Concentration and Mobility of Electrons in ZnO from Electrical

Sep 9, 2010 - As realized already by Thomas and Lander in 19562 and later, ... ZnO based on Thomas and Lander's results appears high, possibly due to ...
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J. Phys. Chem. C 2010, 114, 16785–16792

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Concentration and Mobility of Electrons in ZnO from Electrical Conductivity and Thermoelectric Power in H2 + H2O at High Temperatures Skjalg Erdal, Christian Kjølseth, and Truls Norby* Department of Chemistry, Centre for Materials Science and Nanotechnology, UniVersity of Oslo, Gaustadalleen 21, NO-0349 Oslo, Norway ReceiVed: July 9, 2010; ReVised Manuscript ReceiVed: August 6, 2010

The electrical conductivity and thermoelectric power of ZnO have been examined in hydrogen-containing atmospheres up to 550 °C. The type and concentration of charge carriers (here electrons) and their charge compensating defects (here protons) were determined from the thermoelectric power, and electron charge mobility was evaluated by combination with the measured conductivity. Above approximately 450 °C, the defects are in equilibrium with the surroundings, and the concentration of protons and electrons increases with temperature and is proportional to pH21/4, in accordance with the defect thermodynamics from early literature. Below approximately 450 °C, the concentration of hydrogen appears to become frozen-in. This results in a high internal hydrogen pressure during further cooling, which, for instance, may crack single crystals. The local strain from the presence of frozen-in neutral H2 species is suggested to cause an observed modest reduction in the mobility and conductivity of electrons below the freezing-in temperatures. The levels of defect concentrations and electron mobility are 1 order of magnitude off compared with established literature when based on our thermoelectric power applying standard theory. This discrepancy is in the order of either a reduction in the assumed effective mass of electrons from the commonly used 0.23m0 to 0.075m0 or the removal of the 5/2kB term in the expression used for the entropy of a free electron gas. Introduction ZnO is a semiconductor with a wide direct band gap of about 3.4 eV, receiving much attention for applications such as UV emitting LEDs and UV photo detectors.1 As with most optical and electronic materials, the behavior of ZnO is governed by the various defects present in the material. It is widely accepted that hydrogen acts as a shallow donor in ZnO and that hydrogen is a major source of the observed consistent n-type conductivity.2-9 Hydrogen is omnipresent as H2 and H2O in fabrication processes and during use and is as such an element whose impact on the electrical and optical properties of materials is of great importance to understand. Another important driving force for our investigation is the possibility to combine knowledge of defect concentrations from equilibrium measurements with theoretical models of thermodynamics and transport to improve our understanding of these models and the parameters involved. The combination of the particular defect structure and the broad knowledge-base of ZnO is rare and, thus, a particularly suitable and important case for addressing fundamental questions of this nature. Most experimental work on the electrical properties of ZnO has been conducted at temperatures too low to expect that the concentration of hydrogen in the material can come to an equilibrium with the surroundings and, thus, often neglect the potential effects of hydrogen species.6,10 Implanted hydrogen has been demonstrated to alter the electrical properties of ZnO,11,12 and in general, most work considering hydrogen today applies a physicist’s common view that hydrogen may be present from synthesis or implantation, that it is present at a fixed overall concentration, and that it may possibly be lost by heat treatment. However, the equilibrium defect chemistry established already * To whom correspondence should be addressed. E-mail: truls.norby@ kjemi.uio.no.

in the mid 1950s by Thomas and Lander2 is rarely applied even though it showed beyond doubt that hydrogen dominates the defect structure of ZnO under important conditions. It seems not to have been challenged and still remains the state-of-theart experiment and establishment of the defect chemical parameters. To focus on their important work and to put their defect thermodynamics into a physicochemical context, we will here first review, comment, and interpret their results and then derive theory for the thermoelectric power of ZnO, attempting to uphold the connection with defect thermodynamics. We then interpret thermoelectric power measurements to obtain defect concentrations. Together with simultaneous conductivity measurements, these give rise to charge mobilities of electrons. The comparison with literature and expected behavior allows us to test and discuss the theory and thermodynamic parameters applied. It will become apparent that the freezing-in of hydrogen concentrations beyond the solubility limit seems to affect the electron mobility, a conclusion that may be important for understanding ZnO at lower temperatures. Review of Selected Relevant Literature and Theory Charge Mobility of Electrons in ZnO. We will first review and discuss briefly the charge mobility for electrons in ZnO, with a focus on high temperatures. This is required because we will need it in our treatment of conductivity data and because the vast literature on ZnO may easily leave a confused picture of the prevalent transport and scattering mechanisms. ZnO exhibits n- and p-type electronic conduction, with ionic conductivity being minor. Under the conditions of interest in our work, n-type electronic conduction prevails. The mobility of electrons is characterized by itinerant transport in the conduction band, governed by a range of scattering mechanisms:13 At low temperatures, scattering from ionized and neutral

