2. Thermodynamics of native point defects in GaAs

Transcription

1 2. Thermodynamics o native point deects in The totality o point deects in a crystal comprise those existing in a perectly chemically pure crystal, so called intrinsic deects, and those associated with some oreign atoms. The latter are termed dopants i they are deliberately added or impurities i they were introduced unintentionally. There are six principal intrinsic point deects in : vacancies on both sublattices (V and V ), and interstitials ( i and i ) and antisites ( and ). In ramework o this work, the term native deects is related to the point deects being in thermodynamical equilibrium with a host crystal and not introduced intentionally, e.g. by crystal deormation or irradiation. This deinition comprises intrinsic deects, oreign atoms as well as their complexes. The equilibrium concentration o point deects in binary compound semiconductors such as is a complicated unction depending o temperature, Fermi-level position and o the activity o either one o the constituents (gallium or arsenic), i.e. o crystal stoichiometry. In the ollowing sections, some aspects important or this thesis concerning native deect ormation are briely reviewed. 2.1 system at melting point llium arsenide crystallizes in the zincblendstructure at about 1513 K. The lattice constant is.5653 nm at 3 K (Brozel and Stillman 1996). or any compound, there is a certain thermodynamically allowed region o homogeneity or 1-x x. This is shown on 13 Melt liquidus line 12 s s + () l T / o C 11 1 p Σ = bar solidus line T F 9 s + () l x Fig. 2.1: Binary phase diagram o --System. Data are taken rom Res. (Wenzl et al. 1991; Wenzl et al. 1994) (let) and (Jurisch and Wenzl 22) (right)

2 4 2. Thermodynamics o native point deects in Fig. 2.2: Equilibrium pressures o, 2, 4 and along the -rich (a,a*) and -rich (b,b*) phase boundary o.(from Re. (Arthur 1967)) the binary phase diagram in ig The homogeneity region is represented by the shaded area within the solidus line a*-b*. I concentration o one o the components (e.g., ) exceeds a value allowed by the solidus line b*, the system consists o three phase crystal, melt and gas. Inside the homogeneity region, crystal is in thermal equilibrium with its gas phase. Corresponding vapour pressures determined along the solidus line are shown in ig In case o the arsenic-rich melt, the 4 -pressure dominates while the vapour pressure o gallium is negligibly small. The resulting pressure is then p 4. On the gallium-rich side, 2 is the dominating gas component. The pressure becomes important only at low temperatures (T<8 C). It is important to note that the exact shape o the phase diagram is unknown at present. This subject is still under investigations. According to the recent thermodynamic calculations o Jurisch and Wenzl, there is no crystallization at all possible on the -rich side [Fig. 2.1 (right)]. This may explain the act why gallium precipitates were never observed even in the crystals grown rom a very -rich melt (Kiessling 23). However, the Wenzl s hypothesis needs stronger experimental evidence. Anyway, it is commonly accepted that the congruent point, where the solidus and liquidus lines osculate, lies on the -rich side o the phase diagram (Fig. 2.1). Thereore, growth rom a stoichiometric melt always yields arsenic-rich material (Wenzl et al. 1991, Hurle 1999). The deviation rom stoichiometry s is realized through the ormation o point deects on both sublattices. The deviation can be expressed as s = 1 2x = s 1 s 2, where s 1 and s 2 are determined as: 2 s = [ ] + [ ] [ V ] [ ] [ X ] and 1 2 s = [ ] + [ ] [ V ] [ ] [ X ] 2 i i (2.1) where X and X imply an arbitrary oreign atom on the or lattice site. The brackets denote deect concentration in atomic units. Thus, by measuring the stoichiome-

