Symmetry Lowering Through Surface Engineering and Improved Thermoelectric Properties in MXenes

The ground state structural parameters of M₂CO₂ and MM^′CO₂, (M, M^′)=Ti,Mo,Zr,Hf), are presented in Table 1. The ground state structures are obtained from DFT calculations by calculating total energies corresponding to structural models that differ from each other depending upon the sites of passivation by -O. The models along with their total energies are shown in Figures 1,2 and Table 1, supplementary material. We find that the sites of passivation on the M and M^′ surfaces in the Janus compounds remain unchanged from those in their M₂C counterparts. The calculated lattice constants are in excellent agreement with the available results. Expectedly, the lattice constants of Janus MM^′CO₂ MXenes are in between those of end point MXenes M₂CO₂ and M${}^{\prime}_{2}$CO₂. Since the inversion symmetry of the M₂CO₂ MXenes is broken in Janus MXenes (the space group of M₂CO₂ compounds is $P\bar{3}m1$ while that of Janus MM^′CO₂ is $P3m1$), there is significant fluctuations in the M/M^′-O/C bond lengths, as is evident from Table 1. For Janus compounds with Mo as M^′ element, Mo-C(M-C) and Mo-O(M-C) bonds increase (decrease) slightly in comparison to those in parent Mo₂CO₂(M₂CO₂) compounds. However, the common feature across parent and Janus compounds is that the M/M^′ -O bonds are shorter than the M/M^′-C bonds. The fluctuations in the bond lengths lead to fluctuations in the bond strengths resulting in anharmonicity in the system having an effect on the lattice thermal conductivity $\kappa_{l}$.

An assessment of the bond strengths and it’s correlation with the dispersions in the M/M^′-C/O bond lengths is done by the Crystal Orbital Hamilton Population (COHP) method Deringer et al. (2011) as implemented in LOBSTER codeMaintz et al. (2016). In this method, a measure of the bond strengths are obtained from the energy integrated COHP, that is, from the energy averaged bond-weighted densities of states between a pair of atoms. In Table 2, the strengths between various bonds in parent and Janus compounds are shown. We find that the bond strengths are clearly correlated with the variations in the bond lengths. Among the parent M₂CO₂ MXenes, M-C and M-O bond strengths are comparable in case of Mo₂CO₂. For the other three compounds, M-O bonds are significantly stronger than the M-C bonds. For TiZrCO₂ and TiHfCO₂ Janus compounds, Ti-O and Ti-C bonds weaken considerably while M^′-O and M^′-C bonds strengthen, in comparison to those for their respective parent M₂CO₂ MXenes. For MMoCO₂ Janus, Mo-C and Mo-O bonds weaken only slightly while M-O and M-C bonds strengthen with respect to the respective bond strengths in the corresponding parent compounds.

The band structures of Janus and the parent MXenes are shown in Figure 2 and Figure 3, supplementary material, respectively. Densities of states of the Janus MXenes are shown in Figure 4, supplementary material. In agreement with existing results, we find that except Mo₂CO₂ which is a semi-metal, all M₂CO₂ MXenes considered are semiconductors. However all MM^′CO₂ Janus compounds, even with Mo as M^′ component, are semiconductors. The semiconducting nature of Mo-based Janus MXenes are due to the presence of the other transition metal constituent. The electronic structures in Figure 4, supplementary material, clearly show that in the anti-bonding part of the spectra, hybridisations between $d$ orbitals of Mo and the other transition metals that are located higher in energy, open the gaps in Mo-based Janus compounds. The magnitude of the gap depends on the position of the conduction band of M element in MM^′CO₂ MXene. The band gaps of TiZrCO₂ and TiHfCO₂ decrease considerably in comparison with Zr₂CO₂ and Hf₂CO₂, respectively (Table 1). Once again the reductions are due to the positions of Ti states that are closer to Fermi levels. Analysing the band structures we find that for ZrMoCO₂ and HfMoCO₂, the valence band maxima (VBM) and conduction band minima (CBM) are located at the $\Gamma$ and K points, respectively. In contrast, they are located at $\Gamma$ and M points, respectively, for TiZrCO₂ and TiHfCO₂. The electronic structure of TiMoCO₂ has very distinctive features. The VBM and CBM are located at $\Gamma$ and along K-$\Gamma$ direction,respectively. The band structure shows a flat band close to Fermi level in the conduction region, the presence of which is reflected in the large Van Hove singularity in the densities of states. Presence of such flat band near Fermi level would have profound impact on electronic transport properties as the effective mass of different carriers will be significantly different.