10.1021/jp1063534  2010 American Chemical Society Published on Web 09/09/2010

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impurities and native point defects dominate. Piezoelectric scattering is also prominent in ZnO, but only at low temperatures. At high temperatures, scattering by phonons becomes dominating, and in ZnO, this is in turn dominated by optical (rather than acoustic) phonons. Scattering may also occur in surfaces and grain boundaries, but in normal sintered ZnO bodies with micrometer-sized grains, bulk scattering will dominate. This is confirmed in our own study by single crystal and sintered ZnO having essentially the same conductivity under well-defined equilibrium conditions. As pointed out in the excellent review by Ellmer,13 the mobility of electrons at elevated temperatures (essentially above room temperature) was determined many decades ago in several high quality studies, primarily by Hutson,14 but also others.15-17 The data sets for a variety of samples agree both in magnitude and in their temperature dependencies and we summarize them as

un ≈ 1 × 106T -3/2 cm2 /V s

(1)

This temperature dependency arises from nondegenerate electron levels and thus applies to ZnO at high temperatures. It may be added at this stage that the high temperature optical phonon scattering is expected to be isotropic in ZnO.13 Defect Chemistry of Hydrogen in ZnO. Thomas and Lander2 measured the electrical conductivity of ZnO at high temperatures in hydrogen atmospheres. They applied 4-point measurements on a single crystal whisker, contained in a sacrificial ceramic ZnO container to reduce Zn(g) evaporation. This and a protected thermocouple enabled reliable equilibrium measurements at temperatures up to 700 °C. Finally, pH2 was varied over a large range up to 100 atm. From conductivity transients after pressure steps they obtained the chemical diffusion of hydrogen orthogonal to the c-axis, consisting of ambipolar diffusion of protons and electrons. From this they derived the self-diffusion of the slowest species, namely, protons:

-0.91 eV D*H+ (cm /s) ) 0.0315 exp ) kT 2

0.0315 exp

-88 kJ/mol RT

(2)

By combining their conductivities with electron charge mobilities reported by Hutson,14 they obtained the concentration of electrons and expressed this as n (cm-3) ) 9.6 × 1012σnT 3/2 (assuming the conductivity σn is given in S/cm). From standard relations between conductivity and mobility, σn ) enun ) enu0nT -3/2, we obtain the temperature-dependent mobility that they employed as un (cm2/(V s)) ) u0nT -3/2 ) (1)/(9.6 × 1012e)T -3/2 ) 6.5 × 105T -3/2, which is within the range in eq 1. Next, they made the assumption that hydrogen is dissolved and ionized to protons and electrons according to

H2(g) ) 2Hi0 ) 2Hi+ + 2e-

(3)

The mass action equilibrium constant for the overall reaction would be KTL ) [Hi+]2n2pH2-1. If protons and electrons dominate, the electroneutrality and equilibrium constant yield n ) [Hi+] ) KTL1/4pH21/4. The n-type electronic conductivity followed the predicted pH21/4 dependency, and we evaluate the equilibrium

coefficient KTL based on their plots of concentration versus T and pH2 as

KTL (cm-12 atm-1) ) 2.5 × 1086 exp

-306 kJ/mol RT

(4) It has later been well confirmed experimentally, see for example refs 6 and 18, and theoretically (e.g., by DFT calculations), see for example refs 7 and 8, that the ionization energy of hydrogen interstitial atoms (enthalpy of the last step of reaction 3) is indeed small enough to be negligible at elevated temperatures, and hydrogen is thus referred to as a shallow donor in ZnO. To transform KTL into units more suitable to derive thermodynamics, we may first note that the pre-exponential term can be recalculated using the molar density of ZnO so as to become 8.0 × 10-5 (mol/mol ZnO).4 Standard entropy changes for defect reactions are often assigned based on molar fractions, and the value obtained here based on KTL,0 ) 8.0 × 10-5 ) exp(∆rS0)/ (R) yields ∆rS0 ) -78 J/molK. This is not an unreasonable value for a reaction that loses one mole of gas, but the standard states referred to are not well-founded in this case. As realized already by Thomas and Lander in 19562 and later, for instance, by Wagner for ZrO219 it is clear that the protons in oxides reside in the electron clouds of oxide ions as OHdefects. This does not change the fact that the protons diffuse as free protons jumping between the oxide ions, and the notations of an interstitial proton and of a substitutional hydroxide ion are interchangeable for most purposes. However, to assess the thermodynamics of the hydrogenation reaction, we now proceed to write it as hydroxide defects in the Kro¨ger-Vink notation:

H2(g) + 2OOx ) 2OHO• + 2e

(5)