3 2. Thermodynamics o native point deects in 5 try deviation s, type and concentration o the dominating point deects can be determined. There are two direct methods or estimation o s: 1) the direct determination o the amounts o and atoms in a sample by electrochemical titration o one or both o the species; 2) the determination o the average mass o the unit cell in a sample by measuring its density (which is the mass per unit volume o crystal) and the cube o its lattice parameter (which is volume o the unit cell). The quotient o these two quantities is, thereore, the mass per unit cell. By subtracting rom this the mass o an ideal unit cell (containing one and one atom), the excess mass per unit cell which is related to the deviation rom stoichiometry is obtained. Application o these two approaches to study crystals grown rom - and -rich melt showed that deviations on the sublattice dominate (Hurle 1999). Thus, under -rich conditions, there are large concentrations o i and V grown into the crystals at the melting point. However, deviation rom stoichiometry provides inormation only on dominating deects, which remain in a crystal ater its cooling. Investigation o other point deects present at elevated temperatures with the help o these methods is impossible. There are always some point deects existing in a crystal at thermodynamical equilibrium, which contribute to minimization o the Gibbs ree energy. In elemental semiconductor such as Si, the equilibrium concentration o an individual point deect depends on temperature and the position o the Fermi level. For compound semiconductor, the situation is more diicult since the equilibrium concentrations depend additionally on the crystal stoichiometry (Eq. 2.1). This means that deect concentrations, crystal composition and ambient vapour pressure are in thermodynamical equilibrium and thus depend on each other. In addition, many o the point deects may exist in more than one charge state and interact strongly through their inluence on the Fermi energy. Thus, dopant solubility can be strongly inluenced by charged native point deects grown into the crystal and conversely, doping markedly aects charged native point deect populations. 2.2 Deects chemistry in Deect chemistry represents a chemical approach to the problems dealing with imperections in crystalline solids. This method considers the generation o deects, interaction between them and between deects and ambient as chemical reactions based on the assumption that the crystal is in an equilibrium state (Kröger 1964). Six intrinsic point deects in can be described by the ollowing six thermodynamic reactions (Hurle 1999): 1/4 ( ) i 4 υ = (2.2) = V + (2.3) i

4 6 2. Thermodynamics o native point deects in = V + V (2.4) = + V (2.5) i + V = (2.6) i + V = (2.7) i where the superscript () indicates the neutral state. Only one o the reactions involves the external phase (Eq. 2.2). The external phase may be liquid or vapor. To avoid writing dierent equations or the two cases, the activity o the external phase is expressed in terms o the partial pressure o arsenic dimers which would be in equilibrium either with the vapor or with a liquid which was in equilibrium with that vapor. The arsenic tetramer is chosen, since it is the dominating species in arsenic vapor (Fig. 2.2). Having expressed a condition o equilibrium o one component with the external phase, equilibrium with respect to the other component () is deined as: p p = exp( g / kt) (2.8) 1/4 4 where p is the partial pressure o gallium monomers and g is the ree energy o ormation o rom gaseous gallium monomer and arsenic tetramers at one atmosphere total pressure and temperature T. Substituting (2.2) into (2.3), one obtains the ollowing reaction or V ormation: = V + 1/4 ( υ) (2.9) From (2.4) and (2.9), a similar reaction or the vacancy may be deduced: 1/4 4( ) 4 υ = V + (2.1) Concentrations o vacancies in their neutral states are determined by mass-action law: where K V and [ V ] = K ( T) p (2.11) 1/4 V 4 [ V ] = K ( T) p (2.12) 1/4 V 4 K V are mass-action constants o ormation o gallium and arsenic vacancies, respectively. According to (2.11) and (2.12), the density o V should increase and o V decrease with increasing arsenic pressure. In case o charged vacancies, their ionization reactions and corresponding concentrations are given by the ollowing reactions: Arsenic vacancy V = V + e (2.13) +

5 2. Thermodynamics o native point deects in 7 llium vacancy + [ V ] [ V ]( NC / n)exp( EV / k ) BT = (2.14) V = V + h (2.15) + [ V ] [ V ]( n/ NC )exp( EV / k ) BT = (2.16) V = V + qh (2.17) q + V = V n N qe E k T (2.18) [ q q q ] [ ]( / C ) exp g V / B q where q=1,2,3, E g (T) is the energy gap. E V is the ionization energy o the arsenic vacancy and q E V is the ionization energy o the q charge state o the gallium vacancy. n is the ree electron concentration, N C is the eective density o conduction-band states and k B is Boltzmann s constant. 2.3 Tan model and Fermi-level eect The considerations presented in the previous subsection give a qualitative description o vacancy ormation. However, they cannot be applied quantitatively i the mass-action constants are unknown. Tan et. al. tried to evaluate the concentrations o equilibrium point deects explicitly (Tan et al. 1993; Tan 1994). According to Tan s model, concentrations o neutral arsenic and gallium vacancies are determined by: g V and ( ) ( ) 1/4 = 4 4 δ B [ V ] B / p exp{ [ g ( V ) g ]/ k T} ( ) 1/4 4 4 δ B (2.19) [ V ] = p / B exp{ [ g ( V ) + g ]/ k T} (2.2) g V are the ree energies o ormation o neutral arsenic and gallium ( ) vacancies, respectively. δ g is the dierence: δ g = g ( ) g ( ) (2.21) g ( ) is the ree energy o bonding o an arsenic atom in the crystal and g ( 4) is that o bonding o an arsenic atom in the tetramer molecule. B π m h kt 2 3/2 5/2 = (2 / ) ( ) 4 is the gas constant assuming 4 4 can be described as an ideal gas. m 4 is the mass o the 4 molecule and h Planck s constant. 4