Dynamical stabilities of the Janus compounds are assessed by computing their phonon spectra. The phonon dispersion curves and the phonon partial density of states for MM^′CO₂ (M₂CO₂) Janus (parent) MXenes are shown in Figure 3 (Figure 5, supplementary material).

We find all systems to be dynamically stable in their respective ground states. For M₂CO₂ MXenes, the heavier elements (Ti, Mo, Zr, and Hf) dominate the low-frequency (acoustic and lower optical modes) phonon modes, while O and the lightest C atoms dominate the mid and high-frequency optical modes (Figure 5, supplementary materials). Similarly, in Janus MXenes, M and M^′ atoms majorly contribute in the lower frequency range (Figure 3). Mid-frequency optical modes show significant hybridization between two O atoms attached to M and M^′ surfaces. C atoms are responsible for the high frequency vibrations. We have performed the group theory analysis to identify the Raman and infrared (IR) active modes for Janus(for parent MXenes also; refer to Figure 6, supplementary material) MXenes. According to group theory, these vibrations can be expressed as irreducible representations ($\Gamma(vib)$) of C_3v point group and can be represented as

where $A_{1}$ refers out-of-plane vibrations, and $E$ denotes the in-plane vibrations, that are doubly degenerate at the $\Gamma$ point. For this point group, all modes are Raman and IR active. Vibrational patterns for the modes are schematically shown in Figure 3(f) (Vibration patterns corresponding to all $E$ and $A_{1}$ modes are shown in Figure 6, supplementary material).

In Figures 4, 5 and 6, we show results for variations of Seebeck coefficient ($S$), electrical conductivity ($\sigma$) and electronic part of thermal conductivity ($\kappa_{e}$) with energy, at three different temperatures, respectively. While Equations (1)-(5) solved under CRTA and RBA provide $S,\sigma/\tau$ and $\kappa_{e}/\tau$, $\tau$ is calculated from DP theory. The results suggest that TiZrCO₂ and TiHfCO₂ Janus have significantly higher values of S($\sim$1000$\mu$V/K) in comparison to the ones containing Mo($\sim$250$\mu$V/K). Among the parent MXenes, Hf₂CO₂ and Zr₂CO₂ have large values of S($\sim$1600$\mu$V/K) in comparison with the other two. Our results for parent MXenes agree well with existing results Gandi et al. (2016); Sarikurt et al. (2018); Wong et al. (2020); Khazaei et al. (2014). Moreover, the calculated values of $S$ for the Janus MXenes are either higher or are comparable to those for the established thermoelectric materials Shu et al. (2023); Yu et al. (2019).

The comparative behaviour of the electrical conductivity ($\sigma$), on the other hand, is different for different charge carriers. For p-type carrier, Ti₂CO₂ has significantly large $\sigma$ in comparison to Zr₂CO₂ and Hf₂CO₂, which have comparable values. For n-type carrier, $\sigma$ is highest for Hf₂CO₂, followed by Zr₂CO₂ and Ti₂CO₂. However, all three have comparable values of $\sigma$. Mo₂CO₂ has almost negligible electrical conductivity in comparison to the rest of the parent MXenes, irrespective of the type of charge carrier. In case of Janus compounds too, different trends are observed for different charge carriers. For p-type carrier, Mo based Janus compounds have substantially lower values of $\sigma$ in comparison to TiZrCO₂ and TiHfCO₂ which have almost identical magnitudes. For n-type carrier, the trend is exactly opposite. Like $S$, maximum $\sigma$ for Janus compounds are noticeably smaller in comparison with that of the parent MXenes. The behaviour of electronic thermal conductivity $\kappa_{e}$ follows the trends of $\sigma$. This is consistent with Widemann-Franz relationship $\kappa_{e}\propto\sigma$.