We now apply activities for all species involved in the mass action equilibrium constant. We assume that the activity for protons is their fraction of occupancy of oxide ions and that, for oxide ions, it is the site fraction as such. Moreover, we set the electron activity as the concentration n over the effective density of states Nc. For dilute concentrations of defects and hydrogen as an ideal gas, the equilibrium constant of the hydrogenation reaction in eq 5 is then expressed as

KH )

2 2 aOH •a  O e

aO2 OxaH2(g)

( ) ( ) ( )

[OHO• ] 2 [OHO• ]2 n2 -1 n 2 -1 [O] p ) ) pH ) H2 [OOx] 2 Nc [OOx]2 N2c 2 [O] ∆HS0 -∆HH0 exp (6) exp R RT

where [O] is the concentration of oxide ion sites. The hydrogen partial pressure should be expressed in bar, because 1 bar is the standard condition, but atmospheres (atm) will do as an approximation. From the limiting electroneutrality condition and expression for KH we get 1/4 1/4 n ) [OH•o] ) [OOx]1/2N1/2 c KH pH2

(7)

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We can now use the results of Thomas and Lander via the parameters of KTL to obtain parameters for the new KH. Using [O] ) 4.2 × 1022 cm-3 and Nc ≈ 2 × 1019 cm-3 at high temperatures (assuming an effective mass of 0.23 m0 for conduction band electrons20), we get:

KH )

KTL [O]2N2c

)

KTL 7 × 1083

)

2.5 × 1086 -306 kJ/mol -306 kJ/mol exp ) 360 · exp RT RT 7 × 1083 (8) (As a cross-check, insertion of these parameters for KH into the expression for the concentration of electrons and protons (eq 7) reproduces Thomas and Lander’s results.) If the obtained unitless pre-exponential factor of 360 represents the standard change in entropy via K0,H ) 360 ) exp(∆HS0)/(R), as one might expect, then ∆HS0 ≈ +50 J/mol K. The positive sign is incompatible with the loss of one mole of H2 gas. This cannot be easily explained by approximations in the electron effective mass used in the calculation of effective density of states or in possible degeneracy of electrons or protons (the latter, e.g., because protons can take several positions on each oxide ion) as these make only minor changes of the entropy. It may instead be assigned to the entropy consideration of the free electron gas state assigned to electrons in the conduction band: The reaction does not lose, but gains “gas”. However, the two gases do not have the same reference state, and the discussion of the standard entropy change is thus not trivial. We conclude at this stage only that the concentration of hydrogen and electrons in ZnO based on Thomas and Lander’s results appears high, possibly due to an entropy stabilization from the free electron gas. It is of some interest to compare the hydrogenation of ZnO with that of other materials to get insight into for instance why it is so dominated by hydrogen donors and how protons will dissolve under oxidizing conditions. The reaction

1 H2(g) + O2(g) ) H2O(g) 2

(9)

can be subtracted from reaction 5 to obtain an alternative representation of hydrogenation involving water and oxygen:

1 H2O(g) + 2OOx ) 2OHO• + 2e + O2(g) 2

(10)

From an enthalpy of reaction 9 of -248 kJ/mol at the temperatures here considered and the enthalpy of +306 kJ/mol for reaction 5 we get an enthalpy for reaction 10 of +554 kJ/ mol. From a standard tabulated entropy of reaction 9 of -55 J/mol K at the temperatures at hand and the entropies of -78 or +50 J/mol K referred to above for reaction 5, we get entropies for reaction 10 of -23 or +105 J/mol K. The native defect chemistry of ZnO is under dispute. For instance, it is not clear whether oxygen vacancies or zinc interstitials dominate under reducing conditions. For the further discussion we will assume it is oxygen vacancies. Their formation can be described as

1 OOx ) vO•• + 2e + O2(g) 2

(11)

According to Han et al.,21 the equilibrium coefficient of this reaction can be expressed as Kred ) [vO••](n2)/(NC2)pO21/2 ) 3.87 × 1030 exp(-6.34 eV/kT) cm-3 atm1/2. The enthalpy of 6.34 eV corresponds to 612 kJ/mol. In an alternative and more chemical representation, we may write

KR )

aVO· · ae2aO1/22(g) aOOx

( ) ( )( )

[vO••] [O] ) [OOx ] [O]

[vO••] n2 1/2 n 2 1/2 p ) x 2 pO2 ) Nc O2 [OO] Nc

exp

∆RS0 -∆RH0 exp R RT

(12)

and from dividing the expression by Han et al. by the concentration of oxide ions, we get KR,0 ) 9.2 × 107 ) exp(∆RS0)/(R) and, in turn, ∆RS0 ) +152 J/mol K. We may now subtract reaction 10 from reaction 5 and obtain the reaction commonly known as hydration of oxygen vacancies:

H2O(g) + vO•• + OOx ) 2OHO•

(13)

