Discussion of enthalpy, entropy and free energy of formation of GaN ...

ARTICLE IN PRESSK.T. Jacob, G. Rajitha / Journal of Crystal Growth 311 (2009) 3806–3810 3807ln (P N2/ P°)Peshek "equilibrium"Formation of GaNEquilibrium lineof the slow kinetics of decomposition, the minimum N 2 pressurefor stable GaN would be well below the equilibrium dissociationpressure of N 2 . It can be argued that this upper pressure limitidentified by Peshek et al. [1] as the minimum pressure for stableGaN actually marks the onset of decomposition of metastableGaN. The true equilibrium must lie well above it. At ambientpressure of N 2 (0.1 MPa), Peshek et al. [1] varied the temperatureto identify the limit between ‘‘stability’’ and ‘‘instability’’.Since Peshek et al. [1] experimentally identified two pressuremarkers corresponding to the onset and completion of the GaNdecomposition process and derived their so-called ‘‘equilibrium’’as a mean of these, their data can not be considered reliable. Theyapproached ‘‘equilibrium’’ only in one direction. They did notattempt reversal (formation of GaN). True equilibrium is bestidentified by approaching it from both directions. They actuallymeasured the limit between metastability and instability. For thisreason, their data are systematically displaced to lower pressuresor more negative standard Gibbs energies of formation of GaN.3. Statistical analysis of dataDecomposition of GaN(max. P N2for decomp.)Min. P N2for metastable GaN1 / T (1/K)Fig. 1. Schematic representation of nitrogen partial pressures for formation anddecomposition of GaN as a function of reciprocal of absolute temperature. Trueequilibrium line is drawn assuming symmetrical hysteresis. Pressures correspondingto the measurements of Peshek et al. [1] are illustrated schematically.P 0 =0.1 MPa is the standard pressure.to decompose GaN, pressures significantly below equilibrium haveto be applied as shown schematically in Fig. 1. The N 2 pressurehysteresis reduces with increasing temperature. Assuming that thehysteresis is symmetrical, the equilibrium curve may betentatively placed midway between the pressure required forformation and that corresponding to the decomposition atconstant temperature. This is not necessarily a good assumptionbecause the kinetic constants for formation and decomposition aregenerally expected to be different. Nevertheless, in the absence of aclear understanding of the kinetic processes, the symmetryapproximation is the best that can be made. It is clear from Fig.1 that GaN can be retained metastably below its equilibriumdissociation pressure for periods that exceed the laboratorymeasurement time scales.Peshek et al. [1] did not measure the N 2 pressure required forthe formation of GaN. Rather, they reduced the N 2 pressuregradually at constant temperature, and recorded the minimum N 2pressure at which GaN was stable in the experimental time frame(24–36 h), and the maximum N 2 pressure at which the GaN filmcompletely disappeared. Presumably, in the intervening pressurerange partial decomposition of GaN would have occurred. It is tobe noted that the minimum N 2 pressure for observing stable GaNfilm is considerably higher than the maximum pressure for thedisappearance of the film. Peshek et al. [1] identified equilibriummidway between these two pressures as shown in Fig. 1. BecausePeshek et al. [1] ‘‘conservatively’’ associate the pressure ortemperature range for transiting from ‘‘stable’’ to ‘‘unstable’’domain for GaN, displayed in Fig. 1 of their paper, with a 2suncertainty or 95% confidence interval. It is difficult to understandhow these kinetically determined ranges can be associated withstatistical uncertainty. Peshek et al. [1] do not present numericalvalues for their dissociation pressures or temperature. The dataare presented in their Fig. 1, which is a plot of 0:5 R ln P N2 versusthe reciprocal of absolute temperature. From the slope they0derived DH f,GaN = 1,65,000716,000 J/mol [1]. In view of thenarrow temperature range (120 K) and the relatively largescatter in the data, the uncertainty limit on the enthalpy offormation appears to be low. The least-squares linear regressionanalysis of the data presented in the figure, giving equal weight toall the points, gives the expression:0:5 R lnðP N2 Þð2:179ÞðkJ=mol KÞ1; 65; 035ð33; 440Þ¼ 138:16ð29:18Þ(1)Twhere the uncertainty estimates on each term correspond toeither the standard deviation or the standard error estimates (s).00Hence, DH f,GaN = 1,65,035766,880 J/mol, and DS f,GaN =138.16758.36 J/mol K for 2s confidence