Thermodynamic point defect model of barium titanateand application to the photorefractive effect
B. A. Wechsler and M. B. Klein
Hughes Research Laboratories. 011 Malibu Canyon Road, Malibu, California 90265
Received January 18, 1988; accepted March 29, 1988
We present a model that provides a means for understanding the effects of crystal growth and processing conditionson the photorefractive properties of BaTiO3. The densities of photorefractive centers are calculated from athermodynamic point defect model. This model allows for the existence of transition-metal dopants in multiplevalence states and for the incorporation of more than one species with levels in the band gap. Beam-coupling gainand response time are calculated as a function of temperature and oxygen partial pressure of treatment for variousdopant types and concentrations. The predicted behavior is compared with existing experimental data on BaTiO3crystals, and implications for improving the photorefractive performance are discussed.
1. INTRODUCTION
Photorefractive materials are of interest for a wide variety ofpotential applications, including optical signal processing,holographic storage, and optical phase conjugation. BaTiO3is a particularly promising material because its large electro-optic coefficients lead to correspondingly large values ofbeam-coupling gain and four-wave-mixing reflectivity.However, its applications are limited by the fact that mostas-grown BaTiO3 crystals have a relatively long responsetime-of the order of 0.1 to 1.0 sec at an intensity of 1 W/cm2 .This is approximately 3-4 orders of magnitude slower thanthe theoretical limit for the response time, suggesting thatsubstantial improvements can be achieved through modifiedprocessing techniques. Recent efforts at improving thetransient performance of BaTiO3 by doping and/or heattreatments were only partially successful, owing in part tothe limited understanding of the basic photorefractivemechanism in this material.
The identity and properties of the species responsible forthe photorefractive effect in BaTiO3 have been studied forsome time. Micheron and Bismuthl measured the diffrac-tion efficiency in undoped BaTiO3 and samples doped withFe, Ni, and Nb. They found that Fe and Ni enhanced theefficiency, whereas Nb reduced the efficiency. Kratzig etal.2 found a deterioration of the transport properties ofBaTiO3:Fe compared with undoped samples. Ducharmeand Feinberg3 and Ducharme4 showed that heat treatmentsat successively lower oxygen partial pressures could changethe sign of the dominant photocarrier and produce an in-crease in the gain coefficient. Klein and Schwartz5 showedthat Fe was the most abundant impurity in a group of com-mercial BaTiO3 crystals and that the majority of the Fe wasin the Fe3+ valence state. In our previous studies6 we mea-sured the gain and response time in a BaTiO3 sample afterheat treatments at various oxygen partial pressures. Weobserved a factor-of-10 improvement in response time in asample treated in the most reducing atmosphere, but highconductivity prevented complete poling of this sample.Schunemann and co-workers7 8 measured the diffraction ef-ficiency in BaTiO3 doped with various concentrations of Feand treated at high temperatures in atmospheres with vari-
ous oxygen partial pressures. They found a relatively smallvariation in the diffraction efficiency for wide ranges of Feconcentration and substantial photorefractive behavioreven in a high-purity undoped crystal. Godefroy et al.,9 onthe other hand, reported a strong influence of Fe concentra-tion on diffraction efficiency. This effect was largely attrib-uted to an enhancement in the electro-optic coefficient.
The above results strongly suggest that doping and oxida-tion-reduction treatments can substantially modify thephotorefractive properties of BaTiO3. However, there aremany apparently conflicting observations and little theoret-ical understanding of the effects of materials processingtechniques on the photorefractive centers. A thermody-namic defect model of BaTiO3 developed by Hagemann'0
provides a framework for understanding these effects. Thismodel was used by Wechsler and Klein" and by Schune-mann and co-workers7 8 to examine the effects of oxygenpartial pressure and dopant concentration on the popula-tions of possible photorefractive centers. In this paper weapply Hagemann's defect model to develop a more generalunderstanding of the photorefractive behavior of BaTiO3.The model is used to predict the concentrations of pointdefects as a function of temperature and oxygen pressure ofprocessing for doped and undoped crystals. These concen-trations are then used to define the photorefractive trapdensities and hence to determine the beam-coupling gainand response time. The technique is used to evaluate theeffects of oxidation-reduction treatments on the photore-fractive properties, to compare the behavior of various po-tential dopants, and to assess the possible importance ofintrinsic defects as photorefractive centers.
2. DEFECT MODEL
Because of its importance as a dielectric material, the defectproperties of BaTiO3 have already been extensively studied.Electrical conductivity, thermogravimetric, and other phys-ical and chemical measurements were performed for un-doped,12-14 acceptor-doped,' 0"12, 3"15 - 7 and donor-doped12"13,' 8 19 ceramics and single crystals.2 0 2' These stud-ies elucidated the importance of oxygen vacancies as the
principal defect in these materials and showed that the oxy-gen-vacancy concentration is determined by the dopant/impurity concentration and by the partial pressure of oxy-gen in the surrounding atmosphere.
The defect model we have adopted is essentially that ofHagemann.10 Three types of point defect are considered:oxygen vacancies, acceptor dopants/impurities, and freecharge arriers (electrons and holes). Donor-type dopantscan also be included in the model, but these are generally lessimportant in BaTiO3 and will not be considered here. In thediscussion that follows the defect equilibria are presented,using Kroger-Vink notation.22 Each defect species is iden-tified by a symbol indicating the nature of the defect (e.g., Vfor vacancy, A for acceptor), a subscript indicating the crys-tallographic site upon which the defect is located, and asuperscript denoting the charge of the site relative to that ofthe species normally occupying the site (X, neutral; ', -1;,+1). Thus, for example, the symbol Vo- describes a vacancylocated on an oxygen site with a charge of +2 relative to thenormal lattice with an 02- ion on the oxygen site.
Oxygen vacancies are created by the reaction
00 VOX + 1/202, (1)
where Oo denotes a normally occupied oxygen site. Themass action equation for this reaction relates the oxygen-vacancy concentration to oxygen pressure at a particulartemperature:
[VOX]PO2 KR, (2)
where the square brackets indicate concentration. Theequilibrium constant KR is temperature dependent and isconveniently expressed as
KR = K0' exp(-EO'/kBT), (3)
[VO1 2[VoX]exp[(E+ - EF)/hBT (7)
and
[Vo] = [VOX]exp[(E+ + E++ - 2EF)/kB71- (8)
Here, EF is the Fermi level and E+ and E++ are the ionizationenergies corresponding to reactions (5) and (6), respectively.(In this paper we adopt the convention that energies areexpressed relative to the valence band edge.)
The total acceptor concentration does not vary with tem-perature or oxygen pressure. Here, the term acceptor isused to describe any species that, in its usual valence state,has a negative charge relative to the species that it replaces.This includes, for example, multivalent transition metalssuch as Fe and Mn substituting for Ti4+, fixed valence ionssuch as A13+ substituting for Ti4 +, and barium vacancies.These species are present in three forms: Ax, A', and A",which have a charge of 0, -1, and -2, respectively, relative tothe ideal lattice site. These may correspond to multiplevalence states of a transition-metal dopant substituting forTi4 +, e.g., Fe4+, Fe3 +, and Fe2 +. Their relative populations(in the dark) are determined by thermal ionization of elec-trons or holes, according to the following relations:
Ax + e' =; A' or Ax=A'+h (9)
and
A' + e' it; A' or A'=A'+h',
with the condition that
[A] = [AX] + [A'] + [A"].
