The published thermochemical property data for vapor- and liquid-phase trimethylaluminum (TMA) monomer and dimer species are reviewed in this letter. A regression scheme is developed to estimate the missing data to produce a complete set of Gibbs-free energy of formation values over temperature ranges relevant to predicting the vapor pressure and degree of TMA dimerization within thin-film deposition gas delivery and reactor systems.

## I. INTRODUCTION

Despite being a widely used metal-organic chemical vapor deposition reactant and arguably the most common atomic layer deposition (ALD) precursor, a complete set of thermochemical data for liquid- and gas-phase trimethylaluminum (TMA) does not appear to exist in the literature. In this work, we collect what enthalpy, entropy, heat capacity, and vapor pressure data have been generated over the past 80 years for monomer and dimer TMA in both the gas and liquid phases. Our objective is not to assess which values of $\Delta Ho$ and $So$ are the most accurate, but to find a *self-consistent set* of thermochemical data that best matches available data in a least-squares sense. What motivates this work is the need to be able to accurately predict the precursor state (vapor pressure and the degree of dimer dissociation) as the TMA is transported from deposition system source, through the precursor delivery system, and into the reaction chamber.

## II. DATA AVAILABLE IN THE LITERATURE

The relatively scarce thermochemical property data of TMA are briefly reviewed in this section, making note of any assumptions or transformations made regarding the data. All data have been converted to standard SI units. Thermodynamic quantities all correspond to the standard conditions of $To=298.15$ K and $Po=100$ kPa.

### A. Enthalpy and entropy data

#### 1. Gas-phase monomer $\Delta fHo$ and $So$

Gas-phase TMA monomer enthalpy of formation was obtained from page 5–35 of the CRC Handbook^{1} and is listed in column HBCP of Table I. A range of values is reported in the NIST WebBook;^{2} likewise, the range of values listed in column WJ98 was obtained from the experimentally reported values in Table 6 of Ref. 3. No entropy data could be found for the gas-phase monomer in the literature.

kJ mol^{−1}
. | p . | b . | HBCP . | NIST . | WJ98 . | MT63 . | LG41 . | S72 . | HE67 . | Est . |
---|---|---|---|---|---|---|---|---|---|---|

$\Delta fHo$ $m$ | $g$ | $m$ | $\u2212$74.1 | $\u2212$57.0 $\xb1$ 9.7/$\u2212$88.7 | $\u2212$74.1/$\u2212$87.4 | $\u2212$70.81 | ||||

$\Delta rHo$ $d$ $\u2192$ 2$m$ | $g$ | $d$ | 84.5 $\xb1$ 4.2 | 85.4 | 85.4 $\xb1$ 1.4 | 82.78 | ||||

$\Delta fHo$ $d$ | $g$ | $d$ | $\u2212230/\u2212250$ | $\u2212$224.39 | ||||||

$\Delta vHo$ $l$ $\u2192$ $g$ | $d$ | 43/63.2 $\xb1$ 1.7 | 41.9 $\xb1$ 0.2 | 40.2 $\xb1$ 0.4 | 40.8 | 41.3 $\xb1$ 0.3 | 48.41 | |||

$\Delta fHo$ $d$ | $l$ | $d$ | $\u2212$272.8 | $\u2212$272.8 | ||||||

$\Delta rHo$ $d$ $\u2192$ 2$m$ | $l$ | $d$ | 81.2 $\xb1$ 1.3 | 81.2 $\xb1$ 1.3 | 82.85 | |||||

$\Delta fHo$ $m$ | $l$ | $m$ | $\u2212$95.8 | $\u2212$94.97 | ||||||

$\Delta vHo$ $l$ $\u2192$ $g$ | $m$ | 22.5 | 24.16 |

kJ mol^{−1}
. | p . | b . | HBCP . | NIST . | WJ98 . | MT63 . | LG41 . | S72 . | HE67 . | Est . |
---|---|---|---|---|---|---|---|---|---|---|

$\Delta fHo$ $m$ | $g$ | $m$ | $\u2212$74.1 | $\u2212$57.0 $\xb1$ 9.7/$\u2212$88.7 | $\u2212$74.1/$\u2212$87.4 | $\u2212$70.81 | ||||

$\Delta rHo$ $d$ $\u2192$ 2$m$ | $g$ | $d$ | 84.5 $\xb1$ 4.2 | 85.4 | 85.4 $\xb1$ 1.4 | 82.78 | ||||