6 8 2. Thermodynamics o native point deects in Fig. 2.3: Equilibrium concentrations o 3 V in n-type calculated as a unction o temperature or dierent concentrations o donor atoms, i.e. or dierent positions o the Fermi level. Concentrations o the charged deects in Tan s model are determined similarily to the Eq z ( m= 1 ) = (2.22) [ V z ] [ V ]exp ze E am E F is the Fermi energy, z is the charge and E am are the ionisation energies o the vacancy measured rom the valence band. can be seen rom Eq. (2.22), the concentration o the charged vacancies increases with the Fermi-level position moving towards the conduction band edge. The physical reason is the energy dierence E F -E a gained by a whole system ( crystal) in a process o electron transition to the acceptor level E a, i.e. ormation o an additional acceptor. The dependence o the concentration o a charged point deect in a semiconductor on the charge carrier concentration (i.e. Fermi level position) is called the Fermi-level eect. suming V as the only acceptor, Tan et al. calculated the density o gallium vacancies in n-type (Tan et al. 1991). The pressures corresponded to those depicted in ig The Gibbs ree energies in Eqs. (2.19,2.2) did not include entropy terms, i.e. they were set equal to the corresponding ormation enthalpies h. The values h = 2.59eV (van Vechten 1975) and δ h =.69eV were used. The data or V V ionization energies were taken rom Re. (Bara and Schlüter 1985). The results o the calculations perormed or dierent ree electron densities in n- are presented in ig In intrinsic material, the concentration o vacancies increases with increasing temperature. However, the temperature dependence was ound to be dramatically weakened by doping. At high doping level, E F can be very close to the conduction band at all temperatures. This may result in temperature independence or even a small negative temperature dependence o equilibrium concentrations o 3 V (Tan 1994). The irst experimental evidence o this eect was given by Gebauer et al. in a recent positron annihilation study o Te-doped (Gebauer et al. 23). In their analysis, the ormation en-

7 2. Thermodynamics o native point deects in 9 thalpy h as well as the ormation entropy s was considered. The values V h = (3.2 ±.5) ev and s = (9.6 ± 1) k were determined. V V B V 2.4 Quantum mechanical calculations The calculations given in the previous subsection sensitively depend on the knowledge o deect ormation and ionization energies. Experimental determination o these properties is usually possible only or a constrained number o deects. The charge state, ormation energy E D and equilibrium concentration C D o a certain deect may also be obtained with the help o theoretical models based on quantum mechanical calculations. In equilibrium, C D is given as: with Ns [ µ µ ] C = N exp E (, ) / k T (2.23) D s D e B 22 3 = cm being the density o lattice sites or one o the sublattices. E D is a unction o µ e and µ. µ e is the electrons chemical potential, i.e. Fermi energy. µ is the dierence between chemical potentials o () in and in pure () elements. In other words, µ relects the stoichiometry deviation. can be seen rom Eq. (2.23), the deect with the lowest binding energy should have the highest equilibrium concentration. Since deect ormation energies are usually not exactly known, the accuracy o such kind calculation is oten insuicient. Thus, already small numerical uncertainties in the binding energy can aect the calculated deect density by orders o magnitude (Jansen and Sankey 1989). However, a quantum mechanical approach can be used or qualitative predictions o the type o dominating deects. Fig. 2.4 represents the ormation energy o intrinsic point deects in as a Fig. 2.4: Formation energy o intrinsic deects in -rich (let) and -rich (right) as a unction o Fermi energy.