The trends in the electronic transport parameters $S$ and $\sigma$ can be qualitatively understood from the trends of the effective mass of the carriers, $m^{*}_{h}$ (holes) and $m^{*}_{e}$ (electrons), the band gaps $E_{g}$, the mobilities $\mu_{h}$ and $\mu_{e}$ of electrons and holes, respectively. The effective masses and the mobilities of the carriers for parent and Janus MXenes considered in this work, are computed using DP theory and are presented in Table 2, supplementary material. For parabolic bands and energy independent scattering $S\propto\frac{m^{*}}{n^{2/3}}$ where $m^{*}$ is the effective mass of the carrier and $n$ the carrier density. Considering the parent compounds first, we find that $m^{*}_{h,Ti_{2}CO_{2}}(m^{*}_{e,Ti_{2}CO_{2}})<(>)m^{*}_{h,Zr_{2}CO_{2}}(m^{*}_{e,Zr_{2}CO_{2}})\approx(>)m^{*}_{h,Hf_{2}CO_{2}}(m^{*}_{e,Hf_{2}CO_{2}})$. Since band gap of Ti₂CO₂ is substantially smaller than that of the other two, it has a larger electron density $n$. As a result $S$ for Zr₂CO₂ and Hf₂CO₂ are much larger than Ti₂CO₂ irrespective of the carrier type. The reason for quantitatively smaller (larger) $S$ for Janus TiZrCO₂ and TiHfCO₂ in comparison with Zr₂CO₂, Hf₂CO₂ (Ti₂CO₂) can also be understood in a similar way. The effective masses of Mo-based Janus compounds, irrespective of charge carrier type, are larger than those of the three parent MXenes considered. The band gaps follow exactly the opposite trend implying that the carrier densities of Janus are larger. However, $S$ of Janus ZrMoCO₂ and HfMoCO₂ are significantly smaller in comparison to $S$ of Zr₂CO₂ and Hf₂CO₂. Reduction of such magnitude is probably due to substantial reduction of band gaps in Janus compounds which has a larger effect on $n$, superseding the effect of m^∗. A comparison between $S$ of TiMoCO₂ and Ti₂CO₂ shows that $S_{TiMoCO_{2}}\gtrsim S_{Ti_{2}CO_{2}}$. This is because of two reasons: (a) the reduction in the band gap in TiMoCO₂ in comparison with Ti₂CO₂ is not as large as is the case of the other two sets of Janus-parent MXenes and (b) the increase in the effective mass in TiMoCO₂ in comparison with Ti₂CO₂ is much larger with respect to the other two sets. Such increase in effective mass in TiMoCO₂ is due to flatter bands in comparison to the other systems.

The comparative magnitudes of $S$ for the five Janus MXenes can also be understood by comparing the effective masses of holes and electronic band gaps. From the results presented in Table 1 and Table 2, supplementary material, we find that $m^{*}_{h,TiMoCO_{2}}<m^{*}_{h,ZrMoCO_{2}}<m^{*}_{h,HfMoCO_{2}}>m*_{h,TiZrCO_{2}}=m^{*}_{h,TiHfCO_{2}}$ and $E_{g,TiMoCO_{2}}<E_{g,ZrMoCO_{2}}<E_{g,HfMoCO_{2}}<<E_{g,TiZrCO_{2}}\approx E_{g,TiHfCO_{2}}$. As a result $n_{TiMoCO_{2}}>n_{ZrMoCO_{2}}>n_{HfMoCO_{2}}>>n_{TiZrCO_{2}}\approx n_{TiHfCO_{2}}$. The large electron densities in the Mo-based Janus compounds as compared with TiZrCO₂ and TiHfCO₂ nullifies the effect of larger $m^{*}_{h}$ in Mo-based Janus MXenes. Consequently, the Mo-based Janus MXenes have much smaller Seebeck coefficient as compared to the rest.

