Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX
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Microkinetic Model for Oxygen Interstitial Injection from the ZnO(0001) Surface into the Bulk Ming Li and Edmund G. Seebauer* Department of Chemical and Biomolecular Engineering, University of Illinois, Urbana, Illinois 61801, United States ABSTRACT: Semiconductor surfaces provide efficient pathways for injecting native point defects into the underlying bulk. The present work constructs a quantitative microkinetic model for the injection of oxygen interstitial atoms from the polar Zn-terminated ZnO(0001) surface into the bulk. Rate constants for defect interaction with the surface and in the bulk were determined by a global optimization procedure of simulations fitted to selfdiffusion profiles from isotopic gas−solid exchange experiments. Key activation barriers are 2.0 eV for injection, 0.62 eV for hopping diffusion, and 1.6 eV for lattice exchange. The injection barrier does not differ greatly from that for nonpolar TiO2(110), but the coverage of injectable oxygen increases with temperature, in contrast to the behavior of TiO2 and gas adsorption in general.
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barrier has been reported for injection from TiO2,23 only an estimate from density functional theory (DFT) exists for ZnZnO whose reliability has not been tested directly.21 The present work constructs for Zn-ZnO a microkinetic model for Oi injection and subsequent reaction and diffusion in the bulk that permits determination of an experimental value of the barrier for comparison with both DFT and TiO2. The barrier is lower for Zn-ZnO, but not by a sufficient amount to deconvolve the effects of polarity from the change in substrate material.
INTRODUCTION Semiconductor surfaces offer efficient pathways for the production and destruction1−4 of native point defects in the underlying bulk because fewer bonds need to be broken or formed. For semiconducting metal oxides, such efficiency has been demonstrated in both TiO2 and ZnO for injection of oxygen interstitial defects (Oi) to the point that Oi supplants oxygen vacancies as the majority O-related defect.5,6 Evidence is mounting that the chemistry associated with injection is quite rich. Control of this chemistry would provide means for manipulating defect behavior in important applications7−11 especially where surface-to-volume ratios are high. Foreign element adsorption offers one possible avenue that has already been demonstrated.6,12,13 Surface polarity offers another possibility. When cleaved along certain crystallographic directions, noncentrosymmetric ionic semiconductors expose polar surfaces with special electrostatic properties14,15 arising from the intrinsic thermodynamic instability of the bulk-terminated surface.16 Stabilization occurs through defect formation,17 faceting,18 reconstruction,19 or adsorption.19,20 For example, wurtzite ZnO with bulk termination exposes two polar faces along the [0001] c-axis: Zn-terminated (Zn-ZnO) with excess negative charge and Oterminated (O-ZnO) with excess positive charge. Both faces exhibit complicated sets of reconstructions7,8 that provide thermodynamic stabilization and also support differences in behavior for injection of Oi.21,22 Compared with nonpolar TiO2(110), Zn-ZnO injects Oi more efficientlyprobably because the reconstructions facilitate adsorption of negatively charged O2 that dissociates into charged O that in turn injects with particular ease into the bulk as Oi2−. A quantitative comparison between Zn-ZnO and TiO2(110) of the injection barriers would help in understanding the differences. Although an experimental value of the Oi injection © XXXX American Chemical Society
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CALCULATION METHOD The approach to microkinetic modeling of defect behavior in metal oxides has been described in detail elsewhere,23 so the following sections provide an abbreviated exposition highlighting aspects specific to the current work. The primary data originated from a standard gas−solid exchange method, in which Zn-ZnO(0001) single crystals were annealed (500−600 °C) in isotopically labeled gas (18O2) with the oxygen partial pressure PO2 set between 1 × 10−5 and 1 × 10−4 Torr. Such a regime corresponds to O-rich conditions, where PO2 is sufficiently high for oxygen to provide the dominant contribution to the chemical potential of TiO2 formation. The diffused isotopic O concentration profiles were measured ex situ with secondary ion mass spectrometry (SIMS). Procedural details and examples of primary data appear in ref 13. Differential Equations for Defect Reaction and Diffusion. Prior work21 has shown that Oi carries the Received: October 9, 2017 Revised: November 24, 2017 Published: December 11, 2017 A
