarify the differences in terms of the underlying physics between various transition energies associated with an acceptor calculated using different DFT based approaches.
We first offer some qualitative discussions based on a Hartree–Fock (HF) approximation that is also a many-body theory but seems to be conceptually more transparent than a DFT for illustrating the underlying physics. Within the HF approximation, the total energy difference between the two states of the system, the excited state (one electron has been moved to the impurity state from the VBM) and the ground state (the valence band is fully occupied), is given as[34]
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Similarly, we can write the free-to-bound transition energy as the total energy difference between the two states of the system
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We will next discuss the three representative approaches that can be found in the literature for computing the transition energies associated with the acceptor within the framework of DFT. Rather than trying to judge which method is more accurate, our intent here is to highlight the different meanings of the results obtained from these different approaches. We will use Si:In as a prototype system to illustrate the differences.[4] The comparison is made for the results all obtained within the local density approximation (LDA). Despite the limitation imposed by the LDA in the accuracy of the absolute transition energies, these results are sufficient to serve the purpose—revealing the differences in the underlying physics.
In the literature, the total energy difference δEtot is commonly used or implied as the quantity to be compared with the experimentally derived “acceptor binding energy”
There are actually two different ways to calculate δEtot. The conceptually most straightforward way to evaluate the transition energy of the whole system between the excited and ground state should be, with the total number of the valence electrons (N) fixed, calculating the total energy difference between them with one electron being removed from the VBM and forced to occupy the EI level in the excited state (the so-called constrained DFT or selective occupation). The result is referred to as
Taking Si:In as an example, the DFT-LDA calculations yielded
By performing only the ground state calculation (with N electrons), one can obtain the neutral impurity state EI and its wave function φI. One can go one step further to solve the whole bound exciton problem. This problem is similar to the well-known free exciton problem where the excitonic states can be further calculated after the one-electron band structure is obtained with the system in the ground state. If the Coulomb interaction is relatively weak, then the Coulomb contribution can be described by an effective mass equation with the point charge in Eq. (
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This two-step approach is expected to be a reasonably good approximation for solving the acceptor bound exciton problem for many real systems. One potential shortfall of this approach lies in that it does not account for the difference in the lattice configurations between the excited and ground state. This effect will be examined below.
One may also perform the total energy calculation for an excited state, in particular with the single-electron impurity level being occupied at EI,e, where “e” stands for “excited state”. This approach basically requires doing the same calculation as in the first approach, with either the N + 1 or N electron system, but uses the single-particle state to determine the transition energy. For the N + 1 system, the DFT-LDA calculation yields
In the constrained DFT excited state calculation of the N electron system,
With using either the N + 1 or N electron system, after obtaining the impurity state EI,e, in order to account for those discrete absorption features observed experimentally,[29] one has to go one step further to treat the excitonic problem as in Eq. (
There is clearly a qualitative correlation between the EI energy calculated by DFT and the p-orbital energy of the valence electron with respect to the Si 3p orbital for Si:III,[19,35,37] as shown in Fig.
Table
![]() | Table 1. DFT-LDA results for Si:In (in meV). “e”: excited state, “g”: ground state of the system. The first lines are the results of Ref. [4], the second lines of Ref. [35]. . |
Finally, some brief comments are provided about the subtle differences between the DFT calculations with conserved or non-conserved total electron numbers. It is customary in the literature to change the total number of valence electrons in the system to emulate different charge states. For instance, in the DFT calculation for the transition from a neutral vacancy state V0 to an ionized vacancy state V+, the V+ state is simulated by a system with one less valence electron, plus a uniform negative background charge equivalent to one electron. One may understand the uniform background charge representing the plane wave state of the electron or corresponding to the case where the electron has been excited to the vacuum level, which is perhaps more relevant to the photoemission experiment. An alternative that is more relevant to the interband transition in a semiconductor is to let the electron occupy the CBM, mimicking the photoexcitation from the defect to conduction band transition.[3] In the single-particle picture, the energy levels do not depend on the states involved in the transition. Thus, the two options should not make any major difference. However, in the many-electron self-consistent calculation, the atomic configuration of the impurity does depend on the charge distribution of the neighboring atoms, which is why the two options could potentially make some practical difference for the case of strong lattice relaxation. For the case of A0 to A− transition for an acceptor, taking one electron from the VBM to the impurity state at EI seems to be most relevant to either IR absorption or photo-conductivity measurement for the acceptor. The more commonly adopted approach, adding one extra valence electron plus a uniform positive background charge, could be problematic because, on the one hand, the situation conceptually resembles an electron affinity calculation (if we do not consider the added background charge), which tends to yield a larger transition energy; while on the other hand, if the background charge was viewed as a hole state, then clearly it would not be a good approximation for any real valence band state. The results of Table
The conventional hydrogen model for “shallow” impurities overlooks the impurity state that is typically a highly localized state and instead only focuses on the Coulomb interaction between the ionized impurity core and the excited carrier. The consequence of the Coulomb interaction is mistakenly interpreted as the donor or acceptor binding energy. In the new model, the distinction between the “deep” and “shallow” impurities essentially disappears. They all can be understood under a unified framework of the bound exciton model, although with some subtle differences. This new understanding implies that many existing experimental data in the literature should be re-analyzed and explained, and it can also have a real impact on device design.
The results of different first-principles based impurity calculations may mean different things, depending on which approach is adopted. In the total energy approach, the total energy difference between the excited and ground states gives approximately the transition energy of the bound exciton state, which is smaller than the activation energy of the free carrier electrical conductivity. The single-particle state instead should in principle yield the impurity state or the impurity binding energy that is directly relevant to the free carrier electrical conductivity, which, however, is conceptually irrelevant to what is described by the hydrogen mode of the conventional theory. Furthermore, there are subtle but important differences between using selective occupation and uniform background charge in calculating the defect states.
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