Doping of III−V Arsenide and Phosphide Wurtzite Semiconductors

4 ABSTRACT: The formation energies of nand p-type dopants in 5 III−V arsenide and phosphide semiconductors (GaAs, GaP, and 6 InP) are calculated within a first-principles total energy approach. 7 Our findings indicate thatfor all the considered systemsboth 8 the solubility and the shallowness of the dopant level depend on the 9 crystal phase of the host material (wurtzite or zincblende) and are 10 the result of a complex equilibrium between local structural 11 distortion and electronic charge reorganization. In particular, in the 12 case of acceptors, we demonstrate that impurities are always more 13 stable in the wurtzite lattice with an associated transition energy 14 smaller with respect to the zincblende case. Roughly speaking, this 15 means that it is easier to p-type dope a wurtzite crystal and the charge carrier concentration at a given temperature and doping dose 16 is larger in the wurtzite as well. As for donors, we show that neutral chalcogen impurities have no clear preference for a specific 17 crystal phase, while charged chalcogen impurities favor substitution in the zincblende structure with a transition energy that is 18 smaller when compared to the wurtzite case (thus, charge carriers are more easily thermally excited to the conduction band in the 19 zincblende phase).

The elements of the dielectric tensor that we have used for 174 our Madelung correction have been obtained from a DFPT 175 calculation as detailed above. We do not consider it necessary 176 to correct these dielectric tensor entries with their 177 experimental counterparts, given that the interaction between 178 the point charge replicas will be screened according to our 179 used theory level, that is, LDA. Formation Energy. The formation energy is the central 181 quantity in defect analysis and it tells us how likely it is to 182 observe a defect in a crystal matrix, 59 either in the case of 183 intrinsic imperfections of the crystal lattice orthe case 184 addressed herewhen it comes to an impurity added on 185 purpose to alter in a controlled way the property of a material. 186 The knowledge of the formation energy of a defect delivers 187 some important information concerning the impurity equili-188 brium concentrations, 64−66 the solubilities, 67,68 or the 189 diffusivities. 69,70 Additionally, by comparing the formation 190 energy of a neutral and singly charged defect, one can obtain 191 the transition energy, a quantity of paramount importance in 192 semiconductor physics that tells us which is the energy needed 193 to thermally excite carriers from the dopant state to the 194 conduction or valence band. The formation energy, as 195 introduced by Zhang and Northrup, 64 is written as follows (1) 197 where E tot D is the total energy of the system including the defect, 198 the sum runs over all the chemical species present, and n i and 199 μ i are the number of atoms and chemical potential of species i, 200 respectively. q is the charge state of the defect and μ e is the 201 chemical potential of the electron, which is referred to as E V , 202 the highest occupied eigenvalues of the pristine system. 203 Therefore, μ e varies from 0at the top of the valence 204 bandto E gap at the bottom of the conduction bandthus 205 spanning the whole range of doping conditions. 206 In the case of a compound semiconductor like the ones 207 studied in this work, eq 1 is conveniently reformulated as 65

208
(2) 209 which, for simplicity, is written for the case of GaAs with a 210 generic impurity X. The chemical potentials μ Ga bulk , μ As bulk , and 211 μ GaAs bulk refer to the bulk compound of Ga, As, and GaAs. We 212 computed μ Ga bulk and μ As bulk as the energy per atom of Ga and As 213 in the orthorhombic and trigonal phase, respectively; for μ GaAs bulk , 214 we considered the ZB or WZ crystal phase, depending on the 215 case being addressed. Note that μ Ga and μ As are the chemical 216 potential of Ga and As in GaAs, respectively, and that 217 computing their value is not straightforward. However, one can 218 observe that the chemical potential of bulk GaAs is μ GaAs bulk = 219 μ Ga bulk + μ As bulk − ΔH f , where ΔH f is the heat of formation of 220 GaAs. Now, E form is a function of the bulk chemical potential of 221 Ga and As and of the parameter Δμ that accounts for the 222 difference between the chemical potentials of Ga and As in 223 GaAs and in their respective bulk state. The reformulation of 224 E form in eq 2 has the advantage of expressing it in terms of well-225 defined quantities (the bulk chemical potentials) and of the 226 parameter As Ga bulk As bulk 227 (3) 228 which accounts for the macroscopic stoichiometry conditions 229 of the material. Δμ can vary between −ΔH f , the limit that 230 corresponds to the As-rich conditions, and ΔH f , for the Ga-231 rich material, conditions fixed by the inequalities μ Ga ≤ μ Ga bulk 232 and μ As ≤ μ As bulk . This formalism is also applied to the case of 233 GaP and InP, where we considered the cubic phase for bulk P 234 and the trigonal phase for bulk In to define μ P bulk and μ In bulk , 235 respectively. 236 In the case of the chemical potential of the dopant, μ X (X = it is easy to see in Figure 1a in turn affects the 1,4 atomic interactions (see in Figure 1c,d).   