DOI: 10.1021/acs.jpcc.7b09962 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry C preponderance of the oxygen diffusional flux. This dominance follows from the fact that Oi is the majority O-related defect in wurtzite ZnO under O-rich conditions, and that Oi moves more quickly than Vo. The shapes of the diffusion profiles are determined by the kinetics of Oi injection, migration, and immobilization by sequestration in the lattice. Reaction and diffusion behavior was embodied in continuum equations taking the general form
∂Cj ∂t
=−
∂Jj ∂x
+ Gj
Vo2+ are quite similar under O-rich conditions.26,27,29 Photoluminescence (PL) and electron paramagnetic resonance (EPR) have yielded inconclusive results,36−38 but thermogravimetry34,35,39−41 and conductivity35,40 measurements have generally pointed to Zni2+ in preference to Vo2+. The present model therefore presupposes zinc interstitial Zni2+ to be the majority defect, whose concentration sets n according to n ≈ 2CZni2+. The concentration of Zni2+ needed to compute n may be written as
(1)
⎛ ΔSf,Zn i 2+ ⎞ ⎛ ΔHf,Zn i 2+ ⎞⎛ PO2 ⎞−1/2 C Zni 2+ = CZn,L exp⎜ ⎟ exp⎜ − ⎟⎜ ⎟ kBT ⎠⎝ P 0 ⎠ ⎝ kB ⎠ ⎝
where x and t represent the spatial and temporal coordinates, and Cj, Jj, and Gj represent the concentration, flux, and net generation rate of species j, respectively. The species included two oxygen isotopes (16O and 18O). The flux obeys Fickian diffusion for each mobile species with site-hopping diffusivity Dj. Gj contains expressions for bulk reactions of mobile atoms with the lattice by sequestration in the lattice. The total concentration of oxygen interstitials was assumed to obey the conventional expression24 for native defect concentrations: ⎛ ΔS ⎞ ⎛ ΔHf ⎞⎛ PO2 ⎞1/2 ⎛ p ⎞m CT = C LT exp⎜ f ⎟ exp⎜ − ⎟⎜ ⎟ ⎜ ⎟ ⎝ kB ⎠ ⎝ kBT ⎠⎝ P 0 ⎠ ⎝ NV ⎠
⎛ n ⎞−2 ⎜ ⎟ ⎝ NC ⎠
where NC is the effective density of states in the conduction band. Several experimental measurements exist for the formation enthalpy ΔHf,Zni2+ and entropy ΔSf,Zni2+ from thermogravimetric analysis and electrical measurements.34,35,39,41 We employed the statistical technique of maximum likelihood estimation42 to combine these results into values of 5.2 ± 0.3 eV for ΔHf,Zni2+ and (5.3 ± 0.3) × 10−3 eV/K for ΔSf,Zni2+. Note that the concentrations of both Zni and Oi depend upon the partial pressure of oxygen through the defect equilibrium network, although in opposite directions. The hopping diffusivity Dj of each mobile species is represented23,43 as follows:
(2)
where kB and T respectively denote Boltzmann’s constant and temperature, CT and CTL are respectively the total concentration (both isotopes) of mobile and lattice oxygen, and ΔHf and ΔSf are respectively the standard formation enthalpy and entropy in a hypothetical reference state wherein the Fermi level Ef is located at the valence band maximum (VBM). P0 denotes the reference pressure (atmospheric), and p and NV are the hole concentration and effective density of states in the valence band. The parameter m denotes the interstitial’s charge state, taken to be −2 in accord with quantum calculations in ZnO25−27 and recent near-surface profile observations.28 Other literature29−31 has also indicated the charge state for Oi is m = −2. As the formation enthalpy and entropy are model outputs rather than inputs, computation of CT ultimately began with the electroneutrality condition for the solid, whereby the sum of charges from donors, acceptors, and free carriers must sum to zero. For ZnO, which is well-known to be n-type, the majority donor concentration was computed as indicated below. The electron concentration n was computed from this donor concentration, and p was calculated through the law of mass action for charge carriers with the known band gap and effective densities of states for ZnO. The band gap EG was estimated by the simplified Varshni equation: EG = E0K − βT
(4)
Dj = gl 2 Γ
(5)