The enthalpy comes out as 554 - 612 ) -58 kJ/mol, a moderately exothermic value. To obtain an entropy, we should use the entropy for reaction 10 that is based on the fraction of the concentration of electrons over the effective density of states and thus obtain 105 - 152 ) -47 J/mol K. The negative value is realistic, as the reaction loses one gas molecule, and is exclusive of any electrons after they have been eliminated by subtraction of the two reduction reactions. In all oxides where it is known, the hydration of oxygen vacancies as written in reaction 13 has negative entropies around the value expected for loss of one mole of water vapor; -120 ( 40 J/mol K.22 We thus hold it likely that the entropy assigned to reactions 5 and 10 based on Thomas and Lander’s results is somewhat too large or the entropy assigned to reaction 11 based on the expressions given by Han et al. is somewhat too small. Both reactions refer to electrons of which entropy evaluations may be difficult, but they are electrons in the same oxide, ZnO, so the problem should come out the same and be eliminated by the subtraction to obtain reaction 13. One may thus suspect that experimental data are inaccurate in one or both cases. For instance, Han et al.’s data may have been based on measurements of oxygen nonstoichiometry that did not take the presence of protons into account. Thermoelectric Power and Related Terms. The thermoelectric power Q has contributions from the entropy S of charge carriers and, for itinerant band-type carriers, from the temperature dependence of the mobility, that is, of the relaxation time between scattering events. It may also have contributions from the transported heat, that is, coupling between heat and charge transport, but we neglect that here, as it is commonly considered insignificant.23 Thus, we have

Q)

1 (S - kBr) zq

(14)

where z is the number and sign of the charge of the carrier (-1 for electrons), q is the elementary charge, and kB is the

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Boltzmann’s constant. The factor r is a number characteristic for the scattering mechanism, reflecting the temperature dependency in mobility through µ ) µ0T -r 24 and thus the relation between the relaxation time τm and average energy ε of the charge carrier according to τm ) τ0((ε)/(kBT))r.23 The entropy S of nondegenerate electrons in an ideal electron gas can be expressed as24

S ) kB

(

Nc 5 + ln 2 n

)

(15)

Based on this, we obtain the following expression for the n-type thermoelectric power:

Q)

(

)

kB 5 Nc + ln -r -q 2 n

(16)

The effective density of states for a parabolic conduction band can be expressed as:

(

Nc ) 2

2πm*kBT h2

)

3/

2

(17)

where m* is the effective mass of electrons and h is Planck’s constant. For ZnO, all the experimental high-temperature data on electron charge mobility suggest, as mentioned above, that r ) 3/2, while the effective mass of electrons m*, pertinent to the effective density of states, is typically reported in the range 0.2-0.3 m0. We will here use the value 0.23 m0.20 Under the assumptions made, eqs 16 and 17 enable us to calculate the concentration of conduction band electrons, n, from the measured thermoelectric power, Q. The n-type electronic conductivity σn can be expressed as

σn ) qµnn

(18)

We may now combine eqs 18 and 16 to get an expression for the mobility in terms of the thermoelectric power and conductivity:

µn )

σn ( -qQ -5/2+r) e k -qNc B

Figure 1. Schematic depiction of the hot zone parts of the setup for measurement of thermoelectric power. The ZnO bar sample was held in various temperature gradients established by locating it off-center in the furnace. The gradient was monitored by top and bottom thermocouples (TCT and TCB), whereas the mean temperature was held constant via a centered thermocouple (TCC) connected to the furnace’s temperature controller. The voltage was measured using a Pt-electrode at either end of the sample. The sample and electrode assembly were held between spring-loaded alumina “floor” and “roof” parts fitting the cutout of the alumina support tube of the ProboStat cell. The entire setup was enclosed in an outer closed-end alumina tube (not shown) and inserted in a tubular furnace.

(19)

Experimental Section A polycrystalline bar and a disk sample were used for thermoelectric power and conductivity measurements, respectively. They were produced by uniaxially cold-pressing commercially available ZnO powder (Merck, zur analyze, Germany) and sintering for 6 h at 1000 °C in oxygen. The samples had a pale yellowish color and a relative density of ∼80% after sintering. The conductivity measurements of the polycrystalline samples were corrected for porosity by applying the firstapproximation empirical method of dividing by the square of the relative density.25 Conductivity measurements were also performed in the c-direction on a single crystal disk sample (ZnOrdic AB, Sweden). Two electrodes were attached to the ends/faces of each sample by adding thin successive layers of

Pt-paint, the organic portion of which was burned off in air at 800 °C prior to electrical measurements. The measurements of thermoelectric power were performed by exposing the 3 cm long bar sample to temperature gradient of 5-20 °C, created by shifting the sample holder in the hot zone of the vertical tube furnace and measured by two thermocouples at either end of the sample. A third thermocouple was placed at the center of the bar and used to control the furnace to maintain a constant center temperature. The voltage over the sample was measured with a high-impedance multimeter (Agilent, 34401A) over a pair of Pt wires contacting the two end electrodes by help of spring-loaded alumina parts. Four different temperature gradients were applied for each constant center temperature and the thermoelectric power obtained from the slope dE/dT to eliminate thermocouple offsets. Figure 1 shows the thermoelectric measurement setup.