8 1 2. Thermodynamics o native point deects in Fig. 2.5: Calculated ionization levels o (let) and (right) vacancies. Positive numbers designate values measured rom the valence band, negative rom conduction band. unction o Fermi energy (rom Res. (Zhang and Northrup 1991; Northrup and Zhang 1993; Landman et al. 1997)). In -rich, p-type and semi-insulating, the arsenic antisite has the smallest ormation energy. This is in accordance with the wellknown act that the EL2-centers govern properties o semi-insulating (Brozel and Stillman 1996). In n-conducting -rich, triply negative gallium vacancy has the minimal ormation energy. llium interstitials i should dominate in -rich p-type and gallium antisites in SI and n-. Similar results were obtained in several other theoretical works (Bara and Schlüter 1985; Jansen and Sankey 1989; Seong and Lewis 1995). According to them, gallium vacancies V should appear in rich n-doped. The ormation energy o -interstitials is according to (Landman et al. 1997) comparable with the ormation o V or in -rich material. This is in agreement with the results o mass per unit cell measurements, according to which i is the dominating deect is -rich (Section 2.1). The arsenic vacancy V has a ormation energy much higher than that o V in n-type - and -rich. Consequently, occurrence o V is rather improbable in n-. However in p-conducting material, where the Fermi level is positioned in the lower hal o the band gap, ormation o V can be energetically more avourable compared to V. Knowledge about the deect ionisation levels is very important or the interpretation o PALS-results. The positron being a positively charged particle eels the charge o the deects. The direct consequence is that positively charged deects can not be detected by positrons. Unortunately, ionisation levels o vacancies in are not well-known experimentally. In principle, they can be determined with the help o quantum mechanical calculations. Fig. 2.5 presents ionisation levels o - and -vacancy obtained in dier-

9 2. Thermodynamics o native point deects in 11 ent theoretical works. According to the calculations, the gallium vacancy has ionisation levels in the lower hal o the band gap and thus, it is in n-type and SI triply negative charged. The arsenic vacancy should be positive in p-conducting and semi-insulating material. According to some calculations, V has ionisation levels near the conduction band (Res. (Jansen and Sankey 1989; Puska 1989; Seong and Lewis 1995)). Thus, both V and V must be detectable in n-. However, calculated ionisation levels demonstrate rather big quantitative dierences. Thereore, only qualitative conclusions may be drawn rom them, whereas direct comparison with experimental data seems to be questionable. 2.5 Electrical compensation in n-type having electron conductivity type can be produced by means o doping with the atoms o group IV (Si, Ge, Sn) or group VI (S, Se, Te) o the Periodic Table. The elements o the group VI can be incorporated exclusively on the sublattice orming shallow donors. The atoms o the IV group demonstrate in contrary amphoteric behavior: they behave as shallow donors when incorporated on site and as shallow acceptors on the site. N-doped is always partially electrically compensated. This means that the ree electron density is smaller than the number o introduced dopants. The compensation in doped with the atoms o the IV group can to some extent be explained by simultaneous ormation o donors and acceptors due to the amphoteric dopants incorporation. However, measurements o carrier mobility demonstrated the compensation eect occurring also in S-, Se- or Te- doped material. The degree o compensation N D /N A was ound to be ~.25 at moderate doping levels regardless o the dopant kind. Hurle in Re. (Hurle 1977) supposed that the compensating species was the donor atom complexed with its nearest gallium vacancy to orm an acceptor. Positron annihilation studies o Te- and Si-doped have rather clearly demonstrated the existence o Te V (Krause-Rehberg et al. 1995; Gebauer et al. 1999) and Si V complexes (Laine et al. 1996; Gebauer et al. 1997). Vacancy-like deects were ound also in doped with S, Ge and Sn atoms (Gebauer 2). The complexes were ound to be negatively charged. However, there is still a lack o knowledge on their exact charge state (e.g., -1e or -2e). Nowadays, donor-gallium vacancy complexes are commonly accepted as compensating centers in all kinds o n-. Compared to point-like imperections, inluence o large-scale deects on electrical properties o is much less investigated. Thus, it was ound that the distribution o carrier concentration across silicon-doped waer showed structures corresponding to the arrangement o dislocations (Börner 22). This might be due to the decreased concentration o Si V complexes in dislocation-rich regions o a waer, as was proposed in the recent photoluminescence study o :Si (Baeumler et al. 22).

10 12 2. Thermodynamics o native point deects in However, photoluminescence spectroscopy could provide only qualitative inormation on deect concentrations. In contrary, the method o positron annihilation used in this work is suitable or quantitative analysis o the distribution o Si V complexes.