The electrical conductivity $\sigma\propto\mu n$. For the parent and Janus MXenes considered in this work, we find that the trends in $\sigma$ are largely dictated by $\mu$ of the carriers. In case of parent Mxenes, the hole mobilities are related as $\mu^{h}_{Ti_{2}CO_{2}}>>\mu^{h}_{Zr_{2}CO_{2}}\approx\mu^{h}_{Hf_{2}CO_{2}}$ while the electron mobilities follow $\mu^{e}_{Hf_{2}CO_{2}}>\mu^{e}_{Zr_{2}CO_{2}}\approx\mu^{e}_{Ti_{2}CO_{2}}$. The corresponding $\sigma$ for these compounds exactly follow these trends. Mo₂CO₂ is not considered in this context as it has an extremely low $\sigma$ and it being a semi-metal, mobilities associated with different charge carriers is irrelevant. For Janus compounds, the hole mobilities follow the trend $\mu^{h}_{TiZrCO_{2}}>\mu^{h}_{TiHfCO_{2}}>>\mu^{h}_{TiMOCO_{2}}>\mu^{h}_{ZrMoCO_{2}}\approx\mu^{h}_{HfMoCO_{2}}$ while the electron mobilities are related as $\mu^{e}_{ZrMoCO_{2}}>\mu^{e}_{HfMoCO_{2}}>>\mu^{e}_{TiMoCO_{2}}>\mu^{e}_{TiZrCO_{2}}\approx\mu^{e}_{TiHfCO_{2}}$. For the hole carriers, trend of $\sigma$ exactly follows the trend in carrier mobility. For the electron carriers, $\sigma$ of Mo-based Janus compounds are significantly higher than the remaining two. This cannot be explained by the trend in $\mu^{e}$ alone. The Mo-based Janus have much lower band gaps compared to the other two leading to very high carrier density $n$ resulting in substantially high $\sigma$. Even among the three Mo-based Janus, the qualitative trend of $\sigma$ for electron carriers can be explained only if both $\mu^{e}$ and $n$ are considered. The mobility turns out to be the deciding factor too for explaining the comparative trends between parent and Janus MXenes. For hole carriers, $\mu^{h}_{Zr_{2}CO_{2}}<<\mu^{h}_{TiZrCO_{2}}<<\mu^{h}_{Ti_{2}CO_{2}}$ and $\sigma_{Zr_{2}CO_{2}}<\sigma_{TiZrCO_{2}}<\sigma_{Ti_{2}CO_{2}}$. Same behaviour is observed for TiHfCO₂. For Mo-based Janus MXenes, hole mobilities are much smaller than those of Ti₂CO₂, TiZrCO₂ and TiHfCO₂. This explains the reason behind significantly smaller $\sigma$ of Mo-based Janus compounds in comparison with the other MXenes considered. For electron carriers, electrical conductivity of TiZrCO₂ and TiHfCO₂ are not very different from that of the corresponding parent MXenes. This is consistent with close values of their electron mobilities.

According to DP theory Bardeen and Shockley (1950), carrier mobility $\mu$ depends upon the carrier effective mass $m^{*}$, the in-plane stiffness constant $C_{2D}$ and the DP constant $E_{d}$, a measure of electron-phonon coupling strength, the following way: $\mu\propto\frac{C_{2D}}{|m^{*}|^{2}\left(E_{d}\right)^{2}}$. It is expected that $m^{*}$ and $E_{d}$ will have more profound effects on $\mu$. From the calculated values of these quantities presented in Table 2, supplementary material, we infer that the trends in $\mu$ for different charge carriers are dictated by $m^{*}$ which in turn is decided by the curvatures of the top(bottom) of valence(conduction)bands. For example, a noticeably large value of $m^{*}_{e}$ for TiMoCO₂ in conjunction with small $E_{d}$ is responsible for small $\mu^{e}$. The large value of $m^{*}_{e}$ is due to the flattest bottom of the conduction band among the series of compounds investigated. For TiHfCO₂ and TiZrCO₂ Janus compounds, $m^{*}_{e}(m^{*}_{h})$ increase (decrease) in comparison with parent compounds Hf₂CO₂ and Zr₂CO₂. This behaviour is due to the fact that in Janus compounds bottom of the conduction bands (top of the valence bands) are less(more) dispersive. In case of the three Mo-based Janus, larger values of $m^{*}$, in comparison with Ti₂CO₂, Zr₂CO₂ and Hf₂CO₂,irrespective of charge carrier type, are also an artefact of the band structures near Fermi level.

The electronic transport parameters show two overall trends. First, maximum values of the transport parameters are less upon breaking of inversion symmetry in parent MXenes and second, based upon the quantitative trends, the five Janus compounds can be grouped into two distinct sets: three Mo-based compounds and the remaining two. The trends in the electronic transport parts clearly imply that if only the electronic transport is considered, the inversion symmetry breaking to form Janus MXenes may not lead to higher figure of merit (as compared to parent compounds) as the Seebeck coefficient would be dominant due to its higher power in the expression for $ZT$. Therefore, the contribution of lattice thermal conductivity will be crucial. In the next subsection we explore this in detail.