(10)
(11)
The concentrations of the individual acceptor species arecalculated from
where kB is Boltzmann's constant, T is the absolute tempera-ture, Eo' is the enthalpy change, and KO' is a constant relatedto the entropy change of this reaction. Combining Eqs. (2)and (3) and rearranging, we obtain the concentration ofneutral oxygen vacancies as a function of temperature andoxygen pressure:
[VOX] = KO' exp(-EO'/kBT)Po"-1/2 (4)
In reaction (1) oxygen is removed as a neutral species. Thetwo electrons normally associated with the Q2- ion are leftbehind; thus the vacancy is electrically neutral with respectto the normally occupied site. These neutral oxygen vacan-cies behave as shallow donors as the two electrons are ther-mally ionized:
VoX - VO + e' (5)
and
Vo± V0 + e', (6)
where e' is an electron. The concentrations of singly anddoubly ionized oxygen vacancies at thermal equilibrium aregiven by
[A'] = /2[AX]exp[(EF -E-)/kBT], (13)
and
[A"] = [AX]exp[(2EF - E- - E)/kBT]. (14)
E- and E-- are the ionization energies corresponding toreactions (9) and (10). Equations (7), (8), and (12)-(14) arestandard expressions for the statistics of multicharge cen-ters,2 3 in which the factors of 1/2, 1, and 2 that'appear asmultipliers of the exponential terms are appropriate spin-degeneracy factors.
The product of the free-carrier concentrations is deter-mined by thermal ionization across the band gap and is givenby
nP = NCNV exp(-Eg/kBT), (15)
where n and p are the concentrations of electrons and holes,respectively; NC and N are the densities of states in theconduction and valence bands, respectively; and Eg is theband-gap energy. The individual carrier concentrations areadditionally determined by the position of the Fermi level:
n = NC exp[-(EC - EF)/kB7 (16)
[AX] = (12)
_
B. A. Wechsler and M. B. Klein
Vol. 5, No. 8/August 1988/J. Opt. Soc. Am. B 1713
Table 1. Summary of Defect-Model InputParameters
Parameter Valuea
Density of states, conduction band (NC) 1.6 X 1022 cm-3Density of states, valence band (NV) 1.5 X 1022 cm-3
Preexponential factor (Ko') 2.7 X 1028 cm-3 barl/2Enthalpy of reduction (Eo') 5.65 eVEnergy of conduction band edge (Ec) 3.1 eVEnergy of valence band edge (Ev) 0.0 eVE+ (VoXIV6) 3.05 eVE++ (Vo/Vo-) 2.9 eVE- (Fe3+/Fe4+) 0.8 eVE-- (Fe2+/Fe3+) 2.4 eVE- (Mn3+/Mn 4 +) 1.3 eVE-- (Mn2+/Mn3+) 1.9 eVE- (Vnat/VBaX) 1.2 eVE- (VBa"IVBa') 1.8 eV
a All values are taken from Ref. 10.
and
p =Nv exp [- (EF-EV)/kB71, (17)
where EC and EV are the energies of the conduction andvalence band edges, respectively. Equations (15)-(17) arevalid when the Fermi level is more than a few kBT from theband edges.
One further equation is established by the condition thatoverall charge balance must be maintained. This is ex-pressed by
[A'] + 2[A"] + n = [VO'] + 2[V0 -j + p, (18)
where all species on the left-hand side are charged negativelyand those on the right are charged positively with respect tothe ideal lattice. Note that neutral defect species do notenter into this equation. Also, because several acceptor-type defects may be present simultaneously, the acceptorconcentration in this equation refers to the sum of all suchspecies.
We are interested primarily in the populations of defectspecies in thermal equilibrium at room temperature. Be-cause equilibrium between the crystal and the atmospherecannot be maintained at room temperature, one additionalassumption must be made. The oxygen-vacancy concentra-tion is determined by processing at some elevated tempera-ture (typically in the range 600-1000°C) in an atmospherewith a given partial pressure of oxygen. We assume that thecrystal is cooled quickly enough from the process tempera-ture that this concentration remains fixed. Any reequili-bration during the cooling process will introduce an uncer-tainty into the calculation. However, under most circum-stances this is expected to be small relative to otheruncertainties.
The results presented below were obtained by the simulta-neous solution of Eqs. (4), (7), (8), (12)-(16), and (18). Ourprocedure involved two steps. We first calculated defectpopulations at room temperature and then determined theoxygen partial pressure required to produce the same oxy-gen-vacancy concentration at the process temperature. Be-cause of the lack of sufficient experimental data, we neglectthe temperature dependence of the energy parameters de-scribing the system, including the band-gap energy and theionization energies of oxygen vacancies and acceptor dop-
ants. In addition, the model considers only point defectsand does not account for any defect clusters. We also as-sume that the model parameters determined by Hage-mann10 from high-temperature experiments on cubicBaTiO3 can be applied at room temperature, at which pointthe symmetry is tetragonal. The values of all input parame-ters needed to perform a model calculation were taken fromthe work of Hagemann' 0 and are summarized in Table 1.
3. PHOTOREFRACTIVE MODEL,
We begin by reviewing the model for photorefractive gratingformation that assumes a single species in two valence statesand allows for photoconduction by electrons and holes. Asdiscussed below (Subsection 3.B), the two valence states ofthe photorefractive center can be described in terms of asingle ionization-energy level. We therefore refer to this asthe single-level model. We show later that the same modelmay be applied to a species with three possible charge states(i.e., two ionization levels) as well as to the case of twoionizable species. In both instances only one ionizationlevel contributes to the photorefractive effect in a givencrystal. All other levels are either filled or empty and can-not provide the necessary admixture of valence states.
A. Calculation of Gain and Response TimeSolutions for the steady-state space-charge field were givenfor the single-level electron-hole model by Strohkendl etal.24 and Valley.25 The beam-coupling intensity gain coeffi-cient r is calculated from
27r reff kBT K (C - 1) cos 20,Xnb cos e + (K/K,)2 (C +1)
(19)
where X is the wavelength of the writing beams, nb is thebackground refractive index, kB is Boltzmann's constant, Tis the absolute temperature, e is the electron charge, K is thegrating wave vector, and 20 is the angle between the twowriting beams in the crystal. The effective electro-opticcoefficient when the grating wave vector is along the c axis isgiven by reff = r33ne
4 cos20 - rl3n 0
4 sin2 0. K, is the Debyescreening wave vector defined by
Ks = (e2NE/eokBT)/ 2, (20)
where e is the relative dielectric constant parallel to the c axisand EO is the permittivity of free space. NE, the effectivetrap density, is given by
NE = NAND/(NA + ND)- (21)
ND and NA are the concentrations of photorefractive donorsand acceptors, respectively. Specifically, ND represents thedensity of centers that may act as donors of photoelectronsand traps for holes; NA represents the density of centers thatmay act as sources for photoionized holes and traps forelectrons. The total number of photorefractive centers isNA + ND. This notation is equivalent to that used by Val-ley25 and by Klein2 6 (where the symbols N and N+ have thesame meaning as ND and NA here) but differs slightly fromthe terminology used by Kukhtarev,2 7 Strohkendl et al.,24
and Ducharme and Feinberg. 3
The parameter C accounts for the photoconductivity con-tributions due to electrons and holes and is defined by
B. A. Wechsler and M. B. Klein
1714 J. Opt. Soc. Am. B/Vol. 5, No. 8/August 1988
C Sh (K2
+ Ke 2) NA (22)
Se (K 2 + Kh2) ND
where Sh and Se are the photoionization cross sections forholes and electrons, respectively, and Ke and Kh are theinverse diffusion lengths for electrons and holes.