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Figure 3. Secondary electron micrographs (FEI Quanta 200 FEGESEM) of the surface of the single crystal ZnO sample after being exposed to wet hydrogen up to 600 °C and then cooled in a wet hydrogen atmosphere to room temperature.

Figure 2. Conductivity as a function of inverse temperature for single crystal disk and polycrystalline disk in wet (2.5% H2O) ∼1 atm H2, measured at decreasing temperature. Results from Thomas and Lander2 for a single crystal, also in wet 1 atm H2, are included.

For calculating mobilities from thermoelectric power and conductivity independent of sample history, the conductivity of the bar sample was measured during thermoelectric power measurements using the voltage probes and end electrodes. The conductivities of the disk samples were measured using four Pt wires contacting the two electrodes pairwise. All conductivity measurements were made using an impedance analyzer (HP 4192A) with an applied 10 kHz AC signal of 1 V rms. All measurements were made in a ProboStat sample holder cell (NorECs, Norway) with controlled atmospheres supplied by a gas mixer described in detail elsewhere.26 The thermoelectric power we report here was measured from 550 to 150 °C in a gas mixture of 90% He and 10% H2, wetted by bubbling through water and then through a saturated solution of KBr, both at room temperature, yielding 2.5% relative humidity. The conductivity of the single crystal and polycrystalline disk was measured as a function of temperature in wet hydrogen from 550 to 200 °C, and isothermally at 500 °C as a function of oxygen partial pressure in wetted H2 + Ar or O2 + Ar mixtures. Results and Discussion Electrical Conductivity. Figure 2 shows the electrical conductivity versus inverse temperature in wet hydrogen atmosphere, comprising two of our samples as well as data from Thomas and Lander.2 The two single crystal data sets are both measured in the c direction. The data sets are in general agreement, and the remaining differences may be due to inaccuracies in the geometrical factors applied, including the correction for porosity, and to differing thermal histories and ramp rates. At the highest temperatures, the conductivities increase with temperature as a result of the increasing equilibrium concentration with hydrogen, which dissolves as protons and electrons. Below around 500 °C, the conductivities level out, assumingly because the concentration of electrons levels out. This can be because the concentration of the donors, hydrogen, becomes smaller than that of another, fixed donor concentration, for example, trivalent impurities or a frozen-in native point defect such as oxygen vacancies or zinc interstitials.

Alternatively, it can be a result of freezing-in of the prevailing defects themselves (protons and electrons). The latter is supported by diffusion data for hydrogen in single crystal ZnO by Thomas and Lander2 who reported that reasonable amounts of hydrogen started to diffuse into the single crystal only above ∼450 °C. When we cool the samples below the effective freezing-in temperature, the difference between the solubility and the actual concentration of dissolved hydrogen (as protons and electrons) increases. This will lead to a high internal activity of hydrogen in the sample and some of the protons and electrons will combine into atoms and H2 molecules, the latter being described in ZnO earlier,6,27,28 both theoretically and experimentally. The continued but weak decrease in conductivity at the lower temperatures can therefore be due to the resulting lower concentration of free electrons or due to a decrease in the mobility of electrons stemming from local strain due to the frozen-in neutral hydrogen species or trapping on the neutral (H2) or partially neutralized (H0-H+) hydrogen clusters. The internal hydrogen activity eventually evolves hydrogen gas, as evident from the pores in the surface of the single crystal after having been cooled in a hydrogen atmosphere, as seen in the SEM micrographs in Figure 3. Fast cooling even fractured the crystal, probably for the same reason. Above ∼350 °C, our single crystal and polycrystalline samples exhibited similar conductivity behavior. At lower temperatures, the conductivity of the single crystal decreased more sharply with decreasing temperature, see Figure 2. We believe this may be attributed to higher strains in the single crystal because of the longer diffusion pathways for hydrogen escape. The dependency of the conductivity on oxygen partial pressure measured on the single crystal sample is given in Figure 4 together with a replot of equivalent measurements by Thomas and Lander.2 The conductivity under reducing conditions exhibit a pO2-1/8 dependency which is equivalent to a pH21/4 dependency at constant pH2O. This is in agreement with the defect model with protons and electrons being the dominating defects, as discussed earlier. The fact that the conductivity decreases by 3 orders of magnitude on going to higher pO2 (lower pH2) shows that equilibrium is attained at this temperature. Moreover, it shows that there is no level of other donors than hydrogen that becomes dominating as the concentration of protons decreases, at least within 3 orders of magnitude. The fact that the curve stays with the pO2-1/8 (or pH21/4) dependency and does not get steeper further indicates that we do not hit a level of acceptors. Thus, it appears that the single crystal sample contains concentrations of net aliovalent impurities that are minor compared to the equilibrium

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Figure 4. Log conductivity as a function of log oxygen partial pressure at 500 °C for single crystal ZnO at constant pH2O ) 0.025 atm. Results from Thomas and Lander (T&L)2 at 496 °C are included.