where g represents a geometric factor equal to 1/6 in three dimensions, and Γ denotes the jump rate. The hop length l is on the order of a lattice spacing (2.6 × 10−8 cm44). The jump rate comprises the attempt frequency ν, a standard entropy of migration ΔSdiff,j, and the activation energy for migration Ediff,j: ⎛ ΔS ⎞ ⎛ −E ⎞ Γ = ν exp⎜ diff ⎟ exp⎜ diff ⎟ ⎝ kB ⎠ ⎝ kBT ⎠
(6)
In the present case, D16 = D18 (≡D) because isotopic differences in diffusivity are negligible. The Gj term in eq 1 includes the reaction of injected oxygen interstitials with lattice oxygen atoms OL through sequestration and immobilization by kick-in to the lattice: 18
kki
Oi + 16OL ⇄ 18OL + 16Oi kko
(3)
(7)
where kki and kko denote rate constants having a thermally activated form. Because interstitials of the two isotopes undergo kick-in with equal rate constants, and because of the assumption of thermal equilibrium in the experiment,13,21 the total concentration (summing both isotopes) of Oi is invariant with time. However, the relative balance of isotopes changes with both space and time. Initial and Boundary Conditions. Specimens underwent extended annealing in natural-abundance O2 at the diffusion temperature and pressure before exposure to 18O2. Hence, the initial conditions for the interstitial concentrations assumed that Cj for each isotope was at its natural-abundance equilibrium value.
where E0 K = 3.46 eV and β = 0.365 meV/K.32 Undoped single-crystal ZnO is n-type and oxygen deficient,33−37 with Zni2+ or Vo2+ as the majority point defect under typical uncontrolled surface conditions. Very little experimental literature exists for other possible defect species such as Zn or O antisites, but DFT calculations point to high formation energies that would lead to negligible concentrations.25,26 Recent work in this laboratory13,21 indicates that Oi largely quenches Vo when an atomically clean surface is present, leaving Zni2+ as the majority point defect overall. Even if the surface is not clean, Zni2+ probably still dominates. DFT calculations indicate that the formation energies for Zni2+ or B
DOI: 10.1021/acs.jpcc.7b09962 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry C k inj = vinje−Einj/ kBT
Boundary conditions deep in the bulk and at the surface are required for each mobile species in the differential equations represented by eq 1. Deep within the solid, the concentrations Cj were assumed to remain at their natural-abundance values with no spatial variation. Mathematically, these conditions were implemented by ∂Cj/∂x = 0 as x → ∞. The kinetic expressions for defect injection represents boundary conditions for 16Oi and 18Oi at the surface. The boundary condition of each mobile isotope species j can be written as −D
∂Cj ∂x
with an activation energy Einj and pre-exponential factor vinj. In line with standard estimates from absolute rate theory, vinj was set equal to 1 × 1013 s−1. For 18O, the net flux includes both annihilation and injection according to F18 = −D
(8)
3DjS0 l
Θsat(1 − θ )Cj , x = 0
F16 = −D
=
Φ=
(9)
(10)
With this value of θj, rann,j was computed according to eq 9. The value of rinj was computed according to the first-order expression
rinj, j = k injnsatθj
(13)
=− x=0
3DS0 (1 − θ16)ΘsatC16, x = 0 l
(14)
x
1 [Cexpt(x) − Csim(x)]2 σexpt(x)2
(15)
In this expression, σexpt represents the standard deviation of the experimental concentration at each point. Φ is then computed for arrays of parameter values within a local neighborhood around each parameter; a decrease in Φ signifies an improvement in the parameter. This procedure is iterated for successively smaller changes in each parameter until Φ no longer varies significantly. Iteration continued until the objective function changed by less than a tolerance of 0.1%. Confidence intervals at the conventional level of 67% were computed as described in detail elsewhere.23 The estimates of the parameters for eqs 1−14 were mostly chosen from the WSSE optimized parameters from ref 48. For parameters pertaining to the sequestration and emission of interstitial defects at extended defects in the bulk, the values were chosen from experimental literature pertaining to the association and dissociation of defect clusters in silicon34 as a similar material system. Most initial parameters were determined from preliminary manual fitting of experimental diffusion profiles or quantum calculations from the literature.29,40,41 The necessary values for the enthalpy and entropy of formation for point defects in rutile TiO2 were determined from the literature based upon experimental re-
1 +1
∑∑ n=1
nsat,max lνinje−Einj/ kBT + 3DS0Cj , x = 0