$\Delta fHo$ $d$ | $g$ | $d$ | $\u2212230/\u2212250$ | $\u2212$224.39 | ||||||

$\Delta vHo$ $l$ $\u2192$ $g$ | $d$ | 43/63.2 $\xb1$ 1.7 | 41.9 $\xb1$ 0.2 | 40.2 $\xb1$ 0.4 | 40.8 | 41.3 $\xb1$ 0.3 | 48.41 | |||

$\Delta fHo$ $d$ | $l$ | $d$ | $\u2212$272.8 | $\u2212$272.8 | ||||||

$\Delta rHo$ $d$ $\u2192$ 2$m$ | $l$ | $d$ | 81.2 $\xb1$ 1.3 | 81.2 $\xb1$ 1.3 | 82.85 | |||||

$\Delta fHo$ $m$ | $l$ | $m$ | $\u2212$95.8 | $\u2212$94.97 | ||||||

$\Delta vHo$ $l$ $\u2192$ $g$ | $m$ | 22.5 | 24.16 |

J mol^{−1}K^{−1}
. | p . | b . | HBCP . | NIST . | WJ98 . | MT63 . | LG41 . | S72 . | HE67 . | Est . |
---|---|---|---|---|---|---|---|---|---|---|

$So$ $m$ | $g$ | $m$ | 358.8 | |||||||

$\Delta rSo$ $d$ $\u2192$ 2$m$ | $g$ | $d$ | 180.3 | 173.0 | ||||||

$So$ $d$ | $g$ | $d$ | 524.8 | 544.6 | ||||||

$\Delta vSo$ $l$ $\u2192$ $g$ | $d$ | 102.3 | 125.8 | |||||||

$So$ $d$ | $l$ | $d$ | 418.8 | 418.8 | 418.8 | |||||

$\Delta rSo$ $d$ $\u2192$ 2$m$ | $l$ | $d$ | 122.6 $\xb1$ 1.3 | 127.8 | ||||||

$So$ $m$ | $l$ | $m$ | 270.7 | 273.3 | ||||||

$\Delta vSo$ $l$ $\u2192$ $g$ | $m$ | 79.9 | 85.5 |

J mol^{−1}K^{−1}
. | p . | b . | HBCP . | NIST . | WJ98 . | MT63 . | LG41 . | S72 . | HE67 . | Est . |
---|---|---|---|---|---|---|---|---|---|---|

$So$ $m$ | $g$ | $m$ | 358.8 | |||||||

$\Delta rSo$ $d$ $\u2192$ 2$m$ | $g$ | $d$ | 180.3 | 173.0 | ||||||

$So$ $d$ | $g$ | $d$ | 524.8 | 544.6 | ||||||

$\Delta vSo$ $l$ $\u2192$ $g$ | $d$ | 102.3 | 125.8 | |||||||

$So$ $d$ | $l$ | $d$ | 418.8 | 418.8 | 418.8 | |||||

$\Delta rSo$ $d$ $\u2192$ 2$m$ | $l$ | $d$ | 122.6 $\xb1$ 1.3 | 127.8 | ||||||

$So$ $m$ | $l$ | $m$ | 270.7 | 273.3 | ||||||

$\Delta vSo$ $l$ $\u2192$ $g$ | $m$ | 79.9 | 85.5 |

#### 2. Gas-phase $\Delta rHo$ and $\Delta rSo$

#### 3. Gas-phase dimer $\Delta fHo$ and $So$

The enthalpy of formation range of values for the gas-phase TMA dimer listed in column WJ98 was obtained from experimentally reported values (to two significant figures) in Table 6 of Ref. 3. Entropy of the gas-phase dimer species was calculated using Table V of Ref. 4, in which the measured entropy of the TMA dimer in the liquid phase and the calculated $\Delta fHo$ are used to find $So$. Note that the value listed in column MT63 of Table II corresponds to $Po=1$ bar rather than the value at 1 atm given in the cited work.

#### 4. Dimer $\Delta vHo$ and $\Delta vSo$

The enthalpy of vaporization for the TMA dimer is calculated in the final section of Ref. 4 using vapor pressure data and the Clapeyron equation; the resulting value is listed in column MT63 of Table II. Because Ref. 4 is the source of two of the enthalpy values listed in Ref. 2, that range is listed in column NIST, noting that the upper value of the range is inconsistent with all other reported values, thus casting the accuracy of this upper value in doubt. The value listed in column LG41 was taken from Ref. 5 and the value listed in column S72 was obtained from Table 2 of Ref. 6 using a corresponding entropy value $\Delta Sd(l)o=29.3$ cal mol^{−1} K^{−1}. The sole value for the entropy of vaporization listed in Table II is taken from Ref. 6 based on the same criteria used to select the enthalpy value from that source.