interval. Although the enthalpyand entropy of formations derived from the equation are close tovalues reported by Peshek et al. [1], the uncertainties are morethan four times the values given by them for the same (95%)confidence interval. Given the large uncertainty in the enthalpy offormation, the result cannot be used to differentiate among thecalorimetric data [3–5] as suggested by Peshek et al. [1].4. Comparison of data4.1. Gibbs free energy of formation of GaNThe most direct and reliable thermodynamic quantity that canbe obtained from dissociation pressure measurements is theGibbs free energy of formation of GaN. The values obtained fromthe measurements of Peshek et al. [1] are compared with otherpublished data [2,6–10] in Fig. 2. Their values are the mostnegative compared to all the other direct Gibbs energymeasurements at high temperature for reasons outlined inSection 2. In partial justification of their results, Peshek et al. [1]cited the agreement between the enthalpy of formation of GaN

ARTICLE IN PRESS3808K.T. Jacob, G. Rajitha / Journal of Crystal Growth 311 (2009) 3806–3810ΔG f ° (kJ/mol)100806040200-20-40GaNMadar et al.(dissociation pressure)Unland et al.(dissociationtemperature)Jacob et al.(emf)Karpinski et al.(dissociation pressure)* with fugacity correction* without fugacity correctionRecommended ValuesThurmond and Loganminimum NH 3 for synthesisNH 3 in exit stream duringsynthesisNH 3 produced by reactionPeshek et al. of H 2 with GaN(dissociation pressure)Calorimetric Δ H° 298 of Ranade et al.combined with thermal data.-609001000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000T (K)Fig. 2. Comparison of reported values of standard Gibbs energy of formation of GaN.derived from their measurements and the recent calorimetricmeasurements [4,5]. This agreement is fortuitous; accuratemeasurement of dissociation pressure as a function of temperaturemust give simultaneously correct values for both enthalpy andentropy of formation. One cannot extract the correct value ofenthalpy and incorrect value of entropy.Madar et al. [7] and Karpinski et al. [8] measured thedissociation pressure of GaN at high temperatures and nitrogenpressures well above atmospheric. Madar et al. [7] placed a vapordepositedGaN layer on sapphire in the hot spot of an internallyheated pressure vessel at temperatures up to 1523 K. A series ofannealing treatments were given at different N 2 pressures toidentify the dissociation pressure. Formation of Ga droplets on thesurface signaled decomposition. Using microcrystalline powdersof GaN, Karpinski et al. [8] made more extensive measurements attemperatures up to 1913 K. Karpinski and Porowski [9] derivedGibbs energies for formation of GaN from the high-pressure datausing fugacity correction. In Fig. 2 the standard Gibbs energies offormation of GaN derived from the high-pressure studies ofKarpinski et al. [8] with and without fugacity correction areshown. Since there is significant uncertainty in the fugacity valuesat high pressures, the most reliable Gibbs energy data from themeasurements of Karpinski et al. [8] are in the lower end of theirtemperature range, where the magnitude of fugacity correction isrelatively small. The thin film of GaN on sapphire used by Madaret al. [7] was probably associated with higher free energy than themicrocrystalline powder used by Karpinski et al. [8].Thurmond and Logan [6] used the ratio of partial pressures ofNH 3 to H 2 in gas mixture at ambient pressure as an indicator of N 2partial pressure. They measured the minimum NH 3 pressure in H 2required for the synthesis of GaN from Ga, NH 3 concentration inthe exit stream during synthesis and the concentration of NH 3produced by the reduction of GaN by H 2 . The standard Gibbsenergies of formation of GaN calculated from the three sets of dataare shown by separate symbols in Fig. 2. The measurementsprovide upper and lower limits for free energy. Thurmond andLogan [6] suggested that the values derived reduction measurementswere close to equilibrium. Comparison of their results withthe data of Karpinski et al. [8] in the overlapping temperaturerange supports the suggestion. Peshek et al. [1] claim thatreduction of GaN by H 2 to form NH 3 does not happen in thetemperature range of their experiments contrary to the observationof Thurmond and Logan [6]. Based on our experience thereduction does proceed and NH 3 can be detected when the flowrate of H 2 is low. Perhaps at relatively high flow rates used byPeshek et al. [1], the concentration of NH 3 was below theirdetection