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The calculated defect structure of ZnO
W. Mackrodt, R. Stewart, J. Campbell, I. Hillier
To cite this version:
W. Mackrodt, R. Stewart, J. Campbell, I. Hillier. The calculated defect structure of ZnO. Journal de Physique Colloques, 1980, 41 (C6), pp.C6-64-C6-67. �10.1051/jphyscol:1980617�. �jpa-00220055�
JOURNAL DE PHYSIQUE Colloyue C6, supplkment au no 7, Tome 41, Juillet 1980, page C6-64
The calculated defect structure of ZnO
W. C. Mackrodt and R. F. Stewart
ICI Corporate Laboratory, PO Box 11, The Heath, Runcorn, Cheshire WA74QE, England J. C. Campbell and I. H. Hillier
Chemistry Department, ~ n i v e r i i t ~ of Manchester, Manchester, MI3 9PL, England
RBsumk. - Les energies des dCfauts fondamentaux dans ZnO sont calculees et comparCes avec les resultats experimentaux obtenus par Kroger. L'Cnergie correspondante a la bande de conduction est calculCe en considerant des defauts neutres ou charges. Les effets de dopage avec des impuretes cationiques : L i + , N a + , A13+, ~ a ~ + , In3+ et H + sont analyses. Enfin les energies associees aux centres F f et F sont calculkes theoriquement.
Abstract. - We report the energies of the fundamental defects in ZnO and compare these with values that can be deduced from Kroger's analysis of the experimental data. We consider neutral, singly-charged and doubly-charged defects and from our analysis estimate the edge of the co~lduction band. We.also consider the effect of dopants such as Li+, Na', A13+, G a 3 + , In3+ and H + . Finally we consider the energetics involved in the formation of the F + and F centres.
1. Introduction. - There is now quite substantial evidence which suggests that for ionic solids, or materials that are nearly so, point defect energies can be calculated with an accuracy that in many cases is comparable to experimental error. Thus the defect structure, doping and diffusion properties of the alkali halides [I], the alkaline-earth fluorides [2, 31, the alkaline-earth oxides [4], UO, [5], a-Al,03 [6]
and the transition-metal oxides [7, 81 have all been accounted for to a lesser or greater extent using lattice simulation techniques introduced by Lidiard and Norgett [9] and Norgett [lo, 111. On the other hand, materials that are normally thought of as being rather more covalent in nature, have received less attention, due mainly to difficulties associated with the derivation of suitable interatomic potentials. Recently, however, a procedure based on ab initio molecular orbital methods has been described by the present authors [12] and applied to the calculation of the fundamental defect structures of MgO, MnO, CdO and ZnO. The defect properties of ZnO, in particular are of considerable current interest, and we here extend the discussion of the intrinsic defects in stoi- chiometric zinc oxide to include the nature of the reduced state and aspects of the doping by Li', N a f , A13+, In3+ and H + .
2. Interatomic potentials. - Here, as elsewhere, interatomic forces are assumed to be exclusively two-body with an explicit allowance for electronic polarization by means of a shell-model [13]. For ZnO itself, the relevant potentials are obtained from ab
initio molecular orbital calculations previously des- cribed [12], whereas interactions involving dopants, which are assumed to be predominantly ionic in character, are derived from modified electron-gas potentials [3].
3. Lattice simulation. - The calculations reported here are based on a treatment of the defective lattice originally formulated by Mott and Littleton [14] and developed by Lidiard and Norgett [9] and Nor- gett [lo, 111. For non-cubic materials such as ZnO, further refinements to the theory are necessary, and these, in turn have been described by Catlow, James, Mackrodt and Stewart [6].
4. The defect structure of zinc oxide. - 4.1 FUN-
DAMENTAL DEFECTS. - We begin by reviewing the fundamental defect structure of near-stoichiometric ZnO, the details of which have been given previous- ly [12]. Unlike wide band-gap materials such as MgO and a-Al,03, for example, lattice defects in ZnO, which has a band-gap of approximately 3.5 eV, might be expected to exist in charged states other than doubly-charged, i.e. simple vacancies and interstitials. As table I shows, both theory and expe- riment [15] suggest that this is so, with a marked preference for 'singly-charged defects involving both zinc and oxygen. In view of difficulties that can often arise as to the detailed structure of singly-charged and neutral vacancies, it is important at this stage to emphasize that atomistic calculations of the type reported here, by definition, tend to localize charges;
these, however, are not necessarily point-like since
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980617
T H E CALCULATED DEFECT STRUCTURE O F ZnO C6-65
Table I. - Fundamental defects in near-stoichiometric ZnO.