Erdal et al.

Figure 6. Log concentration of electrons n vs 1/T at 0.1 atm H2. Closed squares calculated on basis of our measured thermoelectric power, assuming m* ) 0.23 m0, r ) 3/2, and treating the conduction band electrons as an ideal gas. Solid line represents concentration calculated for 0.1 atm H2 from Thomas and Lander’s data.

In fact, eqs 16 and 17 can be combined and expanded into

Q)

(

(

Figure 5. Thermoelectric power for polycrystalline ZnO in wet 10% H2 in He. The solid and dotted lines are added as guides-to-the-eye for the behavior in the high (equilibrium) and low (nonequilibrium) temperature regions.

hydrogen concentrations introduced at equilibrium at high temperatures. The small leveling out at high oxygen partial pressure can be taken as the encounter with a background donor-level. It could also be interpreted as a contribution from minority p-type conduction, but that would take a more complex defect model in which some acceptor defect comes into play. Thermoelectric Power, Entropy, and Charge Carrier Concentration. Figure 5 shows the measured thermoelectric power as a function of temperature. It is negative (cold end holds a negative voltage vs the hot end), that is, the charge carriers are negative, in accordance with the assumed n-type conductivity. The absolute value decreases sharply at high temperatures, meaning that the concentration of carriers increases, as expected from the electroneutrality comprising protons and electrons and the equilibrium with hydrogen gas. Below 700 K (or around 450 °C) the measured thermoelectric power becomes troubled by scatter from noise and higher impedances, but it is still clear that it mainly levels out. This is in accord with the conductivity that also levels out around these temperatures, showing that the concentration of electrons becomes roughly constant. As discussed above, this seems to be due to freezing-in of the concentration of protons and charge compensating electrons.

(

)

)

kB 5 kB 2πkBTm* 3/2 NC + + ln -r ) ln 2 -q 2 n -q h2 3kB kB 5 - ln n - r ) A + ln T (20) -q 2 -2q

)

where A is a constant that contains the constant frozen-in concentration n. A plot of Q versus ln T for electrons, therefore, independent of the values of m* and r, has a slope of -3kB/2q ) -1.29 × 10-4 V/K. In rough agreement with this, the linear fit of the points at the six lowest temperatures has a slope of -1 × 10-4, and we may from this state that the thermoelectric power in this region may well represent a constant and most likely frozen-in concentration of electrons, where the variation of the thermoelectric power with temperature is only due to the temperature dependence of the effective density of states. From the thermoelectric power we may now calculate the concentration of electrons, n, under assumptions of m* and r. The results are plotted in Figure 6. At low temperatures (below approximately 450 °C) it appears that the concentration is constant, as anticipated above. At high temperatures, log n is proportional to inverse absolute temperature, as expected from the defect model. A fit to this region gives n (cm-3) ) K1/4pH21/4 ) A exp(-38 (kJ/mol)/RT), with the pre-exponential A only varying with the model used. With pH2 ) 0.1 atm during these measurements, we obtain K (atm-1 cm-12) ) K0 exp(-152 (kJ/ mol)/RT) where K0 ) (A4/0.1). Applying standard theory as described earlier, treating the conduction band electrons as an ideal gas and applying an electron effective mass of 0.23 m0, yields electron concentrations just over an order of magnitude above what was reported by Thomas and Lander.2 It is noted here that this discrepancy is of a magnitude equivalent to 5/2kB, as enters into the expression used for the entropy of the assumed ideal electron gas. Conversely, a reduction in the assumed effective mass of electrons in ZnO (as included in the term used for the effective density of states) from 0.23 m0 to 0.075 m0,

Concentration and Mobility of Electrons in ZnO

Figure 7. Double-logarithmic plot of electron charge mobilities vs absolute temperature. Closed squares are our values from thermoelectric power and conductivity of a polycrystalline bar sample, assuming m* ) 0.23 m0, r ) 3/2, and treating the conduction band electrons as an ideal gas. Solid line represents mobility from Hutson14 as used by Thomas and Lander.2