#### 5. Liquid-phase dimer $\Delta fHo$ and $So$

The enthalpy of formation for the TMA dimer in the liquid phase is found on page 5–35 of the CRC Handbook;^{1} the value given in column HBCP of Table I is double the value listed in the cited source to place it on a per-dimer basis. The same source and the same adjustment were used to obtain the entropy value listed in column HBCP of Table II. The liquid-phase dimer entropy value listed in column MT63 of Table II was taken directly from Table V of Ref. 4.

#### 6. Liquid-phase $\Delta rHo$ and $\Delta rSo$

The enthalpy change associated with dimer dissociation in the liquid phase is given in the NIST WebBook^{2} and Ref. 6 on a per-dimer basis; the values are listed in columns NIST and S72, respectively, of Table I. The molar entropy of dissociation on a per-dimer basis is given in Ref. 6 and is listed in column S72 of Table II.

#### 7. Liquid-phase monomer $\Delta fHo$ and $So$

No values for the enthalpy of formation or the entropy of the TMA monomer in the liquid phase could be found in the literature. However, given the liquid-phase dimerization equilibrium relationship $ln\u2061Kd(l)=14.7444\u22129762.5/T$ from Table 1 of Ref. 6 and the enthalpy and entropy values for the liquid-phase dimer, we can calculate the values presented in column S72.

#### 8. Monomer $\Delta vHo$ and $\Delta vSo$

### B. Heat capacity data

Coefficients for the heat capacity correlations for the liquid- and gas-phase TMA monomer and dimer species are in the form of

where $T$ is in K and $Cp$ is in J mol^{−1} K^{−1}, and are given in Table III.

Species . | Source . | $a$ . | $b(103)$ . | $c(106)$ . | $d(109)$ . | $Tmin$ . | $Tmax$ . |
---|---|---|---|---|---|---|---|

m-TMA (l) | GH82^{a} | 89.71 | 33.45 | 457.2 | 0 | 114 | 262 |

m-TMA (g) | NIST^{b} | 45.41 | 6.345 | 779.9 | $\u2212$754.5 | 200 | 500 |

d-TMA (l) | MT63 | 198,5 | 388.0 | $\u2212$340.9 | 1033 | 290 | 380 |

d-TMA (g) | NIST^{c} | 72.78 | $\u2212$185.9 | 2145 | $\u2212$2026 | 200 | 500 |

Species . | Source . | $a$ . | $b(103)$ . | $c(106)$ . | $d(109)$ . | $Tmin$ . | $Tmax$ . |
---|---|---|---|---|---|---|---|

m-TMA (l) | GH82^{a} | 89.71 | 33.45 | 457.2 | 0 | 114 | 262 |

m-TMA (g) | NIST^{b} | 45.41 | 6.345 | 779.9 | $\u2212$754.5 | 200 | 500 |

d-TMA (l) | MT63 | 198,5 | 388.0 | $\u2212$340.9 | 1033 | 290 | 380 |

d-TMA (g) | NIST^{c} | 72.78 | $\u2212$185.9 | 2145 | $\u2212$2026 | 200 | 500 |

^{a}

Corresponds to an approximation of m-TMA by liquid-phase isobutane with a range of validity spanning the triple- to normal-boiling points.

^{b}

Approximated by gas-phase isobutane.

^{c}

Approximated by gas-phase 1,4-dimethylcyclohexane.

No heat capacity information for the monomer TMA in the liquid phase could be found in the literature; the coefficients listed in Table III correspond to saturated liquid-phase isobutane and were taken from Eq. (8) of Ref. 8 to approximate liquid m-TMA. Likewise, the gas-phase monomer TMA, $Cp$, was approximated by that of gas-phase isobutane; data from the NIST WebBook over the temperature range listed in Table III were regressed to obtain the coefficients listed. The same process was used for the gas-phase TMA dimer, where 1,4-dimethylcyclohexane was used instead of the TMA dimer. Heat capacity data for the liquid-phase TMA dimer was found as in Eq. (1) of Ref. 4; the coefficients translated to a per-dimer basis are listed in Table III.

### C. Vapor pressure data

TMA vapor pressure data were obtained from Table IV of Ref. 4; those data were used to identify the coefficients of a TMA vapor pressure correlation in the form of the Cox equation, given as Eq. (3) of the cited source. The data and correlation are plotted in Fig. 2; it is important to note that the vapor pressure $Pvap(T)$ corresponds to the total pressure of a vapor containing both the monomer and dimer forms of TMA.