Defect
-
(V;., V;) Schottky (Vi,, Vo) Schottky (V;,, Vg) Schottky (Zn;, Vg,) Frenkel (Zni, V;.) Frankel
(q, V;) Frenkel (O;, V;) Frenkel Hole (small polaron) Hole (large polaron) Forbidden band-gap Conduction band-edge Self-trapped exciton (Zni., 0 0 )
Formation energy (eV) Calculated (*) Kroger 1151
- -
5.88 5.67
+
0.2 (")4.76 4.04
5.66 6.29
7.68 -
4.46 4.42
5.77 -
2.51 -
6.54 6.38 (') 6.35-6.10 (b)
- 3.44
- 4.40 ( d )
2.98
(*) Campbell, Hillier, Mackrodt and Stewart [12].
(") Estimated from Kroger's data (see reference above).
(b) Calculation based on a valence band-width of (3.0-3.5) eV.
(') Derived from a combination of Kroger's analysis and the calculated value of the cation vacancy formation energy.
( d ) Derived from the valence band ionization energy and the band
gap.
the use of a shell-model allows some spatial extent.
Thus, Vk,, for example, corresponds to a cation vacancy plus a hole localized on an adjacent oxygen site (i.e. an 0- ion) together with the distorted and polarized surrounding lattice. In the usual notation, therefore, we have :
in which the brackets indicate that the vacancies and appropriate charges are on adjacent lattice sites. Our present concept of the defect structure of near- stoichiometric zinc oxide, therefore, is that for the most part it comprises four basic defects, namely Zni, 01, Vt, and V&. Now both types of vacancy can ionize to give free holes and electrons ; however, as both our own calculations [12] and Kroger's ana- lysis [15] show, electrons are more weakly bound to anion vacancies than holes to cation vacancies by about 1 eV, so that in addition to the four basic defects previously mentioned we have an excess of electrons and doubly-charged anion vacancies, V6. The energy levels of the singly-charged vacancies lie approxima- tely in the middle of the band-gap, while the conduc- tion band-edge is some 2.9 eV below the vacuum.
Now Kroger's analysis does not include oxygen vacancies, for which there is increasing evidence (see, for example, Ref. [16]); but apart from this, there is remarkably good accord between the theore- tical formation energies and those deducted by Kroger (bearing in mind the limitations to the accu- racy of both).
4.2 DEFECTS IN REDUCED ZINC OXIDE. - Our dis- cussion so far has concentrated on near-stoichiome- tric ZnO. We now turn to the reduced state, ZnO,
-,,
in which y is 0.1 or less, and in particular focus on the nature of the defects formed on reduction. What in essence we are concerned with is the process :
0 5
+:
O,(g)+
neutral oxygen vacancy defect, and the various possibilities for the neutral defect. In section 4.1 it was assumed that this defect corres- ponded to an oxygen vacancy with two electrons from the conduction band localized on adjacent zinc sites in the form of Znf ions. Alternatively, electrons can be trapped at the vacancy itself to give F+ and F centres. The formation energies of these defects are listed in table I1 in which the internal energies of the Table 11. - The formation energies of defects i~reduced ZnO.
Defect
-
Vo
+
adjacent Zn;, Vb;+
2 adjacent Zni, F' centreInternal energy of F + centre F centre
Internal energy of F centre F + -Zn& complex
Interaction energy of F + -Zn&, complex
Formation energy (eV)
-
16.9 (16.89) (") 12.7 (13.83) 16.9
4.5 ( b )
17.5 14.8 (') 13.2
- 0.8
(") Values in brackets deduced from Kroger's analysis [15].
( b ) Kinetic energy of the single electron.
(') Kinetic plus electron-electron repulsion energies.