has the equivalent effect of removing the discrepancy between our calculated carrier concentrations and those of literature. The enthalpy of +152 kJ/mol is to be compared to that of Thomas and Lander of +306 kJ/mol. The pre-exponential K0 for the r ) 3/2 (middle) curve is correspondingly 5 orders of magnitude lower than that of Thomas and Lander, making the standard entropies calculated thereupon approximately more negative. Thus, the standard entropies from Thomas and Lander, based on molar fractions of all defects and on fractions referring to sites and effective density of states, -78 and +50 J/mol K, respectively, come out from our work instead as -170 and -42 J/mol K. Combining the last value as before with Han et al.’s data for formation of oxygen vacancies, we obtain an entropy of -139 J/mol K for the hydration of oxygen vacancies. This is more realistic than the -47 J/mol K based on Thomas and Lander’s data. The enthalpy for hydration of oxygen vacancies correspondingly becomes -212 kJ/mol based on our data as compared to the -58 kJ/mol we derived above based on Thomas and Lander’s data. It is important to stress now that we are not claiming that our data are better than those of Tomas and Lander. The data sets, especially ours, are vulnerable to correlations between entropy and enthalpy. The main purpose is to illustrate that the analyses are useful and may lead to prediction of parameters of hydration that may be applicable to acceptor-doped ZnO under oxidizing conditions. Mobility of Electrons. We finally combine the electrical conductivity with the concentration of charge carriers found from thermoelectric measurements to obtain the charge mobility of electrons in ZnO according to eq 19. Figure 7 shows the calculated mobilities based on our data, together with similar mobility data by Hutson,14 as used by Thomas and Lander.2 At high temperatures, where the proton and electron concentrations reflect equilibrium with the surrounding atmospheres, the mobility decreases with increasing temperature. This is in accord with a model of optical phonon scattering, where the mobility expectedly has a T-3/2 dependency. The high temperature mobility of Huston14 employed by Thomas and Lander2 is 1 order of magnitude higher than ours; the conductivities (product of charge, concentration, and mobility) in the two works are similar under the same experimental conditions (same pH2 and T), but the concentrations and mobilities vary by the same

J. Phys. Chem. C, Vol. 114, No. 39, 2010 16791 1 order of magnitude. In parallel with the concentration data, the mobilities at high temperatures coincide with the established literature data if we make similar changes to either entropy expression for the assumed free electron gas or by a reduction in the assumed effective mass of electrons, as mentioned above. We must, however, leave for follow-up investigations to clarify whether this is a coincidence of an experimental or theoretical error or whether it is of some validity. No doubt, the issue would benefit from a direct, preferably equilibrium, measurement of the concentration of hydrogen in ZnO at high temperature in atmospheres of controlled H2 partial pressure, as a complement to Thomas and Lander’s and our concentrations, both obtained indirectly from electrical properties. Regardless of the discussion of the level, our mobilities go through a maximum around the temperature where hydrogen becomes frozen in. The small variations of the mobility (Figure 7) and to some extent also the peak appear to be correlated to the concentrations (Figure 6), so one should be careful not to overinterpret the peak. However, the fact that the conductivities do fall off with decreasing temperature in the frozen-in region, where the concentrations appear to be rather constant, may indicate that the mobility is indeed decreasing below the freezing-in point. We suggest this may be attributed to local strain around hydrogen molecules formed from the oversaturated concentration of protons and electrons, as discussed earlier. Additionally, there could be a certain degree of electron trapping (decreasing the mobility) on hydrogen defect clusters (H2 or H0-H+). These are likely to result from the oversaturation of hydrogen as the temperature is decreased and constitute intermediate steps on the way to evolvement of hydrogen gas. In principle, the electron mobility would be less reduced if the sample contained less hydrogen by cooling from atmospheres with lower hydrogen contents or by cooling at lower rates. However, this would of course also reduce the concentration of ionised hydrogen and electrons. A clear-cut positive effect on conductivity would thus not be expected, and we did not pursue this in our study. Conclusions The conductivity of polycrystalline and single crystal ZnO in hydrogen-containing atmospheres at high temperatures behaves in accord with the measurements and defect structure established by Thomas and Lander2 in the 1950s: Hydrogen dissolves and ionizes fully to protons and electrons. These dominate the defect structure and their concentrations are accordingly proportional to pH21/4. Below ∼450 °C the conductivity levels out. In agreement with proton diffusion data and observations that single crystals tend to crack by further cooling, this suggests that hydrogen effectively freezes-in below these temperatures, creating high internal hydrogen pressures and probably the presence of dissolved neutral hydrogen molecules as well as hydrogen pockets. The thermoelectric power has been measured under the same conditions, giving rise to onward calculations of the entropy, concentration, and charge mobility of electrons. The general behavior of the electrons in terms of an equilibrium concentration at high temperatures and a constant concentration below the freezing-in temperature is verified. However, the level is an order of magnitude too high as compared with Thomas and Lander’s data when we apply established concepts of free electron gas theory for the calculations of the entropy and concentration. Similarly, this gives electron mobilities an order of magnitude too low. It is noted that cancellation of the 5/2 term in the entropy of the free electron gas, or a reduction of