## III. ESTIMATING A CONSISTENT SET OF ENTHALPY AND ENTROPY VALUES

Our strategy for determining $\Delta fH298o$ and $S298o$ for the monomer and dimer TMA in each state consists of (1) fixing the values of each for one of the species in a single phase and (2) minimizing the residual (mean-squared error) between measured and predicted vapor pressures and degrees of dimer dissociation. For the first step and based on the observation that the liquid-phase dimer species has a common literature source for enthalpy of formation and entropy, we set

### A. Liquid-phase TMA dimer dissociation

In Table 4 of Ref. 6, the degree of TMA dimer dissociation in the liquid phase $\alpha l$ is given as a function of temperature ranging from 0 to 180 °C; the equilibrium constant also was derived in the cited source and was given as

where $T$ is in K. With $am,l$ and $ad,l$ as the liquid-phase activities of the TMA monomer and dimer, respectively, and the liquid-phase mole fractions as

taking both liquid-phase activity coefficients as unity gives

thus, $\alpha l2=Ka,l/(4+Ka,l)$. Rearranging and expanding Eq. (5)

and so

Taking $nT=14$ temperature points corresponding to the data points plotted in Fig. 2 from Ref. 4 and evaluating Eq. (6) gives a least-squares problem to solve the values listed in Tables I and II

### B. Pure TMA monomer vapor–liquid equilibrium

In Table 1 of Ref. 6, the vapor pressure of pure TMA monomer is given as

where $T$ is in K and $Pmvap$ is in Torr. We note that one must be careful when using Eq. (7) to extrapolate into the low-temperature range. Equation (7) was identified by the author of Ref. 6 using data from Ref. 4.

With $am,g$ as the gas-phase activity of the TMA monomer and making use of ideal (unity) activity coefficients

because $am,l=1$. Following a procedure similar to that used in the previous section, we ultimately find

Using the same $nT=14$ temperature points described in the previous section and evaluating Eq. (9) give a least-squares problem to solve for the values listed in Tables I and II:

### C. Pure TMA dimer vapor–liquid equilibrium

We can follow the same procedure used for the TMA monomer vapor–liquid equilibrium to identify the gas-phase dimer constants $\Delta fH298d,go$ and $S298d,go$. Alternatively, we can make direct use of the TMA vapor pressure (a vapor that will contain a combination of the TMA monomer and dimer species) by using Raoult’s law

with

which ultimately gives

Using the full set of vapor pressure data from Ref. 4 and plotted in Fig. 2, we find the final gas-phase dimer set of enthalpy of formation and entropy as

## IV. RESULTS AND DISCUSSION

Comparing the predicted enthalpy and entropy values in Tables I and II to those that could be found in the literature reveals a relatively good agreement except for a potentially high predicted value for $\Delta vHl\u2192go$ relative to the majority of literature sources. Further research into this apparent 7–8 kJ mol^{−1} difference is underway.

With the full set of enthalpy of formation, entropy, and heat capacity for each TMA species in each state identified, we can now predict the state of TMA liquid and vapor under conditions relevant to thin-film deposition technologies; two examples are presented in Figs. 3 and 4. The vapor–liquid equilibrium behavior of TMA in the precursor source (bubbler) is determined by first computing the degree of dissociation in the liquid-phase TMA, followed by the use of Raoult’s law to compute the vapor-phase composition and saturated vapor total pressure; results are shown in Fig. 3.

While the accurate match between the experimental and predicted total TMA pressure under the saturated vapor/liquid equilibrium conditions of Fig. 3 is expected, the true value of this model is demonstrated in Fig. 4 where the gas-phase TMA composition is computed as a function of total pressure at a constant $T=373$ K. In this plot, we see that the degree of TMA dimer dissociation grows with decreasing pressure, reaching a value approaching unity under conditions typical of an ALD or CVD reactor system.

## V. CONCLUSIONS

We have developed a consistent set of enthalpy of formation and entropy data for TMA in its states relevant to thin-film processing applications. We presented an example showing vapor-phase TMA existing primarily as a dimer at the precursor source, dissociating to its monomer form by the time it reaches typical reactor conditions. This letter is not intended to be a final say on the thermodynamic properties of TMA, but instead represents an effort to develop a consistent set of thermochemical parameters and to begin the development of a database of such data for other thin-film processing precursor systems.

## ACKNOWLEDGMENTS

The author gratefully acknowledges the support of the U.S. National Science Foundation through Grant No. CBET1438375 and NASA through the Goddard Space Flight Center.

## References

*CRC Handbook of Chemistry and Physics*, 85th ed. edited by D. R. Lide (CRC, New York,

*NIST Chemistry WebBook*, 2018, see https://webbook.nist.gov/chemistry.