Figure 7 shows the lattice thermal conductivity ($\kappa_{l}$) as a function of temperature $T$ in the range 300-800 K for the parent and Janus MXenes considered in this work. For all cases $\kappa_{l}$ varies as $(1/T)$, hallmark of intrinsic three-phonon scattering. Three distinct features are observed in the behaviour of $\kappa_{l}$: (1) Among the parent MXenes, Mo₂CO₂ (Ti₂CO₂) has the highest (lowest) $\kappa_{l}$ with the following quantitative trend: $\kappa_{l}^{Mo_{2}CO_{2}}>>\kappa_{l}^{Hf_{2}CO_{2}}\gtrsim\kappa_{l}^{Zr_{2}CO_{2}}>\kappa_{l}^{Ti_{2}CO_{2}}$, (2) Among the Janus MXenes, the three Mo-based compounds have higher $\kappa_{l}$ as compared to the remaining two with the quantitative trend as $\kappa_{l}^{TiMoCO_{2}}>\kappa_{l}^{ZrMoCO_{2}}\approx\kappa_{l}^{HfMoCO_{2}}>\kappa_{l}^{TiZrCO_{2}}\gtrsim\kappa_{l}^{TiHfCO_{2}}$ and (3) Thermal conductivities of Mo-free Janus compounds are significantly lower than the relevant parent MXenes ($\kappa_{l}^{TiZrCO_{2}}\approx\kappa_{l}^{TiHfCO_{2}}<\kappa_{l}^{Ti_{2}CO_{2}}<<\kappa_{l}^{Zr_{2}CO_{2}}<\kappa_{l}^{Hf_{2}CO_{2}}$) while lattice thermal conductivity of the three Mo-containing Janus are substantially lower than Mo₂CO₂ and greater than the other three parent MXenes Zr₂CO₂, Hf₂CO₂ and Ti₂CO₂. The trends are encouraging and intriguing as well. It, therefore, warrants a detailed analysis which has hitherto been unavailable. For example, though lattice thermal conductivities of Ti₂CO₂, Zr₂CO₂ and Hf₂CO₂ have been calculated Gandi et al. (2016); Sarikurt et al. (2018) and our results agree well with them, any discussion on the trends based upon quantitative analysis was absent.

Lattice thermal conductivity of non-metallic solids has been understood by a simple qualitative model Slack (1973). According to this model, crystal structure, average atomic mass ($\bar{m}$), bond strength and anharmonicity are the factors deciding $\kappa_{l}$. $\bar{m}$ and the bond strengths that comprise of the harmonic effects influencing $\kappa_{l}$ are manifested through a single quantity, the average acoustic Debye temperature $\Theta_{D}$:

$\Theta_{i}=\frac{\hbar\omega^{max}_{i}}{k_{B}}$; $\omega^{max}_{i}$ is the maximum frequency of $i^{th}$ acoustic mode, $ZA,TA,LA$ are the out-of-plane, transverse and longitudinal acoustic modes, respectively. According to Ref Slack (1973), $\Theta_{D}\propto\kappa_{l}$ and heavier $\bar{m}$ and weak bond strengths will lead to lower value of $\Theta_{D}$, and consequently lower $\kappa_{l}$. Heavier mass and weaker bonding will also lead to lower phonon group velocity and lower $\kappa_{l}$. In Figure 8, $\bar{m},\Theta_{D}$ and $\kappa_{l}$ for all MXenes considered in this work are shown. The results depict few contradictions with the qualitative model Slack (1973). According to the model, Ti₂CO₂ should have lattice thermal conductivity larger than Zr₂CO₂ and Hf₂CO₂. This is because its $\bar{m}$ is the lowest while the bond strengths (Table 2) are not drastically lower in comparison with the other two. The calculated values of $\Theta_{D}$ corroborate this anomaly. In Figure 8 we find that the $\Theta_{D}$ decreases as one moves from Ti₂CO₂ to Hf₂CO₂ but $\kappa_{l}$ instead of decreasing in the same direction, increases. Similarly, this model cannot explain why $\kappa_{l}^{Mo_{2}CO_{2}}$ is substantially higher than $\kappa_{l}^{Zr_{2}CO_{2}}$ when $\bar{m}$ of the two compounds are almost same. $\Theta_{D}$ of Mo₂CO₂ turns out to be substantially higher in comparison with $\Theta_{D}$ of Zr₂CO₂ (Figure 8) providing the explanation for behaviour of $\kappa_{l}$. However, this happens inspite of no clear trends in the bond strengths. The phonon spectra of these two MXenes (Figure 5, supplementary material) offer some clue to the behaviour of $\Theta_{D}$. The maximum frequencies of the three acoustic modes are noticeably higher in case of Mo₂CO₂, leading to larger $\Theta_{D}$ and larger $\kappa_{l}$ as a consequence. Moreover, large gaps between the four-fold degenerate optical mode around 400 cm^-1 and the two-fold degenerate optical mode around 550 cm^-1 implies a suppression of phonon-phonon scattering due to decrease in phonon population Ziman (2001) leading to subsequent elevation of $\kappa_{l}$. Nevertheless, these are only indirect evidences and do not provide a definite mechanism with quantitative estimates.