We calculate the response time r from the dielectric relax-ation rate y as follows:
1 1 + (K/K S )2 1 + (K/KS)2
-= 7 = zdi 1 + (K/Ke)2' 1 + (K/Kh)(
where
Se/e el ND (24)ye EeO NA
and
Ydih =ShAh el NA
Yh EeO ND (25)
Here, Ydie and Ydih are the dielectric relaxation rates forelectrons and holes, respectively; A1
e and ,lh are the mobilitiesfor electrons and holes, respectively; Ye and Yh are the re-combination rates for electrons and holes, respectively; and Iis the incident intensity.
B. Energy-Level ModelIn previous work in which the single-level model was appliedit was generally assumed that the photorefractive centerswere present in two valence states (e.g., Fe2+ and Fe3+).Studies of BaTiO3 ceramics, 0,28 however, indicate that threevalence states (e.g., Fe2+, Fe3+, and Fe4+) are possible forcertain transition metals. In this section we show how thispossibility may be accounted for in the photorefractive mod-el and examine its consequences.
Figure 1 depicts the ionization, transport, and recombina-tion of charge carriers from a hypothetical species that maybe present in two valence states, X and X+. The ionizationreactions and associated energies may be written as
X ±= X+ + e (AE) (26)
and
X+ X + h+ (AE2)- (27)
AEj represents the difference in energy between the finalstate of the system (X+ plus one conduction electron) andthe initial state (X and no conduction electron) or, in other
e
II+
Fig. 1. Schematic energy-level model for a single species in twovalence states. Electrons are ionized from X and recombine withX+. Holes are ionized from X+ and recombine with X. AE, andAE2 are the energies associated with these processes.
words, the energy involved in promoting an electron from Xinto the conduction band. The same energy is involved inthe recombination of an electron from the conduction bandwith X+. Similarly, E2 represents the energy involved inthe ionization of a hole from X+ and the recombination of ahole with X. The sum of these two reactions can be repre-sented by
nil e + h+ (28)and
AE1 + AE2 = Eg, (29)
which, as expected, corresponds to the energy required tocreate an electron in the conduction band plus a hole in thevalence band.
The ionization energies AE, and AE2 are experimentallyobservable quantities that may be derived from thermal- oroptical-excitation measurements. (Thermal-ionization andphotoionization energies may differ somewhat; in calculat-ing the populations of species in thermal equilibrium it is thethermal-ionization energies that are important.) The ion-ization energies may also be determined from the absoluteenergies for X and X+ through quantum-mechanical calcu-lations.29 For example, the energy AE2 [reaction (27)] isdetermined by the difference in the absolute energies of Xand X+, with account taken for the different occupationnumbers of the valence band for each state.
The ionization-energy diagram shown in Fig. 1 presentsthe same picture that has been used widely in discussions ofdefects and impurities in semiconductors.2 2 29 30 A varietyof notations has been used in labeling the levels in suchdiagrams. Here we choose to identify each level by thespecies forming the initial and final states in an ionizationreaction. This should not be misconstrued to indicate thatthe two discrete species, X and X+, have the same absoluteenergy or, as shown below, that one particular species (e.g.,Fe3+) may have different energies because it appears at twodifferent levels. Rather, the levels in this diagram should beviewed as giving the difference in energy between defectstates with differing numbers of bound electrons, and thelabel identifies those states that are involved.
In Fig. 2 the ionization levels determined by Hagemann10for oxygen vacancies, Fe3+/Fe4+, and Fe2+/Fe3+ in BaTiO3are shown. Note that both ionization levels for Fe lie withinthe band gap and may contribute to photorefractive behav-ior. The numbers of photorefractive donors and acceptorsare determined by the extent to which each level is filled orempty (with respect to electrons). This in turn depends onthe difference between the Fermi level and the energy for theparticular ionization level. When the Fermi level is belowan ionization level, then that level will be primarily empty;when the Fermi level is higher, the ionization level will beprimarily filled. Thus, referring to the example of Fe shownin Fig. 2, we see that when E is significantly less than E-,most of the Fe will be Fe . When E <EF<E, essential-ly all Fe will be Fe3+. With EF> E--, Fe2+ will be predomi-nant. Note that when EF is near E the upper level iscompletely empty; Fe2+ is essentially absent and can beignored. Likewise, with EF near E-, the lower level iscompletely filled and Fe4+ can be disregarded. For anyparticular value of EF then, there is only one partially filledlevel (in the dark).
According to this explanation, the identity of the photore-
B. A. Wechsler and M. B. Klein
Vol. 5, No. 8/August 1988/J. Opt. Soc. Am. B 1715B. A. Wechsler and M. B. Klein
VX/V- --0 0
E+= 3.05 eV
CONDUCTION BAND
E++= 2.95 eV
- … Fe 2 +/Fe3 +
E--= 2.4 eV Eg= 3.1 eV
-- --e Fe3+/ Fe4+|TE-= 0.8 eV
VALENCE BAND
Fig. 2. Energy levels of principal defect species in Fe-dopedBaTiO3. In the situation depicted here the Fermi level is close tothe Fe3+/Fe 4+ ionization level. All energy values are taken fromRef. 10.
fractive donors and acceptors depends on the Fermi level.When EF is close to E-, Fe3+ acts as a donor and Fe4+ acts asan acceptor. When EFlies close to E--, however, Fe2+ is thedonor center and Fe3+ is the acceptor. Within each of thesetwo regimes, the single-level photorefractive model is ap-plied separately. In each case there is, in general, a competi-tion between hole and electron photoconductivity and apoint at which the two contributions are equal. In keepingwith earlier discussions in the photorefractive literature, wewill refer to this as a compensation point (which differs fromthe common usage in which compensation refers to the pres-ence of equal concentrations of donors and acceptors). Atthe crossover between the two regimes (i.e., when EF liesmidway between E- and E--) essentially only Fe3 + ispresent, and therefore the density of empty traps vanishes.At the two compensation points and the crossover point, thesign of the dominant photocarrier changes.
Thus altering the ratio of photorefractive donors to accep-tors corresponds to adjusting the Fermi level. In BaTiO3oxygen vacancies are the principal electron donors. There-fore the Fermi level is raised by increasing the number ofoxygen vacancies, which is accomplished by treating thecrystal in a reducing atmosphere.
This energy-level scheme closely resembles that proposedby Ducharme and Feinberg,3 who noted that the numbers ofdonors and acceptors could be altered by changing the oxy-gen-vacancy concentration. However, in their model thelower levels act only as acceptors, whereas the upper levelsact only as donors. In the model proposed here both donorsand acceptors are associated with each level; for a givencrystal, only one of these levels is associated with photore-fractive centers. Oxidation and reduction can change notonly the number of filled sites on each level but also theidentity of the donor and acceptor sites by switching fromone level to the other.
C. Choice of Model Parameters
To solve for r and T a total of four parameters must bespecified in addition to the populations ND and NA. Therate and transport properties are not well known, and in ourinitial model we have chosen to make the following assump-tions:
1. The diffusion length for holes in an as-grown crystal isgiven by Kh-1 = 10 nm.24
2. The ratio ND/NA for this measured value of Kh-1 is100, and the total number of photorefractive centers (NA +ND) is 5 X 1018 cm-3 .
3. The photoionization cross sections for holes and elec-trons are equal, i.e., Sh/Se = 1.
4. The ratio R = (SeIeYh)/(ShhYe) = 1
The electron diffusion length in an as-grown crystal is calcu-lated from the expression (Ke/Kh)-2 = (Sh/Se)R(ND/NA)
where ND/NA is the value in the measured crystal. For theconditions in assumptions 1-4, this yields a value of Ke-' =100 nm.