3DS0Cj , x = 0
∂C16 ∂x
Nexpt
3DS0Cj , x = 0
nsat,max lνinje−Einj/ k BT
3DS0 (1 − θ18)ΘsatC18, x = 0 l
Parameter Estimation and Confidence Intervals. Activation energies, pre-exponential factors, and other parameters embedded within the system of equations represented by eq 1 were determined via the least-squares technique of weighted sum of square errors (WSSE). Application of this approach to TiO2 has been described in detail elsewhere,23 and relies upon iterative solution of eq 1 using the FLOOPS simulator.47 In each iteration, the system of differential equations is solved using the FLOOPS simulator together with that iteration’s current estimates of the embedded parameters. The simulations yield computed concentration profiles Csim(x) of 18O for the experimental conditions corresponding for each of the Nexpt available diffusion profiles Cexpt(x), with each profile having its own value of T and PO2. The difference Cexpt(x) − Csim(x) is computed for a set of discrete depths x for each profile, and that large set of values is amalgamated to determine the WSSE objective function Φ according to
where S0 is the zero-coverage sticking probability. The coverage Θ is defined as the fractional coverage of active annihilation/ injection sites, normalized with respect to the total possible concentration nsat,max of sites. This fractional coverage permits incorporation of site poisoning effects from foreign-element adsorption, but was not employed here. At steady state, the injection rate equals the annihilation rate, yielding the following expression for the coverage θj of injectable oxygen: θj =
x=0
For 16O however, the net flux only contains an annihilation part since no 16O2 exists in the gas phase, and can be written as
where rinj,j and rann,j respectively denote the elementary-step injection and annihilation rates. Both bulk−surface exchange and surface−gas exchange were presupposed to be in chemical (not isotopic) equilibrium, with the rates of gas adsorption and desorption much faster than that of interstitial annihilation and injection. Indeed, the characteristic time constants for gas adsorption and desorption on ZnO(0001) based upon published kinetics45,46 are roughly 4 orders of magnitude smaller than the time constants characterizing the Oi exchange. Experimental literature indicates that adsorbed oxygen can exist in both atomic and molecular forms on Zn-terminated ZnO(0001) depending upon conditions, and that dissociation is facile.13,30 The principle of detailed balance requires that the surface injection sites be identical to the annihilation sites, and we assume those sites contain a well-defined concentration θ of injectable oxygen. Annihilation and injection kinetics were treated by direct analogy to adsorption and desorption of gaseous species according to first-order Langmuir-type expressions. By analogy to nondissociative gas adsorption, annihilation was assumed to obey a rate expression incorporating the impinging flux and the sticking probability, written as rann, j =
∂C18 ∂x
= k injθ18nsat −
= rinj, j − rann, j x=0
(12)
(11)
where kinj is the injection rate constant given by C
DOI: 10.1021/acs.jpcc.7b09962 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry C Table 1. Initial and Final Parameters for Oi parameter
equation
initial value source
initial value
WSSE estimate
Ediff Do Eki (Eko) Aki (Ako) ΔHf ΔSf nsat S0 Einj
6, 19 5 11, 12 11, 12 2 2 13 9, 19 12
adapted from DFT literature25,33,53 modified from TiO2 model23 modified from TiO2 model23 modified from TiO2 model23 adapted from DFT26,27,29 modified from TiO2 model23 estimated from ZnO literature44 modified from TiO2 model23 estimated from ZnO literature13,21
0.7 eV 0.08 cm2/s 0.2 eV 1× 10−14 cm3/s 5.0 eV 1.5kB 5 × 1013 cm−2 (0.13 ML) 5 × 10−5 1.4 eV
0.62 ± 0.07 eV 0.1 ± 0.01 cm2/s (ΔSdiff,M = 4.49kB) 1.6 ± 0.2 eV (1.2 ± 0.2) × 10−11 cm3/s 5.6 ± 0.2 eV (5.2 ± 0.6) × 10−4 eV/K (4.5kB) (3.2 ± 0.4) × 1013 cm−2 (0.08 ML) (6.5 ± 0.7) × 10−5 2.0 ± 0.2 eV
sults30,31,35,36,40,42−46 using maximum likelihood estimation.42 Calculation of NC and NV employed effective masses of electrons and holes given in the literature.17,49−52
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RESULTS Table 1 shows the final parameter estimates and associated confidence intervals. Figure 1 shows an example of WSSE
Figure 2. Arrhenius plot of net injection flux F using both analytical and microkinetic approaches.