F + and F centres are taken to be the same as those in MgO [3]. As shown we find identical formation ener- - - -
gies for the F + centre and Vb. The formation energy of the F centre, on the otherhand, is found to be much greater than either Vg or the complex (Ff
+
Zni,).While the present calculations suggest the former to be lower in energy by about half an electron volt, the value for V& deduced from Kroger's analysis [15]
suggests the reverse. The important point, however, is that they are similiar. Now the association energy of a localized electron, in the form of a Znf ion, to V, is small (0.2 eV), so that our description of the defect structure of partially reduced ZnO, -
,
isthat it consists of V;, VG, F+-centres, the complex (F+
+
Zn,,) and conduction-band electrons.4.3 DOPING OF ZnO BY Li +, Na+, A13
+,
Ga3 +AND In3+. - As is the case for many other oxides, dopants can and do influence the properties of ZnO to a considerable extent. Unlike materials such as MgO and A1203, however, the situation is somewhat more complex in view of the possible effects of non- stoichiometry. To simplify matters, therefore, our discussion is primarily concerned with near-stoichio- metric ZnO, though the arguments are still valid for
C6-66 W. C. MACKRODT, R. F. STEWART, J. C. CAMPBELL AND I. H. HILLIER
Table 111. - Calculated solvation energies for cation dopants in ZnO.
Solvation mode Monovalent ions
-
a) X 2 0 + 2 Xi,
+
2 ZnO+
V, b) X 2 0 + 2 X i , + 2 Z n O+
2 F ++
i 02(g)
c) X 2 0 + 2 X,
+
01' d ) X 2 0 + 2 X;+
ZnO+
V'i,e ) X 2 0 -t Xi,
+
Xi+
ZnO f) X 2 0 + (Xi - Vg, - Xi)+
ZnOg) X 2 0
+
3 02(g) + 2 Xi,+
2 ZnO+
2 hEnergy (eV) Li (*) Na (*)
- -
3.9 (1.3) (**) 3.5 (0.9)
TrivaIent ions
h ) cc-A1203 + 2 Al,,
+
3 ZnO+
V", 0.9 (**)i) a-A120, + 2 Ali,
+
2 ZnO+
0: 1 .Oj ) cc-Ga20, + 2 G a i n
+
3 ZnO+
Vi, 1.3k) In203 + 2 In,,
+
3 ZnO+
V$ 2.3(*) Figures in brackets indicate the solvation energy allowing for defect aggregation.
(**) No allowance is made for defect aggregation.
low levels of doping which are nevertheless in excess of the degree or non-stoichiometry. Table I11 lists the calculated energies for various modes of solvation of Li+, N a f , A13+, Ga3+ and In3+. Since we have calculated dopant/host-lattice interactions on a diffe- rent basis from that for the host lattice itself, it is important to emphasize that in table I11 energy difTe- rences are of much more significance than the absolute value for any particular mode of solvation. Our calculations suggest that Li' and Na+ behave in a similar fashion both in the absence and presence of oxygen. Thus we predict lattice substitution compen- sated by anion vacancies and holes, and for Li+, in particular, a trimer of the form { Lii-Vk,-Lii }. Para- magnetic resonance and optical experiments provide good evidence for lattice substitution compensated by holes [19, 201, and in the case of Na+, support for an appreciable interaction between the two [20]. Lit has been shown to lower the free electron concentration quite substantially in reduced ZnO [21] and this could take place by either of the following reactions :
L i 2 0
+
2 e'+ : 02(g) -+ 2 Li;, +
2 ZnO
or
L i 2 0
+
e' -+ 2 Lik,+
2 ZnO+
Vbof which the second follows directly from a ) in table 111.
From our basic defect calculations we predict that Refere
[l] CATLOW, C. R. A., DILLER, K. M. and NORGETT, M. J., J. Phys. C 10 (1977) 1395.
[2] CATLOW, C. R. A,, NORGETT, M. J. and Ross, T. A,, J. Phys. C 10 (1977) 1627.
[3] MACKRODT, W. C. and STEWART, J. Phys. C 12 (1979) 431.
[4] MACKRODT, W. C. and STEWART, J. Phys. C 12 (1979) 5015.
[5] CATLOW, C. R. A,, PYOC. Roy. Soc. A364 (1978) 473.
tri- (and higher) valency cation substitution is compen- sated by both cation vacancies and oxygen interstitials.