The parameter R is related to the concentrations of pho-torefractive donors and acceptors at the compensationpoints by the expression NA/ND = R1/2. Therefore, for thevalue R = 1, the compensation points will occur when ND =
NA, assuming that the grating period is large relative to thecarrier diffusion lengths. We make the assumption that R =1 in order to illustrate the main features of the gain andresponse-time dependence on processing parameters andbecause it simplifies the comparison of different dopants.However, R could differ by several orders of magnitude fromthis value. Furthermore, it is likely that the value of R willchange significantly when the crossover between the Fe3+/Fe4 + and Fe2+/Fe3+ regimes occurs. Although the electronand hole mobilities might not vary greatly as a function ofoxygen pressure, substantial changes in the photoionizationcross sections and/or carrier recombination rates are possi-ble. These rates are probably sensitive to the ionizationenergies, which are quite different for Fe3+/Fe4 + and Fe2 +/Fe3+. In applying this model to the interpretation of exist-ing experimental data on BaTiO3 (Subsection 5.A), we willuse more realistic values of R for Fe3+/Fe4 + and Fe2 +/Fe3+based on observed changes in sign of the dominant photocar-rier as a function of oxygen pressure.
The assumption that Sh/Se = 1 is made necessary by thelack of knowledge of this parameter. Uncertainty in thisratio has a greater impact at small grating periods, where theeffect of carrier diffusion lengths is more important and isnot expected to affect the results presented here seriously.
The calculations reported below were performed using theunclamped values for the electro-optic coefficients3 ' r 3 =
19.5 pm/V and r33 = 97 pm/V, the refractive indices32 n, =2.49 and ne = 2.42, X = 0.5145 gm, E = 135,33 and the gratingperiod Ag = 0.7 ,um.
4. RESULTS
A. Defect Properties of Acceptor-Doped Barium TitanateThe calculated defect concentrations (at room temperature)as a function of oxygen partial pressure are shown in Fig.3(a) for a concentration of 5 X 1018 cm-3 Fe and a processtemperature of 1000°C. For most of the experimentallyaccessible range of oxygen pressure, Fe3+ is the predominantvalence state of Fe. At high oxygen pressure the secondaryspecies of iron is Fe4+. This gradually diminishes in concen-tration as the oxygen pressure is reduced and finally disap-pears suddenly and is replaced by Fe . With further re-duction, the concentration of Fe2+ increases slowly until it isroughly equal to the concentration of Fe3+. The latter spe-cies then is eliminated and all the Fe becomes divalent. InFig. 3(b) the Fermi level for this model is shown togetherwith the ionization levels for the transitions involved. Notethat the Fermi level is essentially pinned to each ionizationlevel in the system until that level is completely filled. Itthen jumps suddenly to the next higher level as the crystal is
v f - - - - C) C,__V'/V-
1716 J. Opt. Soc. Am. B/Vol. 5, No. 8/August 1988
20 I l I I I ular ionization level. However, sudden conductivityFe3+ changes occur when the Fermi level shifts to an adjacent
ionization level.18 V 0O \ vO / ~% Throughout most of the range of oxygen partial pressure,
\ > 4 oxygen vacancies are doubly ionized and are present at one16Fe e half of the Fe3+ concentration. These act as the primary
E \ means of charge compensation for the acceptor impurities.When the crystal is reduced, the Fe3+ is converted to Fe2 +,
4 - _ and the concentration of doubly ionized oxygen vacancies0 2becomes equal to that of Fe2+. When further reduction iscarried out, the oxygen-vacancy population becomes domi-
12 - nated by singly ionized species and the total oxygen vacancyconcentration exceeds that required to balance the acceptor
(a) impurity content. In this regime free electrons compensate10 for the oxygen vacancies. Consequently, the dark conduc-3 tivity is high.
vO v In Fig. 4 the total oxygen-vacancy concentration is shownfor three different charge states of the dopant and two dif-
-~ F 2+/ Fe3+ ferent temperatures. For singly and doubly charged defects(e.g., trivalent or divalent ions substituting for Ti4+), the
2 oxygen-vacancy concentration is governed by the content of> acceptor impurities throughout most of the accessible range> of oxygen partial pressures, owing to the charge-compensa-
tion mechanism. Only at extremely low oxygen pressures1 _ _ _ does the oxygen-vacancy concentration become intrinsically
- Fe3+/ Fe4+ controlled. Of course, for neutral defects (e.g., a tetravalention replacing Ti4+), the oxygen-vacancy concentration is in-dependent of the dopant concentration at all oxygen pres-
0 | i g | (b) sures and is much smaller than in the case of charged defects0 over most of the range of oxygen partial pressures. At all0 but extremely low oxygen pressures, the oxygen-vacancy
concentration will change significantly with reduction only7 if the dopant ion undergoes a valence change, as is possible
? -5 l _ for transition metals. Furthermore, the oxygen-vacancyC; t concentration in this region will never exceed the concentra-
_; _tion of the acceptor ions (except in the case of a monovalent> -lo _ ion substituting for Ti). Thus oxygen vacancies cannot be> n considered independently of transition-metal dopants in in-
-15 terpreting the results of reduction experiments. At low0z0
c -20 - 100-J
(C) 8 10000c-25 ~ ~~I I I 1 I000
-20 -15 -10 -5 0 5 10C',
LOG P.bar E°2 0 6-
Fig. 3. Results of defect-model calculation for Fe-doped BaTiO3. o A"The defect populations are those found at room temperature, as- -suming that the total oxygen-vacancy concentration was established 6 4by processing at 1000'C: total Fe concentration, 5 X 18 cm-3; 1 >atm = 1.01325 bars. (a) Point defect concentrations as a function of A A'oxygen partial pressure. (b) Position of the Fermi level. Horizon- _ tal lines indicate the ionization energies of various defect species. 2 6500c(c) Dark conductivity.
0* ~~~ ~~~~ ~~~~-30 -20 -10 0 1 0reduced further. The dark electrical conductivity calculat- LOG bared for this model is shown in Fig. 3(c). At high oxygen °2partial pressures, where the Fermi level is close to the Va- Fig. 4. Calculated oxygen-vacancy concentration at 10000C (solidlence band, p-type conductivity is predicted. The transi- curves) and 650'C (dashed curve) as a function of oxygen partialtion to n-type conductivity occurs when the Fermi level pressure for three different charge states of an acceptor dopant:total acceptor concentration, 5 X 10 i8 cm-3 . For simplicity, onlycrosses midgap. Note that the conductivity remains rela- the singly charged species is shown for a process temperature oftively constant while the Fermi level is pinned to any partic- 6500C.
B. A. Wechsler and M. B. Klein
Vol. 5, No. 8/August 1988/J. Opt. Soc. Am. B 1717
4
2-
+ Fe3 //Fe4+
CD /Fe 2 /Fe 3 /0
-2
-4 0 500 1000 1500
TEMPERATURE, OC
Fig. 5. Variation of N/N+ with temperature for Fe in BaTiO3.This calculation assumes that the crystal is in equilibrium with anoxygen partial pressure of 0.2 bar at all temperatures.
temperatures a much lower oxygen pressure is required inorder to introduce a large number of oxygen vacancies.
The effect of temperature on the valence state of Fe inBaTiO3 for fixed oxygen partial pressure is shown in Fig. 5.Because of the temperature dependence of the oxygen-va-cancy concentration [Eq. (4)], the valence state of the dop-ant increases as the temperature is lowered as long as thecrystal maintains equilibrium with respect to oxygen in thesurrounding atmosphere. Of course, over some temperatureinterval, the kinetics of oxygen diffusion through the crystalbecome too sluggish, and the oxygen-vacancy concentrationwill become frozen in. Thus the valence state of the dopantmay depend strongly on kinetic factors. This may be onesource of variability in the photorefractive behavior of crys-tals grown under nominally similar conditions. It could alsobe responsible for inhomogeneities within a single boule,since outer regions of the boule will be more nearly in equi-librium with the atmosphere than interior regions.