No such anomaly is encountered if one inspects the two groups of Janus compounds, the Mo-based ones and the remaining two, separately. In both groups, $\bar{m}\propto\frac{1}{\Theta_{D}}\propto\frac{1}{\kappa_{l}}$ is satisfied. The impact of the bond strengths, however, is not clear. The bond strengths of TiZrCO₂ and TiHfCO₂ are hardly different so that $\Theta_{D}$ is completely determined by the differences in their masses. The maximum frequencies of the three acoustic modes too explain the behaviour of $\Theta_{D}$. Same is true for the three Mo-based Janus compounds. The results become unexplainable in terms of the simple model with harmonic parameters only when comparison is made between members belonging to different groups. TiMoCO₂ and TiZrCO₂, despite having identical $\bar{m}$ have very different $\Theta_{D}$. Like the parent compounds Zr₂CO₂ and Mo₂CO₂, $\Theta_{D}$ of TiMoCO₂ is higher, albeit not as substantially as is the case for parent MXenes. This, however, has a profound impact on $\kappa_{l}$; $\kappa_{l}^{TiZrCO_{2}}$ is less than $\kappa_{l}^{TiMoCO_{2}}$ by about 80$\%$ . This trend is found for other compounds too which implies that the quantitative variation cannot be explained in terms of variations in the harmonic parameters alone.

The limitation of the model appears more pronounced when comparison is made between the lattice thermal conductivities of parent and Janus groups. For example, $\bar{m}^{Zr_{2}CO_{2}}>\bar{m}^{TiZrCO_{2}}>\bar{m}^{Ti_{2}CO_{2}}$ but $\Theta_{D}^{Ti_{2}CO_{2}}>\Theta_{D}^{TiZrCO_{2}}<\Theta_{D}^{Zr_{2}CO_{2}}$. Though the trend in $\Theta_{D}$ is consistent with the trend in $\kappa_{l}$ for these three compounds, the trends in $\bar{m}$ and $\Theta_{D}$ contradict the harmonic model. Even the trends in the bond strengths (Table 2) contradict the model. A comparison between Mo₂CO₂ and Mo-based Janus compounds too corroborate this. For this group of MXenes, $\Theta_{D}$ and $\kappa_{l}$ of Mo₂CO₂ are always greater than the Janus compounds inspite of the $\bar{m}$ of Janus compounds being equal to (in case of ZrMoCO₂) or greater (in case of HfMoCO₂) than $\bar{m}$ of Mo₂CO₂. The Janus bonds too are stronger.

One more hindrance in understanding the trends of harmonic parameters and $\kappa_{l}$ across all parents and Janus compounds using the harmonic model Slack (1973) is that the symmetry of the members of two different groups of compounds (one containing Mo and the other consists of the remaining ones) is different due to different site preference of the -O functional group. Therefore, we next compute the phonon group velocities as they are directly proportional to lattice thermal conductivity (Equation (6)). The results are presented in Figures S7 and S8, supplementary material. The results still cannot resolve the anomalies. For example, Hf₂CO₂ has group velocities lower than Ti₂CO₂ and TiHfCO₂, HfMoCO₂ have group velocities comparable to TiZrCO₂. But the trends in their $\kappa_{l}$ do not support this behaviour.

These anomalies, therefore, indicate that anharmonicity plays a key role in understanding the trends in $\kappa_{l}$. Earlier works on parent compounds Ti₂CO₂, Zr₂CO₂ and Hf₂CO₂ Gandi et al. (2016); Sarikurt et al. (2018) indicated this by analysing the phonon dispersions. However, they did not analyse the reasons behind the trend among the compounds , neither did they quantify the extent of anharmonicity. In what follows, we compute various quantities associated with the anharmonicity in the systems. In Figure 8, we show the total Gruneisen parameter $\gamma$ , the measure of the strength of anharmonicity in the system, calculated using phonon BTE Li et al. (2014). Among the parent MXenes, $\gamma$ decreases monotonically from Ti₂CO₂ to Mo₂CO₂ suggesting that largest (smallest) anharmonicity in Ti₂CO₂ (Mo₂CO₂) can be the reason for the smallest(largest) $\kappa_{l}$ in Ti₂CO₂(Mo₂CO₂). The comparable $\gamma$ values of Zr₂CO₂ and Hf₂CO₂ explains their near identical $\kappa_{l}$ that are between Mo₂CO₂ and Ti₂CO₂. Among the Janus MXenes, larger anharmonicity is found in compounds without Mo, with their $\gamma$ values being largest among all parent and Janus considered. $\gamma$ for three Mo-based MXenes are almost identical that fits perfectly into the trends of their $\kappa_{l}$. A comparison between $\gamma$ of parent and Janus MXenes too perfectly correlate with the trends in the lattice thermal conductivities.

