Interacting quasi-band model for electronic states in alloy semiconductors: Relation to average t-matrix approximation and band anticrossing model

A new variational theory is proposed for electronic states in alloy semiconductors with arbitrary diagonal and off-diagonal randomness and any concentration. In AcABcB substitutional alloys, the theory derives the mutual interaction between two quasi-A and -B electronic bands whose effective band widths are proportional to cA and cB, respectively, i.e., an interacting quasi-band model. The model provides satisfactory results near-band-edge states, especially for band bowing. For diagonal randomness, the theory corresponds to the average t-matrix approximation, and in the dilute limit, a formula similar to the band anticrossing model, which has been frequently applied to GaN-related alloys, is obtained.

Let us consider a substitutional binary random alloy A cA B cB (c B = 1 ¹ c A ) of which the electronic Hamiltonian is given by Here, jni denotes a Wannier state localized at the site n located at R n in a regular lattice. Each site is randomly occupied by either an A or B atom with a probability of c A or c B , respectively. The site energy, ¾ n , takes a value of either ¾ A or ¾ B depending on the type of atom occupying the site, and the difference ¦ = ¾ A ¹ ¾ B denotes the strength of the diagonal randomness. The transfer energy t nm between sites n and m depends on the relative positions of R n and R m as well as their atomic occupations: In pure limits (c A = 1 and c B = 0, i.e., pure crystal A or c A = 0 and c B = 1, i.e., pure crystal B), the eigenstates are Bloch-type states that are characterized by the wave vector k, where N is the total number of sites. The corresponding eigenenergies E A and E B in crystals A and B, respectively, are given by In a similar manner, we can write t AB k ¼ ð1=NÞ P n P m t AB nm exp½ÀikðR n À R m Þ and likewise for t BA k . However, note that these two quantities do not correspond to any real situations.
Before presenting the new method, it is instructive to start with a simple cubic lattice in which the transfer energy is only nonzero (¹V) between nearest neighbors, irrespective of the type of atoms (i.e., an absence of off-diagonal randomness). If we apply the virtual crystal approximation, in which the random potentials ¾ n are all replaced by the average value " " ¼ c A " A þ c B " B , then the eigenstates are given by Eq. (3), and the energies are At the bottom of the band, the energy Eðk ¼ 0Þ ¼ " " À 6V linearly changes with c A or c B (= 1 ¹ c A ), and the corresponding wave function is where an electron has equal amplitudes extending over all sites. The effects of the randomness, H 0 ¼ P n jnið" n À " "Þhnj, i.e., the deviation from the average potential, can be taken into account using a second-order perturbation. The correction to the virtual crystal [Eq. (5)] is given by For the bottom of the three-dimensional band, ¦E is always negative finite. By using the lattice Green's function in the simple cubic system, 14) we find that the resulting band bowing is ¦E ³ ¹0.247c A c B (¾ A ¹ ¾ B ) 2 /V. Since the wavefunctions in the virtual crystal are uniformly extended, the band bowing suggests a nonuniform amplitude distribution in the alloy, i.e., if ¾ A > ¾ B , then the lowest state favors the lower-potential B sites. Consider the trial wavefunction j~ k¼0 i ¼ ð1= ffiffiffiffi N p Þ P n n jni, in which the variational parameter ¢ n can take one of two values, ¢ A or ¢ B , depending on the atomic occupation at n. 15) The expectation value of the Hamiltonian h~ k¼0 jHj~ k¼0 i should be minimized under the condition h~ k¼0 j~ k¼0 i ¼ c A j A j 2 þ c B j B j 2 ¼ 1. After a statistical average over random occupations of A and B atoms and introducing the Lagrange multiplier, E, we find that the quantity hh~ k¼0 jH À Ej~ k¼0 ii AV should be minimized with respect to Ã A and Ã B . Thus, we have which has two solutions for nonzero ¢ A and ¢ B values only when Figure 1(a) shows the concentration dependence of the lower branch E ¹ . The deviation from the virtual crystal (straight dotted line) shows asymmetrical bowing with respect to the interchange c A § c B , which is contrary to the second-order perturbation given in Eq. (6). Figure 1(b) shows the probabilities «¢ A « 2 and «¢ B « 2 , and as expected, which is the total participation on all B sites (c B N in total). The large deviation from P B = c B (the virtual crystal) indicates that an electron has amplitudes at lower-potential B sites that are larger than those at A sites even when no spatial localization occurs and the wavefunction extends over the alloy. When c B is very small, a localized state at an isolated B impurity splits off from the bottom of the host A band (¾ A ¹ 6V) when ¦/6V > 0.5 (the correct condition is shown to be ¦/6V > 0.34 based on the Koster-Slater method 14) ).