It should also be recognized that a crystal cooled in air at arate fast enough to freeze in an oxygen-vacancy concentra-tion from relatively high temperature will have a higherconcentration of oxygen vacancies, and therefore a morereduced valence state of the dopant, than a crystal cooledmore slowly. If the rapidly cooled crystal is subsequentlyannealed at a lower temperature, it may well become oxi-dized even though the annealing atmosphere is nominallyreducing, e.g., argon gas with an impurity level of 10-6 atm ofoxygen. It is therefore important to establish the initialequilibrium state of a crystal when comparing the effects ofheat treatments on photorefractive properties.
It is often assumed that the valence states of transition-metal dopants can be varied arbitrarily over a wide range byadjusting the oxygen partial pressure of the atmosphere inan annealing experiment. However, the prediction of thismodel is that the valence states can be controlled only overlimited ranges. The model also predicts that significantvariations in valence behavior can be expected for differentdopants, depending on their ionization energies. In Fig. 6the ratios N/N+ for Fe and Mn in BaTiO3 are shown as afunction of oxygen partial pressure for a process tempera-ture of 10000C. Both dopants switch from 3+/4+ to 2+/3+valence states at nearly the same oxygen pressure. This is a
result of the nearly symmetric placement of the ionizationlevels for these ions within the band gap. According to thismodel, the ratio of Fe3+/Fe4 + can be adjusted to a highervalue than Mn3+/Mn4 +, whereas the ratio Fe2+/Fe3+ can beadjusted to a lower value than Mn2+/Mn 3+, owing to thewider separation in ionization energies for Fe than for Mn.The practical ability to control NIN+ for these ions is limitedto a range of approximately 103. The cutoffs limiting theability to control the valence ratios result from the suddenshift in the Fermi level when a particular ionization levelbecomes filled. A somewhat greater range of control is ob-tained at lower process temperatures.
The concentration of acceptor dopants also influences theratios N/N+ for a given temperature and oxygen pressure, asshown in Fig. 7. Increasing the acceptor concentrationshifts the crossover points between the Fe3+/Fe4 + andFe2+/Fe3 + regimes to lower oxygen pressures and lowers theratio Fe2+/Fe3+ for a fixed oxygen pressure. However, theFe3+/Fe4 + ratio at a given oxygen partial pressure is relative-ly insensitive to total Fe concentration. This result differsfrom the conclusions of Schunemann and co-workers,7 8 who
2
+
0-J
0
-2
-4
-15 -10 -5 0 5 10
LOG P0 2. bar
Fig. 6. Variation of N/N+ for Fe (solid curve) and Mn (dashedcurve) as a function of oxygen partial pressure for a process tem-perature of 1000°C.
4
2
+
z
0-J
0
-2
-4
-15 -10 -5 0
LOG P0 2. bar
5 10
Fig. 7. N/N+ as a function of oxygen partial pressure for Fe atthree different concentrations: solid curve, 1017; dashed curve, 1018;dotted curve, 1019 cm- 3. The process temperature is 10000C.
B. A. Wechsler and M. B. Klein
1718 J. Opt. Soc. Am. B/Vol. 5, No. 8/August 1988
did not consider the effect of varying temperature in theirsolutions for the Fe3+/Fe4+ ratio. Because the Fermi level issensitive to temperature, the populations predicted by thiswork are probably more realistic for crystals characterized atroom temperature.
B. Photorefractive BehaviorUsing values for NA and ND calculated from the defect mod-el and the equations for gain and response time describedabove, we have modeled these properties as a function ofoxygen partial pressure. The result for Fe-doped BaTiO3and a process temperature of 1000'C is shown in Fig. 8(a).The response-time curve has been scaled to a maximum of 1sec, and the gain curve is for the absolute value of the gaincoefficient. There are three minima in the gain curve, eachof which also represents a change in the sign of the dominantphotocarrier. The minima at high and low oxygen pressureare compensation points for which the photoconductivitycontributions of holes and electrons are identical. At thehigh-oxygen-pressure compensation point this correspondsto equal concentrations of Fe3+ and Fe4 +. At the lower-oxygen-pressure compensation point the concentrations ofFe2 + and Fe3+ are equal. The third minimum is broughtabout by a minimum in the photorefractive trap density, atwhich point the minority species (Fe4+ and Fe2 +) drop tonear zero. At the compensation points, on the other hand,the trap densities are at a maximum.
The response time reaches a maximum at each of the twocompensation points and a minimum when the photorefrac-tive trap density is a minimum. The response time is seen todecrease by approximately 2 orders of magnitude from itsmaximum while still maintaining optimum gain, assumingthat the transport properties are invariant with oxygen par-tial pressure. There is a small uncertainty in the response-time curve as a result of insufficient knowledge of the carrierdiffusion lengths. Thus the actual response-time variationcould be somewhat greater than or less than this. On theother hand, the gain curve is essentially unaffected by thediffusion lengths for the grating period used in these calcula-tions. For smaller grating periods, however, the uncertain-ties in both the response time and gain will be greater.
At low oxygen pressures the transition-metal dopant iscompletely converted to its lowest valence state. At thispoint the gain and response time again drop to zero as thephotorefractive trap density vanishes. However, at stilllower oxygen pressures singly ionized oxygen vacancies willbecome significantly populated, and these may act as pho-torefractive centers.
The result for Mn-doped BaTiO3 is shown in Fig. 8(b).The compensation points are offset significantly from thosefor Fe, and the shapes of the response-time curves are con-siderably different. Because we have assumed the same rateand transport parameters as for the case of Fe, these differ-ences are entirely a result of the widely differing ionizationenergies for these two dopants (see Table 1). In addition toindicating the strong effect of the dopant species on theoxidation-reduction behavior, this also implies that any er-rors in the ionization energies can have a significant effect onthe calculated positions of the compensation points. Notealso that these models for Fe and Mn were calculated byassuming that the rate and transport properties satisfy theconditionR = 1 (seeSubsection3.C). LargervaluesofRwillshift the compensation points to higher oxygen pressures,
whereas smaller values will shift them to lower oxygen pres-sures.
C. Role of Barium VacanciesThere is considerable uncertainty over the role of Ba vacan-cies in BaTiO3. Many studies of undoped ceramic and sin-gle-crystal materialsl0.'2- 4 20 have postulated the impor-tance of a background acceptor-type defect species, whichhas been explained as being due to impurities and/or tointrinsic defects such as barium vacancies. In BaTiO3 crys-tals grown by the top-seeded solution growth method it islikely that some excess TiO2 is incorporated during growth.Chan et al.14 and Sharma et al.34 estimated this excess to beno more than approximately 100 parts in 106 (ppm) on thebasis of electrical-conductivity measurements and solid-state synthesis experiments performed at 1400°C, which issimilar to the growth temperature of single crystals. Hage-mann'0 found a background level of acceptor centers in un-doped BaTiO3 ceramics at a level of approximately 2 X 108cm3 or -100 ppm.
Excess TiO2 is most likely to be incorporated in BaTiO3 bythe following mechanism:
BaTiO3 + TiO2 ±; BaTiO3 + VBa" + VO.
2
E
-
0
-1
-2
2
E
CD0
0
-1
-2
-15 -10 -5 0
(30)
CD0
0
5 10
LOG P0 bar02'
Fig. 8. Gain coefficient r (solid curve) and response time T (dashedcurve) as a function of oxygen partial pressure for a process tem-perature of 1000°C and a dopant concentration of 5 X 10l3 cm-3. -
and + refer to the sign of the dominant photocarrier and of the gainwithin each region. Note that the response time is normalized to amaximum of 1 sec. The grating period is 0.7 um. (a) Fe doping.(b) Mn doping.