277 IV semiconductors, that is, Si, Ge, and diamond, where the 278 bond is fully covalent and that accordingly adopt the cubic 279 structure. Then, in III−V semiconductors, the larger the ionic 280 contribution is, the less the ZB phase will be favored over the 281 WZ one, until the latter becomes the ground state as in GaN. 282 Our results agree well with this picture, as we found that the 283 preference for the ZB structure according to our calculations is 284 21.9 meV for GaAs, 17.4 meV for GaP, and 10.6 meV for InP 285 per unit formula (f.u.), which follows a prediction based on the 286 electronegativity differences between the anion and cation 287 according to the Pauling scale, a crude measure of the ionicity 288 of the bond: 0.37 for GaAs, 0.38 for GaP, and 0.41 in InP. 289 Therefore, the larger the electronegativity difference is, the 290 more ionic is the bond and the less favored is the ZB crystal 291 phase. A more refined definition of the ionicity of a bond is the 292 so-called atomic asymmetry parameter (AAS) between a pair 293 of atoms, 74,75 which is known to work well in crystals of the 294 A N B 8−N type. The AAS values for GaAs, GaP, and InP are 295 0.316, 0.371, and 0.506, respectively, which are also in good 296 agreement with the abovementioned energy preferences. Other 297 criteria to estimate the ionic character of the chemical bond are 298 of course possible; see, for example, the ionicity scale based on 299 the centers of maximally localized Wannier functions of Abu-300 Farsakh and Qteish. 76

301
One of the reasons of interest in crystal-phase engineering is 302 the tunability of the electronic properties. Therefore, another 303 issue that we addressed and briefly discuss before moving to 304 the case of extrinsic doping is the dependence of the electronic 305 band gap on the crystal phase. As it is well known, DFT in its 306 local and semilocal approximation of the exchange−correlation 307 energy severely underestimates the band gap. Therefore, we 308 have performed quasiparticle G 0 W 0 calculations that allow 309 bypassing this limitation. The results are reported in Table 1. 310 The electronic properties of bulk ZB GaAs have been 311 investigated from first principles since decades in view of the 312 microelectronic-oriented applications of the material 77−80 and 313 theoretically assessing the band gap of GaAs main polymorphs 314 remains controversial, as a definitive conclusion is still missing 315 (see ref 16 for a detailed discussion). Furthermore, although 316 on one side, there is a large availability of experimental data 317 about ZB GaAs (see, e.g., refs 81, 82), the scarcity of 318 experimental data about WZ GaAs samples, mostly derived by 319 NW structures, makes the comparison with experimental data 320 for this polymorph a quite cumbersome task because of the 321 expected overestimation of the gap due to quantum confine-322 ment effects. Indeed, there are experimental reports for the 323 band gap of WZ-GaAs NWs to be either larger or smaller than 324 the one of ZB-GaAs NWs by few tens of meV.  reported to be slightly larger than that of ZB InP (see for 345 instance refs 85, 86), in agreement with our results (see Table   346 1) and other theoretical calculations. 87   electrons is expected to stabilize the WZ (ZB) crystal phase.
D 397 notation C As @GaAs stands for a C atom substituting an As 398 atom in a GaAs lattice (and likewise for the other cases). In all 399 the cases, we considered the neutral charge state and the −1 400 charge state, which is expected to be the more stable charge 401 state when the Fermi level lies above the dopant level. These 402 are all textbook cases of acceptors, where an atom of the lattice 403 is substituted by an impurity from the group of the periodic 404 table immediately to its left. Impurities from group-IV can, in 405 principle, be both donors and acceptors, depending on the 406 sublattice chosen for the substitution. 89 This is the case of Si@ 407 GaAs, which acts as an acceptor when it substitutes an As atom 408 and as a donor when it substitutes a Ga atom. C could behave 409 similarly, but substitution at the As sublattice is much more 410 stable than substitution at the Ga sublattice (we found a 411 difference of 0.27 and 0.37 eV in ZB and WZ GaAs, 412 respectively), so the latter in practice never occurs. We recall 413 that we carried out our calculations in bulk systems, as an 414 approximation of realistic, large-diameter NWs. For a study of 415 extrinsic defects in GaAs NWs, the interested reader can see, 416 Figure 2); (ii) the increased stability is similar in all the cases; 425 and (iii) the transition energy is (slightly) smaller in the WZ 426 and the impurity state is shallower (see the zoomed view for 427 Si As @GaAs). Simply put, it is easier to p-type dope GaAs in the 428 WZ phase and these dopants will be easier to activate. 429 Following the arguments given above, we now attempt to 430 rationalize the observed behavior. A very important factor to 431 consider in understanding the role of the impurity is the 432 mismatch between the impurity and the host lattice. The four 433    a All energies are given in meV/f.u. We also report the difference in transition energies, ΔE(0/−), between the ZB and WZ structures (a positive value indicates that the impurity state in WZ is shallower).