In order to completely understand the extent of anharmonic effects and develop a microscopic picture, we next look into the phonon modes responsible and the strengths of phonon phonon scattering. To this end, we first look at the frequency range and the corresponding modes that are major contributors to $\kappa_{l}$. In Figure 9, supplementary material, we show the variations in normalised cumulative lattice thermal conductivity ($\kappa_{c}$/$\kappa_{l}$) as a function of frequency. The results are presented for 300K only. The results show that for all systems considered, phonon modes with frequencies upto 200 cm^-1 are the major contributors to $\kappa_{l}$. The mode resolved $\kappa_{l}$, presented in Figure 9, corroborates this. We find that for all systems the three acoustic and first three optical modes (numbered 4,5,6 in the figure) contribute almost entirely to the lattice thermal conductivity. A noticeable feature is that for Mo₂CO₂, the contributions from acoustic and optical modes towards $\kappa_{l}$ are significantly different. This is also true for the Janus compounds containing Mo. On the other hand, Ti₂CO₂, TiZrCO₂ and TiHfCO₂ have near equal contributions from acoustic and optical modes implying strong phonon-phonon interactions.

To connect the behaviour of mode-resolved $\kappa_{l}$ with anharmonicity, we first look at mode-resolved Grüneisen parameters ($\gamma_{\lambda}$) calculated using third-order IFCs Broido et al. (2005). The results are presented in Figure 10, supplementary material. We find that among the parent compounds, Zr₂CO₂ has larger $\gamma_{\lambda}$ than Mo₂CO₂ in the relevant frequency range, implying significant large anharmonicity in the former. This can be correlated to their very different $\kappa_{l}$ with $\kappa_{l}^{Zr_{2}CO_{2}}$ much smaller than that of the other. $\gamma_{\lambda}$ for Zr₂CO₂ and Hf₂CO₂ in the same frequency range are comparable. Ti₂CO₂ has the highest $\gamma_{\lambda}$ among all parents. The $\gamma_{\lambda}$ in non-Mo Janus compounds are 2-3 times higher than Ti₂CO₂. The Mo-based Janus compounds, on the other hand, have extremely low $\gamma_{\lambda}$. These trends nicely explain the behaviour of $\kappa_{l}$. In fact $\gamma_{\lambda}$ turns out to be a better indicator than $\gamma$ for comparing the degree of anharmonicity among compounds. For example, $\gamma$ of Ti₂CO₂ and TiHfCO₂ are same. But $\gamma_{\lambda}$ of the later is much higher in the relevant frequency range. The differences in their $\kappa_{l}$ values, thus, can only be explained by $\gamma_{\lambda}$.

We further understand the degree of anharmonic effects as quantified by $\gamma_{\lambda}$ by calculating the anharmonic scattering rates over the frequency region. The results for Janus (parent) MXenes are shown in Figure 10(Figure 11). Among the parent compounds, Ti₂CO₂ has appreciably large scattering rates in comparison with other three. Mo₂CO₂ has the lowest scattering rate, scattering rates of the other two lie in between these two. The large scattering rate of Ti₂CO₂ finally resolves the anomaly regarding its lowest $\kappa_{l}$ as compared to other parent compounds despite all harmonic parameters indicating the opposite. The lattice thermal conductivity is directly proportional to the group velocity $v_{g}$ and the phonon relaxation time $\tau^{0}$ (Equation 6). Since scattering rate is inversely proportional to $\tau^{0}$, the effect of larger $v_{g}$ of Ti₂CO₂ discussed earlier is nullified by the small $\tau^{0}$, yielding lowest $\kappa_{l}$ among the parent compounds. Among Janus compounds, higher scattering rates are observed for TiZrCO₂ and TiHfCO₂. This explains their higher values of $\gamma_{\lambda}$ and consequently lower $\kappa_{l}$ among the MXenes considered. In Figure 11, supplementary material, comparison between scattering rates in Janus and parent compounds are made. The noticeable trend observed in this comparison is that while the scattering rates of MMoCO₂ Janus compounds lie in between that of M₂CO₂ and Mo₂CO₂, the scattering rates of TiZrCO₂ and TiHfCO₂ are higher than the corresponding parent MXenes. To find the reason, we first look at the accessible phase space for phonon scattering since $\tau^{0}$ ( and thus $\kappa_{l}$) is inversely proportional to the accessible phase space Lindsay and Broido (2008); Pandey et al. (2017)which is determined by the condition of phonon energy and momentum conservation.