B. A. Wechsler and M. B. Klein
1
Vol. 5, No. 8/August 1988/J. Opt. Soc. Am. B 1719
20
19
I?
E
>
co
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0-J
18
17
16
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Fig. 9. Calculateacceptor dopant c(
A mass action eqten as
When acceptor icentration, the oly controlled, atherefore constihowever, one o)charged acceptomust diminish tshown in Fig. 9,plotted as a functor impurities atcrystal growth cconcentration isuncertainty in t]clearly a significtion with increaw
These resultsbarium vacancieproduce photoreof acceptor dopecentration of baicies will diminis]ior is likely to bsome range of doand acceptor do]effect. As discupredominate wi.tions. The ioninot well known.ground acceptorionization energthe valence ban(cies, then this sjthat of Mn dope1.9eV.
l l l I 5. DISCUSSION
The defect/photorefractive model of BaTiO3 can be used tointerpret the results of previously reported oxidation-reduc-tion experiments on BaTiO3 single crystals. Such experi-ments have been performed by Ducharme and Fineberg,3Ducharme,4 Wechsler et al.,6 and Schunemann and co-work-ers.7 8 Although none of these experiments is sufficient toprovide a stringent test of the predictions of the model,qualitative comparisons can be made. This model also hasimplications for the wavelength dependence of the photore-fractive behavior, which is discussed below.
A. Oxidation-Reduction ExperimentsFigure 10 shows calculated gain and response-time curvesfor BaTiO3 doped with Fe at a concentration of 5 X 1018cm-3 , for process temperatures of 1000, 800, and 6500C.These correspond to the conditions used by previous investi-
17 18 19 20 21 gators in their oxidation-reduction experiments. In thesecalculations we have used values of the parameter R (seeLOG [A], cm-3 Subsection 3.C) chosen to satisfy several experimental con-
i barium-vacancy concentration as a function of straints, as discussed below. In the more oxidizing regime)ncentration at 13300C. (i.e., where Fe3+ and Fe4 + are the photorefractive centers),
the value R = 1/2500 was used; for the Fe2+/Fe3+ regime, R=2500 was used.
[uation describing this reaction can be writ- Wechsler et al.6 treated a BaTiO3 crystal (Sanders Asso-ciates) at 10000C in CO/CO2 atmospheres with oxygen pres-
[Va"I [VO1 = K. (31) sures between 10-45 and 10-122 atm (1 atm = 1.01325 bars).They observed one change in sign of the dominant photocar-
mpurities (dopants) are present at low con- rier with reduction and a response time that first increased,xygen-vacancy concentration is intrinsical- and then decreased with increasing reduction. After thend the barium-vacancy concentration is most reducing treatment, the electrical resistivity dropped;nt. At higher acceptor concentrations, substantially, and the crystal could no longer be fully poled.xygen vacancy is created for each doubly The crystal studied had an Fe concentration of -4 X 1018r added; the barium-vacancy concentration cm-3 , which was determined by electron paramagnetic reso-to maintain equilibrium. This behavior is nance analysis to be almost entirely in the Fe3+ state. It waswhere the barium-vacancy concentration is assumed that the minority species of Fe was Fe2 + and thatction of the concentration of divalent accep- reduction produced an increase in the ratio Fe2+/Fe3 +.13300C and 0.2 bar of oxygen pressure (i.e., In light of the present study it seems appropriate to rein-)nditions). The calculated barium-vacancy terpret the results of those experiments. We suggest that insubject to some uncertainty because of the the as-grown condition the minority species of Fe was Fe4+he values of Ki and KR. However, there is rather than Fe2 +. The as-grown crystal had predominantlyant decrease in barium-vacancy concentra- hole photoconductivity. However, the initial cooling condi-sed doping. tions of this crystal were not known. Let us assume that thesuggest that in high-purity BaTiO3 crystals, oxygen-vacancy concentration was representative of a pro-s are present in concentrations sufficient to cess temperature of 800°C in an atmosphere of 0.2 atm of 02-fractive behavior. When the concentration The defect model predicts that Fe3+ should be the majorityints or impurities exceeds the intrinsic con- species under these conditions (and electron paramagneticrium vacancies, however, the barium vacan- resonance measurements confirm that this was the case), but,i in number, and the photorefractive behav- such a crystal would be expected to have hole-dominatedbe dominated by the dopant centers. Over photoconductivity if the ratio R is less than about 1/2500.,pant concentrations, both barium vacancies After treatment at 10000C in a pure CO 2 atmosphere (Po2 =
?ants may participate in the photorefractive 10-45 atm), the crystal was found still to be hole dominated.ssed in Section 5, the particular centers that As can be seen from Fig. 10(a), this is consistent with theil vary depending on the processing condi- defect model because the narrow region of electron-domi-zation energies of the barium vacancies are nated photoconductivity would have been missed in per-
However, Hagemann found that the back- forming this experiment. The photorefractive donor/ac-rs in his undoped material appeared to have ceptor system is predicted to change from Fe3+/Fe4+ to Fe 2+/gies of approximately 1.2 and 1.8 eV about Fe3+ under these conditions. When the crystal was subject-I edge. If these acceptors are barium vacan- ed to a somewhat greater reduction, P0 2 = 10-8.1 atm, theJecies should behave in a manner similar to crystal switched to electron-dominated photoconductivity.ants, whose ionization energies lie at 1.3 and To be consistent with the present model, the ratio R in this
region must be between 102 and 104. When the crystal was
B. A. Wechsler and M. B. Klein
1720 J. Opt. Soc. Am. B/Vol. 5, No. 8/August 1988
E
L.
0-J
.E
L.
0
7
E
0-J0
2
a
-1
-2
2
0
-1
-2
2
0
-1
-30 -25 -20 -15 -10 -5 0 5
LOG P02' bar
Fig. 10. Gain coefficient (solid curve) and response time r(dashed curve) as a function of oxygen partial pressure for threedifferent process temperatures. - and + refer to the sign of thedominant photocarrier and of the gain within each region. Thetotal Fe concentration is 5 X 1018 cm-3, and the grating period is 0.7jim. (a) 1000 C, (b) 800'C, (c) 650'C.
still further reduced, Po = 10-12.2 atm, a substantial de-crease in electrical resistivity was observed. This oxygenpressure is close to that for which the model predicts that allFe would be converted to Fe2+, and the relatively high darkconductivity is due to the proximity of the Fermi level to theconduction band edge in this region [see Fig. 3(c)].
Response-time measurements for these experiments indi-cated an initial small increase followed by a small decreasewith increasing reduction. In the most reduced crystal theresponse time was smaller than in the slowest sample byapproximately a factor of 20. These results are consistentwith the present model, assuming that compensation points
are present, consistent with the values of R discussed in thepreceding paragraph. As can be seen from Fig. 10, thecrystal may always have been fairly close to being compen-sated, except under the most reducing conditions. Signifi-cant improvements in response time can only be achievedunder conditions that are far from the compensation points.
Schunemann and co-workers7 '8 studied crystals of BaTiO3doped with Fe up to a level of 1000 ppm (atomic). Anundoped crystal reportedly had an Fe concentration of 0.3ppm, with all other transition-metal impurities less than0.04 ppm. Diffraction-efficiency measurements were re-ported for crystals in the as-grown state, after treatment at8000C in 1 atm of oxygen and after treatment at 8000C in Ar02 with P02 = 10-4 atm. In doped samples (both as grownand oxidized) the diffraction efficiency decreased with in-creasing Fe concentration; after reduction the diffractionefficiency increased with increasing Fe concentration. Thephotorefractive behavior of an undoped crystal under oxi-dizing conditions appeared not to follow the trend observedfor Fe-doped crystals. For both oxidized and reduced crys-tals the variation in diffraction efficiency was small com-pared with the variation in Fe concentration.