limit. At higher flow rates used by Peshek et al. [1], thegas phase may not even be in thermal equilibrium with thecondensed phases, let alone chemical equilibrium.Jacob et al. [10] used a solid-state electrochemical technique,using single-crystal CaF 2 as the electrolyte, to measure thestandard free energy of formation of GaN with relatively highaccuracy (7465 J/mol) from 875 to 1125 K. Unland et al. [2]measured the dissociation temperature of GaN in pure nitrogen atambient pressure as 1110710 K using two different thermogravimetricmethods, dynamic oscillation thermogravimetric analysis(TGA) and isothermal stepping TGA.As seen in Fig. 2, four independent measurements of the freeenergy of GaN (Jacob et al. [10], Unland et al. [2], decompositionstudies of Thurmond and Logan [6] in H 2 and N 2 dissociationstudies of Karpinski et al. [8] at lower pressures) covering in all alarge temperature range from 875 to 1500 K are in goodagreement. These results taken together can be represented bythe equation:DG o f ð2520ÞðJ=molÞ ¼ 1; 30; 297ð3534Þþ116:8ð3:06Þ T (2)This equation represents the best information on Gibbs energy offormation now available.4.2. Standard entropy and entropy of formation of GaNThe most accurate values of entropy of crystalline GaN isobtained by integrating the curve of (C P /T) versus temperaturefrom 0 K to the desired temperature. Fortunately, both low- andhigh-temperature heat-capacity data are available for GaN. Theavailable data have been critically analyzed by Jacob et al. [10] andthe entropy of GaN evaluated as a function of temperature. LowtemperatureC P data of Demidenko et al. [11] from 55 to 300 K andof Koshchenko et al. [12] from 5 to 60 K were used to evaluate the

ARTICLE IN PRESSK.T. Jacob, G. Rajitha / Journal of Crystal Growth 311 (2009) 3806–3810 3809Table 1Recommended thermodynamic properties of GaN from 298.15 to 1400 K.T (K) C 0 P (J/mol K) H o 0T H 298.15(J/mol) S 0 0T S 298.15 (J/mol K) S 0 T (J/mol K) DS T (J/mol K) DH 0 f (kJ/mol) DG 0 f (kJ/mol)298.15 35.30 0 0 34.120 102.458 129.289 98.741300 35.40 65 0.219 34.339 102.493 129.301 98.553302.9 35.56 168 0.560 34.680 102.547 129.319 98.258302.9 35.56 168 0.560 34.680 121.002 134.909 98.258400 40.00 3848 11.067 45.187 122.231 135.324 86.432500 43.36 8024 20.372 54.492 122.227 135.313 74.199600 45.90 12492 28.512 62.632 121.679 135.010 62.002700 47.86 17184 35.741 69.861 120.900 134.500 49.870800 49.43 22051 42.238 76.358 120.026 133.844 37.823900 50.69 27059 48.135 82.255 119.129 133.080 25.8641000 51.72 32181 53.530 87.650 118.240 132.237 13.9971100 52.57 37397 58.501 92.621 117.372 131.326 2.2171200 53.28 42690 63.106 97.226 116.532 130.363 9.4751300 53.87 48049 67.395 101.515 115.733 129.358 21.0941400 54.37 53462 71.406 105.526 114.960 128.320 32.624ostandard entropy of GaN at 298.15 K (S 298.15 ) as 36.5 J/mol K. Theauthors have now become aware of a more recent lowtemperaturecalorimetric measurement by Danilchenko et al.[13] from 5 to 293 K, which yields a value of 34.12 J/mol K foroS 298.15 when extrapolated to 298.15 K guided by earlier measurements[11,12]. The selected value of C o P at 298.15 K is 35.3 J/mol K,slightly higher than the value of 34.91 J/mol K suggested earlier[10], but in sharp disagreement with the value of 19(714) J/mol Ksuggested by Peshek et al. [1]. Koshchenko et al. [12] performedheat-capacity measurements on powder samples of GaN whereasDanilchenko et al. [13] carried out the measurements onhexagonal single crystals of GaN. Hence, the standard entropybased on the measurements of Danilchenko et al. [13] isconsidered to be the most reliable data available now. The entropyof formation of GaN can be calculated by combining the entropy ofGaN with well-established auxiliary data for Ga metal and N 2gas [14].The empirical equation for C o P above 298.15 K suggested earlier[10] is altered to agree with the new value at 298.15 K. Themodified equation isC o PðJ=mol KÞ ¼80:729 0:00313 Tþð308493=T 2 Þ ð828:2=T 0:5 Þ. (3)Based on this equation, the enthalpy and entropy increments forwurzite GaN are evaluated at regular intervals of temperature anddisplayed in Table 1. The standard entropy of GaN as a function oftemperature evaluated from heat-capacity data using the thirdlaw of thermodynamics is also listed in the table. The entropy offormation is obtained by combining the data on GaN withauxiliary data for N 2 gas and Ga metal from the compilation ofPankratz [14]. The heat-capacity data on GaN now availableestablish its entropy of formation with relatively high