A13+ and Ga3+ have been shown to lead to an increase in the free electron concentration in ZnO [21] for which we have three possible processes :
X20, -+ 2 Xi,
+
2 ZnO+ 4
O,(g)+
2 e'X2O3 -, 2 XZ,
+
3 ZnO+
V;,+
e'and
X2O3 -+ 2 Xi,
+
2 ZnO+
0;+
e'.
The second of these reactions follows from h) together with the formation of an electron-hole pair which combines with Vi, to give Vi,
+
e'. We estimate that the energies for the three processes are within 1 eV of each other.4 . 4 DEFECTS INVOLVING H + . - Finally, we des- cribe, in brief, defects l~lvolving H + , the energies of which are listed in table IV. Unlike MgO, the Table IV. -Energies of defects involving H+ in ZnO.
Defect H,, substitution at cation site Hi interstitial
OH- group along c-axis
Nai, plus adjacent OH- along c-axis Interaction energy of Nak, and OH- VL, plus adjacent OH- group along c-axis Interaction energy of V a and OH- group Vi, plus two adjacent OH- groups
Interaction energy of V', and two OH- groups
Energy (eV) -
15.32 - 5.68 - 8.08 7.25 - 3.5
12.56 - 4.9
2.16 - 7.2
uncompensated OH- group, that is to say, (0;;-Hi) is found to be lower in energy than the unbound interstitial Hi, by about 2 eV. There is an appreciable binding energy to cation vacancies, V'i,, which is only slightly reduced by the presence of a second OH- group. Our calculations, therefore, would seem to preclude the existence of free H + in ZnO. We also find an appreciable interaction between an OH- group and monovalent substituents of the type, Nak,, for which Mollwo, Muller and Zwingel[22] have reported experimental evidence based on esr and optical absorp- tion data. In view of this, we suggest that OH- groups should function as efficient trapping centres for conduction-band electrons in the form of complexes of the type (Og-H;-Zn,J with a binding energy of about 3 eV.
[6] CATLOW, C. R. A., JAMES, R., MACKRODT, W. C. and STEWART, R. F., TO be submitted.
[7] CATLOW, C. R. A,, MACKRODT, W. C., NORGETT, M. J. and STONEHAM, A. M., Phil. Mag. 35 (1977) 177.
[8] CATLOW, C. R. A., MACKRODT, W. C., NORGETT, M. J. and STONEHAM, A. M., Phil. Mag. 40 (1979) 161.
THE CALCULATED DEFECT STRUCTURE OF ZnO C6-67
[9] LIDIARD, A. B: and NORGETT, M. J. in Computational Solid State Physics (1972) 385 (New York and London Plenum).
[lo] NORGETT, M. J., AERE Report (1972) AERE-R7015.
[Ill NORGETT, M. J., AERE Report (1972) AERE-R7650.
[12] CAMPBELL, J. C., HILLIER, I. H., MACKRODT, W. C. and STEWART, R. F., TO be submitted.
[13] DICK, B. G. and OVERHAUSER, A. W., Phys. Rev. 112 (1958) 90.
[I41 MOTT, M. F. and LITTLETON, M. J., Trans. Farad. Soc. 34 (1938) 485.
[I51 KROGER, F. A,, The Chemistry of Imperfect Crystals (North- Holland Publishing Co. - Amsterdam) 1964, p. 691.
[16] HAUSMANN, A. and SCHALLENBERGER, B., Z. Phys. B 31 (1978) 269.
[17] JAMES, R. and CATLOW, C. R. A., J. Phys. C 10 (1977) L-237.
[18] COLBOURN, E. A., KINGERY, W. D. and MACKRODT, W. C., To be submitted.
[19] SCHIRMER, 0. F., J. Phys. Chem. Solids 32 (1971) 449.
[20] ZWINGEL, D. and GARTEN, F., Solid. State Commun. 14 (1974) 45.
[21] UEMATSU, T. and HASHIMOTO, H. J., Catalysis 47 (1977) 48.
[22] MOLLWO, E., MULLER, G. and ZWINGEL, D., Solid State Commun. 15 (1974) 1475.