These observations are consistent with the model pro-posed here, assuming that crystals treated in air and in 1 atmOf 02 are close to being compensated for [see Fig. 10(b)]. Asdiscussed above, the ratio Fe3+/Fe4+ is relatively insensitiveto total Fe concentration. This implies that for crystals inwhich Fe3+ and Fe4 + are the photorefractive donors andacceptors, the oxygen pressure at which the compensationpoint occurs will not vary greatly with total Fe concentra-tion. Therefore it is possible for all the crystals, regardlessof doping level, to be close to compensation. Small changesin the degree of compensation between these crystals couldproduce the observed decrease in gain with increasing Feconcentration. The reduction treatment was sufficient todrive the crystals far from the compensation point, andtherefore the expected trend of increasing diffraction effi-ciency with increasing Fe concentration was found. Thevariation of the diffraction efficiency by only a factor of -3despite a 20-fold increase in Fe concentration (50-1000ppm) is related to the value of the grating period used in themeasurements (2 Mm). The model predicts a linear depen-dence of diffraction efficiency on Fe concentration for smallvalues of Ag but no dependence on Fe concentration for largevalues of Ag. The photorefractive response of an undopedcrystal suggests that another center besides Fe (perhapsbarium vacancies) may be active.
In the experiments reported by Ducharme and Feinberg3
and by Ducharme,4 a Sanders BaTiO3 crystal was annealedat 650'C in high-purity argon (Po2 10-6 atm), air, andoxygen (2 atm). No analysis of the impurity content of thecrystal was given. A minimum in the effective trap densitywas observed for Po2 near 0.2 atm, whereas more reducingconditions resulted in predominantly electron photoconduc-tivity, and more oxidizing conditions produced hole photo-conductivity. On the basis of the model presented here,these results are difficult to reconcile with the hypothesisthat Fe alone is responsible for the photorefractive behavior.The model does predict a change in sign from holes to elec-trons as a result of treatment in 10-6 atm of oxygen [Fig.10(c)]. However, this corresponds to a compensation point;we would expect the minimum in the trap density to occur atoxygen pressures closer to 10-11 atm. Furthermore, the
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B. A. Wechsler and M. B. Klein
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Vol. 5, No. 8/August 1988/J. Opt. Soc. Am. B 1721
model predicts that the dominant photocarrier wouldchange from electrons to holes as the crystal is reducedthrough this minimum, which is the opposite of what wasobserved.
An alternative explanation is that two species are involvedin the photorefractive effect in this crystal. Ducharme andFeinberg suggested that the acceptors might be barium va-cancies and that the donors were oxygen vacancies. Thereduction treatment was expected to increase the oxygen-vacancy concentration without affecting the barium vacan-cies. A problem with this interpretation, however, is thatthe oxygen-vacancy concentration was almost certainlyfixed by the acceptor impurities present in the crystal. Un-der the temperature and oxygen partial-pressure conditionsused, it is doubtful that a significant change in oxygen va-cancies would occur (see Fig. 4).
Using the defect model presented in this paper, we canexamine the consequences of incorporating two species withlevels in the band gap. An example of such a calculation isshown in Fig. 11, for which we assume the presence of Fe at aconcentration of 5 X 1018 cm-3 and a second multivalentspecies, A, at a concentration of 5 X 1017 cm-3. Species Ahas two ionization levels (A//Ax and A'/A'), with energiesbetween those of the two Fe levels. Examples of such spe-cies, based on the ionization energies reported by Hage-mann,' 0 include barium vacancies and Mn.
With increasing reduction, the ionization levels are suc-cessively filled in the order Fe3+/Fe4 +, A'/AX, A"/A', andFe2+/Fe3+. For any particular oxidation-reduction condi-tions only one of these levels is partially filled, provided thatthe levels are separated by a few times kBT. The photore-fractive donors and acceptors are associated with this par-tially filled level. The behavior of the gain and responsetime is shown in Fig.11(b). There are three crossover points(minima in the trap density) and four compensation points,leading to a total of seven changes in sign of the dominantphotocarrier. Note that for each region between the cross-over points, a value for the parameter R (Subsection 3.C)must be specified. For this calculation we assume the samevalues for R that were used above for the case of Fe3+/Fe4+and Fe2+/Fe3+. For the regions in which A'/Ax and A'/A'are the photorefractive systems, we assume that R is inter-mediate between the two values used for the Fe levels andhave, therefore, used R = 1.
An interesting consequence of this two-species model isthat a minimum in the photorefractive trap density is pre-dicted to occur at much higher oxygen pressures than foreither species alone. In the calculation shown in Fig. 11, aminimum occurs at Po2 10-2 atm. In addition, regions ofelectron-dominated photoconductivity are found at oxygenpressures between 10-1 and 10-2 atm and between 10-5 and10-10 atm. These results are all consistent with the observa-tions of Ducharme and Feinberg. It should be noted thatthe change in sign of the dominant photocarrier and theminimum in the trap density that were observed need notoccur simultaneously but could indicate two separate eventsin close proximity to each other.
Ducharme and Feinberg also observed a longer responsetime for an oxygen pressure of 0.2 atm than for an oxygenpressure of 2 atm and a change in slope of the response timeversus intensity after treatment at Po, 10-6 atm. FromEq. (23) we can see that the response time is expected todecrease with the trap density. It is therefore surprising
20
18
I?,E
0-j
16
14
12
10
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7
E
0-J
1
0
-1
-2
-30 -25 -20 -15 -10 -5 0
0
-1
-2 00
-3
-4
LOG P0 2 bar
Fig. 11. Results of model calculations for BaTiO3 doped with twospecies, each having two ionization levels in the band gap. Theprocess temperature is 650°C, the Fe concentration is 5 X 1018 cm-3,and the concentration of species A is 5X 1017 cm 3. (a) Point defectconcentrations as a function of oxygen partial pressure. (b) Beam-coupling gain and response time.
that Ducharme and Feinberg found an increase in responsetime together with a minimum in the trap density. Howev-er, their response-time measurements are consistent with thesuggestion of our model that a compensation point occursnear Po2 = 10-1 atm, corresponding to a maximum in theresponse time, followed by a minimum in the trap density(and response time) at a somewhat lower oxygen pressure[Fig. 11(b)]. The single-level photorefractive model doesnot provide an explanation for the change in the intensitydependence of the response time, and it is likely that partici-pation of secondary levels not considered here is importantin determining this behavior.
The two-species model presented above is not a uniqueexplanation of the results of Ducharme and Feinberg. How-ever, given the available data 0 for species believed to be ofimportance in existing BaTiO3 samples, this model is thesimplest and most reasonable one consistent with the ex-perimental observations. It should be pointed out that theresults of both Wechsler et al.6 and Schunemann and co-workers7 '8 can also be explained with this two-species model,although the data do not require this level of complexity.
B. Wavelength Dependence of PhotoconductivityIn the energy-level model presented above it was shown thatthe populations of defect species with multiple valencestates are determined by thermal ionization from levelswithin the band gap. It was implicitly assumed that both
B. A. Wechsler and M. B. Klein
.