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the effective mass as used in the effective density of states expression, both bring the equilibrium concentration and the mobility of electrons into agreement with the established literature. Below the freezing-in temperature the conductivity decreases weakly with decreasing temperature. This appears to be reflected in the mobility more than in the concentration of electrons. It is speculated that it is due to local strain from hydrogen molecules under the high internal hydrogen pressure arising from the frozen-in defect structure. A discussion of defect thermodynamics shows that we may calculate the hydration thermodynamics for acceptor-doped ZnO under oxidizing conditions based on the existing data from hydrogen atmospheres. Despite uncertainties in this, it is clear that hydration is favorable, meaning for instance that oxygen vacancies charge compensating acceptor dopants will be hydrated to high temperatures, introducing protons and depressing p-type conduction. This is of considerable consequence for the fabrication, characterization, and use of p-type ZnO. Acknowledgment. This work has been funded by the Research Council of Norway (FRINAT 171157/V30 “Hydrogen in Oxides”) and by the FUNMAT@UiO programme. References and Notes (1) Ozgur, U.; Alivov, Y. I.; Liu, C.; Teke, A.; Reshchikov, M. A.; Dogan, S.; Avrutin, V.; Cho, S. J.; Morkoc, H. J. Appl. Phys. 2005, 98, 041301/1. (2) Thomas, D. G.; Lander, J. J. J. Chem. Phys. 1956, 25, 1136. (3) Van de Walle, C. G. Phys. B (Amsterdam, Neth.) 2001, 308-310, 899. (4) Cox, S. F. J.; Lord, J. S.; Cottrell, S. P.; Gil, J. M.; Alberto, H. V.; Piroto Duarte, J.; Vilao, R. C.; Keeble, D. J.; Davis, E. A.; Keren, A.;

Erdal et al. Scheuermann, R.; Stoykov, A.; Charlton, M.; van der Werf, D. P.; Gavartin, J. Phys. B (Amsterdam, Neth.) 2006, 374-375, 379. (5) Look, D. C.; Hemsky, J. W.; Sizelove, J. R. Phys. ReV. Lett. 1999, 82, 2552. (6) McCluskey, M. D.; Jokela, S. J.; Oo, W. M. H. Mater. Res. Soc. Symp. Proc. 2005, 864, 493. (7) Peacock, P. W.; Robertson, J. Appl. Phys. Lett. 2003, 83, 2025. (8) Van de Walle, C. G. Phys. ReV. Lett. 2000, 85, 1012. (9) Van de Walle, C. G.; Neugebauer, J. Nature (London, U.K.) 2003, 423, 626. (10) Kinemuchi, Y.; Ito, C.; Kaga, H.; Aoki, T.; Watari, K. J. Mater. Res. 2007, 22, 1942. (11) Johansen, K. M.; Christensen, J. S.; Monakhov, E. V.; Kuznetsov, A. Y.; Svensson, B. G. Appl. Phys. Lett. 2008, 93, 152109/1. (12) Schifano, R.; Monakhov, E. V.; Christensen, J. S.; Kuznetsov, A. Y.; Svensson, B. G. Phys. Status Solidi A 2008, 205, 1998. (13) Ellmer, K. Springer Series in Materials Science; Springer: New York, 2008; Vol. 104. (14) Hutson, A. R. Phys. ReV. 1957, 108, 222. (15) Wagner, P.; Helbig, R. J. Phys. Chem. Solids 1974, 35, 327. (16) Li, P. W.; Hagemark, K. I. J. Solid State Chem. 1975, 12, 371. (17) Look, D. C.; Reynolds, D. C.; Sizelove, J. R.; Jones, R. L.; Litton, C. W.; Cantwell, G.; Harsch, W. C. Solid State Commun. 1998, 105, 399. (18) Seager, C. H.; Myers, S. M. J. Appl. Phys. 2003, 94, 2888. (19) Wagner, C. Ber. Bunsen-Ges. 1968, 72, 778. (20) Oshikiri, M.; Imanaka, Y.; Aryasetiawan, F.; Kido, G. Phys. B (Amsterdam, Neth.) 2001, 298, 472. (21) Han, J.; Mantas, P. Q.; Senos, A. M. R. J. Eur. Ceram. Soc. 2001, 22, 49. (22) Norby, T.; Widerøe, M.; Glo¨ckner, R.; Larring, Y. Dalton Trans. 2004, 3012. (23) Seeger, K. Semiconductor Physics - An Introduction, 6th ed.; Springer-Verlag: Berlin, 1997. (24) Zeghbroeck, B. V. http://ecee.colorado.edu/∼bart/book/, 2007. (25) Marion, S.; Becerro, A. I.; Norby, T. Ionics 1999, 5, 385. (26) Norby, T. Solid State Ionics 1988, 28-30, 1586. (27) Karazhanov, S. Z.; Ulyashin, A. G. Phys. ReV. B 2008, 78, 085213. (28) Lavrov, E. V. Phys. B 2009, 404, 5075.

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