Let us now return to the general situation of alloys and introduce the following trial function for an electron in a quasi-k state of the Hamiltonian, Eq. (1): where ¢ n can take one of two values, ¢ A or ¢ B , depending on the atomic occupation at n. Following a procedure similar to that shown above and averaging over the statistical occupation of atoms, while neglecting all their correlations, we find that the quantity should be minimized with respect to the two variational parameters Ã A and Ã B , and so which has two solutions for nonzero ¢ A and ¢ B values only when Equation (12) implies that in an A cA B cB alloy there are two quasi-A and -B electronic bands with effective band widths proportional to c A and c B , respectively. These bands mutually interact with c B t AB k and c A t BA k ; hence, we refer to the theory as the interacting quasi-band (IQB) model.  The present theory has the following properties: 1) It provides a useful formula that is symmetric in atoms A and B and can be applied to arbitrary types of diagonal and off-diagonal randomness at any strength and concentration.
2) It is a good approximation for near-band-edge states, i.e., near the bottom of the conduction band and near the top of the valence band. For weak disorder, Eq. (13) gives the correct behavior corresponding to a virtual crystal. Depending on the parameters, positive or negative band bowing appears (see the discussion below). However, the theory is less reliable in the middle of a band because it may give a spurious gap, even in weak disorder cases.
3) As can be seen in Eq. (13), E « (k) depends on k through the values of t AA k , t BB k , t AB k , and t BA k . Hence, if all the maxima or minima of the t ij k 's (i; j ¼ A, B) are given at the same k-point in the Brillouin zone, then the resulting allowed electronic bands have a distribution over the common colored area in Fig. 2, irrespective of the form of t ij k . Fig. 2, the obtained E « (k) is located within the band area of pure crystal A and pure crystal B. Thus, the Saxon-Hutner theorem, 16) which has been proven for diagonal randomness, 17) holds.

4) Except for cases (d) and (j) in
5) The trial wave function, Eq. (10), has two solutions (+/¹) for each k, irrespective of the concentration; hence, the total number of states appears to be doubled (2N). This inconsistency will be resolved in the discussion below.