1722 J. Opt. Soc. Am. B/Vol. 5, No. 8/August 1988
electrons and holes could be generated by photoionizationfrom these levels. However, photoionization occurs onlywhen the incident energy is greater than or equal to theionization energy for a particular process. In the case of Fein BaTiO3 the ionization levels for Fe3+/Fe4+ and Fe2 +/Fe3+lie at 0.8 and 2.4 eV above the valence band edge, and thethermal band gap is 3.1 eV (Ref. 10) (neglecting the tempera-ture dependence of these energies). The optical band gapmay be somewhat larger. To photoionize an electron fromFe3+, the energy required is at least 2.3 eV. Ionization of ahole from Fe3+ requires at least 2.4 eV. The photorefractivebehavior predicted by this model, therefore, is applicableonly when the energy of the writing beams exceeds thesevalues.
Many photorefractive characterization measurements areperformed using writing beams from an Ar-ion laser, with awavelength of 514.5 nm. The corresponding photon energyfor this radiation is approximately 2.4 eV, which is justsufficient to ionize both electrons and holes from Fe3+ inBaTiO3 . For longer wavelengths, photoionization fromFe3+ will be suppressed, and photocarriers will be generatedonly from Fe4 + (holes) or Fe2+ (electrons). In this case,assuming the presence of levels due only to Fe, one wouldexpect to find a single minimum in the gain coefficient and aconcomitant switch from hole to electron photoconductivityas the crystal is reduced. This implies that there is a regionof oxygen partial pressure in which the dominant photocar-rier will change from electrons to holes as the writing-beamwavelength is increased. There is another region, undermore reducing conditions, where the change will be fromholes to electrons with increasing wavelength.
6. CONCLUSIONS
We have used a thermodynamic point defect model to studythe effects of temperature, oxygen partial pressure, dopanttype, and concentration on the populations of species thatmay act as photorefractive donors and acceptors in BaTiO3 .This model predicts that in Fe-doped crystals cooled in air orrelatively oxidizing atmospheres, Fe will be present as Fe3+and Fe4 +. When treated in a reducing atmosphere, the Fewill be converted to Fe2+ and Fe3+. Other transition-metaldopants, such as Mn, show a similar valence behavior, butthe specific effects of process variables on defect populationsdiffer depending on the ionization energies of the species.
The single-level electron-hole model for photorefractivegrating formation may be applied to the case of dopants withmultiple valence states as well as when more than one ioniz-able species is present. For a single species with three possi-ble valence states, the model predicts three changes in signof the dominant photocarrier with reduction. Two of thesechanges are associated with compensation points (points atwhich the electron and hole photoconductivities are equal);the third sign change involves a minimum in the photore-fractive trap density and a change in the identity of thephotorefractive donor/acceptor species. More-complex be-havior is predicted when additional multivalent species areincorporated.
Oxidation-reduction treatments may be used to improvethe response time of BaTiO3 crystals. The model predictsthat room-temperature response times that are approxi-mately 2 orders of magnitude faster than those of existingcrystals can be achieved without sacrificing high gain. This
estimate gives only the variation due to adjustment of thedonor/acceptor ratio and is subject to uncertainty that is dueto limited knowledge of the carrier diffusion lengths. Theoxidation-reduction conditions needed to achieve this opti-mum performance differ for various dopants, but there ap-pears to be relatively little difference in the minimum re-sponse time that can be obtained before the gain decreases.Differences in the photoionization cross sections and recom-bination rates may therefore play a large role in determiningwhich dopant provides the optimum performance. Themodel presented here only describes the donor and acceptorconcentrations in the dark and does not account for behaviorthat may be altered by illumination. Consideration of light-induced charge redistribution may prove fruitful in achiev-ing a fuller understanding of the photorefractive effect.
This model is consistent with most existing experimentaldata on the effects of doping and oxidation-reduction treat-ments on the photorefractive behavior of BaTiO 3. Al-though the complexity predicted by the model has not yetbeen observed, previous work has been limited, and morecontrolled experiments are required to confirm this behav-ior. In principle, this approach can be used to determine theprocessing conditions needed to optimize the gain and re-sponse time. However, better values for the photoioniza-tion cross sections, carrier mobilities, and recombinationrates are needed to obtain more quantitative results and tomake more realistic comparisons between various dopants.Application of the model is also limited by the fact thationization-energy data are available for only a few species.Determination of these values for other species could lead tothe development of optimized processing techniques for awider variety of dopants.
ACKNOWLEDGMENTS
We have benefited from many fruitful discussions with col-leagues, especially G. C. Valley, R. N. Schwartz, D. Rytz, D.M. Smyth, and H.-J. Hagemann.
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1. F. Micheron and G. Bismuth, "Holographic storage, electricalfixing and erasing in doped BaTiO3 crystals," J. Phys. (Paris)33, Suppl. 4, 149-150 (1972).
2. E. Kratzig, F. Welz, R. Orlowski, V. Doormann, and M. Rosen-kranz, "Holographic storage properties of BaTiO 3," Solid StateCommun. 34, 817-819 (1980).
3. S. Ducharme and J. Feinberg, "Altering the photorefractiveproperties of BaTiO3 by reduction and oxidation at 650°C," J.Opt. Soc. Am. B 3, 283-292 (1986).
4. S. Ducharme, "Photorefraction in BaTiO 3," Ph.D. dissertation(University of Southern California, Los Angeles, Calif., 1986).
5. M. B. Klein and R. N. Schwartz, "Photorefractive effect inBaTiO3: microscopic origins," J. Opt. Soc. Am. B 3, 293-305(1986).
6. B. A. Wechsler, M. B. Klein, and D. Rytz, "Crystal growth,processing and characterization of photorefractive BaTiO3 ," inLaser and Nonlinear Optical Materials, L. G. DeShazer, ed.,Proc. Soc. Photo-Opt. Instrum. Eng. 681, 91-100 (1987).
7. P. G. Schunemann, D. A. Temple, R. S. Hathcock, C. Warde, H.L. Tuller, and H. P. Jenssen, "Effects of iron concentration andvalence on the photorefractive properties of BaTiO3," in Digestof Topical Meeting on Photorefractive Materials, Effects, andDevices (Optical Society of America, Washington, D.C., 1987),pp. 23-25.
8. P. G. Schunemann, "Growth and characterization of high-puri-ty and iron-doped photorefractive barium titanate," M.S. thesis
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M. B. KleinB. A. Wechsler was born November 4,1951, in Oceanside, New York.He received the A.B. degree in geological and geophysical sciencesfrom Princeton University, Princeton, New Jersey, in 1973, and theM.S. and Ph.D. degrees in earth and space sciences from the StateUniversity of New York at Stony Brook, Stony Brook, New York, in1975 and 1981, respectively. He was a postdoctoral research asso-ciate in chemistry at Arizona State University, Tempe, Arizona,from 1981 to 1983. In 1983 he joined Hughes Research Laborato-ries, Malibu, California, where he is now a member of the technicalstaff in the Optical Physics Department. His research interestsinclude crystallography, crystal chemistry, thermochemistry, andphase equilibria. He is currently involved in crystal growth andoptimization of photorefractive and nonlinear-optical materials.
M. B. Klein was born in Perth Amboy, New Jersey, in 1942. Hereceived the B.A. and the B.S. degrees (1964) from Brown Universi-ty and the M.S. (1965) and the Ph.D. (1969) degrees in electricalengineering from the University of California, Berkeley. His Ph.D.research was on the kinetics of pulsed and cw ion lasers. From 1969to 1973 he was at Bell Laboratories, Holmdel, New Jersey. Hisresearch interests there included metal-vapor lasers and dye-lasergain spectroscopy. In 1973 he joined the Optical Physics Depart-ment at Hughes Research Laboratories. Since that time he hasconducted research on CO2 laser devices, nonlinear materials for themillimeter spectral region, and degenerate four-wave mixing insemiconductors and photorefractive materials. His current re-search interests include the study of photorefractive materials foradaptive optics and optical data processing.