Figure 12(a)((b)) shows the weighted scattering phase space (possible three-phonon scattering processes weighted by the frequencies) accessible to the parent(Janus) MXenes. Among the Janus MXenes, available phase spaces for Mo-based ones are lower as compared to the others. TiHfCO₂ has more accessible phase space than TiZrCO₂. More phase spaces available for three-phonon scattering, thus, can be the explanation for higher scattering rates seen in these two Janus MXenes. But, this proposition is violated in case of the parent MXenes. We find that Ti₂CO₂ has less accessible phase space compared to Hf₂CO₂ and Zr₂CO₂. Even TiHfCO₂ has less accessible phase space in comparison with Hf₂CO₂ in the frequency range 100-150 cm^-1, the range most responsible for anharmonic scattering. Therefore, the trends in the anharmonic scattering rates cannot be understood in terms of accessible phase space. To resolve this issue, we present the results on the strength of scattering matrix elements $|\phi_{\lambda\lambda^{{}^{\prime}}\lambda^{"}}^{\pm}|^{2}$, which is a measure of the strength of three phonon scattering and depends on the anharmonic IFCs (Equation (10)). Figure 12(c)(Figure 12(d)) show the average scattering matrix elements as a function of frequency for parent(Janus) MXenes. It is clear from Fig.12(c) that throughout the frequency range, the scattering strength in Ti₂CO₂ is more than Hf₂CO₂. Therefore, despite Hf₂CO₂ having more available phase space than Ti₂CO₂, it has lower scattering rate. Among the Janus compounds, TiZrCO₂ has the largest scattering strength with TiHfCO₂ competing closely. Therefore, the degree of anharmonicity among the parent and among the Janus can be finally understood in terms of the anharmonic scattering strengths. Comparison between parents and Janus, however, shows an anomaly : scattering strengths in Ti₂CO₂ in the frequency range of interest is still slightly higher than those in TiZrCO₂. Despite this, larger anharmonicity in TiZrCO₂ can be attributed to higher number of scatterings as can be understood by comparing Figures 12 (c) and (d).

The inversion symmetry breaking in the Janus MXenes produces dispersions in the bond strengths (Table 2). We find that such dispersion can be connected to the degree of anharmonicity in Janus. The phonon densities of states of TiHfCO₂(Figure 3(e)) shows that in the frequency range 100-200 cm^-1, the vibrations are dominated by the Ti atoms. In this Janus Ti-anion bond strengths weaken considerably as compared to the corresponding ones in parent Ti₂CO₂. The Hf-cation bond strengths strengthen, on the other hand, in comparison to those in parent MXene Hf₂CO₂. In Janus TiZrCO₂ ( Figure 3(d)), both Ti and Zr vibrations contribute in the relevant frequency range. Here too we find significant dispersion in bond strengths, similar to TiHfCO₂. In the Mo-based Janus ZrMoCO₂ and HfMoCO₂, these dispersions are much less. Although dispersions are there in TiMoCO₂, the Ti-cation bond strengths have hardly changed in comparison with Ti₂CO₂. Since the vibrations in the frequency range of interest is overwhelmingly dominated by the Ti atoms, the Ti-anion bond strengths decide the extent of anharmonicity.

Figure 13 shows the maximum value of figure of Merit $ZT$ as a function of temperature for all parent and Janus MXenes. Energy window from -1.5 to 1.5 eV and carrier concentration $\sim$ 10²⁰ cm^-3 are used to extract maximum $ZT$. We find that surface engineering by forming Janus greatly enhances the $ZT$. The maximum $ZT$ of 3.5 at 800K is obtained for TiZrCO₂ in case of p-doped systems. This is more than double the maximum $ZT$ obtained in any of the parent MXenes (a maximum $ZT$ value of 1.6 at 300 K is obtained for Ti₂CO₂). p-doped TiHfCO₂ also yields a high $ZT$ of $\sim 3$ at 800K. These two Janus compounds, even when $n$-doped, produce maximum $ZT$ of $2$ while none of the parent compounds and Mo-based Janus could produce a maximum $ZT$ more than $1$. Dominant reason of such high $ZT$ in these two systems, irrespective of type of doping, is due to their extremely low $\kappa_{l}$. High values of $S$ and $\sigma$ for p-doped TiZrCO₂ and TiHfCO₂ are responsible for substantially higher $ZT$ in these systems when they are doped by p-type carriers in comparison to doping by n-type one. Thus, desired values of maximum $ZT$ can be obtained by breaking the inversion symmetry in MXenes.