The Journal of Physical Chemistry C pubs.acs.org/JPCC Article 483 the WZ should be further stabilized over the ZB because of the 484 charging, although the effect is only modest for acceptor 485 impurities. Consequently, our calculations suggest that smaller 486 transition energies will be associated with larger stabilizations 487 of the WZ structure for the neutral impurity. 488 All these considerations are straightforwardly extended to 489 the case of Zn Ga @GaP and Zn In @InP, whose formation f3f4 490 energies are shown in Figures 3 and 4, and thus confirm the 491 generality of the trends discussed. 492 Donors. We considered four different systems doped with a 493 donor: Si Ga @GaAs, S P @GaP, Te P @GaP, and Te P @InP. In all 494 these cases, an atom of the lattice is substituted by an impurity 495 from the group of the periodic table immediately to its right. 496 We studied each impurity in the neutral and +1 charge state, 497 which is expected to be the more stable charge state when the 498 Fermi level lies below the dopant level. As mentioned above, Si 499 is an amphoteric dopant, so although Si As @GaAs is an 500 acceptor, here, we study Si Ga @GaAs that acts as a donor. 501 The results of the formation energy as a function of the 502 chemical potential of the electron for the three compounds   The Journal of Physical Chemistry C pubs.acs.org/JPCC Article 561 stances, the additional structural degree of freedom of the WZ 562 structure becomes considerably less effective and the very 563 isotropic nature of the ZB structure becomes comparable or 564 even slightly preferred. Only for the more expanded lattice of t3 565 InP, the WZ structure is again slightly favored (see Table 3). 566 In contrast, because of the structural mismatch, Si Ga @GaAs 567 behaves in the same way described above for the case where Si 568 was acting as an acceptor; the only difference is that the short 569 distances with the four nearest neighbors are now a bit longer 570 (i.e., 2.372 (×3) and 2.382 Å for Si Ga compared with 2.348 571 (×3) and 2.353 Å for Si As in GaAs WZ). Thus, according to 572 our calculations, Si in GaAs has a preference for WZ 573 irrespective of acting as a donor or an acceptor. In fact, the 574 calculated energy differences are comparable (187 meV/f.u. for 575 Si As and 152 meV/f.u. for Si Ga ). This result emphasizes the key 576 role of the mismatch in enforcing the WZ−ZB preference. 577 Note that among the different impurities studied, charging 578 the impurity always favors the ZB structure even when the 579 impurity is smaller than the host atom replaced (Table 3). This 580 contribution is relatively large and finally determines the 581 preference of all donor impurities studied for the ZB structure. 582 We believe the origin of this result is that, as noted above, 583 removing the electron provided by the neutral impurity 584 decreases the ionicity of the lattice and consequently, the ZB 585 structure is favored. According to our calculations, for donors 586 compressing the lattice around the impurity, the shallowness 587 will increase with the size of the impurity and/or decreasing 588 the cell constants of the pristine lattice. 589 As a final remark, we observe that our computed transition 590 energies, indicating that donor states are shallower in the ZB 591 crystal phase, agree well with the predictions of the hydrogenic 592 model of substitutional impurities within effective mass theory 593 (EMT). Within this simple model, the substitutional impurity 594 form four bonds with the nearest neighbors, with negligible 595 relaxation effects and charge transfer, leaving one unpaired 596 electron whose energy is approximately given by (4) 598 where m* is the effective mass in units of the electron mass, n 599 is the main quantum number, ε is the (relative) static dielectric 600 constant, and Ry is the Rydberg constant. This is the quantum 601 mechanical solution of the hydrogen atom except for the fact 602 that it contains parameters of the bulk host crystal, such as m* 603 and ε. E n is the energy of the unpaired electron relative to the 604 conduction band minimum, so large values of ε and small 605 values of m* both contribute to make the impurity shallower, 606 that is, E n small. If we look at the computed values of the static 607 dielectric constant collected in Table 1, we see that for GaAs,   608 GaP, and InP, when going from the ZB to the WZ, it decreases 609 (with a reduction, i.e., slightly more pronounced for the zz