Let us discuss these properties in detail, starting with band bowing. Various types of band bowing, i.e., deviations from a virtual crystal, can be obtained depending on the character and strength of the randomness. In a purely diagonal random case (t AA , it can be easily shown that E « (k) is a monotonically increasing or decreasing function of the concentration. Anomalous large bowing that violates the Saxon-Hutner theorem, such as min E À ðkÞ < min½E A ðkÞ; E B ðkÞ or max E þ ðkÞ > max½E A ðkÞ; E B ðkÞ, is caused by off-diagonal randomness. In a purely off-diagonal case (¾ A = ¾ B ), a necessary condition for the former is j min t AB k j, j min t BA k j > j min t AA k j, j min t BB k j (a similar inequality exists for the latter). The anomalous large band-gap bowing found in several nitride alloys 18) can be explained if these conditions are fulfilled in at least one of the conduction and valence bands of the alloys. Now, let us consider the relationship between the new model and the ATA. We concentrate on a case in which all t ij k 's (i; j ¼ A; B) have the same dispersion, i.e., proportional to t k : t ij k ¼ T ij Â t k . Figure 3(a) shows the typical behavior of E « (k) obtained from Eq. (13). We can attempt to calculate the density of states per site using the normal procedure as AE ðEÞ ¼ N À1 P k ðE À E AE ðkÞÞ ¼ 0 ðt k ðE AE ÞÞðdE AE =dt k Þ À1 , where 0 ðEÞ ¼ N À1 P k ðE À t k Þ, and t k (E « ) is the inverse function of Eq. (13). In this way, we always have R À ðEÞdE ¼ R þ ðEÞdE ¼ 1 irrespective of the concentration, as mentioned above. On the other hand, it would be plausible to consider that the states j k i within an interval dt k are modified to form j~ À k i and j~ þ k i within an interval dE ¹ and dE + , respectively. Since dE + /dt k + dE ¹ /dt k = (c A T AA + c B T BB ) from Eq. (13), the degrees of freedom should be respectively reduced by the factors (c A T AA + c B T BB ) ¹1 dE ¹ /dt k and (c A T AA + c B T BB ) ¹1 dE + /dt k . Thus, the renormalized density of states will be given bỹ In this case, R ½ À ðEÞ þ þ ðEÞdE ¼ 1 holds, and we recover the correct number of states. However, the sum rule (i) The dark colored regions have a second solution (dE « /dt k < 0). Note that the parameters in (f ) and (i) satisfy the condition t AB self-consistent theory such as CPA is necessary to obtain the sum rule in persistence-type alloys. 19) In pure crystals, for which ¾ A = ¾ B and t AA , E AE ðkÞ ¼ "). Since dE « /dt k = 0 in the latter branch, these improper solutions provide null contributions to the density of states. Thus, the reduction factor dE ¹ /dt k (dE + /dt k ) is important especially for the top (bottom) of E ¹ (k) (E + (k)). Figures 3(b) and 3(c) show the calculated energy density of states using a semi-elliptic form 0 ðt k Þ ¼ ð2=Þ ffiffiffiffiffiffiffiffiffiffiffiffi 1 À t 2 k p for ¹1¯t k¯1 . The figures show that the renormalized density of states (solid line) is exactly the same as that in ATA (refer to Fig. 3 in Ref. 4).
Finally, we consider the dilute and isolated limits. In the dilute limit in which one of the constituent concentrations, say c B , is small, B atoms are isolated in the host crystal A. We can set c A ³ 1 and c B ¹ 1. Then, assuming that t AB k ¼ t BA k ¼ const: ¼ V AB and neglecting the concentration dependence of the diagonal terms in Eq. (12), we obtain an equation similar to Eq. (1) in Ref. 10, which was proposed in regard to the BAM. 10,11) Note that one of the off-diagonal terms in Eq. (12) is proportional to c B while the other is not, but in the BAM both are proportional to ffiffiffiffiffi c B p . The BAM has been used to discuss the behavior of electronic states in dilute GaNrelated alloys, and large band bowing is considered to be generated by anticrossing between band states and a localized state. In contrast, the present theory implies that it would be more plausible to explain large band bowing as a result of mixing host band states and the impurity band states. This picture is consistent with a recent first-principles calculation of GaAs 1¹x N x alloys. 20) In summary, a new variational theory has been proposed to explain the electronic states in alloy semiconductors having arbitrary randomness at any strength for any lattice structure and concentration. The theory captures the overall concentration dependence, while positive or negative band bowing depends on the parameters. The theory provides good approximations for near-band-edge states, i.e., it is applicable to those electronic states near the top of the valence band and the bottom of the conduction band. In the present paper, we have assumed A cA B cB binary alloys with simple unit cell structures, but extensions to compound semiconductors such as ternary In x Ga 1¹x N would not be difficult if we consider A = InN and B = GaN. The present theory is now being studied through applications to various alloys of compound semiconductors by using their real energy dispersions.