accuracy(72.5 J/mol K).4.3. Enthalpy of formation of GaNThe enthalpy of formation of GaN is constrained by theselected values for Gibbs free energy of formation and entropyof GaN, which are now well-established. The constrained value of0enthalpy of formation at 298.15 K is DH f,GaN = 1,29,28972700 J/mol.Values of enthalpy of formation at higher temperatures calculatedusing thermal data are listed in Table 1. The derived value ofenthalpy of formation falls between the two sets of widely differingcalorimetric measurements of Hahn and Juza [3] ( 1,10,000 J/mol),on the one hand, and Ranade et al. [4] ( 1,56,800 J/mol) andZieborak-Tomaszkiewicz [5] ( 1,63,70074200 J/mol), on the otherhand. A critical discussion on the calorimetric measurements hasbeen provided earlier [10]. Gibbs energy of formation of GaNcalculated from the enthalpy of formation reported by Ranade et al.[4] and entropy derived from thermal data is shown in Fig. 2 as afunction of temperature. The comparison clearly illustrates that thenew calorimetric data on enthalpy of formation [4,5] are inconsistentwith available data on Gibbs energy of formation. The valuesof entropy of formation derived from the temperature dependenceof Gibb energy of formation is less reliable and generally notaccepted unless they are in agreement with values obtained fromheat-capacity measurements.5. Optimized thermodynamic data for GaNBased on values of entropy and enthalpy of formation, theGibbs energy of formation has been calculated at regular intervalsof temperature. The values are listed in Table 1. All the data in thetable are internally consistent. The table provides the bestthermodynamic information now available on GaN. In thetemperature range from 800 to 1400 K, the Gibbs energy offormation can be represented by the equation:DG o fðJ=molÞ ¼ 1; 31; 530 þ 117:4 T. (4)The recommended values corresponding to Eq. (4) are comparedwith measurements in Fig. 2. The decomposition temperature ofGaN in pure N 2 at 0.1 MPa calculated from this equation is1120(75) K.AcknowledgementsK.T. Jacob is grateful to the Indian National Academy ofEngineering, New Delhi, for the award of INAE distinguishedProfessorship. G. Rajitha thanks the University Grants Commissionof India for the award of Dr. D.S. Kothari Postdoctoral Fellowship.References[1] T.J. Peshek, J.C. Angus, K. Kash, Experimental investigation of the enthalpy,entropy and free energy of formation of GaN, J. Cryst. Growth 311 (2008)185–189.[2] J. Unland, B. Onderka, A. Davydov, R. Schmid-Fetzer, Thermodynamics andphase stability in the Ga–N system, J. Cryst. Growth 256 (2003) 33–51.[3] H. Hahn, R. Juza, Examinations on the nitrides of cadmium, gallium, indiumand germanium: metal amides and metal nitrides; VIII announcement, Z.Anorg. Allg. Chem. 244 (1940) 111–124.

ARTICLE IN PRESS3810K.T. Jacob, G. Rajitha / Journal of Crystal Growth 311 (2009) 3806–3810[4] M.R. Ranade, F. Tessier, A. Navrotsky, V. Leppert, S.H. Risbud, F.J. DiSalvo, C.M.Balkas, Enthalpy of formation of gallium nitride, J. Phys. Chem. B 104 (2000)4060–4063.[5] I. Zieborak-Tomaszkiewicz, Some thermodynamic aspects of nitrides inmaterials science: fluorine bomb calorimetric study, J. Therm. Anal.Calorimetry 83 (2006) 611–616.[6] C.D. Thurmond, R.A. Logan, The equilibrium pressure of N 2 over the GaN,J. Electrochem. Soc. 119 (1972) 622–626.[7] R. Madar, G. Jacob, J. Hallais, R. Fruchart, High pressure solution growth ofGaN, J. Cryst. Growth 31 (1975) 197–203.[8] J. Karpinski, J. Jun, S. Porowski, Equilibrium pressure of N 2 over GaN and highpressure solution growth of GaN, J. Cryst. Growth 66 (1984) 1–10.[9] J. Karpinski, S. Porowski, High pressure thermodynamics of GaN, J. Cryst.Growth 66 (1984) 11–20.[10] K.T. Jacob, S. Singh, Y. Waseda, Refinement of thermodynamic data on GaN,J. Mater. Res. 22 (2007) 3475–3483.[11] A.F. Demidenko, V.I. Koshchenko, L.D. Sabanova, Yu.M. Gran, Low temperatureheat capacity entropy and enthalpy of aluminum and gallium nitrides, Russ. J.Phys. Chem. (Engl. Transl.) 49 (1975) 940.[12] V.I. Koshchenko, A.F. Demidenko, L.D. Sabanova, V.E. Yachmenev, Yu.M. Gran,A.F. Radchenko, Temperature-dependence of the thermodynamicproperties of gallium nitride at 5–300 degrees K, Inorg. Mater. 15 (1979)1329–1330.[13] B.A. Danilchenko, T. Paszkiewicz, S. Wolski, A. Jezowski, T. Plackowski, Heatcapacity and phonon mean free path of wurtzite GaN, Appl. Phys. Lett. 89(2006) 0619011–0619013.[14] L.B. Pankratz, Thermodynamic properties of elements and oxides, UnitedStates Department of the Interior Bureau of Mines, 1982.