Figure 1. Examples of isotopic diffusion profiles computed based on microkinetic model (lines) compared with corresponding experimental profiles (symbols).
fitting to diffusion profiles at 5 × 10−5 Torr. The simulated profiles reproduce the observed experimental profiles fairly accurately. The upward curvature near the surface of experimental profiles (9 eV25,26). As shown in Figure 11, the simulated coverage θ of injectable oxygen rises with increasing temperature in the range 700−900 K and largely saturates the surface above 1000 K. This upward trend runs opposite to the analogous case of gaseous adsorbates, for which coverage generally decreases with increasing temperature. Such behavior in θ in surface−solid equilibrium represents a qualitative difference from surface−gas equilibrium. The difference originates primarily from the existence in the solid case of temperature-dependent terms for defect hopping and charging. Manipulation of eq 10 yields the following expression for θ in the present case of Oi2−: 1
θ∼
e
−E inj/ k BT
e−Ediff / k BT e−ΔH / k BT
−2
( ) p NV
+1 (20)
The effects of solid hopping enter through Ediff, and those of charging manifest through
−2
( ) p NV
and its multiplying factor
that contains ΔH. These terms have no counterparts in surface−gas equilibrium. Furthermore, although defect annihilation at a surface bears some resemblance to gas adsorption, the diffusional hop that leads directly to an annihilation event necessarily incorporates significant thermal activation. In contrast, barriers to gas adsorption are typically much smaller. In eq 20, only defect injection (through Einj) has a direct counterpart to gas desorption (through the desorption barrier). The numerical values of the other parameters in eq 20 for this ZnO surface combine to yield the following approximate expression for θ:
θ∼
1 e
4.2/ kBT
+1
(21)
This expression describes how θ increases with temperature quite quickly in some ranges. Such behavior does not necessarily generalize to all oxides or semiconductors, as the numerical values of the parameters contributing to eq 20 may conspire in very different ways. A small injection barrier together with large values for the hopping barrier, formation enthalpy, and band gap could combine to make θ increase with T. Equations 10, 20, and 21 also contain an implicit dependence on the oxygen partial pressure in the gas phase, through CT, which contains a pressure-dependent term as shown in eq 2. With respect to pressure, θ behaves much like the gas adsorbate coverage; both quantities increase with PO2 at a given temperature. Figure 11 reflects this behavior. Comparison with Other Semiconductors. Microkinetic models for native defects that include semiconductor surfaces exist only for TiO2(110)23 and Si(100).12 The latter case
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CONCLUSION The present work represents only the second existing attempt to construct a quantitative microkinetic model for the injection of point defects from semiconductor surfaces into the bulk (the first being for TiO2(110)), and the first for a polar surface. Although the injection barrier does not differ greatly from that for nonpolar TiO2(110), the coverage of injectable oxygen increases with temperature. This behavior contrasts with that of TiO2 and gas adsorption in general. Such a variation does not necessarily generalize to all oxides or semiconductors, as the numerical values of the parameters contributing to the coverage may conspire in different ways. A small injection barrier together with large values for the hopping barrier, formation H
DOI: 10.1021/acs.jpcc.7b09962 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry C enthalpy, and band gap combine to make θ increase with T. As suitable data sets for other semiconductor surfaces and other defect types become available over time, it will be instructive to see how widespread such behavior is in this largely unexplored form of surface chemistry.
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AUTHOR INFORMATION
Corresponding Author
*Tel.: (217) 244-9214. Fax: (217) 333-5052. E-mail:
[email protected]. ORCID
Edmund G. Seebauer: 0000-0002-4722-3901 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by NSF (DMR 13-06822 and 1709327). Ming Li gratefully acknowledges fellowship support from the Dow Chemical Co.
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DOI: 10.1021/acs.jpcc.7b09962 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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DOI: 10.1021/acs.jpcc.7b09962 J. Phys. Chem. C XXXX, XXX, XXX−XXX
